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 Permanent Link:
 http://ufdc.ufl.edu/AA00026617/00001
Material Information
 Title:
 Selection of important properties to evaluate the use of geostatistical analysis in selected northwest Florida soils
 Creator:
 Ovalles, Francisco A., 1950
 Publication Date:
 1986
 Language:
 English
 Physical Description:
 xvi, 208 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Dissertations, Academic  Soil Science  UF
Soil Science thesis Ph. D Soils ( fast ) Florida ( fast ) City of Gainesville ( local ) Soil properties ( jstor ) Statistical discrepancies ( jstor ) Soil horizons ( jstor )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1986.
 Bibliography:
 Includes bibliographical references (leaves 197206).
 Additional Physical Form:
 Also available online.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Francisco A. Ovalles.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
 Resource Identifier:
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15376316 ( OCLC )

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SELECTION OF IMPORTANT PROPERTIES TO EVALUATE
THE USE OF GEOSTATISTICAL ANALYSIS IN
SELECTED NORTHWEST FLORIDA SOILS
BY
FRANCISCO A. OVALLES
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1986
This dissertation is dedicated to my wife, Giordana, and my
children, Johanna Fernanda and Pedro Jose.
ACKNOWLEDGMENTS
The author wishes to express his gratitude to Dr. Mary
E. Collins, chairman of the supervisory committee, for her
continuous help, guidance, patience, and personal
friendship throughout the graduate program. Appreciation
is also extended to other members of the committee, Dr.
Gustavo Antonini, Dr. Richard Arnold, Dr. Randall "Randy"
Brown, and Dr. Stewart Fotheringham, for their constructive
reviews of this work, participation on the graduate
supervisory committee, and personal friendship.
Appreciation is expressed to the Consejo Nacional de
Investigaciones Cientificas y Tecnologicas (CONICIT),
Venezuela, for the scholarship which supported the author.
Thanks are extended to Dr. Willie Harris who
introduced me to the Keepit and YT, and always was ready to
answer any of my questions.
Very special thanks are due to Dr. Gregory "Greg"
Gensheimer for lending me the geostatistical program and
his own computer to type this dissertation.
Gratitude is expressed to the staff of the Soil
Characterization Laboratory, for their friendship and
valuable assistance, to other graduate students, staff, and
faculty.
iii
Appreciation is extended to all my friends from all
six continents (America, Africa, Asia, Europe, Oceania, and
Florida) whom I had the pleasure of knowing here.
Finally, but certainly not least, I thank my wife,
Giordana, my daughter, Johanna Fernanda, and my son, Pedro
Jose, for their love and continuous help, encouragement and
patience during this work.
iv
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS.........................................iii
LIST OF TABLES..........................................vii
LIST OF FIGURES..........................................ix
ABBREVIATIONS ...........................................xii
ABSTRACT. ................................................ xiv
INTRODUCTION............................................... 1
LITERATURE REVIEW......................................... 4
Principal Component Analysis ............................ 4
Geostatistics .......................................... 13
Historical Development ............................. 13
Theoretical Bases................................... 15
Practical Use ...................................... 34
Fractals................................................ 53
DESCRIPTION OF STUDY AREA ................................ 59
Location...................... .. ........................ 59
Physiography, Relief, and Drainage..................... 59
Geology. ............................................... 61
Climate.................................................62
Land Use and Vegetation................................ 62
Soils..................................................63
MATERIALS AND METHODS .................................... 66
Data Source ............................................ 66
Location of Pedons...................................... 67
Statistical Analyses.................................... 68
Normality Analysis ................................. 68
Principal Component Analysis........................ 69
Geostatistical Analysis ............................ 71
RESULTS AND DISCUSSION ................................... 75
Test of Normality ......................................75
v
Page
Principal Component Analysis ........................... 83
Principal Component Analysis for Standardized
Weighted Data.................................... 83
Principal Component Analysis for A horizon
Standardized Data................................ 93
Principal Component Analysis by Soil Series.......101
Geostatistics ........................................113
SemiVariograms............................ ......114
Fitting SemiVariograms...........................140
Kriging ........................................... 143
Fractals .......................................... 156
SUMMARY AND CONCLUSIONS..................................165
APPENDIX
A CLASSIFICATION OF SOIL SERIES STUDIED.......... 178
B GEOGRAPHIC COORDINATES OF PEDONS STUDIED....... 181
C SEMIVARIOGRAMS FOR DIRECTIONS WITH
LARGEST VARIABILITY........................... 186
D CONTOUR MAPS FOR DIRECTIONS WITH
LARGEST VARIABILITY........................... 192
E MAP OF PHYSIOGRAPHIC REGIONS IN
NORTHWEST FLORIDA ............................ 196
LITERATURE CITED......................................... 197
BIOGRAPHICAL SKETCH ..................................... 207
vi
LIST OF TABLES
Table Page
1 Order, Great Group, and relative proportion
of pedons studied.................................. 65
2 Statistical moments of soil properties studied
and Kolmogorov test................................ 78
3 Proportion of total variance explained by each
principal component................................ 85
4 Eigenvectors of correlation matrix for
standardized weighted average of soil
properties......................................... 89
5 Tolerance of standardized weighted average of
soil properties by principal component............. 91
6 Correlation coefficients between standardized
weighted average of soil properties and
principal components................................ 92
7 Proportion of total variance explained by each
principal component for standardized A horizon
data ............................................... 94
8 Eigenvectors of correlation matrix for
standardized properties of A horizon............... 95
9 Tolerance of standardized properties of
A horizon by principal component................... 96
10 Correlation coefficient between standardized
properties of A horizon and principal
components.........................................97
11 Correlation coefficient between standardized
properties of Al horizon and principal
components......................................... 99
12 Correlation coefficient between standardized
properties of Ap horizon and principal
components....................................... 100
vii
Table Page
13 Variability of studied soil properties within
and between soil series and between horizons......107
14 Important semivariogram parameters of the
weighted average of selected soil properties......127
15 Important semivariogram parameters of the
A horizon selected properties..................... 136
16 Goodnessoffit values of the weighted average
of selected soil properties.......................142
17 Goodnessoffit values of the A horizon
selected properties............................... 144
18 Fractal dimension (D value) derived from
selected soil property semivariograms............ 158
19 Fractal dimension (D value) derived from
selected soil property semivariograms
for a reduced study area..........................162
viii
LIST OF FIGURES
Figure Page
1 Relation among variance, covariance, and
semivariance ..................................... 20
2 Common semivariogram models...................... 27
3 Equation number 35................................31
4 Equation number 36 (a) and Equation
number 37 (b) ..................................... 32
5 Location of the counties from which
characterization data were available
for pedons selected for study..................... 60
6 Histogram (a) and normal probability plot (b)
of fine sand content.............................. 80
7 Histogram (a) and normal probability plot (b)
of organic carbon content......................... 82
8 Location of standardized weighted average
values of soil properties in the plane of
the first two principal components................ 86
9 Location of standardized weighted average
values of soil properties in the plane of
the rotated first two principal components........88
10 Soil properties with a large contribution
to the total variance by county for the
Albany series.................................... 102
11 Soil properties with a large contribution
to the total variance by county for the
Dothan series.................................... 103
12 Soil properties with a large contribution
to the total variance by county for the
Orangeburg series................................ 104
13 Location of selected soil series in the plane
of the first two principal components............106
ix
Figure Page
14 Location of selected soil series in the plane
of the first two principal components derived
from important soil properties...................110
15 Location of selected pedons in the studied
area............................................. 115
16 Weighted average total sand content first
directionindependent semivariogram ............. 120
17 Weighted average clay content first
directionindependent semivariogram............. 121
18 Weighted average total sand content fitted
directionindependent semivariogram............. 123
19 Weighted average total sand content
directiondependent semivariograms.............. 124
20 Weighted average clay content fitted
directionindependent semivariogram............. 125
21 Weighted average clay content direction
dependent semivariograms........................126
22 Weighted average organic carbon content fitted
directionindependent semivariogram.............131
23 Weighted average organic carbon content
directiondependent semivariograms.............. 132
24 A horizon clay content fitted direction
independent semivariogram....................... 134
25 A horizon clay content directiondependent
semivariograms..................................135
26 A horizon organic carbon content fitted
directionindependent semivariogram............. 138
27 A horizon organic carbon content direction
dependent semivariograms........................ 139
28 Contour map (increment is 10.0%) (a) and diagram
(vertical exaggeration is 18x, azimuth of
viewpoint is 259) (b) of kriged weighted
average total sand content.......................146
29 Contour map (increment is 10.0%) (a) and diagram
(vertical exaggeration is 18x, azimuth of
x
Figure Page
viewpoint is 259) (b) of kriged weighted
average clay content............................. 147
30 Contour map (increment is 2.0%) (a) and diagram
(vertical exaggeration is 18x, azimuth of
viewpoint is 25Q) (b) of kriged A horizon
clay content .....................................148
31 Diagram (vertical exaggeration is 18x, azimuth
of viewpoint is 259) of standard errors of
kriged weighted average total sand content.......153
32 Diagram (vertical exaggeration is 18x, azimuth
of viewpoint is 25Q) of standard errors of
kriged weighted average clay content............. 154
33 Diagram (vertical exaggeration is 18x, azimuth
of viewpoint is 25Q) of standard errors of
kriged A horizon clay content....................155
34 Location of reduced study area...................161
35 Weighted average total sand content fitted
NS semivariogram ............................... 186
36 Weighted average clay content fitted
NS semivariogram.............................. 187
37 Weighted average organic carbon content fitted
NS semivariogram............................... 188
38 A horizon clay content fitted
NWSE semivariogram............................. 189
39 A horizon organic carbon content fitted
NWSE semivariogram............................. 190
40 Contour map (increment is 10.0%) derived from
weighted average total sand content NS
semivariogram...................................192
41 Contour map (increment is 10.0%) derived from
weighted average clay content NS
semivariogram ................................... 193
42 Contour map (increment is 2.0%) derived from
A horizon clay content NWSE semivariogram......194
43 Map of physiographic regions in northwest
Florida (Source: Brooks, 1981b)..................196
xi
ABBREVIATIONS
a = range
ACl = A horizon clay content
AOC = A horizon organic carbon content
BS = Base saturation
C = Coarse sand
c = Sill
Ca = Calcium
CEC = Cation exchange capacity
Co = County
3 = Bay
30 = Holmes
32 = Jackson
33 = Jefferson
37 = Leon
40 = Madison
57 = Santa Rosa
66 = Walton
COV = Covariance
C.V. = Coefficient of variation
EXT = Extractable acidity
F = Fine sand
G(h) = GAMMA = Semivariance
h = Lag distance
K = Potassium
M = Medium sand
xii
Mg = Magnesium
Na = Sodium
OC = Organic carbon
PHI = pHwater
PH2 = pHKCl
Sc = Selection criterion for eigenvectors
T = Tolerance
TB = Total bases
TH = Horizon thickness
TS = Total sand
VAR = Variance
VC = Very coarse sand
VF = Very fine sand
xiii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SELECTION OF IMPORTANT PROPERTIES TO EVALUATE
THE USE OF GEOSTATISTICAL ANALYSIS IN
SELECTED NORTHWEST FLORIDA SOILS
BY
FRANCISCO A. OVALLES
December, 1986
Chairman: M.E. Collins
Major Department: Soil Science
Soil variability is a limiting factor in making
accurate predictions of soil performance at any particular
position on the landscape. A large number of studies have
been made to quantify soil variability, but a large portion
of them ignored the multivariate character of soils and the
geographic aspect of soil variability. Data from 151
pedons in northwest Florida were selected (i) to determine
the important properties affecting soil variability and
(ii) to evaluate the soil variability in the area studied
using geostatistics.
Data were nonnormally distributed but statistical
techniques employed did not require the assumption of
normality. This result could support the presence of
systematic patterns of soils.
xiv
Principal component analysis was used to reduce the
number of soil properties to study the soil variability.
Two sets of data were used: weighted average values of soil
properties, and A horizon properties. Horizon thickness
was used as the weighting criterion. Variables were
standardized to mean zero and variance one. Plots of soil
properties in the plane of the principal components,
varimax rotation, analysis of eigenvalues, eigenvectors,
and collinearity, and calculation of correlation
coefficients between soil properties and principal
components were used to select important properties for
evaluation of soil variability.
A nested analysis of variance indicated that
properties selected by the principal component analysis
were differentiating properties.
Geostatistical analysis was applied to the properties
selected. The withinsoil series variance was used as
criterion to assess stationarity. Drift was present.
Consequently, residuals were used to compute semi
variograms.
Semivariograms of total sand and clay contents showed
structure. Nugget variance was present in all semi
variograms. Ranges varied from 15 to 35 km. Soil
variability was directiondependent. The NS and NWSE
were the directions of maximum variability. Organic carbon
content had a large pointtopoint variation.
xv
All observed semivariograms had a characteristic wave
pattern that indicated a cyclic variation of soil
properties.
Kriged standard error diagrams were functions of the
nugget variance and showed areas where more samples are
required to increase the precision of estimates.
Fractal dimensions indicated the scaledependent
character of soil variability.
xvi
INTRODUCTION
The fundamental purpose of a soil survey is to
estimate the potentials and limitations of soils for many
specific uses. Soil delineations are mapped to be as
homogeneous as possible in order to correlate the
adaptability of soils to various crops, grasses, and trees;
and to predict their behavior and productivity under
different management practices (Soil Survey Staff, 1951;
1981).
Quality of soil surveys has been improved over the
years as a result of improved understanding of soil. But
soil variability remains as one of the main constraints to
reliable soil interpretations and is a limiting factor for
making accurate predictions of soil performance at any
particular position on the landscape.
The study and understanding of soil variability
represents a cornerstone for improving soil surveys.
Belobrov (1976), a Russian soil scientist, pointed out that
"The degree of approximation between the true and the
observed soil variability does not depend on the nature of
the soil cover, but mainly on the methods of investigation"
(p. 147).
1
2
For several years, soil scientists used methods of
investigation which did not consider the "real nature" of
soils, because they ignored the systematic variation of
soils on the landscape and assumed a random variation of
soils in space. On the other hand, despite the fact that
it has been recognized that a soil map unit is imperfect to
varying degrees, depending on the scale of the map and the
nature of the soil (Soil Survey Staff, 1975), most soil
surveys in the U.S.A have accepted an unrealistic model in
which map units encompass soil bodies that form discrete,
internally uniform units, with abrupt boundaries at their
edges (Hole and Campbell, 1985).
Studies of soil variability have not been consistent.
These studies have considered a random variation of soils
and at the same time they have used a limited number of
observations for characterizing map units to establish the
range of variation of observed properties. The assumption
has been that properties measured at a point also represent
the unsampled neighborhood. The extent to which this
assumption is true depends on the degree of spatial
dependence among observations.
The number of studies for quantifying soil variability
has sharply increased in the last 10 years, but
quantification still remains a problem. A large proportion
of quantitative studies are based on untested assumptions,
ignore the multivariate character of soils, or use a biased
3
selection of properties to represent the soil variability,
increasing the risk of erroneous conclusions.
For these reasons a large soil data base was selected
in northwest Florida with the following objectives: (i) to
discover which soil properties most strongly influence the
soil variability in the area studied, and (ii) to study how
geostatistics can be used in evaluating soil variability.
LITERATURE REVIEW
Principal Component Analysis
The multivariate character of soil is well recognized;
a large set of measurements of soil properties
(morphological, chemical, physical, and mineralogical) can
be derived from a single sample. The complete set of
available data is not always used for numerical analyses.
Hole and Campbell (1985) indicated that the selection of
soil properties depends on the objectives of the study, and
also reflects the constraints imposed by cost, time,
effort, and access.
There is no doubt that logically correlated variables,
such as soil pH and base saturation, are generally so
highly covariant that one or the other should not be
included in the analysis. Particlesize fractions (sand,
silt, and clay) always add up to 100%, and therefore, the
whole set of particlesize data should not be included in
the analysis. Consequently, in the process of selecting
soil properties, there is an important question to be
answered: Are the selected soil properties the most
important to represent the variability of the complete set
of data?
4
5
Webster (1977) pointed out that when one soil property
is measured in a set of individual sampling units, the
measured values can be represented by their positions on a
single line. The relation between any pair of individuals
can be represented by the distance between them and the
relations among several individuals can be established
simultaneously from their relative positions on the line.
At the same time, it is almost impossible to visualize
their positions on the line and the relations among more
than two individuals simultaneously. Thus, he indicated
that an alternative way of dealing with multivariate data
is to arrange the individuals along one or more new axes.
This reduction of an arrangement in many dimensions to a
few dimensions is known as ordination.
The two most common methods of ordination are Factor
Analysis (FA) and Principal Components Analysis (PCA).
Shaw and Wheeler (1985) said that in both techniques new
variables are defined as mathematical transformations of
the original data. However, FA assumes that the original
variable is influenced by various determinants: a part
shared by othervariables, known as the common variance;
and a unique variance which consists of both a variance
accounted for by influences specific to each variable and
also a variance relating to measurement error. In
contrast, PCA assumes that statistical variation in the
variables is explained by the variables themselves, in this
6
case by the common variance. PCA is recommended when there
are high correlations between variables, a large number of
variables, and a need for only simple data reduction. The
major objective in PCA is to select a number of components
that explain as much of the total variance as possible,
whereas FA is used to explain the interrelationship among
the original variables (Afifi and Clark, 1984). PCA has
the advantage in that the values of principal components
are relatively simple to compute and interpret.
PCA is a method that has been used to reduce the
number of variables without losing important information
(Webster, 1977). In general, the analysis finds the
principal axes of a multidimensional configuration and
determines the coordinates of each individual in the
population relative to those axes. Then, the data can be
represented in a few dimensions by projecting the points
orthogonally on the principal axes.
The basic idea of PCA is to create new variables
called the principal components (PC) (Afifi and Clark,
1984). Each new variable is a linear combination of the X
variables and can therefore be written as
PC = A11 X1 + A12 X2 + . A.. Xj (1)
where PC = principal component
Aij = coefficient (eigenvector)
X. = variable
7
Coefficients of these linear combinations are chosen
to satisfy the following requirements:
(i) Variance PC1 > variance PC2 > . variance PC.n
(ii) The values of any two PCs are uncorrelated.
(iii) For any PC the sum of the squares of the coefficients
is one.
Cuanalo and Webster (1970) used PCA in a study of
numerical classification and ordination in which
morphological, physical, and chemical soil properties (pH,
clay, silt, fine sand, proportion of stones, consistence,
water tension, color, mottling, and peatiness) were
measured at depths of 13 cm and 38 cm at 85 sites and
randomly sampled within physiographic units near Oxford,
England. The variables were standardized to unit variance
and the population was centered at the origin. It was
found that the first six PCs represented almost 70% of the
total variation presented in the original data. The first
three PCs represented more than 50% of the total variance.
The first component showed large contributions from water
tension, and chroma in both the topsoil and the subsoil.
In the second component, contribution of fine sand in the
topsoil (13 cm) and subsoil (38 cm) was dominant. Hue and
value made large contributions to the third component. The
projection of the population scatter on the plane defined
by the first two PCs gave the most informative display of
relations in the whole space. These authors suggested that
8
when numerical data are available, the data should be
examined first by ordination procedures; then, the data
selected by the ordination procedure can be used with a
numerical classification to decide if such classification
grouped data satisfactorily.
Norris (1972) used PCA to study trends in soil
variation. He described several morphological soil
properties (stage of organic matter decomposition,
percentage of stones, structure, consistence, porosity,
roots, biological activity, and color in terms of presence
or absence of gley or dark colors) in 410 pedons, 307
pedons located in woods and 103 pedons located in farmland.
The firstPC accounted for 39% of the total variance, and
corresponded to a trend from deep, stoneless pedons
developed on a clayey formation to pedons developed on
shallow limestone on steep slopes. The second PC accounted
for 14% of the total variance and separated pedons located
on farmland from those located in the woods. He concluded
that the PCs served as a summary of soil variation in the
area, because they accounted for a known percentage of the
soil variation and were correctly defined in terms of the
properties used to describe the soil.
Webster and Burrough (1972) sampled the first two
horizons from 84 soil pedons and recorded selected soil
properties (soil color, CaCO3 content, depth to CaCO3,
total penetrable soil depth, clay content, organic matter
9
content, cation exchange capacity (CEC), pH, exchangeable
Mg and K contents, and available P content). They used PCA
to reduce the dimensionality of the data, and found that
the first two PCs accounted for 55% of the total variance
(40% the first component and 15% the second component).
Separate contributions to the components were determined by
projecting vectors on the components axes. They
established that those properties determined in the field
(CaCO3 content, depth to CaCO3, clay content, and subsoil
color) were closely correlated and well represented in one
dimension in the first component. The properties measured
in the laboratory (organic matter content, CEC, and
exchangeable Mg content) contributed most to the second
component, indicating differences in management rather than
natural soil differences. Results of the numerical
classification were supported by showing the distribution
of sampling sites in space projected on to the plane of the
first two components and showing the frequency distribution
of the first PC. There was a good agreement among the
results. Therefore, it was concluded that when PCs
represent the variables that explain soil variation the
components can be mapped as isarithms and the maps have
interpretable meaning.
Kyuma and Kawaguchi (1973) employed PCA to grade the
chemical potentiality of 41 Malayan paddy soil samples; 23
physical, chemical, and mineralogical properties were
10
evaluated. The first four PCs accounted for 75% of the
total variance. The first PC was highly positively
correlated with electrical conductivity, exchangeable Ca,
Mg, Na, and K contents, moisture, CEC, available Si
content, and 0.2 M HClsoluble K. The first PC was highly
negatively correlated with the kaolin mineral content. All
of these properties were relevant to the chemical
potentiality of the soil, thus, the first PC was called the
chemical potentiality component. The standardized scores
of the first PC were computed. These scores were used for
grading soils in terms of the chemical potentiality. The
authors stated that the result of grading was reasonable.
Placed at the top of the scale were soils developed on
juvenile marine sediments. Soils having high sand and/or
kaolin content were at the bottom of the scale. The
authors concluded that PCA was useful for comparing the
soil fertility status among soils.
Burrough and Webster (1976) used PCA with Similarity
and Canonical Variate Analyses to improve soil
classification in eastern Malaysia. Morphological and
chemical properties determined by routine analysis were
recorded from 66 randomly selected sites. The first nine
PCs accounted for more than 70% of the total variance.
Scatter diagrams of pairs of components were drawn to
elucidate the population structure. Established classes
that were originally thought to be desirable overlapped
almost completely with respect to morphological and
chemical properties. Dendograms derived from similarity
analysis confirmed the interpretations drawn from the
scatter diagrams.
Williams and Rayner (1977) employed PCA as a method
for grouping soils based on chemical composition (Fe, Ti,
Ca, K, Si, Al, P, Mg, Mn, Ni, Cu, Zn, Ga, As, Br, Rb, Sr,
Y, Zr, and Pb total contents) and other soil properties
such as particle size (sand, silt, and clay), loss on
ignition, CaCO3 content, pH, and soil moisture. The
scatter diagram showed that the first two components
divided the soils into parent material groups. This
grouping was also supported by using dendograms derived
from similarity analysis. It was concluded, on the basis
of the PCA, that the soils sampled came from three parent
materials of different ages.
McBratney and Webster (1981) studied the relationships
between sampling points using PCA. A substantial
proportion (44%) of the total variance was explained by the
first two PCs. The first component represented color.
Varimax rotation was employed to obtain a better
interpretation of the scatter diagram but it produced no
appreciable improvement in interpretability. The scatter
diagram of PC allowed the separation of sampled points into
five different groups.
12
Richardson and Bigler (1984) applied PCA to selected
soil properties (clay content, pH, organic carbon content,
CaCO3 equivalent, electrical conductivity, and soluble Mg,
Ca, and Na contents) which were meaningful to soil
development and plant growth in wetlands in North Dakota.
Four routine measurements useful for characterizing and
classifying wetland soils were identified by PCA
(electrical conductivity, organic carbon content, CaCO3
equivalent, and clay content). Electrical conductivity and
soluble Mg and Na contents were the most important
variables in explaining observable differences in wetland
soils. In addition, the use of PCA allowed the examination
of the interaction of chemical and physical properties with
the landscape position of wetland soils, as well as the
variation in properties among vegetation zones, after the
data were plotted in the plane of the first two PCs.
Edmonds et al. (1985) employed PCA as a first step for
using Cluster and Discriminant Analyses to study taxonomic
variation within three soil map units. Forty different
soil properties were included in the analyses. Variables
with low variance were excluded by the analysis. PCA was
used to reduce the number of dimensions needed to ordinate
pedons in the plane of PCs (character space) and to remove
intercorrelation of soil properties. The use of PC scores
as data for Cluster Analysis avoided distortions in
coordinates of the pedons in the plane of PCs. They
13
compared the results with the taxonomic classification of
soils, and concluded that grouping of pedons by numerical
taxonomy did not correspond to groupings by taxa in Soil
Taxonomy.
Geostatistics
Webster and Burgess (1983) pointed out that to
describe soil variation two features of soil must be taken
into account. The first is that long range trends have no
simple mathematical form; usually, there is not any obvious
repeating pattern; and the larger the area or the more
intensive the sampling the more complex the variation
appears. The second is that the pointtopoint variation
in a sample reflects real soil variation. Only a small
part is the measurement error. In addition, the same
authors indicated that earlier attempts to describe spatial
variation in geology and geography involved fitting
deterministic global equations to data, either exactly or
by least squares approximation. But the two features
mentioned above make the approach inappropiate for soil.
Thus, an alternative was to treat the soil as a random
function and to describe it using geostatistics techniques.
Historical Development
Etymologically, the term geostatistics designates the
statistical study of natural phenomena, and it is defined
as the application of the formalism of random functions to
14
the reconnaissance and estimation of natural phenomena
(Journel and Huijbregts, 1978).
Geostatistics was primarily developed for the mining
industry (Matheron, 1963). Geostatistics was very useful
for engineers and geologists for studying the spacial
distribution of important properties such as grade,
thickness, or accumulation of mineral deposits.
Matheron (1963) considered that, historically,
geostatistics was as old as mining itself. He indicated
that as soon as mining men concerned themselves with
foreseeing results of future work and, in particular, as
soon they started to take and to analyze samples and
compute mean values weighted by corresponding thickness of
deposits and influencezones, geostatistics was born.
Geostatistics started in the early 1950s in South
Africa with D.G. Krige (Olea, 1975). Krige realized that
he could not accurately estimate the gold content of mined
blocks without considering the geometrical setting
(locations and sizes) of the samples. Matheron expanded
Krige's empirical observations into a theory of the
behavior of spatially distributed variables which was
applicable to any phenomenon satisfying certain basic
assumptions, and the variables were not limited by their
physical nature.
15
Theoretical Bases
Classical statistics could not be used for ore
estimation because of their inability to take into account
the spatial aspect of the phenomenon (Matheron, 1963). An
aleatory variable had two essential properties: (i) the
possibility, theoretical at least, of repeating
indefinitely the test that assigned to the variable a
numerical value, and (ii) the independence of each test
from the previous and the next tests. A given oregrade
within a deposit would not have those two properties. The
content of a block of ore was first of all unique, but on
the other hand, two neighboring ore samples were certainly
not independent.
Earth scientists usually deal with complex phenomena
which are the result of the interaction of variables,
through relationships which are in part unknown and in part
very complex (Olea, 1975). Variations are erratic and
often unpredictable from one point to another, but there is
usually an underlying trend in the fluctuations which
precludes regarding the data as resulting from a completely
random process. To characterize variables which are partly
stochastic and partly deterministic in their behavior,
Matheron (1971) introduced the term regionalized variable.
He developed the regionalized variable theory to describe
functions which vary in space with some continuity.
16
A regionalized variable is a continuously distributed
variable having a geographic variation too complex to be
represented by a workable mathematical function (Campbell,
1978). Although the precise nature of the variation of a
regionalized variable is too complex for a complete
description, the average rate of change over distance can
be estimated by the semivariance. Conversely, Olea (1977)
stated that a regionalized variable is a function that
describes a natural phenomenon which has geographic
distribution.
The term geostatistics has come to mean the
specialized body of statistical techniques developed by
Matheron and associates to treat regionalized variables
(Olea, 1984). The theory of regionalized variables has two
branches: the transitive methods and the intrinsic theory
(Matheron, 1969). The first is a highly geometrical
abstraction without probabilistic hypothesis and has little
practical interest. The practical counterpart of those
geometrical abstractions is the intrinsic theory which is a
term for the application of the theory of random variables
to regionalized variables.
Matheron (1969) and Olea (1975) indicated that
regionalized variables are characterized by the following
properties: (i) localization, a regionalized variable is
numerically defined by a value which is associated with a
sample of specific size, shape, and orientation which is
17
called geometrical support. (ii) Continuity, the spatial
variation of a regionalized variable may be extremely large
or very small, depending on the phenomenom studied, but
despite this fact, an average continuity is generally
present, in some cases the average continuity cannot be
confirmed, and then a nugget effect is present.
(iii) Anisotropy, changes may be gradual in one direction
and rapid or irregular in another. These changes are known
as zonalities.
A basic assumption in the intrinsic theory is that a
regionalized variable is a random variate (Matheron, 1969).
The observed values are outcomes following some probability
density function. Henley (1981) considered that a
regionalized variable as a random function which may be
defined in terms of a probability distribution (i.e., it
may be normally distributed with a particular mean and
variance).
Olea (1984) indicated that a spatial function can
either be described by a mathematical model or given by a
relative frequency analysis based on experimentation. The
former approach is not practical because of the complexity
of spatial functions. The latter is seriously limited by
the maximum number of samples that can be collected.
Olea (1975) stated that the difficulty of the relative
frequency approach with a regionalized variable is that a
repeated test cannot be run because each outcome is unique.
18
Since a large number of samples are essential to any
statistical inference, it is not possible to determine the
probability density function which rules the occurrence of
a regionalized variable.
The impossibility of obtaining the probability density
function associated with the variable is not a serious
limitation. Most of the properties of interest depend only
on the structure of the regionalized variable as specified
by its first and second moments (Olea, 1975). A key
assumption is stationarity. Stationarity is a mathematical
way to introduce the restriction that the regionalized
variable must be homogeneous. Stationarity permits
statistical inference. A test can be repeated by assuming
stationarity even though samples must be collected at
different points. All samples are assumed to be drawn from
populations having the same moments.
Several scientists have discussed the assumption of
stationarity (Henley, 1981; Huijbregts, 1975; Journel and
Huijbregts, 1978; Olea, 1975; 1984; Tipper, 1979; Trangmar
et al., 1985; Webster, 1985). Geostatistics invokes a
stationary constraint called the intrinsic hypothesis to
resolve the impossibility of obtaining a probability
distribution. A regionalized variable is called strictly
stationary if it is stationary for any order k = 1, 2, 3,
4, . n. If k is equal to one, the regionalized
variable has firstorder stationarity. Secondorder
19
stationarity also implies firstorder stationarity.
Secondorder stationarity signifies that the first two
moments (covariance and variance) of the difference between
two observations are independent of the location and are a
function only of the distance between them.
In general, for a regionalized variable of order k,
all the moments of order k or less are invariant under
translation. For a stationary variable, the covariance has
the following properties:
(i) COV (0) > ICOV(X2 X1)I (2)
where COV = covariance
(ii) LIM COV(h) = 0, h 4 o (3)
where LIM = limit
(iii) COV(0) = VAR[Y(X)] (4)
where VAR = variance
(iv) COV(X2 X1) = COV(X1 X2) (5)
These relations are better visualized in Figure 1.
For secondorder stationarity, VAR[Y(X)] must be
finite. Then, according to equation (4) COV(0) must be
finite. However, many phenomena in nature are subject to
unlimited dispersion and cannot correctly be described when
they are assigned a finite variance. Thus, to avoid this
restriction, the intrinsic theory assumes what is called
the intrinsic hypothesis. The intrinsic hypothesis is
satisfied if, for any displacement h the first two
moments of the difference [Z(x) Z(x + h)] are independent
G () = VAR
VAR
G (h)
I COV(<=o) 0
0
a
Lag distance h
Figure 1. Relation among variance, covariance, and semivariance.
21
of the location x and are a function only of h:
E [Z(x) Z(x + h)] = M(h) (First moment) (6)
E [{ Z(x) Z(x + h) M(h)}2] = 2 G(h) (Second moment)
(7)
where M(h) and G(h) are referred as the drift and the semi
variance or intrinsic function, respectively. The semi
variance is a measure of the similarity, on the average,
between observations at a given distance apart. The more
alike the observations, the smaller is the semivariance.
The semivariogram (Olea, 1975; Journel and
Huijbregts, 1978), which is the plot of semivariance
against distance h (lag), has all the structural
information needed about a regionalized variable: (i) zone
of influence that provides a precise meaning to the notion
of dependence between samples, (ii) anisotropy when
variability is directiondependent revealing the different
behavior of the semivariogram for different directions,
and (iii) continuity of the variable through space, which
is indicated by the shape and the particular
characteristics of the semivariogram near the origin.
One of the oldest methods of estimating space or time
dependency between neighboring observations is through
autocorrelation (Vieira et al., 1983). Nash (1985) pointed
out that the correlogram (plot of autocorrelation against
22
distance) is the mirror image of the semivariogram.
Vieira et al. (1983) indicated that when interpolation
between measurements is needed, the semivariogram is a
more adequate tool to measure the correlation between
measurements. An infinite dispersion is allowed using
semivariances.
According to Journel and Huijbregts (1978) the
autocorrelation is equal to
f(h) = C(h)/ C(0) (8)
where f(h) = autocorrelation
C(h) = autocovariance or covariance at distance h
C(0) = variance
The relationship between C(h) and C(0) is expressed by
equation (4). When the semivariance changes, it is
assumed that its variations are small with respect to the
working scale. This is a condition of quasi or local
stationarity. When the regionalized variable is weakly
stationary, it also obeys the intrinsic hypothesis. The
semivariance is then given by
G(h) = a2 C(h) (9)
where G(h) = semivariance
a2 = population variance
C(h) = autocovariance
23
The autocorrelation and the semivariance are related
by the following equation:
f(h) = 1 G(h)/ C(0) (10)
Burgess and Webster (1980a) pointed out that the
autocorrelation coefficient depends on the variance
(equation 8), and according to equation (4) the variance
must be finite to fulfill the requirement of stationarity.
It was indicated earlier that many phenomena in nature are
subject to unlimited dispersion. The semivariance is free
of this restriction, and consequently is preferred. They
also indicated that a second advantage of working with
semivariance is that it is easier to take into account
local trends in the property of interest. Residuals are
used when trends are present. Webster and Burgess (1980)
demonstrated that the variance of the residuals from the
mean is not equal to the variance of the difference between
the values when trends are present. Therefore,
autocorrelation is difficult to use.
Webster (1985) classified the semivariograms into
four groups:
Safe models. They are defined for one dimension but
are safe in the sense that they are conditional positive
definite in two and three dimensions. These models are
24
1. The linear model:
G(h) = C0 + wh for h > 0 (11)
G(0) = 0 (12)
where G = semivariance
C0 = intercept or nugget variance
w = slope
h = lag distance
Equation (11) assumes that h has an exponent a = 1. When
the exponent a = 0.5 the model is called root. When a = 2
the model is parabolic.
2. The spherical model:
G(h) = c0 + w [1.5 (h/a) 0.5 (h/a)3] (13)
for 0 < h < a
G(h) = c0 + w for h > a (14)
G(0) = 0 (15)
where a = range
c0 + w = sill
3. The exponential model:
G(h) = c0 + w [1 exp (h/a)] for h > 0 (16)
G(0) = 0 (17)
4. The DeWijsian model:
G(h) = c0 + a ln(h) for h > 0 (18)
G(0) = 0 (19)
5. The Gaussian model:
G(h) = co + w (1 exp (h/a)2) for h > 0 (20)
G(0) = 0 (21)
25
6. The hyperbolic model:
G(h) = h/ a + Ph (22)
where a and P are coefficients of the hyperbola
function.
Risky models. The semivariogram increases to a sill.
1. The circular model:
G(h) = c0 + w [1 2/rt cos(h/a) + 2h/ Ta(1 h2/a2)]
for 0 < h < a (23)
G(h) = c0 + w for h > a (24)
G(0) = 0 (25)
2. The linear model with a sill:
G(h) = c0 + w (h/a) for 0 < h S a (26)
G(h) = c0 + w for h > a (27)
G(0) = 0 (28)
Nested model. The components of variance measure the
amount of variance contributed by each scale.
G(h) = j VAR [Z(x) Z(x+h)] = G0(h) + Gl(h) (29)
where G0(h) = pure nugget semivariance
G1(h) = spatially dependent semivariance
Anisotropic model. Variability is not equal in all
lateral directions.
G(h,8) = c0 + u(8) IhI (30)
where
u(8) = [A2 cos2 (9 a) + B sin2 (9 a)]O (31)
26
where 8 = anisotropy angle
a = direction of maximum variation
A = gradient of semivariogram in direction of
maximum variation
B = gradient in the direction a + j n
The most common semivariograms are showed in Figure 2.
Computing a series of semivariograms and deriving a
model from them is usually not an end in itself. The
objectives of geostatistical studies are to determine the
characteristics of the data and to obtain the best
estimates possible with the available data. The advantage
of using a geostatistical approach is that the computed
values are optimum. The error of estimation is minimized.
The acronyn BLUE (best linear unbiased estimation) is
sometimes used to characterize this method (Green, 1985).
Estimation procedures that incorporate regionalized
variable theory were originally known as kriging, a term
named for D.G Krige (DeGraffenreid, 1982). Kriging is a
distanceweighted moving average estimation procedure that
uses the semivariogram to determine optimal weights.
Kriging depends on computing an accurate semi
variogram from which estimates of semivariance are then
used to obtain the weights applied to the data when
computing the averages, and are presented in the kriging
equation (Burgess and Webster, 1980a).
27
C      
CC 
GC (h) G(h)
a h a h
Spherical Exponential
C  
GCh) G(h)
a h h
Gaussian Linear, Root, Parabolic
Figure 2. Common semivariogram models.
28
When values of soil properties are averaged over point
values, which represent volumes with the same size and
shapes as the volumes of soil on which the original
descriptions were recorded (i.e., pedons), the kriging
procedure is called punctual kriging (Burgess and Webster,
1980a). When an average is made over areas, the procedure
is called block kriging (Burgess and Webster, 1980b).
Block kriging produces smaller estimation variances and
smoother maps.
Burgess and Webster (1980a) and Webster and Burgess
(1983) pointed out that kriging is a means of spatial
prediction that can be used for soil properties. In
kriging, the weights take account of the known spatial
dependence expressed in the semivariogram and the
geometric relationships among the observed points. Kriging
is optimal in the sense that it provides estimates of
values at unrecorded places without bias and with minimum
known variance.
It has been indicated by several scientists
(Huijbregts, 1975; Olea, 1975; 1984; Trangmar et al., 1985;
Webster and Burgess, 1980) that kriging is used only with
regionalized variables that are firstorder stationary. For
variables whose drift is not stationary, but for whose
residuals the intrinsic hypothesis holds, universal kriging
is used.
29
Webster and Burgess (1980) stated that universal
kriging takes account of local trends in data when
minimizing the error associated with estimation. Universal
kriging can be performed after computing suitable
expressions for the drift and corresponding semivariograms
of the residuals.
Olea (1984) said that universal kriging is a linear
estimator of the regionalized variable and has the form
n
Z(x0) =i ri Z(xi) (32)
where Z(x0) = unknown parameter at location x0
r. = weights
1
Z(x.) = value of a property at a point xi
Matheron (1963) stated that suitable weights r.
assigned to each sample are determined by two conditions.
The first condition is that Z (x0) and Z(x ) must have
the same average value within the area of influence, and is
written as
n
ii ,i = 1 (33)
The second condition is that r. have such values that
estimation variance (kriging variance) of Z(x0) and Z(x )
should take the smallest possible value. The unknown r.'s
were found by solution of a system of linear equations
which result from forcing the unbiased estimator to have
minimum variance. The equation is as follows:
30
AX = B (34)
where A, B, and X are given by equations (35), (36), and
(37) (Figures 3 and 4).
In recent years a new method for estimation has been
developed. Vieira et al. (1983) stated that in soil
science, agrometeorology, and remote sensing, very often
some variables are crossrelated with others. In addition,
some of those variables are easier to measure than others.
In such situations estimation of one variable using
information about both itself and another crosscorrelated,
easiertomeasure variable should to be more useful than
the kriging of that variable by itself. This estimation is
easily made using cokriging.
Cokriging has been defined as the estimation of one
spatially distributed variable from values of another
related variable (DeGraffenreid, 1982; Gutjahr, 1984).
Dependence between two variables can be expressed by a
cross semivariogram (McBratney and Webster, 1983a). For
any pair of variables i and j there is a cross semi
variance G (h) at lag hij defined as
G i(h) = E [{Z (x) Z (x+h)} {Z (x) Z (x+h)}] (38)
where Z. and Z. are the values of i and j at places x and
x+h. If i = j then, equation (38) represents the auto
semivariance.
31
G(xl,x ) G(xl,x2)...G(Xl,xk) 1 f(x1) f2(x1)...fn(xl)
G(x2,x1) G(x2,x2)...G(x2,xk) 1 f(x2) f2(x2)...fn(x2)
......................................................
G(xj,x ) G(xj,x2)...G(xj,xk) 1 f(x ) f2(xj)...fn(x )
.....................................................
A = G(xk,x1) G(xk,x2)...G(xk,xk) 1 f(xk) f2(xk)...fn(xk)
1 1 ... 1 0 0 0 ... 0
f(x1) f(x2) ... f(xk) 0 0 0 ... 0
f2(x1) f2(x2) ... f2(xk) 0 0 0 ... 0
fn(xI) fn(x2) ... fn(xk) 0 0 0 ... 0
G(xj ,xk) = Semivariance between two sample elements
located at a distance (x. ,x).
f = Function of x, derived from the drift.
Figure 3. Equation number 35.
32
a) F1 b) G(X1,X0)
F2 G(x2,x0)
Fj G(xj,X0)
. ........
X = Fk B = G(xk,x0)
40 1
41 f(x0)
12 f2(x0)
*. .. .....o
4n fn(x0)
rF = weights.
T(Xk,X0) = semivariance between two sample elements
located at a distance (xk x0).
f (x) = function of x, derived from the drift.
4i = Lagrange multipliers.
Figure 4. Equation number 36 (a) and equation
number 37 (b).
33
The cokriging equation is given by
j nj
Zj(x0) = jl il r1ij Z(xij), for all j (39)
where i, j = variables
Z (xj) = estimated value of variable j at location x0
r.. = weights
To avoid bias the weights have to fulfill two conditions:
nj
(i) ij1 rij = 1 (40)
and
nj
(ii) il Fij = 0 for all i not equal to j. (41)
The first condition, according to McBratney and
Webster (1983a), implies that there must be at least one
observation of the variable j for cokriging to be possible,
and as in kriging equation (34), cokriging can be expressed
in matrix notation for solving the unknown weights.
Trangmar et al. (1985) indicated that cokriging
requires at least one sample point of both the primary
variable and covariable properties within the estimation
neighborhood. If the primary variable and covariable are
present at all sampling sites in the neighborhood, then
cokriging is considered as an autokriging of the primary
variable alone. In such cases, cokriging is unnecessary.
34
Practical Use
Earlier studies in soil science used time series
analysis in which spatial dependence of soil properties was
considered. Webster and Cuanalo (1975) computed
correlograms for clay, silt, pH, CaCO3, colorvalue, and
stoniness for three horizons in pedons located at 10 m
interval along a transect in north Oxfordshire, England.
They observed that the relation between sampling points
weakened steadily over distances from 10 m to about 230 m.
The average spacing between geological boundaries on the
transect was also about 230 m. Outcrop bedrock was
inferred as one of the main sources of soil variation.
They concluded that mappable soil boundaries were likely to
occur on the average every 230 m, and sampling at spacing
closer than 115 m would be needed to detect them.
Lanyon and Hall (1980) used morphological, physical,
and chemical soil properties to test the performance and
value of autocorrelation analysis. Spatial dependence was
determined from observations made every 20 m along a
transect for solum thickness; fineearth fraction of the A,
B, and C horizons; and for soil pH, percent base saturation
(PBS), and exchangeable cations from the deepest horizon.
They found that the range varied from 20 m for solum
thickness and exchangeable K content to 60 m for pH and
exchangeable Mg content. They concluded that auto
correlation analysis emphasized the continuous, orderly
35
nature of soils, and the fact that spatially related
observations may be mutually dependent.
Campbell (1978) was one of the first to use
geostatistics in soil science. He studied the spatial
variation of sand and pH measurements employing the semi
variance. Samples were collected at 10 m intervals on two
sampling grids positioned on two contiguous delineations in
eastern Kansas. There was a contrast in spatial variation
of sand content within the two delineations. Distances of
30 and 40 m were sufficient to encounter full variation of
sand content. Soil pH had a random variation within both
areas. It was concluded that the most important
application of semivariograms was in determining optimum
sample spacing in the design of efficient sampling
strategies.
Gambolati and Volpi (1979) introduced the
determination of the trend a priori, and improved the
process of fitting the observed to a theoretical semi
variogram. They used kriging to describe groundwater flow
near Venice, Italy. They proposed and used a modification
of the kriging technique developed by Matheron (1970) which
aimed at improving the accuracy of the interpolation
procedure. In Matheron's (1970) basic theory, the trend
was not assessed a priori. The trend was considered as a
linear combination of functions with unknown coefficients.
Gambolati and Volpi (1979) considered the trend a priori;
36
therefore the trend had to be determined. Also, they
defined the concept of theoretical consistency in kriging
applications. Theoretical consistency was derived from the
validation of the interpretation model. Validation was
made by suppressing each observation point one at a time,
by providing an estimate in that point using the remaining
(n1) observations, and analyzing the distribution of
errors. They stated that consistency occurred when there
was no systematic error (kriged average error was
approximately zero) and the standard deviation was
consistent with the corresponding error (the average ratio
of theoretical to calculated variance was approximately
equal to one). They found that validation of the
interpretation models selected for study showed that their
approach yielded accurate results, provided the trend was
correctly assessed.
Chirlin and Dagan (1980) modeled water flow through
twodimensional porous formations as a random process using
an approximate formulation of flow physics to obtain an
expression for the Head variogram. The Head variogram
proved markedly anisotropic, with heads differing more
widely on average for a fixed lag parallel to the head
gradient than perpendicular to it. Also they examined a
hypothetical case ignoring anisotropy. It was determined
from their experiment that the kriged standard deviation
37
was overestimated perpendicular to the mean flow and was
underestimated parallel to it.
Hajrasuliha et al. (1980) studied salinity levels in
three different fields in southwest Iran which were
initially sampled on an arbitrarily selected grid of 80 m.
Semivariances were calculated for all three sites to
determine the degree of dependence between observations.
The results from two fields showed that observations were
spatially dependent. Contour lines of isosalinity were
obtained by using kriging. In the third field salinity
observations were found to be spatially independent. Thus,
the number of samples necessary to get fiducial limits and
to identify the number of samples to be taken randomly
across the field for a given probability were obtained by
using classical statistical methods.
Luxmoore et al. (1981) used semivariograms to
characterize spatial variability of infiltration rates into
a weathered shale subsoil. Infiltration rates were
measured using doublering infiltrometers installed at 48
locations on a 2 x 2 m grid after the removal of 1 to 2 m
of soil. A high degree of variability in infiltration
rates was determined. The test for spatial patterning
using the semivariogram approach proved negative.
Therefore, they concluded that if patterning existed at
all, it occurred on a spatial scale less than the 2 m used
38
in the study. As a result of this study, it was determined
that infiltration rate was a randomly distributed property.
Vieira et al. (1981) analyzed the spatial variability
of 1280 fieldmeasured infiltration rates on Typic
Xerorthents. The measurements were made at the nodes of an
irregular grid. The semivariogram showed a range of 50 m.
It was considered that, on the average, samples separated
by 50 m or more were not correlated to each other.
Conversely, they examined the effect of the neighborhood
size on the value kriged and its estimation variance. They
determined that a neighborhood of 14 m was sufficient for
the infiltration data. The estimation variances changed
very little for larger distances. Low mean estimation
error, low variance, and high correlation coefficient
showed that the kriging estimation was exceptionally good.
Finally, it was determined that geostatistics was useful to
redesign the sampling scheme. The large number of measured
values made it possible to calculate the minimum number of
samples necessary to reproduce the infiltration rate
measurements with good precision. It was determined that
128 samples were enough to obtain nearly the same
information as with 1280 samples.
Geostatistics was used for first time to study soil
variability of large areas in Kigali, Rwanda by Vander Zaag
et al. (1981). They studied the spatial variability of
selected soil properties (pH; exchangeable Ca, Mg, K, and
39
Na contents; KClextractable Al content; percent Al
saturation; effective CEC; 4g Psorbed at an equilibrium P
concentration of 0.02 and 0.2 4g/g; extractable P content;
P and Si in the saturation extract; total N, NO3, and NH4;
and extractable S contents) in the whole country of Rwanda.
Semivariograms of soil pH, exchangeable Ca content,
effective CEC, Si in the saturation extract, and
extractable NH4 content showed long range spatial
dependence. The spatial dependence varied from 37.5 km for
soil pH to more than 60 km for extractable NH4. The
information contained in the semivariogram was used to
estimate values of soil properties at unsampled locations
within the range of the semivariogram. Maps of estimation
variance of kriged values were also generated. Such maps
showed that estimation variance of kriged values generally
increased with increasing distance from sample points. It
was indicated that geostatistics could be used to make
quick, low cost assessments of soil variability of large
land areas. In addition, the map of estimation variance
gave an indication of the confidence limits of the
estimated values. The map can be used to locate optimum
sampling sites to lower the estimation variance.
McBratney and Webster (1981) computed semivariograms
of subsoil properties (depth to subsoil, soil color,
particlesize, mineralogy, organic carbon content total N
content, ratio OC/total N, and pH). Samples were taken on
40
a transect at 20 m intervals. Semivariograms showed
spatial dependence extending to about 360 m for some
properties, in particular color and pH. Other subsoil
properties had little or no spatial dependence, notably
particlesize fractions and organic carbon content. The
shape of some semivariograms indicated presence of
different map units on the transect.
Van Kuilenburg et al. (1982) applied three
interpolation techniques (proximal, weighted average, and
kriging) to point data involving soil moisture supply
capacity on a 2 x 2 km grid of cover sand in the eastern
part of the Netherlands. Survey points used for
interpolation were randomly stratified with an average
density of 1.5 per ha. The root mean squared error was
used as a measure of efficiency. The root mean squared
error was large for the proximal method (less efficient)
and there was a negligible difference between root mean
squared errors for weighted average and kriging. Weighted
average had the weakness that possible clusters of survey
points were weighted too heavily. This was avoided in
kriging. Therefore, kriging proved to be the most
efficient for the survey method used.
Yost et al. (1982a) collected samples from 80 sites at
1 to 2 km intervals in Hawaii. Soil samples were taken
from 0 to 15 cm (topsoil) and 30 to 45 cm depths (subsoil).
The former depth represented the nutrient status as
41
influenced by management and the latter depth represented
the natural conditions. Semivariograms for soil pH,
exchangeable cations (Ca, Mg, K, Na), sum of cations, P
requirements, Si and P in saturation extract, extractable P
content, and rainfall were calculated. Ranges were much
greater for soil properties in the 0 to 15 cm depth than
for those in the 30 to 45 cm depth. Semivariograms for
Ca, Mg, K, and P contents based on the 30 to 45 cm depth
samples demonstrated greater variability and had smaller
ranges (Ca, Mg, and K) than those based on the 0 to 15 cm,
or were extremely variable (P). Si in saturation extract
had the same range in the subsoil as in the topsoil.
Subsoil properties were highly variable. Thus, soil
management and rainfall imposed a degree of uniformity on
the surface soil properties not apparent in the subsoil.
Yost et al. (1982a) concluded that soil chemical properties
had spatial dependence and that understanding such
structure may provide new insights into soil behavior over
the landscape. The semivariograms changed at large
distances. These changes suggested that soils should be
grouped to obtain uniform regions of soil properties
suitable for management regimes.
Yost et al. (1982b) used soil data from transects in
Hawaii for estimating soil P sorption over the entire
island by using kriging. The necessity of considering non
stationarity and the use of universal kriging were
42
evaluated. Universal kriging, either by prior polynomial
trend removal or by local polynomial trend removal during
estimation, was not beneficial in spite of widely varying P
sorption and a significant polynomial trend in the data.
The kriged estimates indicated that P sorption properties
of soil obtained from transects could be estimated in an
optimal way and could be displayed in a manner to better
understand the soil properties and genesis, and for
practical purposes, estimating the fertilizer needs and
distribution facilities.
McBratney et al. (1982) sampled 3500 sites to study
the spatial variability of Cu and Co soluble in mild
extractants measured to identify places where these metals
were deficient for animals. Semivariograms for both Cu
and Co were isotropic and appeared to combine three
components of variation: a short range component extending
up to 3 km, a long range or geological component extending
to 15 km, and a nonspatial or nugget component, which
accounted for 32% and 63% of the total variance of Cu and
Co, respectively. Cu showed a greater degree of spatial
dependence than Co. In addition, isarithmic maps
identified areas where Cu and/or Co were deficient. An
error map showed that precision was generally acceptable.
Also, the map identified a few areas in the region in which
sampling was too sparse for confidence.
43
Byers and Stephens (1983) sampled an untilled medium
grained fluvial sand in horizontal and vertical transects
to study the spatial structure of particle size and
saturated hydraulic conductivity. Semivariogram and
kriging analyses indicated that both hydraulic conductivity
and particle size were relatively isotropic in the
horizontal plane but had marked anisotropy in the vertical
plane. There were marked similarities in spatial structure
in the horizontal plane. The spatial distribution of
saturated hydraulic conductivity in the horizontal plane
was estimated reasonably well using an empirical
relationship between particle size and conductivity along
with kriged estimates of the 10% finer particle size.
Ten Berge et al. (1983) studied the spatial
distribution of selected soil properties (moisture content,
moisture tension, bulk density, texture, temperature, and
equivalent surface temperature). Two transects were
sampled at 4 m intervals. Semivariograms for moisture
content and bulk density did not show any range but only a
nugget effect. For other soil properties semivariograms
had a range varying between 80 and more than 120 m (texture
and temperature). Gradual changes in soil characteristics
were expected. The presence of abrupt map unit boundaries
was determined for some properties (e.g., texture). The
spatial structure of the field moisture content was found
only at very shallow depths. Texture introduced
44
differences in hydraulic conductivity, which were thought
to cause differences in topsoil moisture content.
Vauclin et al. (1983) used geostatistics for studying
the variability of particlesize data, available water
content, and water stored at 1/3 bar. The soil samples
were collected within a 70 x 40 m area at the nodes of a 10
m square grid. All semivariograms had a nugget effect
which corresponded to the variability that occurred within
distances shorter than the sampling interval and to
experimental uncertainties. The range varied from 26 m for
water stored at 1/3 bar to 50 m for silt content. Cross
semivariograms were calculated demonstrating that
available water content at 1/3 bar was correlated with sand
content within distances of 43.5 and 30 m. Semivariograms
and cross semivariograms were used to krige and cokrige
additional values of available water content and water
stored at 1/3 bar every 5 m. They indicated that the use
of cokriging was a promising tool whether the principal
objective was the reduction of the estimated variance
compared with kriging or the need to estimate an under
sampled variable by taking into account its spatial
correlation with another more sampled variable.
Spatial variability of nitrates in cotton petioles
was determined employing semivariograms and kriging (Tabor
et al., 1984). Sampling of petioles was of two types, on
transects and from randomly selected sites on a rectangular
45
grid. Nitrates in petioles showed definite spatial
dependence in the field studied. However, for sampling
areas of < 1 m, spatial dependence was insignificant
compared to the inherent variability of the sample and
laboratory analyses. Semivariograms and kriged maps of
nitrates in petioles suggested a strong influence of the
cultural practices such as direction of rows and
irrigation.
Bos et al. (1984) sampled in a rectangular grid 50 x
200 m at 10 m intervals on sandtailings capped with 0 to
> 2 m of stripmine overburden. This was done to present
and discuss the use of semivariograms to study the spatial
variation of extractable P, Na, K, Mg, and Ca contents,
extractable acidity, CEC, total P content, pHwater, pH
KC1, and soluble salts of the topsoil (0 to 25 cm) and
relative elevation in reclaimed Florida phosphate mine
lands. Semivariograms were calculated for data taken
along transects in four different sampling directions and a
combined direction. Some properties (CEC and relative
elevation) did not present structure of spatial variation.
The range was approximately 6 m for the combined and EW
semivariograms. Also, a nugget effect was observed which
represented variability at distances < 10 m. Presence of
anisotropy could not be established because welldefined
sills and ranges could not be determined for directions N
S, NESW, and NWSE. The semivariograms were supported by
46
too few data points at large distances. It was concluded
that semivariograms were useful in determining the spatial
variability of soil properties on reclaimed phosphate mine
lands and in improving sampling design for liming and
fertilization needs.
Xu and Webster (1984) used geostatistics to test how
these techniques could be applied for large areas. Topsoil
of 102 pedons evenly distributed throughout the studied
area in China were sampled. Soil pHwater, organic matter,
sand, total N, total P, and total K contents were measured.
Variation of soil properties appeared to be isotropic.
Soil pH showed the strongest spatial dependence.
Isarithmic mapping of local estimates of pH showed zones of
alkaline soils. Because sampling was sparse, on average
one sample for 3.5 km2, the estimation errors were large.
It was suggested that a more intensive sampling scheme
would increase confidence in the maps. This would also
improve the estimation of semivariograms, especially for
lags in the range of 0.5 to 5 km.
Saddiq et al. (1985) collected data on soil water
tension from 99 tensiometers along a 76 m row planted with
chile pepper and irrigated through trickle tubing placed 5
cm below the soil surface. Semivariograms indicated a
large variability and little spatial dependence in soil
water tension. The range was < 6 cm. Also, it was
determined that variability and spatial dependence were
47
functions of the method and timing of water application and
the magnitude of the soil water tension. When water was
applied through a trickle line, variability was greatest
and spatial dependence was smallest. Variability was low
and spatial dependence high after rain or extensive
flooding.
Rogowski et al. (1985) were probably the first to use
geostatistics to estimate erosion at different scales.
Erosion was measured at nodes of three different size
grids: 225 measurements from a 15 x 22.5 km grid, 25 from a
5 x 7.5 km grid, and 150 from a 1 x 1.5 km grid in west
central Pennsylvania. Erosion at each node was computed
using the universal soil loss equation. Kriging was
employed to map potential erosion. It was determined that
the large grid sampling size smoothed out the variability
by assumming that a fixed slope length and gradient were
applicable to the entire area. It was concluded that
estimation of erosion on a 1 ha basis (small grid) would
likely lead to the optimum prediction capability. This
conclusion was based primarily on the results of structural
analysis of soil loss data which suggested a workable
continuity range of about 0.1 km for an exponential semi
variogram model. The relative dispersion was about the
same for the smaller and the larger areas.
Jim Yeh et al. (1986) measured soil water pressure
with 94 tensiometers permanently installed at 3 m intervals
48
along a 290 m transect at a 0.3 m depth in New Mexico.
Observations showed a gradual increase of soil water
pressure over time and a high degree of spatial
variability. Variations were spatially correlated over
distances at least 6 m and they were dependent upon their
mean value. These data supported the hypothesis obtained
from stochastic analysis that the variation of soil water
pressure was meandependent.
Phillips (1986) applied geostatistics to determine the
spatial structure of the pattern of variability of shore
erosion to identify the important scale of variation.
Shoreline erosion was measured in terms of recession rates
from two sets of aerial photographs taken in 1940 and 1978.
Statistical analysis indicated that variability of erosion
rates was high. The complex alongshore pattern and the
scale of local variability indicated that, despite
significant longrange differences in erosion rates, short
range, local factors were more important in determining
differences in erosion rates. It was also concluded that
two major factors accounted for alongshore differences in
erosion rates. These were (i) a complex pattern of
differential resistance related to marsh fringe morphology
and (ii) a crenulated, irregular shoreline configuration
affecting exposure to wave energy.
Several scientists (Burgess et al., 1981; McBratney
and Webster, 1983a, 1983b; Webster and Burgess, 1984;
49
Webster and Nortcliff, 1984; Russo, 1984) have used
geostatistics for improving sampling techniques. The
classical statistical approach for sampling soils does not
take account of the spatial dependence among the data
within one class. Therefore, it leads to conservative
estimates of precision, with oversampling and unnecessary
cost resulting (Burgess et al., 1981). Burgess et al.
(1981) presented a sampling strategy that depended on
accurately determining the semivariogram of the property,
and then estimation variances could be calculated for any
combination of block size and sampling density by kriging.
By this sampling method, the sampling density needed to
attain a predetermined precision could be obtained, and the
sampling effort needed to achieve the precision desired was
at a minimum.
McBratney and Webster (1983b) stated that the number
of observations needed to achieve a particular acceptable
error depends on the variation of the property in the
region concerned. The assumptions of classical statistics
have required more observations than investigators could
afford to attain the desired precision. These authors used
a method for determining the sample size that depended on
knowing the semivariogram of the property of interest.
The semivariogram information was used in kriging for
estimation of variance in the neighborhood of each
observation point. Variances were pooled to form the
50
global variance from which a standard error could be
calculated. The pooled value was minimized for a given
sample size if all neighborhoods were of the same size.
Therefore, the sampling size required to determine the
semivariogram would be a major part of the task. So, if
the semivariogram had not been known, then the best
strategy was to sample on a regular grid, with the interval
determined by the number of observations that could be
reasonably obtained.
McBratney and Webster (1983a) extended the sampling
principle for each variable to two or more coregionalized
variables. The choice of the strategy was complicated
because not only did the sampling intensities of the main
variable and subsidiary variables differ but also their
relative sampling intensities could be changed.
Conversely, maximum kriging variance did not necessarily
occur at the center of the sampling configuration as it did
with a single variable. It was stated that in attempting
to find an optimal strategy, the maximum kriging variance
must be found by first calculating the variance for a range
of sampled spacings and relative sampling intensities.
Those that matched the maximum tolerable variance were
potentially useful. It was suggested that the optimum
scheme was the one that achieved the desired precision for
least cost.
51
Webster and Burgess (1984) described optimal
rectangular grid sampling configurations by which
estimation variance could be minimized. The geostatistical
approach had the advantage that standard errors would be
much smaller than with the classical approach. It was
stated that even when standard errors were estimated
properly by taking into account known spatial dependence,
the cost of making the desired number of measurements in a
region might still be prohibitive. Under those
circumstances weighting might provide a feasible way of
overcoming this difficulty. The aim of weighting was to
reduce the effort of measuring soil properties within
regions while maintaining the precision of replicated
observations. It was concluded that the most serious
obstacle to using optimal sampling strategies for single
estimates was the need to know the semivariogram in
advance. The main task was the number of samples needed to
determine the semivariogram.
Webster and Nortcliff (1984) used measured values of
extractable Fe, Mn, Cu, and Zn contents to calculate the
sampling effort required to estimate mean values with
specified precision. Semivariograms showed that there was
a substantial dependence for Fe and Mn contents, less for
Zn content, and even less for Cu content. Estimation
variances generated by classical methods and geostatistics
were compared. The largest nugget variance in relation to
52
the total variance in the sample was for Cu. Classical
statistics slightly exaggerated the estimation variance for
Cu. The overestimate was more serious for Zn, Mn, and Fe.
However, the major disadvantage is having to sample
intensively to obtain the semivariogram.
Russo (1984) proposed a method to design an optimal
sampling network for semivariogram estimation. The method
required an initial sampling network. The location of
points could be either systematically or randomly selected.
For a given sample size (n) and using a constant number of
pairs of points for each lag class, the sampling network
criterion for selecting the location of sampling points was
the uniformity of the values of the separating lag distance
(h) within a given lag class, for each of the lag classes
which covered the area of interest in the field. The
method provided a set of scaling factors which were used to
calculate the new locations of the sampling points by an
iterative procedure. Using the aforementioned criterion,
the best set of sampling points was selected. Analysis of
results indicated that by using the proposed method the
variability within and among lag classes was considerably
reduced relative to the situation where the original
locations were used. In addition, sampling points
generated by the method proposed fitted the theoretical
semivariograms better than those which were estimated from
53
data generated on the original coordinates of sampling
points.
Fractals
It has been widely recognized that the perception of
soil variation is a function of the scale of observation.
Fridland (1976) was one of the first soil scientists to
recognize that a series of randomly operating but
interacting spatial processes at different scales could be
combined to give definite soil patterns.
Beckett and Bie (1976) indicated that the variance of
the values of any soil property within a given area is the
sum of all contributions to the soil variability within the
area. Thus, the overall variance within an area of 100 m2
contains contributions from the average variability within
areas of 1 m2, and from that between areas of 1 m2 within
areas of 5 m2, and between areas of 5 m2 within areas of 10
m2, and between areas of 10 m2 within areas of 100 m2. The
partition of the total variance can be performed for any
number of stages.
Wilding and Drees (1978) pointed out that the nature
of soil variability is dependent on the scale of
resolution. They indicated that at a low resolution level
(for example, looking at the earth from the moon) spatial
diversity may be seen as land vs water. With greater
resolution, spatial variabilty can be recognized
54
microscopically and submicroscopically in the systematic
organization of biological, chemical, and mineralogical
composition of hand specimens representative of given
horizons.
Burrough (1983a) stated that each cause of soil
variation may not only operate independently or in
combination with other factors, but also over a wide range
of scales.
Soil variation has been considered to be the result of
a systematic and a random components (Fridland, 1976;
Wilding and Drees, 1978; 1983). The former is related to
features such as landform, geomorphic elements, and soil
forming factors. The latter corresponds to those changes
in soil properties that are not related to a known cause.
Random variability is unresolved.
Burrough (1983b) indicated that the distinction
between systematic variation and noise (random variation)
is entirely scale dependent because increasing the scale of
observation almost always reveals structure in the random
component. He also stated that making allowances for the
artifices of map making, several conclusions can be drawn:
(i) pattern structures, and therefore spatial correlations,
have been recognized at all scales; (ii) the detail
resolved is partly the result of the scale of variation
present and partly due to the resolving power of the map at
the given scale; (iii) the intricacy of the drawn
55
boundaries is not related to scale; and (iv) a feature
regarded as random at one scale can be revealed as
structure at a larger scale. Also, Burrough (1983b)
pointed out that in any given spatial study there may be
many sources and scales of variability present. The
sources and scales of variability come into play
simultaneously and affect observations over all distances
between the resolution of the sampling device and the
largest intersample distance. Therefore, it is necessary
to find a substitute for the noise concept that takes into
account the nested, autocorrelated, and scale dependent
character of unresolved variations. Burrough (1983a;
1983b; 1983c) suggested that the concepts embodied in
fractals appear to offer a solution.
The term fractal was introduced by Mandelbrot (1977)
specifically for temporal and spatial phenomena that were
continuous but not differentiable, and exhibited partial
correlations over many scales. A continuous series, such
as a polynomial, is differentiable because it can be split
into an infinite number of absolutely smooth straight
lines. A nondifferentiable continuous series cannot be
solved. Every attempt to split a nondifferentiable
continuous series into smaller parts results in the
resolution of still more structure or roughness. Fractal
etymologically has the same root as fraction and fragment
56
and means "irregular or fragmented." It also means "to
break."
Fractals have two important characteristics (Burrough,
1983b). They embody the idea of "selfsimilarity," that
is, the manner in which variations at one scale are
repeated at another, and the concept of a fractional
dimension. The concept of fractional dimension is the
source of the name "fractal."
Mandelbrot (1977) defined a fractal curve as one where
the HausdorffBesicovitch dimension (D) strictly exceeds
the topological dimension. The simplest example is a
continuous linear series such as a polynomial which tends
to look more and more like a straight line as the scale at
which it is examined increases. The D value is calculated
using the following equation:
D = log N/log r (42)
where D = HausdorffBesicovitch dimension
N = number of steps used to measure a pattern
r = scale ratio
Burrough (1983a) pointed out that for a linear fractal
curve, D may vary between 1 (completely differentiable) and
2 (noisy). The corresponding range for D lies between 2
(absolutely smooth) and 3 (infinitely crumpled) for
surfaces. It is implicit in the concept of fractal that
57
when fractals are examined at increasingly large scales
increasing amounts of detail are revealed, while at the
same time vestiges of variations persist on the smaller
scale.
Mandelbrot (1977) developed the fractal theory based
on the physical Brownian motion. Burrough (1983b, 1983c)
extended the fractal theory to soils using Brownian and
nonBrownian fractal models and indicated that soil data
were fractals because increasing the scale of mapping
continued to reveal more and more detail. Soil data were
not "ideal" fractals because the data did not possess the
property of selfsimilarity at all scales. Pure fractals
are theoretically infinitely nested structures with
infinite variance.
Burrough (1981, 1983a) demonstrated that the double
logarithmic plot of a semivariogram of a series which can
be represented by a fractional Brownian function was a
straight line of slope:
m = 4 2 D (43)
where m = slope.
D = HausdorffBesicovitch dimension.
Therefore, semivariograms are also useful in
computing the fractal dimension, but despite this fact,
58
fractals have been not used by many scientists, especially
soil scientists.
Burrough (1981) computed D from semivariograms of
different soil properties. D values varied between 1.1 and
1.9. Low values indicated a predominance of a systematic
variation in soil properties studied. Large values
indicated a random variation of soil properties. Most of
the fractal values were between 1.5 and 1.9. Fractals were
also useful in revealing short and longrange variation
when the D dimension was used along the semivariogram
range. Low values of D indicated domination of longrange
variation.
Fractals have been also applied to erosion studies.
Phillips (1986) studying shoreline erosion used the
methodology proposed by Burrough (1981, 1983b). He
calculated a D value of 1.91. This value indicated a very
complex, irregular pattern of erosion which was
statistically random. It also indicated a pattern
dominated by shortrange, local controls which completely
obscured any longrange trends that may have existed.
A negative correlation between adjacent sites was also
found. Phillips (1986) concluded that the complex
landscape revealed by the analysis was probably related to
the dynamic nature of estuaries and coastal wetlands and
the variety of geomorphic, ecological, and human factors
that influenced marsh and shoreline development.
DESCRIPTION OF STUDY AREA
Location
The area studied is located in northwest Florida. It
extends from Santa Rosa County on the west to Madison
County on the east, and comprises most of the Florida
Panhandle (Figure 5).
Physiography, Relief, and Drainage
The study area lies in the Coastal Plain Province
(Duffee et al., 1979, 1984; Sanders, 1981; Sullivan et al.,
1975; Weeks et al., 1980). The landscape is largely the
product of streams and waves acting upon the land surface
over the past 10 to 15 million years (Fernald and Patton,
1984).
The major physiographic divisions in the area are the
Northern Highlands and the Marianna Lowlands. They comprise
the Southern Pine Hills, the Dougherty Karst, the Tifton
Uplands, the Apalachicola Delta, and the Ocala Uplift
physiographic districs. Elevations in the Northern
Highlands range from 16 or less to 114 m above sea level.
Several stream systems have produced a significant
erosional feature called the Marianna Lowlands, which
59
60
JACKSON
WALTON HOLMES
LEON MADISON
SANTA ROSA
BAY JEFFERSON
Study counties
Kilometer
0 50 100 150
Figure 5. Location of the counties from which
characterization data were available
for pedons selected for study.
61
interrupts the continuous span of the highlands across
northwest Florida. Elevations in the Marianna Lowlands
range from 20 to 80 m above sea level (Brooks, 1981a;
Fernald and Patton, 1984).
Topography varies from nearly level to gently
undulating, with slopes ranging from 0 to 35%.
Commonly the gentle slopes terminate in sinks or shallow
depressions.
The drainage system is well organized in streams that
flow southward from Alabama and Georgia. The Chattahoochee
and Flint Rivers combine to form the Apalachicola River,
the largest in this southwardflowing group of rivers.
Some of the drainage is disjointed particularly in the
karst topography of the Marianna Lowlands (Fernald, 1981).
Geology
Soils are mainly underlied by the Citronelle
Formation, the Crystal River Formation, and by
undifferentiated Miocene and Oligocene sediments (Fernald,
1981).
The Citronelle Formation is composed of sand, gravels,
and clays of Plioceneage. The Crystal River Formation
comprises shallow marine limestone of Eoceneage. Miocene
and Oligocene sediments are mainly composed of "silty"
sand, clay, dolomitic limestone, and fossiliferous shallow
62
marine limestone. Some of the materials are part of the
Marianna Limestone Formation.
Climate
The climate of the area is controlled by latitude and
proximity to the Gulf of Mexico. The area studied is
characterized by long, warm summers and short, mild winters
(Bradley, 1972). Maximum and minimum temperatures are
affected by breezes coming from the Gulf of Mexico.
The average annual temperature is approximately 21Q C.
Maxima of about 382 C occur in June to August and minima of
about 10Q C occur in January and February. The average
growing season is approximately 275 days.
The average annual rainfall ranges between 1400 and
1660 mm. Approximately 50% of the average rainfall falls
during a 4month rainy season from June to September. A
second period of relatively high rainfall occurs in the
late winter and early spring. Frequently, a short drought
during the late spring causes considerable moisture stress
to trees, crops, and grasses.
Land Use and Vegetation
The area studied has a considerable extension of prime
farmland that is adequate for producing crops and to
sustain high yields under conditions of high levels of
management (Caldwell, 1980). Most of the acreage is used
for urbanization, field crops, pasture, and forestry. The
63
most common crops are corn (Zea mays), soybean (Glycine
max), peanuts (Arachis hypogaea), watermelon (Citrullus
vulgaris), tobacco (Nicotiana spp), and assorted
vegetables. Livestock operations are also common.
A large part of the area is also covered by forest.
Well drained areas are characterized by the presence of
slash pine (Pinus ellioti var ellioti Engelm.), black jack
oak (Quercus marilandica Munch.), turkey oak (Quercus
laevis Walt), bluejack oak (Quercus incana Bartr.), long
leaf pine (Pinus palustris Mill), and laurel oak (Quercus
hemiphaerica Bartr.). The poorly drained areas,
corresponding to shallow, densely wooded swamps, and river
valley lowlands, are characterized by the presence of saw
palmetto (Serenoa repens Bartr.), sweet gum (Liquidamber
styraciflua L.), and cypress (Cupressus sp. L.) (Duffee et
al., 1979, 1984; Sanders, 1981; Sullivan et al., 1975;
Weeks et al., 1980).
Soils
Soils in the area studied have developed from medium
textured marine sediments. These coastal plain materials
were transported from uplands farther north during
interglacial periods when the present land areas were
inundated by water from the Gulf of Mexico. Most of the
soils in the study area are characterized by a low level of
natural fertility and are susceptible to erosion (Duffee et
64
al., 1979, 1981; Sanders, 1981; Sullivan et al., 1975;
Weeks et al., 1980).
Approximately 83% of the soils are classified as
Ultisols (Table 1). Complete taxonomic classification is
presented in Appendix A. In general, the Typic Hapludults;
and the Typic, Aquic, Plinthic, and Rhodic Paleudults are
well and moderately welldrained, with moderate to low
available water capacity and with moderate to moderately
slow permeability. These soils are acidic, low in organic
matter and nutrient contents. In gently sloping areas,
limitations are moderate for cultivate crops due to the
erosion hazard.
Arenic Hapludults; Arenic, Grossarenic, Arenic
Plinthic, and Grossarenic Plinthic Paleudults; and Typic
Quartzipsamments commonly are well to excessively drained.
Permeability varies from rapid to moderately rapid, and
available water capacity is low to very low. Droughtness
and low water retention capacity are among the principal
limitations for cropping on these soils.
Typic Fluvaquents; Typic Humaquepts; Typic
Ochraqualfs; Ultic Haplaquods; Typic, Arenic, Grossarenic,
Aeric, Plinthic, Umbric, and Arenic Umbric Paleaquults;
Typic Albaquults; and Typic and Aeric Ochraquults are
typically poorly drained. Permeability varies from
moderate to slow. Excessive wetness and flooding are among
the most important limitations for growing crops.
65
Table 1. Order, Great Group, and relative proportion of
pedons studied.
Order Great Group Number of pedons %
studied
Alfisols Hapludalfs 2 1.3
Ochraqualfs 2 1.3
Entisols Quartzipsamments 5 3.3
Others 2 1.3
Inceptisols Dystrochrepts 1 0.7
Humaquepts 1 0.7
Spodosols Haplaquods 2 1.3
Ultisols Hapludults 10 6.6
Paleudults 97 64.5
Paleaquults 15 9.9
Others 3 2.0
.. ......o .............................................
Nondesignated 11 7.1
series *
TOTAL 151 100.0
These pedons have not been classified.
MATERIALS AND METHODS
Data Source
Data from 151 pedons (Calhoun et al.,1974; Carlisle et
al., 1978, 1981, 1985; I.F.A.S. Soil Characterization
Laboratory, unpublished data) were used for the study. In
total, 20 soil properties were selected (horizon thickness;
very coarse, coarse, medium, fine, and very fine sand
fractions; total sand, silt, and clay contents; pHwater;
pHKCl; organic carbon content; Ca, Mg, Na, and K contents
extractable in NH40AC; total bases; extractable acidity;
CEC; and base saturation). The criterion for selection was
that these soil properties had to have been measured for
each horizon of the pedon. The number of horizons per
pedon varied between 4 and 7 horizons. There were 19,820
observations.
Pedon location, description, and sampling were done by
soil scientists from U.S.D.A. Soil Conservation Service and
the I.F.A.S. Soil Science Department. Physical and
chemical analyses of the soils were made by the personnel
of the Soil Characterization Laboratory of the University
of Florida, Gainesville. Procedures used for sampling and
66
67
chemical and physical analysis were outlined by Calhoun et
al. (1974) and by Carlisle et al. (1978, 1981, 1985).
Approximately half of the data was already stored in
an IBM XT microcomputer using the database management
software KeepIT (ITsoftware, 1984). It was necessary to
input approximately half of the data to complete the set of
observations for this study.
Location of Pedons
The pedons selected for study were located for soil
survey purposes using the system of Ranges and Townships
with the Tallahassee Meridian and Base Line as reference.
The program used for spatial analysis requires the location
of pedons expressed by geographic coordinates (Xs and Ys).
Therefore, each pedon was located on topographic maps at
1:24,000 scale according to the system of Ranges and
Townships, and each location was transformed into cartesian
coordinates (longitude and latitude). Elevation above sea
level was also recorded.
The map of physiographic regions of Florida (Brooks,
1981b) at the 1:500,000 scale was used as a base map to
locate the entire set of pedons. Using as a reference the
point 30Q 00' 00'' N and 87Q 24' 18'' W (X = 0 and Y = 0),
X and Y coordinates were determined. This reference point
was used to allow only positive Xs and Ys in the studied
area.
68
Pedon locations were plotted using the POST command of
Surface II software (Sampson, 1978).
Statistical Analyses
Statistical analyses were performed using an IBM XT
microcomputer and IFASVAX and NERDC main frame computers.
Transfer of data between microcomputer and main frame
computers was possible by using the public domain
communication programs Kermit (to link with IFASVAX) and
YT (to link with CMSNERDC).
Statistical Analysis System software (SAS Institute
Inc, 1982a, 1982b) was used for the normality and principal
component analyses and for plotting purposes. The Fortran
program written by Skrivan and Karlinger (1979) was used
for the geostatistical analysis. Surface II software
(Sampson, 1978) was employed to generate isarithmic
(contour) maps and surface diagrams.
Normality Analysis
The UNIVARIATE procedure (SAS Institute Inc., 1982a)
was used to test normality. This test was mainly based on
the study of skewness, kurtosis, the Kolmogorov test, and
cumulative plots.
The NORMAL option was employed to compute a test
statistic for the hypothesis that the input data had a
normal distribution. The Kolmogorov D statistic was
computed because the sample size was greater than 50.
69
The PLOT option was used to plot the data. The CHART
procedure was employed to obtain histograms of the data.
Principal Component Analysis
The PRINCOMP procedure (SAS Institute Inc., 1982b) was
employed for the PCA. Because the soil properties studied
had different measurement scales, there was a risk of
having heterogeneous variances. An important assumption in
this analysis is the homogeneity of variances (Afifi and
Clark, 1984). Therefore, soil properties were standardized
to mean equal to 0 and variance equal to 1. As a result
the PCs were derived from the correlation matrix instead of
the covariance matrix. Eigenvalues (variances) and
eigenvectors (coefficients) of PCs were obtained by using
the PRINCOMP procedure.
The number of PCs was selected by using a rule of
thumb (Afifi and Clark, 1984, p. 322) that the PCs selected
are those that explain at least 100/P percent of the total
variance where P is the number of variables. The PCs
selected had an eigenvalue that represented > 5% of the
total variance. Eigenvectors for each PC were selected on
the basis that they had a value larger than the value
calculated using the following equation:
Sc = 0.5/ (PC eigenvalue)i (44)
where Sc = Selection criterion
70
The PLOT procedure (SAS Institute Inc., 1982a) was
employed to plot eigenvectors. The larger the value and
the closer the eigenvector to the PC axis, the larger the
contribution of the variable to the total variance. A
varimax rotation (orthogonal rotation of axes) was used
because some of the eigenvectors did not show a clear
contribution to a particular PC.
The FACTOR procedure (SAS Institute Inc., 1982b) was
employed for the varimax rotation and to plot the rotated
eigenvectors.
Each PC is a linear combination of standardized
variables having the eigenvectors as coefficients. Due to
this fact, collinearity between variables can be a problem.
It has been reported (SAS Institute Inc., 1982b) that use
of highly correlated variables produces estimates with high
standard errors. These estimates are very sensitive to
slight changes in the data.
The REG procedure (SAS Institute Inc., 1982b) with the
option COLLIN was used for the analysis of collinearity.
Variables with a tolerance lower than 0.01 were not
considered in the analysis (Afifi and Clark, 1984).
Tolerance is defined as:
T = 1 R (45)
where T = tolerance
R = coefficient of multiple correlation
71
Finally, the correlation coefficient between the PCs
and the soil properties was computed using the equation:
rij = aij (VAR PC)I (46)
where r i = correlation coefficient
aij = eigenvector
VAR PC = PC eigenvalue
Soil properties selected for further study were those
having a high (>10.751) correlation coefficient.
Geostatistical Analysis
A Fortran program written by Skrivan and Karlinger
(1979) was employed. The geostatistical analysis had four
parts.
Semivariograms. The X, Y, and Z (soil property)
values were used as input in this step. Before a valid
semivariogram can be calculated, the drift, if present,
must be removed, otherwise the stationarity assumption is
not fulfilled.
Journel and Huijbregts (1978) stated the criterion to
consider when the drift is absent. They indicated that,
considering the semivariogram as a positive definite
function, an experimental semivariogram with an increase
smaller than h2 (where h = modulus of the lag distance)
for large distances h is incompatible with the intrinsic
hypothesis. Such an increase in the semivariogram most
often indicates the presence of a trend or drift. However,
drift can be determined if the semivariogram has already
72
been calculated. Thus, an iterative process (trial and
error) was followed to calculate the semivariogram.
An observed semivariogram based on the data was
calculated. If drift was present, then the information
contained in the observed semivariogram was used to
calculate the drift coefficients and residuals of the
observations relative to the drift function. Then, a new
semivariogram from the residuals could be calculated.
This process was repeated until drift was removed or a
satisfactory semivariogram was obtained.
Five semivariograms were calculated for each
variable: directionindependent and directiondependent
(NS, EW, NESW, NWSE). The semivariogram plots were
obtained by using the Energraphics software (Enertronics,
1983).
Fitting semivariograms. In this step the structural
information (range, lag distance, and slope) was used to
adjust the parameters in the semivariogram until the model
was theoretical consistent (Gambolatti and Volpi, 1979).
Consistency occurred when the kriged average error (KAE)
was approximately zero and the average ratio of theoretical
to calculate variance, called reduced mean square error
(RMSE) was approximately equal to one. These parameters
are represented by the following equations:
n
(i) KAE = 1/n ill(Zi Zi) (47)
73
where n = number of points
Z. = measured value
Z. = kriged value
n
(ii) RMSE = 1/n i (Zi Zi)2/ a2 (48)
where a2 = calculated variance and is equal to
n1 n n1
02 = K(0) ilrii C(h) ili M(h)+igi1i2 Si2 (49)
where K(0) = sill
P. = unknown weighting coefficient
C(h) = covariance based on semivariance and
sill
i. = unknown LaGragian multiplier
M(h) = drift
S 2 = variance of the measurement error
The fitting procedure was based on the jackknife
method developed by Tukey (Sokal and Rohlf, 1981) which is
a useful technique for analyzing statistics if
distributional assumptions are of concern.
The procedure was to split the observed data into
groups (usually of size one) and to compute values of the
statistic with a different group of observations being
ignored each time. The average of these estimates was used
to reduce the bias in the statistic. The variability among
these values was used to estimate the standard error.
74
Gambolati and Volpi (1979) extended the use of this
technique to geostatistics.
Kriging. Universal kriging was the method used in
this investigation. Universal kriging takes into account
local trends in data, minimizing the error associated with
estimation. The kriged Z value for X and Y location and
its associated variance were computed.
The kriged Z values and associated standard errors
were the inputs to the Surface II software to produce
isoline maps of the different values and associated
variances.
Fractals. Statistical Analysis System (SAS Institute
Inc., 1982a, 1982b) was employed for transforming semi
variance and lag distance values into logarithmic values.
The REG procedure was used to obtain the slope of the
line. The HausdorffBesicovitch dimension was computed by
using equation (43).
Finally, this dissertation was written using
WordPerfect software (SSI Software, 1985).
RESULTS AND DISCUSSION
Test of Normality
The assumption of normality is important for most
statistical analyses. Mean and standard deviation are
needed to characterize completely the distribution of
values if the data are normally distributed. When data are
normally distributed, approximately 95% of the values fall
within two standard deviations of the mean (Montgomery,
1976; Snedecor and Cochran, 1980; SAS Institute Inc.,
1982a).
Gower (1966), however, pointed out that in PCA, unlike
other forms of multivariate analyses, no assumptions are
needed about the distribution of the variates, hypothetical
populations, except when significance tests are of
interest. Likewise, Gutjahr (1985) and Olea (1975) have
stated that the assumption of normality is not needed in
geostatistics. Stationarity is the most important
assumption in geostatistics, although Burrough (1983a)
indicated that stationarity is very difficult to achieve.
Normality, therefore, is not required for PCA and
geostatistics. However, the test of normality was
performed because a large number of soil variability
75
76
studies have implicitly assumed a normal distribution of
soil properties without using any statistical test to
justify this assumption. Also, a large data base was
available. Thus, a conclusion such as "data were non
normally distributed because of the small number of
observations" has no validity in this study.
There are two main tests of normality. One is a
graphical method based on histograms or plots of values
measured on probability paper. The other one is based on a
quantitative measure such as the Kolmogorov test. Rao et
al. (1979) indicated that graphical methods have specific
drawbacks. First, they often rely on visual inspection,
and thus are subject to human error. Second, as graphical
methods are not based on quantitative measures, an
objective statistical evaluation of the goodnessoffit of
the theoretical distribution to the measured data is not
possible. Consequently, the normality analysis was based
on more a quantitative measure rather than a graphical
method.
The data were tested against a theoretical normal
distribution with mean and variance equal to the sample
mean and variance. Skewness, kurtosis, the Kolmogorov D
statistic, and plot of data were used to test the null
hypothesis that the input data values were normally
distributed (SAS Institute Inc., 1982a).
77
When the distribution is not symmetric, the skewness
can be positive (skewed to the right) or negative (skewed
to the left). Kurtosis refers to the degree of peakedness
of a frequency distribution (Silk, 1979). A heavy tailed
distribution has positive kurtosis. Flat distributions
with shorttails or when almost all data values appear very
close to the mean have negative kurtosis. The measure of
skewness and kurtosis for a normally distributed population
is zero (SAS Institute Inc., 1982a).
A significance level (a) value of 0.15 was selected as
the criterion for acceptance or rejection of the null
hypothesis (H = Normal). When normality is tested the
interest is in accepting the null hypothesis. This is in
contrast to most situations when the interest is in
rejecting the null hypothesis. For these reason, Rao et
al. (1979) proposed an a value between 0.15 and 0.20 in
order to have a balance between type I and II errors.
Statistical moments for each soil property were
computed (Table 2). Most variables had large coefficients
of variation (C.V.). Soil pH (water and KC1) had the
lowest variation, reflecting uniform condition of pH, in
this case the acidity.
Other soil properties had a large C.V.. Most of these
soil properties are naturally related, and the large C.V.s
were mutually influenced. For example, the amount of
78
Table 2. Statistical moments of soil properties studied
and Kolmogorov test.
* Mean Variance C.V Skewness Kurtosis D:Normal PROB>D
(%)
TH 30.2 365.8 63.3 1.44 2.92 0.12 <.01
VC 1.2 5.4 189.6 4.59 29.3 0.30 <.01
C 6.4 39.5 97.5 1.36 1.53 0.15 <.01
M 17.0 125.6 65.9 0.86 1.11 0.06 <.01
F 32.7 209.2 44.2 0.40 0.10 0.07 <.01
VF 12.9 68.8 64.4 1.11 1.92 0.07 <.01
TS 70.0 354.8 26.9 1.18 1.63 0.08 <.01
Silt 10.7 78.4 83.0 3.92 27.0 0.16 <.01
Clay 19.4 260.9 83.3 1.33 2.11 0.12 <.01
PHI 5.1 0.35 11.6 0.67 13.5 0.12 <.01
PH2 4.2 0.28 12.5 0.19 9.91 0.10 <.01
OC 0.43 0.52 167.8 3.41 14.1 0.28 <.01
Ca 0.94 5.51 250.3 6.15 49.1 0.34 <.01
Mg 0.36 0.81 253.2 10.8 154.3 0.35 <.01
Na 0.03 0.002 130.8 3.62 25.5 0.22 <.01
K 0.06 0.009 170.5 3.72 19.3 0.28 <.01
TB 1.38 8.76 213.7 6.05 49.1 0.32 <.01
EXT 5.61 30.8 98.9 2.79 12.8 0.16 <.01
CEC 7.01 49.1 99.9 2.83 11.0 0.18 <.01
BS 18.8 399.7 106.2 1.78 2.90 0.18 <.01
* See Abbreviations, pp. xiixiii
TH is expressed in cm; VC, C, M, F, VF, TS, silt, clay,
OC, and BS are expressed as %; Ca, Mg, Na, K, TB, EXT,
and CEC are expressed as cmol/kg.
n = 991
79
extractable cations (Ca, Mg, Na,and K) depends largely on
the CEC, which in turn depends on particle size.
The large variation in particle size (very coarse,
coarse, medium, fine, and very fine sand fractions; silt;
and clay contents) was influenced by the diversity of
Paleudults (Appendix A) and the presence of horizons with
quite different textures. Paleudults had variable
thickness of coarsetextured horizons (Typic, Arenic, and
Grossarenic Subgroups) overlying finetextured argillic
horizons.
Most of the soil properties studied did not have
skewness and/or kurtosis close to zero. The exception was
fine sand. Also, the histogram and normal probability plot
(Figure 6) indicated that fine sand values were normally
distributed, but when the Kolmogorov test was performed, it
indicated that fine sand had a large probability of being
nonnormal. The significance probability (PROB>D) of the
Kolmogorov D statistic (D:Normal) was smaller than
a = 0.15. So, the null hypothesis was rejected for fine
sand.
Results of the Kolmogorov test indicated that the soil
properties studied had a nonnormal distribution. Results
of the Kolmogorov test were also supported by the
histograms and normal probability plots. Histograms
revealed that distribution of values by soil property did
not have the characteristic bellshaped curve of a normal
80
HISTOGRAM
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NORMAL PROBABILITY PLOT
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function.
rilRank of the data value.
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Figure 6. Histogram (a) and normal probability
plot (b) of fine sand content.
4,
81
distribution. In addition, normal distribution plots
indicated a lack of correspondence between the observed and
the theoretical distributions, for example organic carbon
content (Figure 7).
Transformations (logarithmic, arcsine, or square root)
were not made on the original data because the objective
was to accept or reject the normal distribution. In
addition, interpretation of transformed data is complex.
These results could support the fact that there were
systematic patterns of soil properties; observations were
not independent but associated within certain distance.
Patterns of soil properties influenced the probability
distribution.
The presence of trends in soil properties associated
with landscape position has been recognized. Walker et al.
(1968) pointed out that such trends suggested that the
analysis of soil data in terms of mean and standard
deviation is questionable, since the assumption of random
variation does not appear valid.
In addition, Hole and Campbell (1985) indicated that
if placetoplace variation occurred at random, without
elements of organization and order, mapping efforts could
proceed only with the greatest difficulty because
information and experience gained at one location would
have little predictive value at new locations. Under such
circumstances each mapping problem would be unique because
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83
of the lack of a consistent geographic order that can be
transferred from previous experience to analogous settings.
Principal Component Analysis
Twenty soil properties were initialy selected to study
the soil spatial variability using geostatistics.
Geostatistical analysis is time consuming and complex.
Conversely, all soil properties do not have the same degree
of importance to quantify the spatial variability of soils.
Therefore, reduction of soil properties was necessary for
further analysis.
PCA was used as an unbiased method to select the most
important soil properties. Important soil properties were
defined as those that explained a large proportion of the
total variance.
Two sets of data were employed for this analysis. One
set was composed by the weighted average of selected soil
properties in individual pedons. Horizon thickness was
used as the weighting criterion. Information is lost when
averages are used. Therefore, a second set of data composed
of selected soil properties from the surface A horizon were
used.
Principal Component Analysis for Standardized Weighted Data
A basic assumption of PCA is that variables have
homogeneous variances (Afifi and Clark, 1984; Webster,
1977). The soil properties studied had different scales of
84
measurement (thickness was measured in cm; particle size,
organic carbon content, and base saturation in %; and
extractable cations, total bases, extractable acidity, and
CEC in cmol/kg). Therefore, it is difficult to compare
them. For this reason, all soil properties were
standardized to mean zero and variance one.
One measure of the amount of information conveyed by
each PC is in its variance (eigenvalue). For this reason,
the PCs are commonly arranged in order of decreasing
variance (Table 3). The most informative PC is the first
and the least informative is the last.
The criterion for selecting PCs was stated in the
Materials and Methods section. The first five PCs were
selected for further analysis. Each of them explained more
than 5% of the total variance (Table 3). The first five
PCs together explained more than 73% of the total variance.
Different interpretative analyses were performed to
select the soil properties that contributed the most to the
total variance. A very informative display of the
relationships between soil properties and PCs were plots
(Figure 8). The most important soil properties were those
with large values located closer to the axis of the PC.
Some properties did not have a clear contribution to
an individual PC, such as coarse and medium sand fractions
and Mg content (Figure 8). The axes of PCs were rotated

Full Text 
PAGE 1
SELECTION OF IMPORTANT PROPERTIES TO EVALUATE THE USE OF GEOSTATI STI C AL ANALYSIS IN SELECTED NORTHWEST FLORIDA SOILS BY FRANCISCO A. OVALLES A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1986
PAGE 2
This dissertation is dedicated to my wife, Giordana, and my children, Johanna Fernanda and Pedro Jose.
PAGE 3
ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. Mary E. Collins, chairman of the supervisory committee, for her continuous help, guidance, patience, and personal friendship throughout the graduate program. Appreciation is also extended to other members of the committee, Dr. Gustavo Antonini, Dr. Richard Arnold, Dr. Randall "Randy" Brown, and Dr. Stewart Fotheringham, for their constructive reviews of this work, participation on the graduate supervisory committee, and personal friendship. Appreciation is expressed to the Consejo Nacional de Investigaciones Cientificas y Tecnologicas (CONICIT), Venezuela, for the scholarship which supported the author. Thanks are extended to Dr. Willie Harris who introduced me to the Keepit and YT, and always was ready to answer any of my guestions. Very special thanks are due to Dr. Gregory "Greg" Gensheimer for lending me the geostatistical program and his own computer to type this dissertation. Gratitude is expressed to the staff of the Soil Characterization Laboratory, for their friendship and valuable assistance, to other graduate students, staff, and faculty. iii
PAGE 4
Appreciation is extended to all my friends from all six continents (America, Africa, Asia, Europe, Oceania, and Florida) whom I had the pleasure of knowing here. Finally, but certainly not least, I thank my wife, Giordana, my daughter, Johanna Fernanda, and my son, Pedro Jose, for their love and continuous help, encouragement and patience during this work. iv
PAGE 5
TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF TABLES vii LIST OF FIGURES ix ABBREVIATIONS x ii ABSTRACT x i v INTRODUCTION 1 LITERATURE REVIEW 4 Principal Component Analysis 4 Geostatistics 13 Historical Development 13 Theoretical Bases 15 Practical Use 34 Fractals 53 DESCRIPTION OF STUDY AREA 59 Location 59 Physiography, Relief, and Drainage '.'.'.'.59 Geology 61 Climate 62 Land Use and Vegetation 62 Soils [53 MATERIALS AND METHODS 66 Data Source 66 Location of Pedons !!!!!!!!!! 67 Statistical Analyses 68 Normality Analysis 68 Principal Component Analysis 69 Geostatistical Analysis '.'.'.11 RESULTS AND DISCUSSION 75 Test of Normality !75 v
PAGE 6
Page Principal Component Analysis 83 Principal Component Analysis for Standardized Weighted Data 83 Principal Component Analysis for A horizon Standardized Data 93 Principal Component Analysis by Soil Series 101 Geostatistics 113 SemiVariograms 114 Fitting SemiVariograms 140 Kriging 143 Fractals 156 SUMMARY AND CONCLUSIONS 165 APPENDIX A CLASSIFICATION OF SOIL SERIES STUDIED 178 B GEOGRAPHIC COORDINATES OF PEDONS STUDIED 181 C SEMIVARIOGRAMS FOR DIRECTIONS WITH LARGEST VARIABILITY 186 D CONTOUR MAPS FOR DIRECTIONS WITH LARGEST VARIABILITY 192 E MAP OF PHYSIOGRAPHIC REGIONS IN NORTHWEST FLORIDA 196 LITERATURE CITED 197 BIOGRAPHICAL SKETCH 207 vi
PAGE 7
LIST OF TABLES Table Page 1 Order, Great Group, and relative proportion of pedons studied 65 2 Statistical moments of soil properties studied and Kolmogorov test 78 3 Proportion of total variance explained by each principal component 85 4 Eigenvectors of correlation matrix for standardized weighted average of soil properties 89 5 Tolerance of standardized weighted average of soil properties by principal component 91 6 Correlation coefficients between standardized weighted average of soil properties and principal components 92 7 Proportion of total variance explained by each principal component for standardized A horizon data 94 8 Eigenvectors of correlation matrix for standardized properties of A horizon 95 9 Tolerance of standardized properties of A horizon by principal component 96 10 Correlation coefficient between standardized properties of A horizon and principal components 97 11 Correlation coefficient between standardized properties of Al horizon and principal components 99 12 Correlation coefficient between standardized properties of Ap horizon and principal components 100 vii
PAGE 8
Table Page 13 Variability of studied soil properties within and between soil series and between horizons 107 14 Important semivariogram parameters of the weighted average of selected soil properties 127 15 Important semivariogram parameters of the A horizon selected properties 136 16 Goodnessoff it values of the weighted average of selected soil properties 142 17 Goodnessoff it values of the A horizon selected properties 144 18 Fractal dimension (D value) derived from selected soil property semivariograms 158 19 Fractal dimension (D value) derived from selected soil property semivariograms for a reduced study area 162 viii
PAGE 9
LIST OF FIGURES Figure Page 1 Relation among variance, covariance, and seraivariance 20 2 Common semivariogram models 27 3 Equation number 35 31 4 Equation number 36 (a) and Equation number 37 (b) 32 5 Location of the counties from which characterization data were available for pedons selected for study 60 6 Histogram (a) and normal probability plot (b) of fine sand content 80 7 Histogram (a) and normal probability plot (b) of organic carbon content 82 8 Location of standardized weighted average values of soil properties in the plane of the first two principal components 86 9 Location of standardized weighted average values of soil properties in the plane of the rotated first two principal components 88 10 Soil properties with a large contribution to the total variance by county for the Albany series 102 11 Soil properties with a large contribution to the total variance by county for the Dothan series 103 12 Soil properties with a large contribution to the total variance by county for the Orangeburg series 104 13 Location of selected soil series in the plane of the first two principal components 106 ix
PAGE 10
Figure Page 14 Location of selected soil series in the plane of the first two principal components derived from important soil properties 110 15 Location of selected pedons in the studied area 115 16 Weighted average total sand content first directionindependent semivariogram 120 17 Weighted average clay content first directionindependent semivariogram 121 18 Weighted average total sand content fitted directionindependent semivariogram 123 19 Weighted average total sand content directiondependent semivariograms 124 20 Weighted average clay content fitted directionindependent semivariogram 125 21 Weighted average clay content directiondependent semivariograms 126 22 Weighted average organic carbon content fitted directionindependent semivariogram 131 23 Weighted average organic carbon content directiondependent semivariograms 132 24 A horizon clay content fitted directionindependent semivariogram 134 25 A horizon clay content directiondependent semivariograms 135 26 A horizon organic carbon content fitted directionindependent semivariogram 138 27 A horizon organic carbon content directiondependent semivariograms 139 28 Contour map (increment is 10.0%) (a) and diagram (vertical exaggeration is 18x, azimuth of viewpoint is 25s) (b) of kriged weighted average total sand content 146 29 Contour map (increment is 10.0%) (a) and diagram (vertical exaggeration is 18x, azimuth of x
PAGE 11
Figure Page viewpoint is 25q) (b) of kriged weighted average clay content 147 30 Contour map (increment is 2.0%) (a) and diagram (vertical exaggeration is 18x, azimuth of viewpoint is 25q) (b) of kriged A horizon clay content 148 31 Diagram (vertical exaggeration is 18x, azimuth of viewpoint is 25s) of standard errors of kriged weighted average total sand content 153 32 Diagram (vertical exaggeration is 18x, azimuth of viewpoint is 25s) of standard errors of kriged weighted average clay content 154 33 Diagram (vertical exaggeration is 18x, azimuth of viewpoint is 25q) of standard errors of kriged A horizon clay content 155 34 Location of reduced study area 161 35 Weighted average total sand content fitted NS semivariogram 186 36 Weighted average clay content fitted NS semivariogram 187 37 Weighted average organic carbon content fitted NS semivariogram 188 38 A horizon clay content fitted NWSE semivariogram 189 39 A horizon organic carbon content fitted NWSE semivariogram 190 40 Contour map (increment is 10.0%) derived from weighted average total sand content NS semivariogram 192 41 Contour map (increment is 10.0%) derived from weighted average clay content NS semivariogram 193 42 Contour map (increment is 2.0%) derived from A horizon clay content NWSE semivariogram 194 43 Map of physiographic regions in northwest Florida (Source: Brooks, 1981b) 196 xi
PAGE 12
ABBREVIATIONS a = range ACl = A horizon clay content AOC = A horizon organic carbon content BS = Base saturation C = Coarse sand c = Sill Ca = Calcium CEC = Cation exchange capacity Co = County 3 = Bay 30 = Holmes 32 = Jackson 33 = Jefferson 37 = Leon 40 = Madison 57 = Santa Rosa 66 = Walton COV = Covariance C.V. = Coefficient of variation EXT = Extractable acidity F = Fine sand G(h) = GAMMA = Semivariance h = Lag distance K = Potassium M = Medium sand xii
PAGE 13
Mg = Magnesium Na = Sodium OC = Organic carbon PHI = pHwater PH2 = pHKCl Sc = Selection criterion for eigenvectors T = Tolerance TB = Total bases TH = Horizon thickness TS = Total sand VAR = Variance VC = Very coarse sand VF = Very fine sand xiii
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SELECTION OF IMPORTANT PROPERTIES TO EVALUATE THE USE OF GEOSTATI STI CAL ANALYSIS IN SELECTED NORTHWEST FLORIDA SOILS BY FRANCISCO A. OVALLES December, 1986 Chairman: M.E. Collins Major Department: Soil Science Soil variability is a limiting factor in making accurate predictions of soil performance at any particular position on the landscape. A large number of studies have been made to quantify soil variability, but a large portion of them ignored the multivariate character of soils and the geographic aspect of soil variability. Data from 151 pedons in northwest Florida were selected (i) to determine the important properties affecting soil variability and (ii) to evaluate the soil variability in the area studied using geostatistics Data were nonnormally distributed but statistical techniques employed did not require the assumption of normality. This result could support the presence of systematic patterns of soils. xiv
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Principal component analysis was used to reduce the number of soil properties to study the soil variability. Two sets of data were used: weighted average values of soil properties, and A horizon properties. Horizon thickness was used as the weighting criterion. Variables were standardized to mean zero and variance one. Plots of soil properties in the plane of the principal components, varimax rotation, analysis of eigenvalues, eigenvectors, and collinearity and calculation of correlation coefficients between soil properties and principal components were used to select important properties for evaluation of soil variability. A nested analysis of variance indicated that properties selected by the principal component analysis were differentiating properties. Geostatistical analysis was applied to the properties selected. The withinsoil series variance was used as criterion to assess stationarity Drift was present. Consequently, residuals were used to compute semivariograms. Semivariograms of total sand and clay contents showed structure. Nugget variance was present in all semivariograms. Ranges varied from 15 to 35 km. Soil variability was directiondependent. The NS and NWSE were the directions of maximum variability. Organic carbon content had a large pointtopoint variation. xv
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All observed semivariograms had a characteristic wave pattern that indicated a cyclic variation of soil properties Kriged standard error diagrams were functions of the nugget variance and showed areas where more samples are required to increase the precision of estimates. Fractal dimensions indicated the scaledependent character of soil variability. xvi
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INTRODUCTION The fundamental purpose of a soil survey is to estimate the potentials and limitations of soils for many specific uses. Soil delineations are mapped to be as homogeneous as possible in order to correlate the adaptability of soils to various crops, grasses, and trees; and to predict their behavior and productivity under different management practices (Soil Survey Staff, 1951; 1981) Quality of soil surveys has been improved over the years as a result of improved understanding of soil. But soil variability remains as one of the main constraints to reliable soil interpretations and is a limiting factor for making accurate predictions of soil performance at any particular position on the landscape. The study and understanding of soil variability represents a cornerstone for improving soil surveys. Belobrov (1976), a Russian soil scientist, pointed out that "The degree of approximation between the true and the observed soil variability does not depend on the nature of the soil cover, but mainly on the methods of investigation" (p. 147). 1
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For several years, soil scientists used methods of investigation which did not consider the "real nature" of soils, because they ignored the systematic variation of soils on the landscape and assumed a random variation of soils in space. On the other hand, despite the fact that it has been recognized that a soil map unit is imperfect to varying degrees, depending on the scale of the map and the nature of the soil (Soil Survey Staff, 1975), most soil surveys in the U.S. A have accepted an unrealistic model in which map units encompass soil bodies that form discrete, internally uniform units, with abrupt boundaries at their edges (Hole and Campbell, 1985). Studies of soil variability have not been consistent. These studies have considered a random variation of soils and at the same time they have used a limited number of observations for characterizing map units to establish the range of variation of observed properties. The assumption has been that properties measured at a point also represent the unsampled neighborhood. The extent to which this assumption is true depends on the degree of spatial dependence among observations. The number of studies for quantifying soil variability has sharply increased in the last 10 years, but quantification still remains a problem. A large proportion of quantitative studies are based on untested assumptions, ignore the multivariate character of soils, or use a biased
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selection of properties to represent the soil variability, increasing the risk of erroneous conclusions. For these reasons a large soil data base was selected in northwest Florida with the following objectives: (i) to discover which soil properties most strongly influence the soil variability in the area studied, and (ii) to study how geostatistics can be used in evaluating soil variability.
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LITERATURE REVIEW Principal Component Analysis The multivariate character of soil is well recognized; a large set of measurements of soil properties (morphological, chemical, physical, and mineralogical ) can be derived from a single sample. The complete set of available data is not always used for numerical analyses. Hole and Campbell (1985) indicated that the selection of soil properties depends on the objectives of the study, and also reflects the constraints imposed by cost, time, effort, and access. There is no doubt that logically correlated variables, such as soil pH and base saturation, are generally so highly covariant that one or the other should not be included in the analysis. Particlesize fractions (sand, silt, and clay) always add up to 100%, and therefore, the whole set of particlesize data should not be included in the analysis. Consequently, in the process of selecting soil properties, there is an important question to be answered: Are the selected soil properties the most important to represent the variability of the complete set of data? 4
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Webster (1977) pointed out that when one soil property is measured in a set of individual sampling units, the measured values can be represented by their positions on a single line. The relation between any pair of individuals can be represented by the distance between them and the relations among several individuals can be established simultaneously from their relative positions on the line. At the same time, it is almost impossible to visualize their positions on the line and the relations among more than two individuals simultaneously. Thus, he indicated that an alternative way of dealing with multivariate data is to arrange the individuals along one or more new axes. This reduction of an arrangement in many dimensions to a few dimensions is known as ordination. The two most common methods of ordination are Factor Analysis (FA) and Principal Components Analysis (PCA). Shaw and Wheeler (1985) said that in both technigues new variables are defined as mathematical transformations of the original data. However, FA assumes that the original variable is influenced by various determinants: a part shared by other variables known as the common variance; and a unigue variance which consists of both a variance accounted for by influences specific to each variable and also a variance relating to measurement error. In contrast, PCA assumes that statistical variation in the variables is explained by the variables themselves, in this
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case by the common variance. PCA is recommended when there are high correlations between variables, a large number of variables, and a need for only simple data reduction. The major objective in PCA is to select a number of components that explain as much of the total variance as possible, whereas FA is used to explain the interrelationship among the original variables (Afifi and Clark, 1984). PCA has the advantage in that the values of principal components are relatively simple to compute and interpret. PCA is a method that has been used to reduce the number of variables without losing important information (Webster, 1977). In general, the analysis finds the principal axes of a multidimensional configuration and determines the coordinates of each individual in the population relative to those axes. Then, the data can be represented in a few dimensions by projecting the points orthogonally on the principal axes. The basic idea of PCA is to create new variables called the principal components (PC) (Afifi and Clark, 1984). Each new variable is a linear combination of the X variables and can therefore be written as PC = A 11 X 1 + A 12 X 2 + (1) where PC = principal component coefficient (eigenvector) variable
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7 Coefficients of these linear combinations are chosen to satisfy the following requirements: (i) Variance PC. > variance PC> variance PC i z n (ii) The values of any two PCs are uncorrelated. (iii) For any PC the sum of the squares of the coefficients is one. Cuanalo and Webster (1970) used PCA in a study of numerical classification and ordination in which morphological, physical, and chemical soil properties (pH, clay, silt, fine sand, proportion of stones, consistence, water tension, color, mottling, and peatiness) were measured at depths of 13 cm and 38 cm at 85 sites and randomly sampled within physiographic units near Oxford, England. The variables were standardized to unit variance and the population was centered at the origin. It was found that the first six PCs represented almost 70% of the total variation presented in the original data. The first three PCs represented more than 50% of the total variance. The first component showed large contributions from water tension, and chroma in both the topsoil and the subsoil. In the second component, contribution of fine sand in the topsoil (13 cm) and subsoil (38 cm) was dominant. Hue and value made large contributions to the third component. The projection of the population scatter on the plane defined by the first two PCs gave the most informative display of relations in the whole space. These authors suggested that
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8 when numerical data are available, the data should be examined first by ordination procedures; then, the data selected by the ordination procedure can be used with a numerical classification to decide if such classification grouped data satisfactorily. Norris (1972) used PCA to study trends in soil variation. He described several morphological soil properties (stage of organic matter decomposition, percentage of stones, structure, consistence, porosity, roots, biological activity, and color in terms of presence or absence of gley or dark colors) in 410 pedons, 307 pedons located in woods and 103 pedons located in farmland. The first PC accounted for 39% of the total variance, and corresponded to a trend from deep, stoneless pedons developed on a clayey formation to pedons developed on shallow limestone on steep slopes. The second PC accounted for 14% of the total variance and separated pedons located on farmland from those located in the woods. He concluded that the PCs served as a summary of soil variation in the area, because they accounted for a known percentage of the soil variation and were correctly defined in terms of the properties used to describe the soil. Webster and Burrough (1972) sampled the first two horizons from 84 soil pedons and recorded selected soil properties (soil color, CaC0 3 content, depth to CaC0 3 total penetrable soil depth, clay content, organic matter
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9 content, cation exchange capacity (CEC), pH, exchangeable Mg and K contents, and available P content). They used PCA to reduce the dimensionality of the data, and found that the first two PCs accounted for 55% of the total variance (40% the first component and 15% the second component). Separate contributions to the components were determined by projecting vectors on the components axes. They established that those properties determined in the field (CaC0 3 content, depth to CaC0 3 clay content, and subsoil color) were closely correlated and well represented in one dimension in the first component. The properties measured in the laboratory (organic matter content, CEC, and exchangeable Mg content) contributed most to the second component, indicating differences in management rather than natural soil differences. Results of the numerical classification were supported by showing the distribution of sampling sites in space projected on to the plane of the first two components and showing the frequency distribution of the first PC. There was a good agreement among the results. Therefore, it was concluded that when PCs represent the variables that explain soil variation the components can be mapped as isarithms and the maps have interpretable meaning. Kyuma and Kawaguchi (1973) employed PCA to grade the chemical potentiality of 41 Malayan paddy soil samples; 23 physical, chemical, and mineralogical properties were
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evaluated. The first four PCs accounted for 75% of the total variance. The first PC was highly positively correlated with electrical conductivity, exchangeable Ca, Mg, Na, and K contents, moisture, CEC, available Si content, and 0.2 M HClsoluble K. The first PC was highly negatively correlated with the kaolin mineral content. All of these properties were relevant to the chemical potentiality of the soil, thus, the first PC was called the chemical potentiality component. The standardized scores of the first PC were computed. These scores were used for grading soils in terms of the chemical potentiality. The authors stated that the result of grading was reasonable. Placed at the top of the scale were soils developed on juvenile marine sediments. Soils having high sand and/or kaolin content were at the bottom of the scale. The authors concluded that PCA was useful for comparing the soil fertility status among soils. Burrough and Webster (1976) used PCA with Similarity and Canonical Variate Analyses to improve soil classification in eastern Malaysia. Morphological and chemical properties determined by routine analysis were recorded from 66 randomly selected sites. The first nine PCs accounted for more than 70% of the total variance. Scatter diagrams of pairs of components were drawn to elucidate the population structure. Established classes that were originally thought to be desirable overlapped
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11 almost completely with respect to morphological and chemical properties. Dendograms derived from similarity analysis confirmed the interpretations drawn from the scatter diagrams. Williams and Rayner (1977) employed PCA as a method for grouping soils based on chemical composition (Fe, Ti, Ca, K, Si, Al, P, Mg, Mn, Ni, Cu, Zn, Ga, As, Br, Rb, Sr, Y, Zr, and Pb total contents) and other soil properties such as particle size (sand, silt, and clay), loss on ignition, CaC0 3 content, pH, and soil moisture. The scatter diagram showed that the first two components divided the soils into parent material groups. This grouping was also supported by using dendograms derived from similarity analysis. It was concluded, on the basis of the PCA, that the soils sampled came from three parent materials of different ages. McBratney and Webster (1981) studied the relationships between sampling points using PCA. A substantial proportion (44%) of the total variance was explained by the first two PCs. The first component represented color. Varimax rotation was employed to obtain a better interpretation of the scatter diagram but it produced no appreciable improvement in interpretability. The scatter diagram of PC allowed the separation of sampled points into five different groups.
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12 Richardson and Bigler (1984) applied PCA to selected soil properties (clay content, pH, organic carbon content, CaC0 3 equivalent, electrical conductivity, and soluble Mg, Ca, and Na contents) which were meaningful to soil development and plant growth in wetlands in North Dakota. Four routine measurements useful for characterizing and classifying wetland soils were identified by PCA (electrical conductivity, organic carbon content, CaC0 3 equivalent, and clay content). Electrical conductivity and soluble Mg and Na contents were the most important variables in explaining observable differences in wetland soils. In addition, the use of PCA allowed the examination of the interaction of chemical and physical properties with the landscape position of wetland soils, as well as the variation in properties among vegetation zones, after the data were plotted in the plane of the first two PCs. Edmonds et al. (1985) employed PCA as a first step for using Cluster and Discriminant Analyses to study taxonomic variation within three soil map units. Forty different soil properties were included in the analyses. Variables with low variance were excluded by the analysis. PCA was used to reduce the number of dimensions needed to ordinate pedons in the plane of PCs (character space) and to remove intercorrelation of soil properties. The use of PC scores as data for Cluster Analysis avoided distortions in coordinates of the pedons in the plane of PCs. They
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13 compared the results with the taxonomic classification of soils, and concluded that grouping of pedons by numerical taxonomy did not correspond to groupings by taxa in Soil Taxonomy Geostatistics Webster and Burgess (1983) pointed out that to describe soil variation two features of soil must be taken into account. The first is that long range trends have no simple mathematical form; usually, there is not any obvious repeating pattern; and the larger the area or the more intensive the sampling the more complex the variation appears. The second is that the pointtopoint variation in a sample reflects real soil variation. Only a small part is the measurement error. In addition, the same authors indicated that earlier attempts to describe spatial variation in geology and geography involved fitting deterministic global eguations to data, either exactly or by least squares approximation. But the two features mentioned above make the approach inappropiate for soil. Thus, an alternative was to treat the soil as a random function and to describe it using geostatistics techniques. Historical Development Etymologically, the term geostatistics designates the statistical study of natural phenomena, and it is defined as the application of the formalism of random functions to
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14 the reconnaissance and estimation of natural phenomena (Journel and Huijbregts, 1978). Geostatistics was primarily developed for the mining industry (Matheron, 1963). Geostatistics was very useful for engineers and geologists for studying the spacial distribution of important properties such as grade, thickness, or accumulation of mineral deposits. Matheron (1963) considered that, historically, geostatistics was as old as mining itself. He indicated that as soon as mining men concerned themselves with foreseeing results of future work and, in particular, as soon they started to take and to analyze samples and compute mean values weighted by corresponding thickness of deposits and influencezones, geostatistics was born. Geostatistics started in the early 1950s in South Africa with D.G. Krige (Olea, 1975). Krige realized that he could not accurately estimate the gold content of mined blocks without considering the geometrical setting (locations and sizes) of the samples. Matheron expanded Krige 's empirical observations into a theory of the behavior of spatially distributed variables which was applicable to any phenomenon satisfying certain basic assumptions, and the variables were not limited by their physical nature.
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15 Theoretical Bases Classical statistics could not be used for ore estimation because of their inability to take into account the spatial aspect of the phenomenon (Matheron, 1963). An aleatory variable had two essential properties: (i) the possibility, theoretical at least, of repeating indefinitely the test that assigned to the variable a numerical value, and (ii) the independence of each test from the previous and the next tests. A given oregrade within a deposit would not have those two properties. The content of a block of ore was first of all unigue, but on the other hand, two neighboring ore samples were certainly not independent. Earth scientists usually deal with complex phenomena which are the result of the interaction of variables, through relationships which are in part unknown and in part very complex (Olea, 1975). Variations are erratic and often unpredictable from one point to another, but there is usually an underlying trend in the fluctuations which precludes regarding the data as resulting from a completely random process. To characterize variables which are partly stochastic and partly deterministic in their behavior, Matheron (1971) introduced the term regionalized variable. He developed the regionalized variable theory to describe functions which vary in space with some continuity.
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16 A regionalized variable is a continuously distributed variable having a geographic variation too complex to be represented by a workable mathematical function (Campbell, 1978). Although the precise nature of the variation of a regionalized variable is too complex for a complete description, the average rate of change over distance can be estimated by the semivariance. Conversely, Olea (1977) stated that a regionalized variable is a function that describes a natural phenomenon which has geographic distribution. The term geostatistics has come to mean the specialized body of statistical techniques developed by Matheron and associates to treat regionalized variables (Olea, 1984). The theory of regionalized variables has two branches: the transitive methods and the intrinsic theory (Matheron, 1969). The first is a highly geometrical abstraction without probabilistic hypothesis and has little practical interest. The practical counterpart of those geometrical abstractions is the intrinsic theory which is a term for the application of the theory of random variables to regionalized variables. Matheron (1969) and Olea (1975) indicated that regionalized variables are characterized by the following properties: (i) localization, a regionalized variable is numerically defined by a value which is associated with a sample of specific size, shape, and orientation which is
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17 called geometrical support. (ii) Continuity, the spatial variation of a regionalized variable may be extremely large or very small, depending on the phenomenom studied, but despite this fact, an average continuity is generally present, in some cases the average continuity cannot be confirmed, and then a nugget effect is present, (iii) Anisotropy, changes may be gradual in one direction and rapid or irregular in another. These changes are known as zonalities. A basic assumption in the intrinsic theory is that a regionalized variable is a random variate (Matheron, 1969). The observed values are outcomes following some probability density function. Henley (1981) considered that a regionalized variable as a random function which may be defined in terms of a probability distribution (i.e., it may be normally distributed with a particular mean and variance) Olea (1984) indicated that a spatial function can either be described by a mathematical model or given by a relative freguency analysis based on experimentation. The former approach is not practical because of the complexity of spatial functions. The latter is seriously limited by the maximum number of samples that can be collected. Olea (1975) stated that the difficulty of the relative freguency approach with a regionalized variable is that a repeated test cannot be run because each outcome is unigue.
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18 Since a large number of samples are essential to any statistical inference, it is not possible to determine the probability density function which rules the occurrence of a regionalized variable. The impossibility of obtaining the probability density function associated with the variable is not a serious limitation. Most of the properties of interest depend only on the structure of the regionalized variable as specified by its first and second moments (Olea, 1975). A key assumption is stationarity. Stationarity is a mathematical way to introduce the restriction that the regionalized variable must be homogeneous Stationarity permits statistical inference. A test can be repeated by assuming stationarity even though samples must be collected at different points. All samples are assumed to be drawn from populations having the same moments. Several scientists have discussed the assumption of stationarity (Henley, 1981; Huijbregts, 1975; Journel and Huijbregts, 1978; Olea, 1975; 1984; Tipper, 1979; Trangmar et al., 1985; Webster, 1985). Geostatistics invokes a stationary constraint called the intrinsic hypothesis to resolve the impossibility of obtaining a probability distribution. A regionalized variable is called strictly stationary if it is stationary for any order k = 1, 2, 3, 4, n. If k is egual to one, the regionalized variable has firstorder stationarity. Secondorder
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stationarity also implies firstorder stationarity. Secondorder stationarity signifies that the first two moments (covariance and variance) of the difference between two observations are independent of the location and are a function only of the distance between them. In general, for a regionalized variable of order k, all the moments of order k or less are invariant under translation. For a stationary variable, the covariance has the following properties: (i) COV (0) > COV(X 2 X x ) (2) where COV = covariance (ii) LIM COV(h) = 0, h * <*> (3) where LIM = limit (iii) COV(0) =VAR[Y(X)] (4) where VAR = variance (iv) COV(X 2 X 1 ) = COV(X 1 x 2 ) (5) These relations are better visualized in Figure 1. For secondorder stationarity, VAR[Y(X)] must be finite. Then, according to equation (4) COV(0) must be finite. However, many phenomena in nature are subject to unlimited dispersion and cannot correctly be described when they are assigned a finite variance. Thus, to avoid this restriction, the intrinsic theory assumes what is called the intrinsic hypothesis. The intrinsic hypothesis is satisfied if, for any displacement h the first two moments of the difference [Z(x) Z(x + h) ] are independent
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20
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21 of the location x and are a function only of h: E [Z(x) Z(x + h)] = M(h) (First moment) (6) E [{ Z(x) Z(x + h) M(h)} 2 ] = 2 G(h) (Second moment) (7) where M(h) and G(h) are referred as the drift and the semivariance or intrinsic function, respectively. The semivariance is a measure of the similarity, on the average, between observations at a given distance apart. The more alike the observations, the smaller is the semivariance. The semivariogram (Olea, 1975; Journel and Huijbregts, 1978), which is the plot of semivariance against distance h (lag), has all the structural information needed about a regionalized variable: (i) zone of influence that provides a precise meaning to the notion of dependence between samples, (ii) anisotropy when variability is directiondependent revealing the different behavior of the semivariogram for different directions, and (iii) continuity of the variable through space, which is indicated by the shape and the particular characteristics of the semivariogram near the origin. One of the oldest methods of estimating space or time dependency between neighboring observations is through autocorrelation (Vieira et al., 1983). Nash (1985) pointed out that the correlogram (plot of autocorrelation against
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distance) is the mirror image of the semivariogram. Vieira et al. (1983) indicated that when interpolation between measurements is needed, the semivariogram is a more adeguate tool to measure the correlation between measurements. An infinite dispersion is allowed using semivariances According to Journel and Huijbregts (1978) the autocorrelation is egual to f(h) = C(h)/ C(0) (8) where f(h) = autocorrelation C(h) = autocovariance or covariance at distance h C(0) = variance The relationship between C(h) and C(0) is expressed by eguation (4). When the semivariance changes, it is assumed that its variations are small with respect to the working scale. This is a condition of guasi or local stationarity. When the regionalized variable is weakly stationary, it also obeys the intrinsic hypothesis. The semivariance is then given by G(h) = a 2 C(h) (9) where G(h) = semivariance a 2 = population variance C(h) = autocovariance
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The autocorrelation and the semivariance are related by the following equation: f(h) = 1 G(h)/ C(0) (10) Burgess and Webster (1980a) pointed out that the autocorrelation coefficient depends on the variance (equation 8), and according to equation (4) the variance must be finite to fulfill the requirement of stationarity It was indicated earlier that many phenomena in nature are subject to unlimited dispersion. The semivariance is free of this restriction, and consequently is preferred. They also indicated that a second advantage of working with semivariance is that it is easier to take into account local trends in the property of interest. Residuals are used when trends are present. Webster and Burgess (1980) demonstrated that the variance of the residuals from the mean is not equal to the variance of the difference between the values when trends are present. Therefore, autocorrelation is difficult to use. Webster (1985) classified the semivariograms into four groups : Safe models They are defined for one dimension but are safe in the sense that they are conditional positive definite in two and three dimensions. These models are
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24 1. The linear model: G(h) = c Q + wh for h > 0 (11) G(0) = 0 (12) where G = semivariance c Q = intercept or nugget variance w = slope h = lag distance Equation (11) assumes that h has an exponent a = 1. When the exponent a = 0.5 the model is called root. When a = 2 the model is parabolic. 2. The spherical model: G(h) = c Q + w [1.5 (h/a) 0.5 (h/a) 3 ] (13) for 0 < h < a G(h) = c Q + w for h > a (14) G(0) = 0 (15) where a = range c Q + w = sill 3. The exponential model: G(h) = c Q + w [1 exp (h/a)] for h > 0 (16) G(0) = 0 (17) 4. The DeWijsian model: G(h) = c 0 + a ln(h) for h > 0 (18) G(0) = 0 (19) 5. The Gaussian model: G(h) = c 0 + w (1 exp (h/a)*) for h > 0 (20) G(0) = 0 ( 2i)
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25 6. The hyperbolic model: G(h) = h/ a + 3h (22) where a and p are coefficients of the hyperbola function. Risky models The semivariogram increases to a sill. 1. The circular model: G(h) = c Q + w [1 2/ti cos(h/a) + 2h/ na(l h 2 /a 2 )*] for 0 < h < a (23) G(h) = c Q + w for h > a (24) G(0) = 0 (25) 2. The linear model with a sill: G(h) = c Q + w (h/a) for 0 < h < a (26) G(h) = c Q + w for h > a (27) G(0) = 0 (28) Nested model The components of variance measure the amount of variance contributed by each scale. G(h) = i VAR [Z(x) Z(x+h)] = G Q (h) + G 1 (h) (29) where G Q (h) = pure nugget semivariance G^fh) = spatially dependent semivariance Anisotropic model Variability is not equal in all lateral directions. G(h,0) = c Q + u(0) h (30) where u(6) = [A 2 cos 2 (9 a) + B sin 2 (9 a)]* (31)
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26 where 9 = anisotropy angle a = direction of maximum variation A = gradient of semivariogram in direction of maximum variation B = gradient in the direction a + Â§ u The most common semivariograms are showed in Figure 2. Computing a series of semivariograms and deriving a model from them is usually not an end in itself. The objectives of geostatistical studies are to determine the characteristics of the data and to obtain the best estimates possible with the available data. The advantage of using a geostatistical approach is that the computed values are optimum. The error of estimation is minimized. The acronyn BLUE (best linear unbiased estimation) is sometimes used to characterize this method (Green, 1985). Estimation procedures that incorporate regionalized variable theory were originally known as kriging, a term named for D.G Krige ( DeGraf f enreid, 1982). Kriging is a distanceweighted moving average estimation procedure that uses the semivariogram to determine optimal weights. Kriging depends on computing an accurate semivariogram from which estimates of semivariance are then used to obtain the weights applied to the data when computing the averages, and are presented in the kriging eguation (Burgess and Webster, 1980a).
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27 Gaussian Linear, Root, Parabolic Figure 2. Common semivariogram models.
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28 When values of soil properties are averaged over point values, which represent volumes with the same size and shapes as the volumes of soil on which the original descriptions were recorded (i.e., pedons ) the kriging procedure is called punctual kriging (Burgess and Webster, 1980a). When an average is made over areas, the procedure is called block kriging (Burgess and Webster, 1980b). Block kriging produces smaller estimation variances and smoother maps. Burgess and Webster (1980a) and Webster and Burgess (1983) pointed out that kriging is a means of spatial prediction that can be used for soil properties. In kriging, the weights take account of the known spatial dependence expressed in the semivariogram and the geometric relationships among the observed points. Kriging is optimal in the sense that it provides estimates of values at unrecorded places without bias and with minimum known variance. It has been indicated by several scientists (Huijbregts, 1975; Olea, 1975; 1984; Trangmar et al., 1985; Webster and Burgess, 1980) that kriging is used only with regionalized variables that are firstorder stationary. For variables whose drift is not stationary, but for whose residuals the intrinsic hypothesis holds, universal kriging is used.
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29 Webster and Burgess (1980) stated that universal kriging takes account of local trends in data when minimizing the error associated with estimation. Universal kriging can be performed after computing suitable expressions for the drift and corresponding semivariograms of the residuals. Olea (1984) said that universal kriging is a linear estimator of the regionalized variable and has the form n Z(x Q ) = i Â£ 1 r i Z( XjL ) (32) where Z(x Q ) = unknown parameter at location x Q r. = weights Z(x i ) = value of a property at a point x i Matheron (1963) stated that suitable weights r. assigned to each sample are determined by two conditions. The first condition is that Z (x Q ) and Zix^ must have the same average value within the area of influence, and is written as ill r i = 1 (33) The second condition is that r. have such values that estimation variance (kriging variance) of Z(x Q ) and Z(x i ) should take the smallest possible value. The unknown I\ 1 s were found by solution of a system of linear eguations which result from forcing the unbiased estimator to have minimum variance. The eguation is as follows:
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AX = B (34) where A, B, and X are given by equations (35), (36), and (37) (Figures 3 and 4) In recent years a new method for estimation has been developed. Vieira et al. (1983) stated that in soil science, agrometeorology, and remote sensing, very often some variables are crossrelated with others. In addition, some of those variables are easier to measure than others. In such situations estimation of one variable using information about both itself and another crosscorrelated, easiertomeasure variable should to be more useful than the kriging of that variable by itself. This estimation is easily made using cokriging. Cokriging has been defined as the estimation of one spatially distributed variable from values of another related variable (DeGraf fenreid, 1982; Gutjahr, 1984). Dependence between two variables can be expressed by a cross semivariogram (McBratney and Webster, 1983a). For any pair of variables i and j there is a cross semivariance G (h) at lag h^. defined as G ij (h) = E [{Z (x) Z^x+h)} {Zj(x) Z.(x+h)}] (38) where Z and Z. are the values of i and j at places x and x+h. If i = j then, equation (38) represents the auto semivariance.
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31 G(x 1 ,x 1 ) G(X 1 ,X 2 ) 1 f (x 1 ) f a (x 1 ) .fn(x 1 ) G(x 2 ,x 1 ) G(x 2 ,x 2 ) ..G(x 2 ,x k ) 1 f(x 2 ) fMx 2 ). .fn(x 2 ) G(x. ,x 1 ) G(x ,X) .G(x. ,x, ) 1 f (x.) J f 2 (x.) j ..fn(x.) j G(x R ,x 1 ) G(x k ,x 2 ). ..G(x k ,x k ) 1 f(x k ) fMx k ). ..fn(x k ) 1 1 1 0 0 0 0 f (x x ) f(x 2 ) f(x k ) 0 0 0 0 f 2 (x 1 ) f 2 (x 2 ) f*(x k ) 0 0 0 0 fn(x 1 ) fn(x 2 ) fn(x k ) 0 0 0 0 G(x. ,x k ) = Semivariance between two sample elements located at a distance (x. ,x v ). D X f 1 = Function of x, derived from the drift. Figure 3. Equation number 35.
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32 b) G(X 1 ,X Q ) G(x 2 ,x Q ) G(Xj ,x Q ) B = G(x k ,X Q ) 1 f(x Q ) fMx Q ) n fTl(X Q ) Tj = weights. T(x k ,x Q ) = semivariance between two sample elements located at a distance (x k x Q ). f 1 (x) = function of x, derived from the drift. p.. = Lagrange multipliers. Figure 4. Eguation number 36 (a) and equation number 37 (b)
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33 The cokriging equation is given by j nj W = jll iÂ£l r ij z < x ij)' for a11 3 (39) where i, j = variables Z (Xj) = estimated value of variable j at location x Q r ij = we ^9 nts To avoid bias the weights have to fulfill two conditions: nj (i) r j = 1 (40) and nj (ii) i Â£ 1 r A j = 0 for all i not equal to j (41) The first condition, according to McBratney and Webster (1983a), implies that there must be at least one observation of the variable j for cokriging to be possible, and as in kriging equation (34), cokriging can be expressed in matrix notation for solving the unknown weights. Trangmar et al. (1985) indicated that cokriging requires at least one sample point of both the primary variable and covariable properties within the estimation neighborhood. If the primary variable and covariable are present at all sampling sites in the neighborhood, then cokriging is considered as an autokriging of the primary variable alone. In such cases, cokriging is unnecessary.
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34 Practical Use Earlier studies in soil science used time series analysis in which spatial dependence of soil properties was considered. Webster and Cuanalo (1975) computed correlograms for clay, silt, pH, CaC0 3 colorvalue, and stoniness for three horizons in pedons located at 10 m interval along a transect in north Oxfordshire, England. They observed that the relation between sampling points weakened steadily over distances from 10 m to about 230 m. The average spacing between geological boundaries on the transect was also about 230 m. Outcrop bedrock was inferred as one of the main sources of soil variation. They concluded that mappable soil boundaries were likely to occur on the average every 230 m, and sampling at spacing closer than 115 m would be needed to detect them. Lanyon and Hall (1980) used morphological, physical, and chemical soil properties to test the performance and value of autocorrelation analysis. Spatial dependence was determined from observations made every 20 m along a transect for solum thickness; fineearth fraction of the A, B, and C horizons; and for soil pH, percent base saturation (PBS), and exchangeable cations from the deepest horizon. They found that the range varied from 20 m for solum thickness and exchangeable K content to 60 m for pH and exchangeable Mg content. They concluded that autocorrelation analysis emphasized the continuous, orderly
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35 nature of soils, and the fact that spatially related observations may be mutually dependent. Campbell (1978) was one of the first to use geostatistics in soil science. He studied the spatial variation of sand and pH measurements employing the semivariance. Samples were collected at 10 m intervals on two sampling grids positioned on two contiguous delineations in eastern Kansas. There was a contrast in spatial variation of sand content within the two delineations. Distances of 30 and 40 m were sufficient to encounter full variation of sand content. Soil pH had a random variation within both areas. It was concluded that the most important application of semivariograms was in determining optimum sample spacing in the design of efficient sampling strategies. Gambolati and Volpi (1979) introduced the determination of the trend a priori, and improved the process of fitting the observed to a theoretical semivariogram. They used kriging to describe groundwater flow near Venice, Italy. They proposed and used a modification of the kriging technique developed by Matheron (1970) which aimed at improving the accuracy of the interpolation procedure. In Matheron' s (1970) basic theory, the trend was not assessed a priori. The trend was considered as a linear combination of functions with unknown coefficients. Gambolati and Volpi (1979) considered the trend a priori;
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36 therefore the trend had to be determined. Also, they defined the concept of theoretical consistency in kriging applications. Theoretical consistency was derived from the validation of the interpretation model. Validation was made by suppressing each observation point one at a time, by providing an estimate in that point using the remaining (n1) observations, and analyzing the distribution of errors. They stated that consistency occurred when there was no systematic error (kriged average error was approximately zero) and the standard deviation was consistent with the corresponding error (the average ratio of theoretical to calculated variance was approximately equal to one). They found that validation of the interpretation models selected for study showed that their approach yielded accurate results, provided the trend was correctly assessed. Chirlin and Dagan (1980) modeled water flow through twodimensional porous formations as a random process using an approximate formulation of flow physics to obtain an expression for the Head variogram. The Head variogram proved markedly anisotropic, with heads differing more widely on average for a fixed lag parallel to the head gradient than perpendicular to it. Also they examined a hypothetical case ignoring anisotropy. it was determined from their experiment that the kriged standard deviation
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37 was overestimated perpendicular to the mean flow and was underestimated parallel to it. Hajrasuliha et al. (1980) studied salinity levels in three different fields in southwest Iran which were initially sampled on an arbitrarily selected grid of 80 m. Semivariances were calculated for all three sites to determine the degree of dependence between observations. The results from two fields showed that observations were spatially dependent. Contour lines of isosalinity were obtained by using kriging. In the third field salinity observations were found to be spatially independent. Thus, the number of samples necessary to get fiducial limits and to identify the number of samples to be taken randomly across the field for a given probability were obtained by using classical statistical methods. Luxmoore et al. (1981) used semivariograms to characterize spatial variability of infiltration rates into a weathered shale subsoil. Infiltration rates were measured using doublering inf iltrometers installed at 48 locations on a 2 x 2 m grid after the removal of 1 to 2 m of soil. A high degree of variability in infiltration rates was determined. The test for spatial patterning using the semivariogram approach proved negative. Therefore, they concluded that if patterning existed at all, it occurred on a spatial scale less than the 2 m used
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38 in the study. As a result of this study, it was determined that infiltration rate was a randomly distributed property. Vieira et al. (1981) analyzed the spatial variability of 1280 fieldmeasured infiltration rates on Typic Xerorthents. The measurements were made at the nodes of an irregular grid. The semivariogram showed a range of 50 m. It was considered that, on the average, samples separated by 50 m or more were not correlated to each other. Conversely, they examined the effect of the neighborhood size on the value kriged and its estimation variance. They determined that a neighborhood of 14 m was sufficient for the infiltration data. The estimation variances changed very little for larger distances. Low mean estimation error, low variance, and high correlation coefficient showed that the kriging estimation was exceptionally good. Finally, it was determined that geostatistics was useful to redesign the sampling scheme. The large number of measured values made it possible to calculate the minimum number of samples necessary to reproduce the infiltration rate measurements with good precision. It was determined that 128 samples were enough to obtain nearly the same information as with 1280 samples. Geostatistics was used for first time to study soil variability of large areas in Kigali, Rwanda by Vander Zaag et al. (1981). They studied the spatial variability of selected soil properties (pH; exchangeable Ca, Mg, K, and
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Na contents; KClextractable Al content; percent Alsaturation; effective CEC; ug Psorbed at an equilibrium P concentration of 0.02 and 0.2 ug/g; extractable P content; P and Si in the saturation extract; total N, NO3 and NH4; and extractable S contents) in the whole country of Rwanda. Semivariograms of soil pH, exchangeable Ca content, effective CEC, Si in the saturation extract, and extractable NH4 content showed long range spatial dependence. The spatial dependence varied from 37.5 km for soil pH to more than 60 km for extractable NH 4 The information contained in the semivariogram was used to estimate values of soil properties at unsampled locations within the range of the semivariogram. Maps of estimation variance of kriged values were also generated. Such maps showed that estimation variance of kriged values generally increased with increasing distance from sample points. It was indicated that geostatistics could be used to make quick, low cost assessments of soil variability of large land areas. In addition, the map of estimation variance gave an indication of the confidence limits of the estimated values. The map can be used to locate optimum sampling sites to lower the estimation variance. McBratney and Webster (1981) computed semivariograms of subsoil properties (depth to subsoil, soil color, particlesize, mineralogy, organic carbon content total N content, ratio OC/ total N, and pH) Samples were taken on
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40 a transect at 20 m intervals. Semivariograms showed spatial dependence extending to about 360 m for some properties, in particular color and pH. Other subsoil properties had little or no spatial dependence, notably particlesize fractions and organic carbon content. The shape of some semivariograms indicated presence of different map units on the transect. Van Kuilenburg et al. (1982) applied three interpolation technigues (proximal, weighted average, and kriging) to point data involving soil moisture supply capacity on a 2 x 2 km grid of cover sand in the eastern part of the Netherlands. Survey points used for interpolation were randomly stratified with an average density of 1.5 per ha. The root mean sguared error was used as a measure of efficiency. The root mean sguared error was large for the proximal method (less efficient) and there was a negligible difference between root mean sguared errors for weighted average and kriging. Weighted average had the weakness that possible clusters of survey points were weighted too heavily. This was avoided in kriging. Therefore, kriging proved to be the most efficient for the survey method used. Yost et al. (1982a) collected samples from 80 sites at 1 to 2 km intervals in Hawaii. Soil samples were taken from 0 to 15 cm (topsoil) and 30 to 45 cm depths (subsoil). The former depth represented the nutrient status as
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influenced by management and the latter depth represented the natural conditions. Semivariograms for soil pH, exchangeable cations (Ca, Mg, K, Na) sum of cations, P reguirements Si and P in saturation extract, extractable P content, and rainfall were calculated. Ranges were much greater for soil properties in the 0 to 15 cm depth than for those in the 30 to 45 cm depth. Semivariograms for Ca, Mg, K, and P contents based on the 30 to 45 cm depth samples demonstrated greater variability and had smaller ranges (Ca, Mg, and K) than those based on the 0 to 15 cm, or were extremely variable (P). Si in saturation extract had the same range in the subsoil as in the topsoil. Subsoil properties were highly variable. Thus, soil management and rainfall imposed a degree of uniformity on the surface soil properties not apparent in the subsoil. Yost et al. (1982a) concluded that soil chemical properties had spatial dependence and that understanding such structure may provide new insights into soil behavior over the landscape. The semivariograms changed at large distances. These changes suggested that soils should be grouped to obtain uniform regions of soil properties suitable for management regimes. Yost et al. (1982b) used soil data from transects in Hawaii for estimating soil P sorption over the entire island by using kriging. The necessity of considering nonstationarity and the use of universal kriging were
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evaluated. Universal kriging, either by prior polynomial trend removal or by local polynomial trend removal during estimation, was not beneficial in spite of widely varying P sorption and a significant polynomial trend in the data. The kriged estimates indicated that P sorption properties of soil obtained from transects could be estimated in an optimal way and could be displayed in a manner to better understand the soil properties and genesis, and for practical purposes, estimating the fertilizer needs and distribution facilities. McBratney et al. (1982) sampled 3500 sites to study the spatial variability of Cu and Co soluble in mild extractants measured to identify places where these metals were deficient for animals. Semivariograms for both Cu and Co were isotropic and appeared to combine three components of variation: a short range component extending up to 3 km, a long range or geological component extending to 15 km, and a nonspatial or nugget component, which accounted for 32% and 63% of the total variance of Cu and Co, respectively. Cu showed a greater degree of spatial dependence than Co. In addition, isarithmic maps identified areas where Cu and/or Co were deficient. An error map showed that precision was generally acceptable. Also, the map identified a few areas in the region in which sampling was too sparse for confidence.
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43 Byers and Stephens (1983) sampled an untilled mediumgrained fluvial sand in horizontal and vertical transects to study the spatial structure of particle size and saturated hydraulic conductivity. Semivariogram and kriging analyses indicated that both hydraulic conductivity and particle size were relatively isotropic in the horizontal plane but had marked anisotropy in the vertical plane. There were marked similarities in spatial structure in the horizontal plane. The spatial distribution of saturated hydraulic conductivity in the horizontal plane was estimated reasonably well using an empirical relationship between particle size and conductivity along with kriged estimates of the 10% finer particle size. Ten Berge et al. (1983) studied the spatial distribution of selected soil properties (moisture content, moisture tension, bulk density, texture, temperature, and equivalent surface temperature). Two transects were sampled at 4 m intervals. Semivariograms for moisture content and bulk density did not show any range but only a nugget effect. For other soil properties semivariograms had a range varying between 80 and more than 120 m (texture and temperature). Gradual changes in soil characteristics were expected. The presence of abrupt map unit boundaries was determined for some properties (e.g., texture). The spatial structure of the field moisture content was found only at very shallow depths. Texture introduced
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44 differences in hydraulic conductivity, which were thought to cause differences in topsoil moisture content. Vauclin et al. (1983) used geostatistics for studying the variability of particlesize data, available water content, and water stored at 1/3 bar. The soil samples were collected within a 70 x 40 m area at the nodes of a 10 m square grid. All semivariograms had a nugget effect which corresponded to the variability that occurred within distances shorter than the sampling interval and to experimental uncertainties. The range varied from 26 m for water stored at 1/3 bar to 50 m for silt content. Cross semivariograms were calculated demonstrating that available water content at 1/3 bar was correlated with sand content within distances of 43.5 and 30 m. Semivariograms and cross semivariograms were used to krige and cokrige additional values of available water content and water stored at 1/3 bar every 5 m. They indicated that the use of cokriging was a promising tool whether the principal objective was the reduction of the estimated variance compared with kriging or the need to estimate an undersampled variable by taking into account its spatial correlation with another more sampled variable. Spatial variability of nitrates in cotton petioles was determined employing semivariograms and kriging (Tabor et al., 1984). Sampling of petioles was of two types, on transects and from randomly selected sites on a rectangular
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grid. Nitrates in petioles showed definite spatial dependence in the field studied. However, for sampling areas of < 1 m, spatial dependence was insignificant compared to the inherent variability of the sample and laboratory analyses. Semivariograms and kriged maps of nitrates in petioles suggested a strong influence of the cultural practices such as direction of rows and irrigation. Bos et al. (1984) sampled in a rectangular grid 50 x 200 m at 10 m intervals on sandtailings capped with 0 to > 2 m of stripmine overburden. This was done to present and discuss the use of semivariograms to study the spatial variation of extractable P, Na, K, Mg, and Ca contents, extractable acidity, CEC, total P content, pHwater, pHKC1, and soluble salts of the topsoil (0 to 25 cm) and relative elevation in reclaimed Florida phosphate mine lands. Semivariograms were calculated for data taken along transects in four different sampling directions and a combined direction. Some properties (CEC and relative elevation) did not present structure of spatial variation. The range was approximately 6 m for the combined and EW semivariograms. Also, a nugget effect was observed which represented variability at distances < 10 m. Presence of anisotropy could not be established because welldefined sills and ranges could not be determined for directions NS, NESW, and NWSE. The semivariograms were supported by
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too few data points at large distances. It was concluded that semivariograras were useful in determining the spatial variability of soil properties on reclaimed phosphate mine lands and in improving sampling design for liming and fertilization needs. Xu and Webster (1984) used geostatistics to test how these technigues could be applied for large areas. Topsoil of 102 pedons evenly distributed throughout the studied area in China were sampled. Soil pHwater, organic matter, sand, total N, total P, and total K contents were measured. Variation of soil properties appeared to be isotropic. Soil pH showed the strongest spatial dependence. Isarithmic mapping of local estimates of pH showed zones of alkaline soils. Because sampling was sparse, on average one sample for 3.5 km 2 the estimation errors were large. It was suggested that a more intensive sampling scheme would increase confidence in the maps. This would also improve the estimation of semivariograms especially for lags in the range of 0.5 to 5 km. Saddig et al. (1985) collected data on soil water tension from 99 tensiometers along a 76 m row planted with chile pepper and irrigated through trickle tubing placed 5 cm below the soil surface. Semivariograms indicated a large variability and little spatial dependence in soil water tension. The range was < 6 cm. Also, it was determined that variability and spatial dependence were
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47 functions of the method and timing of water application and the magnitude of the soil water tension. When water was applied through a trickle line, variability was greatest and spatial dependence was smallest. Variability was low and spatial dependence high after rain or extensive flooding. Rogowski et al. (1985) were probably the first to use geostatistics to estimate erosion at different scales. Erosion was measured at nodes of three different size grids: 225 measurements from a 15 x 22.5 km grid, 25 from a 5 x 7.5 km grid, and 150 from a 1 x 1.5 km grid in west central Pennsylvania. Erosion at each node was computed using the universal soil loss eguation. Kriging was employed to map potential erosion. It was determined that the large grid sampling size smoothed out the variability by assumming that a fixed slope length and gradient were applicable to the entire area. It was concluded that estimation of erosion on a 1 ha basis (small grid) would likely lead to the optimum prediction capability. This conclusion was based primarily on the results of structural analysis of soil loss data which suggested a workable continuity range of about 0.1 km for an exponential semivariogram model. The relative dispersion was about the same for the smaller and the larger areas. Jim Yeh et al. (1986) measured soil water pressure with 94 tensiometers permanently installed at 3 m intervals
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48 along a 290 m transect at a 0.3 m depth in New Mexico. Observations showed a gradual increase of soil water pressure over time and a high degree of spatial variability. Variations were spatially correlated over distances at least 6 m and they were dependent upon their mean value. These data supported the hypothesis obtained from stochastic analysis that the variation of soil water pressure was meandependent. Phillips (1986) applied geostatistics to determine the spatial structure of the pattern of variability of shore erosion to identify the important scale of variation. Shoreline erosion was measured in terms of recession rates from two sets of aerial photographs taken in 1940 and 1978. Statistical analysis indicated that variability of erosion rates was high. The complex alongshore pattern and the scale of local variability indicated that, despite significant longrange differences in erosion rates, shortrange, local factors were more important in determining differences in erosion rates. It was also concluded that two major factors accounted for alongshore differences in erosion rates. These were (i) a complex pattern of differential resistance related to marsh fringe morphology and (ii) a crenulated, irregular shoreline configuration affecting exposure to wave energy. Several scientists (Burgess et al., 1981; McBratney and Webster, 1983a, 1983b; Webster and Burgess, 1984;
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49 Webster and Nortcliff 1984; Russo, 1984) have used geostatistics for improving sampling technigues. The classical statistical approach for sampling soils does not take account of the spatial dependence among the data within one class. Therefore, it leads to conservative estimates of precision, with over sampling and unnecessary cost resulting (Burgess et al., 1981). Burgess et al. (1981) presented a sampling strategy that depended on accurately determining the semivariogram of the property, and then estimation variances could be calculated for any combination of block size and sampling density by kriging. By this sampling method, the sampling density needed to attain a predetermined precision could be obtained, and the sampling effort needed to achieve the precision desired was at a minimum. McBratney and Webster (1983b) stated that the number of observations needed to achieve a particular acceptable error depends on the variation of the property in the region concerned. The assumptions of classical statistics have reguired more observations than investigators could afford to attain the desired precision. These authors used a method for determining the sample size that depended on knowing the semivariogram of the property of interest. The semivariogram information was used in kriging for estimation of variance in the neighborhood of each observation point. Variances were pooled to form the
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50 global variance from which a standard error could be calculated. The pooled value was minimized for a given sample size if all neighborhoods were of the same size. Therefore, the sampling size reguired to determine the semivariogram would be a major part of the task. So, if the semivariogram had not been known, then the best strategy was to sample on a regular grid, with the interval determined by the number of observations that could be reasonably obtained. McBratney and Webster (1983a) extended the sampling principle for each variable to two or more coregionalized variables. The choice of the strategy was complicated because not only did the sampling intensities of the main variable and subsidiary variables differ but also their relative sampling intensities could be changed. Conversely, maximum kriging variance did not necessarily occur at the center of the sampling configuration as it did with a single variable. It was stated that in attempting to find an optimal strategy, the maximum kriging variance must be found by first calculating the variance for a range of sampled spacings and relative sampling intensities. Those that matched the maximum tolerable variance were potentially useful. It was suggested that the optimum scheme was the one that achieved the desired precision for least cost.
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51 Webster and Burgess (1984) described optimal rectangular grid sampling configurations by which estimation variance could be minimized. The geostatistical approach had the advantage that standard errors would be much smaller than with the classical approach. It was stated that even when standard errors were estimated properly by taking into account known spatial dependence, the cost of making the desired number of measurements in a region might still be prohibitive. Under those circumstances weighting might provide a feasible way of overcoming this difficulty. The aim of weighting was to reduce the effort of measuring soil properties within regions while maintaining the precision of replicated observations. It was concluded that the most serious obstacle to using optimal sampling strategies for single estimates was the need to know the semivariogram in advance. The main task was the number of samples needed to determine the semivariogram. Webster and Nortcliff (1984) used measured values of extractable Fe, Mn, Cu, and Zn contents to calculate the sampling effort reguired to estimate mean values with specified precision. Semivariograms showed that there was a substantial dependence for Fe and Mn contents, less for Zn content, and even less for Cu content. Estimation variances generated by classical methods and geostatistics were compared. The largest nugget variance in relation to
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52 the total variance in the sample was for Cu. Classical statistics slightly exaggerated the estimation variance for Cu. The overestimate was more serious for Zn, Mn, and Fe. However, the major disadvantage is having to sample intensively to obtain the semivariogram. Russo (1984) proposed a method to design an optimal sampling network for semivariogram estimation. The method required an initial sampling network. The location of points could be either systematically or randomly selected. For a given sample size (n) and using a constant number of pairs of points for each lag class, the sampling network criterion for selecting the location of sampling points was the uniformity of the values of the separating lag distance (h) within a given lag class, for each of the lag classes which covered the area of interest in the field. The method provided a set of scaling factors which were used to calculate the new locations of the sampling points by an iterative procedure. Using the aforementioned criterion, the best set of sampling points was selected. Analysis of results indicated that by using the proposed method the variability within and among lag classes was considerably reduced relative to the situation where the original locations were used. In addition, sampling points generated by the method proposed fitted the theoretical semivariograms better than those which were estimated from
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53 data generated on the original coordinates of sampling points Fractals It has been widely recognized that the perception of soil variation is a function of the scale of observation. Fridland (1976) was one of the first soil scientists to recognize that a series of randomly operating but interacting spatial processes at different scales could be combined to give definite soil patterns. Beckett and Bie (1976) indicated that the variance of the values of any soil property within a given area is the sum of all contributions to the soil variability within the area. Thus, the overall variance within an area of 100 m 2 contains contributions from the average variability within areas of 1 m 2 and from that between areas of 1 m 2 within areas of 5m 2 and between areas of 5 m 2 within areas of 10 m 2 and between areas of 10 m 2 within areas of 100 m 2 The partition of the total variance can be performed for any number of stages. Wilding and Drees (1978) pointed out that the nature of soil variability is dependent on the scale of resolution. They indicated that at a low resolution level (for example, looking at the earth from the moon) spatial diversity may be seen as land vs water. With greater resolution, spatial variabilty can be recognized
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54 microscopically and submicroscopically in the systematic organization of biological, chemical, and mineralogical composition of hand specimens representative of given horizons Burrough (1983a) stated that each cause of soil variation may not only operate independently or in combination with other factors, but also over a wide range of scales. Soil variation has been considered to be the result of a systematic and a random components (Fridland, 1976; Wilding and Drees, 1978; 1983). The former is related to features such as landform, geomorphic elements, and soil forming factors. The latter corresponds to those changes in soil properties that are not related to a known cause. Random variability is unresolved. Burrough (1983b) indicated that the distinction between systematic variation and noise (random variation) is entirely scale dependent because increasing the scale of observation almost always reveals structure in the random component. He also stated that making allowances for the artifices of map making, several conclusions can be drawn: (i) pattern structures, and therefore spatial correlations, have been recognized at all scales; (ii) the detail resolved is partly the result of the scale of variation present and partly due to the resolving power of the map at the given scale; (iii) the intricacy of the drawn
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55 boundaries is not related to scale; and (iv) a feature regarded as random at one scale can be revealed as structure at a larger scale. Also, Burrough (1983b) pointed out that in any given spatial study there may be many sources and scales of variability present. The sources and scales of variability come into play simultaneously and affect observations over all distances between the resolution of the sampling device and the largest intersample distance. Therefore, it is necessary to find a substitute for the noise concept that takes into account the nested, autocorrelated, and scale dependent character of unresolved variations. Burrough (1983a; 1983b; 1983c) suggested that the concepts embodied in fractals appear to offer a solution. The term fractal was introduced by Mandelbrot (1977) specifically for temporal and spatial phenomena that were continuous but not dif f erentiable and exhibited partial correlations over many scales. A continuous series, such as a polynomial, is dif f erentiable because it can be split into an infinite number of absolutely smooth straight lines. A nondiff erentiable continuous series cannot be solved. Every attempt to split a nondiff erentiable continuous series into smaller parts results in the resolution of still more structure or roughness. Fractal etymologically has the same root as fraction and fragment
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56 and means "irregular or fragmented." It also means "to break." Fractals have two important characteristics (Burrough, 1983b). They embody the idea of "self similarity that is, the manner in which variations at one scale are repeated at another, and the concept of a fractional dimension. The concept of fractional dimension is the source of the name "fractal." Mandelbrot (1977) defined a fractal curve as one where the Hausdorf fBesicovitch dimension (D) strictly exceeds the topological dimension. The simplest example is a continuous linear series such as a polynomial which tends to look more and more like a straight line as the scale at which it is examined increases. The D value is calculated using the following equation: D = log N/log r (42) where D = Hausdorf fBesicovitch dimension N = number of steps used to measure a pattern r = scale ratio Burrough (1983a) pointed out that for a linear fractal curve, D may vary between 1 (completely dif ferentiable) and 2 (noisy). The corresponding range for D lies between 2 (absolutely smooth) and 3 (infinitely crumpled) for surfaces. It is implicit in the concept of fractal that
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57 when fractals are examined at increasingly large scales increasing amounts of detail are revealed, while at the same time vestiges of variations persist on the smaller scale. Mandelbrot (1977) developed the fractal theory based on the physical Brownian motion. Burrough (1983b, 1983c) extended the fractal theory to soils using Brownian and nonBrownian fractal models and indicated that soil data were fractals because increasing the scale of mapping continued to reveal more and more detail. Soil data were not "ideal" fractals because the data did not possess the property of self similarity at all scales. Pure fractals are theoretically infinitely nested structures with infinite variance. Burrough (1981, 1983a) demonstrated that the double logarithmic plot of a semivariogram of a series which can be represented by a fractional Brownian function was a straight line of slope: m = 4 2 D (43) where m = slope. D = Hausdorf f Besicovitch dimension. Therefore, semivariograms are also useful in computing the fractal dimension, but despite this fact,
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58 fractals have been not used by many scientists, especially soil scientists. Burrough (1981) computed D from semivariograms of different soil properties. D values varied between 1.1 and 1.9. Low values indicated a predominance of a systematic variation in soil properties studied. Large values indicated a random variation of soil properties. Most of the fractal values were between 1.5 and 1.9. Fractals were also useful in revealing shortand longrange variation when the D dimension was used along the semivariogram range. Low values of D indicated domination of longrange variation. Fractals have been also applied to erosion studies. Phillips (1986) studying shoreline erosion used the methodology proposed by Burrough (1981, 1983b). He calculated a D value of 1.91. This value indicated a very complex, irregular pattern of erosion which was statistically random. It also indicated a pattern dominated by shortrange, local controls which completely obscured any longrange trends that may have existed. A negative correlation between adjacent sites was also found. Phillips (1986) concluded that the complex landscape revealed by the analysis was probably related to the dynamic nature of estuaries and coastal wetlands and the variety of geomorphic, ecological, and human factors that influenced marsh and shoreline development.
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DESCRIPTION OF STUDY AREA Location The area studied is located in northwest Florida. It extends from Santa Rosa County on the west to Madison County on the east, and comprises most of the Florida Panhandle (Figure 5). Physiography, Relief, and Drainage The study area lies in the Coastal Plain Province (Duffee et al., 1979, 1984; Sanders, 1981; Sullivan et al., 1975; Weeks et al. 1980). The landscape is largely the product of streams and waves acting upon the land surface over the past 10 to 15 million years (Fernald and Patton, 1984) The major physiographic divisions in the area are the Northern Highlands and the Marianna Lowlands. They comprise the Southern Pine Hills, the Dougherty Karst, the Tifton Uplands, the Apalachicola Delta, and the Ocala Uplift physiographic districs. Elevations in the Northern Highlands range from 16 or less to 114 m above sea level. Several stream systems have produced a significant erosional feature called the Marianna Lowlands, which 59
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60 Figure 5. Location of the counties from which characterization data were available for pedons selected for study.
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61 interrupts the continuous span of the highlands across northwest Florida. Elevations in the Marianna Lowlands range from 20 to 80 m above sea level (Brooks, 1981a; Fernald and Patton, 1984). Topography varies from nearly level to gently undulating, with slopes ranging from 0 to 35%. Commonly the gentle slopes terminate in sinks or shallow depressions The drainage system is well organized in streams that flow southward from Alabama and Georgia. The Chattahoochee and Flint Rivers combine to form the Apalachicola River, the largest in this southwardflowing group of rivers. Some of the drainage is disjointed particularly in the karst topography of the Marianna Lowlands (Fernald, 1981). Geology Soils are mainly underlied by the Citronelle Formation, the Crystal River Formation, and by undifferentiated Miocene and Oligocene sediments (Fernald, 1981) The Citronelle Formation is composed of sand, gravels, and clays of Plioceneage. The Crystal River Formation comprises shallow marine limestone of Eoceneage. Miocene and Oligocene sediments are mainly composed of "silty" sand, clay, dolomitic limestone, and f ossilif erous shallow
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62 marine limestone. Some of the materials are part of the Marianna Limestone Formation. Climate The climate of the area is controlled by latitude and proximity to the Gulf of Mexico. The area studied is characterized by long, warm summers and short, mild winters (Bradley, 1972). Maximum and minimum temperatures are affected by breezes coming from the Gulf of Mexico. The average annual temperature is approximately 21s C. Maxima of about 38q c occur in June to August and minima of about 10q C occur in January and February. The average growing season is approximately 275 days. The average annual rainfall ranges between 1400 and 1660 mm. Approximately 50% of the average rainfall falls during a 4month rainy season from June to September. A second period of relatively high rainfall occurs in the late winter and early spring. Frequently, a short drought during the late spring causes considerable moisture stress to trees, crops, and grasses. Land Use and Vegetation The area studied has a considerable extension of prime farmland that is adequate for producing crops and to sustain high yields under conditions of high levels of management (Caldwell, 1980). Most of the acreage is used for urbanization, field crops, pasture, and forestry. The
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63 most common crops are corn ( Zea mays ) soybean ( Glycine max), peanuts (Arachis hypogaea ) watermelon ( Citrullus vulgaris ) tobacco ( Nicotiana spp) and assorted vegetables. Livestock operations are also common. A large part of the area is also covered by forest. Well drained areas are characterized by the presence of slash pine (Pinus ellioti var ellioti Engelm. ) black jack oak ( Quercus marilandica Munch. ) turkey oak ( Quercus laevis Walt), blue jack oak ( Quercus incana Bartr.), long leaf pine (Pinus palustris Mill), and laurel oak ( Quercus hemiphaerica Bartr.). The poorly drained areas, corresponding to shallow, densely wooded swamps, and river valley lowlands, are characterized by the presence of saw palmetto ( Serenoa repens Bartr.), sweet gum ( Liquidamber styracif lua L. ) and cypress ( Cupressus sp. L. ) (Duffee et al., 1979, 1984; Sanders, 1981; Sullivan et al., 1975; Weeks et al. 1980) Soils Soils in the area studied have developed from mediumtextured marine sediments. These coastal plain materials were transported from uplands farther north during interglacial periods when the present land areas were inundated by water from the Gulf of Mexico. Most of the soils in the study area are characterized by a low level of natural fertility and are susceptible to erosion (Duffee et
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64 al., 1979, 1981; Sanders, 1981; Sullivan et al., 1975; Weeks et al. 1980 ) Approximately 83% of the soils are classified as Ultisols (Table 1). Complete taxonomic classification is presented in Appendix A. In general, the Typic Hapludults; and the Typic, Aquic, Plinthic, and Rhodic Paleudults are well and moderately welldrained, with moderate to low available water capacity and with moderate to moderately slow permeability. These soils are acidic, low in organic matter and nutrient contents. In gently sloping areas, limitations are moderate for cultivate crops due to the erosion hazard. Arenic Hapludults; Arenic, Grossarenic, Arenic Plinthic, and Grossarenic Plinthic Paleudults; and Typic Quartz ipsamments commonly are well to excessively drained. Permeability varies from rapid to moderately rapid, and available water capacity is low to very low. Droughtness and low water retention capacity are among the principal limitations for cropping on these soils. Typic Fluvaguents; Typic Humaquepts; Typic Ochraqualfs; Ultic Haplaquods; Typic, Arenic, Grossarenic, Aerie, Plinthic, Umbric, and Arenic Umbric Paleaquults; Typic Albaguults; and Typic and Aerie Ochraquults are typically poorly drained. Permeability varies from moderate to slow. Excessive wetness and flooding are among the most important limitations for growing crops.
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65 Table 1. Order, Great Group, and relative proportion of pedons studied. Order Great Group Number of pedons studied % Alf isols Hapludalf s 2 1.3 Ochragualf s 2 1.3 Entisols Quartz ipsamments 5 3.3 Others 2 1.3 Inceptisols Dystrochrepts 1 0.7 Humaquepts 1 0.7 Spodosols Haplaquods 2 1.3 Ultisols Hapludults 10 6.6 Paleudults 97 64.5 Paleaquults 15 9.9 Others 3 2.0 Nondesignated series 11 7.1 TOTAL 151 100.0 These pedons have not been classified.
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MATERIALS AND METHODS Data Source Data from 151 pedons (Calhoun et al.,1974; Carlisle et al., 1978, 1981, 1985; I.F.A.S. Soil Characterization Laboratory, unpublished data) were used for the study. In total, 20 soil properties were selected (horizon thickness; very coarse, coarse, medium, fine, and very fine sand fractions; total sand, silt, and clay contents; pHwater; pHKCl; organic carbon content; Ca, Mg, Na, and K contents extractable in NH 4 OAC; total bases; extr actable acidity; CEC; and base saturation). The criterion for selection was that these soil properties had to have been measured for each horizon of the pedon. The number of horizons per pedon varied between 4 and 7 horizons. There were 19,820 observations Pedon location, description, and sampling were done by soil scientists from U.S.D.A. Soil Conservation Service and the I.F.A.S. Soil Science Department. Physical and chemical analyses of the soils were made by the personnel of the Soil Characterization Laboratory of the University of Florida, Gainesville. Procedures used for sampling and 66
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67 chemical and physical analysis were outlined by Calhoun et al. (1974) and by Carlisle et al. (1978, 1981, 1985). Approximately half of the data was already stored in an IBM XT microcomputer using the database management software KeepIT (ITsoftware, 1984). It was necessary to input approximately half of the data to complete the set of observations for this study. Location of Pedons The pedons selected for study were located for soil survey purposes using the system of Ranges and Townships with the Tallahassee Meridian and Base Line as reference. The program used for spatial analysis requires the location of pedons expressed by geographic coordinates (Xs and Ys). Therefore, each pedon was located on topographic maps at 1:24,000 scale according to the system of Ranges and Townships, and each location was transformed into cartesian coordinates (longitude and latitude). Elevation above sea level was also recorded. The map of physiographic regions of Florida (Brooks, 1981b) at the 1:500,000 scale was used as a base map to locate the entire set of pedons. Using as a reference the point 30q 00* 00'' N and 87q 24' 18" W (X = 0 and Y = 0), X and Y coordinates were determined. This reference point was used to allow only positive Xs and Ys in the studied area.
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' 68 Pedon locations were plotted using the POST command of Surface II software (Sampson, 1978). Statistical Analyses Statistical analyses were performed using an IBM XT microcomputer and IFASVAX and NERDC main frame computers. Transfer of data between microcomputer and main frame computers was possible by using the public domain communication programs Kermit (to link with IFASVAX) and YT (to link with CMS NERDC ) Statistical Analysis System software (SAS Institute Inc, 1982a, 1982b) was used for the normality and principal component analyses and for plotting purposes. The Fortran program written by Skrivan and Karlinger (1979) was used for the geostatistical analysis. Surface II software (Sampson, 1978) was employed to generate isarithmic (contour) maps and surface diagrams. Normality Analysis The UNIVARIATE procedure (SAS Institute Inc., 1982a) was used to test normality. This test was mainly based on the study of skewness, kurtosis, the Kolmogorov test, and cumulative plots. The NORMAL option was employed to compute a test statistic for the hypothesis that the input data had a normal distribution. The Kolmogorov D statistic was computed because the sample size was greater than 50.
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69 The PLOT option was used to plot the data. The CHART procedure was employed to obtain histograms of the data. Principal Component Analysis The PRINCOMP procedure (SAS Institute Inc., 1982b) was employed for the PCA. Because the soil properties studied had different measurement scales, there was a risk of having heterogeneous variances. An important assumption in this analysis is the homogeneity of variances (Afifi and Clark, 1984). Therefore, soil properties were standardized to mean egual to 0 and variance equal to 1. As a result the PCs were derived from the correlation matrix instead of the covariance matrix. Eigenvalues (variances) and eigenvectors (coefficients) of PCs were obtained by using the PRINCOMP procedure. The number of PCs was selected by using a rule of thumb (Afifi and Clark, 1984, p. 322) that the PCs selected are those that explain at least 100/P percent of the total variance where P is the number of variables. The PCs selected had an eigenvalue that represented > 5% of the total variance. Eigenvectors for each PC were selected on the basis that they had a value larger than the value calculated using the following equation: Sc = 0.5/ (PC eigenvalue)^ (44) where Sc = Selection criterion
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70 The PLOT procedure (SAS Institute Inc., 1982a) was employed to plot eigenvectors. The larger the value and the closer the eigenvector to the PC axis, the larger the contribution of the variable to the total variance. A varimax rotation (orthogonal rotation of axes) was used because some of the eigenvectors did not show a clear contribution to a particular PC. The FACTOR procedure (SAS Institute Inc., 1982b) was employed for the varimax rotation and to plot the rotated eigenvectors Each PC is a linear combination of standardized variables having the eigenvectors as coefficients. Due to this fact, collinearity between variables can be a problem. It has been reported (SAS Institute Inc., 1982b) that use of highly correlated variables produces estimates with high standard errors. These estimates are very sensitive to slight changes in the data. The REG procedure (SAS Institute Inc., 1982b) with the option COLLIN was used for the analysis of collinearity. Variables with a tolerance lower than 0.01 were not considered in the analysis (Afifi and Clark, 1984). Tolerance is defined as: T = 1 R where T = tolerance R = coefficient of multiple correlation (45)
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Finally, the correlation coefficient between the PCs and the soil properties was computed using the equation: r ij = a ij (VAR PC) (46) where r^j = correlation coefficient a^j = eigenvector VAR PC = PC eigenvalue Soil properties selected for further study were those having a high (>0.75) correlation coefficient. Geostatistical Analysis A Fortran program written by Skrivan and Karlinger (1979) was employed. The geostatistical analysis had four parts Semivariograms The X, Y, and Z (soil property) values were used as input in this step. Before a valid semivariogram can be calculated, the drift, if present, must be removed, otherwise the stationarity assumption is not fulfilled. Journel and Huijbregts (1978) stated the criterion to consider when the drift is absent. They indicated that, considering the semivariogram as a positive definite function, an experimental semivariogram with an increase smaller than h 2 (where h = modulus of the lag distance) for large distances h is incompatible with the intrinsic hypothesis. Such an increase in the semivariogram most often indicates the presence of a trend or drift. However, drift can be determined if the semivariogram has already
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72 been calculated. Thus, an iterative process (trial and error) was followed to calculate the semivariogram. An observed semivariogram based on the data was calculated. If drift was present, then the information contained in the observed semivariogram was used to calculate the drift coefficients and residuals of the observations relative to the drift function. Then, a new semivariogram from the residuals could be calculated. This process was repeated until drift was removed or a satisfactory semivariogram was obtained. Five semivariograms were calculated for each variable: directionindependent and directiondependent (NS, EW, NESW, NWSE). The semivariogram plots were obtained by using the Energraphics software (Enertronics, 1983) Fitting semivariograms In this step the structural information (range, lag distance, and slope) was used to adjust the parameters in the semivariogram until the model was theoretical consistent (Gambolatti and Volpi, 1979). Consistency occurred when the kriged average error (KAE) was approximately zero and the average ratio of theoretical to calculate variance, called reduced mean square error (RMSE) was approximately equal to one. These parameters are represented by the following equations: n (i) KAE = 1/n i fi 1 (Z i Z ) (47)
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73 where n = number of points = measured value Z^ = kriged value n (ii) RMSE = 1/n i Â£ 1 (Z i Z ) 2 / a 2 (48) where a 2 = calculated variance and is equal to n1 n n1 a 2 = k(0) i Z 1 r i c(h)i Â£ 1 u i M(h)+ i s 1 r i 2 S L 2 (49) where K(0) = sill I\ = unknown weighting coefficient C(h) = covariance based on semivariance and sill = unknown LaGragian multiplier M(h) = drift S^ 2 = variance of the measurement error The fitting procedure was based on the jackknife method developed by Tukey (Sokal and Rohlf, 1981) which is a useful technique for analyzing statistics if distributional assumptions are of concern. The procedure was to split the observed data into groups (usually of size one) and to compute values of the statistic with a different group of observations being ignored each time. The average of these estimates was used to reduce the bias in the statistic. The variability among these values was used to estimate the standard error.
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74 Gambolati and Volpi (1979) extended the use of this technique to geostatistics Kriging Universal kriging was the method used in this investigation. Universal kriging takes into account local trends in data, minimizing the error associated with estimation. The kriged Z value for X and Y location and its associated variance were computed. The kriged Z values and associated standard errors were the inputs to the Surface II software to produce isoline maps of the different values and associated variances Fractals Statistical Analysis System (SAS Institute Inc., 1982a, 1982b) was employed for transforming semivariance and lag distance values into logarithmic values. The REG procedure was used to obtain the slope of the line. The Hausdorf f Besicovitch dimension was computed by using equation (43). Finally, this dissertation was written using WordPerfect software (SSI Software, 1985).
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RESULTS AND DISCUSSION Test of Normality The assumption of normality is important for most statistical analyses. Mean and standard deviation are needed to characterize completely the distribution of values if the data are normally distributed. When data are normally distributed, approximately 95% of the values fall within two standard deviations of the mean (Montgomery, 1976; Snedecor and Cochran, 1980; SAS Institute Inc., 1982a) Gower (1966), however, pointed out that in PCA, unlike other forms of multivariate analyses, no assumptions are needed about the distribution of the variates, hypothetical populations, except when significance tests are of interest. Likewise, Gutjahr (1985) and Olea (1975) have stated that the assumption of normality is not needed in geostatistics Stationarity is the most important assumption in geostatistics, although Burrough (1983a) indicated that stationarity is very difficult to achieve. Normality, therefore, is not required for PCA and geostatistics. However, the test of normality was performed because a large number of soil variability 75
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76 studies have implicitly assumed a normal distribution of soil properties without using any statistical test to justify this assumption. Also, a large data base was available. Thus, a conclusion such as "data were nonnormally distributed because of the small number of observations" has no validity in this study. There are two main tests of normality. One is a graphical method based on histograms or plots of values measured on probability paper. The other one is based on a guantitative measure such as the Kolmogorov test. Rao et al. (1979) indicated that graphical methods have specific drawbacks. First, they often rely on visual inspection, and thus are subject to human error. Second, as graphical methods are not based on guantitative measures, an objective statistical evaluation of the goodnessoff it of the theoretical distribution to the measured data is not possible. Conseguently, the normality analysis was based on more a guantitative measure rather than a graphical method. The data were tested against a theoretical normal distribution with mean and variance egual to the sample mean and variance. Skewness, kurtosis, the Kolmogorov D statistic, and plot of data were used to test the null hypothesis that the input data values were normally distributed (SAS Institute Inc., 1982a).
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77 When the distribution is not symmetric, the skewness can be positive (skewed to the right) or negative (skewed to the left). Kurtosis refers to the degree of peakedness of a freguency distribution (Silk, 1979). A heavy tailed distribution has positive kurtosis. Flat distributions with shorttails or when almost all data values appear very close to the mean have negative kurtosis. The measure of skewness and kurtosis for a normally distributed population is zero (SAS Institute Inc., 1982a). A significance level (a) value of 0.15 was selected as the criterion for acceptance or rejection of the null hypothesis (H = Normal). When normality is tested the interest is in accepting the null hypothesis. This is in contrast to most situations when the interest is in rejecting the null hypothesis. For these reason, Rao et al. (1979) proposed an a value between 0.15 and 0.20 in order to have a balance between type I and II errors. Statistical moments for each soil property were computed (Table 2). Most variables had large coefficients of variation (C.V.). Soil pH (water and KC1) had the lowest variation, reflecting uniform condition of pH, in this case the acidity. Other soil properties had a large C.V.. Most of these soil properties are naturally related, and the large C.V.s were mutually influenced. For example, the amount of
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78 Table 2. Statistical moments of soil properties studied and Kolmogorov test. Mean Variance C.V Skewness Kurtosis D: Normal PROB>D (%) TH 30.2 365.8 63 3 1 /I A 1.44 "1 ft A 2.92 0 1 ft 12 < ft 1 01 VC Â•1 ft 1 2 5 4 ion 189 6 4.59 29.3 U *5 A 30 < A 1 01 C /A 6 4 39.5 97 5 "1 C 1.36 1.53 0 15 < A 1 .01 M 1 *1 ft 17 0 125.6 65 9 0.86 1 11 u A C Ob < A 1 .01 F 32.7 209.2 A A 44 2 ft >1 ft 0.40 A 1 A U 1U A 0 A *7 0 / < A 1 .01 VF 12 9 68.8 64 4 1.11 1 ft ft 1.92 0 ft "7 07 < 01 TS 70.0 354.8 26. 9 1.18 1.63 0. 08 < .01 Silt 10.7 78.4 83. 0 3.92 27.0 0. 16 < .01 Clay 19.4 260.9 83. 3 1.33 2.11 0. 12 < .01 PHI 5.1 0.35 11. 6 0.67 13.5 0. 12 < .01 PH2 4.2 0.28 12. 5 0.19 9.91 0. 10 < .01 OC 0.43 0.52 167. 8 3.41 14.1 0. 28 < .01 Ca 0.94 5.51 250. 3 6.15 49.1 0. 34 < .01 Mg 0.36 0.81 253 2 10.8 154.3 0. 35 < .01 Na 0.03 0.002 130. 8 3.62 25.5 0. 22 < .01 K 0.06 0.009 170. 5 3.72 19.3 0. 28 < .01 TB 1.38 8.76 213. 7 6.05 49.1 0. 32 < .01 EXT 5.61 30.8 98. 9 2.79 12.8 0. 16 < .01 CEC 7.01 49.1 99. 9 2.83 11.0 0. 18 < .01 BS 18.8 399.7 106. 2 1.78 2.90 0. 18 < .01 See Abbreviations, pp. xiixiii TH is expressed in cm; VC, C, M, F, VF, TS, silt, clay, OC, and BS are expressed as %; Ca, Mg, Na, K, TB, EXT, and CEC are expressed as cmol/kg. n = 991
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79 extractable cations (Ca, Mg, Na,and K) depends largely on the CEC, which in turn depends on particle size. The large variation in particle size (very coarse, coarse, medium, fine, and very fine sand fractions; silt; and clay contents) was influenced by the diversity of Paleudults (Appendix A) and the presence of horizons with guite different textures. Paleudults had variable thickness of coarsetextured horizons (Typic, Arenic, and Grossarenic Subgroups) overlying finetextured argillic horizons Most of the soil properties studied did not have skewness and/or kurtosis close to zero. The exception was fine sand. Also, the histogram and normal probability plot (Figure 6) indicated that fine sand values were normally distributed, but when the Kolmogorov test was performed, it indicated that fine sand had a large probability of being nonnormal. The significance probability (PROB>D) of the Kolmogorov D statistic (DiNormal) was smaller than a = 0.15. So, the null hypothesis was rejected for fine sand. Results of the Kolmogorov test indicated that the soil properties studied had a nonnormal distribution. Results of the Kolmogorov test were also supported by the histograms and normal probability plots. Histograms revealed that distribution of values by soil property did not have the characteristic bellshaped curve of a normal
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80 HISTOGRAM $ 17. 5+* l Â•* 2 .** 6 W .^^^^^sS^^^^^i^^^r^JS^^^r^^^^^t^:^; 101la, ,5S5:<;^:**^:**5S*n:t555*^*5S***^**5X:**^^^<: 14.3 2.5+***** 13 MAY RE PRESENT UP TO COUNTS* 87.5 + NORMAL PR08A3ILITY PLOT : ^ ~ ^ ^ ^ + 2 1 0 lAv Invrs of th* standard normal distribution function. rl'Dank of ths data value. nNuabr of nonalasing values. ^ThaoTstical distribution. Â•sSaapls distribution. Figure 6. Histogram (a) and normal probability plot (b) of fine sand content.
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81 distribution. In addition, normal distribution plots indicated a lack of correspondence between the observed and the theoretical distributions, for example organic carbon content (Figure 7). Transformations (logarithmic, arcsine, or sguare root) were not made on the original data because the objective was to accept or reject the normal distribution. In addition, interpretation of transformed data is complex. These results could support the fact that there were systematic patterns of soil properties; observations were not independent but associated within certain distance. Patterns of soil properties influenced the probability distribution. The presence of trends in soil properties associated with landscape position has been recognized. Walker et al. (1968) pointed out that such trends suggested that the analysis of soil data in terms of mean and standard deviation is guestionable since the assumption of random variation does not appear valid. In addition, Hole and Campbell (1985) indicated that if placetoplace variation occurred at random, without elements of organization and order, mapping efforts could proceed only with the greatest difficulty because information and experience gained at one location would have little predictive value at new locations. Under such circumstances each mapping problem would be unigue because
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82 5.3+* HISTOGRAM 2 2.7 ** ^ * 1, *T T^.,s.vV r. Â— SI ^ ~ ~ t 1~ + Â— K Â—.Â—Â—.Â— => V A Y REPRESENT UP TO 13 COUNTS l 2 l i + i 3 3 2 2 I 3 3 ^ 12 11 10 lh 12 31 37 39 53 505 5.3+ 2.7+ NORMAL PROBABILITY PLOT lvInvera. of the standard normal distribution function. rl'Rantc of the data Talus. n.NuBjber of nonmissing values. > = Th8oretlcal distribution. 'sSaapls distribution. ** ** + *** ++ ++ *+++ **+ ++** +++Â• ** +++ *** +++ ***** 0. 1+**************************** 2 1 0~ ~* +T" inv(ri3/8)/(n+ /4) +2 Figure 7. Histogram (a) and normal probability plot (b) of organic carbon content.
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83 of the lack of a consistent geographic order that can be transferred from previous experience to analogous settings. Principal Component Analysis Twenty soil properties were initialy selected to study the soil spatial variability using geostatistics Geostatistical analysis is time consuming and complex. Conversely, all soil properties do not have the same degree of importance to quantify the spatial variability of soils. Therefore, reduction of soil properties was necessary for further analysis. PCA was used as an unbiased method to select the most important soil properties. Important soil properties were defined as those that explained a large proportion of the total variance. Two sets of data were employed for this analysis. One set was composed by the weighted average of selected soil properties in individual pedons. Horizon thickness was used as the weighting criterion. Information is lost when averages are used. Therefore, a second set of data composed of selected soil properties from the surface A horizon were used. Principal Component Analysis for Standardized Weighted Data A basic assumption of PCA is that variables have homogeneous variances (Afifi and Clark, 1984; Webster, 1977). The soil properties studied had different scales of
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84 measurement (thickness was measured in cm; particle size, organic carbon content, and base saturation in %; and extractable cations, total bases, extractable acidity, and CEC in cmol/kg) Therefore, it is difficult to compare them. For this reason, all soil properties were standardized to mean zero and variance one. One measure of the amount of information conveyed by each PC is in its variance (eigenvalue). For this reason, the PCs are commonly arranged in order of decreasing variance (Table 3). The most informative PC is the first and the least informative is the last. The criterion for selecting PCs was stated in the Materials and Methods section. The first five PCs were selected for further analysis. Each of them explained more than 5% of the total variance (Table 3). The first five PCs together explained more than 73% of the total variance. Different interpretative analyses were performed to select the soil properties that contributed the most to the total variance. A very informative display of the relationships between soil properties and PCs were plots (Figure 8). The most important soil properties were those with large values located closer to the axis of the PC. Some properties did not have a clear contribution to an individual PC, such as coarse and medium sand fractions and Mg content (Figure 8). The axes of PCs were rotated
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85 Table 3 Proportion of total variance explained by each principal component. Principal Component Eigenvalue Proportion (%) Cumulative Proportion 1 5, .9119 29, .56 29 .56 2 3, .0450 15, .23 44 .79 3 2. .5310 12, .65 57 .44 4 1, .9153 9, .57 67, .01 5 1, .2385 6, .19 73, .20 6 0, .8040 4, ,02 77, .22 7 0. .7824 3, ,91 81, .13 8 0, .6933 3, ,47 84, .60 9 0, ,6382 3. ,19 87, ,79 10 0, ,5872 2. ,94 90, .73 11 0, ,4871 2. ,44 93, ,17 12 0. ,4377 2, ,19 95, ,36 13 0. ,3458 1, ,73 97, ,09 14 0. ,2393 1. ,20 98. ,29 15 0. ,2020 1. ,01 99. ,30 16 0. ,1209 0. ,60 99. ,90 17 0. ,0168 0. 08 99. ,98 18 0. ,0037 0. 02 100. ,00 19 0. 0002 0. 00 100. ,00 20 0. 0000 0. 00 100. 00 Proportion of the total variance.
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86 CO rO O o I Â• O I O C\J o o o o I CM o I rO o I d i o I CM Id Z o CL O o < o Q. n 0) Â•H +J M CD Ci 0 a h Â•H 0 n m Â• 0 p c cu CD c 1 Â— 1 0 .1! a > 6 0 CD 0 CP id rH n > Â•H (0 u c n3 Â•H CD +J a X, cr> o H 0) 5 n CO N H H 41 M CD +J C n3 4J 4J 0 CD 41 c 0 id H c 0 H cd d 41 0 0 G 3 H iN3N0dW00 "IVdlONIdd CO CU u CT> Â•H fa
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87 toward clusters of those soil properties with no clear contribution to an individual PC. An orthogonal rotation (Varimax rotation) was employed (Figure 9). For this specific example varimax rotation showed that those soil properties with initially no clear contribution to an individual PC were closer to the axis of the principal component 1 (PCI). This analysis was complemented with a guantitative selection of eigenvectors (coefficients of the linear combination of soil variables). Eigenvectors were calculated for each PC (Table 4). The criterion for selecting important eigenvectors was also stated in the Materials and Methods section. Selected eigenvectors for PCI had an absolute value larger than the selection criterion value (Sc) 0.2056. Soil properties selected as important constituents of the PCI, based on the Sc value, were medium and total sand contents, clay content, Ca, Mg, Na, and K contents, total bases, extractable acidity, and CEC. Eigenvectors selected for principal component 2 (PC2), principal component 3 (PC3), principal component 4 (PC4), and principal component 5 (PC5) had absolute values larger than 0.2865, 0.3143, 0.3613, and 0.4493, respectively. Each PC is defined as a linear combination of the standardized variables, but collinearity among variables may be a problem. An analysis of collinearity was
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88 05 o ro IT) o O Ld O a. o o _l < ql o cr Q. O I rO O un o o o I s o CM o o CM o I o o i 0) H 4> U CD ft 0 M Â• Cu tn +J i Â— i c H 0 c 0 41 s 0 0 o BO O C 0) Â•H CP M fl ft M CD 0 > R3 4) 0) w 4) u XS Â•H Cn tw Â•H 3 CU 41 Q 0 4J N 0 Â•H u T3 M fl s: T3 p C fl +J 0
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89 Table 4. Eigenvectors of correlation matrix for standardized weighted soil properties. Soil Principal Component Property 12 3 4 5 rpTT in n U U z a z a U uuy y a U 1 jib u A 1 A 1 4.341 u 1 Jz4 VL a u 1 O A "3 n U Ub /Z a u j b y d A U ^ "7 53 C z / ob u z b lb L A u 1 o b b n U i "7 c n 1 /bU A u 4bJb A U lb /u A U A C *3 Q Ub Jo TVT M A u zjy4 a (J 1 o o c u 0 ~> ^ 0 1 z j J A u uy uu A U zobU El r u iby 4 a u Ulb / A U 4b J b A u U4bl A U A o O C zoob Vr a U Ulzl a u 1 Q Â£ 1 iy bi A U 4U1 / A u 1 C C A lbbu A u jo /o mo TS a u u i a C i 1 ZD 1 A U lb Jo A u iiy i A U z /4y Silt 0. 1623 0. 1796 0. 0478 0. 4111 0. 3917 Clay 0. 3109 0. 0604 0. 1687 0. 3341 0. 1507 PHI 0 1816 0 3867 0 I860 0 0321 0 0877 PH2 0. 1018 0. 4156 0. 1352 0. 1275 0. 1812 OC 0. 1027 0. 0814 0. 0655 0. 5700 0. 1678 Ca 0. 2654 0. 3372 0. 0558 0. 0649 0. 0258 Mg 0. 2552 0. 2061 0. 0143 0. 0635 0. 1237 Na 0. 2590 0. 0205 0. 0103 0. 0525 0. 2423 K 0. 2577 0. 1255 0. 0119 0. 0861 0. 1200 TB 0. 3007 0. 3353 0. 0407 0. 0343 0. 0267 EXT 0. 3131 0. 2098 0. 0584 0. 0920 0. 2795 CEC 0. 3755 0. 0274 0. 0307 0. 0981 0. 2194 BS 0. 0829 0. 4214 0. 0785 0. 0198 0. 2434 Sc** 0. 2056 0. 2865 0. 3143 0. 3613 0. 4493 See Abbreviations, pp. xiixiii. \ ** Sc = 0.5 + (Principal Component eigenvalue) All underlined values had an absolute value larger than its corresponding Sc. Underlined values were selected for further study.
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90 performed for those soil properties previously selected. Soil properties with a Tolerance (T) < 0.01 were considered to be highly intercorrelated and were also excluded (Table 5). PC5 was not included in the analysis of collinearity because all eigenvectors had an absolute value smaller than Sc. According to this criterion Ca, Mg, Na, and K contents, total bases, extractable acidity, and CEC were highly intercorrelated for PCI. Similar reduction of variables was applied to other PCs. A final reduction was made by calculating correlation coefficients between soil properties and PCs (Table 6). A large correlation coefficient (0.75) was initially selected as criterion to the reduce even more the number of soil properties. Based on the correlation coefficient, fine sand, total sand, clay, and organic carbon contents were selected. Other soil properties also had a large correlation coefficient, but they were previously eliminated because of the small eigenvectors or the low tolerance. In summary, fine sand, total sand, clay, and organic carbon contents were selected for further analysis. The selection was based on analyses of PCs plots, PCs rotated axes plots, guantitative selection of larger eigenvectors, collinearity tests, and computation of correlation coefficients between soil properties and PCs. The selected
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91 Table 5. Tolerance of standardized weighted soil properties by principal component. Principal Component 1 2 3 4 T T T T M 0.69 PHI 0.61 vc 0.45 TH 0.90 TS 0.11 PH2 0.54 c 0.22 Silt 0.73 Clay 0.14 Ca 0.08 M 0.36 OC 0.73 Ca <.01 TB 0.08 F 0.78 Ma < 01 RC u o U .J O VP" v r Na <.01 K <.01 TB <.01 EXT <.01 CEC <.01 See Abbreviations, pp. xiixiii. T = 1 R (R = coefficient of multiple correlation)
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92 Table 6. Correlation coefficients between standardized weighted soil properties and principal components Soil Principal Component Property 12 3 4 5 TH 0.0686 ft ft 1 T ft 0 0173 0 2157 0.6008 0 1473 VC 0.2538 ft ft ft ft ft 0.0998 /Â• C ft ^1 ft 0 5878 0.3856 0 2911 c ft, A C 1 ft 0.4513 rt i /" ft, 0.3069 ft 1 "\ *7 yl 0 .7374 0.2173 0 0599 M ft c ft ft *i 0.5821 ft *^ 4 /* J 0 .3464 0 5143 0 .1246 0 3183 F /Â•N A ^ t ft, 0 4119 0.0274 ft ~7 y*" ft A 0.7694 0.0624 0 3212 VF ft ft ft ft a 0.0294 0.3422 0.6391 0 2145 0 4315 TS 0.8357 0.2183 0.2606 0.1648 0. 3058 Silt 0.3946 0.3134 ft a i r i 0.0757 0 .5689 0 4359 Clay 0.7559 0.1054 0.2684 0.4624 0. 1677 PHI 0.0443 0.6748 0.2959 0.0444 0. 0976 PH2 0.2475 0.7252 0.2151 0.1765 0. 2017 oc 0.2497 0.1420 0.1042 0 7889 0 1867 Ca 0.6453 0.5884 0.0888 0.0898 0. 0287 Mg 0.6205 0.3596 0.0227 0.0879 0. 1377 Na 0.6297 0.0358 0.0164 0.0727 0. 2697 K 0.6266 0.2188 0.0191 0.1192 0. 1335 TB 0.7311 0.5851 0.0647 0.0475 0. 0297 EXT 0.7613 0.3661 0.0929 0.1275 0. 3109 CEC 0.9130 0.0478 0.0488 0.1358 0. 2442 BS 0.2016 0.7353 0.1249 0.0274 0. 2708 See Abbreviations, pp. xii xiii. All underlined values had an absolute value > 0.75
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93 soil properties were those that explained most of the variance of the total set of data. Principal Component Analysis for A Horizon Standardized Data The first five PCs explained approximately 74% of the total variance for the A horizon (Table 7). Similar analyses of plots as indicated earlier for standardized weighted average values were used. Eigenvectors with absolute values larger than a Sc value of 0.2126, 0.2553, 0.3070, 0.3857, and 0.4605 for PCI, PC2, PC3, PC4, and PC5, respectively, were selected (Table 8) Analysis of collinearity showed that only total sand had a T value < 0.01 (Table 9), therefore, A horizon total sand was eliminated for further analysis. Silt and clay also had low T values indicating some correlation among those properties. After computing the correlation coefficient between soil properties and PCs (Table 10), clay content and CEC were selected. They had a correlation coefficient larger than 0.75. While organic carbon content in the A horizon is an important property it was not selected by the PCA. Therefore, it may be concluded that organic carbon content was not as important as clay and CEC in explaining the total variance. Two kinds of A horizons were present (Ap and Al). The Ap horizon is influenced by management conditions and the
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94 Table 7. Proportion of total variance explained by each principal component for standardized A horizon data. Principal Eigenvalue Proportion Cumulative Component (%) Proportion 1 5.5308 27.65 27.65 2 3.8348 19.17 46.82 3 2.6524 13.26 60.08 4 1.6807 8.40 68.48 5 1.0856 5.43 73.91 6 0.9913 4.96 78.87 7 0.9273 4.64 83.51 8 0.7856 3.93 87.44 9 0.6752 3.08 90.52 10 0.3676 1.84 92.36 11 0.3144 1.57 93.93 12 0.2913 1.46 95.39 13 0.2078 1.04 96.43 14 0.1913 0.96 97.39 15 0.1796 0.90 98.29 16 0.1335 0.72 99.01 17 0.0783 0.62 99.63 18 0.0653 0.33 99.96 19 0.0073 0.04 100.00 20 0.0000 0.00 100.00 Proportion of the total variance.
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95 Table 8. Eigenvectors of correlation matrix for standardized properties of A horizon. Soil Principal Component Property 12 3 4 5 TH 0 0509 0 0935 0 0047 0 2287 0 0146 VC 0 1167 0 2425 0 3346 0 0244 0. 1879 c 0 1566 0 2945 0 3872 0 0516 0. 0386 M 0 2206 0 2533 0 2541 0 1803 0 2296 F 0 1573 0 2129 0 4110 0 2365 0 0394 VF 0 0461 0 2672 0 2912 0. 1316 0 5020 TS 0 3654 0 0104 0 1294 0. 3020 0 0999 Sil 0. 3093 0. 1332 0. 1973 0. 2671 0. 0016 Cla 0. 3280 0. 1182 0. 0222 0. 2658 0. 1782 jrn n u Â• n r 1 n u jU JO u j Â£. y U u U j / ? U UU1U PH2 0. 0958 0. 3251 0. 2977 0. 1084 0. 1614 OC 0. 2990 0. 1663 0. 0832 0. 0868 0. 1278 Ca 0. 2397 0. 2909 0. 2653 0. 0275 0. 1489 Mg 0. 2157 0. 2739 0. 2156 0. 1050 0. 0385 Na 0. 0251 0. 1989 0. 0893 0. 2417 0. 6454 K 0. 2722 0. 0417 0. 0078 0. 4142 0. 0353 TB 0. 1644 0. 2427 0. 0529 0. 4085 0. 3198 EXT 0. 2051 0. 3312 0. 1259 0. 0017 0. 0302 CEC 0. 3238 0. 1661 0. 1199 0. 2168 0. 0189 BS 0. 2971 0. 0946 0. 0441 0. 3565 0. 1662 Sc ** 0. 2126 0. 2553 0. 3070 0. 3857 0. 4605 See Abbreviations, pp. xiixiii. i ** Sc = 0.5 + (Principal Component eigenvalue) 2 All underlined values had an absolute value larger than its corresponding Sc. Underlined values were selected for further study.
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96 Table 9. Tolerance of standardized properties of A horizon by principal component. Principal Component 1 2 3 4 5 T T T T T M u by C a Â£ i 0.61 vc 0 a n 27 K 0.64 VF A A A 0.99 TS < 01 VF 0 67 c 0 A >l 24 TB 0.64 Na A A A 0.99 C* A "1 4Silt r\ a a 0.09 PHI 0.38 F 0 .73 Clay a a a 0.03 PH2 0.36 PHI 0 96 OC 0.23 Ca 0.23 Ca 0.28 Mg 0.39 Mg 0.35 EXT 0.36 K 0.56 CEC 0.14 BS 0.32 See Abbreviations, pp. xiixiii. T = 1 R (R = coefficient of multiple correlation).
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97 Table 10. Correlation coefficient between standardized properties of A horizon and principal component. Soil Principal Component Property 12 3 4 5 TH 0. 1197 0. 1831 0. 0077 0. 2964 0. 0152 VC 0. 2744 0. 4748 0. 5450 0. 0316 0. 1957 C 0. 3683 0. 5768 0. 6306 0. 0669 0. 0403 M 0. 5189 0. 4961 0. 4138 0. 2337 0. 2393 F 0. 3699 0. 4169 0. 6693 0. 3066 0. 0411 VF 0. 1084 0. 5233 0. 4743 0. 1706 0. 5230 TS 0. 8593 0. 0203 0. 2107 0. 3915 0. 1042 Silt 0. 7274 0. 2609 0. 3213 0. 3462 0. 0017 Clay 0. 7714 0. 2315 0. 0362 0. 3446 0. 1857 PHI 0. 2025 0. 5988 0. 5358 0. 0751 0. 0010 PH2 0. 2254 0. 6367 0. 4849 0. 1406 0. 1681 OC 0. 7031 0. 3256 0. 1355 0. 1126 0. 1332 Ca 0. 5639 0. 5696 0. 4321 0. 0357 0. 1551 Mg 0. 5073 0. 5364 0. 3511 0. 1362 0. 0402 Na 0. 0589 0. 3895 0. 1454 0. 3134 0. 6724 K 0. 6403 0. 0817 0. 0127 0. 5371 0. 0368 TB 0. 3865 0. 4753 0. 0861 0. 5296 0. 3332 EXT 0. 4824 0. 6487 0. 2051 0. 0023 0. 0315 CEC 0. 7615 0. 3253 0. 1954 0. 2811 0. 0197 See Abbreviations, pp. xiixiii. All underlined values had an absolute value > 0.75.
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98 Al horizon is found in relatively natural conditions. Thus, the PCA was employed separately on these two classes of A horizons. All steps previously described were followed in the PCA for these two groups of A horizons. The final selection of soil properties by correlation coefficients (Tables 11 and 12) revealed that organic carbon content and extractable acidity were two important properties of the Al horizon for PCI (these soil properties represented approximately 39% of the total variance). The PCA revealed the importance of organic carbon content and the natural acidic conditions reflected by the extractable acidity values. Base saturation was the most important property of the Ap horizon for PCI (Table 12). Base saturation represented approximately 24% of the total variance. PCA revealed, therefore, the influenced of management conditions (liming) on the Ap horizons. The PC2 for the Ap horizon also indicated organic carbon content was an important property. For this reason organic carbon content was also selected. Other soil properties were not selected because they were previously excluded by the PCA. Organic carbon and clay contents were selected as important soil properties for both sets of data, weighted average and A horizon values.
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99 Table 11. Correlation coefficient between standardized properties of Al horizon and principal components Soil Principal Component Property 12 3 4 5 in 0 0 1 R S \J _L O J 99RR n 9 1 Q R z _L y o A u a ^ "5 1 u ") 7 Q A z. / y 4 vr V 0 ^RQQ j o y y 1 9R7 i. 6. 0 / n u 67 ^ A n u 16 / Z 0 ^96.8 n u Â• ID J J n /DUD u m 9 a 1 "3 C A M D n \j 9 9fi6 u Z D / 1 u i a i i TT J. 0 u Â• 0 u Â• I 1 II J J 764,4 1 O 4 ft UUUl u z y y b VF 0 1727 0. 3194 0 6310 0 2714 0 5134 TS 0. 9064 0. 0054 0. 1940 0. 0038 0. 1558 Silt 0. 7851 0. 2490 0. 1689 0. 0236 0. 3054 Clay 0. 8310 0. 2766 0. 1819 0. 0906 0. 0348 PHI 0. 2737 0. 6579 0. 2146 0. 2636 0. 0405 PH2 0. 3744 0. 6891 0. 0656 0. 3259 0. 1152 OC 0. 7648 0. 3696 0. 0709 0. 0410 0. 1346 Ca 0. 6051 0. 7386 0. 0351 0. 0318 0. 0022 Mg 0. 6332 0. 5895 0. 0881 0. 1155 0. 0607 Na 0. 7245 0. 0781 0. 0625 0. 0307 0. 1225 K 0. 8072 0. 3281 0. 0261 0. 1239 0. 1096 TB 0. 6458 0. 7187 0. 0411 0. 0035 0. 0138 EXT 0. 7792 0. 4500 0. 1489 0. 1139 0. 1982 CEC 0. 8902 0. 2024 0. 1232 0. 1042 0. 1845 BS 0. 0278 0. 7952 0. 1907 0. 0247 0. 2019 See Abbreviations, pp. xiixiii. All underlined values had an absolute value > 0.75.
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100 Table 12. Correlation coefficient between standardized properties of Ap horizon and principal components Soil Principal Component Property 12 3 4 5 TH 0.0867 0 2165 0 1594 0 0704 0 8748 VC 0.2069 0 6088 0 1 r\ f\ o 1908 0 3882 0 "1 A 1 '"4 1432 C 0.2684 0 /inn 6478 0 3617 0 5061 0 f\ A A f 0446 M 0.0718 0 /** A A O 6448 0 2155 0 5817 0 0229 c Â— n A1AA u n u fil 9 4 UU4U u 9 (177 VF 0.2499 0. 4575 0. 3277 0. 5642 0. 1930 TS 0.5737 0. 0372 0. 6245 0. 4727 0. 0997 Silt 0.2532 0. 0964 0. 6603 0. 4205 0. 1907 Clay 0.6416 0. 0231 0. 4217 0. 3842 0. 0219 PHI 0.4305 0. 1677 0. 6459 0. 1684 0. 1371 PH2 0.5942 0. 1764 0. 5754 0. 0675 0. 1712 OC 0.2859 0. 7581 0. 2388 0. 3986 0. 0544 Ca 0.8269 0. 3231 0. 2853 0. 1766 0. 1710 Mg 0.8090 0. 1242 0. 2352 0. 0532 0. 0411 Na 0.1085 0. 4697 0. 0495 0. 5145 0. 2272 K 0.6178 0. 2221 0. 1573 0. 1801 0. 0943 TB 0.8668 0. 3063 0. 2655 0. 1490 0. 1304 EXT 0.0919 0. 7915 0. 3943 0. 3386 0. 0087 CEC 0.2515 0. 8271 0. 2509 0. 3608 0. 0581 See Abbreviations, pp. xiixiii. All underlined values had an absolute value > 0.75.
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101 Principal Component Analysis by Soil Series This analysis was included to determine how this technique can be used to select important soil properties by soil series, to evaluate the variability of similar soils, and to evaluate the correct placement of pedons within the soil classification system. Theoretically, each PC may explain approximately the same proportion of the total variance for similar soils. To evaluate this assumption, soil series with the largest number of observations were selected and analyzed. These were the Albany, Dothan, and Orangeburg series. Results of this analysis are presented in Figures 10, 11, and 12. The proportion of the total variance explained by the first PC varied widely. The proportion varied between 35.7% and 71.5% for the Albany series, from 30.3% to 66.3% for the Dothan series, and from 39.3% to 82.8% for the Orangeburg series. There was a wide average difference (38%) between the minimum and maximum values for the three soils. The degree of importance of the soil properties varied from one county to another for the same soil series. For example, total sand content was an important property to explain the total variation of the Albany series in Jackson and Leon Counties but was not important in Santa Rosa County. Similar examples can be observed with other soil properties between different counties in each soil series selected.
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102 CO CO lice: c\l X Q. co CO rO rO ro ro rO CO ro 1^ro CD rm to CD CD 6 o O O c iQo CO o E u CO co > o c o QJ 0 c id Â•H M > (13 +J O 4J CU X! +J o 4J c o H 4> Â•H in 4J C O u , c rO id si H +Â• C H Â£ CD J3 P M O CM to 0) H 4J M i o *j M c o H O H O >i CO .Q 0 H H
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103 CO CD O LU C X UJ CD 2 O o o X Cl co co o < > o CD CD CM rO if) CO O rO CO ID CD CD o i_ Cl o CO o a. E e o O Cu >> a. o c CO CO > CO o a) u c Â•H P id > rH ia p o P a) p o p c o H P P P C C u 0) tr U to Si p Â•H a) H p p a) o o p a H o CO CO 0) rl P CD CQ c (13 P 0 Q i P c o 0 p 3 tr H
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104 v. cr C Ll c h~ X Ui 10 C c cvj !TTT l:::: a.:::: x Q. > C +Â•* co CO 6 o O 0>> c > c > 0) 0 a rd H !l > H id p 0 a) 4J C c 0 H +J X! Â• H P CD +J Â•H C P 0 QJ u tn o tr & p. M a) D id c id P. P o Â•H QJ w P 0 H 11 4) 0 P. iw a) 0 >l 0 4J p c a. 3 0 H 0 Â•H 0 CO ^3 o CO cr < > 10 o CO CM GO CM co CD oi o CM ID CO CO a. E CN 0) P c H
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105 A large variation existed in the proportion of the total variance explained by the first PC between counties in each soil series. In addition, the degree of importance of each soil property varied between counties in each soil series. For these reasons, the three soil series were plotted in the plane of the first two PCs (Figure 13) to visualize the relationship between pedons in each soil series. A large degree of dispersion was observed in each soil series. A clear grouping of pedons by individual soil series did not exist. Thus, a nested analysis of the variance was used to created a clearer understanding of the variation among pedons within each series. Soil series, pedons within each series, and horizons within each pedon were considered as sources of soil variation (Table 13). Theoretically, a larger variation may occur between soil series (e.g., between Albany and Dothan) and between horizons within pedons belonging to the same series (e.g., between A and B horizons in the Albany series ) A large part of the total variation was explained by differences between pedons belonging to the same series. More than 30% of the variability in all sand fractions (except total sand), silt, pHwater, K content, and CEC was explained by the differences between pedons within the same soil series.
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106 e o o <4 O z o i o 8 < a. o z K a I CD P UJ 0 0) eÂ— 10 1Â— 1 fcij CD +j c H (0 CD Â• H 4J G C cn CD C i Â— i 0 H ft 0 B en 0 u na rc o CD Â•H rH U CD c 09 Â•H H ft 0 o c 0 *J H +J +J CO en 0 M 0 H 2 iN3NOdHOO IVdlONIUd CD U 3 tJi Â•H fa
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107 Table 13. Variability of studied soil properties within and between soil series and between horizons. Soil Source of Variation Property Soil Series Pedon ** Horizon Error % rTlT T TH A 1 U U J.J Q 0 A yz 4 VC U b b 4 Z 0.1 J Z X C Z j u z 1 a n 1 0 u M U U Q "3 "7 O.J a n 0 Â• u F Q "7 O / R "7 T lo Z 1 ^ a VF 0 0 y l i J 1 c; a J o TS 51.8 0.0 34.0 14.2 Silt 7.0 34.6 21.5 36.9 Clay 36.7 0.0 43.1 20.2 PHI 7.2 36.9 4.5 51.4 PH2 0.0 26.0 9.2 64.8 OC 1.4 0.0 95.0 3.6 Ca 12.7 0.0 77.5 9.7 Mg 26.7 0.0 61.5 11.8 Na 7.0 0.0 88.8 4.2 K 0.0 30.2 25.6 44.2 TB 5.7 23.9 27.9 42.4 EXT 0.0 0.0 81.0 19.0 CEC 0.0 37.3 48.5 14.2 BS 9.8 22.4 23.4 44.3 See Abbreviations, pp. xiixiii. ** Pedon within soil series.
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108 A large part of the variability in total sand and clay contents was explained by the differences between soil series. More than 40% of the variability in clay, organic carbon, Ca, Mg, and Na contents; extractable acidity; and CEC was explained by the difference between horizons. Some soil properties (horizon thickness, very coarse sand and silt contents, pHKCl, K content, total bases, and base saturation) had a large unexplained variability (error). Total sand and clay contents fulfilled the initial hypothesis which stated that a large part of the variability was explained by differences among soil series. Organic carbon content also fulfilled the initial hypothesis that a large part of the variability was explained by differences among soil horizons within similar soil series. These results validated the conclusions of the PCA, for both standardized weighted data and standardized A horizon data. Total sand, clay, and organic carbon contents were selected by the PCA as soil properties which were important in explaining the total variance. Fine sand content and CEC were also selected by PCA, but according to the nested analysis of variance, a large part of their variability was explained by the differences among pedons within soil series. Therefore, fine sand and CEC were not included in the geostatistical analysis.
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109 In addition, these results also validated the use of a large correlation coefficient (0.75) because this coefficient allowed the selection of those variables with large variability between soil series and horizons. Both PCA and nested analysis of variance were very useful in selecting important soil properties (total sand, clay, and organic carbon contents) for further analysis. PCA reduced the large number of soil properties selected initially. The nested analysis of variance demonstrated that most of the soil properties selected by the PCA were important as differentiating properties between soil series and/or horizons. Likewise, the selected soil properties are important to determine specific soil potentials (e.g., fertility and irrigation). Thus, the variability of the selected soil properties affect the accuracy of the predictions for these specific performances. For a final validation, soils were plotted in the plane of the first two PCs, considering only the important selected soil properties (Figure 14). In this a slightly better grouping of soils by series was observed compared to Figure 13. An important conclusion from these analyses is that because of the multivariate character of soils, the selection of variables must be based on some quantitative method. Otherwise the biased selection of variables can introduce a large source of error in the results.
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110 P P O Ml (U < Q o o <4 S O M o Â— h o a. 3 o < a. 3 z Â£ a. I 2 iN3N0d00 IVdIONiad CD (1) CO c H O a h C "H Â•h u co t3 CO U 0) co ItJ a H Â• U CO Â•H >l P Ml Oj P C 5 O P Â•H P P (C CO u o (J 41 0) QJ O 0,
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Ill Conversely, the use of the complete set of data would add more complexity to the analysis. A large number of soil properties had a large proportion of the variability either explained by differences among pedons belonging to the same soil series and/or unexplained soil variability. It is believed that the possible causes of the variability are: (i) Soil properties relevant to define series, such as morphological properties, were not considered in the analyses Variability of total sand, clay, and organic carbon contents was successfully explained by differences between soil series and/or horizons because these soil properties are related to morphological properties of a given horizon. Total sand content is related to the coarsetextured surface horizon, clay content is related to the argillic horizon, and organic carbon content is related to the surface A horizon. (ii) Sampling errors by assuming an erroneous concept of soil variability. Sampling errors are introduced if soil scientists assume that the sampling unit is completely uniform when it is not so. It seems very difficult to have a completely uniform sampling unit. Variability has been recognized at all scales. Soil variability has been widely recognized at macroscopic scale (Beckett and Webster, 1971; Beckett and
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112 Bie, 1976). In addition, variability can be recognized microscopically and submicroscopically (Wilding and Drees, 1978, 1983). If soil variability is considered as the sum of variability at all scales, then, it is very difficult to have uniform soils. The reality is that "uniform" soils are those in which the internal variability ("within" variability) is lower than the variability compared to the surrounding soils ("between" variability). (iii) A large source of variation was introduced because of lack of emphasis by soil scientists on soil and landscape relationships. Descriptions of the geomorphic environment are very ambiguous for some soil series. For example, the geographic setting of Orangeburg series is described as follows: Orangeburg soils are on nearly level to strongly sloping uplands of the Coastal Plain. Slopes range from 0 to 20% (National Cooperative Soil Survey, 1982). The geomorphic environment was described as gently sloping uplands with 4% gradient for an individual pedon of Orangeburg series (Carlisle et al., 1985; p. 192). Many soil investigations in the U.S. have involved geomorphic surfaces. Ruhe (1969) defined a geomorphic surface as a portion of the landscape specifically defined in space and time. The surface is a mappable unit that has no size limit and may include a number of landforms and landscapes. According to this concept only time for a
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113 geomorphic surface is uniform; other geomorphic features related to space (i.e., physiography) can have large variations. In addition, a low degree of accuracy in the pedon location descriptions was evident while locating the selected pedons on topographic maps. More emphasis has been placed on the descriptive aspect of the soil series than in the geographical aspect. Importance of the geographical aspect of soil was pointed out by Bie (1984). He indicated that the accurate location of pedons by X and Y coordinates would be a great contribution to soil science. (iv) Possible errors in soil correlation. The large degree of pedon dispersion within individual soil series may be the result of incorrect placement of individual pedons into the soil classification system. Soil correlation was beyond the scope of this investigation, but PCA may be a useful guantitative method to indicate problems in soil correlation. Geostatistics The variability of soil properties is a limiting factor for reliable soil interpretations and for making accurate predictions of soil performance at any particular location on the landscape.
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114 A large number of studies have been made to quantify soil variability, but they have not taken into consideration the geographic character of soil variability. Conversely, geostatistical analysis is based on the geographic location of the individual observations. Therefore, geostatistical techniques can offer a solution to some of the unsolved problems of spatial variability of soils The 151 pedons studied were located by a system of X and Y coordinates (Appendix C). Pedons were irregularly distributed in an approximately 380 x 100 km grid (Figure 15) Geostatistical analysis can be used for horizontal and vertical directions, but using both these directions adds more complexity to the analysis. Therefore, the data were selected to represent the variability of soils in the horizontal plane. Two sets of data were analyzed. One set was composed by weighted average values of total sand, clay, and organic carbon contents. The other set of data included clay and organic carbon contents from the A horizon. SemiVarioqrams The first step in the geostatistical analysis was to calculate the semivariance. The number of pedons provided sufficient pairs of observations for reliable estimates of semivariograms. The total number of pairs was calculated
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115
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116 from the combinatorial equation: Ns of Combinations =n!/n!(nr)! (48) where n = Total number of pedons r = Number of pedons taken at one time When r = 2, equation (48) reduces to Ns of Pairs = n (n 1) / 2 (49) According to equation (49), 151 pedons provided 11325 pairs. Directionindependent and directiondependent semivariances were calculated for each soil property studied. Semivariances for directionindependent and EW, NESW, NWSE, and NS directions were supported by 11,325; 8,298; 924; 1,450; and 653 pairs of observations, respectively. A reliable semivariogram is obtained when intervals are chosen such that the number of pairs is large enough to ensure accurate definition of each point on the semivariogram. A rule of thumb is to use intervals such that the minimum number of pairs of observations in each interval is about 50 (Skrivan and Karlinger, 1979). Likewise, the maximum lag distance to provide reliable semivariograms is a half of the total length (Journel and Huijbregts, 1978). The total length was approximately 380 km in the EW direction and 100 km in the NS direction. A
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117 lag distance of 10 km was selected to calculate the semivariance up to 190 km in the EW direction and 50 km in the NS direction. Stationarity is an important assumption to consider in geostatistics The criterion used to determine the validity of this assumption was explained by Journel and Huijbregts (1978). They indicated that when the semivariance increase is larger than h 2  ( h = modulus of the lag distance) for large distances h, the increase is incompatible with the intrinsic hypothesis. Such an increase in the semivariance often indicates the presence of drift. Statisticians established the constraint of stationarity because each sample was considered unigue when geostatistics was developed. Conseguently statistical inference about the population could not be made. Geostatistics was developed in the mining industry. Sampling procedure in mining is guite different from sampling procedures applied in soils. Sampling ore deposits involves large volumes of individual samples, large sampling time, and high costs. It is very difficult to take sample replications in mining. Sampling soils is a completely different situation. Most soil samples are taken within 2 m from the soil surface. In addition, soil samples can be taken at
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118 distances varying from a few cm to several km apart at relatively low cost. Soil stationarity was assumed before geostatistics could be used in soil science. Soil scientists have assumed stationarity when they take replications to increase the precision of the results. Stationarity of soils is assumed when the placement of soils in the classification system is tested. Stationarity of soils has been also implicitly assumed when a map unit is delineated by a soil survey. Observations are not unique in soil science. It is possible to take relatively homogeneous replications of soil samples. Stationarity is not as serious a problem in soils as it is in mining. Therefore, the criterion to determine soil stationarity needs to be defined. Stationarity is important within the area in which a large degree of similarity and dependence in soil property values exits. The similarity and dependence of soil properties values are large within a map unit. The degree of dependence decreases when soil properties are measured in different map units up to the point in which soil properties values are no longer related. The withinunit (WU) variability and the betweenunit (BU) variability are important in order to know the degree of uniformity of map units, of individual pedons, or of soil properties. The variability WU is expected to be
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119 smaller than the variability BU. Although, where different levels of management have been applied, WU variability may exceed the BU variability (Beckett and Webster, 1971; McCormack and Wilding, 1969). In general, the WU variability gives us the degree of uniformity or variability of a map unit or individual soil properties. The WU variance could then be used as a criterion to establish stationarity for those soil properties less affected by management (e.g., total sand and clay contents). When twice the value of semivariance (G) is larger than the WU variance ( 2G = variance) in the area in which the soil properties are supposed to be related, then, stationarity is absent. Data were grouped by soil series. The WU variability was represented by the WU variance of the soil series. Total sand and clay contents had a WU variance of 33.8 and 14.3 respectively. The first semivariogram for total sand content (Figure 16) had a G value that increased from 198.4 at 5 km distance to 249.1 at 15 km distance. The increase in distance represented an increase of 2 in the modulus of the lag distance. The semivariogram for clay content had a G value that increased from 153.5 at 5 km distance to 180.1 at 15 km distance (Figure 17). Therefore, semivariograms of weighted average total sand and clay contents had drift. If the information contained in the semivariogram is to be used for making optimal unbiased
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122 estimates (kriging) of the selected soil properties at unsampled locations, the drift must be removed. An iterative procedure, explained in the Materials and Methods section, was used to remove the drift. The observed drift in total sand and clay content semivariograms was reduced, but it was not completely removed. A reason for this may be the presence of a short range variability in the soil properties. Pedons with large differences in soil properties were located at short distances. Different pedons were compared when semivariograms were calculated with lag increments of 10 km. These results are supported by previous works. Burrough (1983a) pointed out that it seems impossible to achieve stationarity. Olea (1975) could not eliminate the drift in the data; then, he used universal kriging to produce maps that indicated trends in data variability. Total sand and clay content semivariograms were characterized by the presence of structure. Semivariogram structure occurs when there is an increase of the semivariance to a maximum value (Figures 18, 19, 20, and 21). These semivariograms had characteristics nugget variances (intercept), ranges, and sills (Table 14). Theoretically, the semivariogram should pass through the origin when the distance h = 0 (h is the lag distance). However, total sand and clay contents had nonzero semivariances as h decrease to zero. This is called nugget
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123
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125
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126 VWWV9
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127 Table 14. Important semivariogram parameters of the weighted average of selected soil properties. Semivariogram Range (km) Sill Nugget variance g. 0 Total sand content Directionindependent 34.7 324.0 173.6 53 6 EastWest 24.8 295.8 160.1 54. 1 NortheastSouthwest 34.7 292.4 182.7 62. 5 NorthwestSoutheast 25.4 287.3 220.8 76. 9 NorthSouth 30.0 352.5 120.5 34. 2 Clay content Directionindependent 34 .7 230.2 135.9 59. 0 EastWest 34 .6 211.3 123.2 58. 3 NortheastSouthwest 34 .7 206.5 143.3 69. 4 NorthwestSoutheast 15 .6 210.3 153.2 72. 9 NorthSouth 34 .7 299.5 99.7 33. 3 Organic < carbon content Directionindependent <10 .0 0.120 0.120 100. 0 EastWest <10 .0 0.119 0.119 100. 0 NortheastSouthwest <10 .0 0.069 0.069 100. 0 NorthwestSoutheast <10 .0 0.052 0.052 100. 0 NorthSouth <10 .0 0.201 0.201 100. 0 % of the sill represented by the nugget variance
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128 variance or nugget effect (Journel and Huijbregts, 1978). Nugget variance represents the unexplained variance, often caused by measurement error or variability of the soil properties that could not be identified at the scale employed. The intercept, which is the estimate of G at h = 0, provided an indication of the variation at a distance shorter than 10 km. The range of the semivariogram is the distance at which G attains the maximum value (sill). The range can be interpreted as the diameter of the zone of influence which represents the average maximum distance over which observations are related. They are dependent. At a distance larger than the range, observations are no longer related. They are independent. At distances less than the range, measured properties (e.g., total sand and clay contents) of two samples become more alike with decreasing distance between them. Thus, the range provides an estimate of the areas of similarity. The range also represents the average minimum distance at which maximum variation occurs. The maximum semivariance value is called the sill. The sill is egual to the sum of the nugget variance and the spatial covariance (Co + C). Often, the sill is approximately egual to the sample variance (Journel and Huijbregts, 1978).
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129 Total sand and clay content semivariograms were anisotropic, indicating that the variability of selected soil properties changed with direction (Table 14). The longest range was of approximately 35 km for both total sand and clay contents. The longest range was for directionindependent and NESW semivariograms for total sand content; and for directionindependent, NESW, and NS semivariograms for clay content. The largest variation (sill) occurred in the NS direction for both total sand and clay contents. The largest proportion of the unexplained variation occurred in the NWSE direction for both total sand and clay contents. Differences between directiondependent semivariograms for the soil properties selected by the PCA could be the result of differences in geology and topography. Organic carbon content is a soil property influenced by management. The WU variance was smaller than the BU variance (0.0 and 0.07 respectively). Organic carbon content had very low WU and BU variances because the largest variation occurred between horizons (Table 13). Stationarity in organic carbon values was present when the wu variance was used to determine stationarity; but stationarity was absent if the semivariance increment compared to the lag distance increment was used. This situation resulted because semivariance organic carbon values were very small (generally less than one),
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130 therefore, any increment in lag distance always resulted in the absence of drift. The absence of drift, when lag distance was used as the criterion, was a problem related to the measurement scale. This was supported by the fact that organic carbon had the largest C.V. (Table 2) among the three soil properties used for the geostatistical analysis. Consequently, the WU variance was used as the criterion to determine stationarity Semivariogram for organic carbon content did not have any structure (Figures 22 and 23); there was no increase to a maximum value. A pure nugget effect was observed, indicating a shortrange variability in organic carbon content. Organic carbon content had a large point to point variation at short distances of separation and an absence of spatial correlation at the scale used. The range of the organic carbon content semivariogram was a distance smaller than 10 km. Directiondependent semivariograms (Figure 23) showed an anisotropic variation in organic carbon content. The largest variation ocurred in the NS direction (Table 14). The anisotropic variation of organic carbon indicated that the factors which influence the organic carbon content (e.g., vegetation, moisture, drainage, relief, management) are different in different directions with the largest variability in the NS direction.
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133 Semivariograms of A horizon soil properties (clay and organic carbon contents) also indicated presence of drift. The WU variances were used as criterion to determine stationarity WU variances were 10.0 and 0.0 for the A horizon clay and organic carbon contents, respectively. A large part of the drift was removed for semivariograms of the A horizon soil properties by using the residuals, but it was not completely removed. A reason for this can be related to the presence of different pedons within short distances. Semivariograms of the A horizon clay content indicated presence of structure (Figures 24 and 25). The maximum variance (sill) was reached within distances varying from 20 to 35 km (Table 15). The maximum variation occurred in the NWSE direction. Variation of the A horizon clay content was smaller than the variation of the weighted average clay content. This was due to the fact that weighted average data included contrasting horizons in clay content such as A and B horizons. The direction of maximum variation of the A horizon clay content corresponded to the direction in which the weighted average clay content had the largest nugget variance. Therefore, it is probable that the large variation in the A horizon clay content in the NWSE direction was one of the causes of the large unexplained
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134
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135 VWWV9
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136 Table 15. Important semivariogram parameters of the A horizon selected properties. Semivariogram Range Sill Nugget o "O (1cm) variance Clay content Directionindependent 34.8 75.8 24.4 32. 2 EastWest 45.0 96.9 25.8 26. 6 NortheastSouthwest 25.0 55.6 28.6 51. 4 NorthwestSoutheast 35.1 118.0 7.5 6. 4 NorthSouth 20.1 53.0 22.3 42. 1 Organic carbon content Directiondependent <10.0 1.048 1.048 100. 0 EastWest <10.0 1.045 1.045 100. 0 NortheastSouthwest <10.0 1.014 1.014 100. 0 NorthwestSoutheast <10.0 1.089 1.089 100. 0 NorthSouth <10.0 0.955 0.955 100. 0 % of the sill represented by the nugget variance.
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137 variation in the same direction for the weighted average clay content. Semivariogram of A horizon organic carbon indicated, as did the semivariogram of weighted average organic carbon content, a pure nugget effect (Figures 26 and 27). A reason for this is that the A horizon organic carbon content had a large point to point variation at short distances. Variation of the A horizon organic carbon content was larger than that for the weighted average organic carbon content. This could be due to the fact that some of the A horizons were affected by management conditions (Ap) and other A horizons were under relatively natural conditions (Al). This result seems to be in contradiction with results obtained with the nested analysis of variance (Table 13), but two aspects need to be considered. First, the nested analysis of variance did not considered the pedon location. Second, the nested analysis of variance included surface and subsurface A horizons. Therefore, a masking of the differences between surface A horizons (Ap and Al) could have occurred when the nested analysis of variance was used. All observed semivariograms had a characteristic wave pattern, indicating a cyclic variation in the studied soil properties
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140 Fitting SemiVariograms The process of fitting the observed semivariogram to a theoretical model is another important step in the geostatistical analysis. It is important to choose the appropriate model for estimating the semivariogram because each model yields quite different values for the nugget variance and range, both of which are critical parameters for kriging. The process of fitting observed semivariograms to theoretical models was time consuming. Therefore, directionindependent and directiondependent semivariograms with the largest variation were selected. Olea (1984) stated that there is no single solution to curve fitting. The user must decide what part of the semivariogram should be fitted and what part should be regarded as anomalous. Points located within distances varying from 0 to 50 km were selected because the range was included and there was a large semivariogram reliability within these distances. The choice of the model was governed by the general graphic appearance of the observed semivariogram. The curve fitting procedure described in the Material and Methods section was used. The objective of the fitting procedure was to adjust the parameters in the semivariogram until the model was theoretically consistent. Consistency is reached when the kriged average error (KAE)
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141 is approximately zero and the kriged reduced mean square error (KRMSE) is approximately equal to one. KAE and KRMSE are defined by equations (47) and (48). KAE gives the average of the difference between the observed and the theoretical (estimated) values, KRMSE represents the ratio between the theoretical and the calculated variance (sill). Models selected had KAE and KRMSE values very close to zero and one, respectively. Kriged mean square errors (KMSE) were computed according to the following equation: KMSE = [1/n (Z j Z i )2 ] 1/2 (52 ) where n = number of points Z^= measured value Z i = kriged value KMSE gives an idea of the dispersion of the measured values respect to the kriged values. Because of these values (Table 16), directionindependent semivariograms for weighted average values of total sand and clay contents were fitted by the DeWijsian (logarithmic) model (Figures 18 and 20). Total sand and clay content NS semivariograms were fitted by the Spherical model (Figures 35 and 36). Organic carbon content directionindependent and NS semivariograms were fitted by the Linear model (Figures 22 and 37).
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142 Table 16. Goodnessof f it values of the weighted average of selected soil properties. Semivariogram Model KAE* KMSE+ KRMSE** Total sand content Directionindependent DeWijsian 0.0589 16.8347 1.0670 NorthSouth Spherical 0.0699 16.7292 1.3826 Clay content Directionindependent DeWijsian 0.0666 13.7246 1.0001 NorthSouth Spherical 0.0363 13.6306 1.1479 Organic carbon content Directionindependent Linear 0.0001 0.3754 1.0535 NorthSouth Linear 0.0004 0.3778 0.8321 + KAE = Kriged Average Error KMSE = Kriged Mean Sguare Error KRMSE = Kriged Reduced Mean Square Error
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143 The KAE and KRMSE values for the A horizon soil properties (Table 17) indicated that directionindependent and the NWSE semivariograms for clay content were fitted by the Spherical and Root models, respectively (Figures 24 and 38). The A horizon directionindependent and the NWSE semivariograms for organic carbon content were both fitted by the Linear model (Figures 26 and 39). Kriging One of the prime reasons for obtaining a semivariogram is to use it for estimation. Soil survey recognizes two main kinds of estimates (Webster, 1985). One is the average value of a soil property within some defined region. The other is the prediction of values of a property at unsampled places (interpolation). The information derived from the fitted semivariograms was used to generate contour maps of kriged values of soil properties (interpolated values). Contour maps were produced by using universal kriging because trends were present in the data. Kriging is a technigue of making optimal, unbiased estimates of regionalized variables at unsampled locations using the information contained in the semivariogram (range, nugget effect, theoretical model). Kriging is optimal because it reduces the estimation variance and is unbiased because KAE is zero.
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144 Table 17. Goodnessof f it values for the A horizon selected properties. Semivariogram Model KAE* KMSE+ KRMSE** Clay content Directionindependent Spherical 0.1071 6.3258 1.0175 NorthwestSoutheast Root 0.2497 8.1069 1.4189 Organic carbon content Directionindependent Linear 0.0000 1.0212 0.9883 NorthwestLinear 0.0003 1.0336 1.0017 Southeast + ** KAE = Kriged Average Error KMSE = Kriged Mean Sguare Error KRMSE = Kriged Reduced Mean Sguare Error
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145 Contour maps were generated for total sand and clay (weighted average and A horizon) contents derived from the directionindependent and directiondependent with largest variability semivariograms (Figures 28, 29, and 30). Contour maps were better interpreted when used with diagrams. Organic carbon content (weighted average and A horizon values) was not used for contouring maps because of the large nugget variance that produced large variance estimates influencing the reliability of the map. Areas with discontinuous contour lines (i.e., lower left hand side and upper right hand side of the Figures used) corresponded to zones with no sampled pedons (Figure 15). Contour maps were influenced by the nugget and the minimum variance criterion (withinunit variance). The latter ensured that the interpolated value at a sampling point was the observed value there. The presence of a nugget variance indicated that the semivariogram was composed by two functions (except organic carbon), one describing the spatial dependence, and the other a purely random variation that influenced the boundary between delineations. Clay content weighted average had a nugget variance larger in proportion to the sill than that for total sand content (Table 14). This situation could indicate that the map of of kriged values of clay content weighted average encompassed more variable units than the map of kriged
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149 values total sand content. In general, there was a very good correspondence between the two maps. Because both soil properties were important components of the same PC, in addition the silt content was very low. Another aspect is that PCA not only allowed the selection of important soil properties for separating different soil series, but also the soil properties selected were spatially related. An important objective was to find a physical meaning of the contour maps and diagrams generated by the geostatistical analysis. For this reason contour maps and diagrams were compared with the map of physiographic regions of Florida (Figure 43). All contour maps have X and Y axes represented by the values of the geographic coordinates, which were very useful in locating the physiographic regions. Pedons studied were located in five physiographic regions: Southern Pine Hills, located between the 0 and 29 X coordinates and the 8 and 22 Y coordinates; Dougherty Karst, located between the 29 and 49 X coordinates and the 8 and 22 Y coordinates; Apalachicola Delta, located between the 24 and 58 X coordinates and the 0 and 12 Y coordinates; Tifton Uplands, located between the 48 and 58 X coordinates and the 11 and 15 Y coordinates; and Ocala Uplift, located between the 58 and 80 X coordinates and the 0 and 15 Y coordinates
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150 In general, maximum total sand content corresponded to minimum clay content (Figures 28 and 29). These soil properties were naturally related because of the presence of argillic horizons. Total sand content had a large depression between the 47 and 54 X coordinates, which corresponded to an area among the Apalachicola Delta, the Dougherty Karst, and the Tifton Uplands physiographic regions. The total sand content diagram indicated predominance of high values across the Panhandle, but there was a break in the continuity of the high values because of the presence of the Apalachicola Delta. Diagrams of total sand and clay contents (Figures 28 and 29) also indicated large number of small depressions between the 22 and 45 X coordinates and the 58 and 70 X coordinates, these areas corresponded to the Dougherty Karst and the Ocala Uplift, respectively. The presence of depressions seems to indicate a large variability in total sand and clay contents. The A horizon clay content (Figure 30) also had large variability in the Dougherty Karst physiographic region. The variability in soil properties in the Dougherty Karst is apparently related to the large variability in the karst topography. The variability in the Ocala Uplift can be the result of local differences in geology and topography including karst. Diagrams of clay content (Figures 29 and 30) indicated that clay content tend to decrease from the north to the south.
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151 Contour maps derived from directiondependent semivariograms with maximum variation (Figures 40, 41, and 42) did not differ from those maps derived using directionindependent semivariograms. This fact indicated that the geostatistical program was not capable of generating contour maps derived from directiondependent semivariograms. A reason for this is that the kriging subroutine of the geostatistical program did not reguire the direction of the semivariogram as input data. Therefore, improvement of the geostatistical program to take into account the directiondependent semivariograms is recommended. Despite the fact that the program was not capable of generating contour maps derived from directiondependent semivariograms, one guestion that needs to be answered is: Are there significant differences among the directiondependent variances? If there are no significant differences among directiondependent variances, there should be no differences between contour maps derived from directionindependent and directiondependent semivariograms for individual soil properties. An important advantage of kriging was that this interpolation technigue provided estimates of the estimation variance for each observation. These estimates can be displayed in the form of standard error (reliability) maps or diagrams. Reliability estimates indicate the precision of the kriged values and alternately
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152 could indicate where more samples would provide more information. Reliability diagrams (Figures 31, 32 and 33) were produced based on the standard errors of the kriged values The standard errors varied from 14.95 to 21.11 for total sand and from 13.10 to 16.33 for clay content weighted averages, and from 4.54 to 6.05 for A horizon clay content The standard error is a function of the nugget variance. The larger the nugget variance compared to the sill value, the larger the standard error compared to the kriged value Clay content weighted average had the largest standard error and A horizon clay content had the smallest. Reliability diagrams indicated areas of large and small standard error. Generally, areas with the smallest standard errors corresponded to areas with no sampled pedons (Figure 15). All three reliability diagrams coincided in indicating the eastern part of the Dougherty Karst physiographic region as the area with largest standard error. These reliability diagrams can be very useful in the design of new sampling strategies. Areas with large standard error require an increase in sampling intensity to increase the precision of the estimates. Geostatistical techniques were useful in evaluating the spatial variability of soils and to indicate zones where more intensive sampling is required. Geostatistical techniques require more investigations in order to better
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156 define the assumption of stationarity and anisotropy. Anisotropy indicated that the variability of soil properties was directiondependent. But an important question to be answered is: is there a significant difference among the directiondependent variances? Fractals Soil variation can also be a function of the scale of observation. The components of the variance measure the amount of variance contributed by each scale, and by accumulating them, it may possible to show how variance increases with increasing distance. The ranges of the semivariograms studied varied from "15 to 35 km. The random variation corresponded almost always to a large proportion of the total variance within these range distances (Tables 14 and 15). Therefore, the objective of this section was to evaluate quantitatively how semivariograms can be used to indicate the scale dependence of soil variability in the study area. All semivariograms had a nugget component which represented the random variability. Organic carbon content (weighted average and A horizon values) had a pure nugget effect indicating a shortrange variation. Other soil properties studied had longrange variation. Soil properties with longrange variation had nugget variances that varied from approximately 6% to about 73% of the total variance (Tables 14 and 15).
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157 The Hausdorf f Besicovitch dimension or fractal dimension (D) was calculated according to equation (43), to determine if the scale of study was appropriated to resolve the random component in the variability of the soil properties studied. Theoretically, the D value ranges from 1 to 2. The value of 1 indicates a systematic variation of soil properties and also indicates an appropriate scale of study. A value of 2 indicates a random variation and the need of increasing the scale of the study. If soil variability is scaledependent, a decrease in the D value is expected when the distance between sampling points is decreased. For this reason, fractal dimensions were computed from the semivariograms studied using a lag distances of 10 and 5 km (Table 18). Organic carbon content (weighted average and A horizon values) had a D value of 2 in all directions and for both lag distances. This result was supported by the semivariogram analysis which indicated a pure nugget effect. In addition, organic carbon content had also the largest C.V. (Table 2). These results indicated the shortrange variation in organic carbon content. The variation in organic carbon content was random at the scale used, thus, an increase in the scale of study used is recommended to explain the variability of this soil property. Weighted average total sand content had D values that varied from 1.80 to 1.92 (Table 18) for the 10 km lag
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158 Table 18. Fractal dimension (D value) derived from selected soil property semivariograms Soil Property Semivariogram Lag distance (km) TS CL or D value APT AOC Directionindependent 10 1 87 1.90 2.00 1.77 2.00 5 1 84 1.72 2.00 1.74 2.00 Ew 10 1 87 1 90 2.00 1.76 2.00 r 5 1.86 1 77 2.00 1.85 2.00 XTT? CT.T NESW 10 1 90 1 .93 2.00 1.86 2. 00 5 1 88 1 .73 2.00 1.80 2.00 WWÂ— CTT i y z 1 Oft 2.00 1.63 2.00 5 1.92 1.80 2.00 1.66 2.00 NS 10 1.80 1.82 2.00 1.84 2.00 5 1.66 1.51 2.00 1.77 2.00 See abbreviations, pp. xiixiii
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159 distance. The high D values indicated that is necessary to increase the scale of the study to explain better the variability and to reduce the random component in the variability of the total sand content. The D values of total sand content calculated for a lag distance of 5 km were always smaller than or egual to the D values calculated for the 10 km lag distance. Therefore, it can be concluded that the variability in total sand content is scaledependent. The decrease in the D value was a function of the proportion of the nugget variance (random variability) present. There was no decrease for the direction NWSE which had the largest proportion of nugget variance (Table 14). This fact may indicate a complex variability in the total sand content in the NWSE direction because of a complex variability in geology and topography in this directions. It is necessary to increase the scale of the study not only to reduce the random variability but also to find a physical meaning to the variability. Fractal analysis of weighted average clay content (Table 18) gave similar results as the analysis of weighted average total sand content. The D values calculated for the A horizon clay content were almost always smaller than the D values calculated for other soil properties studied. This can be related to the fact that the A horizon clay content had smaller
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160 proportions of nugget variance (Table 15) than those for other soil properties studied (Tables 14 and 15). The results of the fractal analysis were not in contrast with the results obtained with total sand and clay (weighted average and A horizon) content semivariograms Semivariograms indicated the presence of a systematic and a random variability within the range. The D values suggested that the random component of the variance was large because of the small scale of the study. The long distance between observations could influence the presence of a large random variation. When distance between pedons is long, local variations in parent materials and/or topography may increase the complexity in the variability of the soil properties studied. Therefore, an increase in scale and small distance between sampling locations (pedons) is necessary to reduce the random variability. Despite the fact that D values for 5 km lag distance were smaller than those for 10 km lag distance, the semivariograms for 5 km lag distance were only reliable for short distances (1/3 to 1/2 of the total length). Therefore, a smaller area with greater pedons density, shorter distance between pedons, and with a more uniform physiography was selected (Figure 34). The reduced area was located on the Ocala Uplift physiographic region. In general, the D values were reduced (Table 19) for weighted average total sand and clay contents if compared
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162 Table 19. Fractal dimension (D value) derived from selected soil property semivariograms for a reduced study area. Soil Property Semivariogram Lag TS CL OC ACL AOC distance (km) D value Direction10 1.80 1.60 2.00 1.74 2.00 independent 5 1.76 1.68 1.97 1.72 2.00 EW 10 1.87 1.86 2.00 1.75 2.00 5 1.95 1.66 2.00 1.72 1.96 NESW 10 1.66 1.70 1.90 1.86 2.00 5 1.22 1.33 1.90 1.86 2.00 NWSE 10 1.96 1.91 2.00 1.62 2.00 5 2.00 1.98 2.00 1.78 2.00 NS 10 1.76 1.85 1.59 1.74 2.00 5 1.68 1.79 1.59 1.40 2.00 See Abbreviations, pp. xiixiii
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to the D values for the entire studied area (Table 18). Some of the D values for organic carbon content decreased. The D values for the A horizon clay content remained approximately the same or increased. These results indicate that when the scale of study is increased, soil properties with a large proportion of nugget variance (e.g., total sand) had a larger decrease in the random variation than soil properties with small proportion of nugget variance (e.g., the A horizon clay content). Some D values for weighted average total sand and clay contents and A horizon clay content increased for 5 km lag distances indicating a complex variation within this distance. Thus, if random variation is to be reduced, the distance between sampling locations has to be smaller than 5 km. The large D values ( larger than 1.5) have been found to be common in soils (Burrough, 1983b). For future planning of small scale studies in the area, it is necessary to consider that even 5 km distance between sampling locations give large D values which indicates a large proportion of random variability. The geostatistical analysis allowed the separation of the systematic and the random components of the variance. The fractal analysis indicated that it is necessary to increase the scale of study if the random component is to be reduced. Then, the variability of the soil properties
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164 studied could be explained better as a result of their specific location with respect to their parent material, topography, vegetation, and climate.
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SUMMARY AND CONCLUSIONS One hundred fifty one pedons were selected to determine the important soil properties affecting the spatial variability of soils in northwest Florida. Each pedon was located by a system of geographic coordinates (X and Y) Twenty soil properties (horizon thickness; very coarse, coarse, medium, fine, and very fine sand fractions; total sand, silt, and clay contents; pHwater; pHKCl; organic carbon content; Ca, Mg, Na, and K contents extractable in NH 4 OAC; total bases; extractable acidity; cation exchange capacity; and base saturation) were initially selected for this study. Principal component analysis (PCA) and geostatistics were used in addition to other statistical analyses. All properties were nonnormally distributed, based on the Kolmogorov test. This result could be influenced by systematic patterns of soil properties. Observations are not independent. For example, the clay content in the argillic horizon is not independent of the clay content in the eluvial horizon. Argillic horizons developed because clay is translocated from the upper horizons and is deposited in the lower horizons. Argillic horizons are 165
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166 developed under specific conditions. The process is not random. Values of soil properties were not independent but associated, and this fact could influence the probabilistic distribution. Individual soil properties have different degrees of importance in influencing the spatial variability of soils. In addition, geostatistical analysis is time consuming and complex. Therefore, PCA was used as an unbiased method to reduce the number of soil properties initially selected for study with geostatistics Two sets of data were used. One set was composed of weighted average of soil properties of individual pedons. Horizon thickness was used as the weighting criterion. Information is lost when averages are used; therefore, a second set of data composed of soil properties from the surface A horizon were used. All soil properties were standardized to mean zero and variance one. Each principal component (PC) selected explained at least 5% of the total variance of each set of data. Selection of soil properties was based on plots of soil properties in the plane of the first two PCs, orthogonal rotation of PC's axes, guantitative selection of large eigenvectors, analysis of collinearity and correlation coefficient between soil properties and PC.
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167 Weighted average total sand, clay, and organic carbon contents and A horizon clay and organic carbon contents were selected by the PCA for the geostatistical analysis. Results of the PCA were supported by a nested analysis of variance. Soil properties selected by the PCA were important in explaining the variability within and between soil series and between horizons as shown by the nested analysis of variance. PCA and the nested analysis of variance proved to be useful statistical technigues to select important soil properties to study soil variability. The nested analysis of variance not only validated the results of the PCA but also indicated that the selected soil properties were differentiating properties. Therefore, both analyses can be used together for a guantitative determination of differentiating soil properties. PCA also can be useful to determine the correct placement of pedons into the soil classification system. Selected soil properties were employed to study soil variability using geostatistical analysis. The geostatistical analysis had four parts: semivariogram calculation, fitting of semivariograms kriging, and use of fractals. Directionindependent and dependent (EW, NESW, NWSE, and NS) semivariograms were calculated for each selected soil property on a 380 x 100 km irregular grid.
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168 Within series variance was used as the criterion to assess stationarity of soil property values. The first calculated semivariogram (directiondependent and independent) indicated presence of drift in the soil property values. Drift was reduced by using residuals, but was not completely removed. A reason for this may the presence of a shortrange or a cyclic variation in soil properties. Weighted average total sand and clay contents and A horizon clay content were characterized by the presence of structure. A nugget variance was also present. The semivariogram range varied from 15 to 35 km. Variability of soil properties was directiondependent. Weighted average values had the largest variability in the NS direction. The A horizon clay content had the largest variability in the NWSE direction. Differences between directiondependent semivariograms could be the result of differences in geology and topography. Weighted average and A horizon organic carbon contents had pure nugget effects, indicating that organic carbon contents had a large point to point variation at short distances All observed semivariograms had a wave pattern that indicated the presence of cyclic variations in the studied soil properties.
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169 Observed semivariograms ( directionindependent and directiondependent with largest sill) were fitted to theoretical models. Spherical, Dewijsian, Linear, and Root models were selected. The information derived from the fitted semivariogram was used to produce contour maps and diagrams of kriged soil properties. Contour maps were generated using universal kriging because of the presence of drift in the data. Contour maps for the direction with largest variation did not differ from those derived from directionindependent semivariograms. A reason for this is that the geostatistical program does not take into account the direction of semivariograms. Contour maps of weighted average clay and sand contents were similar. This similarity was due to the low silt content and that these two variables were members of the same principal component. Therefore, the use of principal components as variables in the geostatistical analysis can generate individual contour maps that would represent all soil properties included in the principal component. More investigations in the use of principal components as individual variables for the geostatistical analysis are recommended. It is also recommended that such results should be compared with those obtained using cokriging. The use of principal components as individual
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170 variables may have the advantage that a single map can represent the variability of a group of soil properties which have two characteristics: first, they are important in explaining the total soil variability. Second, they are differentiating properties. Diagrams of kriged standard errors were also produced. They indicated that soil properties with large nugget variance had large standard errors. Likewise, kriged standard error diagrams can be very useful in the design of new sampling strategies. These diagrams identified areas that require an increase in sampling intensity to improve the precision of the estimates. The diagrams indicated an area located in the northeastern part of the Dougherty karst as the one with the largest standard errors. A possible reason for this may be the irregular topography of the limestone in the area. Kriged standard error diagram and the plot of pedon location can be very useful for planning future sampling strategies. Standard error diagrams indicate areas that require an increase in sampling. A plot of pedon location indicates specific places within the areas with large standard error where additional samples need to be taken. Results of the geostatistical analysis supported the fact that values of soil properties are not independent. Total sand and clay content semivariograms had ranges that varied from 10 to 35 km. Values of studied soil properties
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171 were related (i.e., they were not independent) within the range distances. Organic carbon content semivariograms did not have any range, but had a characteristic wave pattern that also indicated a degree of dependence among the values of organic carbon content. Finally, the fractal dimension was derived from the semivariograms. In general, the fractal dimension was large ( larger than 1.5). These large values indicated that the scale of the study needs to be increased. A fractal dimension was also calculated for a reduced area. The reduced area had a larger pedon density, and therefore shorter distances between pedons than those for the area initially studied. The fractal dimension of soil properties was reduced, indicating the scaledependent character of soil variability. Results of the fractal analysis were as expected because the studied area is a large area with a large variation in geology and topography. Ranges obtained from semivariograms can be used to determine the grid size necessary to study the spatial variability of new areas in northwest Florida. The guantif ication of soil variability has two aspects. First, the conditions in which statistical analyses are used. Second, the statistical analyses that are employed.
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172 Soils in their natural environment do not follow randomized block or latin square patterns but soil scientists have used statistical experimental designs to study specific soil properties in laboratory or in greenhouse experiments. The greenhouse experiments have been performed to study how management (fertilizer, liming, or irrigation) influences the soil properties of interest. Results of greenhouse experiments have been validated by specific experimental design under field conditions. For soil scientists involved in the study of pedogenesis, soil variability, or soil geography, is very difficult to apply the same statistical experimental designs or to apply similar statistical analyses because of the difficulty of fulfilling the required assumptions. Soil properties that are nonnormally distributed cannot be forced to normality. Dependent values of soil properties cannot be forced to be independent. A biased sampling of typical pedons cannot be forced to be a random sampling procedure. But controlled condition are required to guarantee some standard conditions for quantitative analysis and for further application in field conditions. The opinion of this author is that soils have two scales of variability. One, the large scale (vertical direction) limited by the root system of crops, grasses, trees, or specific engineering uses (e.g., septic tanks). The other, the small scale (horizontal direction) is
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173 limited by the extension of the study or the land use. Soil scientists have been relatively successful in studying the vertical variability because they have been capable of recognizing soil horizons. Soil properties are then related to specific horizons. But soil scientists have been less sucessful in studying the variability in the horizontal direction because of the lack of emphasis in the geographic aspect of soils. Polypedons have a geographic connotation. More emphasis has been placed on their morphologic description than on their geographic relations. Therefore, the use of polypedons as geographic entities is limited. External environmental features (e.g., landform, vegetation) which are consistently recognized and mappable have been helpful in delineating soils. Therefore, this author believes that landscape position can be used as a geographic entity to study soil variability. Landscape position takes into account geographic aspects. The close relationship between soil and landscape position has been discussed by many pedologists. Landscape position can represent the "greenhouse" to test quantitative methods to study variability of soil in natural conditions. The other important aspect of the quantitative analysis of soil variability is related to the statistical analyses themselves. Normality, independence, and
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174 homogeneous variance are basic assumptions for classical statistical analyses. This study showed that the set of data, selected for this study, sampled in northwest Florida since 1967, was nonnormally distributed, and values of individual soil properties were not independent. In addition, sampling locations were not selected randomly. They were selected, as often is the case in a soil survey, to be modal pedons. This sampling procedure contradicts the criterion of randomness important in determining the degrees of freedom in classical statistics. Statistical analyses need to be divided into two groups: (i) those methods that can be used to analyze data in an artificial context (i.e, laboratory or greenhouse), and (ii) those techniques that can be used to analyze data derived from studies in the natural environment (i.e., data obtained from a soil survey) In natural conditions it may be difficult to satisfy the assumptions required for classical statistical analyses. Testing the assumptions is recommended; otherwise erroneous conclusions can be stated. When assumptions are not achieved it is necessary to study how this fact affects the results of the analysis, when assumptions are not achieved, it is necessary to employ alternative analyses (i.e, nonparametric analysis), or not to use inferential statistics.
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175 For example, in this study a classical statistical technique, nested analysis of variance, was employed to support results of the soil survey. But the hypothesis testing was not performed; thus, assumptions of this analysis were not required. The assumption of PCA is related with homogeneity of the variances. This assumption was fulfilled by standardizing the data. Results of the PCA were validated by the results of the nested analysis of variance on the raw data. The geostatistical analysis has the assumption of stationarity. This assumption was not fulfilled. Thus, universal kriging, which takes into account the presence of drift, was used as an interpolation method. Statistical analyses are needed to support and to improve the results of soil surveys. The nested analysis of variance is a technique that can be used very easily to determine the extent of the withinmap unit variance. PCA is useful to select the important soil properties to study the soil variability. PCA reduced the number of variables selected for additional study. But a hiatus remains between the results of the PCA and the results of the soil survey. Soil scientists classified several pedons to a specific soil series. But the PCA indicated a large degree of dispersion within the soil series. This author believes that this result is mainly due to the lack of emphasis of
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176 soil and landscape relationships, and to the fact that morphological soil properties were not considered in the PCA. Therefore, more investigations are recommended in the use of morphological properties, not only in PCA but also in common statistical techniques used in quantitative analysis. The other statistical analysis used, geostatistics has two advantages over classical statistical techniques. First, geostatistics takes into account the location of the observations. Second, geostatistics not only considers the observation values but also their geometric support (i.e., soil as a volume). Results of the geostatistical analysis need to be validated by comparison of the isarithmic map with the soil survey map. The use of individual PCs to obtain the isarithmic map is one way to do so. More investigations are recommended.
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APPENDIX A CLASSIFICATION OF SERIES STUDIED
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Soil Series Taxonomic classification (Soil temperature is thermic)* Alaga Albany Angie Apalachee Ardilla Bethera Blanton Bonif ay Bonneau Cantey Chipley Chipola Compass Cowarts Coxville Dothan Duplin Esto Escambia Faceville Fuquay Garcon Greenville Goldsboro Hornsville Iuka Kenansville Kinston Lakeland Leef ield Lucy Lyerly Lynchburg Malbis Mulat Nankin Norfolk Ocilla Oktibbeha Coated Typic Quartz ipsamments Loamy, siliceous Grossarenic Paleudults. Clayey, mixed Aquic Paleudults. Very fine, montmorillonitic Fluvaquentic Dystrochrepts Fineloamy, siliceous Fragiaquic Paleudults Clayey, mixed Typic Paleaguults. Loamy, siliceous Grossarenic Paleudults. Loamy, siliceous Grossarenic Plinthic Paleudults Loamy, siliceous Arenic Paleudults. Clayey, kaolinitic Typic Albaquults. Coated Typic Quartz ipsamments Loamy, siliceous Arenic Hapludults. Coarseloamy, siliceous Plinthic Paleudults Fineloamy, siliceous Typic Hapludults. Clayey, kaolinitic Typic Paleaquults. Fineloamy, siliceous Plinthic Paleudults. Clayey, kaolinitic Aguic Paleudults. Clayey, kaolinitic Typic Paleudults. Coarseloamy, siliceous Plinthaguic Paleudults. Clayey, kaolinitic Typic Paleudults. Loamy, siliceous Arenic Plinthic Paleudults. Loamy, siliceous Arenic Hapludults. Clayey, kaolinitic Rhodic Paleudults. Fineloamy, siliceous Aquic Paleudults. Clayey, kaolinitic Typic Hapludults. Coarseloamy, siliceous, acid Aguic Udif luvents Loamy, siliceous Arenic Hapludults. Fineloamy, siliceous, acid Typic Fluvaguents Uncoated Typic Quartz ipsamments. Loamy, siliceous Arenic Plinthaguic Paleudults. Loamy, siliceous Arenic Paleudults. Veryfine, montmorillonitic Vertic Hapludalf s Fineloamy, siliceous Aerie Paleaguults. Fineloamy, siliceous Plinthic Paleudults. Coarseloamy, siliceous Typic Ochraguults. Clayey, kaolinitic Typic Hapludults. Fineloamy, siliceous Typic Paleudults. Loamy, siliceous Aguic Arenic Paleudults. Veryfine, montmorillonitic Vertic 178
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179 Orangeburg Pansey Pantego Pelham Plummer Rains Redbay Rutlege Sapelo Shubuta Stilson Surrency Tifton Troup Wagram Yemasse Yonges Hapludalf s. Fineloamy, siliceous Typic Paleudults. Fineloamy, siliceous Plinthic Paleaquult Fineloamy siliceous Umbric Paleaquults. Loamy, siliceous Arenic Paleaquults. Loamy, siliceous Grossarenic Paleaquults. Fineloamy, siliceous Typic Paleaquults. Fineloamy, siliceous Rhodic Paleudults. Sandy, siliceous Typic humaquepts. Sandy, siliceous Ultic Haplaquods. Clayey, mixed Typic Paleudults. Loamy, siliceous Arenic Plinthic Paleudults. Loamy, siliceous Arenic Umbric Paleaquult Coarseloamy, siliceous Plinthic Paleudults Loamy, siliceous Grossarenic Paleudults. Loamy, siliceous, Arenic Paleudults. Fineloamy, mixed Aerie Ochraquults. Fineloamy, mixed Typic Ochraqualfs. Source: Calhoun et al., 1974; Carlisle et al., 1978, 1981, 1985; I.F.A.S. Soil Characterization Laboratory, unpublished data.
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APPENDIX B GEOGRAPHIC COORDINATES OF PEDONS STUDIED
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Pedon Laboratory number number X Coordinate (4.8 km/X) Y Coordinate (4.6 km/Y) 1 153159 42.50 18.75 2 140146 43.10 17.40 3 133139 43.80 21.85 4 147152 42.00 19.00 5 160166 40.50 13.10 6 167172 3.60 17.15 7 101106 5.25 16.60 8 107111 3.40 16.70 9 397401 5.80 17.45 10 409414 3.90 17.45 11 402408 29.90 19.15 12 8389 34.20 19.20 13 9095 9.40 21.55 14 389396 10.10 21.60 15 9699 33.70 19.20 16 379383 43.80 15.75 17 384388 7.50 17.85 18 327332 43.50 14.50 19 271277 32.90 17.30 20 295301 42.55 16.45 21 10721080 42.20 17.55 22 322326 43.35 17.60 23 333337 7.65 21.50 24 338342 7.25 21.60 25 278281 7.50 21.60 26 282284 9.90 21.60 27 285287 7.40 21.60 28 288291 41.90 16.50 29 292294 5.20 16.80 30 10651071 40.50 13.85 31 316321 41.40 16.30 32 10561064 43.30 18.30 33 302307 18.20 43.00 34 18951899 64.25 9.85 35 14751479 59.25 11.55 36 14271433 46.80 20.15 37 14461453 64.80 10.95 38 14541460 37.50 20.70 39 14691474 37.25 20.50 40 18871894 42.60 17.40 41 18801886 37.90 20.35 42 18721879 62.65 8.90 43 14341441 59.00 11.40 44 19001905 44.50 0.30 45 14611468 59.30 10.85 46 14801484 37.80 20.10 181
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182 47 20492055 48 13751379 49 15751581 50 628633 51 20562064 52 20652071 53 15611567 54 16091614 55 15821588 56 15971602 57 634637 58 19061910 59 16031608 60 22622268 61 28232827 62 23822387 63 22992306 64 28422847 65 32833288 66 23602364 67 23652369 68 23772381 69 23542359 70 33223328 71 28662874 72 23702376 73 28352841 74 32683273 75 22912298 76 28572865 77 23472353 78 28752880 79 22762283 80 26172623 81 23072312 82 28282834 83 26242630 84 22842290 85 26372644 86 28102817 87 23882392 88 26312636 89 26452652 90 26532658 91 32743282 92 34173425 93 43034309 94 40644069 95 40704077 96 42764283 97 40564063 58 .90 11. 50 65 00 9.30 21 15 21. 35 22.40 21.20 20 90 21. 25 23 10 20 00 23 95 18 40 22.00 21.10 23 .15 20.85 65. 10 10.15 64 90 9 55 68.60 10.95 68 85 12.70 66 90 10. 40 25 25 18 00 23 .55 20 70 66 50 12.80 65.20 8 .70 38 10 7.00 65.35 10.15 68 50 11 95 67.75 8.75 65 30 9.45 68.70 13 50 64 90 10.15 79 70 11.55 75 30 10.85 67 .30 13.35 67. 95 9.70 66 65 11.45 72 45 13 40 65.70 4 60 43 .95 21 .80 40 40 18.15 40.45 11.30 43.70 15.90 40. 10 11.25 6 40 15.00 40.05 19.70 40.80 15.60 43.95 15.85 9.35 17.45 3.00 17.20 5.80 14.85 10 95 18.10 11.95 14.25 3.10 16.40 3.40 14.00 5.55 16.00 5.45 16.80 34.60 15.50
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98 3395 3401 99 4086 4092 100 3388 3394 101 4101 4107 102 4270 4275 103 3410 3416 104 4284 4291 105 4108 4114 106 4078 4085 107 4263 4269 108 4298 4302 109 4050 4055 110 4619 4623 111 4756 4763 112 4847 4853 113 4478 4486 114 4788 4795 115 4874 4881 116 4510 4516 117 5214 5217 118 4796 4803 119 4504 4509 120 4991 4997 121 4517 4522 122 4750 4755 123 4764 4773 124 4499 4503 125 5131 5138 126 4493' 4498 127 5139' 5147 128 5148' 5157 129 4898' 4903 130 5027' 5032 131 4470' 4477 132 5062' 5066 133 4804' 4810 134 48364839 135 50215026 136 46324639 137 49985003 138 52075213 139 44884492 140 48924897 141 57265731 142 57325738 143 57145719 144 55115518 145 57205725 146 54935498 147 54995504 148 55055510 183 28.00 15.25 7.20 17.70 6.60 20.60 5.10 16.50 5.15 14.10 42.10 18.40 11.45 21.55 49.15 14.75 64.70 10.00 65.15 10.55 41.75 18.55 58.70 11.65 56.60 8.50 62.60 12.20 46.60 19.80 46.95 20.10 65.00 11.35 7.35 12.60 47.00 19.85 65.10 12.70 7.65 19.95 60.90 10.90 60.80 11.40 59.40 8.40 60.60 10.50 24.00 14.55 21.00 15.70 22.05 21.35 29.80 14.70 23.15 16.10 60.60 13.20 60.75 10.80 22.30 21.35 63.70 12.55 26.60 19.20 36.95 4.90 20.25 20.90 65.60 7.10 27.00 15.80 38.50 8.60 23.90 14.70 67.55 12.50 25.15 20.60 28.80 10.20 29.00 9.60 67.40 9.20 20.20 14.60 68.30 12.15 38.75 4.00 38.45 11.55 78.10 10.00
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184 149 53965402 77.80 10.70 150 55245528 27.65 14.45 151 54035410 26.55 14.00 Coordinates are relative to a point of origin 30s 00' 00'' N and 87s 24' 18'' W (X = 0 and Y = 0) chosen to ensure that all coordinates would be positive.
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APPENDIX C SEMI VARIOGRAMS FOR DIRECTIONS WITH LARGEST VARIABILITY
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186
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187
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188 w i 0) P u H 41 4J C 0) p c o u c o A u u c id U o VWWV9 a; id g H (0 CD H > id o Â•H CD 03 Â•P > .C I Â•H Â£ CD CD m 0) P Di Â•H
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189 g res U o H U > I Â•H g 0) cn W CO I Z T3 0 +J +J Â•H m +j c OJ 4J c o o >1 ia H o c o N Â•H 2 CO n a) M CP VWWV9
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APPENDIX D CONTOUR MAPS FOR DIRECTIONS WITH LARGEST VARIABILITY
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192
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193 4 o o Â£ O 00 rO >i 0 CD Di rO H CU > Â•H M 01 O E 03 (tf Â•H U Cn O Â•H M id > i H e
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194
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APPENDIX E MAP OF PHYSIOGRAPHIC REGIONS IN NORTHWEST FLORIDA
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id Â•H u o 4> K to I M Â§ C H CO G O Â•H CP Â• CD M XI E rH O CO O Â•H CTi 00 JS h 10 ft id U co 0 o Â•H 0 CO M 05 >iCQ rH CU Â•Â• H H 0) d) 41 B3 *U O Q CO CO O M U CD T3 4J C C a> o rH 41 H 05 0 H^H O >irH a Â•r) 4J C D s: u o u Â• O CD
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LITERATURE CITED Afifi, A. A., and V. Clark. 1984. Computeraided multivariate analysis. Lifetime Learning Publications, Belmont, California. Beckett, P.H.T., and S.W. Bie. 1976. Reconaissance for soil survey. II. Presurvey estimates of the intricacy of the soil pattern. J. Soil Sci. 27:101110. Beckett, P.H.T., and R. Webster. 1971. Soil variability: A review. Soils and Fert. 34:115. Belobrov, V.P. 1976. Variation in some chemical and morphological properties of Sodpodzolic soils within the boundaries of elementary soil areals and taxonomic groups. In V.M. Fridland (ed.), Soil combinations and their genesis. Amerind Pub. Co. Pvt. Ltd. New Delhi. Bie, W.S. 1984. Soil data in digital space. In P. A. Burrough and S.W Bie (eds.), Soil information system technology. Proc. 6th meeting ISSS, Soil information system working group, Pudoc, Wageningen. Bos, J., M.E. Collins, G.J. Gensheimer, and R.B. Brown. 1984. Spatial variability for one type of phosphate mine land in central Florida. Soil Sci. Soc. Am. J. 48:11201125. Bradley, T.J. 1972. The climate of Florida. In U.S. Dept. of Commerce. Climates of the states. Oceanic and Atmospheric Administration. Water Information Center Inc., Syosset, New York. Brooks, H.K. 1981a. Guide to the physiographic divisions of Florida. IFAS. Univ. of Florida, Gainesville. Brooks, H.K. 1981b. Map of physiographic divisions of Florida. IFAS. Cooperative Extension Service, Univ. of Florida, Gainesville. Burgess, T.M. and R. Webster. 1980a. Optimal interpolation and isarithmic mapping of soil properties. I. The semivariogram and punctual kriging. J. Soil Sci. 31:315331. 197
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198 Burgess, T.M. and R. Webster. 1980b. Optimal interpolation and isarithmic mapping of soil properties. II. Block kriging. J. Soil Sci. 31:333341. Burgess, T.M. R. Webster, and A.B. McBratney. 1981. Optimal interpolation and isarithmic mapping of soil properties. IV. Sampling strategy. J. Soil Sci. 32:643659. Burrough, P. A. 1981. Fractal dimensions of landscapes and another environmental data. Nature 294:240242. Burrough, P. A. 1983a. Problems of superimposed effects in the statistical study of the spatial variation of soil. Agr. Water Manag. 6:123143. Burrough, P. A. 1983b. Multiscale sources of spatial variation in soil. I. The application of fractal concepts to nested levels of soil variation. J. Soil Sci. 34:577597. Burrough, P. A. 1983c. Multiscale sources of spatial variation in soil. II. A nonBrownian fractal model and its application in soil survey. J. Soil Sci. 34: 599620. Burrough, P. A. and R. Webster. 1976. Improving a reconnaissance soil classification by multivariate methods. J. Soil Sci. 27:534571. Byers, E. and D.B. Stephens. 1983. Statistical and stochastic analyses of hydraulic conductivity and particle size in a fluvial sand. Soil Sci. Soc. Am. J. 47:10721081. Caldwell, R.E. 1980. Major land resources areas in Florida. Soil Crop Sci. Soc. Fla. Proc. 39:3840. Calhoun, F.G, V.W. Carlisle, R.E. Caldwell, L.W. Zelazny, L.C. Hammond, and H.L Breland. 1974. Characterization data for selected Florida soils. Soil Sci. Dept. Research Report No. 741. IFAS, Univ. of Florida, Gainesville Campbell, J.B. 1978. Spatial variation of sand content and pH within single contiguous delineations of two soil mapping units. Soil Sci. Soc. Am. J. 42:460464. Carlisle, V.W. and R.B. Brown. 1982. Florida soil identification handbook. Soil Sci. Dept. IFAS. Univ. of Florida, Gainesville.
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199 Carlisle, V.W. R.E. Caldwell, F. Sodek III, L.C. Hammond, F.G. Calhoun, M.A. Granger, and H.L. Breland. 1978. Characterization data for selected Florida soils. Soil Sci. Dept. Research Report No. 781. IFAS. Univ. of Florida, Gainesville. Carlisle, V.W. M.E. Collins, F. Sodek III, and L.C Hammond. 1985. Characterization data for selected Florida soils. Soil Sci. Dept. Research Report No 851. IFAS. Univ. of Florida, Gainesville. Carlisle, V.W. C.T. Hallmark, F. Sodek III, R.E. Caldwell, L.C. Hammond, and V.E. Berkheiser. 1981. Characterization data for selected Florida soils. Soil Sci. Dept. Research Report No. 811. IFAS. Univ. of Florida, Gainesville. Chirlin, G.R. and G. Dagan. 1980. Theoretical head variograms for steady flow in statistical homogeneous aquifers. Water Res. Res. 16:10011015. Cuanalo, H.E., and R. Webster. 1970. A comparative study of numerical classification and ordination of soil profiles in a locality near Oxford. I. Analysis of 85 sites. J. Soil Sci. 23:6275. DeGraffenreid, J. A. 1982. Timeand spacedependent data in the earth sciences. Kansas Geol. Survey, Series in Spatial Analysis No. 6. Univ. of Kansas, Lawrence. Duffee, E.M., W.J. Allen, and H.C. Ammons. 1979. Soil survey of Jackson County, Florida. U.S.D.A., U.S. Govt. Printing Office, Washington, D.C. Duffee, E.M., R.A. Baldwin, D.L. Lewis, and W.B. Warmack. 1984. Soil survey of Bay County, Florida. U.S.D.A., U.S. Govt. Printing Office, Washington, D.C. Edmonds, W. J. J.B. Campbell, and M. Lentner. 1985. Taxonomic variation within three mapping units in Virginia. Soil Sci. Soc. Am. J. 49:394401. Enertronics. 1983. Energraphics Version 1.3. Enertronics Research Inc., St Louis, Missouri. Fernald, E.A. 1981. Atlas of Florida. Florida St. Univ., Tallahassee. Fernald, E.A. and D.J. Patton. 1984. Water resources, Atlas of Florida. Florida St. Univ., Tallahassee.
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BIOGRAPHICAL SKETCH Francisco Antonio Ovalles Viani was born in Caracas, Venezuela, on August 1, 1950. He is son of the late Dr. Pedro Jose Ovalles and Mrs. Alba Viani de Ovalles. He received the degree of Ingeniero Agronomo in the Universidad Central de Venezuela in February, 1976. In March, 1976, he joined the Ministerio de Obras Publicas, Direccion General de Recursos Hidraulicos in the Division de Edafologia, Region Central. In April, the former official institution became Ministerio del Ambiente y de los Recursos Naturales Renovables. From January, 1978, to December, 1980, he was also a member of the Soil Science Department of the Facultad de Agronomia, Universidad Central de Venezuela. In August 1982, he received a scholarship from Consejo Nacional de Investigaciones Cientificas y Tecnologicas (CONICIT) to pursue graduate studies. He was accepted for a graduate program in the Soil Science Department, University of Florida, in August, 1982, under the guidance of Dr. Mary E. Collins. He received the degree of Master of Science from the University of Florida in December, 1984. 207
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208 He is member of Colegio de Ingenieros de Venezuela, Sociedad Venezolana de Ingenieros Agronomos, Sociedad Venezolana de la Ciencia del Suelo, American Society of Agronomy, Soil Science Society of America, International Soil Science Society, and honor societies Sigma Xi and Gamma Sigma Delta. He served as secretary of Sociedad Venezolana de la Ciencia del Suelo in 1982. He is married to Giordana de Ovalles; they have two children, Johanna Fernanda and Pedro Jose.
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. M. E.^Â£ollins, Chairman Associate Professor of Soil Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Antonini Professor of Geography I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. R. W. Arnold Professor of Soil Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. if MB 3 R. B. Brown Associate Professor of Soil Science
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. A~. S Fotherwigham Associate Professor of Geography This dissertation was submitted to the Graduate Faculty of the College of Agriculture and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1986 g^A of. Jl^f Dean, College of Agi#culture Dean, Graduate School
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