• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Copyright
 Table of Contents
 List of Tables
 Figures
 Annexes
 Summary
 Acknowledgement
 Interpolation of acronyms...
 Introduction
 Interpolation techniques
 Reviewing interpolation techni...
 A case study for Jalisco,...
 Conclusions for the study area
 Conclusions and recommendation...
 Reference
 Annex 1 : Description of applying...
 Annex 2 : Dataset comparison for...
 Annex 3 : Interpolated monthly...
 Annex 4 : Basic surface charac...
 Annex 5 : Prediction error surfaces...
 Back Cover






Group Title: Geographic information systems series - Natural Resources Group - 99-01
Title: Interpolation techniques for climate variables
CITATION PDF VIEWER PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00077518/00001
 Material Information
Title: Interpolation techniques for climate variables
Series Title: Geographic information systems series
Physical Description: vi, 26 p. : ill. (some col.), maps (some col.) ; 27 cm.
Language: English
Creator: Hartkamp, A. Dewi
International Maize and Wheat Improvement Center
Publisher: CIMMYT
Place of Publication: Mexico
Publication Date: 1999
 Subjects
Subject: Meteorology, Agricultural -- Mathematical models -- Mexico -- Jalisco   ( lcsh )
Climatic changes -- Mathematical models -- Mexico -- Jalisco   ( lcsh )
Interpolation -- Mathematical models   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Bibliography: Includes bibliographical references (p. 16-17).
Statement of Responsibility: A. Dewi Hartkamp ... et al..
 Record Information
Bibliographic ID: UF00077518
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 45569667
issn - 1405-7484 ;

Downloads

This item has the following downloads:

NRGGIS%2099_01 ( PDF )


Table of Contents
    Front Cover
        Front cover
    Title Page
        Page i
    Copyright
        Page ii
    Table of Contents
        Page iii
    List of Tables
        Page iv
    Figures
        Page iv
    Annexes
        Page iv
    Summary
        Page v
    Acknowledgement
        Page v
    Interpolation of acronyms and terminology
        Page vi
    Introduction
        Page 1
    Interpolation techniques
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Reviewing interpolation techniques
        Page 8
    A case study for Jalisco, Mexico
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
    Conclusions for the study area
        Page 15
    Conclusions and recommendation for further work
        Page 15
    Reference
        Page 16
        Page 17
    Annex 1 : Description of applying the linear model of co-regionalization
        Page 18
    Annex 2 : Dataset comparison for precipitation
        Page 19
    Annex 3 : Interpolated monthly precipitation surfaces from IDWA, Splining, and Co-kriging for April and August
        Page 20
        Page 21
        Page 22
    Annex 4 : Basic surface characteristics
        Page 23
    Annex 5 : Prediction error surfaces for precipitation interpolated by Splining and Co-kriging
        Page 24
        Page 25
        Page 26
    Back Cover
        Back cover
Full Text





II
CIMMYT
Sustainable
Maize and Wheat
Systems for the Poor

Interpolation Techniques for
Climate Variables

A. Dewi Hartkamp, Kirsten De Beurs,


Alfred Stein, Jeffrey W. White


NR.G
Natural Resources Group

Geographic Information Systems
Series 99-01









II
CIMMYT
Sustainable
Maize and Wheat
Systems for the Poor


Interpolation Techniques for

Climate Variables


A. Dewi Hartkamp,1 Kirsten De Beurs,2


Alfred Stein,2 Jeffrey W. White1












NKO
Natural Resources Group

Geographic Information Systems Series 99-01

1 CIMMYT Natural Resources Group.
2 Sub-department Soil Science and Geology, Department of Environmental Sciences,
Wageningen Agricultural University, PO Box 37, 6700 AAWageningen, The Netherlands












CIMMYT (www.cimmvt.mx or www.cimmvt.caiar.orq) is an internationally funded, nonprofit scientific
research and training organization. Headquartered in Mexico, the Center works with agricultural
research institutions worldwide to improve the productivity, profitability, and sustainability of maize and
wheat systems for poor farmers in developing countries. It is one of 16 similar centers supported by the
Consultative Group on International Agricultural Research (CGIAR). The CGIAR comprises over 55
partner countries, international and regional organizations, and private foundations. It is co-sponsored
by the Food and Agriculture Organization (FAO) of the United Nations, the International Bank for
Reconstruction and Development (World Bank), the United Nations Development Programme (UNDP),
and the United Nations Environment Programme (UNEP). Financial support for CIMMYT's research
agenda also comes from many other sources, including foundations, development banks, and public
and private agencies.


CIMMYT supports Future Harvest, a public awareness campaign that builds understanding about the
importance of agricultural issues and international agricultural research. Future Harvest links respected
research institutions, influential public figures, and leading agricultural scientists to underscore the
wider social benefits of improved agriculture: peace, prosperity, environmental renewal, health, and the
alleviation of human suffering (www.futureharvest.orq).


International Maize and Wheat Improvement Center (CIMMYT) 1999. Responsibility for this publica-
tion rests solely with CIMMYT. The designations employed in the presentation of material in this
publication do not imply the expressions of any opinion whatsoever on the part of CIMMYT or contribu-
tory organizations concerning the legal status of any country, territory, city, or area, or of its authorities,
or concerning the delimitation of its frontiers or boundaries.


Printed in Mexico.


Correct citation: Hartkamp, A.D., K. De Beurs, A. Stein, and J.W. White. 1999. Interpolation Tech-
niques for Climate Variables. NRG-GIS Series 99-01. Mexico, D.F.: CIMMYT.


Abstract: This paper examines statistical approaches for interpolating climatic data over large regions,
providing a brief introduction to interpolation techniques for climate variables of use in agricultural
research, as well as general recommendations for future research to assess interpolation techniques.
Three approaches-1) inverse distance weighted averaging (IDWA), 2) thin plate smoothing splines
and 3) co-kriging-were evaluated for a 20,000 km2 square area covering the state of Jalisco, Mexico.
Taking into account valued error prediction, data assumptions, and computational simplicity, we
recommend use of thin-plate smoothing splines for interpolating climate variables.


ISSN: 1405-7484
AGROVOC descriptors: Climatic factors; Climatic change; Meteorological observations; Weather
data; Statistical methods; Agriculture; Natural resources; Resource management; Research;
Precipitation; Jalisco; Mexico
AGRIS category codes: P40 Meteorology and Climatology
Dewey decimal classification: 551.6











Contents


iv List of Tables, Figures, Annexes
v Summary
v Acknowledgments
vi Interpolation acronyms and terminology
1 Introduction
1 Interpolation techniques
8 Reviewing interpolation techniques
9 A case study for Jalisco, Mexico
15 Conclusions for the study area
15 Conclusion and recommendations for further work
16 References


















































--










Tables

3 Table 1. A comparison of interpolation techniques.
10 Table 2. Stations for which geographic coordinates were changed to INIFAP values.
10 Table 3. Station numbers with identical geographic coordinates.
12 Table 4. Correlation coefficients between prediction variables: precipitation (P), maximum temperature
(Tmax), and the co-variable (elevation).
12 Table 5. Variogram and cross-variogram values for the linear model of co-regionalization for
precipitation.
13 Table 6. Variogram and cross-variogram values for the linear model of co-regionalization for maximum
temperature.
15 Table 7. Validation statistics for four monthly precipitation surfaces.
15 Table 8. Validation statistics for four maximum temperature surfaces.



Figures

2 Figure 1. An example of interpolation using Thiessen polygons and inverse distance weighted
averaging to predict precipitation.
6 Figure 2. An example of a semi-variogram with range, nugget, and sill.
7 Figure 3. Examples of most commonly used variogram models a) spherical, b) exponential, c)
Gaussian, and d) linear.
11 Figure 4. Validation selection areas and two validation sets of 25 points each for precipitation.
14 Figure 5. Frequency distribution of precipitation values after splining and co-kriging for two months.
14 Figure 6. Frequency distribution of elevation, the co-variable for interpolation in this study, for Jalisco.




Annexes

18 Annex 1. Description of applying a linear model of co-regionalization.
19 Annex 2. Dataset comparison for precipitation.
20 Annex 3. Interpolated monthly precipitation surfaces from IDWA, splining and co-kriging for the months
April, May, August and September.
23 Annex 4. Basic surface characteristics.
24 Annex 5. Prediction error surfaces for precipitation interpolated by splining and by co-kriging.










Tables

3 Table 1. A comparison of interpolation techniques.
10 Table 2. Stations for which geographic coordinates were changed to INIFAP values.
10 Table 3. Station numbers with identical geographic coordinates.
12 Table 4. Correlation coefficients between prediction variables: precipitation (P), maximum temperature
(Tmax), and the co-variable (elevation).
12 Table 5. Variogram and cross-variogram values for the linear model of co-regionalization for
precipitation.
13 Table 6. Variogram and cross-variogram values for the linear model of co-regionalization for maximum
temperature.
15 Table 7. Validation statistics for four monthly precipitation surfaces.
15 Table 8. Validation statistics for four maximum temperature surfaces.



Figures

2 Figure 1. An example of interpolation using Thiessen polygons and inverse distance weighted
averaging to predict precipitation.
6 Figure 2. An example of a semi-variogram with range, nugget, and sill.
7 Figure 3. Examples of most commonly used variogram models a) spherical, b) exponential, c)
Gaussian, and d) linear.
11 Figure 4. Validation selection areas and two validation sets of 25 points each for precipitation.
14 Figure 5. Frequency distribution of precipitation values after splining and co-kriging for two months.
14 Figure 6. Frequency distribution of elevation, the co-variable for interpolation in this study, for Jalisco.




Annexes

18 Annex 1. Description of applying a linear model of co-regionalization.
19 Annex 2. Dataset comparison for precipitation.
20 Annex 3. Interpolated monthly precipitation surfaces from IDWA, splining and co-kriging for the months
April, May, August and September.
23 Annex 4. Basic surface characteristics.
24 Annex 5. Prediction error surfaces for precipitation interpolated by splining and by co-kriging.










Tables

3 Table 1. A comparison of interpolation techniques.
10 Table 2. Stations for which geographic coordinates were changed to INIFAP values.
10 Table 3. Station numbers with identical geographic coordinates.
12 Table 4. Correlation coefficients between prediction variables: precipitation (P), maximum temperature
(Tmax), and the co-variable (elevation).
12 Table 5. Variogram and cross-variogram values for the linear model of co-regionalization for
precipitation.
13 Table 6. Variogram and cross-variogram values for the linear model of co-regionalization for maximum
temperature.
15 Table 7. Validation statistics for four monthly precipitation surfaces.
15 Table 8. Validation statistics for four maximum temperature surfaces.



Figures

2 Figure 1. An example of interpolation using Thiessen polygons and inverse distance weighted
averaging to predict precipitation.
6 Figure 2. An example of a semi-variogram with range, nugget, and sill.
7 Figure 3. Examples of most commonly used variogram models a) spherical, b) exponential, c)
Gaussian, and d) linear.
11 Figure 4. Validation selection areas and two validation sets of 25 points each for precipitation.
14 Figure 5. Frequency distribution of precipitation values after splining and co-kriging for two months.
14 Figure 6. Frequency distribution of elevation, the co-variable for interpolation in this study, for Jalisco.




Annexes

18 Annex 1. Description of applying a linear model of co-regionalization.
19 Annex 2. Dataset comparison for precipitation.
20 Annex 3. Interpolated monthly precipitation surfaces from IDWA, splining and co-kriging for the months
April, May, August and September.
23 Annex 4. Basic surface characteristics.
24 Annex 5. Prediction error surfaces for precipitation interpolated by splining and by co-kriging.










Summary

Understanding spatial variation in climatic conditions is key to many agricultural and natural
resource management activities. However, the most common source of climatic data is
meteorological stations, which provide data only for single locations. This paper examines statistical
approaches for interpolating climatic data over large regions, providing a brief introduction to
interpolation techniques for climate variables of use in agricultural research, as well as general
recommendations for future research to assess interpolation techniques. Three approaches-1)
inverse distance weighted averaging (IDWA), 2) thin plate smoothing splines and 3) co-kriging-
were evaluated for a 20,000 km2 square area covering the state of Jalisco, Mexico. Monthly mean
data were generated for 200 meteorological stations and a digital elevation model (DEM) based on
1 km2 grid cells was used. Due to low correlation coefficients between the prediction variable
(precipitation) and the co-variable (elevation), interpolation using co-kriging was carried out for only
four months. Validation of the surfaces using two independent sets of test data showed no
difference among the three techniques for predicting precipitation. For maximum temperature,
splining performed best. IDWA does not provide an error surface and therefore splining and co-
kriging were preferred. However, the rigid prerequisites of co-kriging regarding the statistical
properties of the data used (e.g., normal distribution, non-stationarity), along with its computational
demands, may put this approach at a disadvantage. Taking into account error prediction, data
assumptions, and computational simplicity, we recommend use of thin-plate smoothing splines for
interpolating climate variables.


Acknowledgments

The authors would like to thank Ariel Ruiz Corral (INIFAP, Guadalajara, Mexico), Mike Hutchinson
(University of Canberra, Australia), Aad van Eijnsbergen (Wageningen Agricultural University, The
Netherlands) and Edzer Pebesma (University of Utrecht, The Netherlands) for assistance in various
aspects of this research. Finally, we are indebted to CIMMYT science writer Mike Listman and
designer Juan Jos6 Joven for their editing and production assistance.






















-0










Summary

Understanding spatial variation in climatic conditions is key to many agricultural and natural
resource management activities. However, the most common source of climatic data is
meteorological stations, which provide data only for single locations. This paper examines statistical
approaches for interpolating climatic data over large regions, providing a brief introduction to
interpolation techniques for climate variables of use in agricultural research, as well as general
recommendations for future research to assess interpolation techniques. Three approaches-1)
inverse distance weighted averaging (IDWA), 2) thin plate smoothing splines and 3) co-kriging-
were evaluated for a 20,000 km2 square area covering the state of Jalisco, Mexico. Monthly mean
data were generated for 200 meteorological stations and a digital elevation model (DEM) based on
1 km2 grid cells was used. Due to low correlation coefficients between the prediction variable
(precipitation) and the co-variable (elevation), interpolation using co-kriging was carried out for only
four months. Validation of the surfaces using two independent sets of test data showed no
difference among the three techniques for predicting precipitation. For maximum temperature,
splining performed best. IDWA does not provide an error surface and therefore splining and co-
kriging were preferred. However, the rigid prerequisites of co-kriging regarding the statistical
properties of the data used (e.g., normal distribution, non-stationarity), along with its computational
demands, may put this approach at a disadvantage. Taking into account error prediction, data
assumptions, and computational simplicity, we recommend use of thin-plate smoothing splines for
interpolating climate variables.


Acknowledgments

The authors would like to thank Ariel Ruiz Corral (INIFAP, Guadalajara, Mexico), Mike Hutchinson
(University of Canberra, Australia), Aad van Eijnsbergen (Wageningen Agricultural University, The
Netherlands) and Edzer Pebesma (University of Utrecht, The Netherlands) for assistance in various
aspects of this research. Finally, we are indebted to CIMMYT science writer Mike Listman and
designer Juan Jos6 Joven for their editing and production assistance.






















-0










Interpolation Acronyms and Terminology


CIMMYT International Maize and Wheat Improvement Center.
DEM Digital elevation model; a digital description of a terrain in the shape of data and algorithms.
ERIC Extractor Rapido de Informaci6n Climatol6gica.
GCV Generalized cross validation. A measure of the predictive error of the fitted surface which is
calculated by removing each data point, one by one, and calculating the square of the
difference between each removed data point from a surface fitted to all the other points.
IDWA Inverse distance weighted averaging.
IMTA Instituto Mexicano de Tecnologia del Agua.
INIFAP Mexican National Institute of Forestry, Agriculture, and Livestock Research (Instituto Nacional
de Investigaciones Forestales y Agropecuarias).
Interpolation The procedure of estimating the value of properties at unsampled sites within an area covered
by sampled points, using the values of properties from those points.









































----------------











Interpolation Techniques


for Climate Variables


Introduction


Geographic information systems (GIS) and modeling
are becoming powerful tools in agricultural research
and natural resource management. Spatially distrib-
uted estimates of environmental variables are increas-
ingly required for use in GIS and models (Collins and
Bolstad 1996). This usually implies that the quality of
agricultural research depends more and more on
methods to deal with crop and soil variability, weather
generators (computer applications that produce
simulated weather data using climate profiles), and
spatial interpolation-the estimation of the value of
properties at unsampled sites within an area covered
by sampled points, using the data from those points
(Bouman et al. 1996). Especially in developing
countries, there is a need for accurate and inexpen-
sive quantitative approaches to spatial data acquisition
and interpolation (Mallawaarachchi et al. 1996).

Most data for environmental variables (soil properties,
weather) are collected from point sources. The spatial
array of these data may enable a more precise
estimation of the value of properties at unsampled
sites than simple averaging between sampled points.
The value of a property between data points can be
interpolated by fitting a suitable model to account for
the expected variation.

A key issue is the choice of interpolation approach for
a given set of input data (Burrough and McDonnell
1998). This is especially true for areas such as
mountainous regions, where data collection is sparse
and measurements for given variables may differ
significantly even at relatively reduced spatial scales


(Collins and Bolstad 1996). Burrough and McDonnell
(1998) state that when data are abundant most
interpolation techniques give similar results. When
data are sparse, the underlying assumptions about the
variation among sampled points may differ and the
choice of interpolation method and parameters may
become critical.

With the increasing number of applications for
environmental data, there is also a growing concern
about accuracy and precision. Results of spatial
interpolation contain a certain degree of error, and this
error is sometimes measurable. Understanding the
accuracy of spatial interpolation techniques is a first
step toward identifying sources of error and qualifying
results based on sound statistical judgments.


Interpolation Techniques

One of the most simple techniques is interpolation by
drawing boundaries-for example Thiessen (or
Dirichlet) polygons, which are drawn according to the
distribution of the sampled data points, with one
polygon per data point and the data point located in
the center of the polygon (Fig. 1). This technique, also
referred to as the "nearest neighbor" method, predicts
the attributes of unsampled points based on those of
the nearest sampled point and is best for qualitative
(nominal) data, where other interpolation methods are
not applicable. Another example is the use of nearest
available weather station data, in absence of other
local data (Burrough and McDonnell 1998). In contrast
to this discrete method, all other methods embody a


a











Interpolation Techniques


for Climate Variables


Introduction


Geographic information systems (GIS) and modeling
are becoming powerful tools in agricultural research
and natural resource management. Spatially distrib-
uted estimates of environmental variables are increas-
ingly required for use in GIS and models (Collins and
Bolstad 1996). This usually implies that the quality of
agricultural research depends more and more on
methods to deal with crop and soil variability, weather
generators (computer applications that produce
simulated weather data using climate profiles), and
spatial interpolation-the estimation of the value of
properties at unsampled sites within an area covered
by sampled points, using the data from those points
(Bouman et al. 1996). Especially in developing
countries, there is a need for accurate and inexpen-
sive quantitative approaches to spatial data acquisition
and interpolation (Mallawaarachchi et al. 1996).

Most data for environmental variables (soil properties,
weather) are collected from point sources. The spatial
array of these data may enable a more precise
estimation of the value of properties at unsampled
sites than simple averaging between sampled points.
The value of a property between data points can be
interpolated by fitting a suitable model to account for
the expected variation.

A key issue is the choice of interpolation approach for
a given set of input data (Burrough and McDonnell
1998). This is especially true for areas such as
mountainous regions, where data collection is sparse
and measurements for given variables may differ
significantly even at relatively reduced spatial scales


(Collins and Bolstad 1996). Burrough and McDonnell
(1998) state that when data are abundant most
interpolation techniques give similar results. When
data are sparse, the underlying assumptions about the
variation among sampled points may differ and the
choice of interpolation method and parameters may
become critical.

With the increasing number of applications for
environmental data, there is also a growing concern
about accuracy and precision. Results of spatial
interpolation contain a certain degree of error, and this
error is sometimes measurable. Understanding the
accuracy of spatial interpolation techniques is a first
step toward identifying sources of error and qualifying
results based on sound statistical judgments.


Interpolation Techniques

One of the most simple techniques is interpolation by
drawing boundaries-for example Thiessen (or
Dirichlet) polygons, which are drawn according to the
distribution of the sampled data points, with one
polygon per data point and the data point located in
the center of the polygon (Fig. 1). This technique, also
referred to as the "nearest neighbor" method, predicts
the attributes of unsampled points based on those of
the nearest sampled point and is best for qualitative
(nominal) data, where other interpolation methods are
not applicable. Another example is the use of nearest
available weather station data, in absence of other
local data (Burrough and McDonnell 1998). In contrast
to this discrete method, all other methods embody a


a











model of continuous spatial change of data, which can
be described by a smooth, mathematically delineated
surface.


Methods that produce smooth surfaces include various
approaches that may combine regression analyses and
distance-based weighted averages. As explained in
more detail below, a key difference among these
approaches is the criteria used to weight values in
relation to distance. Criteria may include simple
distance relations (e.g., inverse distance methods),
minimization of variance (e.g., kriging and co-kriging),
minimization of curvature, and enforcement of
smoothness criteria (splining). On the basis of how
weights are chosen, methods are deterministicc" or
"stochastic." Stochastic methods use statistical criteria
to determine weight factors. Examples of each include:


* Deterministic techniques: Thiessen polygons, inverse
distance weighted averaging.
* Stochastic techniques: polynomial regression, trend
surface analysis, and (co)kriging.


Thiessen polygons :?? is closest to 141, therefore ??=141 mm.
Inverse distance weighted averaging: The value for ?? is
calculated by weighting the values of all 5 points by the
inverse of their distance squared to point ??. After
interpolation, ?? = 126 mm. The number of neighbors taken
into account is a choice in this interpolation procedure.

Figure 1. An example of interpolation using Thiessen
polygons and inverse distance weighted averaging to
predict precipitation.


Interpolation techniques can be "exact" or "inexact." The
former term is used in the case of an interpolation
method that, for an attribute at a given, unsampled
point, assigns a value identical to a measured value
from a sampled point. All other interpolation methods
are described as "inexact." Statistics for the differences
between measured and predicted values at data points
are often used to assess the performance of inexact
interpolators.


Interpolation methods can also be described as "global"
or "local." Global techniques (e.g. inverse distance
weighted averaging, IDWA) fit a model through the
prediction variable over all points in the study area.
Typically, global techniques do not accommodate local
features well and are most often used for modeling
long-range variations. Local techniques, such as
splining, estimate values for an unsampled point from a
specific number of neighboring points. Consequently,
local anomalies can be accommodated without affecting
the value of interpolation at other points on the surface
(Burrough 1986). Splining, for example, can be
described as deterministic with a local stochastic
component (Burrough and McDonnell 1998; Fig. 1).


For soil data, popular methods include kriging, co-
kriging, and trend surface analysis (McBratney and
Webster 1983; Yates and Warrick 1987; Stein et al.
1988a, 1989a,1989b). In climatology, IDWA, splining,
polynomial regression, trend surface analysis, kriging,
and co-kriging are common approaches (Collins and
Bolstad 1996; Hutchinson and Corbett 1995; Phillips et
al. 1992; Hutchinson 1991; Tabios and Salas 1985). For
temperature interpolations, methods often allow for an
effect of the adiabatic lapse rate (decrease in tempera-
ture with elevation) (e.g. Jones 1996). An overview and
comparison of interpolation techniques, their assump-
tions, and their limitations is presented in Table 1.


In the following section, three interpolation techniques
commonly used in interpolating climate data-IDWA,
splining and (co)kriging-are described in more detail.


Q













Inverse distance weighted averaging-IDWA is a

deterministic estimation method whereby values at

unsampled points are determined by a linear

combination of values at known sampled points.

Weighting of nearby points is strictly a function of

distance-no other criteria are considered. This

approach combines ideas of proximity, such as

Thiessen polygons, with a gradual change of the trend

surface. The assumption is that values closer to the

unsampled location are more representative of the


value to be estimated than values from samples

further away. Weights change according to the linear

distance of the samples from the unsampled point; in

other words, nearby observations have a heavier

weight. The spatial arrangement of the samples does

not affect the weights. This approach has been applied

extensively in the mining industry, because of its ease

of use (Collins and Bolstad 1996). Distance-based

weighting methods have been used to interpolate

climatic data (Legates and Willmont 1990; Stallings et


Table 1. A comparison of interpolation techniques.
Assumptions
Deterministic/ Transitions Exact Computing Output data of interpolation
Method stochastic Local/global (abruptlgradual) interpolator Limitationsof theprocedure Bestfor load structure model
Classification Deterministic Global Abrupt if No Delineation of areas and classes Quick assessments when data are Small Classified Homogeneity
'soft' information used alone may be subjective. Error assessment sparse. polygons within boundaries
limited to within-class standard Removing systematic differences before
derivations, continuous interpolation from data points.
Trend surfaces Essentially Global Gradual No Physical meaning of trend may be Quick assessment and removal of Small Continuous, Phenomeno
deterministic unclear. Outliers and edge effects spatial trends gridded surface logical
(empirical) may distort surface. Error explanation of
assessment limited to goodness of fit trend, normally
distributed data
Regression Essentially Global with Gradual if No Results depend on the fit of the Simple numerical modeling of expensive Small Polygons or Phenomenological
models deterministic local inputs have regression model and the quality and data when better methods are not continuous, gridded explanation of
(empirical refinements gradual detail of the input data surfaces. available or budgets are limited surface regression model
statistical) variation Error assessment possible if input
errors are known.
Thiessen Deterministic Local Abrupt Yes No errors assessment, only one Nominal data from point observations Small Polygons or Best local
polygons data point per polygon. Tessellation gridded surface predictor is
proximall pattern depends on distribution nearest data point
mapping) of data.
Pycnophylatic Deterministic Local Gradual No, but Data inputs are counts or densities Transforming step-wise patterns of Small Gridded surface Continous,
interpolation conserves population counts to continous surfaces moderate or contours smooth variation is
volumes better than ad hoc
areas
Linear Deterministic Local Gradual Yes No error assessments Interpolating from point data when data Small Gridded surface Data densities are
interpolation densities are high, as in converting so large that linear
gridded data from one project to another approximation is
no problem
Moving Deterministic Local Gradual Not with No error assessments. Results Quick interpolation from sparse data on Small Gridded surface Underlying surface
averages regular depend on size of search window regular grid or irregularly spaced samples is smooth
and inverse smoothing and choice of weighting parameter.
distance window, Poor choice of window can give
weighting but can be artifacts when used with high data
forced densities such as digitized contours
Thin plate Deterministic Local Gradual Yes, within Goodness of fit possible, but within Quick interpolation (univariate or Small Gridded surface, Underlying surface
splines with local smoothing the assumption that the fitted multivariate) of digital elevation data and contour lines is smooth
stochasatic limits surface is perfectly smooth. related attributes to create DEMs from everywhere
component moderately detailed data.
Kriging Stochastic Local with Gradual Yes Error assessment depends on When data are sufficient to compute Moderate Gridded surface Interpolated
global varlogram and distribution of data varlograms, kriging provides a good surface is smooth.
variograms. points and size of interpolated blocks, interpolator for sparse data. Binary and Statistical
Local with local Requires care when modeling nominal data can be interpolated with stationarity and
variograms spatial correlation structures. indicator kriging. Soft information can also the intrinsic
when stratified, be incorporated as trends or stratification. hyphotesis.
Local with Multivariate data can be interpolated
global trends with co-kriging.
Conditional Stochastic Local with Irregular No Understanding of underlying Provides an excellent estimate of the Moderate Gridded surfaces Statistical
simulation global stochastic process and models is range of possible values of an attribute at heavy stationarity and
varlograms. necessary. unsampled locations that are necessary the intrinsic
Local with local for Monte Carlo analysis of numerical hypothesis
variograms when models, also for error assessments that do
stratified, not depend on distribution of the data but
Local with global on local values.
trends.

Source: Based on Burrough and McDonnell 1998.


a










al. 1992). The choice of power parameter (exponential
degree) in IDWA can significantly affect the
interpolation results. At higher powers, IDWA
approaches the nearest neighbor interpolation
method, in which the interpolated value simply takes
on the value of the closest sample point. IDWA
interpolators are of the form:


(x)=XI y(x,)

where:
1 = the weights for the individual locations.
y(x,) = the variables evaluated in the observation
locations.


The sum of the weights is equal to 1. Weights are
assigned proportional to the inverse of the distance
between the sampled and prediction point. So the
larger the distance between sampled point and
prediction point, the smaller the weight given to the
value at the sampled point.


Splining-This is a deterministic, locally stochastic
interpolation technique that represents two
dimensional curves on three dimensional surfaces
(Eckstein 1989; Hutchinson and Gessler 1994).
Splining may be thought of as the mathematical
equivalent of fitting a long flexible ruler to a series of
data points. Like its physical counterpart, the
mathematical spline function is constrained at defined
points.


The polynomial functions fitted through the sampled
points are of degree m or less. Aterm rdenotes the
constraints on the spline. Therefore:
* When r= 0, there are no constraints on the function.
* When r= 1, the only constraint is that the function is
continuous.
* When r= m+1, constraints depend on the degree m.


For example, if m = 1 there are two constraints
(r=2):
* The function has to be continuous.


* The first derivative of the function has to be
continuous at each point.


For m = 2, the second derivative must also be
continuous at each point. And so on for m= 3 and more.
Normally a spline with m =1 is called a "linear spline", a
spline with m = 2 is called a "quadratic spline," and a
spline with m = 3 is called a "cubic spline". Rarely, the
term "bicubic" is used for the three-dimensional situation
where surfaces instead of lines need to be interpolated
(Burrough and McDonnell 1998).


Thin plate smoothing splines-Splining can be used for
exact interpolation or for "smoothing." Smoothing splines
attempt to recover a spatially coherent-i.e.,
consistent-signal and remove the noise (Hutchinson
and Gessler 1994). Thin plate smoothing splines,
formerly known as "laplacian smoothing splines," were
developed principally by Wahba and Wendelberger
(1980) and Wahba (1990). Applications in climatology
have been implemented by Hutchinson (1991),
Hutchinson (1995), and Hutchinson and Corbett (1995).
Hutchinson (1991) presents a model for partial thin plate
smoothing splines with two independent spline variables:


P
q, =f(x, y,)+iY p + 3 (i
j=1


l,......,n)


where:


f(x,y):


unknown smooth function


/3- set of unknown parameters

x,y, l/j = independent variables

= independent random errors with
zero mean and variance d 02

d = known weights

The smoothing function f and the parameters 13 are
estimated by minimizing:


[(q,-f,, ( ,y jn /d


G










where:


J (f) = a measure of the smoothness of defined
in terms of mth order derivates of f
= a positive number called the smoothing
parameter


The solution to this partial thin plate spline becomes
an ordinary thin plate spline, when there is no
parametric sub-model (i.e.; when p=0).


The smoothing parameterX is calculated by
minimizing the generalized cross validation function
(GCV). This technique is considered relatively robust,
since the method of minimizing of the GCV directly
addresses the predictive accuracy and is less
dependent on the veracity of the underlying statistical
model (Hutchinson 1995).


Co-kriging and fitting variogram models-Named
after its first practitioner, the south-African mining
engineer Krige (1951), kriging is a stochastic
technique similar to IDWA, in that it uses a linear
combination of weights at known points to estimate
the value at an unknown point. The general formula
for kriging was developed by Matheron (1970). The
most commonly applied form of kriging uses a "semi-
variogram"-a measure of spatial correlation between
pairs of points describing the variance over a distance
or lag h. Weights change according to the spatial
arrangement of the samples. The linear combination
of weights are of the form:

D Y,
where:

y = the variables evaluated in the observation
locations

= the kriging weights


Kriging also provides a measure of the error or
uncertainty of the estimated surface.


The semi-variogram and model fitting-The semi-
variogram is an essential step for determining the
spatial variation in the sampled variable. It provides
useful information for interpolation, sampling density,
determining spatial patterns, and spatial simulation.
The semi-variogram is of the form:


y (h)= E(y(x)- y(x+h))2

where:


y (h) = semi-variogram, dependent on
lag or distance h
(x,x+h) = pair of points with distance
vector h
y(x) = regionalized variable y at point x
y(x)-y(x+h) = difference of the variable at two
points separated by h
E = mathematical expectation


Two assumptions need to be met to apply kriging:
stationarity and isotropy. Stationarity for spatial
correlation (necessary for kriging and co-kriging) is
based on the assumption that the variables are
stationary. When there is stationarity, y (h) does not
depend on x, where x is the point location and h is the
distance between the points. So the semi-variogram
depends only on the distance between the
measurements and not on the location of the
measurements. Unfortunately, there are often
problems of non-stationarity in real-world datasets
(Collins and Bollstad 1996; Burrough 1986). Stein et
al. (1991a) propose several equations to deal with this
issue. In other cases the study area may be stratified
into more homogeneous units before co-kriging
(Goovaerts 1997); e.g., using soil maps (Stein et al.
1988b).


When there is isotropy for spatial correlation, then y
(h) depends only on h. So the semi-variogram
depends only on the magnitude of h and not on its
direction. For example, it is highly likely that the
amount of groundwater increases when approaching a


0











river. In this case there is anisotropy, because the
semi-variogram will depend on the direction of h.
Usually, stationarity is also necessary for the
expectation Ey(x), to ensure that the expectation
doesn't depend on X and is constant.


From the semi-variogram (Fig 2.), various properties
of the data are determined: the sill (A), the range (r),
the nugget (Co), the sill/nugget ratio, and the ratio of
the square sum of deviance to the total sum of
squares (SSD/SST). The nugget is the intercept of the
semi-variogram with the vertical axis. It is the non-
spatial variability of the variable and is determined
when h approaches 0. The nugget effect can be
caused by variability at very short distances for which
no pairs of observations are available, sampling
inaccuracy, or inaccuracy in the instruments used for
measurement. In an ideal case (e.g., where there is no
measurement error), the nugget value is zero. The
range of the semi-variogram is the distance h beyond
which the variance no longer shows spatial
dependence. At h, the sill value is reached.
Observations separated by a distance larger than the
range are spatially independent observations. To
obtain an indication of the part of the semi-variogram
that shows spatial dependence, the sill:nugget ratio
can be determined. If this ratio is close to 1, then most
of the variability is non-spatial.

sill




C1
SI range
-------- ----------- *

Co nugget


distance (h)
Figure 2. An example of a semi-variogram with range,
nugget, and sill.


Normally a "variogram" model is fitted through the
empirical semi-variogram values for the distance
classes or lag classes. The variogram properties-the
sill, range and nugget-can provide insights on which
model will fit best (Cressie 1993; Burrough and
McDonnell 1998). The most common models are the
linear model, the spherical model, the exponential
model, and the Gaussian model (Fig. 3). When the
nugget variance is important but not large and there is
a clear range and sill, a curve known as the spherical
model often fits the variogram well.


Spherical model:


y(h)=Co+A*( h)-_ ()3) for he O,r]


=Co+A for h>r


(where = ris the range, h is lag or distance, and Co+A
is the sill )


If there is a clear nugget and sill but only a gradual
approach to the range, the exponential model is often
preferred.


Exponential model:


y(h)=Co+A*(1-e) for h>0


If the variation is very smooth and the nugget variance
is very small compared to the spatially random
variation, then the variogram can often best be fitted
by a curve having an inflection such as the Gaussian
model:


y(h)=Co+A*(1-e-)') for


h>0


All these models are known as "transitive" variograms,
because the spatial correlation structure varies with
the distance h. Non-transitive variograms have no sill










Exponential variogram


Co nugget Co nugget

distance (h) distance (h)



Gaussian variogram Linear variogram
(with sill)
ilsill


-range C1 '



Co nugget range

distance (h) distance (h)

Figure 3. Examples of most commonly used variogram models a) spherical, b) exponential, c) Gaussian,
and d) linear.


within the sampled area and may be represented by
the linear model:


y(h)=Co+bh

However, linear models with sill also exist and are in
the form of:


(h) =A* for he (o,r]

The ratio of the square sum of deviance (SSD) to the
total sum of squares (SST) indicates which model best
fits the semi-variogram. If the model fits the semi-
variogram well, the SSD/SST ratio is low; otherwise,
SSD/SST will approach 1. To test for anisotropy, the
semi-variogram needs to be determined in a different
direction than h. To ensure isotropy, the semi-
variogram model should be unaffected by the direction
in which h is taken.


Co-kriging-Co-kriging is a form of kriging that uses
additional covariates, usually more intensely sampled
than the prediction variable, to assist in prediction. Co-
kriging is most effective when the covariate is highly
correlated with the prediction variable. To apply co-
kriging one needs to model the relationship between
the prediction variable and a co-variable. This is done
by fitting a model through the cross-variogram.
Estimation of the cross-variogram is carried out
similarly to estimation of the semi-variogram:


y1,2(h)= E ((y (x)-y,(x+h))(y,(x)-y2(x+h)))

High cross-variogram values correspond to a low
covariance between pairs of observations as a
function of the distance h. When interpolating with co-
kriging, the variogram models have to fit the "linear


0


Spherical variogram











model of co-regionalization" as described by Journel
and Huijbregts (1978) and Goulard and Voltz (1992).
(See Annex 1 for a description of the model.) To have
positive definiteness, the semi-variograms and the
cross-variogram have to obey the following
relationship:


Y,,2(h)- yl(h)y2(h)


This relationship should hold for all h.


The actual fitting of a variogram model is an
interactive process that requires considerable
judgment and skill (Burrough and McDonnell 1998).



Reviewing Interpolation Techniques

Early reviews of interpolation techniques (Lam 1983;
Ripley 1981) often provided little information on their
efficacy and did not evaluate them quantitatively.
Recent studies, however, have focused on efficacy
and quantitative criteria, through comparisons using
datasets (Stein et al. 1989a; Stein et al. 1989b;
Hutchinson and Gessler 1994; Laslett 1994; Collins
and Bolstad 1996). Collins and Bolstad (1996)
compared eight spatial interpolators across two
regions for two temperature variables (maximum and
minimum) at three temporal scales. They found that
several variable characteristics (range, variance,
correlation with other variables) can influence the
choice of a spatial interpolation technique. Spatial
scale and relative spatial density and distribution of
sampling stations can also be determinant factors.
MacEachren and Davidson (1987) concluded that data
measurement accuracy, data density, data distribution
and spatial variability have the greatest influence on
the accuracy of interpolation. Burrough and McDonnell
(1998) concluded that most interpolation techniques
give similar results when data are abundant. For
sparse data the underlying assumptions about the
variation among sampled points differ and, therefore,
the choice of interpolation method and parameters
becomes critical.


The most common debate regards the choice of
kriging or co-kriging as opposed to splining (Dubrule
1983; Hutchinson 1989; Hutchinson 1991; Stein and
Corsten 1991; Hutchinson and Gessler 1994; Laslett
1994). Kriging has the disadvantage of high
computational requirements (Burrough and McDonnell
1998). Modeling tools to overcome some of the
problems include those developed by Pannatier
(1996). However, the success of kriging depends upon
the validity of assumptions about the statistical nature
of variation. Several studies conclude that the best
quantitative and accurate results are obtained by
kriging (Dubrule 1983; Burrough and McDonnell 1998;
Stein and Corsten 1991; Laslett 1994). Cristobal-
Acevedo (1993) evaluated thin splines, inverse
distance weighting, and kriging for soil parameters.
His conclusion was that thin splines were the less
exact of the three. Collins and Bolstad (1996) confirm
what has been said before: splining has the
disadvantage of providing no error estimates and of
masking uncertainty. Also, it performs much better
when dense, regularly-spaced data are available; it is
not recommended for irregular spaced data. Martinez
Cob and Faci Gonzalez (1994) compared co-kriging to
kriging for evapotranspiration and rainfall. Predictions
with co-kriging were not as good for evaporation but
better for precipitation. However, prediction error was
less with co-kriging in both cases.


The debate does not end there. For example,
Hutchinson and Gessler (1994) pointed out that most
of the aforementioned comparisons of interpolation
methods did not examine high-order splines and that
data smoothing in splining is achieved in a statistically
rigorous fashion by minimizing the generalized cross
validation (GCV). Thus, thin plate smooth splining
does provide a measure of spatial accuracy (Wahba
and Wendelberger 1980; Hutchinson 1995).


There appears to be no simple answer regarding
choice of an appropriate spatial interpolator. Method
performance depends on the variable under study, the
spatial configuration of the data, and the underlying











assumptions of the methods. Therefore a method is "best"
only for specific situations (Isaaks and Srivastava 1989).



A Case Study for Jalisco, Mexico

The GIS/Modeling Lab of the CIMMYT Natural Resources
Group (NRG) is interfacing GIS and crop simulation
models to address temporal and spatial issues
simultaneously. A GIS is used to store the large volumes
of spatial data that serve as inputs to the crop models.
Interfacing crop models with a GIS requires detailed
spatial climate information. Interpolated climate surfaces
are used to create grid-cell-size climate files for use in
crop modeling. Prior to the creation of climate surfaces,
we evaluated different interpolation techniques-including
inverse distance weighting averaging (IDWA), thin plate
smoothing splines, and co-kriging-for climate variables
for 20,000 km2 roughly covering the state of Jalisco in
northwest Mexico. While splining and co-kriging have
been described as formally similar (Dubrule 1983; Watson
1984), this study aimed to evaluate practical use of
related techniques and software.


Material and methods-Regarding software, the
ArcView spatial analyst (ESRI 1998) was used for inverse
distance weighting interpolation. For thin plate smoothing
splines, the ANUSPLIN 3.2 multi-module package
(Hutchinson 1997) was used. The first module or program
(either SPLINAA or SPLINA1) is used to fit different partial
thin plate smoothing spline functions for more
independent variables. Inputs to the module are a point
data file and a covariate grid. The program yields several
output files:


* A large residual file which is used to check for data
errors.
* An optimization parameter file containing parameters
used to calculate the optimum smoothing parameterss.


* A file containing the coefficients defining the fitted
surfaces that are used to calculate values of the
surfaces by LAPPNT and LAPGRD.
* A file that contains a list of data and fitted values
with Bayesian standard error estimates (useful for
detecting data errors).
* A file that contains an error covariance matrix of
fitted surface coefficients. This is used by ERRPNT
and ERRGRD to calculate standard error estimates
for the fitted surfaces.


The program LAPGRD produces the prediction
variable surface grid. It uses the surface coefficients
file from the SPLINAA program and the co-variable
grid, in this case the DEM. The program ERRGRD
calculates the error grid, which depicts the standard
predictive error.


For co-kriging, the packages SPATANAL and CROSS
(Staritsky and Stein 1993), WLSFIT (Heuvelink 1992),
and GSTAT (Pebesma 1997) were used. The
SPATANAL and CROSS programs were used to
create semi-variograms and cross-variograms
respectively from ASCII input data files. The WLSFIT
program was used to get an initial model fit to the
semi-variogram and cross-variogram. GSTAT was
used to improve the model. GSTAT produces a
prediction surface grid and a prediction variance grid.
A grid of the prediction error can be produced from the
prediction variance grid using the map calculation
procedure in the ArcView Spatial analyst.


Data-The following sources were consulted:


* Digital elevation model (DEM): 1 km2 (USGS 1997).
* Daily precipitation and temperature data from 1940
to 1990, Instituto Mexicano de Tecnologia del Agua
(IMTA; 868 stations/20,000 km2).


1 This depends on the type of variable to be predicted. The SPLINAA program uses year to year monthly variances to weigh sampled
points and is more suitable for precipitation, the SPLINA program uses month to month variance to weigh sampling points and is more
suitable for temperature.


K











* Monthly precipitation data from 1940 to 1996,
Institute Nacional de Investigaciones Forestales y
Agropecuarias (INIFAP; 100 stations/20,000 km2).


In this study, daily precipitation and temperature
(maximum and minimum) data were extracted from
the Extractor Rapido de Informaci6n Climatol6gica
(ERIC, IMTA 1996). We selected a square (1060W; -
101o W; 180 N; 230 N) that covered the state of Jalisco,
northwest Mexico, encompassing approximately
20,000 km2 (Fig. 4). A subset of station data from
1965-1990 was "cleaned up" using the Pascal
program and the following criteria:


* If more then 10 days were missing from a month,
the month was discarded.
* If more then 2 months were missing from a year, the
year was discarded.
* If fewer than 19 or 16 years were available for a
station, the station was discarded.


Data for monthly precipitation from 180 stations were
provided by INIFAP. There were 70 data points with
station numbers identical to some in ERIC (IMTA
1996). The coordinates from these station numbers
were compared and, in a few cases, were different.
INIFAP had verified the locations for Jalisco stations
using a geographic positioning system, so the INIFAP
coordinates were used instead of those from ERIC,
wherever there were differences of more than 10 km
(Table 2). For the other states in the selected area, we
used ERIC data. In four cases stations had identical

Table 2. Stations for which geographic coordinates
were changed to INIFAP values.

Station ERIC ERIC INIFAP INIFAP
NR. Name latitude longitude latitude longitude
14089 La Vega,
Teuchitlan 20.58 -103.75 20.595 -103.844
14073 Ixtlahuacan
del Rio 20.87 -103.33 20.863 -103.241
14043 Ejutla,
Ejutla 19.97 -104.03 19.90 -104.167
14006 Ajojucar,
Teocaltiche 21.42 -102.40 21.568 -102.435


coordinates (Table 3), and the second station was
removed from the dataset that was to be used for
interpolation.


Table 3. Station numbers with identical geographic
coordinates (stations in bold were kept for
interpolation).

Station NR. Latitude Longitude
16164 19.42 102.07
16165 19.42 102.07
16072 19.57 102.58
16073 19.57 102.58
18002 21.05 104.48
18040 21.05 104.48



Daily data were used to calculate the monthly means
per year and consequently the station means using
SAS 6.12 (SAS Institute 1997). The monthly means by
station yielded the following files:


* Monthly precipitation based on 19 years or more for
194 stations.
* Monthly precipitation based on 16 years or more for
316 stations.
* Monthly mean maximum temperature based on 19
years for 140 stations.
* Monthly mean minimum temperature based on 19
years for 175 stations.


Validation sets-To evaluate whether splining or co-
kriging was best for interpolating climate variables for
the selected area, we determined the precision of
prediction of each using test sets. These sets contain
randomly selected data points from the available
observations. They are not used for prediction nor
variogram estimation, so it is possible to compare
predicted points with independent observations. In this
study two test sets were used.


First five smaller, almost equal sub-areas were defined
(Fig. 4). For precipitation, 10 stations were randomly
selected from each. These 50 points were divided into
two sets. Each dataset had 25 validation points and
169 interpolation points. The benefit of working with


































Figure 4. Validation selection areas and two
validation sets of 25 points each for precipitation.


two datasets of 169 points each is that all 194 points
are used for analysis and interpolation, but the
validation stations are still independent of the dataset.


The interpolation techniques were tested as well for
maximum temperature. Because only 140 stations
were available, only 6 validation points were randomly
selected from each square. Therefore, interpolation for
maximum temperature was executed using 125 points
and 15 points were kept independent as a validation
set.


Exploratory data analysis and co-kriging
requirements-An exploratory data analysis was
conducted prior to interpolation to consider the need
for transformation of precipitation data, the
characteristics of the dataset to be used, and the
correlation coefficients between the prediction variable
and the co-variable "elevation." Log transformation is
commonly applied to give precipitation data a more
normal distribution. However, back-transforming the
precipitation values can be problematic because
exponentiation tends to exaggerate any interpolation-
related error (Goovaerts 1997).


The two precipitation datasets were compared to see if
the dataset from 194 stations (19 years or more) had
greater precision than that from 320 stations (16 years
or more). This was done by comparing the nugget
effects of the variograms. As an indication of
measurement accuracy, if the nugget of the large
dataset is larger than the nugget of the small dataset,
then the large dataset is probably less accurate. For
each variogram, the number of lags and the lag
distance were kept at 20 and 0.2 respectively. The
model type fitted through the variogram was also the
same for each dataset. This allowed a relatively
unbiased comparison of the two nugget values,
because the nugget difference is independent of model,
number of lags, and lag distance. Variogram fitting was
done with the WLSFIT program (Heuvelink 1992). The
nugget difference can be calculated as:


(nugget of the 320 station dataset) (nugget of the 194
station dataset).


Thus, the relative nugget difference can be presented
as:


. -nugget320


Results-In the exploratory data analysis, precipitation
data for all months showed an asymmetric distribution.
The difference between the non-transformed surface
and the transformed surface was high only in areas
without stations. In most areas, the difference was
smaller than the prediction error. We therefore decided
not to transform the precipitation data for interpolation.
The temperature data did not show an asymmetric
distribution, so it was not necessary to test
transformation (De Beurs 1998).


The relative nugget difference of the large precipitation
dataset (320 stations, 16 years of data) was compared
to that for the small dataset (194 stations, 19 years of
data). For every month except July and November, the
relative nugget difference was less then 30% (Annex 2)











and, in two cases, the nugget value was smaller for
the small dataset. Because the difference in accuracy
between the two datasets was not large, the small
dataset of monthly means based on more than 19
years was used.


Co-kriging works best when there is a high absolute
correlation between the co-variable and the prediction
variable. In general, during the dry season
precipitation shows a positive correlation with altitude,
whereas during the wet season there is a negative
correlation. The correlation between each variable to
be interpolated (precipitation and maximum
temperature) and the co-variable (elevation) were
determined. For the selected area, April, May, August
and September had acceptable correlation coefficients
between precipitation and elevation (Table 4). May to


Table 4. Correlation coefficients between prediction
variables: precipitation (P), maximum temperature
(Tmax), and the co-variable (elevation).

Month Correlation Correlation
P*Elevation Tmax*Elevation
January -0.26 -0.82
February 0.26 -0.80
March 0.20 -0.71
April 0.68 -0.63
May 0.59 -0.63
June -0.02 -0.74
July -0.36 -0.84
August -0.52 -0.84
September -0.59 -0.84
October -0.39 -0.85
November -0.37 -0.85
December -0.39 -0.84


October had the highest precipitation values. The lack
of a correlation between precipitation and elevation for
June may be because it rains everywhere, making co-
kriging difficult for that month. There is little
precipitation in the other months.


Maximum temperature showed a greater absolute
correlation with elevation, so the interpolation methods
were evaluated for the same months (April, May,
August and September). April and May had the lowest
and August and September the highest correlation
coefficients.


Semi-variogram fitting for the co-kriging technique-
Variograms were made and models fitted to them. For
months with a negative correlation, cross-variogram
values were also negative. To fit a rough model with
the WLSFIT program (Heuvelink 1992), it was
necessary to make the correlation values positive,
because WLSFIT does not accept negative
correlations. This first round of model fitting was used
to obtain an initial impression. The final model was
then fitted using GSTAT (Pebesma 1997). Linear
models of co-regionalization were determined only for
the months April, May, August and September (Table 5
and 6). A linear model of co-regionalization occurs
when the variogram and the cross-variogram are
given the same basic structures and the co-
regionalization matrices are positive semi-definite
(Annex 1). For precipitation the other months had
correlation coefficients that were too low for


Table 5. Variogram and cross-variogram values for the linear model of co-regionalization for precipitation.

Semi-variogram Cross -variogram
Month Variable Model Nugget Sill Range Model Nugget Sill Range
April Precip. Exponential 3.20 46.3 2.10 Exponential 17.8 3720 2.10
Elevation Exponential 5050 565000 2.10
May Precip. Exponential 38.4 256 2.10 Exponential 0 8680 2.10
Elevation Exponential 5050 565000 2.10
August Precip. Gaussian 1110 43400 5.65 Gaussian 2330 -198000 5.65
Elevation Gaussian 62300 2990000 5.65
September Precip. Gaussian 1260 64300 7.00 Gaussian 1560 -365000 7.00
Elevation Gaussian 63500 4400000 7.00











Table 6. Variograms and cross-variograms for the linear model of co-regionalization for maximum temperature.

Semi-variogram Cross-variogram
Month Variable Model Nugget Sill Range Model Nugget Sill Range
April Tmax Exponential 1.40 9.69 0.60 Exponential 22.4 -1310 0.60
Elevation Exponential 360 303000 0.60
May Tmax Exponential 1.10 9.81 0.60 Exponential 20.3 -1280 0.60
Elevation Exponential 380 303000 0.60
August Tmax Spherical 0.926 10.7 1.50 Spherical -11.2 -1640 1.50
Elevation Spherical 2810 309000 1.50
September Tmax Spherical 1.13 10.1 1.50 Spherical -20.2 -1580 1.50
Elevation Spherical 2810 309000 1.50


satisfactory co-kriging. The final ASCII surfaces
interpolated at 30 arc seconds were created with
GSTAT


Surface characteristics and surface validation-
Splining and co-kriging technique results were
truncated to zero to avoid unrealistic, negative
precipitation values. Interpolated monthly precipitation
surfaces are displayed forApril, May, August, and
September in Annex 3. Surfaces were also created
with IDWA, splining, and co-kriging (not shown). The
IDWA surfaces show clear "bubbles" around the actual
station points. Visually, the co-kriging surfaces follow
the IDWA surfaces very well. The splined surfaces are
similar to the DEM surface but appear more precise.


Basic characteristics of the DEM, monthly
precipitation, and temperature surfaces created
through IDWA, co-kriging, and splining are presented
in Annex 4. Maximum elevation as reported in the
stations is 2,361 m. Maximum elevation from the DEM
was 4,019 m-much higher than the elevation of the
highest station. Therefore precipitation and maximum
temperature were estimated at elevations higher than
elevations of the stations. It is not possible to validate
these values because there are no measured values
for such high elevations. However, the extreme values
of the interpolated surfaces can be evaluated. For
precipitation, it is difficult to know whether values at
high elevations were reasonable estimates, because
there is no generic association with elevation as


occurs with temperature. The maximum value of the
splined surfaces was smaller than the maximum
measured value from the station.


Measured precipitation data have a distribution that is
skewed to the right. A frequency distribution of
precipitation after interpolation (Fig. 5) provides
another means of comparing the effects of
interpolation methods. The interpolated surfaces were
clipped to the area of Jalisco to avoid side effects.
Depending on the month, splining and co-kriging
produced contrasting distributions. In May, splining
indicated that 77% or more grid cells had less than
30 mm precipitation, whereas co-kriging allocated
70% of cells to this precipitation range. For
September, co-kriging showed over 28% of the cells
had from 138 to 161 mm precipitation, whereas
splining assigned 24.5% of the cells to this
precipitation class. In both cases co-kriging gave a
wider precipitation range. The frequency of the co-
variable "elevation" within Jalisco is not normally
distributed either (Fig. 6). Considering that there was a
positive correlation between precipitation and altitude
in May and a negative correlation in September,
splining seemed to follow the distribution of elevation
more than co-kriging. However, in the absence of
more extensive validation data, it is not possible to
state that one method was superior to the other based
on resulting frequency distributions.


Usually, temperature decreases 5 to 6C per 1,000-m
increase in elevation, depending on relative humidity
and starting temperature (Monteith and Unsworth


O












Cokriging May



SnHHH~n_


o Lf 0oL 0 Ln 0 Ln 0 Lfl
c:) C:) C^ o r Lf: Lf: C: C)
LN 0 Lf 0 Ln 0 Lnl 0 Ll 0
c:iN C) o Lm L) C
Precipitation (mm)


Cr 0 C D IN L cO
co 0 Co Lfl r C) IN cs rCD
CD rS 0 CM ) CM 1. IN Lfl
CD c 0 C) LfC rM C) IN M
T CDJ MJ M


Precipitation (mm)


25
20-
15-
10-
U-


r : 0o o C c Ll M co
0 C: LM I CM IN r (0
i : r M 0 C^ CD C1 i M
C 0 C Ln) CM I
T- (\ (i \i (\ ciM


Co) Lfl c
M) CO
C)


IPrecipitation (mm)

Figure 5. Frequency distribution of precipitation values after splining and co-kriging for two months.


DEM clipped to Jalisco
35-

30-

25-

S20-

15-
C -
u 10-

5



C C -J (0O C0j CNJ (0 0
0 0 iio i ''


I*- CD CD CD CD CD CD CD CD
co (=j -I*- co Cj Q0

Elevation (masl)

Figure 6. Frequency distribution of elevation, the co-
variable for interpolation in this study, for Jalisco.


1990; see also Linacre and Hobbs 1977). The
difference between the maximum elevation of the
stations and the maximum elevation in the DEM was

1,658 m. Thus, the estimated maximum temperature

should be approximately 11 degrees below values
measured from the stations. This can be seen for the

minimum value of the spline-interpolated values,
which were about 9-10 degrees below measured

values (Annex 4). The range of the co-krige and IDWA
interpolated values was almost the same as that for

the measured data. Therefore, at higher elevations
splining appeared to predict the maximum

temperature better than co-kriging.


The surfaces were validated using the two
independent test sets. For precipitation, the IDWA

appeared to perform better than the other techniques
(Table 7), but the difference was not significant

(statistical analysis not shown). There was little
difference between splining and co-kriging, but we

could apply the latter only for the four months when


Ln 0 Ln f 0n Lf Ln 0 Lf 0 Lf 0
CD CN C- Csj c LD LC CD CD

Precipitation (mm)


Splining September








ol I 1. P n n


LfDl0
0
Lf 0


Cokriging September
"n .r


" 20I
oC
- 10.
,-


A 0Hnnnnnnn


I. 1 1 1 1 1 1 1 1 1


Cv) Lfl
e~n
C-)


Splining May











there was a high correlation with the
co-variable. The predictions for August
and September using the second
interpolation set were less accurate
than those obtained using the first set.


Validation showed that splining
performed better for all months for
maximum temperature (Table 8). There
was no difference between co-kriging
and IDWA predictions (statistical
analysis not shown).


Prediction uncertainty (GCV)-
Prediction uncertainty or'error'
surfaces were produced with the
splining and co-kriging techniques.


Annex 5 shows this for precipitation. The prediction
error from splining was more constant across months.
The co-kriging error surfaces showed greater
variability spatially and between months.



Conclusions for the Study Area

IDWA gave the best results for precipitation, though its
superiority was not significant over results obtained
through the other methods. There was no gain from
using elevation as a co-variable to interpolate
precipitation. Distance to sea was another co-variable
checked. However, the correlation was local and not
always present (De Beurs 1998). Other co-variables
were not readily available.


For maximum temperature there was a higher
correlation with elevation and interpolation improved
when this co-variable was used. Interpolation of
maximum temperature was better handled by splining
than by co-kriging or IDWA.


Table 7. Validation statistics for four monthly precipitation surfaces.

Precipitation Mean absolute difference (mm) Relative difference (%)
April May August September April May August September
Validation 1
IDWA 2.0 5.4 23.9 25.9 31 23 12 17
Splining 2.5 5.5 35.9 31.3 38 23 18 20
Co-kriging 2.0 5.5 33.6 32.5 31 23 17 21
Validation 2
IDWA 1.9 6.6 41.2 41.7 41 31 20 24
Splining 2.2 6.1 55.1 47.3 46 28 26 27
Co-kriging 1.7 6.4 39.1 40.9 37 30 19 23


Table 8. Validation statistics for four maximum temperature surfaces.

Tmax Mean absolute difference (C) Relative difference (%)
April May August September April May August September
IDWA 2.7 2.5 2.0 1.9 8.6 7.6 7.0 6.7
Splining 1.6 1.4 1.2 1.1 5.0 4.5 4.2 3.9
Co-kriging 2.6 2.3 1.8 1.9 8.4 7.3 6.6 6.5


Conclusions and Recommendations
for Further Work

Conclusions of this work apply to this case study only,
but several general recommendations can be made
for future case studies:

* Splining and co-kriging should be preferred over the
IDWAtechnique, because the former provide
prediction uncertainty or "error" surfaces that
describe the spatial quality of the prediction
surfaces. Co-kriging was possible for only four
months for precipitation in the study area, due to the
data prerequisites for this technique. Spline
interpolation was preferred over co-kriging because
it is faster and easier to use, as also noted in other
studies (e.g., Hutchinson and Gessler 1994).

* For all techniques interpolation can be improved by
using more stations.

* For splining and co-kriging, interpolation can be
improved by using more independent co-variables
that are strongly correlated with the prediction
variable.


O











there was a high correlation with the
co-variable. The predictions for August
and September using the second
interpolation set were less accurate
than those obtained using the first set.


Validation showed that splining
performed better for all months for
maximum temperature (Table 8). There
was no difference between co-kriging
and IDWA predictions (statistical
analysis not shown).


Prediction uncertainty (GCV)-
Prediction uncertainty or'error'
surfaces were produced with the
splining and co-kriging techniques.


Annex 5 shows this for precipitation. The prediction
error from splining was more constant across months.
The co-kriging error surfaces showed greater
variability spatially and between months.



Conclusions for the Study Area

IDWA gave the best results for precipitation, though its
superiority was not significant over results obtained
through the other methods. There was no gain from
using elevation as a co-variable to interpolate
precipitation. Distance to sea was another co-variable
checked. However, the correlation was local and not
always present (De Beurs 1998). Other co-variables
were not readily available.


For maximum temperature there was a higher
correlation with elevation and interpolation improved
when this co-variable was used. Interpolation of
maximum temperature was better handled by splining
than by co-kriging or IDWA.


Table 7. Validation statistics for four monthly precipitation surfaces.

Precipitation Mean absolute difference (mm) Relative difference (%)
April May August September April May August September
Validation 1
IDWA 2.0 5.4 23.9 25.9 31 23 12 17
Splining 2.5 5.5 35.9 31.3 38 23 18 20
Co-kriging 2.0 5.5 33.6 32.5 31 23 17 21
Validation 2
IDWA 1.9 6.6 41.2 41.7 41 31 20 24
Splining 2.2 6.1 55.1 47.3 46 28 26 27
Co-kriging 1.7 6.4 39.1 40.9 37 30 19 23


Table 8. Validation statistics for four maximum temperature surfaces.

Tmax Mean absolute difference (C) Relative difference (%)
April May August September April May August September
IDWA 2.7 2.5 2.0 1.9 8.6 7.6 7.0 6.7
Splining 1.6 1.4 1.2 1.1 5.0 4.5 4.2 3.9
Co-kriging 2.6 2.3 1.8 1.9 8.4 7.3 6.6 6.5


Conclusions and Recommendations
for Further Work

Conclusions of this work apply to this case study only,
but several general recommendations can be made
for future case studies:

* Splining and co-kriging should be preferred over the
IDWAtechnique, because the former provide
prediction uncertainty or "error" surfaces that
describe the spatial quality of the prediction
surfaces. Co-kriging was possible for only four
months for precipitation in the study area, due to the
data prerequisites for this technique. Spline
interpolation was preferred over co-kriging because
it is faster and easier to use, as also noted in other
studies (e.g., Hutchinson and Gessler 1994).

* For all techniques interpolation can be improved by
using more stations.

* For splining and co-kriging, interpolation can be
improved by using more independent co-variables
that are strongly correlated with the prediction
variable.


O











* Preferably, all surfaces for one environmental
variable should be produced using only one
technique.
* Interested readers might wish to evaluate kriging
with external drift, where the trend is modeled as a
linear function of smoothly varying secondary
(external) variables, or regression kriging, which
looks very much like co-kriging with more variables.
In regression kriging there is no need to estimate
the cross-variogram of each co-variable
individually-all co-variables are incorporated into
one factor.
Taking into account error prediction, data
assumptions, and computational simplicity, we would
recommend use of thin-plate smoothing splines for
interpolating climate variables.



References

Bouman, B.A.M., H. van Keulen, H.H. van Laar and R.
Rabbinge. 1996. The 'School of de Wit' crop growth
simulation models: A pedigree and historical
overview. Agricultural Systems 52:171-198.
Burrough, P.A. 1986. Principles of Geographical
Information Systems for Land Resource
Assessment. New York: Oxford University Press.
Burrough, P.A., and R.A. McDonnell. 1998. Principles
of Geographical Information Systems. New York:
Oxford University Press.
Collins, F.C., and PV. Bolstad. 1996. Comparison of
spatial interpolation techniques in temperature
estimation. In: Proceedings of the Third International
Conference/Workshop on Integrating GIS and
Environmental Modeling, Santa Fe, New Mexico,
January 21-25, 1996. Santa Barbara, California:
National Center for Geographic Information Analysis
(NCGIA). CD-ROM.
Cressie, N.O.A. 1993. Statistics for spatial data.
Second revised edition. New York: John Wiley and
Sons. 900pp.
Cristobal-Acevedo, D. 1993. Comparaci6n de
m6todos de interpolaci6n en variables hidricas del
suelo. Montecillo, Mexico: Colegio de
Postgraduados, 111p.
De Beurs, K. 1998. Evaluation of spatial interpolation
techniques for climate variables: Case study of
Jalisco, Mexico. MSc thesis. Department of
Statistics and Department of Soil Science and
Geology, Wageningen Agricultural University, The
Netherlands.


Dubrule, O. 1983. Two methods with different
objectives: Splines and Kriging. Mathematical
Geology 15:245-257.
Eckstein. B.A. 1989. Evaluation of spline and weighted
average interpolation algorithms. Computers and
Geoscience 15:79-94.
ESRI-Environmental Systems Research Institute.
1998. ArcView Spatial Analyst Version 1.1. INC.
Redlands, California. CD-ROM.
Goovaerts, P. 1997. Geostatistics for Natural
Resources Evaluation. A.G. Journel (ed.)Applied
Geostatistics Series. New York: Oxford University
Press. 483 pp.
Goulard, M., and M. Voltz. 1992. Linear co-
regionalization model: Tools for estimation and
choice of cross-variogram matrix. Mathematical
Geology 24:269-286.
Heuvelink, G.B.M. 1992. WLSFIT- Weighted least
squares fitting of variograms. Version 3.5.
Geographic Institute, Rijks Universiteit, Utrecht, The
Netherlands.
Hutchinson, M.F. 1989. A new objective method for
spatial interpolation of meteorological variables from
irregular networks applied to the estimation of
monthly mean solar radiation, temperature,
precipitation and windrun. Technical Report 89-5,
Division of Water Resources, the Commonwealth
Scientific and Industrial Research Organization
(CSIRO), Canberra, Australia, pp. 95-104.
Hutchinson, M.F. 1991. The application of thin plate
smoothing splines to content-wide data assimilation.
BMRC Research Report Series. Melbourne,
Australia, Bureau of Meteorology 27:104-113.
Hutchinson, M.F., and PE. Gessler. 1994. Splines-
more than just a smooth interpolator. Geoderma
62:45-67.
Hutchinson, M.F. 1995. Stochastic space-time weather
model from ground-based data. Agricultural and
Forest Meteorology 73:237-264.
Hutchinson, M.F., and J.D. Corbett. 1995. Spatial
interpolation of climate data using thin plate
smoothing splines. In: Coordination and
harmonization of databases and software for
agroclimatic applications. Agroclimatology Working

paper Series, no. 13. Rome: FAO.
Hutchinson, M.F. 1997. ANUSPLIN version 3.2. Center
for resource and environmental studies, the
Australian National University. Canberra. Australia.
CD-ROM.
IMTA. 1996. Extractor Rapido de Informaci6n
Climatol6gica, ERIC, Instituto Mexicano de
Tecnologia delAgua, Morelos Mexico. CD-ROM.
Isaaks, E.H., and R.M. Srivastava. 1989. Applied
Geostatistics. New York: Oxford University Press.
561 pp.
Jones PG. 1996. Interpolated climate surfaces for
Honduras (30 arc second). Version 1.01. CIAT, Cali,
Colombia.











Journel, A.G., and C.J. Huijbregts.1978. Mining
Geostatistics. London: Academic Press. 600 pp.
Krige, D.G. 1951. A statistical approach to some mine
valuations problems at the Witwatersrand. Journal
of the Chemical, Metallurgical and Mining Society of
South Africa 52:119-138.
Lam, N.S. 1983. Spatial interpolation methods: A
review. American Cartography 10:129-149.
Laslett, G.M. 1994. Kriging and splines: An empirical
comparison of their predictive performance in some
applications. Journal of the American Statistical
Association 89:391-400.
Legates, D.R., and C.J. Willmont. 1990. Mean
seasonal and spatial variability in global surface air
temperature. TheoreticalApplication in Climatology
41:11-21.
Linacre, E., and J. Hobbs. 1977. The Australian
Climatic Environment. Brisbane, Australia: John
Wiley and Sons. 354 pp.
MacEachren, A.M., and J.V. Davidson. 1987.
Sampling and isometric mapping of continuous
geographic surfaces. The American Cartographer
14:299-320.
Mallawaarachchi, T, P.A. Walker, M.D. Young, R.E
Smyth, H.S. Lynch, and G. Dudgeon. 1996. GIS-
based integrated modelling systems for natural
resource management. Agricultural Systems
50:169-189.
Martinez Cob, A., and J.M. Faci Gonzalez. 1994.
Analisis geostadistico multivariante, una soluci6n
para la interpolati6n espacial de la
evapotranspiraci6n y la precipitaci6n. Riegos y
Drenaje 2178:15-21.
Matheron, G. 1970 La th6orie des variables
r6gionalis6es et ses applications. Fascicule 5. Les
Cahiers du Centre de Morphologie Math6matique
de Fontainebleau. Paris: Ecole Nationale
Sup6rieure des Mines. 212 pp.
McBratney, A.B., and R. Webster. 1983. Optimal
interpolation and isarithmic mapping of soil
properties, V. Co-regionalization and multiple
sampling strategy. Journal of Soil Science 34:137-
162.
Monteith, J.L., and M.H. Unsworth. 1990. Principles of
Environmental Physics, Second Edition. London:
Edward Arnold.
Pannatier, Y 1996. VARIOWIN: Software for Spatial
Data Analysis in 2-D. New York: Springer Verlag.
Pebesma, E.J. 1997. GSTAT version 2.0. Center for
Geo-ecological Research ICG. Faculty of
Environmental Sciences, University of Amsterdam,
The Netherlands.
Phillips, D.L, J. Dolph, and D. Marks. 1992. A
comparison of geostatistical procedures for spatial
analysis of precipitation in mountainous terrain.
Agricultural and Forest Meterology 58:119-141.
Ripley, B. 1981. Spatial Statistics. New York: John
Wiley & Sons. 252 pp.


SAS Institute. 1997. SAS Proprietary Software
Release 6.12, SAS Institute INC., Cary, NC.
Stallings, C., R.L. Huffman, S. Khorram, and Z. Guo.
1992. Linking Gleams and GIS. ASAE Paper 92-
3613. St. Joseph, Michigan: American Society of
Agricultural Engineers.
Staritsky, I.G., and A. Stein. 1993. SPATANAL,
CROSS and MAPIT. Manual for the geostatistical
programs. Wageningen Agricultural University. The
Netherlands.
Stein, A., W van Dooremolen, J. Bouma, and A.K.
Bregt. 1988a. Cokriging point data on moisture
deficit. Soil Science ofAmerica Journal 52:1418-
1423.
Stein, A., M. Hoogerwerf, and J. Bouma. 1988b. Use
of soil map delineations to improve (co)kriging of
point data on moisture deficit. Geoderma 43:163-
177.
Stein, A., J. Bouma, S.B. Kroonenberg, and S.
Cobben. 1989a. Sequential sampling to measure
the infiltration rate within relatively homogeneous
soil units. Catena 16:91-100.
Stein, A., J. Bouma, M.A. Mulders, and M.H.W.
Weterings. 1989b. Using cokriging in variability
studies to predict physical land qualities of a level
river terrace. Soil Technology 2:385-402.
Stein, A., and L.C.A. Corsten. 1991. Universal kriging
and cokriging as regression procedures. Biometrics
47: 575 -587.
Stein, A., A.C. van Eijnsbergen, and L.G. Barendregt.
1991a. Cokriging Nonstationary Data. Mathematical
Geology 23(5): 703-719.
Stein, A., I.G. Staritsky, J. Bouma, A.C. van
Eijnsbergen, and A.K. Bregt. 1991b. Simulation of
moisture deficits and areal interpolation by universal
cokriging. Water Resources Research 27:1963-
1973.
Tabios, G.Q., and J.D. Salas. 1985. A comparative
analysis of techniques for spatial interpolation of
precipitation. Water Resources Bulletin 21:365-380.
USGS. 1997. GTOPO-30: Global digital elevation
model at 30 arc second scale. USGS EROS Data
Center, Sioux Falls, South Dakota.
Wahba, G. 1990. Spline models for observational
data. CBMS-NSF. Regional Conference Series in
Mathematics, 59. SIAM, Philadelphia.
Wahba, G., and J. Wendelberger. 1980. Some new
mathematical methods for variational objective
analysis using splines and cross-validation. Monthly
Weather Review 108:1122-1145.
Watson, G.S. 1984. Smoothing and Interpolation by
Kriging and with Splines. Mathematical Geology 16:
601-615
Yates, S.R., and A.W. Warrick. 1987. Estimating soil
water content using Cokriging. Soil Science Society
of America Journal 51:23-30.











Annex 1.


Description of Applying the Linear


Model of Co-regionalization2


The linear model of co-regionalization is a model that
ensures that estimates derived from co-kriging have
positive or zero variance. For example there is the
following model:

(y,,(h) y,2(h) bl b0 b0 b0
F(h)= 1 12 t 11 )go(h)+ 11 ()g/h)
y21(h) y22(h) 2b b22 b 2 22

where:
F(h) = the semi-variogram matrix, and
gl(h)= Ith basic variogram model in the linear model of
co-regionalization.

So the basic variogram models are the same for every
variogram, or cross-variogram. In this case, g0(h) is
the nugget model and gl(h) is the sill model.

For a linear model of co-regionalization, all of the co-
regionalization matrices (B1) should be positive
definite. Asymmetric matrix is positive semi-definite if
its determinants and all its principal minor
determinants are non-negative. If Nv = 2, as in the
example or with the precipitation data:


Thus, when fitting the basic structure g,(h) in the linear
model of co-regionalization, these four general rules
should be considered:

bl +0 -b' #0 and b 10
b1 = 0 O- b = 0 Vj
b' may be equal to zero
bl 0 and b 0-- b' =0 or bI 0.

To fit a linear model of co-regionalization:
* Take the smallest set of semi-variogram models
g,(h) that captures the major features of all Nv.
* Estimate the sill and the slope of the semi-
variogram models g,(h) while taking care that the co-
regionalization matrices are positive definite.
* Evaluate the "goodness" of fit of all models. When a
compromise is necessary, then the priority lies in
fitting a model to the variogram of the variable to be
predicted, as opposed to the variogram of the co-
variable or cross-variogram.


b, > 0 and b2
blb2 b',b 0 b:1, b


2 As described by Goulard and Voltz, 1992.











Annex 2.


Dataset Comparison


for Precipitation



Comparison between the dataset with 320 points and the dataset with 194 points.

Variable Model Nugget Sill Range ssd/sst nugget diff.1 rel. nugget diff. (%)2
198 January Exponential 0.0254 0.0724 1.36 0.086 0.0071 21.8
320 January Exponential 0.0325 0.0785 4.06 0.025
198 February Spherical 0.0099 0.0269 1.07 0.155 0.00213 17.8
320 February Spherical 0.0120 0.0183 1.10 0.236
198 March Gaussian 0.0226 0.0678 9.47 0.487 -0.0019 -9.2
320 March Gaussian 0.0207 0.0041 2.65 0.631
198 April Gaussian 0.0175 0.192 9.47 0.073 -0.0014 -8.7
320 April Gaussian 0.0161 0.243 9.47 0.039
198 May Spherical 0.0786 3.41 9.47 0.055 0.0174 -0.3
320 May Spherical 0.0594 0.322 9.47 0.066
198 June Spherical 0.434 5.40 5.33 0.019 0.121 21.8
320 June Spherical 0.555 5.09 6.09 0.014
198 July Spherical 0.993 8.94 6.00 0.022 0.452 36.7
320 July Spherical 1.23 6.47 6.00 0.034
198 August Spherical 0.778 13.7 7.00 0.063 0.292 27.3
320 August Spherical 1.07 10.3 7.00 0.094
198 September Gaussian 1.85 20.5 3.58 0.061 0.02 1.1
320 September Gaussian 1.87 95.3 9.47 0.050
198 October Gaussian 0.475 4.98 5.98 0.017 -0.052 -12.3
320 October Gaussian 0.423 10.2 8.95 0.025
198 November Exponential 0.0129 0.170 1.38 0.038 0.0162 55.7
320 November Exponential 0.0291 0.428 7.00 0.037
198 December Spherical 0.0058 0.179 9.47 0.019 0.0015 20.5
320 December Spherical 0.0073 0.139 9.47 0.075

1. Nugget difference = nugget of the big dataset nugget of the small dataset.

2. Relative nugget difference is: /nugget320-nugget94 100
nugget320 ) *100






Annex 3. Interpolated Monthly
Precipitation Surfaces from
IDWA, Splining, and Co-kriging
for April and August





Cokrige interpolation August


N 37-65
Spline interpolation August 65 92
92 120
120-150
150-175

t 200-230
M. 230-260

M 285-315
31 5-340
340-365
365-395
S395-425


0 200


IDW interpolation August


400 Kibmeters






Cork rige i;nterpolation April


Spline inerpolation April


Precipitation (mm)
0-5


S-- 1 0
10- 15
15- 20
20 c25





i .


Ii


IDW interpolation April


*1 I 1 1." :I_ ri r 1











Annex 4.


Basic Surface


Characteristics



Station values and DEM surface values for elevation Measured values and interpolated surfaces for
maximum temperature (oC).
Elevation Measured (m) DEM (m)
Minimum 27 1 Measured IDWA Spline Co-krige
Maximum 2361 4019 April oC oC oC "C
Mean 1396.56 1455.321 Minimum 25.4 24.7 14.9 27.5
Maximum 40.1 40.9 40.0 38.8
Mean 31.9 32.0 31.0 32.0
Measured values and interpolated surface values for Measured IDWA Spline Co-krige
precipitation. May oC oC oC oC
Minimum 26.9 25.4 15.7 28.0
Measured IDW Spline Co-krige Maximum 41.2 41.2 41.2 40.1
April (mm) (mm) (mm) (mm) Mean 32.9 33.0 31.9 33.2
Minimum 0 0.0 -1.4 -1.5 Measured IDWA Spline Co-krige
Maximum 20.3 24.9 15.6 20.4 August "C oC oC "C
Mean 5.9 5.2 284.7 6.2
Mean 5.9 5.2 284.7 6.2 Minimum 21.3 20.9 12.3 22.6
Measured IDW Spline Co-krige Maximum 35.2 35.5 36.0 34.7
May (mm) (mm) (mm) (mm) Mean 28.3 29.4 28.1 28.8
Minimum 3.8 3.7 -11 1.3 Measured IDWA Spline Co-krige
Maximum 60.8 75.1 57.1 57.1 September oC "C "C C
Mean 27.2 22.6 24.0 24.9
Mean 27.2 22.6 24.0 24.9 Minimum 21.3 21.3 12.1 22.5
Measured IDW Spline Co-krige Maximum 35.5 35.1 35.6 35.4
August (mm) (mm) (mm) (mm) Mean 28.2 29.1 27.8 28.7
Minimum 76.8 47.0 55.7 38.0
Maximum 426.8 495.7 317.3 423.1
Mean 197.6 216.7 175.4 198.3
Measured IDW Spline Co-krige
September (mm) (mm) (mm) (mm)
Minimum 95.8 44.0 28.9 44.5
Maximum 429.6 472.4 280.0 382.8
Mean 166.4 186.7 143.6 164.4






Annex 5. Prediction Error Surfaces
for Precipitation Interpolated by
Splining and Co-kriging









Spline error April
P*

























Spline error August
r

>1 i i


Error (mm)


' *



?-^


4 m
o


Spline error May


7.4 -7.8
7.8 -8.1
8.1 -8.5
8.5-8.9
89-9,3
9.3 9.6
9.5-10
10- 10.4
10.4- 10.7
10.7-11.1
11.1 11.5
11.5-11.7
11,7- 12.2
122-12.6


Error (mm)

7.4 7.8
7.8 .1
8.1 -8.5
Se8,5-8,9
9.3
9.3 9.6
E 95 10
10- 1D.4
10.4- 10.7
^. *107 -111
11.1- 11.5
S 11.5-11.7
e 11.7- 12.2
m 12.2 12.6


Spline error September










I


400 Kilometers


200


A


* 1 A











k-]". e. -n r'F: e 'ic \13


El I I- I "ii' l I


S. .




-.




1 -4



ii 52
: 4


, 7.4

S.4

-4


* I
4-



i .- I





.......


I 'i '


Jlill Ir fri


A~l


II, rl~:1E ~r BLUTO~ae~ i'' F ~li


S -I ; I *- I I ** .?'! ., I


GI










ISSN: 1405-7484


International Maize and Wheat Improvement Center
Centro Internacional de Mejoramiento de Maiz y Trigo
SLisboa 27, Apartado postal 6-641, 06600 M6xico, D.F., M6xico




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs