A model of error propagation from digital elevation models to viewsheds


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A model of error propagation from digital elevation models to viewsheds
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ix, 178 leaves : ill. ; 29 cm.
Ruiz, Marilyn O'Hara, 1959-
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Geography thesis, Ph. D
Dissertations, Academic -- Geography -- UF
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Thesis (Ph. D.)--University of Florida, 1995.
Includes bibliographical references (leaves 167-177).
Statement of Responsibility:
by Marilyn O'Hara Ruiz.
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Two years ago, as I began in earnest to carry out this dissertation, I wondered why

people wrote such long acknowledgments, thanking various people for their help. It is

only after having carried this task to completion that I fully appreciate that this project

was not performed by myself, alone. It has been achieved only through the cooperation

and support of my family and colleagues. The Department of Geography at the

University of Florida has been very supportive. I was also greatly helped by numerous

people at Fort Knox, Kentucky, and at the U.S. Army Construction Engineering Research

Lab (USACERL), where the GRASS GIS was developed. In particular, I must offer

thanks to several people.

First, I thank Ivan, my husband, whose ability to carry on in my absence made

it possible to finish this project. This work is his as much as it is my own. I am also

grateful to Emilio and Nathaniel, our children. Their presence in my life has been an

anchor to reality through the years of graduate school. I thank my parents, Jim and Ruth

O'Hara, who had the courage to load seven children in a station wagon and travel around

the country, instilling in me an appreciation of new places and the joy of discovery.

Lorenzo (Chucho) and Lydia Ruiz also deserve special thanks for caring for my "men"

for many days at a time, as I finished this project.

I thank the members of my graduate committee for their comments and

suggestions. In particular, Grant Thrall has offered encouragement and advice repeatedly

through the years, and Bon DeWitt and John Dunkle provided critical appraisal of my

work, which led to significant improvements.

The data for this study were supplied by USACERL, in Champaign, Illinois. It

was through work with the Lab that I became familiar with the problems of digital

elevation models, and it was there that the original idea for this project was formed. I

thank, in particular, Bill Goran, under whose leadership I first learned about GIS and



ACKNOWLEDGMENTS ................

LIST OF FIGURES ....................

ABSTRACT .........................


I INTRODUCTION ...........

Background ................
Problem Statement ...........
Research Objectives ..........
Presentation of the Research .....


Concepts, Terms and Definitions .
Applications of DEMs .........
Conclusion ................


Photogrammetry and GIS ......
Key Factors in DEM Production ..
DEM Production at the USGS ...


What is accuracy? ...........
Fractals, Scale and DEM Accuracy

Spatial Autocorrelation and DEM Accuracy ............ 47
Ways to Assess DEM Accuracy .................... 49
Error in DEMs from Various Production Methods ........ 60


. .ii

. vi





Summ ary ................................... 66

V DATA AND METHODS ......................... 68

The Study Area ............................... 68
The Geographic Information System ................. 69
The Digital Elevation Models ................... ... 70
Data Processing ............................... 76
Other Variables in the Analysis ................... 79
Path Analysis ................................. 81

AND PATTERN .............................. 84

Visual Assessment ............................. 85
Root Mean Square Error and Differences .............. 93
EVTV Ratios ................................. 96
Spatial Patterns of EVTV Ratios .................... 98
Summary of the Models ........................ 107


Overview of the Analysis ....................... 110
Viewshed Classification Accuracy ......... ........ Ill
From Error in DEMs to Error in Viewsheds ........... 133
Conclusion ................................. 147


Summary of the Findings ....................... 149
Recommendations for Future Research .............. 155


MODELS .............. ......... ............ 158

REFERENCES ........................................ 167

BIOGRAPHICAL SKETCH ............................... 178


Figure page
2-1 DEM Data Structures ................ .. ......... 12

4-1 Error Matrix ...... ..... ..... ........... 58

5-1 Fort Knox, Kentucky ... ...................... 69

5-2 The Study Area ........................... 72

6-1 DEMC Shaded Relief ........................... 86

6-2 DEMT1 Shaded Relief .......................... 87

6-3 DEMT2 Shaded Relief .......................... 89

6-4 DEMT3 Shaded Relief ............................ 90

6-5 DEMC Shaded Relief Subregion .................... 92

6-6 DEMT4 Shaded Relief .......................... 92

6-7 DEMT1 EVTV Ratios .......................... 100

6-8 DEMT3 EVTV Ratios .......................... 101

6-9 DEMT2 EVTV Ratios .......................... 104

6-10 DEMT4 EVTV Ratios .......................... 105

7-1 Subregion I 30 Observation Points
with 500 Meter Buffers .................. 112

7-2 Subregion II 30 Observation Points
with 500 Meter Buffers ................... 112

7-3 Entire Study Area 61 Observation Points
with 500 Meter Buffers ................... 113

7-4 Four Classes in Viewshed Comparisons .............. 114

7-5 Error Propagation Model for DEMTI ............... 143

7-6 Error Propagation Model for DEMT3 ............... 144

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



Marilyn O'Hara Ruiz

May, 1995

Chairman: Professor Grant Ian Thrall
Major Department: Geography

The purpose of this research is to develop a model of error propagation from

digital elevation models (DEMs) to viewsheds. The hypothesis that guides the research

is that the error in the viewsheds is a function of the error in the DEM, the terrain

conditions, and the height of the observation point. Further, it is hypothesized that the

error in the DEM is dependent, in part, on the method of production used to create the

DEM. This is tested by comparing several test DEMs for the same study area with a

control model that serves to represent truth.

The use of the control DEM makes it possible to determine both the magnitude

and the pattern of the error in the test DEMs. Further, it is used to create viewsheds, or

areas of visibility, that are compared with the viewsheds created with the test DEMs. A

comparison of the differing ability of the test DEMs to produce accurate viewsheds is the

major emphasis of this research. It is found that the contour-interpolated viewshed has

the lowest magnitude of error. The DEM created from high altitude aerial photography

using an automated correlation technique performs next best. The DEM from the United

States Geological Survey is the third, and the DEM derived from SPOT (Systeme Pour

l'Observation de la Terre) imagery is fourth most accurate.

It is also found that errors of omission and errors of commission in the viewsheds

are significantly different for the different DEMs. Path diagrams illustrate the proposed

models of error propagation from the DEMs to viewsheds. The error models account for

the sources of DEM error, the degree to which these contribute to errors of omission and

commission in the viewsheds, and the manner in which the type of viewshed error affects

the validity of the viewsheds.



A digital elevation model (DEM) is a digital representation of the shape of the

earth's surface. The DEM in a geographic information system (GIS) plays the role

contour lines and shaded relief renderings play on traditional paper maps. All DEMs

contain error, but the magnitude and spatial distribution of the error vary. This research

focuses on error in several gridded DEMs when the DEMs are produced by different

methods. The analysis investigates the causes of the error and the consequences of the

error when the DEM is used in a GIS to estimate a viewshed.

The test DEMs evaluated here were

1) A DEM derived from 10 meter digital SPOT imagery
2) An experimental high density DEM from high altitude aerial
3) Five United States Geological Survey 30 meter DEMs
4) A contour interpolated DEM produced with GIS software.

A reference model served as the basis for "truth." Accuracy is defined as conformance

to true values, so the reference model provided a proxy for true values. The reference

model, itself, was considered of higher accuracy than the test models, due to the scale of

its source data and manner in which it was produced. The reference was produced with

1:24,000 scale air photos and traditional photogrammetric methods.

From the time that digital elevation models were first introduced in the late 1950s

until the present, tremendous changes have occurred in digital mapping. Geographic

Information Systems (GIS) have become a focal point for many of the developments of

techniques to display, store, transform and analyze digital spatial data. A DEM is a key

component of many GISs intended for natural resource management. In many cases, the

DEM provides the backbone for the analysis. Without a valid DEM, the output of many

a GIS analysis may be put in the garbage heap with the other "garbage in" that becomes

"garbage out."

A DEM within a GIS capable of full utilization of the DEM represents a powerful

tool. The technology required to create this tool has been a fairly recent achievement, and

its full potential has not been exploited. This combination has been realized only after

a multitude of small contributions from a number of disciplines -- notably,

photogrammetry, remote sensing, geography, and computer science. As these inter-related

disciplines have worked on the problem of DEM production, they have, also, increasingly,

become aware of the DEM accuracy problem. Because of the enormous potential for data

misuse that a GIS allows, the context of a GIS is particularly challenging.

Problem Statement

A variety of DEMs from different sources and using different production methods

are available to the GIS analyst. Intermediate-scale mapping at a scale of about 1:24,000

to 1:100,000 is often appropriate for natural resource based GIS projects. Even if the

DEM market is confined to intermediate-scale mapping the number of methods to

produce a DEM can be difficult to understand. This is especially true when the GIS user

lacks a background in the mapping sciences. How can the GIS analyst make an informed

decision about the best DEM for their application? In particular, how can the GIS analyst

discuss the problem with a data vendor or producer in an intelligent manner in order to

learn the information they will need to make that judgement? These questions were the

impetus for this research.

It is difficult for the user of a DEM to assess the nature of the problem of error

(Bolstad and Smith, 1992). This is true, in part, because information about the error in

DEMs has not been documented in a way that is useful to the user. The error calculation

most commonly reported, the root mean square error (RMSE), tells something about the

average error, but reveals nothing about the location and absolute magnitude of the error.

Also, the RMSE varies with the location and number of check points used in its

calculations (Li, 1991). The error in rugged terrain and forested areas is almost always

greater than in other areas, so if check point locations do not represent those areas in

proportion to their occurrence, the RMS error will not be representative of the model

(Torlegard, Ostman, and Lindgren, 1984).

A further problem is simply a lack of information about the error. As Openshaw

(1989) noted, surprisingly little information about error in GIS databases is available to

the GIS analyst, and the tendency is to ignore the problem. Fisher (1992: 1322) wrote

about DEMs that "[h]ow the error is distributed across the area of any one [DEM] is

currently unknown, and factors that may affect the distribution of error are largely

unresearched." The spatial pattern of the error will be most apparent when the entire test

surface is compared to a control of higher accuracy. This approach was followed in this


A better understanding of the nature of the error in DEMs will provide the user

of a DEM information about the potential error in their analyses involving the DEM.

Error in elevation models has a significant effect on the validity of models derived from

the DEMs (Lee, Snyder and Fisher, 1992; Fisher, 1992; Walsh, Lightfoot and Butler,

1987). Many GIS analyses rely on estimates of elevation. They are useful to delineate

watersheds and to determine flood plains. In addition, they are a key component in soils

mapping, in vegetation studies, and in many facilities location analyses. How does the

error in the DEM affect the outcome of such analyses?

This research focused on the DEM-intensive application of viewshed calculation.

Viewsheds, or visibility regions, have been useful to military planners and landscape

architects, in particular, but also have a broader usefulness and are a common tool in the

GIS toolbox. The accuracy of the DEM is a key component in the validity of the

viewshed calculation.

Research Objectives

This research had two major objectives:

1) Assess the location and magnitude of error in DEM test models created
by common production methods compared to a high accuracy control

2) Determine the relationship between the error in the test models and the
error in viewsheds calculated from those models, and develop a conceptual
model to summarize that relationship.

In all cases, the test models were compared to a higher accuracy control model.

In mapping, money buys truth. All else being equal, the more money, the more truth.

The cost of the control model was considerable, and the level of "truth" was

correspondingly high. The relative amount of truth available in the test models was

revealed through the accuracy assessment. The degree to which the lack of truth affects

the validity of the GIS viewshed operation was the final product of this investigation.

Presentation of the Research

Following this introductory chapter, Chapters II, III, and IV provide an overview

of the literature related to the accuracy of DEMs. The terms and definitions used in this

literature have not been consistent, and Chapter II is devoted to tracing the source of the

usage of several terms over the past thirty years. In addition, it establishes the definitions

and concepts applicable to this research.

Chapter III looks specifically at DEM production methods, with a focus on those

represented among the test DEMs for this study. In addition, it examines the relationship

between photogrammetry and GIS, and the role of photogrammetry within the GIS

industry today. Chapter IV then discusses accuracy issues related to the DEMs described


in Chapter III. Chapter IV also provides an overview of results of past empirical studies

related to DEM error from various methods.

Chapters V, VI, and VII report on the results of the empirical research which is

the main object of this dissertation. Chapter V describes the data set used for the tests.

Chapter VI is an overview of the error in the test models. This corresponds to the first

research objective given above. The results of the second research objective are found

in Chapter VII, where the propagation of error in the test models is analyzed in view of

its role in the viewshed calculations. The model of error propagation is presented at the

end of Chapter VII. Chapter VIII is a summary and conclusion of this work, and it

provides suggestions for future research in this area.


Concepts. Terms and Definitions


A digital elevation model (DEM) is a digital representation of the earth's relief for

a specified geographic region. The term "digital terrain model," or DTM, has been used

interchangeably with DEM, and common usage of the terms is not consistent. DTM was

the earlier term. It first appeared in the late 1950s, when transportation planners at the

Massachusetts Institute of Technology defined it as:

a statistical representation of the continuous surface of the ground by a
large number of selected points with known xyz coordinates in an arbitrary
coordinate field (Miller and Laflamme, 1958:435).

DEM often describes the same concept as given above for DTM, though DTM is

more common than DEM. Other terms have been used for similar models including

"digital height model," "digital ground model," "digital surface model" and "digital terrain

elevation data" (DTED). The usage tends to reflect the national origin or place of

education of the person using the term (Li, 1990). For example Ackermann (1980), from

Germany, wrote about digital height models, and in Britain, it is common to use the term

digital ground model (Li, 1990).

Both DEM and DTM are common terms used in the United States. Exact

definitions of these two terms have not been clearly established. Carter (1988), chose a

his/her approach, calling the models DTM/DEM. This is cumbersome, and it sidesteps

the possibility of a meaningful distinction between the two terms. A useful distinction

may be found by an examination of two general meanings of the older term, DTM.

These two meanings provide a way to relate DEMs with DTMs and to define the way

elevation models fit within the context of GIS data bases today.

In 1976, Makarovic described two views of DTMs; one broad and one narrow.

In the narrow, sense, he wrote, "a DTM represents terrain relief data, but with distinct

morphological data included" (1976:57). In the broad sense, "a DTM includes both

planimetric detail, and terrain relief data, in digital form." In addition, he suggested that

a "DTM system" would be capable of handling a DTM at all production stages. These

three concepts continue to have merit, and their use in the context of this thesis follows.

The Term "DEM"

The term DEM will be used to express the narrow view of DTM. This means that

the use of DEM will be consistent with the definition by Miller and Laflamme (above)

for DTM. Helava, in a discussion in 1978, went so far as to reject the broader view,

stating that though he agreed that "there is a great attraction to a wider definition [of

DTMs, t]he origin of our concept was dealing simply with terrain elevations expressed

against their position" (Mikhail et al., 1978:1488). He argued that when the database

included other information than elevation, the system should have some name other than


Burrough expressed his view on the matter, stating that because the word "terrain"

implies other aspects of the landscape beyond elevation, "the term DEM is preferred for

models containing only elevation data" (Burrough, 1987:39). Makarovic' stipulation that

distinct morphological data be included will not be followed strictly. The section in this

chapter on data structures addresses this issue in more detail.

The Term "DTM"

Burrough did not elaborate on what constituted a terrain model. Help with this

issue comes from Collins and Moon (1981), who considered a DEM to be a sub-unit of

a DTM. They stated that a "Digital Terrain Model is an array of numbers that represents

the spatial distribution of a set of properties of the terrain" (1981:71). The DEM

contained only information on elevation and the DTM included a DEM, as well as other

information, such as soil type, land use, or hydrographic features.

When the term DTM is used in this research, it will refer to the more inclusive

view just described. This view reflected the understanding that it was in combining

elevation with other map information that its analytical potential could be realized. Most

of the work presented here deals only with DEMs, so the exact properties that constitute

a true terrain model are left for others to consider.

DTM Systems and Geographic Information Systems

Data alone do not solve problems. They must be in a useful context. In 1976,

Makarovic described a DTM system that focused on the production of DTM. It would

be capable of handling all aspects of DTM creation, but would not necessarily be used for

analysis. This fits in well with the photogrammetric tradition of which he was a part.

Frederiksen, Jacobi and Kubik, from the same tradition, elaborated on this idea, and

defined a DTM [system] as "a program package consisting of routines for data storage,

data retrieval, editing, interpolation and contouring" (1985:101). Analysis was still


More recently, Weibel and Heller (1990b) described a DTM system that would

include the ability to generate, manipulate, interpret, visualize, and apply a DEM. Their

system included the analysis aspect of DEMs. The use of "DEM" in this case reflects the

usage established for this research. The analogous concept used by Weibel and Heller

was a "structured model," which was a second generation product produced by

manipulating the original elevation values measured from images into the particular format

required by analysis software.

The analysis software of Weibel and Heller is very similar to a geographic

information system (GIS). Geographic information systems are computer software and

hardware for acquisition, preprocessing, management, manipulation and analysis, and

output of spatial data (Star and Estes, 1990). Other important components of a GIS are

the personnel who develop and use it, the institutional setting and the geographic data

base, itself. The functions of a GIS are similar to Weibel and Heller's DTM system

functions. For this research, a DTM system is considered as a special type of GIS, which

is especially designed for handling elevation data. All GISs do not have, and should not

be expected to have, the ability to generate DEMs, nor will all GIS data bases include all

aspects of a DTM. However, the GIS that is expected to perform analyses, in which the

DEM plays a critical role, should be a DTM system.

Scientific applications of DEMs have roots in SYMAP, short for the Synagraphic

Mapping System, conceived and developed by Howard Fisher, while on the faculty at

Northwestern University in 1963. Fisher later moved to the Harvard University

Laboratory for Computer Graphics and Spatial Analysis, the Lab to which SYMAP is

usually attributed. One of Fisher's principal goals with SYMAP, was to allow the

cartographic analyst to create a regular grid of Z-values from irregularly spaced points

(Sheehan, 1979). This was an important precedent that led to the current use of DEMs

in GIS analysis.

DEMs and Data Structures

The definition of DEM as the digital representation of elevation expressed against

its geographic location includes many distinctly different data structures. A data structure

defines the way in which the elevation values are stored in the data base. Digital contour

lines, irregular triangulations, regular grids, and various mathematically derived models

all could serve the purpose of representing the earth's surface elevation (Burrough, 1987;

Mark, 1978). Figure 2-1 illustrates these four data structures.

140 142 145 147 150
142 144 147 150 152
139 140 143 147 150
135 136 135 140 145
134 132 130 135 140
135 134 133 134 136
136 137 134 134 134
Grid DEM

Triangulated Irregular Network

Contour Unes

E{fY=200-4x' -3x--3x x

Mathematical Model

Figure 2-1
DEM Data Structures

All data structures are not equally amenable to use as an analytical model of

elevation in a GIS analysis. For GIS applications, the data structures that have been used

the most frequently are the regular grid models and the models created with irregular

triangulations (Burrough, 1987). Digital contour lines and mathematical models have been

used less frequently in GIS analysis for several reasons discussed below. The grid DEM

is the most common form of DEM.

Grid DEMs

In this research, grid DEMs produced by different methods are the only DEMs

under consideration. For this reason, when the term DEM is used without qualification,

a grid DEM is implied. Grid DEMs are organized in rows and columns of height values.

Each equally-spaced value--also called a z-value--represents a square portion of the earth's

surface. Each of these squares is called a grid cell. The z-value associated with a cell

is the estimated height of the cell. This is measured as either the average value for the

cell or the height estimated for the point at the center of the cell. The horizontal

resolution of a grid DEM refers to the length of the sides of each square grid cell. This

type of data structure is often called raster data (Burrough, 1987, provides a good


The format of a grid DEM does not require that explicit information concerning

the relationships among the various cells be stored with the z-values. Those relationships

are implied by the location of the cells within the matrix. This simple data structure is

relatively easy to use in computer programs, adding much to its popularity among

programmers. Generally, spatial analysis and digital map overlays with digital data are

simpler with raster data (Burrough, 1987). In addition, grid data are appropriate for

terrain visualization (Weibel & Heller, 1990a). The most serious shortcoming of grid

DEMs is their inability to store critical features explicitly, such as ridges and breaklines.

A brief discussion of the competing data structures follows.

Digital contour lines

Contour lines are the traditional mainstay of topographic mapping. They are well-

understood by map users, and they are used extensively for topographic interpretation.

Digital contour lines are frequently included in a GIS land base and remain useful

graphically. However, they are cumbersome for computer processing in GIS analysis, and

usually, they are not used as input for a quantitative GIS analysis. A major use of

contour lines in GIS data bases is as a data source from which gridded DEMs and TINs

are created.

Mathematical models

Surface elevation has been represented by local methods, including piecewise

approximations and by global methods, such as Fourier series and multiquadratic

polynomials (Burrough, 1987). The smooth, highly abstract nature of mathematical

models is more appropriate for depicting surfaces abstractly than for depicting elevation

at a particular place. In the opinion of Yoeli, for example "[t]he earth's surface, being

completely irregular, is prima facie, devoid of mathematical characteristics (Yoeli,

1984:141). Although mathematical models are used successfully to simulate hypothetical

terrain types and are used in computer-aided design systems for modelling complex

surfaces (Burrough, 1987), they have been less useful as method to store an easily

accessible DEM in a GIS. Mathematical functions tend to be visually unrealistic when

portrayed graphically, because they are often "endowed with peculiarities which are

unlikely to appear in reality" (Wolf, 1991:202).


After grid DEMs, the second most common DEMs are the triangulated models.

The triangulated models are frequently called triangulated irregular networks, or TINs

(Peuker et al., 1978). TINs were developed as an alternative data structure to DEMs, and

they have several advantages over gridded models. TINs store values more efficiently

than grid DEMs, because the density of points in the TIN varies with the roughness of

the terrain. With a grid, if the horizontal grid spacing is based on the roughest areas,

there will be many redundant points in smooth areas. Another advantage of TINs is that

critical points, such as peaks, pits, passes and breaklines, may be stored at their exact


A disadvantage of TINs is their complexity. The topology, or definitions of the

spatial relationships among the TIN data objects, requires complex computing and

considerable computer memory. Further, the algorithms for TINs tend to be much more

complicated, for both the programmer and user, than those that deal with grids. In

addition, the advantage of accuracy of TINs over grid DEMs, which has long been

assumed, was challenged recently in work by Kumler (1992).

Kumler tested the accuracy of two gridded DEMs and six TINs for 25 different

study areas, representing a variety of terrain types. He used two different interpolants to

create the two gridded DEMs from the same set of digital contour lines. For the TINs,

he used three different methods to select points from digitized contours and three other

methods that involved a selection of a subsets of a grid. In selecting the points, the TINs

were constrained to be about the same size (in terms of computer memory) as the gridded

models. These eight models were produced for 25 different study areas, representing a

variety of terrain types. The error in the competing models was tested using three

different sets of test points and several error statistics. To his surprise, the TINs did not

perform better in terms of accuracy than the grid DEMs that he tested. The accuracy of

the grid DEMs was generally of about the same or better accuracy than the TINs.

Some Final Thoughts on DEM Concepts

To complete this section on the definition of digital elevation models, several

minor issues are addressed. Two terms used in the United States to describe elevation

models have taken on special meaning. Digital Terrain Elevation Data (DTED) and DEM

are both trade names of a sort. DTED is the name of the elevation model developed and

distributed by the United States Defense Mapping Agency (DMA, 1986). DEM is the

term used by the United States Geological Survey for the models that comprise the major

part of a national digital elevation data base (USGS, 1990). USGS DEMs have been so

widely used in the United States that they have become a standard against which other

DEM products are compared. DEMs distributed by the USGS will be called USGS


Goodchild has noted the difference between a model and a data structure. In a

model, he says, every point on the surface is known, as opposed to only discrete elevation

values. The elevation values and their geographic relationships are only data structures,

not models (Goodchild, 1992). For the purpose of this research it is assumed that the

DEMs are a part of a geographic information system. Thus, though the models,

themselves, only include information on elevation, GIS interpolation algorithms make it

possible to estimate an elevation value at any point on the surface using the values stored

explicitly in the DEMs. The GIS can also be used to derive information on slope,

direction of slope (aspect), and visibility regions from the DEM. A DEM, in this context

should be considered a true model.

As a final point, the data storage methods and the GIS programs applied to DEMs

could also be applied to other surfaces. These could include abstract surfaces portraying

land values, pollution, ore distributions or population densities. Though extensions could

be made from this research to these other types of surfaces, the elevation of the earth is

distinctly different from these other surfaces (Carter, 1988). The surface of the earth is

not an abstraction, it is truly continuous and it is a surface about which information can

be measured for virtually any location. These factors along with the legacy of

photogrammetry and the key role that DEMs play in many GIS analyses makes the focus

on DEMs justifiably different from the other surfaces.

Applications of DEMs

In 1978, Helava commented that practical applications of DEMs had not kept up

with theoretical progress. The large volume of data required for DEMs and inefficient

digital storage methods led to high costs and hampered usability. This limitation has been

greatly attenuated as advances in both hardware and software have made it possible to

realize many practical applications of DEMs. In particular, DEMs have become more

widely used with the increasing use of geographic information systems and other

technology that facilitates modelling of the natural environment.

Many early DEMs were data without a mission. Some of the earliest DEMs were

by-products of the production of orthophotographs (Allder, 1983). An orthophotograph

is a vertical photograph from which all distortion, including that introduced by changes

in relief across the extent of the photograph, has been removed. In order to remove relief

distortion, elevation needs to be measured. As digital methods of orthophoto production

became common, the digital elevation values required for the orthophoto were saved

(Case, 1981; Hoffman et al., 1982). These elevation values were among the first

commonly available DEMs. The availability of DEMs gave rise to applications. Only

as applications have become more firmly established and users of DEMs become more

aware of the potential problems with DEMs has DEM production become more structured

and focused on particular outcomes. Most applications have been related to engineering,

scientific investigations, and military operations.

Transportation planners at the Massachusetts Institute of Technology were among

the first to explore the functionality of DEMs (Miller and Laflamme, 1958). The

traditional, nondigital methods to store elevation data, such as topographic maps, terrain

profiles and physical terrain models were useful for human interpretation, they argued; but

a DEM, in digital form, was superior for doing numerical interpretation with the digital

computer. A more recent example of the centrality of DEMs in transportation planning

is found in Christenson (1988).

The military has also used DEMs (Weibel and Heller, 1990b). Topographic

mapping as an aid to strategic planning has been improved by using digital techniques.

Visibility regions determined with the DEM have been of particular interest to military

planners. Another important use of DEMs has been to guide missiles for long distances.

With a DEM on board, the missile can find its path by repeatedly checking for changes

in elevation below, and comparing the changes with what is known about the elevation

at that point as recorded in the DEM.

One group of scientific applications of DEMs has centered on the areas of

geomorphology and hydrology. Examples include Mark (1984), Hutchinson (1989), and

Jenson and Domingue (1988). In these applications, the effects of errors in the elevation

models are particularly apparent. Mark focused on a method of extracting a drainage

network from DEMs, with the goal being the digital drainage network. Hutchinson

focused on improving the DEM by actually including the drainage network in the input

data for the DEM. His purpose was to create a DEM that was geomorphologically based.

Jenson and Domingue described software that "conditioned" the DEM by removing pits

and by more fully describing the topographic structure inherent to, but not explicitly

contained in the DEM. The results of that conditioning could then be used to delineate

watersheds, map drainage networks or for other geomorphic purposes.

Many other applications of DEMs in GIS analyses can be cited. Of particular

interest in this research are applications of DEMs to produce viewsheds, or visibility

regions. Landscape architects have been doing visibility analysis for 30 years, with the

digital applications being a more recent permutation on a long tradition (Palmer and

Felleman, 1991). In one example, a DEM helped landscape architects to determine the

path of a road which was not visible from a visitors' center in Great Basin National Park

(Waggoner, 1989). Goodchild and Lee (1989) considered the necessary parameters to

determine the best location and optimum number of fire towers required to ensure full

coverage of a forested area in rugged terrain. Burrough discussed the need for estimating

the visibility of a tall building in the Netherlands, where surface features, such as trees,

play a more important role than does topographic variation (Burrough, 1987).


In this chapter, I discussed the lack of consistent terminology surrounding DEMs.

After reviewing the use of the terms DTM and DEM, I presented the concepts that will

be used in this research when using those two terms. The move toward DTM systems

underscores the relationship between the history of DEMs and the history of GIS. I also

described various data structures used to store elevation information digitally. Finally, I

stressed the importance of DEMs in GIS analysis by providing examples of DEM's

usefulness in several broad classes of analysis.

Viewsheds are of particular concern in this work, and several examples were also

given of applications of DEMs in determining viewshed areas. The more recent

appreciation of the subtleties of the problems with error in DEMs and its effect on the

validity of results has come about as result of the combination of the applications based

and the production based traditions. It is within this context that the present research



Photogrammetry and GIS

Photogrammetry is firmly behind the production of DEMs. According to the

American Society for Photogrammetry and Remote Sensing (ASPRS), photogrammetry


the art, science, and technology of obtaining reliable information about
physical objects and the environment through processes of recording,
measuring, and interpreting photographic images and patterns of recorded
radiant electromagnetic energy and other phenomena (Wolf 1983:1).

Mapping, especially topographic mapping, and making precise measurements have

been primary functions of photogrammetrists. Assessing the accuracy of the

measurements has been another important function. Analysis has tended to focus on

interpretation of the resultant images. An appreciation of the need to tie data production

with data use has been a more recent development.

Photogrammetry and the closely related field of surveying have been steady, if

indirect, contributors to the development of geographic information systems (Burrough,

1987). One major contribution has been highly accurate mapped information and a

myriad of methods to facilitate capturing and storing digital map data. Digital elevation

models have been an important component of this contribution. In addition,

photogrammetry is related to remote sensing, which has contributed numerous digital

analysis techniques that are useful within a GIS.

Although closely related in theory; in practice, there has been a gap between

photogrammetrists and GIS analysts. Many GIS data bases, especially those intended

primarily for analysis, rather than mapping, have been developed from existing maps,

without the direct aid of a photogrammetric mapping firm. GIS managers frequently have

not been aware of the inaccuracies inherent in this method of developing data bases.

They have tended to lack knowledge of the sophisticated techniques used by

photogrammetrists and of the long pursuit of accuracy in mapping by the photogrammetric


The data producers have also contributed to this gap. Some photogrammetrists

saw the early geographic information systems as outside the realm of their concern. For

example, at the DTM Symposium, sponsored by the American Society for

Photogrammetry in 1978, a discussion panelist mentioned the "exotic" databases "where

all possible data can be collected and organized according to their position on the Earth's

surface" as an area in which "[g]eographers [were] particularly involved" (Mikhail,

1978:1488). It was not, apparently, an area in which that photogrammetrist was involved

at that time.

At the same time, interest in GIS was expanding among photogrammetrists. By

1987, Merchant and Ripple wrote that "[i]t seems clear that, in the future, the interests of

members of the American Society of Photogrammetry and Remote Sensing (ASPRS)

increasingly will be linked, integrated, and applied through GIS" (Merchant & Ripple,

1987:1359). The link between photogrammetry and GIS has been strengthened through

1) the entrepreneurial spirit of photogrammetric firms; 2) the ongoing need for the

development of GIS data bases; and 3) the links between remote sensing and


Photogrammetric firms have been increasingly aware of the financial potential of

involvement in the GIS industry. They have noticed the demand by GIS users for higher

quality digital maps and products such as digital orthophotography and gridded DEMs.

Constantine & Trunkwalter (1991) noted that "in the early stages of GIS implementation

... the photogrammetric community had to lobby hard to make the GIS user community

aware of [their] unique role in this new field [of GIS]" (p. 44). The authors went on to

challenge photogrammetrists to understand the depth of change that complex GIS data

bases required in their approach to mapping "that only the rudimentary procedures are the

same, and the overall data collection strategy for a GIS has become an involved and

challenging, if somewhat elusive, undertaking" (Constantine & Trunkwalter, 1991:45).

The gap between photogrammetry and GIS has been bridged somewhat through

the field of remote sensing. The official journal of the ASPRS is Photogrammetric

Engineering and Remote Sensing. The "Remote Sensing" was added to the name in 1974

to reflect the "belief that the quantitative and qualitative aspects of photogrammetry should

not develop independently, but in close coordination" (Steakley, 1975:38). The qualitative

aspect was mostly interpretation of remotely sensed images. The first conference that

ASPRS sponsored related to GIS was in 1981, when ASPRS co-sponsored the "Pecora

VII Symposium on Remote Sensing: An Input to Geographic Information Systems in the

1980's". The emphasis was remote sensing as data input.

Originally, remote sensing was mostly a means to acquire data. Over time, remote

sensing came to signify not only data collection, but also data analysis (Colwell, 1984).

Further, remote sensing has focused on raster analysis. The need to remove distortion

from images, to filter out noise, and to accurately classify objects has provided the

impetus to develop numerous digital and statistical tools, which today are among the

important historic precedents for statistical analysis of raster data (Cressie, 1993). This

tradition has contributed to the analysis presented in this thesis.

Another contribution of the field of photogrammetry to GIS has been the pursuit

of accuracy and the methods used to obtain and measure it. Since GIS users are all,

potentially, map creators, they can benefit from understanding error in digital maps. The

photogrammetrists have been leaders in this area. The conversion to digital data has

posed an important challenge to the understanding of map accuracy (Slonecker & Tosta,

1992). But photogrammetry has a long tradition of quantitative map accuracy research

from which to draw. As multi-disciplinary work becomes the norm, the

photogrammetrists would be logical partners in the pursuit of accurate DEMs.

Meanwhile the GIS community has been broadly aware of the issue of accuracy

more recently (Goodchild & Gopal, 1989). A recent issue of The Journal of Forestry, for

example, included a feature on errors in GIS (Bolstad & Smith, 1992). This article

included much information that is basic knowledge to photogrammetrists. The lack of

understanding of map error among GIS analysts has hindered research on map accuracy

issues in GIS, especially those issues that affect the user of the data directly. The present

research helps to close the gap between data user and data producer.

Key Factors in DEM Production

This section provides both background on DEM production and justification for

the data base collected for this research. GIS data developers who purchase or produce

DEMs should understand the available production options and the cost parameters

involved in the decisions in DEM production. GIS analysts who use DEMs should

understand the potential problems with the DEMs in their data base. The data base used

for this research was intended to represent common options available to a GIS analyst in

need of an intermediate-scale DEM.

The dynamic nature of the computer industry is reflected in the state of affairs in

DEM production and application. With advances in computer science, the production of

DEMs has evolved toward increased automation since the 1950s. Increasingly, methods

have been developed that are purely digital, and sources of data have become more

diverse. Softcopy photogrammetry, where all data are digital from start to finish, is an

important trend (Clarke, 1990). The January, 1992, issue of PERS was devoted to articles

about "Softcopy Photogrammetric Workstations". The expansion of data sources and

methods of production have both contributed toward the goal of automation (Ayeni, 1982;

Makarovic, 1984; Mikhail, 1978; Tewinkel, 1961).

The development of geographic information systems has served as a focal point

for the integration of systems functionality required to perform all aspects of DEM

creation and use (McCullagh, 1988). Traditionally, terrain modelling has been in the

purview of photogrammetrists, topographic mapping specialists, and land surveyors. As

GIS analysts have required DEMs for an ever-increasing range of applications,

workstation or PC-based systems have brought DEM production to the scientist's desk

(Dowman, 1991; Males & Gates, 1980; McKinney, 1990). Increasingly, GISs provide

functions that facilitate image processing of raster data elements. These functions include

those that allow spatial interpolation from digitized contour lines, the creation of an

irregular triangulated network from an altitude matrix, and visualization of three

dimensional surfaces.

The changes in DEM production over the past thirty years have been profound.

At the same time, certain aspects of DEM production remain important in assessing the

quality of the DEMs. By quality is meant the level of detail, the accuracy, and the

geometric fidelity of DEMs. Production parameters that most affect DEM quality are the

scale and accuracy of the source data, the pattern, density and method of acquisition of

the elevation values acquired from the selected source, and data processing, including

spatial interpolation and choice of data structure to create the final elevation model. Each

of these topics is discussed below.

Data Sources for DEM Production

While photogrammetric stereomodels collected from airborne craft are the most

common source of data when new mapping is performed, other sources are also used.

Ground surveys are one method to collect data directly from the earth. Alternatively,

remote systems may be used to collect elevation data. These include sonar or radar

scanning and altimeters carried on aircraft (Doyle, 1978; Makarovic, 1976). Much of the

equipment and methods used in DEM production have been developed for photographic

sources (Wolf, 1983).

From photographs or maps

Helava (1978) observed that DEMs may be generated either by digitizing existing

map materials or by new mapping directly from photography. New mapping, in his

opinion, was the better choice in order to avoid the pitfalls associated with the unequal

or unknown quality of existing maps. On the other hand, when map quality is relatively

high and well known, the decision to use existing maps may be well justified. The United

States Geological Survey, in its national program of 7.5' DEM creation, has changed from

using photographic stereomodels for DEM productions to digitizing the contours from

existing maps. This has proved to produce more accurate DEMs at a 30 meter resolution

than the photographic sources in use since the 1970s (Allder, 1983; Clarke, Gruen &

Loon, 1982a; Kumler, 1992; USGS, 1990).

Topographic maps at a variety of scales provide a rich source of elevation

information. In order to capture the data and produce a DEM from it, the lines must first

be digitized and then an interpolation procedure applied to fill in information between the

contour lines. Digitizing contours manually is tedious, difficult, and prone to error. The

amount of labor involved commonly makes it prohibitively expensive. Use of digitally

scanned contour data is often the most reasonable choice when creating a DEM from

contour lines (Doyle, 1978). Hydrographic information and spot elevations provide

further detail for inclusion in the DEM prior to interpolation.

From digital images

Since the late 1950s, automated photogrammetric systems have been analogue,

digital or a mixture of the two. All-digital photogrammetric systems have been in use

since 1976 (Makarovic, 1984). Full automation of the techniques using digital data still

presents some difficulties and research has continued to focus on ways to make data

capture and processing more efficient and reliable in a variety of conditions.

A fully digital and automatic system requires digital input. Digital images may

be created in several ways. With a digital camera, images may be captured directly from

the earth in digital form. This technology has advanced to a point where the images that

are captured by digital cameras are metrically useful (Stefanidis, Agouris & Schenk,

1990). Digital images may also be created from film images through automatic digital

scanning of photo image densities. Scanning microdensitometers record and store in

digital form the grey tones on the film (Wolf, 1983). A third form of digital image may

be acquired through a variety of multi-spectral or panchromatic scanners that sense the

emitted and reflected properties of objects and record them in digital form. These

scanners may be loaded on both air- and spacecraft.

Satellite image stereopairs are relatively new sources of data for the production of

DEMs (Konecny et al., 1987; Raggam, Buchroithner, & Mansberger, 1989; Rodriguez et

al., 1988). DEMs produced from satellite image stereopairs are relatively cheap when

large, remote areas require mapping. Stereopairs can be created conveniently from SPOT


The French satellite, SPOT, produces panchromatic images at a 10 meter

resolution. When two images are taken of the same area from appropriate angles and

widely separated positions, it is possible to apply modified photogrammetric techniques

to the images. SPOT imagery is especially appropriate for the collection of imagery for

stereo viewing, because the satellite's sensor can be pointed to the side of the orbital path,

to provide a side-looking view. This makes it possible to view the same area on the

earth's surface from different perspectives. Thus, SPOT imagery can be used to produce

stereopairs for DEM production similar to those based on aerial photo stereopairs.

Mapping Scale and DEM Resolution

The map scale of the DEM data source is important whether the methods and

images are digital or not. A large scale source is required for a highly accurate and

detailed DEM. It can be confusing when comparing sources where "scale" has different

implications. The scale of aerial photography, for example, must be considered in light

of the fact that it may be magnified under zoom optics, thus allowing considerably more

detail to be seen in the photos than was available at the smaller scale. A hard copy map,

on the other hand, is much more static. Enlargement will not change the data content of

the map.

The proper relationship between the horizontal resolution of the DEM and the map

scale of the data source has not been resolved in a systematic manner. The analogous

situation in traditional topographic mapping are the related issues of contour intervals and

map and photo scales for a mapping project. Appropriate mapping scale for topographic

mapping from aerial photography depends on a variety of factors. These include the

instruments used, the type of terrain, and the accuracy requirements.

The resolution of a DEM would involve similar considerations, with the important

addition of data storage. The cost of setting too high a resolution is disk overhead and

memory usage. The cost of using too low a resolution is lack of possibly critical detail

(Aronoff, 1982a; Mitchell, 1981). The major source of available information on the issue

comes from common practice. Here are two examples. The USGS has used 1:80,000

scale photography to produce 30 meter resolution DEMs. They have also used larger

scale photography at 1:40,000 for 30 meter DEMs, they and have digitized contour lines

from 1:24,000 scale maps for the same type of DEM (USGS, 1986, 1990). In mapping

project which resulted in the creation of reference DEM used for this research, 1:24,000

photography was used for mapping at 1:4800 scale. The DEM produced from that project

from the same photography had a 10 meter resolution. The decisions leading up to the

setting the proper relationship between mapping scale and DEM resolution varies with the

quality of the photography and the type of terrain. It is also a somewhat subjective


Spatial Sampling for DEM Production

In addition to the scale and origin of the data source, the decisions about the

spatial properties of the points to use as reference data for DEM production play a key

role in the accuracy of the final DEM. The density and location of elevation values

collected will affect the accuracy and level of detail (map scale) of the final DEM. The

decisions about density and spatial location of reference points should be appropriate for

the type of interpolation and the instruments used to collect the data, as well as the terrain

characteristics. In addition, they should minimize operator fatigue and the cost of

production. Li (1990) has provided an excellent review of sampling strategy and its

importance in DEM construction.

The sampling of terrain information for the creation of a terrain model is a special

case of sampling. Unlike most other phenomena, it is possible to know almost everything

about the population from which the sample is taken. The problem is in the

understanding of the surface from which the sample is taken so that when the reference

points are used to produce a DEM, the resultant DEM is a valid rendition of the earth's

surface. Economy is important. Spatial sampling decisions greatly affect the time

required to create a DEM. The cost of data acquisition is the largest part of the cost of

DEM production (Stefanovic, Radwan & Tempfli, 1977). From a technical stand

point, the primary function of the original elevation points cannot be understated. As

Chapman noted (1987:42-43), "geometric fidelity can only be achieved by acquiring

[elevation data] at appropriate locations. More specifically, this implies that the form,

character and other geometric qualities of the terrain must be captured in order to convey

a 'true' picture of the landscape." Ackerman (1980:2) concluded that "[t]he primary

factor deciding the attainable accuracy of a [DEM] is data acquisition."

Photogrammetric methods

For photogrammetric data acquisition, the pattern of the reference points is either

random, at nodes on a grid, or in the form of contours, profiles, or other morphological

lines and points (Stefanovic, Radwan & Tempfli, 1977). The most suitable pattern is

dictated by the application intended for the DEM, the type of terrain and various

operational questions, such the necessity of speed and the need to minimize operator

discomfort. Generally, there is a yet unresolved debate in the field over whether it is

better to sample carefully selected highly accurate points or to collect enormous numbers

of points of lower accuracy, and discard those which do not meet required standards

(Ackerman, 1980).

Closely related to the pattern of the reference points is the density of the points.

Makarovic (1973, 1976, 1979) has examined automated methods of determining optimal

elevation samples. A major goal of his research has been to achieve fully automated data

acquisition that preserves geometric fidelity. He proposed the use of progressive

sampling, whereby points are sampled in a coarse grid, which is then refined in areas of

greater relief roughness until a specified accuracy is attained. This strategy tends to over-

sample places with abrupt changes in the terrain. It can be improved by first selectively

sampling places where the terrain makes an abrupt change and then using progressive

sampling on the rest of the area.

Ayeni (1982) compared seven sampling schemes based on the well known

classifications of random, stratified and systematic sampling patterns. He used a measure

of terrain roughness to determine the optimal sample size, using a parameter that

incorporated roughness of one area as a compared to the entire area. With all of the

terrain types that he tested, the "unaligned systematic stratified random" pattern proved,

overall, most efficient (see Ayeni, 1982:1694 for a description).

Balce (1987) compared four different computer programs designed to determine

the optimal sampling interval of reference points from photogrammetric sources. He

found that in rough terrain, any of the four programs was suitable. In fact, though the

programs exhibited systematic tendencies to over or under sample in certain conditions,

the differences in the results of the different programs was small enough that any of them

could be defended. The one program that performed slightly better, overall, was based

on the concept of self-similarity. In areas where elevation values are the similar to each

other, less sampling needs to be done to get adequate data to represent the area.

Automated methods

When DEMs are created with automatic correlation techniques from digital

imagery, the elevation reference points are usually collected in "patches" (Panton, 1978;

Hannah, 1981; Vincent, Pleitner & Oshel, 1984). These are small areas of the left and

right stereo images for which the same features may be identified automatically on both

of the two images. Once this correlation has been achieved, the calculation of the

elevation of points within the patch may be made. The patches vary in size depending

on the quality of the image, the features available to make the correlation between images

possible, and the roughness of the terrain.

Sampling schemes are less important when the cost per point of data collection is

low. This is the case with the automatic methods using digital input. In this case, it is

common to calculate pixel by pixel height values and then get rid of those that are

obviously in error. For example, a single pixel that has a z value that is dramatically

different from its neighbors would be suspect. When automatic image correlation

techniques are used, the spot size selected for digital scanning of photographic images

impacts the density of subsequent elevation sampling. A photograph at a scale of

1:68,000 scanned with a 30 micrometer (photo measurement) spot size yields a digital

image with a 2.04 meter (ground measurement) pixel size. Elevation samples taken from

the digital stereomodel will be limited by the density of the pixels in the imagery.

Likewise, a satellite image stereomodel with a pixel size of 10 meters (such as SPOT

panchromatic imagery) will provide elevation detail only to the extent possible given the

detail in the imagery.

Data from existing maps

When the data are acquired from an existing map, they are often in the form of

contour lines supplemented by spot elevations. An alternative to this is to overlay

manually, a regular grid on the topographic map. Then, reference elevations may be

estimated visually from the map at the intersections of the grid lines. The directional

pattern of contour lines can make interpolation more difficult, so the use of a grid is better

in some cases (Wood & Fisher, 1993). On the other hand, the grid method requires a

person's subjective effort, and cannot be automated, as can the digital tracing of existing

contour lines.

Spatial Interpolation

Interpolation is a means to estimate unknown values using the information from

known points. Spatial interpolation is the "estimation of surface elevations in areas with

data values nearby, on more than one side" (Monmonier, 1982:199). The known points

are often called "reference points." Two items are important with regard to spatial

interpolation and DEMs in the present context. These are the role of interpolation and

the choice of method of interpolation.

First, in terms of the quality of DEM, the role of the method of interpolation is

secondary to the role of the density and accuracy of the original elevation values sampled

from the source data (Weibel & Heller, 1990; Yoeli, 1984). "Even the simplest

interpolation method is useful if the density of reference points is sufficiently great"

(Schut, 1976:390). On the other hand, a complex interpolation routine is not a magic cure

for insufficient original data.

Though secondary to data sampling, interpolation still plays a role in DEM quality.

Thus, the second issue is the question of the differences among interpolation methods and

their relative suitability for DEMs. Monmonier noted that "[i]nterpolation is a highly

subjective process, and an estimation procedure is not right or wrong, but merely plausible

or absurd" (Monmonier, 1982:61). Thrall and his colleagues provided an empirical

example of the effect of using different types of interpolation on the same reference points

(Thrall et al. 1993). Methods of interpolation abound, but they are not all equally

suitable for DEM production. Some, such as the areal methods, for example, are geared

toward data stored as polygons, which is often the case of data on socio-economic topics.


They are not useful in the case of topography (Lam, 1983). Yoeli (1984) discussed some

differences between surface models of earth elevation and surface models of other

thematic features.

Over the past several decades, a vast body of literature, from numerous academic

fields, has developed covering the question of spatial interpolation. These have included

the fields of photogrammetry, remote sensing, cartography, geography, geostatistics,

statistics, and computer science. The related topics of triangulation, spatial smoothing and

spatial filtering could be included to expand the possibilities still more. The present

discussion has been narrowed considerably by taking into account only those methods

used for creating a grid DEM from reference data.

Reviews of interpolation methods have focused on several areas, related to, or

dealing directly with, DEMs. Leberl (1975) considered interpolation in the broader

context of photogrammetry, with DEMs being one major component. Schut (1976)

provided the best review of methods designed specifically for DEM production. Though

many individual interpolation methods have been documented since Schut did his work,

his framework (presented below) is still useful to place even the newer methods in

context. Lam (1983) was concerned with cartographic interpolation, and contours were

the output from the methods she described. The grid DEM was an intermediate product

in many of the methods included in her review. Burrough (1987) also provided an

important review in his chapter on interpolation methods. Though set in the context of

GIS in general, rather than DEMs in particular, Burrough's assessment of such factors as

the assumptions, the computing load, and limitations of various interpolation methods is

useful in choosing appropriate methods.

From a practical view, the method of interpolation needs to match the form of the

reference data. Reference data may be randomly distributed in space, they may fall along

contour lines, or they may be in some other regular pattern. Elevation values in a random

pattern could be considered the dominant form of reference data for a DEM. Many point

methods, such as defined by Lam in her review, may be applied to random data points.

Schut also distinguished between those methods that are best used for random points or

for other patterns. A second important pattern is a set of digital contour lines. This is

a form of data of special interest to the GIS analyst who wishes to create a DEM from

paper maps. Schut discusses some methods to do this. Several other authors have written

of issues related to interpolation from digital contour lines (Wood_& Fisher, 1993; Craig

& Adams, 1991; Yoeli, 1984; and Clarke, Gruen & Loon, 1982b).

One difficulty facing the GIS analyst who wants to better understand the role of

interpolation in a particular DEM is lack of information about the available interpolation

programs. Programmers who write interpolation algorithms do not necessarily read the

academic literature, and it is usually up to the user to relate the actual algorithm to the

various frameworks. Programmers also do not always document the procedure used for

the program. Two examples illustrate some of the problems.

Kumler described his attempt to determine the interpolant used in the

LATTICESPOT program (Environmental Systems Research Institute, Redlands,

California). He found that the number of points from which the interpolation was

performed was not clear and that the ESRI technical staff could not provide complete

information. He concluded that what he had been told about the interpolant did not

appear to be true and that his tests showed "evidence of something very mysterious"

(Kumler, 1992:81).

In another example, Fandrich (1995) related her attempts to understand the output

of the kriging interpolation option used in the PC-based SURFER program (Golden

Software, Golden, CO) program. Kriging is a powerful estimation tool developed and

applied mostly in geostatistics (Isaaks & Srivastava, 1989; Journel, 1986; Burrough,

1987). One important characteristic of kriging is its ability to answer the question "what

are the errors (uncertainties) associated with the interpolated values" (Burrough,

1987:155). The SURFER kriging output does not include this error estimation. Without

the error estimation map, the value that kriging offers is seriously lessened.

DEM Production at the USGS

The U.S. Geological Survey National Mapping Division distributes various digital

map products, including DEMs. It is one of the major producers and suppliers of DEMs

for use in GISs in the United States. As such, the USGS has set a standard against which

other products are compared. The USGS distributes several different gridded DEM

products, each comparable in coverage and detail to an existing USGS map series. They

also distribute digital contour lines as Digital Line Graphs (DLGs), which are digitized

from 1:24,000 scale maps.

The largest scale USGS DEM is distributed in coverages and detail similar to the

USGS 7.5' map series (USGS, 1990). Allder (1983) summarized the development of this

product at the USGS. The first DEMs, produced in the early 1970's were related to

orthophoto production. The z-values required to make the orthophotos were saved, in

digital form, and were the basis of the first USGS grid-DEMs. During the 1970s, USGS

developed a standard format for grid-DEMs, which is still in use today.

The USGS 7.5 minute DEMs provide elevation estimates in a grid pattern with a

30 meter horizontal resolution. The primary source of data for the early 7.5' DEMs was

aerial photography from the National High Altitude Photography program (NHAP).

Larger scale aerial photography and DLG contours are used for DEM production


Four methods have been employed in the collection of elevation points for the 30

meter USGS models (USGS, 1986). The USGS has made extensive use of the Gestalt

Photomapper (GPM-2). This instrument allowed automatic correlation of the stereo model

for one patch at time (Allam, 1978). The patch size varied with the terrain, with larger

patches for flatter terrain. Photogrammetric profiling was another early method of USGS

DEM production. Profiles, or regularly spaced rows of elevation points, were collected

from aerial photography using stereoplotters.

Though the vast majority of USGS 7.5' DEMs have been produced by either

manual profiling or with the GPM-2, more recently produced USGS models have been

created using interpolation from digital contour lines. The contour lines have been

digitized photogrammetrically or have been digitized from existing 7.5' maps. The

contour to grid interpolation procedure, CTOG, was developed to interpolate a regular

grid from digitized contours. These later DEMs are of better quality than the early ones,

but are also more costly to produce (Kumler, 1992).

This chapter has outlined the relevant pieces of a vast body of literature related to

DEM production. Photogrammetry has been a key component in the roots of DEM

production, but the trend recently has been to diversify. Methods of data capture,

processing, and analysis as well as sources of data have been developed for and by a

broad community from various disciplines. GIS has been the impetus behind this

expansion. The USGS has been an important player in the development of DEMs for the

GIS marketplace, so a section of this chapter outlined the DEM program at the USGS.


What is Accuracy?

Accuracy implies a conformity to true values. Accuracy of a DEM is expressed

by some measure of the degree to which the values in the DEM reflect true ground

values. When elevation values are not accurate, they contain error. In terms of GIS

analysis, the result of a system query supplies the user with estimated elevation values.

Error is measured as the difference between the values generated by the system and the

true values on the ground (Goodchild, 1990). In the present research, truth is represented

by the reference model. Error is the difference between values in the test models

compared to values in the reference model.

DEM error has been classified is several ways. The USGS classifies DEM error

as systematic, random or a blunder (USGS, 1986; Wolf, 1983). Systematic error occurs

in a pattern related to some aspect of the production process or the natural phenomenon

measured. Random error is unpredictable and generally a small part of overall error.

Blunders are often large and usually obvious, and relatively easy to correct.

The spatial distribution of systematic error, in some cases, can be measured, and

thus, corrected. Often, however, systematic error is very complex, making a global

correction impossible (Hannah, 1981). Random error is normally distributed, so the

positive and negative errors tend to compensate for each other when measured globally.

Large blunders should not pose any great problem with DEMs, because with adequate

post-production editing of the DEM, blunders, as defined above, should not be present in

the final DEM (Ackermann, 1982). Blunders that result in error of a lesser magnitude are

more insidious and are very difficult to detect or correct.

Horizontal and vertical accuracy of DEM data are separate concepts. Horizontal

error results in z values that are misplaced in space, but are accurate relative to each

other. Vertical error implies that an elevation value is correctly located planimetrically,

but is not the correct height. However, in terms of DEMs, "an implicit link exists

between horizontal and vertical accuracy wherein the predominate measure of accuracy

is represented almost totally by tests of the vertical dimension" (USGS, 1986: 2-4).

Virtually all work related to DEM height accuracy has dealt with vertical accuracy (Li,


Accuracy and scale are closely related in cartography. Map scale is a measure of

the relationship between map distance and ground distance. The National Map Accuracy

Standards (NMAS), written in the 1940s, focused on acceptable amounts of error in

various scales of topographic maps. To meet the NMAS, it is necessary, for example

"that not more than 10 percent of the elevations tested shall be in error more than one-half

the contour interval" (U.S. Bureau of the Budget, 1947). The contour interval is closely

associated with the scale of the map. Selection of the contour interval for a project is an

important part of the "art" of map making (Wolf, 1983).

The NMAS have proved to be inadequate when dealing with digital products

(Slonecker and Tosta, 1992). For example, how does a contour interval relate to a

gridded DEM produced from satellite imagery? Intuitively, the detail present in a satellite

image and the potential information content in the image are related to the scale of a map

produced from the image. The conceptual understanding of digital map data has lagged

behind the technology that has radically changed spatial information handling. Some

recent initiatives have made progress in this area, notably, the Spatial Data Transfer

Standards of the U.S. National Institute of Standards and Technology. To date, however,

there is not a straight forward, commonly used formula to express quantitatively the

accuracy standards for digital map products.

For a given area, and assuming proper production standards, larger scale DEMs

will provide more information, and will be more accurate than smaller scale models.

Large scale DEMs are not necessarily the solution to solving the problem of accuracy,

however, since data storage and handling costs are major issues with data-intensive grid

DEMs. Intermediate scale DEMs of the sort tested here are useful when used within their

intended context. The emphasis in this research is not quantity of error, but rather the

result of that error and the usefulness of the DEM for a particular purpose. The American

Congress on Surveying and Mapping's National Committee for Digital Cartographic Data

Standards emphasized that "the foundation of data quality is to communicate information

from the producer to the user so that the user can make an informed judgement on the

fitness of the data for a particular use" (Chrisman, 1984:81).

Fractals. Scale and DEM Accuracy

Mandelbrot's (1983) theory of fractals is an important thread in DEM research

papers. It has a logical place in a discussion of error in DEMs. The fractal dimension

(D) of a topographic surface is between the Euclidean dimensions of 2 and 3. The

dimension increases with increased roughness, and has a linear relationship to roughness

as perceived subjectively (Pentland, 1984). An important characteristic of a fractal surface

is that it is self-similar. A surface (or line) is self similar "if any part of the feature,

appropriately enlarged, is indistinguishable from the feature as a whole" (Goodchild &

Mark, 1987:268).

If the fractal model is appropriate for a topographic surface, then map scale ceases

to be an issue. The characteristics of the original DEM could be applied to a subsection

of the model and it could essentially fill in the pieces required for an enlarged view. This

was the approach of Yokoya and his colleagues (1989). But the random element

introduced to the simulated portion, usually through a fractional Brownian motion (fBm)

model can never replicate terrain. Terrain surfaces have trends which pose a major

problem for the fractal model.

Currently, the theory and methods surrounding the use of the fractal model for

topography are in flux. When Polidori, Chorowicz, and Guillande (1991) assumed the

validity of the fractal model (using the fBm model) of terrain in assessing interpolation

error in a DEM, Goodchild and Tate (1992) objected. They questioned the authors'

conclusions that a difference in the estimate of D at different spatial lags implied evidence

of error in the DEM. "The authors assume that the real surface is characterized by

fBm despite the fact that there is ample evidence in the literature that real topographic

surfaces are not perfectly modeled by fBm and, more seriously, without knowledge of

how much their surface deviates from fBm" (p. 1569).

Michael Goodchild has led geography in questioning the usefulness of fractals to

model real terrain. In addition, he has questioned the existence of self-similarity in many

cases. More than a decade ago, Goodchild (1980) was critical of earlier work by

Richardson (1961) which focused on line length, map scale, and the concept of self-

similarity. "Although Richardson's data clearly support the existence of self-similarity,

in reality the application of the concept to the natural terrain is limited" (Goodchild,


Goodchild established two major problems with fractal theory and DEMs. One

is with the measurement of the fractal dimension. Estimates of D using different

estimation methods on the same DEM, yield different results and are sensitive to the type

of terrain being measured (Klinkenberg & Goodchild, 1992). A second problem is with

the assumption that a stochastic model is appropriate as a model of terrain. Others have

echoed this concern, among them Xia, Clarke and Plews (1991) who found that among

the 200 DEMs that they measured, "very few ... [were] truly self-similar and the fractal

model is only applicable to most data sets within limited scale ranges or in certain

directions" (Xia, Clarke, & Plews, 1991:336).

The fractal model has been useful to create realistic simulated terrain, and it can

be useful to distinguish among certain geomorphic structures, but it should be used with

caution as a model of real terrain. Goodchild and Mark (1987:275) concluded that:

The fractional Brownian process has been used as a convenient way of
generating self-similar surfaces, and certainly such surfaces more closely
resemble some types of real topography than do the results of any other
available method of simulation. We have argued that ... fBm offers a
unique tool to geomorphology as a null hypothesis terrain. Its self-
similarity gives it the appearance of rawness or lack of geomorphic
modification, suggesting further application as an initial form for
simulation of process.

In spite of its currency, the question of the applicability of the fractal model to

topography is sufficiently great to warrant an entire separate line of research to establish

the degree to which a fractal model is appropriate for real terrain. This task is left to

others. This thesis is concerned with the error in a particular set of DEMs, and since a

complete reference set of higher quality is available, there is no need to simulate reference

data. Of some related interest is the question of the difference in the estimate of the

fractal dimension when DEMs are created by different methods. Estimates of D are

sensitive to data and methods of analysis, so it is possible that the subtle patterns of error

in DEMs is responsible for some of the variation measured by D. This is an item of

potential future research.

Spatial Autocorrelation and DEM Accuracy

Spatial autocorrelation is another important topic in DEM research. "Spatial

autocorrelation exists whenever a variable exhibits a regular pattern over space in which

its values at a set of locations depend on values of the same variable at other locations"

(Odland, 1988:7). This concept is often called spatial dependence. The term "correlation"

as used commonly in statistics is most often in reference to the co-relation of two

variables. Autocorrelation, on the other hand, occurs when a single variable is related to

itself when pairs of observations are compared with each other. Griffith concluded that

spatial autocorrelation (SA) is "the relationship among values of some variable that is

attributable to the manner in which the corresponding areal units are ordered or arranged

on a planar surface" (Griffith, 1987:11).

Spatial autocorrelation may be positive or negative. With positive autocorrelation,

like values tend to cluster. With negative autocorrelation, values are found predominantly

near unlike values. Positive autocorrelation is the most common case, and all subsequent

discussion of SA may be assumed to be about positive SA.

Spatial autocorrelation is important to a study of error in DEMs for at least two

reasons. First, is the question of violation of the assumption of independence required of

variables used with inferential statistical methods. If this assumption is not met, then

certain parameters, in particular the standard error estimates used in hypothesis testing,

will be biased (Odland, 1988; Clark & Hosking, 1986; Griffith, 1992). Interpretation of

linear regression involving geographic data should take into account this potential factor

in order to avoid errors in hypothesis testing.

Spatial autoregressions are a common topic in discussions of statistics that address

issues related to spatial data (Ripley, 1981; Clark & Hosking, 1986; Cressie, 1993). As

Odland (1988) explained, the result of spatial dependence in variables used for regression

analysis is that the values of tests of significance of the regression coefficients are

inflated. This could lead the researcher to mistakenly believe that variables are related,

when in fact, they are not (Type I error). The way around this is to include a spatial term

in the model. An example of this approach is found in Griffith (1992).

A second reason that SA is important to a discussion about DEM error is that it

is a way to express quantitatively the degree to which a variable is clustered in space.

Neither the mean nor the standard deviation can express this. When a description of the

pattern of the data is the goal of the research, the SA is not a problem, it is a variable of

interest. Congalton (1988a) provided a good example of the use of SA to describe a

pattern of error in image classification. He used an SA measure to express, quantitatively,

the different patterns of error in image classifications for landcover with different spatial


Ways to Assess DEM Accuracy

Given the complexity of issue of error in DEMs, it is not surprising that numerous

methods have been employed to assess the error and, in turn, to judge the quality of a

DEM. In addition to the (questionable) fractal method just discussed, the methods have

included visual assessment, measurement of root mean square error, and assessment of

measurable derivative products such as a drainage network or viewshed.

The Root Mean Square Error

The most common measure of accuracy of DEMs is the Root Mean Square Error

(RMSE) (Li, 1988). This is a measure of the difference between discrete elevation values

estimated by the DEM and the value of the true elevation at those points (Table 4-1).

The USGS employs this method and reports the RMSE for the DEMs that have been thus

assessed in the header file of DEM digital products (USGS, 1990). The USGS classifies

their DEMs according to the accuracy thus measured.

Table 4-1
The Root Mean Square Error


where Z, = the value of a test value cell
Zc = the value of a control value cell
N = the sample size

Based on the RMSE calculated from at least 28 test points, USGS 7.5 minute

DEMs are divided into three different accuracy levels. The lowest level is level 1, for

which 7 meters is the target RMSE. DEMs produced with high altitude photography

(1:40,000 scale or smaller) and DEMs acquired photogrammetrically with manual

profiling or the GPM-2 are restricted to level 1. This includes almost all DEMs available

currently (USGS, 1990:15). Level 2 DEMs are produced by interpolation from contour

lines digitized from air photos or hardcopy maps. Following the NMAS convention of

linking contour interval with accuracy, level 2 DEMs must have an RMSE within one-half

of the contour interval. So, a DEM interpolated from two meter (6.56 feet) contours

should have an RMSE of one meter or less. Level 3 DEMs should have an RMSE of less

than one-third of the contour interval.

The RMSE estimates of DEM accuracy have three weaknesses. First, in a

statistical sense, the RMSE calculated from a discrete set of check points is not sufficient.

Li (1991) demonstrated that the reliability of the RMSE measured at discrete points was

related to the location, accuracy and number of control (check) points used in calculating

the RMSE. The number of control points required to get a reliable RMSE was related

to the amount of variation of the difference in elevation between the predicted and

expected values. Li recommended that the location of check points should be a random

distribution of points. He also noted the need "to extend [DEM accuracy] analyses to a

complete surface" (Li, 1990:7).

The second weakness is that while the RMSE tells something about the overall

magnitude of error, it does not give information about the error in slope calculations.

Slope, or steepness of the terrain, is frequently more important to GIS analyses than is

absolute elevation. Goodchild noted that "to know the reliability of estimates of slope and

other parameters, we must have knowledge of the spatial dependence of error, particularly

between adjacent measurements in the DEM" (Goodchild & Tate, 1992:1569). The ttdi

weakness of the RMSE when calculated from a limited number of points, is that it is not

able to detect horizontal shifts in the DEMs. The question of horizontal error, which is

very difficult to detect, was not considered explicitly in this research. An assessment was

made, however, of the potential for a regular, and relatively obvious shift in the test

DEMs, compared to the control DEM. This was done by selecting a set of profiles,

extracting the elevation values from each of the DEMs, and plotting the elevation values,

using a spreadsheet graphics option. None was apparent in these tests.

In the current research, these two weaknesses of the RMSE were virtually

eliminated through the availability of a control DEM which provided true elevation values

for the entire study area, not only at discrete points. An additional weakness of the

RMSE became apparent in the course of this research. The difference between the true

value (in the control) and the estimated value (in the test models) is the basis for the

RMSE. For individual cells, this may result in negative numbers, which are unwieldy for

some statistical analysis. Logistic transformations, for example, are not possible on

negative numbers. To overcome this problem, I used the ratio between the control and

the test values calculated for each cell to measure the magnitude of the error in the test


The use of such ratios is documented in social science research. The use of ratios

in the real estate/public finance and urban geography literature is most relevant to this

dissertation. One frequently used application is in the calculation of the ratio of assessed

value (the estimated value) to the market value (true value) of housing. Davies (1978)

provides a good summary of the literature on property tax assessment quality, particularly

ratios of estimated to known, or true, values. Also see Thrall (1979a, 1979b).

The test DEMs are analogous to the assessed value and the control DEM is

analogous to the market value. The estimate may be made for individual pixels, the mean

control-test (EVTV) ratio for subregions can be calculated, or a root mean square error

may be calculated from the ratios. The mean EVTV ratio, expressed as a percentage is

calculated by dividing the ratio by the sample size and multiplying by 100 (Table 4-2).

The multiplier of 100 is used to minimize rounding errors common with computers and

small decimal values. It also makes the resultant calculation more intuitively


Table 4-2
The Estimated Value True Value Ratio

( ZC
MeanEVTV= x100

where Z, = the estimated (DEMT) cell value
Z, = the true (DEMC) cell value
N = the sample size

It was necessary to measure the magnitude of error in the test DEMs using ratios

in a manner similar to the way the way the RMSE was calculated (Table 4-1) for

differences. This was accomplished by calculating the root mean ratio squared error

(RMSRE). Table 4-3 shows the calculation for the root mean square ratio error.

The RMSRE measures the average absolute difference of the test model values from the

control model values. A value of 100 for EVTV indicates that the estimated and true

values were the same. Thus, the difference between the EVTV and 100 is calculated for

each grid cell. The square root of squared average difference of the EVTV from 100 is

the RMSRE.

Table 4-3
The Root Mean Square Ratio Error

t (EVTV-100) 2)

where EVTV = the cell EVTV ratio

Visual Assessment

A visual assessment of shaded relief renditions of DEMs provides valuable insight

into the nature of DEM error. An example of a shaded relief map is provided in Figure

6-1. Acevedo (1991), for example, assessed USGS 30-minute DEMs in this way. An

image of the 30-minute DEM compared with one from a 1-degree DEM of the same area

clearly demonstrated the difference in detail in products of different scales. Artifacts

introduced from the gridding routine were also apparent, appearing as "flattopped ridges

and valleys" (Avecedo, 1991:7) in the 1-degree product. Interpolation in flatter areas

created visible artifacts in the 30-minute DEMs.

Wood and Fisher (1993) took a more structured approach to using visualization

to detect errors in DEMs. Their work was of particular interest in the current context

because they focused on the differences in the pattern of error in DEMs related to the

different interpolation algorithms. They created their DEMs through interpolation from

digital contour lines, using four different interpolation methods. They also noted the

deficiency of knowing only the RMSE in terms of assessing the usefulness of a DEM for

a particular application. The major thrust of their work was to illustrate how viewing the

maps of shaded relief, aspect, the result of an edge detection filter, and profile convexity

all revealed artifacts of the interpolation process that would otherwise be hidden.

Error in Derivative Products

Another way to assess the quality of a DEM is to measure the error in a product

derived from the DEM. Many applications of GIS are quite complex, and the propagation

of error from the data to the final product is equally complex (Heuvelink, Burrough, &

Stein, 1989). Fisher (1992) considered the effect of DEM error on the calculation of

viewsheds and Lee, Snyder and Fisher (1992), its effect on floodplain delineation. Both

of these projects involved using a USGS 7.5-minute DEM, to which random and

systematic error was added in a controlled manner in order to assess its effect on the

derived product. The original DEM was used as the control. In both cases, the addition

of error within a threshold of 7 meters was found to have a significant effect on the

validity of the products derived from the DEMs. The magnitude and pattern of the error

should be considered when evaluating the result of a GIS procedure using DEMs.

Fisher's (1992) work on viewshed uncertainty was especially important as a

precedent for the current research. The major thrust of his work was to test the null

hypothesis that "the viewshed area in the original DEM is not a member of the set of

viewshed areas in elevation models generated by [a perturbation algorithm]" (Fisher,

1992:1323). The original DEM represented truth, or reference data. His basis for

inclusion in the original viewshed was comparing the number of cells in the viewsheds

from 19 realizations of the perturbations with the number determined by area, or number

of cells, of the viewsheds.

This method of determining inclusion in a set of viewsheds is expedient, but

imperfect. The method used in this dissertation allows a full accounting of type of error

(see next session for a discussion of this) and thus leads to a better understanding of the

error in the viewsheds.

A second important aspect of Fisher's work was his preliminary analysis of

determinants of viewshed error. He used correlation analysis and stepwise regression to

assess the ability of the elevation of the view point and the minimum and maximum

elevations within 1000m (a measure of terrain roughness) of the view point to predict

viewshed area. The elevation of the view point was the only variable with predictive

power. He also observed that more spatially autocorrelated error seemed to produce a

better rendition of the viewshed.

Error in Cell Classifications

The process of producing a viewshed from a DEM is, in essence, a matter of

classifying cells as visible or not visible. Raster image classification is an important

component among space remote sensing techniques. Satellite images need to be classified

in order to be useful in quantitative analysis. A cohesive set of literature has accrued over

the past 15 years concerned with the error in image classifications (Aronoff, 1982a,

1982b, 1985; Congalton, 1988a, 1988b, 1991; Rosenfield & Fitzpatrick-Lins, 1986; Story

& Congalton, 1986). One particular thread of that research is related to the use of error

matrices, or contingency tables, which have now become a standard part of remote

sensing (Lillesand & Kiefer, 1987). The understanding of error gained from the

experience in image classification is applicable to better understanding the accuracy of

viewsheds derived from grid DEMs.

The recent review by Congalton (1991) provides an overview of the problems

associated with accuracy assessment of remotely sensed data. The error matrix is a key

to the concepts described in the review (Figure 4-1). In an error matrix, the reference

data occupies the horizontal axis and the test data, the vertical axis. The diagonal cells

indicate the number of cells properly classified. The cells off the diagonals indicate the

number of cells misclassified and also provide the user with information on how the

misclassification came about. In the case of viewsheds, for example, a visible area may

have been classified as not visible or vice versa.

The overall accuracy of the classification is the sum of the properly classified cells

divided by the total number of cells (Figure 4-1). The value is usually expressed as the

percent correct. Additional information can be found by using the row sums and the

columns sums and determining the percent correctly classified as a particular group, e.g.

the percent correctly classified as visible. The classic article by Story and Congalton

(1986) illustrated that the diagonal divided by the row sum measures errors of

commission, while the diagonal divided by the column sum measures errors of omission.

Visible Not Visible
Total Accuracy
(29 + 184)/300 = .71
29 16 45
Errors of Omission
S29/100 = .29
184/200 = .92
SErrors of Commission
S71 184 255 29/45 = .64
184/255 = .72

100 200 300

Values are in numbers of cells

Figure 4-1
Error Matrix

An analogy illustrates the significance of omission and commission errors in terms

of viewshed errors. In a court of law, the witness is asked to tell the:

1) the truth -- overall accuracy
2) the whole truth -- no errors of omission
3) and nothing but the truth -- no errors of commission

Overall accuracy does not distinguish among the sorts of error that might occur. Using

the example from Figure 4-1, the overall accuracy is 74%. But not only do we want to

know if an answer is "true," we also want to know that we have the whole truth. How

much of the visible area was omitted from the viewshed? In some cases, cells that should

have been included in the visible area were classified as not visible, and some not visible

were classified as visible. These would be called errors of omission. The column totals

and the diagonals allow us to calculate that errors of omission resulted in only 29% of the

truly visible region as being properly classified. At the same time, 92% of the truly not

visible region was classified correctly.

Now, consider errors of commission. We also want to know that the truth is not

mixed up with some "untruth." With the viewsheds in Figure 4-1, some of the area that

was classified as visible, was actually not visible. Using calculations from the row totals

and the diagonals, 47% of the area identified as visible, was truly visible. Of the area

identified as not visible, 87% percent was truly not visible.

The distinction between errors of omission and commission is especially important

in terms of evaluating DEMs according to their suitability for a particular application.

For example, the Forest Service might be concerned with protecting scenic views from

logging activity. From the point of view of the Forest Service, the viewshed calculation

should be relatively free from errors of omission in terms of the visible area, i.e. all the

visible area would be included within the viewshed. If some of the area classified as

visible was actually not visible (an error of commission), that would present a problem

only insofar as the logging company might be concerned, since some trees that could have

been safely logged were excluded.

Error in DEMs from Various Production Methods

Error in Photogrammetrically Sampled DEMs

The instruments used for stereoscopic plotting are one important limiting factor

in DEM accuracy. Other factors are well known and usually expressed in terms of

contour accuracy. "Contour accuracy depends .. also upon the nature of the terrain, the

camera and its calibration, the quality of the photography, the density and quality of

ground control, and the capability of the plotter operator" (Wolf, 1983:418).

Error in DEMs created photogrammetrically from aerial photographs has been

defined the most comprehensively and by numerous sources (e.g., Wolf, 1983;

Hakkarainen, Kilpela & Savolainen, 1982; Doyle, 1978). Torlegard, Ostman and

Lindgren (1984) reported for the International Society of Photogrammetry and Remote

Sensing (ISPRS) on a test that revealed the pattern of error in large-scale DEMs created


In the ISPRS test, the researchers requested DEMs from a variety of

photogrammetric mapping firms. The participants were provided the air photos for six

test areas and asked to create DEMs from them. In addition, the participants recorded

information on the production process and were asked to assess the accuracy of the

DEMs. DEMs created from larger scale photography was used as reference data to assess

the test DEMs. The project leaders came to several conclusions. The experience of the

operator played an important role in DEM quality. Almost all of the firms over-estimated

the accuracy of their products, especially for a particularly rugged study area. Forested

areas were not handled consistently, and were another area of generally poor quality.

The trend toward automation, as discussed above, has led to methods of extracting

elevation information from digital images without the need for an operator to make the

stereoscopic measurements. Automated methods are fast, but they create special problems

with accuracy of DEMs. Problems associated with collecting data from areas of rugged

terrain, dense vegetation and featureless areas are common to all DEM production

methods. These problems have been especially difficult to overcome with automated


The major problems with automated methods have been known for decades. An

early edition of the Manual of Photogrammetry (Amer. Society of Photogrammetry, 1966)

noted that the performance of the automated systems were not equally predictable in a

variety of terrains. This was because "[t]he machine does not recognize forms, search for

shapes or memorize objects. Thus, when the automatic instrument is confronted with a

situation where the two corresponding images differ considerably in appearance it can no

longer function properly" (Amer. Society of Photogrammetry, 1966: 764). Steep terrain

was one such instance. Further, the instruments tended to "get lost" when the correlation

between the two images was not possible at some point. Recovery required an operator.

Another problem was water (Carter, 1987). Water bodies reflect the sun and make the

mechanical correlation impossible.

The Gestalt Photomapper (GPM-2) was one of the first DEM production systems

that allowed automation of the production of DEMs (Kelly, McConnell & Mildenberger,

1977; Allam, 1978). It was used extensively by the USGS. The system was based on

the principle that like-looking areas on two stereopair images could be identified by

finding areas of image correlation. When the best correlation between the two images

was found for a particular point, the elevation for that point was computed based on the

parallax between the two images. An operator needed to be on hand to get the GPM-2

back on track when it became "lost" in the terrain. The GPM-2 was a significant step

forward in the automation of DEM production, but the early models often had obvious

patterns associated with the patches used for the correlations (Carter, 1987; O'Callaghan

& Mark, 1984).

Another problem associated with automated techniques is the production of false

elevation values from the presence of trees, buildings, an other protrusions on the earth's

surface. Automated techniques cannot detect whether an elevation value sits on the

ground or on a tree top. A clearing in the forest would appear as a pit in the DEM. For

some applications, such as visual terrain simulations, this effect may be desirable. In

other cases, such as when the DEM is used for extraction of drainage networks, these

errors need to be recognized and eliminated from the DEM during editing to produce

more accurate DEMs (O'Callaghan & Mark, 1984; Hutchinson, 1989).

A fairly simple global digital filtering can eliminate false pits and systematic

patterns in some instances (Allam, 1982; O'Callaghan and Mark, 1984, Stitt, 1990). More

complex local filtering can correct some of the error in DEMs produced with automated

correlation techniques. The filtering will also result in smoothing of the data, however,

and this should be avoided to preserve topographic detail (Hannah, 1981). The effect of

tree cover could also be reduced during DEM production through identification of forested

areas and then subsequent subtraction of tree height from the estimated elevation.

Vincent (1987) described three problems and his solutions associated with

automatic correlation. Featureless areas, such as bodies of water or ice, leave nothing for

the correlators to work with. He has used nearest-neighbor interpolation to fill in

featureless areas. Steep areas are distorted because of parallax effects. This problem can

be reduced by doing correlation searches over larger areas at one time. When search

areas are too large, however, the automatic correlation systems tend to get lost. Vincent

used a digital filter to remove unrealistic parallaxes, eliminating the need for an operator.

Error in SPOT Image Source DEMs

Several studies have focused on SPOT derived DEMs. The methods used to create

the DEMs varied somewhat. In many cases, digital imagery was used to collect the

elevation data. In earlier examples, a film product produced from a digital image

provided the input data. Both manual and automated methods have been employed in the

data collection process. Notably, the reported error and estimated map scale of the SPOT

derived DEMs varied considerably among the studies.

Konecny et al. (1987) used SPOT image film products with photogrammetric

instruments to develop a rigorous method to produce a DEM given the geometry of the

SPOT image. Unlike aerial photography, for which all parts of an image are recorded

simultaneously, SPOT imagery is collected line by line as the satellite orbits. The fact

that the satellite and the earth are moving as the image is being collected makes it

necessary to compensate for these additional variables. These variables are systematic,

and they can be rectified with relatively little difficulty. Konecny et al. (1987) were able

to create a DEM from this imagery with an elevation RMS error of 7.5 meters. They

concluded that the SPOT imagery was sufficient to create topographic information at a

scale of 1:25,000.

Rodriguez et al. (1988) examined the accuracy of SPOT DEMs using different

image geometries and spectral modes. Like Konecny et al. (1987), they created DEMs

from photographic products using stereo plotters. They found that elevation errors

averaged 7 meters. The effects of different base/height ratios on the accuracy of the

DEMs were examined. The base is the distance between the satellite locations at the time

when the images were obtained. This varies, while the height remains constant at about

832 km. A ratio of about 0.6 is recommended for stereoscopic SPOT images (SPOT

image order form). Rodriguez et al. calculated a vertical accuracy of 3.5 meters for a B/H

ratio of 1, and an accuracy of 7 meters was achieved for a B/H ratio of 0.5. The

orientation of the imagery has a strong impact on the accuracy of the DEM. Gugan

and Dowman (1988) discussed the problem of image quality. Cloud cover on SPOT

images affects the degree to which elevation values may be sampled. For some places,

"cloud-free" conditions are impossible. They also noted that the process used to create

the film product from the SPOT digital image impacted image quality. They examined

the film products produced from digital imagery from several suppliers. The film

products varied greatly in the degree to which they accurately replicated the original

image. They also found that the experience of the operator played an important role in

DEM quality. In particular, the operator needed experience with interpreting very small

scale images. Gugan and Dowman were rather more critical of the potential of SPOT

derived DEMs than the previous two examples. They concluded that SPOT was suitable

for mapping at 1:50,000 scale or smaller.

Ley (1988) used three different stereo plotting systems to extract elevation

information from both digital and film SPOT images (depending on the system). One

was a totally digital environment, one was a traditional analytical plotter that used a film


product, and the third was a hybrid. Ley found that areas of steep terrain and poor

contrast had the lowest accuracy, and gentle terrain with high contrast, the highest. Even

with optimal terrain and contrast, the base-height ratio of the two image limited the

accuracy of measurements. On average, errors were 10 to 20 meters.

The error in SPOT derived DEMs documented in the above examples estimated

average error between 3.5 and 20 meters. The number of different parameters that

affected image and DEM quality provide a clue to the reasons for the differences. The

question of operator experience is an important one, which should become less of a

problem in production environments, but which will remain an important factor for in-

house desktop workstations capable of stereoscopic data collection. The methods and

expectations of data quality with SPOT DEM are still evolving. Expected error may vary

greatly from one DEM to another.

Error in DEMs from Contour Interpolation

Several problems are associated with the use of interpolation techniques. The most

accurate part of the DEM is closest to the contour lines. As with contour maps, the area

between contours is assumed to change more or less predictably, but naturally, it does not.

J. Carter (1987) found spikes and a "false butte" in a USGS contour interpolated DEM.

The spikes were associated with DEM points that fell on a spot elevation. Carter also

noted the difficulty in accurately digitizing areas where contours crowd together on the

topographic map. This occurs frequently in areas of very steep terrain. Other problems

arise due to the mass and complexity of the data. It is difficult to develop computer

programs that deal effectively with all possible terrain configurations.

Another artifact common to contour interpolated DEMs is banding that corresponds

with the location of the original contour line. The pattern may be visible in a graphic

display of the DEM or be visible as spikes in a histogram of the elevation values. It

occurs when contour lines are widely spaced and many cells are assigned the same value

as the original contour line. This effect has been noted with bicubic spline interpolation,

which was used for the 1:250,000 equivalent DEM distributed by the USGS (Isaaks &

Srivastava, 1989).


In this chapter, I summarized the common typologies of error in DEMs. These

are important for an understanding of the goals of research focused on DEM error. Next,

two current topics of interest in DEM error research were considered. These were first,

fractals and secondly, spatial autocorrelation. Each one of these topics, in the context of

DEM accuracy could constitute a major research effort of its own. The questionable

applicability of fractal theory to real terrain surfaces led to it being put aside for the

present. Spatial autocorrelation is certainly related to the present research, and it proves

to be an important factor in the research results that follow in later chapters.

The second half of this chapter was concerned with the methods of assessing error

in DEMs. The methods reviewed here were subsequently applied in the assessments

described in Chapters VI and VII. Finally, previous research on DEM error from various

methods was reviewed. The results from this dissertation research may be compared, to

some extent, with other studies. An additional element in the current research, missing

from most previous work in DEM accuracy is the assessment of the quality of a derivative

product, the viewshed, and its relationship to the pattern of error in the original DEMs.

The work in remote sensing related to error matrices and analyses of patterns of error in

image classifications will be directly applicable when considering viewshed error.


The Study Area

The DEMs used for the error assessments were all collected for the same area in

Fort Knox, Kentucky. Fort Knox is located in North Central Kentucky, just south of

Louisville, Kentucky (Figure 5-1). It is an area of karst topography, characterized by "an

irregular terrain punctuated with many depressions, called sinkholes, or sinks" (Tarbuck

and Lutgens, 1990:275).

The rolling hills, with some very steep slopes, are covered with dense forests

interspersed with short grass and shrub-covered plains. The elevation above mean sea

level ranges from 116 to 301 meters. This area is part of the physiographic region known

as the interior low plateaus province (Thorbury, 1965).

The study area was chosen for two reasons. First, data availability was enhanced

by intensive mapping of this region by the United States Army for a variety of projects.

Second, the tree cover, the steep slopes and the presence of standing water make this

fairly difficult terrain to map. The complexity of the terrain added to the usefulness of

the DEM comparisons. The study area comprised about 120,200 acres. Some of the test

DEMs were for only a portion of this area.

Figure 5-1
Fort Knox, Kentucky

The Geographic Information System

The GIS functions for this project were performed on a UNIX-based workstation

from SUN Microsystems (Mountain View, California). The workstation was the model

SUN4c, also known as an IPC. The machine was provided by Professor Grant Thrall in

his research lab. The GIS software was the Geographic Resources Analysis Support

System (GRASS). GRASS is a public domain software package developed by the U.S.

Army Construction Engineering Research Laboratory in cooperation with many federal

agencies, universities and private organizations.

GRASS is particularly well-suited for raster analysis. It was designed originally

for this type of analysis, and it includes a large suite of tools for manipulation of raster

data (USACERL, 1991). Since all the DEMs in this project are in raster format, this

functionality fulfilled project requirements. A statistical package supplemented the GIS.

Data from the GIS were output as ASCII files and then read into the database for the

statistical package.

The Digital Elevation Models

Five digital elevation models were included in the assessments. One was a control

model, the other four were test models. Two of the test models covered a much larger

area than did the other two. These two were tested more extensively than were the others.

Table 5-1 summarizes the characteristics of the models. Figure 5-2 shows their extent

and relative locations within the study area. The models are each described below.

The Control Model

The DEM of the highest accuracy served as the control, or reference, model. It

was used to represent true elevation values. Engineering accuracy standards dictate that

check points used to check the accuracy of mapping are of "adequate accuracy" if the

check points are estimated to be of equal or greater accuracy than the map to be assessed

(Merchant, 1983). This DEM was judged to meet that criteria, and the DEM was deemed

adequate for the purpose of assessing the test models.

The use of an entire DEM for reference data rather than discrete points provided

by a ground survey, was an important aspect of this project. The reference DEM allowed

the entire range of values in the test DEMs to be assessed. It was this which provided

a picture of the spatial pattern of error in the test models which would otherwise not have

been apparent. To duplicate this with ground survey points would have been prohibitively


The control DEM was produced by the photogrammetric firm, Aerometric

Engineering, in Sheboygan, Wisconsin. This firm was selected on the basis of the

quality of previous work and a competitive bidding process to produce a DEM with a

10 meter horizontal resolution for Fort Knox, Kentucky. The DEM was produced for the

entire 120,278 acre study area. The vertical precision of the elevation grid was in

decimeters. The values were stored in the database as integers for this and all other raster

maps. In order to preserve precision, the values in decimeters were multiplied by ten and

then stored.

The DEM was created photogrammetrically by collecting mass points and critical

elevation values from 1:24,000 scale photography, flown at 12,000 feet. An operator

directed the spatial distribution and the quantity of the elevation sample points using a

variant of progressive sampling. The aerial photography had been used for a previous

mapping project for mapping at a scale of 1" = 400'.


Figure 5-2
The Study Area


The SPOT-Derived Test Model

The first test DEM (DEMTI, Table 5-1) was derived from panchromatic imagery

from the French satellite Systeme Pour I'Observation de la Terre (SPOT). Although

acquisition of the two scenes was requested for Spring, 1990, cloud cover prevented

acquisition of the second scene during that season. The second scene was acquired in

Spring, 1991. The one year difference in the dates of these scenes was not ideal, but this

problem has been typical of SPOT-derived DEMs. The test data collected were not

Table 5-1
An Overview of the DEMs

DEM Type/Source Res. Acres
DEMC Traditional photogrammetry
1:24,000 air photos 10 m 120,278
DEMT1 Automated method from digital
SPOT panchromatic imagery 10 m 120,278
DEMT2 Automated method from scanned
1:80,000 air photos (NHAP) 3 m 2,222
DEMT3 USGS 7.5' DEMs from NHAP photos
GPM-2 and manual profiling 30 m 120,278
DEMT3a USGS 7.5' DEM Fort Knox quad
NHAP w/GPM-2 30 m 37,490
DEMT3b USGS 7.5' DEM Rock Haven quad
NHAP w/manual profiling 30 m 15,193
DEMT4 Contour interpolated inverse
distance weighted from 1:24,000
digital contour lines 10 m 935

intended to be representative rather than ideal. They were meant to represent the data

available to a GIS user for intermediate-scale mapping.

The SPOT-derived DEM was purchased through the SPOT Image Corporation

(Reston, Virginia). The creation of the DEM was subcontracted to a value-added firm,

but the quality assurance came from SPOT. The DEM had a horizontal resolution of 10

meters and a vertical precision of 1 meter. The elevation data were collected using an

automated image correlation process from the two digital images. The control points used

for image rectification came from USGS 1:24,000 quad sheets. The overlap area of the

two images was 2647 square kilometers (654,050 acres), about 75% of the size of the

typical, 60 km by 60 km, SPOT scene. The 120,278 acres of the study region was cut

out of this larger DEM and used for the test.

The NHAP Test Model

The second test model (DEMT2, Table 5-1) was derived from high altitude aerial

photography from the United States National High Altitude Aerial Photography program

(NHAP, now called NAPP). Geospectra Corporation (Ann Arbor, Michigan) digitally

scanned the 1:80,000 scale black and white photographs at a 25 micrometer spot size.

This resulted in a pixel size of 2 meters (ground units), which were subsequently

resampled (by Geospectra) to 3 meters. The elevation values were calculated from this

very dense grid using proprietary software called ATOM (Vincent, Pleitner, and Oshel,

1984). The resultant DEM comprised 2222 acres. Horizontal resolution was 3 meters and

vertical precision was 1 meter. Ground control for geo-processing came from 1:24,000

scale quad maps.

The USGS Test Models

The third test DEM (DEMT3 in Table 5-1) was a set of models purchased from

the U.S. Geological Survey. These were in the 30 meter (horizontal resolution) series,

corresponding with 1:24,000 scale map coverages. Of the five USGS DEMs included in

the study area, one was produced using manual profiling with NHAP photography, and

the other four were products of NHAP and the Gestalt Photomapper (GPM-2). The

elevation values of these DEMs were in meters above local mean sea level. The five

quads, or parts of quads were trimmed to correspond with the 120,278 acres of DEMC.

Two different production methods were represented in the five USGS DEMS.

Four of the DEMs were produced by the GPM-2. One of them was produced through

manual profiling. All were from NHAP photography. The four GPM-2 DEMs were

produced in the late 1980's. No date was provided with the manual profiling DEM. All

of the USGS DEMs had the Level 1 error classification. The header file included with

the DEMs indicated that three of the GPM-2 DEMs had relative accuracy of 7 or 6 meters

RMSE. The fourth GPM-2 DEM did not include accuracy information in the header file.

The manual profiling DEM header file reported an accuracy of 4 meters.

The Contour Interpolated DEM

The fourth test DEM (DEMT4 in Table 5-1) was interpolated from digital contour

lines digitized manually from a 1:24,000 scale USGS map. The contours were at a 20

foot interval with supplemental 10 foot contours. The contour lines were digitized as

vectors and then rasterized to a 10 meter resolution grid. The elevation values were

stored in decimeters. The area of DEMT4 was about 1000 acres.

An inverse distance weighted interpolation program was used to fill in the final

DEM from the rasterized contour lines (Isaaks and Srivastava, 1989; USACERL, 1991).

In this algorithm, the z-values are calculated to give greater weight to closer known points

than to those that are more distant. For the interpolation of this DEM, the 12 closest

original elevation points determined the interpolated values of the output map.

Data Processing

Datum and Coordinate System

The change in datums with the "new" 1983 North American Datum was not an

issue in this project, because the North American Datum of 1927 was the reference datum

for the horizontal locations in all of the DEMs. All elevations were calculated in terms

of their distance in meters above average mean sea level as determined by the National

Geodetic Vertical Datum of 1929.

The DEMs were all projected in the Universal Transverse Mercator (UTM)

projection. The x and y coordinates were all determined by the UTM grid system for

zone 16. All of the DEMs were brought into the data base with the GRASS software.

The DEMs were all stored in their original forms, but they required some resampling

prior to the assessments. This was necessary because the DEMs were stored with

different geographic extents and resolutions.


GRASS uses the concept of a "region" in which data are stored or analyses are

performed. Each GRASS raster file includes a header with information on the geographic

extent (north, south, east and west boundaries) and spatial resolution of the map. In the

GRASS database, each individual DEM has its own region defined in its header file.

When two DEMs with different resolutions or extent are used in the same analysis,

resampling of data from at least one DEM was required. For the first part of the research,

in which differences and ratios were calculated, the test DEM provided the region

information. The control was resampled to match the test using the default resampling

algorithm, which was nearest-neighbor resampling.

The comparison of values between the control and test models occurred at the

center point of each corresponding grid cell of the two models. This was not as precise

as a point by point comparison using control points identified by x, y and z coordinate

values. The large number of cells compared, however, was an important aspect of this

research. The comparison of whole surfaces rather than discrete points or profiles gave

a look at the "big picture." Li (1990: 7), for example was particularly aware of the need

"... to extend [DEM accuracy] analyses to a complete surface." Although grid cells are,

theoretically, discrete points; they are conceptually similar to a surface.

For the second part of the research, in which the viewsheds from the DEMs were

compared, a different logic was used regarding resampling. For this part, all of the DEMs

that were not stored at a ten meter resolution were resampled to match the control DEM.

All of the viewshed calculations and the measurements of slope, tree cover and the

selection of viewpoints were performed for a set of regions defined by the control DEM.

The sheer volume of data processing required for these comparisons made this

simplification necessary.

Viewshed Calculations

Each of the DEMs was used to derive a set of viewsheds. GRASS was used for

this task. The GRASS viewshed algorithm identifies all of the grid cells that are visible

from a specified point. The observation point provides the location of the viewer in three

dimensions, both the x-y location and the height above the ground of the observer. For

this analysis, the height of the observation was set at 1.75 meters above the elevation

found at the point at which the observation was made.

In order to reduce the number of data values around each observation point, a limit

of 500 meters from the observation point was set, with the viewshed being calculated only

up to that limit, in all directions from the observation point. The location of the

observation points were chosen using a random point generator available in the GRASS

GIS. This type of random number selection is actually "pseudo-random" since the same

series of numbers is used each time the program runs. It was deemed adequate for the

present purpose.

The random selection of points was performed three times. Each of the two

smaller areas, covered by DEMT2 and DEMT4, were treated as individual subregions for

the viewshed assessment. A set of 30 observation points were selected at random in each

of the subregions. A third set of 61 observation points were selected for the entire study

area as described below.

Viewsheds were calculated at 61 random observation points spread around the

study area. The points were selected using a stratified random selection strategy. The

USGS 7.5' quad coverage areas were used as the stratum. The number of points selected

from each quad area was determined by the area of the quad that was within the study

area. The number of sampled points was proportional to the percentage of the total study

area represented by a particular quad or partial quad.

After the number of observation points for each quad was determined, a simple

random sampling was performed from all the possible locations within that quad. Two

additional constraints were placed on the location of the observation points. First, the 500

meter buffer area around the point could not overlap with the area of another point.

Second, the 500 meter buffer area was not allowed to overlap into an adjacent quad.

Points that did not meet these criteria were discarded, and new points were selected to

replace them.

Other Variables in the Analyses

In addition to the viewshed and the DEM accuracy variables, a number of other

variables served as explanatory variables. The measurements and methods used to create

those independent variables are described below.

Slope Steepness

The steepness of slope incline plays an important part in explaining the variation

in the accuracy of DEMs. For this variable, the control DEM was used. A GRASS

program calculates slope in degrees. This was reclassified into percent rise, which is the

tangent of the slope value in degrees. Each pixel in the slope map thus contained the

value of the percent incline of that pixel.

Tree Cover

Tree cover data was classified from SPOT 20 meter multispectral imagery. Cloud

cover was a problem, an ortho-corrected image from 10 meter SPOT imagery provided

data on the areas covered by clouds. The image classification was performed using

GRASS classification programs. For the 61 viewshed areas used in the final analysis,

each 500 meter radius circle was examined individually to improve the tree cover

classification for those areas. The major source for this was the SPOT panchromatic



Several height variables were used in this analysis. For the DEM accuracy

assessments, height was found to be a significant factor in explaining the variation in

accuracy. When height classes were used to aid in this assessment, the classes were based

on the true height of the area, as determined by the control DEM. The height of the

observation point was also a factor in terms of viewshed accuracy. The average height

of each viewshed area was a third height variable. These variables were, once again,

measured from the control DEM.

Diversity Measure

The measure of diversity has been used in remote sensing. The measure used in

this research was based on a GRASS filtering program. The program allowed the

calculation of the number of cells of different values in an n-cell neighborhood. This was

used to measure the diversity of the EVTV ratio values of the DEMs. A nine cell

neighborhood was used. The neighborhood was passed across the original EVTV maps

and a new map was created with values ranging from 1, meaning all values in the

corresponding neighborhood were the same, to 9, meaning that all values in the

neighborhood were different.

The diversity measure was used as a proxy for an estimate of spatial

autocorrelation. The spatial autocorrelation measure, Moran's I, was used to measure this

dimension of the ratio maps. It was found that with a 25 cell neighborhood, the average

I value of the EVTV ratio in the 61 viewshed areas was over .9 in all cases. Basically,

the error was highly autocorrelated. The difference between .91 and .93 in a measure that

is not particularly sensitive, was not deemed useful. The diversity measure was used


Path Analysis

The conceptual model that was developed at the end of this research was guided

by the concepts of path analysis, as defined by Loehlin (1987). This method has been

used by social scientists to help to bring order to a large number of inter-related variables.

Path diagrams are an important tool for this effort. The path diagram uses straight lines

with arrows between variable names to represent a causal relationship between two

variables. Curved lines with arrows represent a correlation between two variables (Figures

7-5 and 7-6).

The variables may be either observed or latent. Latent variables are those which

may not be fully measured, but instead are represented by a number of observable

variables. Latent variables are often called factors, and are depicted as ovals or circles

in the path diagram. In this analysis, DEM error was considered a latent variable, as

measured by RMSRE, diversity and slope. Factor analysis is often used to determine

which observable variables are related and may be measuring a latent variable.

Observed variables are depicted as squares or rectangles. Variables in a path

diagram are also defined as either exogenous or endogenous. The exogenous variables

are the independent variables. In path analysis, other variables within the path diagram

do not explain exogenous variables. This means that causal arrows may leave them, but

not go toward them. The endogenous variables are those which are explained by the

variables within the path diagram. These are also known as downstream variables.

Endogenous variables have causal arrows that point toward them.

This method of organizing the variables was very helpful for this research. It

allowed the complexity of the data to be more fully exploited in a manner that would

have been easily overlooked using a series of multiple regression models. The solution

to the problem was found to be a two level process. First, there was the question of the

factors that affected DEM quality. Then, there was the question of how these factors and

DEM quality affected the viewshed outcomes. These questions could not be handled

adequately with a straight regression analysis. The final assignment of weights to the

paths, which is the final goal of path analysis, was not performed for this research. This

task was left for a future effort.

In this chapter, I provided details of the data collection, processing and analysis

used in the research. This chapter provided basic background information on the Fort

Knox study area. I introduced the GRASS GIS and the statistical software for the project.

The control model and each of the test DEMs were described in some depth. The

sampling of the observations points for the viewsheds was a key point in the methodology

of the research. Finally, the path diagram techniques used in Chapter VII were explained

and justified.



This chapter summarizes the results of several descriptive assessments of the test

DEMs. Shaded relief renditions of the DEMs were first evaluated visually, with each test

model being compared with the control model as well as with the other test models. The

second assessment involved a calculation and comparison of the root mean square error

of each of the test models, using the control model as a reference. The difference maps

created for the calculation of the RMSE provided a further means to compare and contrast

the test models.

Limits in the use of differences, which can be negative numbers, led to the

adoption of a ratio of the estimated value to the true value as a third means to evaluate

the test DEMS. The spatial pattern of the error is visible in maps of the ratios. Finally,

the spatial distribution of slope steepness, elevation height, water and tree cover were

compared to the differences in the ratio measures. To complete this section, the diversity

of error ratio values were calculated and compared to error magnitude values.

Visual Assessment

Shaded relief renditions for each of the test DEMs and for the control DEM are

illustrated in Figures 6-1 through 6-6. Each map was printed at 1:24,000, thus creating

comparable views of each one. These figures were used for the visual assessment that is

described in this section. The test DEMs were compared with the control DEM in terms

of level of detail evidence of artifacts related to interpolation or the data collection

process. Interpolation artifacts are often visible in the shaded relief map, though they are

not apparent in the map of elevation (Wood & Fisher, 1993).

DEMC -- The Control Model

The shaded relief rendition of the control DEM (Figure 6-1 and Figure 6-5)

illustrates graphically the potential of a highly detailed DEM. The topographic features

are clearly defined and there is little visual evidence of features that do not appear to be

part of the natural surface. These two figures will be used to compare and contrast with

the other figures.

DEMT1 -- The SPOT Model

Figure 6-2 shows the shaded relief produced from the SPOT DEM (DEMT1).

This is the same area as shown in Figure 6-1 for the control DEM. The most obvious

feature of DEMT1 is its lack of detail. Though the model was produced at a resolution

of 10 meters -- the same resolution as the control DEM -- the 10 meter control model is

clearly of a higher level of detail. A user who assumed that all 10 meter DEMs could be

used interchangeably may be disappointed by the information content of DEMT1.

1 kilometer

Figure 6-1
DEMC Shaded Relief

1 kilometer

Figure 6-2
DEMT1 Shaded Relief

DEMT2 -- The High Density Model

Figure 6-3 is from DEMT2, the high density DEM produced with automated

correlation and NHAP photography. This DEM has a horizontal resolution of 3 meters.

The area depicted in this figure is upper three-fifths of the area shown in Figure 6-1.

DEMT2 provides more detail than DEMT1. The areas with high relief differences are

especially well rendered. The flat areas, however, are muddled by an indistinct salt and

pepper pattern. The irregular flat area that appears about half way up the east side of the

DEM is a lake.

DEMT3 The USGS Models

The shaded relief from DEMT3 is found in Figure 6-4. The detail in this

depiction is once again considerably less than in Figure 6-1. The 30 meter cells of the

USGS DEM are more clearly apparent than are the cells in the other test models, creating

a patchwork quilt pattern in the southeast quadrant of the map. At 30 meters, this DEM

had the lowest resolution of all of the test models. The hilly area in the north and west

of this area are depicted more realistically by DEMT2 (Figure 6-3) than by this DEM.

The hilly area is easily perceived as hills, though they are not very distinct in the

northwest corner of the area. DEMT3 performs better in terms of overall information

content, than does DEMT1. The stream that flows through the southeast part of this

region is visible in DEMT3, while it was only vaguely visible in DEMT2 and absent in


1 kilometer

Figure 6-3
DEMT2 Shaded Relief

1 kilometer

Figure 6-4
DEMT3 Shaded Relief

DEMT4 -- The Contour Interpolated Model

Figure 6-5 is another view of the control DEM, and Figure 6-6 depicts DEMT4,

the contour interpolated DEM. The scale is 1:24,000 for each, and the two figures are

for the same area. The detail of DEMT4, which was stored at a ten meter horizontal

resolution, compares favorably with DEMC. The terracing that is present in the flatter

areas is typical of an inverse distance weighted interpolation. DEMT4 missed the detail

of the lower area in the western part of DEM, because this detail was not apparent in the

original contours. In order to portray the missing detail, the original input data would

need to be made more dense. The interpolation was not necessarily at fault. In spite of

its weaknesses, DEMT4 provides the best shaded relief rendition of the four test models.

Summary of the Visual Assessment

The detail of the four test DEMs was markedly different among these models.

None of the tests has as much detail as the control DEM. DEMT4, the contour

interpolated DEM, was the most detailed of the test models, DEMT2 and DEMT3 were

similar in content, and DEMT1 was, by far, the least detailed. From this simple

comparison, it is clear that the horizontal resolution of a DEM is not necessarily indicative

of its information content. Judging a DEM by its horizontal resolution would be similar

to judging the content of a book by the number of pages it contained. The amount of

information contained in the data source and the way in which the information was

processed affected the information available. For the DEMs examined in this research,

the horizontal resolution and the information content of the grid DEM are not consistently


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