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University Microfilms International 300 North Zeeb Road Ann Arbor, Michigan 48106 USA St. John's Road, Tyler's Green High Wycombe, Bucks, England HP10 SHR LQDM, JOSEPH COLM&N CÀRTDGRAPHTC GENERALIZATION OF DIGITAL TERRAIM MODELS. THE OHIO STATE UNiVERSITY, PH.D., 1976

" S m s IrrtematKXVU 300 n. zeeb road, ann arbor, mi asiob

® Copyright by Joseph Colman Loon 1978 CARTOGRAPHIC GENERALIZATION OF

DIGITAL TERRAIN MODELS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

Joseph Colman Loon, B.Sc., M.Sc., M.S.

* * * *

The Ohio State University

1978

Reading Committee:

Dr. U. A. Uotila, Chairman Dr. R. H. Rapp U y A . Uotila, ^ v i s e r Dr. J. N. Rayner Apartment of/ceodetic Science ACKNOWLEDGMENTS

I am indebted to Dr. U. A. Uotila, for his support during the whole

period of my graduate studies at the Ohio State

University and for his encouragement and ideas which

led to the study presented here; to Drs. I. I. Mueller, H. J. Steward and R. H. Rapp,

for allowing me the use of their extensive personal

libraries; to my reading committee, Drs. U. A. Uotila, R. H. Rapp

and J. N. Rayner, for their patience and advice; to the Faculty, Staff and Students of the Department

of Geodetic Science, for contributing to an atmos­

phere and experience which has made a lasting im­

pression on me; and to my wife, Zilla, for undertaking m m e r o u s difficult

tasks including that of mother and budget director,

while I immersed myself in the search.

My thanks to Diana Howes, who typed this disserta­

tion with good humor and efficiency. November 2, 1931 . . . . Born— Parys, South Africa

1954 ...... B.Sc. (Land Surveying), University of Cape Town, South Africa

1956 ...... L (S.A). Registered Land Surveyor of South Africa.

1956-1960...... Junior Lecturer and Assistant, Department of Land Surveying, University of Cape Town, South Africa

1961-1967...... Lecturer and Senior Lecturer, Division of Technicians, Cape Technical College, Cape Town, South Africa

1967 ...... Part-time Lecturer, Department of Land Surveying, University of Cape Town, South Africa

1967 ...... M.Sc. (Land Surveying), University of Cape Town, South Africa

1969-1973...... Lecturer, Division of Geodesy, Technion— Israel Institute of Technology, Haifa, Israel

1974-1978...... Teaching and Research Associate, Department of Geodetic Science, The Ohio State University, Columbus, Ohio

1976 ...... M.S., The Ohio State University, Columbus, Ohio PUBLICATIONS

"Isostasy." Journal of the University of Cape Town Engi­ neering and Scientific Society, Vol. 1, No. 3, 1955.

"Photogrammetry." The Lens— Journal of the Cape Technical College, Cape Town, South Africa, 1961.

Photogrammetry. Course Book, published by the Cape Technical College, Cape Town, South Africa.

Surveying for the National Technical Certificate 3. Course Book published by the Cape Technical College, Cape Town, South Africa, 1963.

"Gravimetric Geodesy." Proceedings of the Third South African National Survey Conference, Johannesburg, South Africa, 1967.

"Survey Technicians." Proceedings of the Third South African National Survey Conference, Johannesburg, South Africa, 1967.

With P. Yoeli. Map Symbols and Lettering; A Two Part Investigation. Part I— An examination of map symbols in their most elementary forms. Part II— The logic of auto­ mated map lettering. Final Technical Report. European Research Office of the United States Army. (AD-741" 834) , 1972.

FIELDS OF STUDY

Major Field; Geodetic Science

Studies in Cartography. Professors H. J. Steward and H. Moellering

Studies in Adjustments and Geodesy. Professors U. A. Uotila and I. I. Mueller

Studies in Photogrammetry and Remote Sensing. Professors S. K. Ghosh, D. C. Merchant and 0. W. Mintzer

Studies in Statistics and Numerical Methods. Professors W. A. T. Archambault, P. Feder, J. S. Rustagi, D. R. Whitney and R. LaRue

iv TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS...... Ü

VITA ...... iii

LIST OF TABLES ...... viii

LIST OF FIGURES...... X

LIST OF SYMBOLS AND NOTATION USED...... xiv

Chapter

1 CARTOGRAPHIC GENERALIZATION.

1.1 Introduc t ion...... 1.2 Generalization of Topographic Maps. . 1.2.1 Why Should Topographic Maps Be Generalized ...... 1.2.2 Generalization of Topographic Map Elements Other Than Relief 1.2.3 Generalization of Relief . . . 10 1.3 Purpose of This Study ...... 15 1.3.1 Computer-Assisted Cartography. 15 1.3.2 Problems of Generalization in Computer-Assisted Cartography. 16 1.3.3 Needed— A Flexible Method. . . 18 1.4 Hypothesis and Subsidiary Research Questions ...... 1.4.1 The Least Squares Collocation Algorithm...... 19 1.4.2 Subsidiary Research Questions. 20 1.5 Assumptions and Scope of This Study . 21 1.6 Organisation of This Study...... 24 Page

Chapter

2 REVIEW OF RELATED RESEARCH AND METHODS . . . 26

2.1 State of the Art of the Cartographical Generalization of the Relief Continuum on Topographical M a p s ...... 26 2.1.1 Manual M e t h o d s ...... 28 2.1.2 Automated Methods...... 32 2.2 Summary and How This Study Will Make a Contribution...... 35

3 THE CHARACTER OF THE TERRAIN ...... 39

3.1 Introduction...... 39 3.2 Some Terrain Descriptors...... 41 3.3 Description of DTM and Contouring Program Used in This Study...... 44 3.4 Double Fourier (Trigonometric) Series . 48 3.5 Harmonic Surfaces ...... 52 3.6 The Empirical Covariance Function .. . 59 3.6.1 Theoretical Considerations .. . 59 3.6.2 The Empirical Covariance F u n c t i o n ...... 63 3.6.3 Reference surfaces ...... 66 3.6.4 Some Results ...... 77 3.7 Some Quantitative Measures Related to the Change in the Character of the T e r r a i n ...... 84 3.8 Summary ...... 93

4 LEAST SQUARES COLLOCATION...... 95

4.1 Introduct ion...... 95 4.2 Overview of the Least Squares Collocation Method...... 96 4.3 Proposed Use of Least Squares Collocation Concepts in Cartographic Generalization...... 103 4.4 S u m m a r y ...... 109 Page

Chapter

TESTING THE HYPOTHESIS...... HO

5.1 Preliminary Investigations Using Diagonal Matrices Only...... HO 5.2 Using the Least Squares Collocation Algorithm— With a Limited Distance of Influence...... 116 5.3 Constraints and Varying Degrees of Generalization...... 124 5.4 S u m m a r y ...... 128

DESIGN OF A GENERALIZATION FILTER...... 133

6.1 The Nines Filter...... 133 6.2 Requirements of a Filter...... 137 6.3 Design of Nines Filter Using the Gaussian Function ...... 138 6.4 The Gaussian Function Nines Filter Compared With Other Filters ...... 143 6.5 Frequency Response...... 150 6.6 Results and Analysis of Experiments . . 154 6.7 S u m m a r y ...... 165

PROPOSED SYSTEM FOR THE CARTOGRAPHIC GENERALIZATION OF THE RELIEF CONTINUUM ON TOPOGRAPHIC MAPS......

7.1 Outline of the System ...... 7.1.1 Filter Selection and Generalization ...... 164 7.1.2 Analysis of the Generalized Map 169 7.2 Proposed System Applied to a Digitized Portion of a Topographic Map...... 170 7.3 Summary and Analysis...... 178

8 SUMMARY AND SUGGESTIONS FOR ADDITIONAL RESEARCH...... 182

BIBLIOGRAPHY ...... 190

A. References Cited ...... 190 B. References Consulted But Not Cited ...... 194 LIST OF TABLES ^ .

Table Page

3.1 Digital Terrain Model Used in This Study. . . 45

3.2 Number of Grid Distances Used in Figure 3/9 . 65

3.3 First Table of Quantitative Measures for Maps 1, 2, 3, 4, 6, 7, 8, 9 ...... 90

3.4 Second Table of Quantitative Measures for Maps 1, 2, 3, 4, 6, 7, 8, 9 ...... , 91

5.1 First Table of Quantitative Measures for Maps 11, 12, 13, 1 4 ...... 114

5.2 Second Table of Quantitative Measures for Maps 11, 12, 13, 1 4 ...... 115

5.3 First Table of Quantitative Measures for Maps 18, 34, 36, 38, 40 ...... 122

5.4 Second Table of Quantitative Measures for Maps 18, 34, 36, 38, 40 ...... 123

5.5 First Table of Quantitative Measures for Maps 41, 4 2 ...... 127

5.6 Second Table of Quantitative Measures for Maps 41, 4 2 ...... 127

6.1 Nines Filter Weights Using the Gaussian Function...... 141

6.2 Discrete One-Dimensional Filters Derived From the Gaussian Function With Various Values of a ...... 146

6.3 First Table of Quantitative Measures for Maps 43 Through 54...... 158 Table Page

6.4 Second Table of Quantitative Measures for Maps 43 Through 5 4 ...... 159

Relationship Between Profiles and Maps 43 to 54...... 162

Elevations of Selected Data Points in Profile Shown in Figure 6/21...... 164

6.7 Filter With 25 Weights...... 165

7.1 Quantitative Measures (Part One)— Alma 175

7.2 Quantitative Measures (Part Two)— Alma 176 LIST OF FIGURES

Figure Page

1/1 Original Map...... 11

1/2 Generalized M a p ...... 11

1/3 Over-Generalized Map...... 13

2/1 Flow Diagram For Map D e s i g n ...... 27

2/2 Correct Representation of an A r e a ...... 29

2/3 Incorrect Representation of an A r e a ...... 29

2/4 Emphasis on Planes or Ridge Lines With 10 m Vertical Interval...... 31

2/5 Generalized Terrain With 20 m Vertical Interval...... 31

2/6 Rounding Results in Loss of Original Character...... 31

3/1 The Original Digital Terrain Model as a Contoured Map. Center Point Weight = 1.0 . . . 46

3/2 The Original Digital Terrain Model as a Contoured Map. Center Point Weight = 3.0 . . . 47

3/3 The Two-Dimensional Power Spectrum (Showing Only the First Five m and n Harmonics)...... 53

3/4 First Harmonic Surface...... 55

3/5 Second Harmonic Surface ...... 56

3/6 Third Harmonic Surface...... 57

3/7 Fourth Harmonic Surface ...... 58 Figure Page

3/8 The Covariance Function ...... 60

3/9 Covariance Function, Original Data With Mean Plane as Reference Surface ...... 64

3/10 4-Term Bilinear Polynomial Surface Fitted to Original Data...... 68

3/11 8-Term Biquadratic Polynomial Surface Fitted to Original D a t a ...... 70

3/12 12-Term Bicubic Polynomial Surface Fitted to Original Data...... 71

3/13 16-Term Bicubic Polynomial Surface Fitted to Original Data...... 72

3/14 Residual Ma p ...... 74

3/15 16-Term Bicubic Polynomial Fitted to Smoothed D a t a ...... 75

3/16 Residual Map...... 76

3/17 Covariance Function, Original Data With 4-Term Bilinear Polynomial Reference Surface. . 78

3/18 Covariance Function, Original Data With 16-Term Bicubic Polynomial Reference Surface. . 79

3/19 Covariance Function, Binomially Once Smoothed Data With Mean Plane as Reference S u r f a c e ...... 81

3/20 Covariance Function, Binomially Twice Smoothed Data With Mean Plane as Reference S u r f a c e ...... 82

3/21 Covariance Function, Binomially Three Times Smoothed Data With Mean Plane as Reference Surface ...... 83 Figure Page

3 / 2 2 3/23 Generalizations of Original Data— 3/24 Scale Approx. 1 in 250,000...... 88 3 / 2 5

3/26 3/27 Generalizations of Original Data— 3/28 Scale Approx. 1 in 250,000...... 89 3/29

5/1 5/2 Generalizations of Original Data— 5/3 Scale Approx. 1 in 250,000 . . . 113 5/4

5/5 Generalizations of Original Data— Scale Approx. 1 in 250,000...... 120

5 / 6 5/7 Generalizations of Original Data— 5/8 Scale Approx. 1 in 250,000...... 121 5/9

5/10 Generalizations of Original Data— 5/11 Scale Approx. 1 in 250,000...... 126

6/1 The Nines Filter...... 134

6/2 A Corner F i l t e r ...... 135

6/3 An Edge Filter...... 136

6/4 Filters Based on Gaussian Function With a = 0 . 8 3 ...... 142

6/5 Tobler's Binomially Weighted Smoothing Filter . 144

6/6 Nines Filter Based on Stegena's Low Pass Filter For Lines ...... 147

6/7 Nines Filter Used by Robinson, Sale and Morrison (1978) ...... 148

6/8 Frequency Response Curves ...... 153 Figure Page

6 / 9 6/10 Generalizations Based on Use of a Nines 6/11 Filter— Scale Approx. 1 in 250,000...... 155 6/12

6/13 6/14 Generalizations Based on Use of a Nines 6/15 Filter— Scale Approx. 1 in 250,000...... 156 6 / 1 6

6/17 6/18 Generalizations Based on Use of a Nines 6/19 Filter— Scale Approx. 1 in 250,000...... 157 6/20

6/21 P r o f i l e s ...... 161

7/1 The Alma Digital Terrain Model...... 171

7/2 7/3 7/4 Generalization of Alma Model— 7/5 Scale Approx. 1 in 276,000...... 173 7 / 6 7/7

7/8 Alma Generalized, a = 0.04...... 177

7/9 From USGS Sheet NL 15-12...... 177

7/10 Alma Generalized, a = 0.83...... 177

7/11 River System From Rayner (1972) Superimposed on Alma Generalization With a = 0.04 Twice. . . 181

7/12 River System From Rayner (1972) Superimposed on Alma Generalization With a = 0.83...... 181 LIST OF SYMBOLS AND NOTATION USED

A coefficient (or sensitivity or design) matrix

Cyy covariance matrix for the element Y

cross-covariance matrix for the elements X and Y

C(d) covariance as a function of distance, d

E the statistical expectancy

I unit matrix

N a vector, being the (random) noise on the observations

P vector of parameters

S the random signal at the observation point

S' the random signal at the predicted point

( )^ superscript, the m

W the weight matrix

Z vector of known quantities or observations (usually elevations)

data variance ag variance of unit weight

( )”^ superscript, the matrix inverse, for example, C“^ CHAPTER ONE

CARTOGRAPHIC GENERALIZATION

1.1 Introduction

This study is in the nature of an exploratory probe, into part of the field of cartographic generaliza­ tion, with tools which have been used in other fields.

The main tools used here are concepts from the method of least squares collocation as used in geodesy and some aspects of spectral analysis as used in geophysics and related sciences.

Cartographic generalization is here considered as

"generalization" in the wide meaning of the word, with the subjective intervention of the cartographer. This subjective intervention will determine the "character" of the final map. Aspects of the "character of the terrain" will be discussed in Chapter Three.

In cartographic generalization, the cartographer is trying to communicate with the map perceiver by telling him something about the character of the data being symbolized. For example, these data could be elevation (of part of the earth's surface) contained in a digital

terrain model (DTM). In cartographic generalization,

the individual cartographer uses the digital terrain

model or base data, and subjectively determines the

character of the final result to suit a particular purpose.

The final result could be a contour map at a particular

scale. Or, the subjective intervention could be the

result of an administrative policy for the cartographic

generalization of a map series or for the cartographic

generalization of different terrain types.

In order to get some idea of the complexity of

generalization in cartography, the following quotations will serve as a guide.

Robinson (1960) writes about the difficulty of

setting down a consistent set of rules which will pre­

scribe what should be done in each particular case. He

goes on to state that "it seems likely that cartographic generalization will remain forever an essentially creative process. . . . It is helpful to distinguish in this

creative process between what may be called intellectual

generalization as opposed to visual generalization." He points out, however, that in practice these may overlap.

Robinson defines intellectual generalization as "that part of the process that involves the selection and portrayal of map items in the manner which satisfies the purpose of the map," (Robinson's emphasis) and in visual

generalization "the cartographer is concerned with the

visual effect of the character of the line on the view­

er. ..." Moreover, Robinson points out that a mere

smoothing out of lines is not enough. "Basic shapes and

outlines must be related and emphasized in their sim­

plicity. . . . Good generalization requires many qualities

on the part of the cartographer, chief among which are a

thorough knowledge of the subject matter, a clear under­

standing of the purpose of his map and essential intel­

lectual honesty. The latter is particularly important. . ."

Robinson and Sale (1969) assert that "cartographic

generalization is born of the necessity to communicate"

and they define the "elements of cartographic generaliza­

tion" as simplification, symbolization, classification

and induction, while the "controls of cartographic general­

ization" are given as the objective, the scale, the graphic

limits and the quality of the data.

Morrison (1974) prefers to call the "elements" mentioned above as "processes," and this is probably a better word as it evokes a series of considerations/

actions which eventually bring about the desired result,

namely, generalization.

Steward (1974) states that "cartographic generaliza­

tion may be identified as an interpretive process, executed by individuals, and characterized by subjective varia­

tion. ..." He also points out that

. . . in recent years, the growth and poten­ tial of cartographic automation has sharpened the focus of this search for a demand for more explicit formulations. Computer metho­ dology requires that objective instructions for changes-with-scale be given in discrete form. This, in turn, means that a clear understanding of the /reasons for such instructions should accompany the purely technical manipula­ tions. . . . First there is a need to com­ pare suitable algorithmic solutions for the multivariate map array. Secondly, to relate these algorithms to the psychophysical con­ straints of the human perceptual process; thirdly to make the theoretical possibilities operational by applying them to the mappable data of the 'perceived' world; and lastly, to place the whole in the context of technical economic and logistic possibilities.

Angwin (1959) stated that

Cartographic generalization is often re­ garded as a matter for the ad hoc exercise of individual skills and experience and there is no doubt that it is not an exact science and cannot be according to rigid rules. But, in a large organisation, if individual cartographers are given freedom to generalize without any form of guidance, there will always be failure in represen­ tation and often even absolute misrepre­ sentation .

The usual procedure in cartographic generalization

is to resolve the total system to be generalized, into partial systems. That is, one generalizes small landscape units (or feature complexes) while taking into account that they belong to larger landscape units of a certain character. This means, for example, that a river system can be generalized first. It also means that it is im­ portant to be able to apply generalization procedures uni­

formly within a map sheet or a map series— that is, things of the same kind must be generalized in the same way. These ideas are especially important in the genera­ lization of maps in a series where some of the maps may be produced by different agencies.

The above concepts will also lead to an important problem found in generalization— that is, the degree of generalization should be the same for all the feature complexes making up the whole map and it is not always an easy task to achieve this.

Topfer (1974) has said that:

Every conscientious generalization, and most of all a generalization by scientific principles, requires coordination of the various generalization procedures for in­ dividual map elements. All the procedures have as their goal the optimum reproduc­ tion of the situation and of the geo­ graphical milieu.

The attainment of this goal is made complicated by many factors. For example, when considering the signifi­ cance of a particular feature one would have to consider its topographic significance, its geographic-morphological significance and its political-economic significance. And significance or importance is usually a subjective decision. However, striving for the above mentioned goal

is what makes cartographic generalization an exciting

and ever changing field of study.

The above brief introduction to generalization

should begin to give some idea why Robinson and

Petchenik (1976) refer to generalization as "elusive."

So it is within the complexities stated (and

alluded to) above, that this study is undertaken.

1.2 Generalization of Topographic Maps

1.2.1 Why Should Topographic Maps Be Generalized?

An important constraint imposed on this study is

that it is directed towards the generalization of topo­

graphic maps. The intended use of a topographic map is

as a general reference map. Topographic maps provide a general description of a region by showing all the features

and landscape phenomena which are considered to be impor­ tant at the published map scale. Therefore, in the

generalization of topographic maps, the cartographer is not concerned with providing a detailed analysis (as might be done in thematic cartography), but rather in providing, on a general reference map, a basis for an analysis.

The purpose of the original map (or base map) and that of the generalized map (or derived map) are the same but there has been a change in the mapping scale. The purpose of a topographic map is to give a comprehensive description of a particular region and to do this, all the so-called "topographic map elements" (Topfer, 1974) must be shown. These elements include localities (like buildings and populated places), the transportation net­ work, the hydrography, the vegetation and the relief.

In addition, features like boundaries and control points and items like names, the co-ordinate grid and marginal information make up the complete topographic map content.

In this study, we are dealing with one of the above elements only, namely, the relief as represented by con­ tour lines.

Every map represents the feature complexes of the

Earth's surface in a reduced manner. So that as the map scale decreases (that is, the amount of paper surface for representing these feature complexes gets smaller), there is an increase in the density of the map contents at the smaller scales. In addition, there is a limit to the accuity of the human eye. This and the increase in the map contents— with the required decrease in line widths— demands a generalization of the map contents.

Accepting the fact that topographic maps need to be generalized, the objective of this generalization process in topographic maps has been well stated by

Imhof (1965): The objective of generalization [of topographic maps] is the highest possible accuracy in accordance with the map scale, good geometric informative power, good characterisation of the element and forms, the greatest possible similarity to nature in the forms and colors, clarity and good legibility, simplicity and ex- plicitness of the graphical expression and coordination of the different elements.

In other words, a not too easy task!

1.2.2 Generalization of Topographic Map Elements Other Than Relief

In 1.2.1 mention was made of the elements of the topographic map, and considering some of these elements purely as graphical elements we could group them as follows: planimetry; hydrography; vegetation; contours.

Most of planimetry on topographic maps is made up of man-made features like boundaries, cities, towns, roads, railroads, etc. Generalization of planimetric features will concentrate mainly on positional accuracy of the selected number of points, roads, built-up areas, etc. In this case particular attention is also paid to the black-white ratio or, as it is sometimes called,

"the graphic feature density" (Topfer, 1974).

Hydrography is an important element in defining the shape and natural divisions of the earth's surface.

In the generalization of hydrography, positional accuracy is also an important factor. Other factors are the de­

scription of the "true" shape or form of lines and again,

the graphic feature density. Certain unpredictable

factors enter into the generalization of hydrography.

For example, a small stream may be omitted on a small

scale map if it is surrounded by many other streams, but

if a stream of the same size is the only stream in a

large area then it takes on an added significance and

would therefore have to be shown on the derived map.

In the generalization of vegetation, positional

accuracy, especially of isolated vegetation (hedges,

trees, etc.), is an important factor. One also has to pay attention to the shape of the area outline.

All the above elements are usually generalized before the contours (that is, the relief). This means

that whatever the degree of generalization of these ele­ ments has been, the generalization of the contours must harmonise with these previously generalized elements.

Difficulties arise as it is not always possible to de­

termine the degree of generalization in a purely mathe­ matical way (SGK, 1977). 1.2.3 Generalization of Relief

If continua are defined as continuous areal phenomena, then the topographic map has only one continuum, the relief. Based on measured observed values, continua are usually depicted by isarithms. "Isarithm" is a generic term and in the case of relief these isarithms are called "contours." Topfer (1974) has written that:

"Contour lines are supposed to reproduce the terrain relief visible in nature, be measurable in height and plan position and depict the individual terrain forms with optimum clarity."

In thematic cartography one may map many other kinds of continua, for example, temperature or precipi­ tation or even pseudo-continua like population density.

This study deals only with the continuum represented on the topographic map. However, some of the results of this study may apply, in varying degrees, to the continua or pseudo-continua used in thematic cartography.

To illustrate the specific field of this study, consider the following figures. n

Figure 1/1 Figure 1/2

Original Map Generalized Map

Figure 1/1 represents a contour drawing from the original base data. Figure 1/2 represents a generaliza­ tion of' these contours. This could be done in a number of ways, for example, using various line smoothing tech­ niques. Now the contours in Figure 1/1 can be digitized 12 as well as those in Figure 1/2, giving the original digital terrain model and the final digital terrain model.

In this study methods are proposed which enable the cartographer to go from the original digital terrain model directly to the final digital terrain model. In other words, the cartographic generalization is carried out on the original digital terrain model itself. The assumption is also made that the data are equally spaced.

For irregularly spaced data, there are a number of tech­ niques which can be utilized to obtain equally spaced data. (For example, see Davis, 1973, and Morrison, 1971).

For this study we are assuming that the "truth" is the original digital terrain model or the base map, and it is against this "truth" that we will make com­ parisons.

The generalized map is usually at a smaller scale than the original map, but, as is usually done when com­ paring generalizations, the base map and the derived map are drawn at the same scale to make comparisons easier.

This practice will be followed in this study.

If the area depicted in Figure 1/1 was generalized as shown in Figure 1/3 below, then the cartographer may decide that the derived map is unacceptable. Figure 1/3

Over-Generailzed Map

What has happened.in Figure 1/3 is that, besides the quantitative change in the original data, a quali­ tative change has taken place. The original data

(Figure 1/1.) could be described as "mountain range with hilltops," but in Figure 1/3 this quality has been changed to the quality "spur." It is, however, possible that this qualitative change may be acceptable at certain 14 scales. This illustration serves as an introduction to an important aspect of generalization in topographic mapping, namely, the requirement to preserve the character of the terrain, which will be further discussed in

Chapter Three.

Mention was made previously that the degree of generalization of the relief must fit in with the degree of generalization of the other topographic map elements.

To illustrate this point, consider the following hypo­ thetical case;

An area has 300 settlements, 800 km of roads and a river system of 100 km of rivers all appearing on a map of a scale of 1 in X. To generalize this map to a scale of 1 in Y (where Y is larger than X) we somehow arrive at the figures of 100 settlements, 300 km of roads and 80 km of rivers. Then the relief has to be genera­ lized to fit in with the above. Now assume that the same area, at scale 1 in X had 500 settlements, 1800 km of roads and 200 km of rivers. In this case, for generalization at a scale of 1 in Y, we somehow arrive at 100 settlements, 300 km of roads, and 80 km of rivers.

Then the relief would have to be generalized to fit in with the new situation— and this degree of generalization would not be the same as in the previous case. The above illustration shows how difficult it can

be to determine in a purely mathematical way the degree

of generalization which has to be applied to the relief

continuum.

1.3 Purpose of This Study

1.3.1 Computer-Assisted Cartography

Computer-assisted methods have been used in car­

tography for about two decades and the changes wrought

have been immense. As the above title implies, the com­ puter has come to the aid of the cartographer who has

adapted the basic cartographic procedures so as to take

as much advantage as possible of the current computer

technology. Robinson, Sale and Morrison (1978) mention

that "... cartography has only succeeded in computer-

assisting the production of 50% of a map product." This

is based on the fact that lettering on maps has not been

successfully automated and that lettering accounts for

about 40% of map production cost.

In a United Nations report (see United Nations,

1977) the following comment, on a paper entitled: "Instru­ mentation and techniques for modern cartography" submitted by the United States of America, appears: "Though 16 automated map production currently represented only a small percentage of total production, a steady and sub­ stantial trend towards digitization was forseen."

It is clear that whatever the estimate of the percent contribution of computer-assisted cartography is to map production, we are in a period of rapid development leading to an ever greater contribution.

Computer assisted cartography is currently widely used in the construction of a map base (that is, the cal­ culation and plotting of the graticule— or a grid— on a chosen map projection); also in compilation of base maps and derived maps.

1.3.2 Problems of Generalization in Computer-Assisted Cartography

In the previous sections of this chapter, a de­ scription of cartographic generalization has been given.

It is clear that prior to the computer, generalization was a subjective process done by (usually experienced) car­ tographers. As Robinson, Sale and Morrison (1978) put it: "Often an ability to 'feel' the correct generaliza­ tion was obtained by practicing mapmakers, and a good mapmaker's rendition was indisputable."

The advent of the computer and its use in carto­ graphy demanded that a scheme for generalization be com­ pletely defined. This will lead to a more consistent 17 generalization on a map sheet or a map series. In this new world of computer assisted systems, the question arises: "Can cartographic generalization, and relief generalization on topographic maps in particular ever be completely automated?" Robinson, Sale and Morrison

(1978) make the point that . . it is desirable to have available manual overrides for all computer-assisted generalization." In other words, in this particular car­ tographic process, computer-assisted generalization, the cartographer should be able to intervene when problems arise or when decisions can be made more effectively by such subjective intervention. A similar point has been made by Gottschalk (1978) who said that

. . . it is impossible to define a com­ puter program . . . able to do all gen­ eralization work. It is understood that the program does the routine work of the cartographer, whereas the cartographer corrects the results of the work of the program by means of interactive editing facilities.

One should bear in mind that cartography is a communication system and that a map is a means of com­ munication. In discussing the impact of computer-assisted methods on the generalization process, Robinson, Sale and

Morrison (1978) make the interesting observation that:

"sometimes in cartography inconsistency is a virtue, and now the cartographer must plan desired inconsistencies. and that in an ideal situation

. . . a cartographer should be able to watch the production of each segment so that as problems arise it is possible to intercede, or if a chance to be sub­ jectively inconsistent will add to the map's communicative effectiveness, the production can be interrupted and the desired effect created.

For this study we are assuming that the subjective intervention of the cartographer is needed in some stage of the cartographic generalization of relief on topographic maps. One of the aims of this study is to find some quantitative measures (which the cartographer can use with any subjective intervention) which will assist in the making of decisions.

1.3.3 Needed— A Flexible Method

The purpose of this study is to arrive at a pro­ posal for an efficient and flexible method for the carto­ graphic generalization of relief on topographic maps, whereby computer-assisted techniques will help the car­ tographer in making decisions.

The method should contain the elements needed for determining (with the subjective intervention of the car­ tographer, if required) the appropriate degree of genera­ lization of the relief continuum to fit in with the 19 previously generalized topographic map element. This method should be able to be used by an experienced car­ tographer to determine the degree of generalization empir­

ically for a map or map series and the method should also be able to be used by an inexperienced cartographer in carrying out previously determined policies.

1.4 Hypothesis and Subsidiary Research Questions

1.4.1 The Least Squares Collocation Algorithm

The method of least squares collocation (which is described in Chapter Four) has been used in geodesy for, among other purposes, the prediction of values at points where observations are not available. For this study, the hypothesis is that the least squares collocation al­ gorithm can be used for the efficient cartographic gen­ eralization of relief in topographic maps, with the choice of the covariance function as the point of subjective intervention. The covariance function, which in a sense describes the "character" of the original terrain, could perhaps be used to control the character of the generalized terrain and hence the degree of generalization.

That is, in the application being considered here, one would be "predicting" the generalized values at points where observations (that is, elevations in the original 20

digital terrain model) on a regular grid are available.

If this can be done, the efficiency of the method in the

topographic mapping environment will also have to be

considered.

Although the least squares collocation algorithm

can be used as a "local operation"— that is, assuming

that the covariance function applies over very limited distances only, it is usually used as a "global operation."

1.4.2 Subsidiary Research Questions

Parallel with the testing of the above hypothesis, this study will be concerned with answering the following questions on the process of the cartographic generalization of the relief continuum on topographic maps:

A. How can the degree of generalization

be best quantified?

B. What quantitative measures are most suitable

for describing the changes in the character

of the terrain from the original to the

generalized map?

C. How can a covariance type function be effi­

ciently used as the basis for the design of

a digital filter? This last question is considered as important as any in this study, because we will note in Section 2.1.2 that most methods of "automated" smoothing to date use a digital filter— that is, a limited local effect rather than broad regional or global smoothing as mentioned in

Section 1.4.1 above.

1.5 Assumptions and Scope of This Study

The initial assumption made is that the relief continuum to be generalized has been digitized on a regular grid. That is, this original digital terrain model has elevations on a regular grid and the generalized digital terrain model will consist of the same number of points located at the planimetric coordinates of the original points. Methods of generalization based on de­ creasing the number of elevation points are not being considered in this study.

It is clear that even the original terrain model is a generalization of reality, for this study we are assuming that the relief continuum on the large scale

(or base) map— as defined by the original digital terrain model— is the "truth." That is, the generalized model will always be compared with the original digital terrain model. 22

As mentioned in Section 1.1 above, this study is in the nature of an exploratory probe, and for this reason the experiments being carried out will be restricted to small digital terrain models.

It is further being assumed that all the elements of the topographic map— except for the relief— have been generalized and the generalization of the relief continuum has to "fit" into this situation.

Although the degree of generalization of some of the elements of the map (for example, settlements, rivers) can be relatively easily quantified— this degree cannot easily (if at all) be carried over into the generalization of the relief continuum by a mathematical formula. So for this study we are assuming that the subjective inter­ vention of the cartographer is required. It is possible that a stage may be reached where the subjective inter­ vention will not be needed, everything will then be com­ pletely automated— but the assumption made in this study is that this stage (age?) has not arrived yet and the subjective intervention of the cartographer is needed.

It is possible that eventually complete automation of generalization of the relief continuum may turn out to be expedient and acceptable although not altogether correct.

The above concepts are included in the following scenario set for this study: 23

The cartographer has available a digital terrain model with equally spaced data and the production of a generalized contour map of all or part of the digital terrain model is required at a smaller scale. In other words, morphological features as expressed by contour lines are being considered in this study. This contour map will be generalized so as to "fit in" with the other elements of the topographic map which have been previously generalized. The cartographer will be using a method which will have the flexibility to vary the degree of generalization in a controlled, predictable and measurable manner. The quantities to be generalized are the ele­ vations in the regularly spaced original digital terrain model. This original digital terrain model (which may be part of a large data base system) is not destroyed in the generalization process, so that it may be used repeatedly for generalizations of varying degrees. The subjective intervention done by the cartographer may or may not in­ clude a visual analysis and the cartographer will have available some quantitative measures which will assist in the making of any decisions.

In this study, application is made of methods used in other fields. A high degree of sophistication is usually needed in these fields, where accuracy is usually a prime concern. As the final product being considered 24 here is a contour map, it should be pointed out that on account of the change in the contour interval from the original to the derived (or generalized) map, a certain latitude exists in the application of any of the methods used. Iitihof (1965) gives, for example, the "ideal" contour interval for a 1:100,000 map as being 47 m and for a 1:200,000 map as being 75 m. In practice, contour intervals used for these scales may be 50 m and 100 m

(or even 100 m and 200 m), respectively. According to the U.S. National Map Accuracy Standards "not more than

10% of the heights tested shall be in error more than one-half the contour interval" (ACIC, 1971). Therefore, in the example given above, for a final map to be a scale of 1:200,000 with a contour interval of between 100 m and 200 m, "errors" of a few tens of meters are acceptable from a map accuracy standards point of view. It may there­

fore be possible to relax certain accepted rules and re­

strictions which are usually applied to the procedures used in this study.

1.6 Organisation of This Study

In Chapter Two a brief review of related research and methods is given, as well as a summary of the state of the art of the cartographic generalization of the relief continuum on topographic maps. Included is a 25

statement of the contribution which it is intended for

this study to make to cartographic generalization.

Chapter Three deals with aspects of the character

of the terrain. The point will be made that we need to

compare the character of the terrain as defined by the

original digital terrain model with the character defined

by the generalized digital terrain model. Such a compari­

son will enable the cartographer to decide on how much

the character of the terrain has been changed by the

generalization, bearing in mind that a basic principle

of generalization is to "preserve the character." Some

quantitative measures related to the character of the

terrain will be discussed.

In Chapter Four the method of least squares col­

location is given in brief outline as well as the proposed

use of least squares collocation concepts in cartographic

generalization.

In Chapter Five, the hypothesis mentioned in

Section 1.4 is tested.

Chapter Six deals with the design of a generaliza­

tion digital filter based on the Gaussian function.

In Chapter Seven a system for the cartographic

generalization of the relief continuum on topographic maps is proposed and tested on a sample area.

Chapter Eight is a summary of this study and sug­

gestions for additional research are made. CHAPTER TWO

REVIEW OF RELATED RESEARCH & METHODS

2.1 State of the Art of the Cartographical Generalization of the Relief on Topographical Maps

To give this chapter its proper perspective and to tie it in with the previous chapter, we are reminded that "... the aim of generalization [of topographic maps] is to produce a map image which is clearly legible and interpretable from a super abundance of information."

"Therefore the map contents should first be re­ duced to what is necessary and possible and secondly, the most important emphasized and the less important re­ pressed" (SGK, 1977).

This necessarily implies an interaction between a number of processes/operations. These interactions are clearly shown in the following diagram from SGK

(1977). In this diagram the factors influencing generali­

zation are shown in ellipses. Iterative interaction be­ tween steps four and five in Figure 2/1, with the subjec­ tive intervention of the cartographer (including a visual

26 START: Need for a map

Step I: Ring of Meeific require» Aim . atslorchvRiap

forlegsbihnr

Step 2; Conceptual generaiisadon

Step 3: _ Drewingup of ^ b o l spceifipadons . ^ Graphic realisadon QejeaidJincluding colour choics M

Sizcandfona

Step 4: inginto considsranon all theînfluencâag especially selection of thcapprepinte ■ Graphic generalisation obiecatrom the source nurenal and represen* tanon mchin a posiBonai. ssleianee cezrespon» Step 5: Assessment of the provisionally cheated image and mdica- ___ ting or carrying out any ' necessary corrections

END: Fair drawing for reproduction

Figure 2/1*

Flow Diagram For Map Design (SGK, 1977)

Reproduced with permission from "Cartographic General­ ization— Topographic Maps, Swiss Society of Cartography, Cartographic Publication Series No. 2, 1977." PLEASE NOTE:

Dissertation contains small and indistinct p rint. Filmed as received.

UNIVERSITY microfilms. judgment) will, in theory, result in "a good, acceptable map image" (SGK, 1977).

2.1.1 Manual Methods

At the present time major topographic map produc­ tion organisations are performing the type of generaliza­ tion being discussed in this study mainly by purely manual means (Adler, 1978).

The mode of operation is as follows; The base map is photographically reduced to the derived map scale and an experienced cartographer performs the generaliza­ tion of the contour lines on an overlay by the "eye-ball" method.

For the execution of manual generalization, it is enough to give an experienced cartographer only general guidelines and instructions. Instructions, for example, like: "Preserve the landscape character by maintaining the relative density within the whole map series," or

"Retain whatever is most important." As Topfer (1974) has pointed out, one accepts in this situation that,

"... for the smallest shapes, one cartographer will call the left corner more important and preserve its position, while another cartographer will so treat the right corner." 29

In a recent publication on the cartographic

generalization of topographic maps (SGK, 1977) many

beautiful examples are made available to a cartographer

on what should be considered correct representation and what incorrect representation of the generalized, relief

continuum. For example Figure 2/2 shows the correct

representation of an area and Figure 2/3 shows— according

to SGK (1977)— the incorrect representation.

Figure 2/2* Figure 2/3* Correct Representation Incorrect Representation of an Area of an Area

Reproduced with permission from "Cartographic Generalization— Topographic Maps,-Swiss Society of Cartog­ raphy, Cartographic Publication Series No. 2, 1977."

The comment given in SGK (1977) on these figures is: "A

small vertical interval [Figure 2/2] allows more land

forms to be shown than a larger vertical interval. When the vertical interval is larger, [Figure 2/3] tiny details appear unconnected with each other." 30

Also from the same publication (SGK, 1977), the

following three figures demonstrate clearly the concept

of "the change in the character of the terrain."

Figures 2/2 through 2/6 indicate the sort of knowledge which an experienced cartographer is expected

to bring to bear on a task of manual generalization.

How does one build this experience or knowledge

into an algorithm for completely automating generalization?

Topfer (1974) has stated that: "In automatic generaliza­ tion, the necessary generalization procedures must be un­ ambiguously specified and executed on the basis of suitable criteria." For the generalization of map elements other than relief this may be attainable. But generalization principles like "preserve the character of the terrain" or "fit the relief generalization in with the degree of generalization of the other elements" are not easy to define for computer-assisted methods and it seems that the subjective intervention of the cartographer will be needed for quite some time. Figure 2/4*

Emphasis on Planes or Ridge Lines with 10 m Vertical Interval

Figure 2/5*

Generalized Terrain With 20 m Vertical Interval Figure 2/6* Rounding Results in Loss of Original Character

*Reproduced with permission from "Cartographic Gener­ alization-Topographic Maps, Swiss Society of Cartography, Cartographic Publication Series No. 2, 1977." 2.1.2 Automated Methods

In this section/ an overview is given of automated methods which have been or can be applied to generaliza­ tion. Most of these methods are discussed in detail in

Section 6.4.

Topfer (1974) dealt with various one-dimensional filters for line smoothing. He used a moving average and gave examples at scales of 1:50,000, 1:100,000 and 1:200,000.

He stated that "... this method yields usable results which resemble the manual generalization of experienced cartographers..," For the generalization or smoothing of relief, he found that "a moving average of nines with constant weight appears especially suitable."

It does appear, that for topographic relief mapping the most used method of generalization is that which uses the moving average— see also Gottschalk (1978).

A number of other methods suitable for computer- assisted generalization (which have not necessarily been used for the relief continuum on topographic maps) appear in the literature and they will be considered below.

For smoothing line elements, Holloway (1958) sug­ gested the following method. The smoothing function weights are made proportional to the ordinates of the normal probability curve and he proposed the following ”... continuous analytic form of this smoothing func­

tion. . .

W(d) = {2-no) exp(-d^/2a^) (2-1)

where d is the distance

0 is "the normal curve dispersion parameter"

Further, Holloway (1958) stated that "... the

filtering interval is taken to 6a, for beyond 3a from the origin the normal curve ordinates have negligible value."

Tobler (1966) discussed a nines filter (see Chapter

Six for details on Nines filters) and the weights of this

filter were determined by a binomial weighting function.

This filter will be discussed in Section 6.4.

Hardy (1972 and subsequent papers) proposed the method of multiquadric surfaces to represent, among other things, the topography. These multiquadric surfaces could be used for generalization by decreasing the number

of points defining the original topography. This was done by Gottschalk (1972),

Gottschalk (1972) generalized the isarithms of a

surface by determining a subset of points derived from the original surface. The subset of points, being less

in number than the original data point, was obtained by 34 a method for reducing the original information content.

The final surface, based on a lesser number of points than the original surface was represented by Hardy's multiquadric equations.

Stegena (1973) used a low pass filter for smoothing

"linear elements (or curves) of topography. . ." and

”... any thematic element (shorelines, contour lines, roads, isolines)." The weighting function used was

V(d) = -- exp(-dV4K2) (2-2)

where K is ". . . a value characteristic of the band passed by the filter." This filter will be further dis­ cussed in Section 6.4.

Robinson, Sale and Morrison (1978) give examples of surface smoothing using a constant weight moving average nines filter and another, apparently non-isotropic, filter. This will be discussed in Section 6.4.

Allam (1978) discusses a method of moving averages for smoothing digital terrain models obtained from the

Gestalt Photomapper GPM-2/3. The problems to be en­ countered in practice are well brought out— that is, one digital terrain model consists of about one million ele­ vations and one 1:50,000 topographic map consists of 25 models, giving a total of 25 million elevation points per map. This method of smoothing will be discussed in

Section 6.4.

The desire to completely automate cartographic generalization can be understood. Topfer (1974) has stated that:

The use of numerical criteria makes it possible in all important problems to state clearly what is correctly general­ ized and what incorrectly. Thus general­ ization has become testable. This kind of generalization can be learned and is not reserved to particularly talented cartographers. . . . The goal of gen­ eralization by rule is the regular re­ production of objective reality on the map, i.e., the optimum, clear, and ef­ fective reproduction of the details and character of the landscape units.

It is possible, however, that these ideals may still take some time to be achieved, especially in the cartographic generalization of the relief continuum.

2.2 Summary and How This Study Will Make a Contribution

The state of the art of the cartographic generali­ zation of the relief continuum on topographic maps could be summarized as follows:

Mostly manual methods are used with a few com­ puter-assisted (including interactive) methods. It also seems clear that topographic cartography is in a period of change— a change towards the use of more computer-

assisted methods.

Hoinkes (1973), in describing the Swiss concept of an automated cartography system, said:

. . . we do not expect the day when a whole topographic map can be reasonably generalized automatically in the near future. Until eventually it comes (if ever) we rather plan to develop our system towards more efficient man- machine communication.

It would be inadvisable, at the present time, to take a digital terrain model obtained, say, from a

1:20,000 topographic map and automatically (that is, using some automated procedure) produce contours for a

1:200,000 topographic map without the cartographer study­ ing the original contour configuration and analysing the final contour configuration. The subjective intervention of the cartographer is still a factor in the generaliza­ tion process.

Mention was made in Chapter One of the fact that generalization is a creative process. And this fact has not been ignored by Hoinkes (1977). He discussed the setting up of "a complete system for digitizing, editing and automatic drafting of cartographic data," and he pointed out that: Interactive— rather than automatic— processing of the data entered into the system is absolutely necessary if the system is to support a creative process such as the design of a map including generalization.

In an overview article on some developments in

automation in cartography, van Zuijlen (1978) mentions

that with the digital information of the topographic

elements one can, with the help of the computer, carry

out most of the processes involved in generalization. But

he points out that "such an objective method of generali­

zation has the danger that strong characteristic points

of elements do not show up in the generalized image." He

also points out that in some countries, like West Germany,

. . a strong objective generalization [is preferred],

rather than a subjective [generalization] as is still

performed manually by a cartographer." On the practical

side, van Zuijlen (1978) reminds us that "for generaliza­

tion in an automated manner it is essential to have avail­

able very large computers to be able to store the huge

quantities of data, which will always have to be handled

in cartography, ..." He also makes the interesting

remark that "... the production of maps by automated means does not have any financial advantage. The present

experience is that it is about as costly as conventional map production methods." However, it should be pointed 38

out that, with the cartographic data base, very much more than only topographic mapping can be achieved and

that these data bases will pay dividends at a later stage when maps are updated or changed in some way or another.

This study will make a contribution to the field of the cartographic generalization of the relief con­ tinuum on topographic maps by proposing a system (in

Chapter Seven) which will enable the cartographer to control and analyse (with the aid of the computer) the achievement of efficient cartographic generalization.

The cartographer will be able to choose a filter to give the required degree of generalization which in many cases would be done by an empirical process. Although this

study is directed towards the generalization of a digital terrain model on a regular grid, indications will be given for dealing with data arranged differently. Some quantitative measures will also be available to the cartographer for the comparison of the generalized map with the original map (or for comparing representative parts of each map).

Another contribution of this study will be the gathering together of literature from various fields which in some way can be related to cartographic generalization of the relief continuum. This information will be divided into references cited in this study and references read but not cited. CHAPTER THREE

THE CHARACTER OF THE TERRAIN

3.1 Introduction

In a discussion on the preservation of landscape

character in generalization, Topfer (1972) writes that:

Most publications on generalization contain the requirement to preserve or maintain the character of the landscape, to pay atten­ tion to the properties and peculiarities of the geographical milieu, to maintain the geographical characteristics, etc. All these requirements have the same aim. Map layout and generalization must main­ tain a course that goes beyond the stage of pure fact recording and stresses the map's effectiveness in expressing the landscapes. Nothing but a correct, effective expression of the character of the landscape unit provides a true concept of reality.

It is clear that the sort of map generalization which

depicts all the details in an area of few details, and

only selected details in an area of dense details, would be falsifying the landscape character to the extent that

the generalized map may show a homogeneous area when in

fact it is not so. To maintain the landscape character,

the degree of generalization should be the same in all 40 parts of the area. In other words, one tries to maintain the relative density of the map elements within a parti­ cular map or map series.

Again we quote Topfer (1974) who stated that one should pay attention to "the correctness of the feature complexes and landscape units, especially as regards their characteristic attributes," and that "... what matters primarily is to plot just enough small features to pro­ vide a real expression for the character of each land­ scape unit."

Statements such as the above give some idea of the difficulties involved in trying to precisely define

"the character of the terrain."

However, in this study we are not so much con­ cerned with describing the actual character of the terrain as we are in finding out how the character of the terrain has changed from the original terrain model (which is being regarded as the "truth") to the generalized model.

We will therefore not attempt in this study to define precisely "character of the terrain" but we will rather transfer the X, Y and Z co-ordinates of the digital terrain model to the frequency domain and then compare frequencies (or functions of frequencies) in order to ascertain the change in the character of the terrain. 41

The method used will be to compare the power (or variance) spectra of the original and the generalized

models. Transferring the data into the frequency domain simply means that we are rearranging the data according to frequency instead of according to space sequence, that is, location.

In an attempt to translate "character of the terrain" into a more concise mathematical type description, we could say that we are looking for some sort of multi­ variate geometric signature of the terrain. By going into the frequency domain, it seems that we can recover a large part of this signature or, at least, we can recover enough of the signature for the purpose of this study— that is, comparing the original terrain with the generalized terrain.

3.2 Some Terrain Descriptions

In describing terrain, we are in a sense analysing features or giving attributes to various feature types.

These attributes could be divided into qualitative attributes, quantitative attributes, the feature dis­ tribution and the feature significance (see Topfer, 1974).

When describing the qualitative attributes, the particular landform is described, for example, coastal plain, plateau, mountain, valley and so on. In this case, form is the basis for classifying the type of

feature.

The quantitative attributes describe the feature

size and configuration. Some examples of quantitative attributes are: maximum elevation, minimum elevation, average elevation, maximum difference in elevation, slope or gradient, average slope (in a certain direction).

Slopes or gradients can also be described in a qualitative way, for example, steep, gentle, intermediate, convex, concave, uniform, etc.

A description of the feature distribution would give information on the location(s) of the feature within the terrain being considered.

The above attributes, and other considerations, would contribute to a description or determination of the feature significance. In the context of generaliza­ tion, a decision would have to be made as to the impor­ tance of a feature. Importance is a subjective quality and this is another factor making the complete automation of cartographic generalization a difficult matter.

A partial description of certain aspects of the terrain may also be given by an autocorrelation function which will be discussed in Section 3.6.

The first three attributes mentioned above— quali­ tative, quantitative and feature distribution— could perhaps be described by a double Fourier series. The double Fourier series is probably the best way of de­ scribing the physical characteristics of terrain and this will be discussed in Section 3.4. A topographical map, being a general purpose map, is essentially a visual statement on the area being depicted. Wingert (1974) has suggested that a frequency parameter may be connected with the visual response to details on a map and this is what is achieved with the double Fourier series— the locational information (that is, the digital terrain model) is transformed into the frequency domain.

Pike and Rozema (1975) point out that "inasmuch as a single 'magic number' such as average slope or rela­ tive relief cannot express the shape of the land ade­ quately for most taxonomic purposes, terrain signatures require several parameters."

To use the double Fourier series to describe or classify terrain is a study on its own and will not be attempted here. In this study, the interest lies more in trying to get a quantitative estimate of the change in the character of the terrain as mentioned in Section

3.1 above. 3.3 Description of DTM and Con­ touring Program Used in This Study

The original digital terrain model was taken from

Tobler (1966). The model used in this study is an 18 x 18 regular grid model. The grid interval is 0.5 miles and elevations, given in Table 3.1, are in feet. In this study, all the initial investigations have made use of this relatively small model of about 8 1/2 miles by

8 1/2 miles. The usual practice in topographic mapping is to generalize relatively small areas at a time.

Topfer (1974) calls these areas "Landschaftseinheiten." or "landscape units." Although the practice is to generalize these landscape units separately, the carto­ grapher takes into account the fact that they belong to a larger region and the generalization process, therefore, also takes into account the character of the whole region.

The model mentioned above is shown in Figure 3/1 with contours plotted at 100 foot intervals using the contour program "of Snowden (1969). Snowden's program uses a weighted moving average for the contour interpo­ lation with the user specifying the center point weight.

(The weights of the other two points are equal to unity.)

Figure 3/1 was drawn with the center point weight = 1.0 and Figure 3/2 was drawn with the center point weight = 3.0.

There is no difference between the two figures, probably due to the relatively large grid spacing. Center §§SsliS5g§§Siëggii

iiSsgiêssggssgëSgi giigisHëssgÊsiigS î

•° s IggïiiiggëgggigSsi

gggggggggggggSgggg

iuësiHSsssgsgissg Ggggssgg'iggggsggsg

ssasSssgsi'isiHsgg ggggigsSggs'ÏÏilfiss il sgsiSggISsgiïilëiS

ggiasilSSgggisïisg

lëgsêsSgggëgssïHg

gigggisiiiSsggriïg

ii“g|lisa§|gg§igrf

sgIëssisasëSssgiri

gs§alSggiggg3gsggl iggssslsiggggggsgs “Isggigsgissgggsss 4G ORIGINAL DATA

m

ftppanx SCALE i't t^aaa CONTOUa IMTEaVAL = 100 FEET 1.0

Figure 3/1

The Original Digital Terrain Model.as a . Contoured Map

Center Point Weight =1.0 ORIGINAL DATA

G§P f

RPPROX SCALE 1 : 125000 CONTOUR INTERVAL = 100 FEET 3.0

Figure 3/2

The Original Digital Terrain . Model as a Contoured Map

Center Point Weight = 3.0

Approx. scale 1:250,000 48 point weights from 1.0 to 10.0 were used and no differences were detected for this particular digital terrain model.

All the remaining contoured figures in this study use a center point weight =3.0.

Both Figures 3/1 and 3/2 show the locations of the 324 points of the model— marked with crosses. In addition. Figure 3/2 shows an inset of the same model at half the scale without the crosses.

3.4 Double Fourier (Trigonometric) Series

The following overview is based on Bath (1974),

Davis (1973), Esler and Preston (1967), Harbaugh and

Sackin (1968), James (1966) and the reader should consult these references for more details than will be given below.

The Fourier theorem deals with a function which can be expressed as a sum of an infinite number of sinu­ soidal terms. This function should satisfy certain con­ ditions (the Dirichlet conditions) which need not be discussed here as ". . . all known functions encountered in the physical world fulfill the more restrictive

Dirichlet conditions" (Bath, 1974).

Any complex surface— for example, the relief con­ tinuum— can be thought of as the sum of two interacting groups of two-dimensional sinusoidal wave forms and we assume that Z, the dependent variable, can be represented by a double Fourier series as follows:

Z = F(X,Y)

= - L ¥

+ =mn ¥

+ dmn ¥ > <3-l>

Z = the dependent variable— in this case,

the elevation

F(X,Y) = the Fourier approximation at grid point X,Y

m = harmonic number in the X direction

n = harmonic number in the Y direction

a,b,c,d = Fourier coefficients of degree m and n

M = specified maximum degree of terms in the

X direction

N = specified maximum degree of terms in the

Y direction

L = half of sampling length in X direction H = half of sampling length in Y direction

The coefficients of any harmonic can be found as follows;

M mTTX. nïïY. Z — T T - i=l j=l

K ^ r niTTX. nïïY. ^mn = m 'ii —

^ ^ mirX. m r Y . K "mn MN .1^ ^ij L " H

M N mïïX. mrY. K Z. . sin “rtm - MN ^ ^ ^ij T - T ~ (3-2) i=l j=l

where :

K = 1 if m = 0 and n = 0

K = 2 if m = 0 or n = 0

K = 4 if m > 0 and n > 0 51

The two-dimensional variance spectrum is given by

(3-3)

This spectrum measures the variance contributed by the

m^^ and n^^ harmonics to the total variance of and

S" percentage contribution = • 100% (3-4)

where = total variance of

As the elevations of the original digital terrain model are given, S| can be calculated from the usual for­ mula for data variance.

We should note that in the double Fourier series when m = 0 and n = 0 then

^ M N ^OO = MN ^ : ^ij i=l j=l

= mean terrain elevation (3-5)

The procedure adopted in this study is to determine

the change in the "character of the terrain" from the 52 original to the generalized terrain model without attempting a precise definition of this "character." This is done by comparing the variance spectrum of the original data set with the variance spectrum of the generalized data set as will be explained later.

An important point should be noted here. In the usual Fourier analysis one first removes the trend before making an analysis. (Trend is that part of the data which varies smoothly and systematically and behaves in a predictable manner.) Moreover, in this study we are not using the variance spectrum for landform analysis.

That is, we are not taking the landform as represented by the digital terrain model and analysing by investigating the elements of the topographic geometry. We are simply comparing a "before" and "after" situation and the most appropriate method seems to be a ratio comparison of the variance spectra. No attempt has been made in this study to describe exactly what there is in terms of the topo­ graphy but rather what has changed.

3.5 Harmonic Surfaces

The two dimensional variance spectrum given by the sum of the squares of the Fourier coefficients— see equation

(3-3)— can be written in tabular form as shown in Figure 3/3.

(See James, 1966, and Davis, 1973.) Figure 3/3

The Two Dimensional Power Spectrum

(Showing Only the First Five m & n Harmonics)

The harmonics in area A give the mean plane.

The harmonics in area B give wavelength 1.

The harmonics in area C give wavelength 2.

The harmonics in area D give wavelength 3.

The harmonics in area E give wavelength 4.

And further

A + B gives the 1st harmonic surface

A + B + C gives the 2nd harmonic surface

A + B + C + D gives the 3rd harmonic surface

A + B + C + D + I5 gives the 4th harmonic surface 54

In spectral analysis one usually smooths the raw

spectrum obtained from equation (3-3)— see Bath (1974),

Davis (1973), Rayner (1971)— but, as previously explained, we are investigating a "before-and-after" situation and the raw spectrum was therefore used in this study.

The spectral analysis done in this study is based on a program given in Davis (1973) . This program was modified to give the additional elements required for this study.

The concept described above will be used mainly to compute wavelength ratios and harmonic surface ratios between the original terrain and the generalized terrain.

These ratios may then be used as an indication of how the terrain has been changed by the generalization process.

The following four figures give the first, second, third and fourth harmonic surfaces of the original terrain described in Section 3.3. The contour interval is 100 feet in each case. HARMONIC SURFACES FROM DOUBLE FOURIER SERIES

1 HARMONIC SURFACES

Figure 3/4

First Harmonic Surface HARMONIC SURFACES FROM DOUBLE FOURIER SERIES

600/8 0 0

2 HARMONIC SURFACES

Figure 3/5

Second Harmonic Surface HRRMONIC SURFACES FROM ” DOUBLE FOURIER SERIES

3 HRRMONIC SURFACES

Figure 3/6

Third Harmonic Surface 58 HRRMONIC SURFACES FROM DOUBLE FOURIER SERIES

o

11 HRRMONIC SURFACES

Figure 3/7

Fourth Harmonic Surface 3.6 The Empirical Covariance Function

3.6.1 Theoretical Considerations

A statistical correlation between, say, elevations, is called the autocorrelation between the elevations.

The autocorrelation between these elevations can be characterized by a covariance function. Assuming isotropy and homogeneity, then this covariance function will be a function of the plane distance between elevation points.

If the mean of the elevations is constant and if the auto­ correlation does not depend on absolute location, then the elevations field is called homogeneous. If the auto­ correlations do not depend on direction (that is, they depend only on distance irrespective of the direction) then the elevations field is also isotropic. In this study, homogeneity and isotropy are assumed for the digital terrain models.

Let Z* refer to the difference of the elevations of points from the mean elevation of the digital terrain model, so that E[Z*] = 0.

Using the term "heights" for Z* values, we have that the average product of pairs of heights separated by a constant distance, d, is called the covariance of the heights for this particular distance.

Cov^CZ*) = E[Z*, Z*] (3-6) This covariance gives the statistical correlation of the heights, that is, their tendency to have more or less the same value.

Considering the covariance as a function of the distance, d, we get the covariance function

C(d) = Cov^(Z*) = EIZŸ, Z*].j=a (3-7)

and when d = 0, then

C(0) = E[Z*2] (3-8)

Therefore, the covariance when d = 0 is the variance of the elevations.

The covariance function will usually be of the form shown in Figure 3/8.

C(d)

C(0)

Figure 3/8

The Covariance Function 61

So that for small distances we will have high cor­

relation and for large distances, we have small correlation.

And for a realistic covariance function

C(0) > |C(d) I for d 5^ 0

Moritz (1976) has stated that stationary and iso­

tropic covariance functions analytic in a plane can be

expressed as a power series, namely

C(d) = - bjd^ + bgd^ - b^d® + . . .

Z (-1)^ b d^^ (3-9) k=0 ^

Because of symmetry there are only even powers of d and

convergence is obtained "for sufficiently small" d

(Moritz, 1976).

Moritz (1976) shows that covariance matrices should be positive definite. He has also shown that one can set

up a basic general condition for the positive definiteness which the coefficients of the series in equation (3-9)

must satisfy, namely,

> (2n + 1) (2n + 2)b;k 2 n+1 (3-10) (2n + 3)(2n + 4)b 62

Therefore in order to test a covariance function for posi­

tive definiteness, we express it as a series expansion

and test it against the condition above, that is, equation

(3-10) .

For example, the Gaussian function, used as a

covariance function, can be expressed in the following form:

C(d) = (3-11)

Simplifying by putting a = 1 we get

= : - TT + § r + 3 T (3-12)

where ~ (see equation 3-9) .

Then equation (3-10), the general condition for positive

definiteness, then becomes

(2n + 1) (2n + 2) Çn -f 1) J (2n + 3) (2n + 4) which can be simplified to

, > 2n + 1 (3-13) 63 which is satisfied by all n, and therefore this particular

covariance function will give positive definite covariance matrices.

3.6.2 The Empirical Covariance Function

Empirical covariances can be calculated using the following equation.

= ° '3-14)

where d = distance, say, grid units in a digital

terrain model

N = number of points

R = normalized residuals

Figure 3/9 shows the empirical covariance function obtained from the original data (given in Section 3.3) using normalized residuals with the mean plane as the reference surface. The abscissa in Figure 3/9 is in grid units, each grid unit being 0.5 miles.

The covariances were calculated using distances in north-south and east-west directions and Table 3.2 shows the number of distances used in Figure 3/9.

The empirical covariance function shown in Figure 3/9 is based on residuals from the mean plane. But instead of ÜJ U Z cc 0= Œ > o u

GRID DISTANCE

Figure 3/9

Covariance Function, Original Data With Mean Plane as Reference Surface TABLE 3.2

NUMBER OF GRID DISTANCES USED IN FIGURE 3/9

d Number of Distances

1 612 2 576 3 540 4 504 5 468 6 432 7 396 8 360 9 324 10 288 11 252 12 216 13 180 14 144 15 108 16 72 17 36 66 the mean plane, various other reference surfaces could be used and these will be considered in the next section.

3.6.3 Reference Surfaces

In addition to the mean plane as a reference surface, one could use a number of polynomial surfaces.

A number of investigations were carried out with various polynomial surface fits and these will be dis­ cussed below.

A 16-term bicubic polynomial surface can be de­ fined as follows (Schut, 1976):

^ " =00 + ^01^ + =02^^ + ^03^^ + =10^ +

+ + a^^^x^y

+ a^gX^y^ + + =30==^ +

+ ajjX^^ + aj j X ^ ^ (3-15)

To obtain a 12-term bicubic polynomial surface make

=22 = =23 “ =32 = =33 “ ° '3-16) in equation (3-15). 67

For an 8-term biquadratic polynomial surface, in addition to equation (3-16), make

(3-17) in equation (3-15).

And, finally, for a 4-term bilinear polynomial surface use

^ = =00 + =01^ + =10== + =11^ <3-18)

Each of the above polynomials can be used to obtain a least squares solution for a particular digital terrain model or for a part of the digital terrain model. The polynomial coefficents could be calculated using the least squares solution with equal weights, that is,

(3-19) where P is the matrix of the polynomial coefficient;

A is the design matrix obtained by a partial

differentiation of the polynomial with re­

spect to the polynomial coefficient; and

Z is the matrix of the elevations of the digital

terrain model.

The polynomial coefficients obtained in P are then used to generate a surface. Figure 3/10 shows such a LEAST SQUARES POLTMOMIAL SURFACE

ORIGIMAL DATA ll-TERH BILINEAR PGLTNGMIfiL % GOODNESS FIT = 23.1

Figure 3/10

4-Term Bilinear -Polynomial Surface Fitted to Original Data 69

surface using the original digital terrain model (as

described in Section 3.3) and a 4-term bilinear polynomial.

The percentage goodness of fit indicated on the

figure was calculated from equation (3-20) which is based

on a formula given by Fox (1975) for the goodness of fit

of a smoothed curve to the original curve

EXJ - (EX )Vn Percentage goodness of fit = ------(3-20) EX^ - (EX ) V n

where X^ = values on trend surface (that is, the fitted

polynomial surface) at location of data

points

Xq = observed (or original) data values

n = number of data values

E indicates the summation

It can be seen that equation (3-20) is simply the percent ratio of the trend variance to the variance of the original data.

Figure 3/11 shows the 8-term biquadratic poly­ nomial fitted to the original surface.

Figure 3/12 shows a 12-term bicubic polynomial

fitted to the original data and Figure 3/13 a 16-term bicubic polynomial LEAST SQUARES POLYNOMIAL SURFACE

ORIGINAL DATA 8-TERH BIQiJSORATIC POLYNOMIAL % GOODNESS FIT = 31.6

Figure 3/11

8-Term Biquadratic Polynomial Surface Fitted to Original Data LEAST SQUARES POLYNOMIAL SURFACE

ORIGINAL DATA 12-TERH BICUBIC POLYNOMIAL % GOODNESS FIT = 11.0

Figure 3/12

12-Term Bicubic Polynomial Surface Fitted to Original Data LEAST SQUARES POLYNOMIAL SURFACE

ORIGINAL DATA 16-TERM BICUBIC FOLTMOüIAL % GOODNESS FIT = U9.9

Figure 3/13

16-Term Bicubic Polynomial Surface Fitted to Original Data Figure 3/14 shows the residuals between the 16- term bicubic polynomial surface and the original surface and exhibits a systematic trend.

Figure 3/15 shows the 16-term bicubic polynomial surface obtained from the least squares fit of the three times binomially smoothed data (see Sections 2.1.2 and

6.4). The residuals shown in Figure 3/16 show a slightly less systematic trend present in this case, indicating a better fit of these two surfaces than was obtained pre­ viously with the original data.

One would probably get a smaller systematic trend by fitting the polynomial surface to small sections (or patches) of the original model RESIDUAL MRP fl

ORIGINAL DATA 16-TERM BICUBIC PGLYNGMIAL % GOODNESS FIT = Lg.g

Figure 3/14

Residual Map LEAST SQUARES POLYNOMIAL SURFACE

BINOMIAL HT 3 16-TEAM BICUBIC POLYNOMIAL % GOODNESS FIT = 81.9

Figure 3/15

16-Term Bicubic Polynomial Fitted to Smoothed Data RESÎDLjRL MRP

BINOMIAL NT 3 16-TERM BICUBIC POLYNOMIAL % GOODNESS FIT = 81.S

Figure 3/16

Residual Map 3.6.4 Some Results

The following figures show the results of using various reference surfaces with the original and with smoothed data to get the empirical covariance function using equation (3-14). The original data were smoothed using a binomially weighted moving average— see Sections

2.1.2 and 6.4 for details.

In Figure 3/9 we notice that the covariance function crosses the zero covariance line at about 4.2 grid units when the mean plane was used as a reference surface.

Figure 3/17 shows the covariance function for the same data (that is, the original digital terrain model) but here the reference surface is the 4-term bilinear poly­ nomial surface and in this case C(d) = 0 for d equal to approximately 2.8 grid units. In Figure 3/18, where the

16-term bicubic polynomial surface is the reference surface, C(d) = 0 at about 1,8 grid units. These figures clearly illustrate how the empirical covariance function depends on the reference surface used.

The following questions arise: How does the co- variance function behave when the terrain is smoothed, and is it feasible to use the empirical covariance func­ tion as an indicator of the amount of smoothing which has taken place? LU CJ Z Œ OC Œ > D O

GRID DISTANCE

Figure 3/17

Covariance Function^ Original Data With 4-Term Bilinear Polynomial Reference Surface LU O z ΠOC OC > O o

15 GRID DISTANCE

Figure 3/18 -

Covariance Function, Original Data With 16-Term Bicubic Polynomial Reference Surface Figures 3/19 through 3/21 show how the empirical

covariance function behaves with the once smoothed data

(C(d)=0 at about 4.9 grid units); the twice smoothed data

(C(d)=0 at about 5.1 grid units); and the three times

smoothed data (C(d)=0 at about 5.7 grid units).

Figures 3/19 through 3/21 show that as the original

data are smoothed, the resulting terrain points become more correlated with each other, as one would expect.

These investigations lead to the conclusion that

although the empirical covariance function could be used

as a sort of "signature” of the terrain and an indication of the degree of smoothing, it is not a particularly

sensitive indicator. ÜJ CJ z cc cc cc > D CJ

GRID DISTANCE

Figure 3/19

Covariance Function, Binomially Once Smoothed Data With Mean Plane as Reference Surface LU O Z CE CC CE > O (_)

GRID DISTANCE

Figure 3/20

Covariance Function, Binomially Twice Smoothed Data With Mean Plane as Reference Surface Lü O Z CC co en > D U

GRID DISTANCE

Figure 3/21

Covariance Function, Binomially Three Times Smoothed Data With Mean Plane as Reference Surface Some Quantitative Measures Related to the Change in the Character of the Terrain

Instead of investigating the "degree of generaliza­ tion" it seems to be more appropriate for this study to investigate the "change in the character of the terrain."

There is obviously a relationship between the two con­ cepts. In the cartographic generalization of the relief continuum on topographic maps, one would not blindly generalize to some "degree" but rather fit a particular generalization in to the previously generalized topo­ graphic map elements while investigating the change in the character of the terrain and then deciding if the generalization is suitable. Having decided on the suitability of the generalization one has obviously achieved a certain "degree" of generalization. Whether one can exactly quantify this "degree" is something which will not be attempted in this study.

One of the aims of this study, as mentioned in

Chapter One, is to attempt to find some quantitative measures which could be used, for example, with a visual analysis in order to judge the "quality" of a cartographic generalization of a digital terrain model.

A number of generalizations were therefore performed on the original data in order to explore the above ideas. In each case the contour map of the generalized terrain was obtained as well as two tables with information as described below.

The first table of quantitative measures;

Column 1: Map number

Column 2: Mean elevation of generalized values

Column 3: Data variance (of generalized values)

Column 4: Root Mean Square (R.M.S) of elevations

Column 5: R.M.S of residuals (Residual = original

elevation - generalized elevation)

Column Mean of residuals

Col\amn Spectrum variance as obtained from equation

(3-3), with m and n each taking values from

0 to 9.

All units are based on feet.

The second table of quantitative measures :

Column 1; Map number

Column 2; Wavelength ratios (that is, in relation to the

wavelength of the original data)

Column 3: Harmonic surface ratios (that is, the ratios—

in relation to the original data— of the sum

of the particular harmonics making up the

surface) Column 4: Cumulative percentage contribution of each

harmonic surface to the data variance given

in Column 3 of the first table.

All units are based on feet.

From all the maps produced in this study 37 were selected for presentation. These are noted by numbers from 1 to 65, with the missing numbers relating to maps not selected for presentation.

The first group of eight maps is made up as follows:

Map No. 1: The original data, with 200 feet subtracted

from each elevation (200 feet being an ar­

bitrarily chosen value).

Map No. 2: The original model smoothed once using the

binomially weighted filter described in

Tobler (1966) and Tobler (1970). This filter

will be discussed in Chapter 6.

Map No. 3: Similar to No. 2, but smoothed twice.

Map No. 4: Similar to No. 2, but smoothed three times.

Map No. 6: The first harmonic surface of the original

data.

Map No. 7: The second harmonic surface of the original

data. Map No. 8; The third harmonic surface of the original

data.

Map No. 9; The fourth harmonic surface of the original

data.

These maps are given in Figures 3/22 through 3/29 and the quantitative measures relating to them are given in Tables 3.3 and 3.4.

Figures 3/23, 3/24 and 3/25 show, respectively, once, twice and three times binomially weighted smoothing.

From a visual point of view we could decide that the once smoothed terrain is an acceptable generalization.

On the other hand, the twice and three times smoothed terrain may be considered unacceptable because the character of the original terrain ("mountain range with hill tops") has been qualitatively changed (to "spur").

Disregarding the contour values and looking only at the form of the topography. Map No. 1 (Figure 3/22) could be taken as the original terrain model.

Examination of Tables 3.3 and 3.4, in connection with Maps 2, 3 and 4, indicates that as one smooths or generalizes the terrain then

a) the data variance decreases;

b) the R.M.S of the elevations decreases; Figure 3/22 Figure 3/23

Figure 3/24 Figure 3/25

Generalizations of Original Data

Scale approx. 1 in 250,000 ^ / / f / / / / ^ (7/ I/o { ( ^

6 7 Figure 3/26 Figure 3/27

(iyj Mr, ) / h o

Figure 3/28 Figure 3/29

Generalizations of Original Data

Scale approx. 1 in 250,000 FIRST TABLE OF QUANTITATIVE MEASURES FOR MAPS 1, 2, 3, 4, 6, 7, 8, 9*

Map Data R.M.S. R.M.S. Spectrum No. Elevation Variance Elevation Residuals Residuals Variance

Original 603.9 39775 635.9 35804

1 403.9 39775 450.3 200.0 200.0 35804

2 597.7 26097 619.1 73.4 6.1 22239

3 591.8 21275 609.4 94.5 12.1 17662

4 585.9 18252 601.2 107.6 17.9 14833

603.8 14642 615.8 158.4 0.0 11924

603.8 24067 623.4 125.2 0.0 20663

603.8 29234 627.5 102.6 0.0 25387

603.8 32488 630.1 85.4 0.0 28461

See page 85 for full description. TABLE 3.4

SECOND TABLE OF QUANTITATIVE MEASURES FOR MAPS 1, 2, 3, 4, 6, 7, 8, 9*

Cumulative % Wavelength Ratios Harmonie Surface Ratios Contribution of Map No. Harmonie Surfaces

Original 30 52 64 72

1 1.0 1.0 1.0 1.0 0.511 0.549 0.568 0.579 30 52 64 72

2 0.904 0.732 0.531 0.330 0.971 0.954 0.935 0.920 41 75 79

3 0.820 0.546 0.318 0.171 0.944 0.913 0.888 0.870 46 76 78

4 0.746 0.415 0.215 0.124 0.919 0.880 0.853 0.834 49 69 74 76

6 1.0 1.0 0.922 0.884 0.862 81 81 81 81

7 1.0 1.0 1.0 1.0 0.950 0.935 50 86 86 86

1.0 1.0 1.0 1.0 1.0 1.0 0.974 41 71 87 87

1.0 1.0 1.0 1.0 1.0 1.0 1.0 37 64 78 78

See page 85 for full description. c) the R.M.S of the residuals increases;

d) the spectrum variance decreases;

e) the wavelength ratio decreases;

f) the harmonic surface ratios decrease.

All the above are not unexpected. The conclusion to be drawn from the above comments is that by studying only the quantitative measures given in the tables. we would not be able to come to a definite conclusion as to the "goodness" of a generalization. But with a visual analysis, the quantitative measures act as a confirmation and could indicate more precisely than a visual inspection the greater degree of generalization. For example, the quantitative measures indicate that Map No. 4 has been generalized to a greater degree than Map No. 3 because the ratios are smaller in the case of Map No. 4.

The four harmonic surfaces are given in Maps

6, 7, 8 and 9, and the quantitative measures in both tables would seem to check the method used for determining the harmonic surfaces and the program written for the analysis of the surface and the wavelengths. Examining the quanti­ tative measures for the fourth harmonic surface (Map No. 9) may lead us to conclude that we have almost a replica of the original terrain— but a visual analysis does not com­ pletely confirm this conclusion. What the quantitative measures do not show clearly is that there has been a sort of "broadening out" of the contours in the fourth harmonic surface.

3.8 Summary

In this chapterf ways of quantifying aspects of the character of the terrain have been discussed.

In Section 3.2 some terrain descriptors were men­ tioned, and, included in the quantitative measures evalu­ ated in Section 3.7 were some based on the variance spectrum. As Pike and Rozema (1975) point out, the spectrum contains information which characterizes the topography and the landforms. So that by using the quantitative measures in Section 3.7 we are in a sense quantifying the change in the character of the terrain.

The analysis of the eight maps in Section 3.7 shows that, in addition to a visual analysis, the following quantitative measures seem to give a fairly good indica­ tion of the change in the character of the terrain:

a) the mean elevation;

b) the variance of the elevations;

c) the R.M.S of the elevations;

d) the R.M.S of the residuals

e) the spectrum variance; f) the wavelength ratio;

g) the harmonic surface ratios.

As mentioned in Section 3.2, there is no one

"magic number" to describe the terrain— and therefore the change in the character of the terrain— but rather several parameters.

We have not investigated exhaustively what these parameters are, nor have we tried to classify, in order of importance, the parameters or measures used above.

But we are stating that the quantitative measures given in the tables (described in Section 3.7)— together with a visual analysis— form a strong basis

a) for estimating, in qualitative terms,

the change in the character of the

terrain; and

b) when given a number of generalized maps,

for determining which map has been

generalized to a greater or lesser degree.

This means that no matter what system or method is used for the generalization of the terrain, the con­ cepts embodied in this chapter can assist in judging the result. CHAPTER FOUR

LEAST SQUARES COLLOCATION

4.1 Introduction

The method of least squares collocation as used in geodesy is well documented in publications and other sources, for example, Moritz (1972 and 1973) ; Moritz and Schwarz (1973); Schwarz (1975); Rapp (1976); Rummel

(1976).

As an introduction we can consider simple linear regression and the conventional least squares model.

In simple linear regression, we have the model

Z = AP + V (4-1)

where Z is the vector of observations;

P is the vector of parameters;

A is a matrix of constants; and

V is the vector of the independent random variable. The solution gives the least squares estimator, namely

(4-2)

In this solution the observations are weighted equally and the assumption is that there are no modeling errors.

In the conventional least squares model, weights can be given to the observations and the solution is then

(4-3)

where W is the weight matrix equal to ^;

Og is the a priori variance of unit weight;

Z is the variance-covariance matrix of the

observations.

In this solution, too, no modeling errors are assumed.

4.2 Overview of the Least Squares Collocation Method

To introduce the concept of least squares colloca­ tion, we can say that the principle of collocation is to fit some representation to the data and the method of least squares collocation includes a least squares condi­ tion of minimum interpolation errors at all points

(Moritz, 1976).

Rummel (1976) identifies two least squares collo­ cation models (only one will be discussed here) . The so- called Model I is

Z = AP + S' + N (4-4)

where Z is an (nxl) vector of observations— that is,

there are n observations ;

P is a (pxl) vector of unknown parameters;

A is an (nxp) coefficient (or sensitivity or

design) matrix which relates the observations

to the parameters;

S' is an (nxl) vector, being the random signal

part of Z;

N is an (nxl) vector, called the (random) "noise"

on the observations.

The following assumptions are made in this model:

(1) E[S'] = 0

(2) E[S'N

(3) E[N] 98

The following definitions will be used:

E[S'S'^] = , which is the covariance

matrix for the signal S';

E[NN^] = , which is the covariance

matrix for the noise K.

In the model being considered, we want to predict the signal, S, at some point. Note that S' is the signal at the observation point and S is the signal at the point to be predicted.

Equation (4-4) can be rewritten as

(4-5)

■ [ ; ] ... (4-6)

Attention should be drawn to the similarity between equation (4-5) above and the linearized equation used in the least squares adjustment of a model combining obser­ vation and condition equations. See Ubtila (1967) and

Rampai (1975). The covariance matrix Q of the vector V can be written as

Q =

where is the covariance matrix of the signal S; ^SS is the covariance matrix of the measurement Z ^ZZ is the cross-covariance between S and Z; ^SZ

^ZS

It can be shown^ see Moritz (19 73) that

I Q =

A solution for the unknown parameters, P, is obtained by minimizing V^Q~^V.

Let C = Cg,g, + then the parameter solution is

= (A^C "^A) ^A^C ^ Z (4-7) and the solution for V is

s S3' (Z - AP) (4-8) S' + N _c _

The solution for the estimated values of the

signals at the points where we want predictions will be

T-1 (4-9)

The solution using the method of least squares

collocation consists of equations (4-7) and (4-9) given

above and some of the properties of this solution are:

(1) The solution of the parameters P is inde­

pendent of the number of signals which we

estimate;

(2) The prediction of any signal is independent

of the number of signals being predicted;

(3) The observations may be considered errorless

or in error, that is, N = 0 or N 0.

The "total" or "complete" signal at the points where we want predictions is then given by

(4-10) points where predictions are required.

It turns out that the original model used by

Moritz (1962 and later publications) could be described as a simplified version of Model I above. For example,

Lachapele (1975) used this simplification of Model I:

Z* = S' + N (4-11)

where Z* is a vector of known quantities;

S' and N are as described for Model I with the

same assumptions but in this case the added

assumption is that Z* consists of random

quantities with E[Z*] = 0. That is, Z*, S',

and N are all random vectors with zero average.

This simplified Model I could be called the non­ par ame trie case of least squares collocation. That is, in Model I let AP = 0 and then from equation (4-9) we

S = Cgg,C"^Z* (4-12)

which is the solution for the signal vector in this case. 102

It should be noted here that an important part of the

least squares collocation method is an a priori knowledge

of the covariance function used to set up Cg,g, (or Cgg) and

the use of an incorrect or "wrong" covariance function

should be considered. Theoretically if one knows the true

covariance function then there is no need for least squares

collocation for prediction— the surface, where the predic­

tions are required, would then be defined.

It is clear that the right or true covariance func­

tion of the elevations is never known and in practice it is approximated by an analytical covariance function. Moritz

(1976) has stated that:

The use of an incorrect covariance function for interpolation does not greatly change the interpolated value, but that for calculating accuracies the very best avail­ able covariances should be used.

The practical result, for this study, of the above

discussion is that we can approximate the empirically de­

termined covariance function and not greatly change the

"interpolated" value, which occurs at the original data point.

The concept of generalization as generally accepted,

and as used in this study, involves an increase in the

contour interval from the original map to the final gener­

alized map. This fact also adds to the latitude of using

a "bad" covariance function. 4.3 Proposed Use of Least Squares Collocation Concepts in Cartographic Generalization

A liberal interpretation of some of the covariance matrices described in Section 4.2, as applied to digital terrain models, is that C^,g, is related to the character of the original terrain and Cgg, is the relationship be­ tween the character of the original terrain and that of the derived (or generalized) terrain.

Using the above concepts, we arrive at equations

for cartographic generalization based on the least squares collocation algorithm described previously:

(4-13)

Sg = G(D + F)“^(Z - AP) (4-14)

where A is related to the interpolation model used;

F is related to the error in the obser^^ations,

(or elevations, Z), i.e., "noise";

D is related to the "nature" or "character" of the

data points;

G is related to the kind or degree of generalization

desired;

G is controllable by the cartographer and therefore rep­ resents the subjective intervention. Using the simplified version of the model de­

scribed in Section 4.3, that is equation (4-13), we have

for cartographic generalization

Sg = G(D + F)‘'^Z* (4-15)

where Z* could be equal to (Z - ml), m being the mean elevation. In this case, AP = ml.

Corresponding to the "total" signal given by equation (4-10), the generalized values of the derived digital terrain model will then be

Sg (4-16)

In this case, A refers to all the points of the original digital terrain model.

Using the non-parametric model, AP will be equal to some surface which makes E[Z*] = 0 and Z* is a random quantity.

Some sort of smoothing could result by manipulating

D and F or only F (see Kraus, 1973). This would involve a matrix inversion for every degree of generalization, which may not always be convenient.

For cartographic generalization, it may be more appropriate to achieve this manipulation by controlling G. 105

In our case, we could easily assume a small and negligible observation error, then F = 0. However, we should re­ member that the signal, S, could also contain a small

(undetermined) amount of "model noise."

Some of the matrices mentioned above will now be examined, bearing in mind that the original model is

Z = AP + S + N (4-17)

and the generalized model (that is, the derived digital terrain) is given by equation (4-16) above.

The Z matrix;

(4-18)

where Z, , Z~ Z^ are the n-elevations of the

digital terrain model. 106

The A matrix:

This matrix is sometimes called the "design" or

"sensitivity" matrix. It is obtained from the particular interpolation model used.

For example, if the interpolation model chosen is the 8-term biquadratic polynomial, then

iX + + b^y + b_y^ (4-19)

+ c.xy + c„x y + c_xy

The size of the A matrix is (nxp) where n = total number of observations (elevations) and p = number of parameters in the interpolation model.

For example, if the 8-term biquadratic polynomial is used as the interpolation model then the A matrix would take the following form:

(4-20) 4 yi yi ='1^1 vl

^2 ^2 ^2 ^2 ^ 2^2 ^ 2^2 '=2^2 where the x's and y's are the planimetric coordinates of

the points in the digital terrain model.

The P matrix:

This is the solution vector for the parameters and

the number of parameters of course depends on the parti­

cular interpolation model used

The F matrix:

This is the matrix related to the observational

errors, that is, errors in the elevations . . . Z^.

Assuming no statistical correlation, F will be a diagonal matrix with the elements of the diagonal equal to the variances of the errors of the observations. As mentioned

previously, the observational errors would be small when

compared with the contour interval of the generalized

data and we can therefore neglect F completely.

The S matrix :

This is the signal vector, of size (mxl), where

m = number of points being predicted. In this study,

m = n, the number of grid points in the digital terrain

model. The D matrix:

This is the covariance matrix of the signal part

of Z and together with F, the covariance matrix of the

observational errors, it forms the covariance matrix of

the observations, i.e. (D + F).

The general form of the D matrix is

(4-21)

where is the variance of the elevations;

is the covariance between points 1 and 2; and

so o n .

The covariance between the points is calculated by a covariance function. The distance for which covariances will be calculated will obviously be limited and most of the covariances will therefore be approximately equal to 4.4 Summary

In this chapter we have indicated some of the ways in which the concepts of least squares collocation can be applied to cartographic generalization. From a purely theoretical point of view, the "true" covariance function will only be known when the whole surface is known but, in the usual application of the method, the covariance function is used to define the surface. We have indicated in this chapter that, for the purpose of this study, the use of an approximate covariance function can be tolerated.

One of the attractions of the least squares collo­ cation method for generalization is the apparent possi­ bility in this method of dealing with the concept of the character of the original terrain and change in this character resulting from the generalization process. This was discussed in Section 4.3.

We should also note that, conceptually, we could con­ sider the covariance function in the least squares colloca­ tion method as part of a transfer function and it may therefore be possible to use a covariance model as the basis for a filter used for generalization as described in

Chapter Two. This idea-will be developed in Chapter Six. CHAPTER FIVE

TESTING THE HYPOTHESIS

5.1. Preliminary Investigations Using Diagonal Matrices Only

Some preliminary investigations were made using the equations described in Chapter Four, but with diagonal matrices only— that is, no covariance function was used in setting up the matrices.

The purpose of the maps in this section— together with those discussed in Section 3.7— was to build up a repertoire of experiences for the visual examination of various generalizations together with an examination of the quantitative measures described in Section 3.7.

For the following group of four maps, equation

(4-16), namely

was used for the generalized values. For this group, equation (4-14), namely

Sg = G(D + F)"^(Z " AP) was used for the "signal" where

G = cl

I = the unit matrix

D = I

F = 0

Z = elevations of original terrain

c takes on different values and AP represents a

reference surface, (Z - AP) therefore represents the

residuals between the original surface and the reference

surface. The rationale for the above procedure is that we have a simple method (but not theoretically defensible)

for "generalizing" by scaling the residuals. What is

actually happening is that the residuals are being

"squashed" towards the reference surface if c<1.0, and

"stretched" away from the reference surface if c > 1.0.

In the four maps to be described, all the residuals

are being scaled by the same amount— not a procedure to

preserve the character of the terrain. A variation of

the above procedure would be to scale the residuals by a

factor depending on the size of the residual.

The following details refer to the individual

maps of this group:

Map No. 11: AP = mean plane

c = 1,5 Map No. 12: AP = third harmonic surface

c = 1. 5

Map No. 13: AP = fourth harmonic surface

c = 1.5

Map No. 14: AP = fourth harmonic surface

c = 0.5

The maps belonging to this group are shown in

Figures 5/1 to 5/4 and the quantitative measures relating

to these maps are given in Tables 5.1 and 5.2.

As one would expect, Maps 11, 12 and 13 show an

increased data variance (that is, an increase in the data variance over the original data variance— see Table 5.1,

column 3) because in each case c > 1.0. We also notice that if the reference surface approximates to the original terrain (for example. Maps 13 and 14 have the fourth harmonic surface as the reference surface) then the wave­

length and harmonic surface ratios (see Table 5.2) seems

to indicate no change in the character of the terrain, however, the quantitative measures in Table 5.1 (data

variance, R.M.S elevation and spectrum variance) indicate that some changes have occurred. The "worst" generaliza­

tion seems to have been obtained in Map No. 11 where the mean plane was used as a reference surface and c = 1.5.

The wavelengths and harmonic surface ratios in this case

(Table 5.2) seem to indicate that irregularities have Figure 5/1 Figure 5/2

Figure 5/3 Figure 5/4

Generalizations of Original Data

Scale approx. 1 in 250,000 FIRST TABLE OF QUANTITATIVE MEASURES FOR MAPS 11, 12, 13, 14*

Map Mean Data R.M.S. R.M.S. Mean spectrum No. Elevation Variance Elevation Residuals Residuals Variance

Original 603.9 39775 635.9 -- 35804

11 603.9 89493 673.7 99.6 0.0 80556

12 603.8 52999 646.1 51.3 0.0 48820

13 603.9 48932 642.9 42.7 0.0 44979

14 603.8 34317 631.5 52.7 0.0 30295

See page 85 for full description. TABLE 5.2

SECOND TABLE OF QUANTITATIVE MEASURES FORMATS 11, 12, 13, 14*

Cumulative % Wavelength Ratios Harmonie Surface Contribution of Map No. Harmonie Surfaces

1 2 3 4 1 2 3 4 1 2 3 4

Original ------30 52 64 72

11 2.249 2.249 2.249 2.249 1.145 1.231 1.272 1.297 30 52 64 72

12 1.0 1.0 1.0 2.249 1.0 1.0 1.0 1.032 23 39 48 61

13 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 24 42 52 58

14 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 35 60 74 83

See page 85 for full description. 116 been accentuated rather than attenuated. The very high data and spectrum variances in Table 5.1 would seem to confirm this.

The maps examined so far have given us a repertoire of experiences for judging generalizations and would seem to confirm the comment made in Section 3.7, that for the best information as to the generalization of the original terrain, one should perform a visual analysis in addition to an examination of the quantitative measures given in both tables.

5.2 Using the Least Squares Collocation Algorithm— With a Limited Distance of Influence

Map No. 18:

For this map, equation (4-16), namely

was used for the generalized values and equation (4-14) , namely

Sg = G(D + F)"^(Z - AP) was used for the signal. AP, the reference surface, is the fourth harmonic surface in this case. The covariance function

(5-1) was used to set up the D matrix and F = 0. For the D matrix a = 2.0, K = 1.0 and d takes on the values 0 and 1 only, that is, the distance of influence in this case is taken as 1 grid unit (= 0.5 mile). For this map, G = 1.51

Map No. 34:

For this map, equation (4-16), namely

Zg = AP + was used for the generalized values, and equation (4-15), namely.

was used for the signal, where

Z* = Z - ml

m = mean elevation of original terrain

Z = elevation of original terrain

I = the unit matrix

F = 0 118

The D matrix was set up using the covariance function given in equation (5-1) above with the following values:

a = 0.6

K = 1.5 for d = 0

K = 1.0 for d = 1, 2, 3 and 4

That is for this map a distance of influence of four grid units (that is, 2 miles) was used.

For the G matrix, the covariance function given in equation (5-1) was also used, but with the following values:

a = 0.6

K = 1.21 for d = 0 , 1, 2, 3, and 4

Map No. 36 :

For this map, the same equations and values were used as in Map 35, except for the G matrix the following change was made

K = 1.47 for d = 0, 1, 2, 3, and 4

By way of comparison, the following two maps were obtained by using diagonal matrices for the D and G matrices in the above mentioned equations. Map No. 38 : D = 1.51

G = 1.211

Map No. 40: D = 1.51

G = 1.471

Map No. 18, where we have assumed correlation of elevations up to only one grid distance, is very close to the original terrain as can be noted from Table 5.4. How­ ever, the data variance, R.M.S of elevations and the spec­ trum variance given in Table 5.3 would seem to indicate a slight accentuation of irregularities. Visual examination confirms the closeness of this map to the original data.

Map No, 18 cannot, therefore, be considered a significant generalization, a result probably due to the relatively large value used for a (that is, 2.0) in the covariance function.

Maps 34 to 40 are discussed in Section 5.4. Figure 5/5

Generalization of Original Data

Scale approx. 1 in 250,000 36 Figure 5/6 Figure 5/7

^

Figure 5/8 Figure 5/9

Generalizations of Original Data

Scale approx. 1 in 250,000 FIRST TABLE OF QUANTITATIVE MEASURES FOR MAPS 18, 34, 36, 38, 40*

Map R.M.S. R.M.S. Spectrum No. Elevation Variance Elevation Residuals Residuals Variance

Original 603.9 39775 635.9 - - 35804

18 603.8 48973 643.0 42.8 0.0 45019

34 603.8 37804 634.3 72.7 0.0 34895

36 603.8 73422 661.7 239.8 0.0 80293

38 603.8 26045 625.0 38.1 0.0 23425

40 603.8 38057 634.5 4.4 0.0 34234

See page 85 for full description. TABLE 5.4

SECOND TABLE OF QUANTITATIVE MEASURES FOR MAPS 18, 34, 36, 38, 40*

Cumulative % Wavelength Ratios Harmonie Surface Ratios Contribution of Map No. Harmonie Surfaces

Original 30 52 64 72

18 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 24 42 52 58

34 0.967 0.960 0.941 0.903 0.996 0.993 0.991 0.989 31 53 65 72

36 1.064 1.080 1.119 1.203 1.007 1.013 1.017 1.033 17 30 37 42

38 0.654 0.654 0.654 0.654 0.960 0.936 0.925 0.918 30 52 64 72

40 0.956 0.956 0.956 0.956 0.995 0.992 0.990 0.990 30 52 64 72

See page 85 for full description. 5.3 Constraints and Varying Degrees of Generalization

The following two maps indicate how elevations can be constrained or generalized to different degrees in the

same area.

For both these maps the concepts outlined in

Section 5.1 were used— that is, only diagonal matrices were considered. Equation (4-16), namely,

was used for the generalized values. Equation (4-14) , namely,

Sg = G(D + F)"^(Z - AP) was used for the signal, where

G = cl

D = I

F = 0

AP = mean plane

Z = elevations of original terrain

Details relating to each map are given below. Map No. 41:

c = 1.0 for all elevations greater than 1200 ft

c = 0.9 for all other elevations

Therefore, all elevations greater than 1200 feet were constrained at their original values.

Map No. 42:

For this map we performed what could be called

"generalization slicing," that is, instead of generalizing certain areas to varying degrees, different horizontal

"slices" of the terrain were generalized to varying degrees.

For this map,

c = 1.0 for elevations less than 800 ft and

greater than 1200 ft;

c = 0.867 for elevations between 800 ft and

1000 ft;

c = 0.933 for elevations between 1001 ft and

1200 ft.

That is, elevations below 800 ft and above 1200 ft were constrained to their original values and between these elevations, two "slices" were generalized to dif­ ferent, arbitrarily chosen degrees.

The two maps described above are shown in Figures

5/10 and 5/11 and the quantitative measures relating to these generalizations are given in Tables 5.5 and 5.6. Figure 5/10 Figure 5/11

Generalizations of Original Data

Scale approx. 1 in 250,000 TABLE 5.5 FIRST TABLE OF QUANTITATIVE MEASURES FOR MAPS 41, 42*

Map Data R.M.S. R.M.S. Spectrum No. Elevation Variance Elevation Residuals Residuals Variance

Original 603.9 39775 635.9 - - 35804 41 606.9 36495 636.2 13.1 -3.1 33010 . 42 593.6 35078 622.3 26.6 10.3 31644

TABLE 5.6 SECOND TABLE OF QUANTITATIVE MEASURES FOR MAPS 41, 42*

Cumulative % Wavelength Ratios Harmonie Surface Ratios Contribution of Map No. Harmonie Surfaces

1 2 3 4 1 2 3 4 1 2 3 4

Original ------30 52 64 72 41 0.883 0.913 0.938 0.942 0.996 0.989 0.987 0.986 29 51 63 71 42 0.826 0.835 0.881 0.987 0.950 0.941 0.939 0.938 28 49 61 69

See page 85 for full description. 128

Visual examination of Maps 41 and 42 seems to reveal that there are not large differences between the maps, but examination of Tables 5.5 and 5.6 clearly reveals these differences. As expected. Map 41 is closer to the original map and also as expected, the mean elevation of the original area has not been retained by Map 41 or Map 42.

These two maps explored some possibilities which could be applied to the cartographic generalization of topographic maps. Although only diagonal matrices were used in the above exercise, the concepts could be applied to D and G matrices set up with a covariance function as described in Section 5.2.

5.4 Summary

Maps 34 and 36 (Figures 5/6 and 5/7) were obtained by using the non-parametric least squares collocation solution as the basis for generalization. The basic equations used were (4-15) and (4-16). This involved the inversion of the D matrix (F = 0), a 324 x 324 square type matrix and correlation was assumed up to 4 grid distances.

The matrix inversion of the D matrix for both the above mentioned maps was relatively expensive. We should note that the 18 x 18 model, used in this study, is a very 129

small model when compared with what would be expected in practice.

Maps 38 and 40 (Figures 5/8 and 5/9) used only the diagonal elements for the D and G matrices of the two previously described maps.

Analysing the above mentioned four maps, visually and via the elements given in Tables 5.3 and 5.4, we come to the conclusion that there is no advantage in using least

squares collocation algorithm for the purpose of generali­

zation, when simple diagonal matrices for D and G give ap­ parently visually acceptable results. But, as was pointed out previously, there is no sound basis for the way Maps-

38 and 40 were generalized— what was being investigated was a broad method of generalization. Something that may perhaps need further investigation is a theoretically sound way of generalizing the residuals within the general scheme used for Maps 38 and 40— but this will not be done in this study.

In Maps 38 and 40, the diagonal elements of the G matrix were all the same. For these two maps the D matrix was also a diagonal matrix. Now if the respective elements on the G and D matrices were equal, then the elevation at this point would retain its original value. That is, the original elevation would be constrained. This concept was used to obtain Maps 41 and 42, as described previously. 130

We conclude that, although the concept of generalization

slicing is attractive, as used here it would be difficult

to predict the effects and to control the generalization.

For example, we would not easily be able to retain in the generalized map, the mean elevation of the original terrain.

The methods for cartographic generalization used in this chapter do not seem to provide us with the flexi­ bility we are seeking— that is, an efficient method where some

control can be easily applied by an a priori knowledge of some of the effects of the method on the generalized model. It could also be correctly argued, that by multi­ plying the residuals by a factor less than unity, which was the case in Maps 38 and 40, we are not really genera­ lizing. We are simply scaling down the original terrain shape to the reference surface. A better method would perhaps be not to use the mean elevation of the whole terrain model in order to obtain the residuals (assuming that the reference surface is the mean plane), but rather to use the mean elevations of smaller regions within the whole digital terrain model. Using such a limited area is an approach to the sort of filter mentioned by Tobler

(1966), Topfer (1974) and others (see Sections 2.1.2 and

6.4). One could also devise a method whereby each residual is multiplied by a factor depending on the size of the 131 residual, which would give a smoothing effect. But a disadvantage of using residuals is that one first has to perform calculations using all the relevant elevations in order to eventually obtain the required residuals. For large digital terrain models, which would be the rule rather than the exception in practice, it would be more efficient to be able to input the digital terrain model, say, a few rows at a time and then to be able to genera­ lize this input immediately. A method for achieving this is suggested in Chapter Six,

Most methods for the computer-assisted cartographic generalization of the relief continuum on topographic maps use some sort of a filter— as described in Chapter Two.

In other words a small local area at a time is considered.

Even if one used least squares collocation for small areas, one may still have to preprocess all the data (or one would have to use a reliable method for classifying the terrain) in order to arrive at the covariance function to be used in the D matrix.

As a result of the investigation described so far, we conclude that the least squares collocation algorithm (as used in Maps 34 and 36) although feasible, would not be the most efficient method for cartographic generalization.

For interpolation or densification of a digital terrain 132 model (that is, manipulations on the original surface), the least squares collocation algorithm would seem to hold more promise. However, for deriving another surface (in our case the generalized surface), the least squares collocation algorithm has not been altogether efficient and we will next examine the use of a nines filter. CHAPTER SIX

DESIGN OF A GENERALIZATION FILTER

6.1 The Nines Filter

Generalization of a digital terrain model using a digital filter involves a linear transformation. This transformation is performed by a digital (or numerical) filter (or operator) consisting of a series of weights.

The filter has a principal (or main) weight— which operates on the elevation being generalized— and a number of other weights which operate on the surrounding eleva­ tions. The weights of the filter specify the proportion which each elevation will contribute to the final value of the elevation being generalized.

The nines filter has a total of nine weights, that is, the principal weight and eight other weights surrounding it which operate on the eight elevations nearest to the elevation being generalized.

Therefore, without the need to analyse the original data in any way, the nines filter (once its composition or design is decided upon) operates on a total of only nine points at a time.

133 The form of the nines filter is shown in Figure 6/1

^i-l.i-1 V l , j

^i,i-l

^i+1,j-1 *'i+l,i ^i+l,j+l

Figure 6/1

The Nines Filter

j is the principal or main weight. Let the elevation of the original terrain at point i,j be defined by Zi j = f(X^ j , j) . Assuming that the grid interval is equal to unity in both the X and Y directions and let

Z| j be the generalized value, then the nines filter is used as follows as a weighted moving average.

p=l q=l

p=il q/-i "i+P,j+q^i+P,j+q 2* _ (6-1) p=l q=l

i+Pfj+q P= -1 g= -1 If one uses the nines filter on a set of evenly

spaced data, there will be a loss of data along the edges

equal to half the width of the filter. (For example, if

j is on the extreme left edge of the area, then one

cannot calculate the generalized value according to

equation (6-1) above because half the filter width— namely,

the whole (j-l)th column of the filter— will not "cover” any data.) To overcome this, one can design a filter for the corners and another for the edges as shown in

Figures 6/2 and 6/3.

Figure 6/2

A Corner Filter V i , j V i , j + i

V i , j ^i+l,j+l

Figure 6/3

An Edge Filter

For topographical maps ; the practical procedure would be to use an area slightly larger all round than the area which is to be generalized. That is, use is made of an overlap where the minimum width of the overlap would be equal to half the width of the filter. In this

study, where a before-and-after situation is being com­ pared, corner and edge filters will be used.

In the previous chapter, least squares collocation was applied to generalization. A distance of influence of four grid units was used, that is, to generalize an elevation the elevations within about 4 grid units

(suitably weighted) were used. In the nines filter we are

assuming a distance of influence of /2~ grid units. In 137

the digital terrain model being used for this study this

amounts to 0.7 miles; That is, C(o) = 0.7 miles, which

is the order of the value for C(o) one would get by using

a high order polynomial surface fitted to the data— see

Section 3.6.4.

One could use a larger filter, say with 25 weights,

where the distance of influence would be /8~" grid units—

or even larger filters. All that follows with regard to

the nines filter can also be applied to the larger filters.

By choosing a particular grid interval, one has

actually filtered the data to some extent and it seems

reasonable to limit the distance of influence for the

cartographic generalization of the relief continuum on

topographic maps as has been described by Topfer (1974).

A justification for the use of the nines filter (rather

than a larger filter) will be discussed in Section 6.7.

6.2 Requirements of a Filter

In order to preserve the mean value of the original

elevations being generalized, the weights of the filter

(the nines filter or the corner or edge filters) must sum

to unity (Holloway, 1958).

A phase shift in the frequencies contained in the

original model shows up as a spatial displacement of the elevations (and therefore the surface features which they represent) relative to the grid. To be phase-distortion­ less the two-dimensional filters must be isotropic, that is, the filter elements (or weights) in the nines filter must be obtained as a function of distance and not direc­ tion (Martin, 19 75). This means that the weights of the filter must be symmetrical about the principal weight.

6.3 Design of Nines Filter Using the Gaussian Function

In Section 5.3, the covariance function, given by equation (5-1) , namely,

C (d) = K exp (-a^d^ ) was used for setting up the D and G matrices. These two matrices combined to act on the elevations (or residuals) as a sort of generalizing or smoothing matrix. We now propose to use the nines filter as a generalizing filter and it is reasonable to investigate how the above covariance function can be used to set up the weights of a filter.

Taking K = 1.0 in the above equation we get the so called

Gaussian function, namely,

C(d) = exp(-a^d^) (6-2) which could be used to calculate the elements (or weights) of a filter where

a = some predetermined constant;

d = the distance from the point being

generalized to the point of influence.

Let (i,j) refer to the central position of the nines filter as shown in Figure 6/1. Then

(6-3) because d = 0 and

C(d)^^l^j = exp (-a^) (6-4) because d = 1. And further

'=<'*>1-1,j-i = '=<'*> 1-1,i+i ' ‘=''*>i+i,j-i =

(6-5) '=<'*> i+l,j+l = because d = .

Let CO refer to the weights from equation (6-3),

Cl refer to the weights from equation (6-4) and C2 refer to the weights from equation (6-5), then in order to pre­ serve the mean value of the original terrain being 140

generalized, we have the condition that

CO + 4 (Cl) + 4(C2) = 1.0 (6-6)

Using equation (6-2) and the notation and condition of equation (6-6) we get, for various selected values of a, the weights given in Table 6.1.

For example, for a = 0.83 we get the nines filter

shown in Figure 6/4. Also shown in Figure 6/4 are the corner and edge filters, the data for which does not appear in Table 6.1, but the principles for obtaining their elements are similar to the above description of the nines filter.

In all the above we have assumed a regular grid, that is AX = AY and this is one of the assumptions in this study. But the above method for designing a nines

filter could easily be adapted to the situation where

AX / AY but where AX = constant and AY = another constant.

The Gaussian function adapted to this situation would be

C(d) = exp[-a^(AX^ + AY^)]

= exp(-a^AX^) • exp(-a^AY^) (6-7) TABLE 5.1

NINES FILTER WEIGHTS USING THE GAUSSIAN FUNCTION

a CO Cl C2 Map. No.*

0.04 O.III O.III O.III 43

0.30 0.125 0.II4 0.104 44

0.62 0.179 0.122 0.083 45

0.64 0.185 0.123 0.081 46

0.66 0.190 0.123 0.080 47

0.82 0.245 0.125 0.064 48

0.83 0.249 0.125 0.063 49

0.84 0.253 0.125 0.062 50

I.O 0.332 0.122 0.045 51

1.5 0.682 0.072 0.008 52

2.0 0.931 0.017 0.000 53

4.0 1.0 0.0 0.0 54

The map numbers refer to maps appearing in Section 6.6. 0.063 0.125 0.063

0.125 0.249 0.125 Nines Filter

0.063 0.125 0.063

0.443 0.223 Corner Filter 0.223 0.112

0.167 0.084

0.332 0.167 Edges Filter

0.167 0.084

Figure 6/4

Filters Based on Gaussian Function With a = 0.83 143

Carrying this argument a step further we could, for example, design a "dynamic filter" using the Gaussian

function. In this case, the generalized value of any ele­ vation would be calculated by, say, the eight nearest points and the weights of these points would be calculated by a predetermined value for a and the relevant value for d, the distance. One could also build into this "dynamic filter" some constraints as regards the directions of the eight points. For example, one may not want all eight points in the same direction.

In any of the above ideas, the concept of "genera­ lization slicing" mentioned in Section 5.4 could be achieved by selecting different values of a (in equation (6-2)) for various elevation ranges.

But, as has been previously pointed out, in this study we will deal only with a regular grid.

6.4 The Gaussian Function Nines Filter Compared With Other Filters

Tobler (1966 and 1970) used a binomially weighted smoothing filter as shown in Figure 6/5.

Comparing Figure 6/5 with Figure 6/4 shows that the nines filter based on the Gaussian function with a=0.83

(Figure 6/4) is almost exactly equal to Tobler*s binomially weighted filter. 0.0625 0.125 0.0625

0.125 0.25 0.125 Nines Filter

0.0625 0.125 0.0625

0.44 0.22 Corner Filter 0.22 0.11

0,167 0.08

0.33 0.167 Edge Filter

0.167 0.08

Figure 6/5

Tobler's Binomially Weighted Smoothing Filter 145

Successive smoothings using this filter are shown in Figures 5/2, 5/3 and 5/4.

A point of interest is that, for a = 0.83 in the

Gaussian function, we get the one-dimensional filter shown in Table 6.2 (1/4, 1/2, 1/4) which is exactly the same as the one-dimensional binomially weighted filter used by

Tobler (1966) and the so called Hanning filter described by Martin (19 75).

The low pass filter used by Stegena (1973) was de­ scribed in Section 2.1.2. If we apply values of K = l/ir,

2/ïï/ 3/it, 4/tt in equation (2-2), namely.

— ^ exp(-dV4K^) 4ttK^ to set up a nines filter then we obtain the filters shown in Figure 6/6.

The values shown in Figure 6/6 for K = 1/tt, 2/ir,

3/ïï, 4/tt can be obtained from the Gaussian function using a = 1.573, 0.785, 0.521 and 0.39, respectively. 146

DISCRETE ONE-DIMENSIONAL FILTERS DERIVED FROM THE GAUSSIAN FUNCTION WITH VARIOUS VALUES OF a

Two Symmetrical Value of Weights Of Principal This Value Weight

0.04 0.333 . 0.334

0.30 0.323 0.354

0.64 0.285 0.430

0.83 0.250 0.500

1.00 0.212 0.576

2.00 0.017 0.965

4.00 0 1.0 0.005 0.062 0.005

0.062 0.732 0.062 K = 1/TT

0.005 0.062 0.005

0.067 0.125 0.067

0.125 0.231 0.125 K = 2 / tt

0.067 0.125 0.067

0.091 0 . 1 2 0 0.091 .

0 . 1 2 0 0.158 0 . 1 2 0 K = 3 / tt

0.091 0 . 1 2 0 0.091

0 . 1 0 0 0.116 0 . 1 0 0

0.116 0.136 0.116 K = 4/TT

0 . 1 0 0 0.116 0 . 1 0 0

Figure 6/6

Nines Filter Based on Stegena's Low Pass Filter For Lines Robinson, Sale and Morrison (1978) used the

following nines filter;

0.0625 0.0625 0.0625

0.0625 0.5 0.0625

0.0625 0.0625 0.0625

Figure 6/7

Nines Filter Used by Robinson, Sale & Morrison

(1978)

The above filter is apparently non-isotropic and can be obtained by using a = 1.442 in the Gaussian function and only d = 0 and d = 1. That is, for the corners of the nines filter the correct distance of d = ^Tî~ is not used, therefore this filter is not considered suitable for the cartographic generalization of the relief continuum.

It is interesting to note that Robinson, Sale and

Morrison (1978) used a three term equally weighted moving average to smooth linear data and their results showed that in 6 crests of the original there were 4 cases of polarity reversal (see Section 6.5) and in 6 troughs of the original data there were 5 cases of polarity reversal. 149

Allam (1978) describes a smothing filter used on digital terrain models obtained from a Gestalt Photomapper.

This filter is not used for generalization but rather for the smoothing of elevations in a sort of editing and verification process. As mentioned in Section 2.1.2, one is dealing with millions of data points in this situation.

Allam (1978) states that "the filtering of the millions of elevations in the . . . grid could be tedious if smoothing is done by Fourier, least square polynomial or min-max methods." The method of filtering described by

Allam (1978) is performed on the grid of elevations

". . .in two-dimensions by applying a one-dimension operator in each dimension. The equation used is

n.i = VKIK3(2..2,3 + +

+ 2.^1,3> + (6-8)

where

ZŸ j is the smoothed elevation

j is the original elevation

K = K 3 + 2(K^ + Kg) and Kg^, Kg/ Kg depend on the degree of smoothing required. 150

Allam (1978) further states that "for a five point operation the ratio of 4:2:1 for and respectively

[sic] produced a smoothing operator that damps only the physically significant elevation data,"

Using the condition for preserving the mean ele­ vation in the above five point operation, then one obtains the weights 0.1, 0.2, 0.4, 0.2, 0.1 for the five points.

It is interesting to note here that if one uses a = 0.83

in the Guassian function one gets the same weights given above. This indicates that Allam's five point weighting operation (with mean preserved) is the same as a five point binomially weighted smoothing operator as described previously.

Equation (6- 8 ) above appears to neglect a number of points close to the elevation being smoothed. For example, ^^^2 j-1 ' ^i+ 1 j+1 ^re neglected and this method is therefore not considered suitable for the cartographic generalization of the relief continuum on topographic maps.

6.5 Frequency Response

The frequency response of a filter is the ratio of the amplitude of a wave at this frequency after filtering to the original amplitude before filtering (Holloway, 1958).

Therefore if a filter is used for smoothing, we would 151

expect the value of the frequency response to be near

unity for low frequencies and to be near zero for high

frequencies. It is possible to get a value larger than

unity for the frequency response and in this case one

could be emphasizing irregularities (see Topfer, 1974).

If we were to get a negative frequency response at a particular frequency, then we have what Holloway (1958) calls "unfortunate polarity reversals." In this study of generalization of digital terrain models, this negative

frequency response would indicate the possibility of turning a crest terrain configuration into a trough terrain configuration.

For this discussion, we will use the one-dimensional equivalent of some of the two dimensional filters given previously. These one dimensional filters are given in

Table 6.2. The values of the principal weight and the two symmetrical weights were obtained in a similar fashion as described previously for the two-dimensional case.

In the two-dimensional filter, the frequency re­ sponse is sometimes called the wavenumber response

(Martin, 1975). For one principal weight and two other equal weights, the equation for the frequency response given by Martin (1975) could be simplified to obtain

R(f) = P» + 2P, cos w (6-9) where R(f) = frequency response

w = frequency at which the ratio (or

frequency response) is being measured

Applying the above equation (6-9) to the filters given in Table 6.2, with values of w from zero to ir, we obtain the set of frequency response functions given in

Figure 6/ 8 .

Figure 6/8 illustrates the effect of a given filter on waves of specified angular frequency. We notice that for values of a less than 0.83, we obtain filters which give negative responses at some frequencies. This means that terrain crest formations may be turned into terrain trough formations. Clearly an undesirable effect in cartographic generalization. Values of a greater than

0.83 do not give filters which exhibit polarity reversal.

Frequency response curves for one-dimensional filters have been examined. For two-dimensional filters the response "curves" would be "volumes" obtained by rotating the one-dimensional response curves 360“ about the axis marked R(f) in Figure 6/ 8 . □ AO.ou O AO.30 A A0.6U + AO.83 X Al.OO A2.0D ^ AU.00 VS PI ü_ CC

ANGULAR FREQUENCY X PI

Figure 6 /8 FREQUENCY RESPONSE CURVES 6.6 Results and Analysis of Experiments

The following group of generalized maps were ob­ tained by using two-dimensional phase-distortionless filters based on the Gaussian function. The generalized maps are numbered 43 through 54 and the filters used have been given in Table 6.1.

Figures 6/9 through 6/20 below show these generalized maps. Each map also indicates the a value used in the Gaussian function to design the filter used for the generalization. The contour interval for each map is 100 feet.

The first table of quantitative measures is given in Table 6.3 and the second table of quantitative measures is given in Table 6.4.

From a visual point of view all the maps in the above group, except Maps 52, 53 and 54, could be considered reasonable generalizations. These last mentioned maps are too close to the original map to be considered as generalizations. In fact, for the purposes of comparison.

Map 54 can be regarded as the original data. Further,

Maps 43, 44, and 45 could perhaps be considered as not altogether acceptable as cartographic generalizations, because the character "mountain ridge with hill tops

(or saddles)"— the qualitative description of the original terrain— only begins to appear in Map 46. ii3 a = 0.04 liti a = 0.30 Figure 6/9 Figure 6/10

«5 a = 0.62 a = 0.64 Figure 6/11 Figure 6/12

Generalizations Based on Use of a Nines Filter

Scale approx. 1 in 250,000 156

U7 a = 0 .66 H8 a = 0.82 Figure 6/13 Figure 6/14

19 a = 0.83 0.84 Figure 6/15 Figure 6/16

Generalizations Based on Use of a Nines Filter

Scale approx. 1 in 250,000 51 1.0 1.5 Figure 6/17 Figure 6/18

2.0 51 4.0 Figure 6/19 Figure 6/20

Generalizations Based on Use of a Nines Filter

Scale approx. 1 in 250,000 158

FIRST TABLE OF QUANTITATIVE MEASURES FOR MAPS 43 THROUGH 54*

Map Data R.M.S. R.M.S. Spectrum No. Elevation Variance Elevation Residuals Residuals Variance

Original 603.9 39775 635.9 - - 35804

43 603.4 24346 623.2 88.3 0.7 20521

44 602.2 24489 622.2 86.7 1.9 20667

45 603.5 25459 624.2 80.5 0.6 21577

46 604.2 25614 624.9 79.8 ”0.1 21714

47 604.8 25734 625.6 79.2 -0.7 21825

48 604.2 26532 625.7 73.4 -0.1 22592

49 604.5 26603 626.0 73.0 -0.4 22661

50 604.2 26654 625.8 72.6 -0.1 22707

51 603.9 27808 626.5 64.5 0.2 23826

52 605.0 33588 632.1 30.3 -0.9 29499

53 603.4 38285 634.2 6.6 0.7 34268

54 603.9 39775 635.9 0.0 0.0 35804

See page 85 for full description. TABLE 6.4 SECOND TABLE OF QUANTITATIVE MEASURES FOR MAPS 43 THROUGH 54*

Cumulative % Wavelength Harmonic Surface Ratios Contribution of Map No. Harmonic Surfaces

1 2 3 4 1 2 3 4 1 2 3 4

Original ------30 52 64 72 43 0.892 0.675 0.431 0.234 0.986 0.962 0.940 0.922 44 68 76 79 44 0.892 0.680 0.441 0.243 0.983 0.959 0.938 0.920 43 68 76 79 45 0.907 0.712 0.487 0.283 0.988 0.967 0.947 0.930 43 67 76 79 46 0.912 0.716 0.492 0.286 0.991 0.969 0.950 0.933 43 67 76 79 47 0.915 0.720 9.497 0.290 0.993 0.972 0.952 0.935 42 67 76 79 48 0.923 0.744 0.537 0.330 0.992 0.973 0.955 0.939 42 66 76 79 49 0.923 0.746 0.540 0.333 0.993 0.974 0.956 0.940 41 66 76 79 50 0.924 0.748 0.542 0.336 0.992 0.973 0.956 0.940 41 66 75 79 51 0.934 0.780 0.596 0.396 0.993 0.976 0.961 0.946 40 65 75 79 52 0.977 0.907 0.816 0.691 1.001 0.993 0.986 0.979 35 58 70 76 53 0,993 0.979 0.961 0.930 0.998 0.996 0.995 0.993 31 53 65 73 54 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 30 52 64 72

See page 85 for full description. 160

Tables 5.3 and 6.4 show the expected increase or

decrease in the respective elements as we go from a large

degree of generalization in Map 43 to no generalization

in Map 54.

In choosing the degree of cartographic generaliza­ tion desired; we should bear in mind the danger of choosing

a filter with a negative frequency response as shown in

Figure 6 / 8 Values of a below 0.83, in the examples used above, exhibit this tendency.

To examine further the results of using the

filters described above, we examine a profile on the digital terrain model, and note how the profile changes with the various degrees of generalization. The profile chosen was the one from row 15, column 1 to row 1, column 15 on the digital terrain model. The profiles ob­ tained by using filters derived from various values of

a in the Gaussian function are shown in Figure 6/21.

The inset in the top left hand corner of Figujre

6 / 2 1 shows the.location of the profile on the original

data with a 200 ft contour interval.

The relationship between the filters and maps

given in Table 6.1 and the profiles in Figure 6/21 are

shown in Table 6.5. Inset

161

x' = profile

O B

X H

1 COLUMN NUMBER

Figure 6/21 Profiles TABLE 6.5

RELATIONSHIP BETIVEEN PROFILES AND MAPS 43 to 54

Profile in Figure 6/18 a Map No.

A original data See Figure 6/2l'

B 0.04 43

c 0.30 44

D 0.62 45

E 0.64 46

F 0.66 47

G 0.82 48

H 0.83 49

I 0.84 50

J 1.00 51

K 1.50 52

L 2.00 53

M 4.00 54 163

We draw the reader's attention to columns 13, 14, and 15 of the profiles, that is, the last three data points on the right of each profile in Figure 6/21. Where the original data gives a trough for these points, all the. profiles B through F show a definite crest. The trough in columns 1 0 , 1 1 , and 1 2 , however, remains a trough throughout. The highest point of the profile, in column 8 , is displaced to column 9 in all the profiles from B through I. The possibility of polarity reversal, mentioned earlier, is therefore realised in at least one part of the generalized model. The profile chosen here has illustrated the point that the possibility of polarity reversal has in fact occurred and therefore no other profiles are deemed necessary.

Table 6.6 gives the elevation of the data points for columns 13, 14, and 15 of the selected profile.

Table 6,6 illustrates that fears of possible polarity reversals should be tempered by practical con­ siderations. Maps 43 to 54 are at an approximate scale of 1:240,000 with a contour interval of 100 feet. This means that the trough in columns 13 to 15 may not be able to be depicted at this scale with the degrees of generali­ zation shown in Maps 43 to 51. ELEVATIONS OF SELECTED DATA POINTS IN PROFILE SHOWN IN FIGURE 6/23

Elevations (feet) in columns Height Profile Range Map No. 13 14 15

A 1300.0 995.0 1210.0 305 original data

B 1097.0 1131.0 1050.0 81 43

C 1099.0 1125.0 1056.0 69 44

D 1117.0 1115.0 1082.0 35 45

E 1021.0 1116.0 1081.0 40 46

F 1123.0 1116.0 1084.0 39 47

G 1136.0 1104.0 1102.0 34 48

H 1137.0 1104.0 1105.0 33 49

I 1138.0 1103.0 1104.0 35 50

J 1156.0 1088.0 1125.0 68 51

K 1237.0 1038.0 1181.0 199 52

L 1286.0 1002.0 1203.0 284 53

M 1300.0 995.0 1210.0 305 54 6.7 Summary

In this chapter we have shown that all the quoted

published methods which can be used for the design of

a nines filter can be obtained by a suitable selection

of the value of a in the Gaussian function. The value of

a is clearly related to the correlation distance of the covariance function.

The Gaussian function, with the constraint men­

tioned in Section 6.3, is being proposed in this study as

the basis for the design of a nines filter.

We have also shown the importance of considering

the frequency response curves of the filters so that

possible polarity reversal can be avoided.

In Section 6.1 the possible use of a filter larger

than a nines filter was discussed. Using the method out­

lined in this chapter we could design a filter with 25 weights based on a value of a of, say, 0.83. we would

then get the filter shown in Table 6.7.

In Table 6.7 we clearly see that the effect of the

outer weights is very small. And the same would apply to

larger filters.

The above example would justify the use of the

nines filter as an efficient and practical filter for the

cartographic generalization of the relief continuum on

topographic maps. 166

TABLE 6,7

FILTER WITH 25 WEIGHTS

0 . 0 0 1 0.008 0.015 0.008 0 . 0 0 1

0.008 0.056 0.109 0.056 0.008

0.015 0.109 0 . 2 1 2 0.109 0.015

0.008 0.056 0.109 0.056 0.008

0 . 0 0 1 0.008 0.015 0.008 0 . 0 0 1

The concepts discussed in this chapter will be used to outline a system for generalization in the next chapter. CHAPTER SEVEN

PROPOSED SYSTEM FOR THE CARTOGRAPHIC

GENERALIZATION OF THE RELIEF

CONTINUUM ON TOPOGRAPHIC MAPS

7.1 Outline of the System

7.1.1 Filter Selection and Generalization

As mentioned in Chapter Six, the recommended filter is the nines filter. Again the assumption is that all the elements, except for the relief, have been genera­ lized.

If the area is not too large— "large" will depend on the computer facilities available— then a number of filters can be used on the whole area in order to find the most suitable. "Most suitable" means the filter which gives the degree of generalization which seems to fit in with the degree of generalization of the other map elements. If the area is too large, selected represen­ tative sections of the terrain can be chosen.

Account should be taken of the frequency response of the filter and for this. Figure 6/10 is especially 168 important. If it turns out that a filter having a nega­ tive response in Figure 6/10 is to be chosen— for example, a filter with a = 0.04— then one may decide to use some check profiles to examine possible cases of polarity reversal, as was done in Section 6 .6 .

It is practical to generalize an area slightly larger than required so as to eliminate the problem of loss of data along the edges. If this is not possible, then - corner and edge filters can be employed as described in

Chapter Six.

The above will involve a number of empirical tests.

In the practical implementation of this proposal, these tests can be carried out most efficiently in some sort of an interactive system and preferably a system with interactive graphics. All the proposals made in this study can be adapted to an interactive mode or they can be quite easily used in batch mode.

The flexibility of this system lies in the easy use of different filters over representative areas so that the most suitable filter can be chosen. Once this is done then this chosen filter can be applied to the whole map or the whole map series. 7.1.2 Analysis of the Generalized Map

The quantitative measures related to the character of the terrain were discussed in Chapter Three and some of these measures were selected for the so called first and second tables of quantitative measures.

In generalization one wants to retain the approxi­ mate mean elevation of the original terrain. So it is important to compare the mean elevations of the original and generalized terrains.

As a digital terrain model is generalized the following measures will usually become smaller:

data variance;

spectrum variance;

R.M.S of elevations;

wavelength ratios;

harmonic surface ratios.

So that by examining these measures one can keep a

"quantitative check" on the degree of generalization.

We again draw attention to Figures 5/1 (Map 11) and 5/2

(Map 12) which visually may appear to be generalized to some extent, but an examination of Tables 5.1 and 5.2 show that some of the above mentioned measures have not become smaller, thus indicating a sort of "hypo-generalization." 170

It is therefore clear that the analysis of the quantitative measures and the visual analysis must go hand in hand and must be regarded as complementary parts of the total analysis of the generalized map.

7.2 Proposed System Applied to a Digitized Portion of a Topographic Map

The area chosen is described in Tobler (1968),

The original digital terrain model is taken from the 1 in

62,500 USGS Alma, Wisconsin sheet with 20 ft contours. A portion of this topographic map was digitized every 0.1 inch on the map (that is 520.8 ft on the ground). The digital terrain model used in this section is a 70 rows by 70 columns model taken from the original 80 x 80 model prepared by Tobler (1968). The contours of this model are shown in Figure 7/1 at scales of approximately 1 in

138,000 and approximately 1 in 276,000 with a contour in­ terval of 200 ft. The nature of the terrain is such that the contours have not been labeled within the map, A few lables are given outside the neat line.

The map at approximately 1 in 138,000 is regarded as the "truth" and we are assuming that the generalized map is to be a half this scale, that is, approximately

1 in 276,000. The scale of lin 138,000was chosen because this area was also used by Rayner (1972) at this scale and some comraents on this will be made in Section 7.3. ALMA - Wise 800 1000

1000 1200

1 2 0 0 800

APPROX SCALE I IN 138000 CONTOUR INTERVAL =■ 200 FEET

Inset Approx. scale 1 in 276,000

Figure 7/1

The Alma Digital Terrain Model For the initial generalizations of this map, filters were used based on the following values of a in the

Gaussian function (see Chapter Six): 0.04 (moving average with equal weights); 0.30; 0.64; 0.83 (binomially weighted filter); 1.00. The frequency response curves of the one­ dimensional case of these filters have all been previously discussed in Section 6.5. In addition, the moving average with equal weights was used twice on the original area.

The results are given in Figures 7/2 to 7/7.

The flexibility of the system can be demonstrated as follows. Considering only the contours at the scale of approximately 1 in 138,000 (Figure 7/1)— that is, dis­ regarding all other topographic map elements— we can observe that a one half reduction of this map, to a scale of approximately 1 in 276,000 (Figure 7/1), seems to be a bit cluttered. One could decide that, visually.

Figures 7/5, 7/6 and 7/7 would be acceptable generaliza­ tions. One may decide that, for this scale. Figures 7/2 and 7/3 are over generalized.

But all the above observations may have to be changed when the other topographic map elements are taken

into account. With these elements to consider. Figure

7/3 may turn out to be the best generalization for this

scale. The flexibility of the proposed system is therefore

shown to be in the relative ease with which one can carry 55 a = 0.04 twice 60 ~ 0.04

Figure 7/2 Figure 7/3

61 a = 0. 30

Figure 7/4 Figure 7/5

, 63 a = 0.83

Figure 7/6 Figure 7/7

Generalizations of Alma Model Scale approx, 1 in 276,000 174

out a number of empirical investigations in arriving at

the "correct" generalization of the relief continuum.

Figures 7/2 to 7/7 have been arranged in order—

from the most generalized to the least generalized. This

is also evident in Tables 7.1 and 7.2 which are based on

the first and second tables of quantitative measures

given in earlier chapters.

"Akward" scales have been used in this example in

order to make some comparisons in Section 7.3 with pre­

viously published work. However, the principle adopted

here is that the generalized map is to be half the scale

of the "true" map. But if we assume that a digital terrain

model at a scale of 1 in 62,500 is available of the whole

area to be covered by, say, a 1 in 250,000 sheet, then we must realize that the area shown in all the figures

in this chapter represents about 1/142 of the area of

the final 1 in 250,000 map sheet. So that in this situ­

ation the Alma sheet would not necessarily be a represen­

tative area.

Figures 7/8, 7/9, 7/10 show the difficulties en­

countered when comparing the results of investigations with

existing maps. All these figures are at a scale of 1 in

250,000. The contour interval in each case is 50 feet.

Figure 7/8 is the generalized map of the Alma area using

a filter with a - 0.04. Figure 7/9 is the relevant section TABLE 7.1*

QUANTITATIVE MEASURES (PART ONE) — ALMA

R.M.S Spectrum Map No. Elevation Variance Elevation Variance

65 0.04 twice 1,029 13,763 1,035 10,852

60 0.04 1,030 15,467 1,037 13,150

61 0.30 1,028 15,501 1,035 11,953

62 0.64 1,032 15,933 1,039 12,214

63 0.83 1,032 16,244 1,040 12,345

64 1.00 1,031 16,618 1,039 12,488

Alma— Original - 1,031 20,120 1,040 13,441

Ail units based on feet. TABLE 7.2* .

QUANTITATIVE MEASURES (PART TWO) — ALMA

Cumulative % Map No. Wavelength Ratios Harmonic Surface Ratios Contribution of Harmonic Surfaces

1 2 3 4 1 2 3 4 1 2 3 4

65 0.04 Twice 0.982 0.947 0.899 0.830 0.006 0.996 0.995 0.994 19 31 49 59

60 0.04 0.990 0.972 0.948 0.910 0.998 0.998 0.997 0.997 17 28 45 54

61 0.30 0.986 0.969 0.945 0.908 0.994 0.994 0.994 0.993 17 28 45 54

62 0.64 0.998 0.982 0.960 0.924 1.000 1.000 1.000 1.000 17 27 44 53

63 0.83 0.997 0.984 0.964 0.934 1.000 1.000 1.000 1.000 16 27 43 52

64 1.00 0.997 0.985 0.968 0.942 1.000 1.000 1.000 1.000 16 26 42 51

A l m a - Original 13 22 36 44

All units based < Figure 7/8* Figure 7/9* Figure 7/10* '

Alma Generalized From USGS Sheet Alma Generalized a = 0.04 NL 15-12 a = 0.83.

Scale 1 in 250,000 with 50 ft contour interval 178

from the 1:250,000 USGS EAU CLAIRE sheet NL-15-12. Figure

7/10 is the generalized map using a filter with a = 0.83.

The EAU CLAIRE sheet was compiled in 1954 by photogrammet- ric methods and the generalization was probably done by manual methods as described in Chapter Two. As a genera­ lization at a scale of 1 in 250,000 using Figure 7/9 as a criterion. Figure 7/10 seehis to be too dense in parts.

Figure 7/8 seems to give acceptable results and is in all likelihood more accurate than Figure 7/9.

7.3 Summary and Analysis

The proposed system for the cartographic generaliza­ tion of the relief continuum on topographic maps has been demonstrated and applied to a section of the Alma Quad­ rangle. In this application, we have not given a solution to a problem— because information on the generalization of the other topographic map elements is not available.

But the flexibility of the system has been demonstrated— that is, the cartographer has at his disposal a series of visual images and quantitative measures which help in the decision making process essential in all relief generalization.

In Section 7.2 mention was made that the Alma area was used as an example by Rayner (1972). The original 179 terrain in Rayner (1971 and 1972) agrees exactly with

Figure 7/1. In these above mentioned publications a smoothed version of the original terrain is given with certain frequencies removed ", . . as an illustration of frequency rejection as a method of trend surface mapping"

(Rayner, 1972), And it is a good example of what can be done by transforming a digital terrain model into the frequency domain and then smoothing by retransformation using only selected frequencies, Rayner (1972) does not propose this method as an example of the generalization of the relief continuum on topographic maps. It is, however, clear that the retransformed (smoothed) model

(given in Rayner, 19 71 and 1972) is not an acceptable generalization of the area suitable for topographic maps.

The reason for this last statement is that a visual analysis of the area (a subjective criterion), shows that the shapes (or forms or character) of the original model are not reproduced consistent with the mapped scale. Or to put it another way, the positional accuracy of the smoothed contours is not good when compared with the original model. And as a result, the river system of the original terrain (given in Rayner, 1972) does not fit the smoothed model very well.

However, the river system mentioned above has a very good fit to all the generalizations given in Figures 180

7/2 to 1/1, which fact can be regarded as additional evidence that the proposed system, as demonstrated in this

study, seems to be highly suitable for the cartographic generalization of the relief continuum on topographic maps.

The river system given in Rayner (1972) is shown in Figures 7/11 and 7/12 superimposed on the generaliza­ tions given in Figures 7/2 and 7/6. In cartographic generalization, when the river system is combined with the contours one would expect a small amount of editing to bring the two topographic map elements into absolute coincidence. 65 0.04 twice

Figure '7/11*

River System from Rayner (1972) Superimposed On Alma Generalization With a = 0.04 Twice

63 0.83

Figure 7/12*

River System From Rayner (1972) Superimposed On Alma Generalization With a = 0.83

Approx. scale 1 in 218,000 CHAPTER EIGHT

SUMMARY AND SUGGESTIONS FOR

ADDITIONAL RESEARCH

General aspects of cartographic generalization have been outlined and applications to the cartographic generalization of the relief continuum on topographic maps has been discussed. In a review of the present state of the art, the conclusion was that mostly manual methods are used but with computer-assisted methods being used more and more. Instead of defining "the character of the terrain" emphasis was placed on trying to determine the change in the character of the terrain and some quanti­ tative measures for doing this were gathered together.

The basic concepts of least squares collocation were de­ scribed with a proposal for the application of this method in the cartographic generalization of the relief continuum on topographic maps.

The initial hypothesis, stated in Section 1.4.1, that the least squares collocation algorithm could be used for cartographic generalization, has been shown to be possible but not efficient,

182 183

Subsidiary research questions posed in Section 1.4.2 can be answered as follows:

A. A covariance type function— the Gaussian

function can be used in the design of a

digital filter for generalization.

B. The degree of generalization can best be

quantified— not by one "magic number"—

but by a group of quantitative measures.

C. These quantitative measures, which can

also be used to describe the change in

the character of the terrain, include:

the mean elevation;

the data variance;

the spectrum variance;

the R.M.S of the elevations;

the R.M.S of the residuals (from

the reference plane);

the wavelength ratios;

the harmonic surface ratios.

Elements of spectral analysis have been used as a supplement (by providing some quantitative measures) to visual observation. In the decision making process neces­ sary in generalization, these quantitative measures have been shown to be useful in providing information on the 184

generalization not easily available from a visual analysis

only. For. example, these quantitative measures can be

used as a check that irregularities have not been accen­

tuated— which may not be easily noticed visually. These measures can also be used to determine which map has been

generalized to a greater degree-which may also not be

apparent from a visual examination. It does seem, however,

that a visual examination will finally be the deciding

factor in determining if the generalization of the relief

continuum "fits in" with the generalization of the other

elements.

By selecting values for the a-index of the Gaussian

function, it has been shown that the degree of generaliza­

tion can be predetermined and controlled. It has also been

shown that the Gaussian function, used as the basis for

the design of a nines filter, can be applied to obtain most of the quoted published filters which could be used

for the generalization of the relief continuum on topo­

graphic maps.

Instead of a nines filter one could use a larger

filter as mentioned in Section 6.7, but for topographic

maps this may not be a practical idea because the weights

in the filter furthest from the principal weight are small

and will therefore not have much effect on the final result

which is the contour line. 185

The nines filter has been shown to be a practical tool for the generalization of the relief continuum and • can give results probably more accurate than some pub­ lished maps— see Sections 7.2 and 7.3.

In this study we have used the nines filter for gridded data. In Section 6.3, mention was made of the use of a filter designed for the case where the AX and AY grid intervals are not equal and for non-gridded data the use of a "dynamic filter" was also mentioned. Although the least squares collocation method has been found to be in­ efficient in this study for the generalization of the relief continuum, it can be pointed out that this method may be most suitable for deriving gridded data from non- gridded data— with the advantage that the value obtained for each grid intersection will be a unique value. This is an idea which needs further research.

A proposed system for the cartographic generaliza­ tion of the relief continuum was described in Chapter Seven and applied to a small area originally digitized from a

1 in 62,500 quadrangle topographic map.

This study has dealt with the cartographic generali­ zation of the relief continuum for topographic maps at derived scales. In almost all the cases dealt with in this study, the derived scale has been one-half of the scale of the original digital terrain model (the "truth"). 186

In one case (see Section 7.2), the derived scale was one-quarter of the scale of the original digital terrain model. It seems clear that it is not practical to use the

same digital terrain model directly for all derived scales because the grid interval at the derived scale may approach an unplottable quantity— this means that the original digital terrain model is too dense. For example, if the derived scale is to be one-tenth of the scale of the original digital terrain model, then it may be more prac­ tical to use some filtering or generalization process to

"derive" a digital terrain model with a larger absolute grid interval (that is, a larger interval on the earth surface) for use with a certain group of maps at various derived scales. This seems to be a fruitful area for further investigation.

What has been achieved in this study is the outline of a method which enables the cartographer to undertake

"generalization by measure and number" as advocated by

Topfer (1974). These measures and numbers add to the car­ tographer's visual evaluation. The method arrived at in this study enables the cartographer to control the degree of generalization which he wants and assists him in judging the quality and suitability of the generalization. The use of nines filter, as described in this study, gives the cartographer an easy way of selecting and varying the degree 187

of generalization which he judges to fit in with the degree

of generalization of the other contents of the topographic

map. The nines filter will also ensure efficient handling

of large amounts of data, as small portions of the total

area can be generalized at a time and one does not have

to process the whole data set to arrive at parameters

for the generalisation.

It seems that, for the relief continuum at least, cartographical generalization will not be able to be done

in a completely automated manner by using only "measure and number" in the initial stages. Visual judgment is an important part of the process and an advance preview of aspects of the final product would seem to be as important as map editing.

An additional area for further research is the use of an interactive graphics capability with the proposed system. This was briefly mentioned in Chapter Seven.

In addition to the above mentioned suggestions for the cartographic generalization of the relief continuum, the results of this, and the investigations of the methods in the previous chapters, lead us to the following addi­ tional suggestions for further study. Although the method of least squares collocation has not been found to be the most efficient method in this case, there would seem to be possibilities of using the method in the following cartographic situations. 1) Given a digital terrain model, it may be possible

to represent the surface by a lesser number of

points than the original points by using a "good"

covariance function obtained from considering all

points.

2) Conversely, one could density points in a digital

terrain model using least squares collocation methods.

This is equivalent to predicting values on a surface

at points other than the data points. This same

principle could be used in the situation where we

have random elevations in an area and we wish to

predict values at grid intersections as mentioned

above. Using least squares collocation it is

possible to get unique values for the grid inter­

sections rather than values which depend on how

many points surrounding the grid intersections are

being used.

3) It may also be possible to apply the above idea to

the development of a contouring algorithm.

The aspects of spectral analysis, as used in this study, could possibly be applied in the following situation:

4) The measurement or comparison of morphological

features (or volumes or shapes) on the same or

different maps. To end with a suggestion which may seem rather esoteric, we suggest that

5) it may be possible to use the method of least

squares collocation in conjunction with aspects

of spectral analysis to interpolate a whole statis­

tical surface between, say, two or more epochs.

For example, population isopleths between two

censuses, or to extrapolate a moving changing

surface like a weather system. The system could

be tracked for a certain time period and then

extrapolated for some date in the immediate future.

What has become clear during the course of this study is that there are a number of scientific methods used in various fields which should be brought to the attention of cartographers and it is hoped that this exploratory probe into some aspects of cartographic generalization has done this for the methods used. BIBLIOGRAPHY

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