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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 76-24,552 AYENI, Olubodun Olarewaju, 1941- CONSIDERATIONS FOR AUTOMATED DIGITAL TERRAIN MODaS WITH APPLICATIONS IN DIFFERENTIAL PHOTO MAPPING. The Ohio State University, Ph.D., 1976 Geodesy

Xerox University Microfilms r Ann Arbor, Michigan 48106 CONSIDERATIONS FOR AUTOMATED DIGITAL TERRAIN MODELS WITH APPLICATIONS IN DIFFERENTIAL PHOTO MAPPING

-DISSERTATION

Presented In Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy In the Graduate

School of The Ohio State University

By

Olubodun Olarewaju Ayenl, B.A., M.Sc., M.S.

* Vc * *

•The Ohio State University

1976

Reading Committee: Approved By

Professor S. K. Ghosh (Adviser) Professor D. C. Merchant Professor J. S. Rustagi Adviser Department of Geodetic Science Dedicated to the Blessed Memory

of my Beloved Father

ZACCHEUS OGUNLADE AINA AYENI

To whom I owe so much not only as

a Father hut also as a Teacher ACKNOWLEDGEMENTS

I wish to acknowledge the invaluable support provided by my academic adviser, Prof. S. K. Ghosh, throughout my academic career at Ohio State.

In particular, I am grateful to him for his prompt guidance and unfailing technical advice at every stage of this research. I also wish to express my gratitude to the other members of the reading committee, Prof. D.C.

Merchant and Prof. J.S. Rustagi for their instructive corrections and suggestions. I .received initial inspiration for this research through the

Master's theses of Messrs. Jiwalai and El-Ghazali. I am therefore grate­ ful to them. The assistance received from Prof. Rayner on Spectral Ana­ lysis in 2 dimensions is also gratefully acknowledged.

I would like to acknowledge with gratitude the financial support pro­ vided by the Nigerian Federal Government and Prof. Uotila, Chairman, Geo- detrlc Science Department, without which the successful completion of my studies would have been impossible. I also gratefully acknowledge exten­ sive use of the facilities of the Instruction and Research Computer Center of the Ohio State University.

Last, but by no means least, I wish to express my profound gratitude to my beloved wife, Aduke and to my lovely sons Tayo and Kunle for their patience and endurance when I had to devote inordinate amounts of time to this research at their expense. ______

i VITA

Dec. 31, 1 9 4 1 ...... Born, Ikole-Ekfiti, Nigeria.

1968 ...... B.A. (Honors) Geography, University of Ibadan, Ibadan, Nigeria.

1968-70 ...... Senior Mathematics and Geography Teacher, St. Mary's Girls' High School, Ikole-Ekiti, Nigeria.

1972 ...... M.Sc. Engineering Surveying, University of Lagos, Lagos, Nigeria.

1972 ...... Assistant Lecturer, Univeristy of Lagos, Lagos, Nigeria.

1974 ...... Graduate Research Associate, Ohio State University, Columbus, Ohio.

1975 ...... M.Sc. Geodetic Science, Ohio State University, Columbus, Ohio

1975-76 , ...... Graduate Research/Teaching Associate, Ohio State University, Columbus, Ohio.

PUBLICATIONS

1. Ayeni 0. 0. "A Nigerian Grid System in 2° T.M. Zones". M. Sc. Thesis, University of Lagos, Lagos, Nigeria, 1972.

2. Ayeni 0. 0. "Computer-Assisted Close-range Photogrammetrlc Mapping of Cows". M.Sc. Thesis, Ohio State University, Columbus, Ohio, 1975.

3. Ayeni 0.. 0. "Computer-Assisted Close-range Photogrammetrlc Mapping of Cows for Genetic Studies", Instrumentation Society of America Award, 1975.

4. Prof. Ghosh S. and Ayeni 0. "Procedures for Computer-Assisted Close- range Photogrammetrlc Mapping", I.S.P./A.S.P. Symposium, July/Aug. 1975, Illinois.

5. Ayeni 0. 0. "Objective Terrain Description and Classification for Digital Terrain Models". Presented paper XIII I.S.P. Congress, Helsinki, Finland, 1976.

ii PUBLICATIONS (Continued)

6. Ayeni 0. 0. "Optimum Sampling for Digital Terrain Models; A trend towards automation", XIII I.S.P. Congress, Helsinki, Finland, 1976.

* * * *

FIELDS OF STUDY

Major Field: Geodetic Science

Studies in General Photogrammetry Professor Sanjib K. Ghosh

Studies in Analytical Photogrammetry Professor Dean C. Merchant

Studies in Adjustment Computation Professor Urho A. Uotila

Studies in General Geodesy and Geodetic Astronomy Professor Ivan I. Mueller

Studies in Physciaal Geodesy and Surveying Professor Gabriel F. Obenson

Studies in Mathematics and Mathematical Projections Professor Richard H. Rapp

Studies in Statistics Professor J. S. Rustagi

iii TABLE OF CONTENTS

Page ACKNOWLEDGMENTS ...... i

VITA ...... ii

LIST OF TABLES...... vii

LIST OF FIGURES ...... viii

Chapter

I. SCOPE AND OBJECTIVE...... 1

1.1 Introduction...... 1 1.2 Scope and Objective ...... 5

II. TERRAIN DESCRIPTION AND CLASSIFICATION ...... 10

2.1 Introduction...... 10 2.2 Parameters of Surface Roughness ...... H 2.21 Slope, Gradient, and Curvature...... H 2.22 Direction Cosine and Eigen Vector Method. . . 12 2.23 Bump Frequency Distribution ...... 14 2.24 Distribution of Planes...... 14 2.25 Surface Area...... 17 2.26 Breaklines...... 18 2.27 The Harmonic Vector Magnitude ...... 19 2.28 Amplitude Power Spectrum...... , 22 2.281 Spectial Analysis in 2-Dimension...... 24 2.3 Parameters for Describing Spatial Distribution of Surface Irregularities. . . 27 2.31 Autocorrelation in one-dimension...... 27 2.32 Autocorrelation in two-dimension...... 29 2.4 Computer Program for Terrain Analysis .... 30 2.5 Terrain Classification...... 34 2.51 Statistical Method Used for Terrain Classification...... 36 2.52 Computer Program for Evaluation and Improvement of Classification ...... 38

III OPTIMUM TERRAIN SAMPLING FOR DIGITAL TERRAIN MODELS 41

3.1 Introduction...... 41 3.2 Optimum Sample Size ...... 42 3.21 Harmonic Vector Magnitude Method...... 42

iv 3.22 Multiple Linear Regression Equation Method...... 46 3.3 Optimum Sample Type ...... 59 3.31 Major Sampling Patterns ...... 60 3.32 Das' Theorems for Relative Efficiency of Sampling Patterns ...... 64 3.33 Theoretical Expectation and Emperical Result of the Relative Efficiency of Sampling Patterns 66 3.4 Automatic Optimum Sampling for D.T.M...... 71

IV INTERPOLATION PROCEDURES FOR DIGITAL TERRAIN MODELS. . . 74

4.1 Definition of Basic Terminologies ...... 74 4.2 Interpolation Formulas...... 75 4.21 Numerical Polynomial Interpolation Formulas .. . 75 4.211 Newton’s Forward Interpolation Formula...... 77 4.212 Spline Interpolation Formula...... 77 4.213 Aitken-Neville Formula...... 78 4.214 Stirling's Interpolation Formula...... 79 4.215 Lagrange Formula ...... 80 4.216 Divided Difference Formula...... 80 4.22 Polynomial Interpolation Formulas ...... 8 1 4.23 Logarithmic Formulas...... 82 4.24 Exponential Formulas...... 82 4.25 Fourier Series Formulas ...... 83 4.26 Multiguadric Formulas ...... 83 4.3 Optimum Interpolation and Least Squares Method. . 84 4.31 Stepwise Regression Procedures for Optimum Interpolation ...... 87 4.32 Least Squares Collocation for Interpolation .. . 96 4.33 Patchwise Interpolation ...... -101 4.4 Interpolation, Sampling Pattern, Samping Error, and Accuracy...... 101 4.41 Interpolation and Sampling P a t t e r n ...... 101 4.42 Interpolation and Sampling Error...... 103 4.43 Interpolation and Accuracy Considerations . . . .103 4.5 Concept of Automated D.T.M...... 105

V DIFFERENTIAL PHOTO APPLICATIONS...... 110

5.1 Introduction...... 110 5.2 Differential Mapping Procedures ...... Ill 5.21 Procedures for Constructing the Map of a Cow. . .111 5.21 Procedures for Constructing the Map of an Average C o w ...... 117 5.3 Interpolation and Volume Computations ...... 123 5.4 Comparisons between two Objects ...... 126

VI 6.1 Conclusions ...... 131 6.2 Recommendations...... 134

v APPENCIDES Page

A Surfaces for Classification ...... 137 B Sampling Patterns ...... 148 C Computer Programs for ATODTM...... 158

BIBLIOGRAPHY ...... 193

vi LIST OF FIGURES

Figures Page

1.1 Interaction between D.T.M. Components...... 8

2.1 Bump Frequency ...... 15

2.2 Distribution of Planes ...... 16

2.3 Breaklines ...... 19

3.1 Seven Sampling Patterns...... 63

4.1 Least Squares Collocation Quantities ...... 97

4.2 ATODTM Flow Chart...... 109

5.1 Set up of Photograph ...... 112

5.2 Mapping Procedures ...... 113

5.3 Map of Cow #50 ...... 116

5.4 Map of Cow #51 ...... 121

5.5 Map of Average Co w ...... 122

5.6 Multi-Dimensional Plot of Cows 50,51 and 50/51...... 130

Al-15 Terrains for Classification...... 137

vii LIST OF TABLES

Digital Terrain Model Systems ...... 2

Quantitative Characteristics of Fifteen Terrains...... 32 Terrain Classification...... 40

Optimum Sample Size using HVM Criteria...... 45 Dependent and Independent Variables for Regression Equations...... 47 Difference Between Optimum Sample Size by HMV and Regression Equations...... 57

Predictor Variables for Optimum Sample Size ...... 58 Theoretical Expectation and Emperical Results of the Relative Efficiency of Sampling Patterns...... 70

Forward Difference Table...... 76 Stepwise Regression: Composite Modes . . . 94 Interpolation and Sampling Patterns ...... 102 Interpolation and Sampling Errors ...... 105 Interpolation Accuracy...... 106 Classification of Interpolation Methods ...... 107

Interpolation Results ...... 122 Volume Computation...... 126 Table for Comparison...... 128

Empirical Relative Efficiency of 7 Sample Patterns for Interpolation ...... 148

viii CHAPTER I

SCOPE AND OBJECTIVE

1.1 INTRODUCTION

In his pioneering work at the Massachusetts Insititute of

Technology, C. L. Miller [5] in 1958 defines Digital Terrain Model

(D.T.M.) as "A statistical representation of a continuous surface of

the ground by a large number of selected pointes with known X, Y, Z

coordinates." Miller developed D.T.M. techniques as a response to a wide variety of terrain problems in civil engineering notably the eval­

uation of an unlimited number of alternative routes and volume of earth

computations for highway designs and construction. As a result of this

pioneering work, the following decade (1959-69) ushered in a new era

for photogrammetry; an era in. which D.T.M. concepts, were successfully

employed to achieve automatic road location and design systems. Table

1.1 shows the summary of the salient features of the various D.T.M.

systems developed in Europe and U.S.A. during the first decade of its

existence (see [17], [1], and [18] for details). The most significant

characteristic of all these systems is that elevations are measured at

some significant points on the terrain and stored in the computer and

and the elevation of new points needed for design work are interpolated

thus eliminating the necessity to remeasure. TABLE 1.1 DIGITAL TERRAIN MODEL SYSTEM 1959-69 TYPE OF METHOD OF YEAR/TYPE OF Marne COUNTRYINSTITUTION SAMPLING INTERPOLATION COMPUTERAPPLICATION

MIT-Model U.S.A Mass. Constant or 3rd degree 1961 Road Design Institute of Variable Polynomial IBM No Automatic Technology spacing along Road Alignment X or Y scan- lines EGI - France Service Purposeful Cubic Parabola 1965 Automatic System Speciale des Sampling for group of IBM 7094 Road Location Aucoroutes 10 or more and Design SSAR/BCOM Pan's Reference Points Card- Great Paris 1965 Road Design System Britain Elliot Auto­ Elliot 803 without mation Company Elliot 503 Automatic Align­ Bf>ehanwood ment Variation Terra- Great Scott, Wilson & Random 1966 Automatic Road System Britain Kirkpatrick & Sampling Remington R Location and Partners Con­ AO points per UNIVAC 1107 Design sulting Co. acre typical London VIATEK- Finland Viatek-Con- Regular Bilinear 1966 Automatic Road Model sulting Co. Grid Pattern Interpolation IBM 360-50 Location and Helsinli 20-400p/10ha Design Volume Comp. W- Sweden National Rd. Combination Moving Average 1967 Automatic Road Model Administration of Rand, and or by Second SAAB Location Stockholm Breakline pts. Function 100-150/ DTM- German Technische Purposeful Linear 1967 Automatic Stuttgact Hochschule Sampling Interpolation Teletunken Road Location Stuttgart along TR 4 "terrain lines" TABLE 1.1 (Cont'd) DIGITAL TERRAIN MODEL SYSTEMS 1959-69 TYPE OF YEAR/TYPE OF NAME COUNTRY INSTITUTION SAMPLING INTERPOLATION COMPUTER APPLICATION

Nakamira Japan University of Grid Pattern Patchwise 1968 Models Tokyo Interpolation TS-1 with 3rd degree polynomial

TS-2 Japan University of Sampling Fourier Tokyo along Series Contours

CS-Model Czechos­ Czecho Research Regular Grid 2nd Degree 1969 lovakian Institute of and Break- polynomial Geodesy line Topography Sampling and Photo'gram- metry

DELFT The Ministry of Patchwise with 1969 Models Netherlands Public Work one or 3rd Degree poly or Bilinear Interpolation The second decade (1969-79) of the existence of D.T.M. has

so far witnessed substantial progress in two major areas. The first

area is the extension of D.T.M. applications to some new terrain

problems other than highway constructions. The reason for this is not difficult to realize. Once the digital data have been generated by (i) digitizing a stereo-model in a stereo-plotting instrument, (il) measuring photo-image coordinates in a fully-analytical solution and

(iii) by digitizing existing topo-maps, the photogrammetrist is pro­

vided with a data-bank which has unlimited flexibility and from where

further numerical, statistical, and graphical analyses can be made

through the use of the highspeed computer. For example, computation

of surface area, volume of earthwork, slope, curvature, distances,

angles, planes, velocity, perimeter, center of gravity, and graphical

outputs such as contours cross-sections and perspective drawings, can be easily derived from D.T.M. data bank. It is, therefore, not sur­

prising that D.T.M. concepts have found ready applications in geology, metallurgy, animal science, and biometrical studies. Such applications are exemplified par excellence by the research efforts of Duncan et al

[11] and Kratky [6] on limbs replica in orthopaedics, Goulet et al [10] and Takasaki [7] on measurements of the human body; Uppert et al [12] on studies in musculaskeletal systems, Karara [13] on aortic heart valve geometry, Ayeni [57] on measurements on the body of a cow, Jancaitis et al [44] on contouring, Bradenberger [45] on the deformation of dams and

Hardy [22] on the earth's topography. The other area in which the second decade has witnessed development has to do with research into the efficiency of D.T.M. It is, however, surprising that detailed research is sparse in this direction. For example, the only research attempt on terrain classi­ fication was done by Silar [15] in 1969 in which he identified four major classes of terrains by using both subjective and objective methods of classification. In most D.T.M. studies researchers often describe the terrains of their investigation with subjective terms such as "flat," "undulating," and "mountainous" thus, precluding adequate evaluation of the optimum number of data points and their associated distribution (scatter) as well as the reactions of various interpolation techniques to different classes of terrains. The only detailed research on adequate sampling for D.T.M. is credited to

Makarovic [3] and Ghazali [2] in 1973 and 1974 respectively working at I.T.C. On the efficiency of interpolation on a square grid data, the work of Jiwalai [1] 1972 and Leberl [4] 1973 also at I.T.C. constitute a notable contribution to attaining an efficient D.T.M.

1.2 SCOPE AND OBJECTIVE

The apparent apathy towards detailed research into various components of D.T.M. may be explained by the inadequate view expressed by photogrammetrists of what constitute Digital Terrain Models.

Contrary to Miller's view [5] D.T.M. is much more than a "statis­ tical representation of a continuous surface — by a large number of points." Linkwitz's definition [14] of D.T.M. as "the analytical numerical description of an irregular surface based on the measurement of discrete points on this surface,” hardly provides an

improvement over that of his predecessor. Jiwalai [1] 1972 considers

D.T.M. as "the terra used to describe a geometric representation of a part of the earth’s surface which can be stored in the memory of a computer in such a way that the height of any point on that surface can be automatically derived provided its planimetric coordinates are

given." These definitions although good in themselves, are considered inadequate because they tend to ignore the problem areas of D.T.M.

The only definition in literature which accords a fairly good recog­ nition to these problem areas was given by Blaschke [17] who defines

D.T.M. as "the term for the representation of TERRAIN SECTION for

PLANNING and DESIGN purposes, by means of storing measured coordinates,

X, Y, Z of CHARACTERISTIC POINTS, in a computer in SUFFICIENT QUANTITY and SIGNIFICANCE so that by AUTOMATIC INTERPOLATION the elevation of any point determined by its coordinates X, Y can be obtained with

NECESSARY ACCURACY and GOOD ADAPTATION to the NATURAL TERRAIN

SURFACE." (Capitals give by this author). A more precise definition may be given as this: Digital Terrain Model is the NUMERICAL (or digi­ tal) and MATHEMATICAL REPRESENTATION of a TERRAIN by making use of

ADEQUATE ELEVATION and PLANIMETRIC MEASUREMENTS, COMPATIBLE in NUMBER and DISTRIBUTION with that terrain, so that one dimension of any other point of known other dimensions can be AUTOMATICALLY INTERPOLATED with

REQUIRED or SPECIFIED ACCURACY for any given APPLICATION.

From such a comprehensive definition as this we can identify certain characteristics of an efficient D.T.M. viz: 1. An efficient D.T.M. should faithfully represent the

terrain numerically as well as mathematically.

2. Acquisition of terrain data should satisfy the condition

that the digital terrain's data should adequately repre­

sent the terrain both in number and spatial distribution.

3. The mathematical representation should be such that it

approximates the terrain with sufficiently high accuracy

commensurate with the accuracy required for any given

application. The numerical or digital representation

constitutes a bridge between the terrain and the math­

ematical representation which is also a bridge between

the numerical representation and the terrain (see

Figure 1.1). It should be clear that the efficiency of

D.T.M. depends on five factors viz:

(a) the type of terrain

(b) the density of sample terrain data

(c) the distribution of sample data,

(d) the desired and obtainable accuracy, and

(e) the type of interpolation method.

The interactions between these five factors are depicted by arrows in Figure (1.1) where double arrows indicate mutual interactions.

Accordingly, the first part of this research will be geared toward find­ ing solutions to the following D.T.M. problems:

1. The problem of quantitatively describing a terrain

• as well as objectively classifying terrain types

2. The problem of finding the optimum sample size of

data for any given terrain type 3. The problem of determining the optimum distribution of

sample data compatible with any given terrain

4. The problem of finding the optimum interpolation formula

that will meet the requirements and limitations imposed

by 2 and 3 above

5. The possibility of automating the procedures involved

in 1, 2, 3, and 4 above so that the stereo-plotter

operator could generate and analyze D.T.M. data with

maximum efficiency and minimum effort.

FIG. 1.1 Interraction Between D.T.M. Components

REAL WORLD

Terrain

Computer World (Medium 2) Interpolation „ Procedure Sample Size

Sample ACCURACY Pattern

Photogrammetric World (medium 1) 9

The second part of this research will focus attention on

applications which are many and varied. However, the particular application of interest here relates to differential mapping with close-range photogrammetry. The problem of mapping an object relative

to another of the same kind will be Investigated and application of automated D.T.M. will be exemplified therefrom. CHAPTER II

TERRAIN DESCRIPTION AND CLASSIFICATION

2.1 INTRODUCTION

There are many quantitative parameters which could be used to describe a terrain since the terrain itself is a complex entity with multivariate characteristics. Any parameter used to describe the terrain must, however, satisfy certain conditions some of which were recognized by Hubson [26]:

(i) The parameter should be conceptually descriptive. That is

it should conjure a mental image of the physical

characteristics of the terrain as much as possible.

(ii) The parameter should be easily measurably in digital form.

(iii) The parameter should be suitable for further numerical and

statistical analyses.

(iv) The parameter should be comparable at different scales.

Quantitative parameters for terrain description may be

classified into two major categories:

(a) those which describe surface "roughness" and

(b) those which describe the spatial distribution of surface

terrain irregularities. Because of the complex

character of the terrain its "roughness" should be

10 11

regarded as a vector in multi-dimensional space.

This makes it almost impossible to give a single concise

definition of surface roughness. The statement that

this surface is rougher than the other is only true in

respect of the roughness parameter being considered.

In general one may assert that roughness parameters

depict the degree and magnitude of surface irregular­

ities and they are not related to the spatial

distribution of surface irregularities.

2.2 PARAMETERS OF SURFACE ROUGHNESS

2.21 SLOPE, GRADIENT, AND CURVATURE

Slope is defined mathematically as the angle which a line on a surface makes with the horizontal and gradient is the tangent of that angle. This means that the gradient of a surface at any point ranges from zero (flat surface) to infinity (precipitous surface). If the gradient is computed at various points on a surface, it may indi­ cate ’the variability of slope all over the surface.

The gradient of a surface is a good measure of surface roughness although it is difficult to compute because the topographic surface Is not easily represented by a mathematical function.

However, the spatial derivative of a surface may be computed by the formula

S ax 12

Where S = magnitude of the gradient obtained as a resultant

of a pair of orthogonal partial derivatives of Z. The above expression

can be approximated by finite difference formulas as demonstrated by

Tobler [39] and McCormick et al [58].

The curvature which is the rate of change of gradient with

respect to distance may be obtained by differentiating the above

expression (eqn (2.1))

Again we appeal to finite difference formula for obtaining

C. Theoretical curvature indicates convexity of slope. The parameters

of surface roughness are then the means and variances of gradient and

curvature. One obvious disadvantage of the above formula is that gradi­

ent and curvature are always positive which implies that no distinction

is made between negative and positive slopes and also between convex and

concave slopes.

2.22 DIRECTION COSINE AND EIGENVECTOR METHOD

The use of direction cosine in conjunction with eigenvectors

and eigen values are first proposed by Loudon [27] for describing a

surface. Whitten [25] made use of Loudon’s concept of direction

cosines to analyze subsurface fold geometry in geology. The method

used here is a slight modification of Loudon's concept which is briefly described below. 13

First the direction cosines between any two points are computed in X, Y, and Z directions. This forms a Matrix A (Nx3) where

N is the number of points measured on the terrain with known X, Y, Z.

The covariance Matrix B (3x3) of direction cosines is obtained from

A as

£piqi £piri N N

B = £qiri (2.3) N ~N~ N

Sr? Lrlpl ^rlqi N N N

Where p^, q^, r^ are the direction cosines of the ith point referred to X, Y, and Z axes. In order to provide suitable internal reference axes which can be used for comparing other surfaces the original Matrix A is transformed to refer to the principal axes by multiplying the eigen vector Matrix R of Matrix B with Matrix A as follows:

C = AR (2.4) R 11 R 12 R13 where R = = eigen vector Matrix of B so ro R 21 U) R23

R 31 R32 R 33

The eigen values of the resultant Matrix C are the variances of the transformed direction cosines and they constitute a measure of the variation of slope on a surface. According to Loudon [27] the 3rd 14

moment (skewness) measures the degree of symmetry of slope about an

axis while the 4th moment (Kurtosis) indicates the relative abundance

of steep or gentle slopes. In other words, a low value of Kurtosis

suggests that there is no wide variety of slopes on the surface.

2.23 "BUMP FREQUENCY" DISTRIBUTION

The method of "bump frequency" as a measure of surface

roughness was proposed by Hobson [26] who suggested three ways

of computing bump frequency of a surface. One way is to compute the vertical distance of measured elevation points from a horizontal

surface (see Figure 2.1). The second way is to compute the distances

from elevation points on the terrain to best-fit planar surface in a

direction normal to the latter (see Figure 2.1). The third way is to

compute the vertical distances from the elevation points to the best-

fit planar surface (see Figure 2.1). The bump frequency parameters

are then the mean and variance statistics which are good indicators

of surface irregularities. The data required for computing bump

frequency statistics are the X, Y, Z readings on the surface.

2.24 DISTRIBUTION OF PLANES

Hobson [26] has described how distribution of planes

can be used to determine surface roughness. First a set of triangular

intersecting planes are fitted to adjacent groups of three elevation

readings (see Figure 2.2A). Normals to these planes and their

corresponding unit vectors are then computed (see Figure 2.2B). Vector Figure 2.1 A) Three possible orientations of Bumb Frequency; B) Array of elevation readings and their geographic coordinates for Bump Frequency; After Hobson [26] 16

Figure 2.2 A) Intersecting planar surfaces defined by adjacent groups of three elevation readings; B) Area with similar elevations producing high vector strength and low vector dispersion; C) Nonsystematic elevation changes yielding low vector strength and high vector dispersion; After Hobson [26] 17

strength (which is the square root of the sum of squares of the

direction cosines divided by the number of unit vectors), vector

dispersion (which is the variance of the unit vectors) and mean and variance of dip of the triangular planes are computed. According to

Hubson [26], Vector Strength is usually high and vector dispersion

low in areas characterized by similar elevation or equal rates of elevation changes, whereas non-systematic elevation changes yield low vector strength and high vector dispersion (see Figure 2.2C). The

data required for computing these parameters consists of elevation

readings at regularly spaced points.

2.25 SURFACE AREA

The idea of using surface area as a measure of surface

roughness is based on the hypothesis that surface area increases with surface irregularities. Hobson [26] has proposed the use of the ratio

(A1/A) where A* represents the surface area and A the corresponding planar area. The computation of A is obtained simply by the product of length and breadth while the A' can be obtained by a number of methods the most popular being that of partitioning the area concerned into small rectangular or square segments whose area can be easily computed.

The number of such segments will depend on the degree of roughness of the surface. Other methods of finding surface area are discussed in

Section 5.3 of this dissertation. 18

2.26 BREAKLINES

A Terrain Breakline has been defined as a line where there

is a sudden or abrupt change in slope. Breaklines represent, mathemat­

ically, lines where the spatial derivatives are discontinuous.

Physically they are identified with the edges of ditches, dikes,

cliffs, and ridge lines. On a stereo-model they are not easy to

recognize although breaklines can be used as a roughness parameter.

In contrast to contour lines which are dimensional curvillinear rep­ resentation of a 3-dimensional topography, breaklines are strictly spatial curves which are difficult to trace with a floating mark on a stereo-plotting instrument and are equally difficult to represent mathematically. The problem is then to find an objective means of

detecting breaklines. The method used in this research is based on

the concept of slope profile analysis by fitting polynomials to various sections of the profile progressively. For example, if a parabolic curve

Z = a + bX + cX2 (2.5) is fitted by least squares method progressively to portions between

A and D, B and E, C and F, D and G, E and H and then F and I (see

Figure 2.3) it will be discovered that there is significant change in the coefficients of the parabola between F and I. This will correspond

to the change in the form of the profile at H. This method, however, assumes that the surface is covered with sufficient density of points along Y-on X-scanlines as to include breaklines. One other method which could be used, is to set up a slope difference criterion for 19 detecting the presence of a breakline. For example, the criterion might be that where the slope of two successive sections of the profile differs by more than 10° indicates a breakline. Extreme breaklines can also be detected by recording points where the components of spatial derivatives along X and Y change from positive to negative or vice versa.

FIGURE 2.3 BREAKLINES

r" "' ' ' '

PROFILE

2.27 HARMONIC VECTOR MAGNITUDE

The method of using the Harmonic Vector Magnitude to describe the roughness of the terrain is based on the following assumptions:

(i) The total variability of a surface can be separated into

two major components: (a) the systematic trend and

(b) the noise and other random components in the digital

data representing the surface.

(il) If the method of least squares Is used in surface fitting

of a mathematical function (polynomials, double fourier

function, etc.) in two Independent directions to the surface 20

data, the resulting trend surface, which is a generalization

of the original surface, is adequately described by co­

efficients of the mathematical function.

(iii) Data which contain a linear trend distorts the character of

the coefficients which describe the trend surface. It

should be evident that the first assumption is taken for

granted each time a mathematical function is fitted to a

surface by least squares method. The second assumption

echoes the concept of "coefficient space" used by Krumbein

[32] as a method of map generalization. The validity of the

third assumption has been demonstrated by Pierson [46] and

others like Preston [64], Hardy [22], and Hubson [26]. The

mathematical function whose coefficients is used to describe

a topographic surface is given by

Z = a0 + a^x + a2y + f (x,,y) (2.6)

m n

,m. n.

C C . . CS.. SC.., and S., are the coefficients of the i j » i j » ij ij double Fourier Series, and the data is such that 21

1 = 0, 1, 2,...... ,m;

j = 0* 1> 2, >nj

xQ= -M and xm = +M

yQ= -N and Yn = +N

M = fundamental wave-length in x- direction

N = fundamental wave-length in y- direction

m = maximum harmonic in x- direction

n = maximum harmonic in y- direction

It has been shown by Tolstov [65] that

1/A for i=j=o

> i j =

A least squares fit of eqn (2.7) requires at least A [ (m + 1)

(n + 1)] - [ (m + 1) + ( n + 1) ] observations. In using eqn (2.6) for at least squares surface fitting we are merely trying to approximate the deviations of the surface from a plane defined as Z, = a + a, x + a- y., c 1 o 1 2 J by a double fourier series. The fourler coefficients according to assump­ tion (ii) provides a direct measure of the configuration of the topographic t surface. Furthermore, the fourier coefficients provide a special meaning in that the global fit assumes that the surface oscillates harmonically in two mutually perpendicular directions. The fourier coefficients are, therefore, the amplitudes of the oscillations* The harmonic vector magni­ tude is defined as the square root of the sums of the squares of fourier coefficients for terms of specified m and n harmonic. The harmonic vector magnitude then represents the contribution of the fourier coefficients in 22

explaining variability of the deviations of the topographic surface from the linear plane. A relatively flat terrain will be depicted by small harmonic vector magnitude and a rough terrain by a large harmonic vector magnitude.

2.28 AMPLITUDE POWER SPECTRUM

The method of amplitude spectrum is based on all of the three assumptions, made for the harmonic vector magnitude method of describing surface roughness. The difference is that instead of working with the fourier coefficients in the harmonic "coefficient space" we use the fast fourier "coefficient space." Detailed des­ cription of the mathematical theory of the fast fourier transform which is outlined here can be found in Rayner [41], Tukey [48], .

Blackman et al [47] and Jenkins et al [38].

First we start with some familiar expressions usually derived by Talyor series expansion

exp(it) = (l-t^/2 + t4/^j + ....) + i +

(2.8)

and exp (-it) = cost-isint (2.9)

exp (+it) = cost + isint (2.1C

We recall the expression for f (x,y) in eqn (2.7) and we now equate the Z values to an approximate function

Z = P(x,y)^ f (x,y) (2.11)

where 23 M — CC \ f(x,y) Cos f 1inx % \ Cos f :\ ny "j dxdy (2.12) ij

J 1 V M J -MM J-s' N _ Cos ^illxj Sln^j Hy^ j dxdy (2.13) f(x,y) c s ^ = ^ M . -N

M o N Sin dxdy (2.14) f (x,y) (^ ) I f )

f(x,y) Sin O ) Sln(^ dxdy (2.15)

Since the form of the function Z - f(x,y) is not known an analytical integration of eqns (2.11 - 2.15) is not feasible. We, therefore, appeal to the concept of fourier transform for a solution to the problem.

Using the relations (2.9), (2.10) and the familiar

trigonometrical expansions for gin (A ± B) and cos (A * B). We may express the relation in eqn (2.7) in exponential form as

n. iC(k^,k2 ) exp ziTf k2 " - 2/2 W1 ~"n l/2 V I n2 JJ (2.16)

where C(k1,k2) = ^aXk^kg) - ib (k.^ /2

a's and b's are coefficients or amplitudes of cosine and

sine functions. They represent the cosine and sine transforms for the

even and odd function of eqn (2.16). Put in fourier transform expression

Z(x,y) is the fourier (spectrum) transform of C(k^,k2) and vice versa. 24 rl/2 AY p 1/2 £X Z(X,Y) ==1 1 C(k k„) exp i 211 /xk. dki ‘dk_ J-l/2 AyJ-X/ 2 jflX 1 Z — + — I " 1 ^ 2 . 1 7 )

and

/'T2/2 f Tl/2 c(kr k2) = Z(x,y) exp - i 2n xk, yk dx dy

^”T2/2J Tl/2 (2.18)

where C(ic^»k2 ) is the amplitude density at the frequency k^, kj •

2.281 SPECTRAL ANALYSIS IN 2-DIMENSIONS

When the coefficients of a double fourier function are ordered or sequenced according to the units of frequency or wave length we refer to C(k^,k2) as spectrum coefficients or simply spectrum. Spectral analysis has been defined by Rayner [41] as the process of calculating and interpreting a spectrum. In two dimensions each spectrum represents a band of frequency. The steps involved in making a spectral analysis of D1 x D2 data array Z(J1,J2) say, are outlined below. (See Rayner [41] for details.)

Step l i Remove means and trends Step 2: Prewhitten if necessary with

Z (J1,J2) = Z(J1,J2) - Ej^ [Z(J1 + 1, J2) + Z(J1-1, J2) ]. -

E2 [Z(J1, J2 + l) + Z(J1, J2-1) ] (2.19) where o

Step 3: Taper with the window function h(Jl,J2)

Z(J1,J2) = Z1(J1,J2) h.(Jl,J2) (2.20) 25 where

h (J1.J2) - | |^1 - Cos ^ njijj , 0 < J1 < GX

= •- 1 - Cos / nJ2^~[ , 0 _< J2 £ L I G2)J G 2

1 G1 < J1 < D1 - Gl, G2 < J2 < D2 - G2

1 — * ** = 2 jj. - Cos ^ n(Dl(D1 -- JlJl)))Jj , D1 - Gl <. J1 < D1

= J Q - cos^ n(02 - J2)jj , D2 - G2 _< J2 < D2

where G1 and G2 are the numbers of columns and rows tapered.

Step 4: Add zeros for the beginning and/or end colums and

rows so that the array Z (jl, J2) has dimension n^ x n ^.

Step 5: Calculate the 2-dimensional fourier (spectrum transform

n n coefficient a (k^jk^) and b(k^,k2) using the fast fourier

transform algorithm eqns (2.16) (see Gentlemen and Sande [92])

for details.

Step 6: Calculate the variance (amplitude) spectrum from the

following relationships

A(ki»k2) =J^?(k^,k2) + b^(k.[ ,kg) j = amplitude (2.21)

( M k ^ ^ ) “ A^(k^,k2T j = Variance (2.22)

It should be noted that Step 1 is necessitated by the

presence of linear trend in the data while Steps 2, 3, and 4 apply 26 when data sample is from a non-periodic process, to get a reliable statistical estimate (see Rayner [42]).

The variance spectrum like the harmonic vector magnitude can be used to describe the degree of surface roughness. The same information is contained in the elevation readings, as in the spectrum but in a different way amenable to easy interpretation which is almost impossible to make from the original data. Also as in the case of the double fourier function the topographic surface is assumed to be oscillatory or undulating in two mutually perpendicular directions

(X and Y directions). Since each spectrum defines a frequency band for the oscillations the power or variance spectrum, therefore, measures the relative contribution of each frequency to the variability of the surface which is represented by the oscillations. By way of contrast, the amplitudes computed from spectral analysis define a frequency band and are ordered or sequenced according to the units of frequency, while the amplitudes obtained from double fourier function define a spatial band and are ordered according to the units of input space distance. In both cases, however, points on the bands so defined do not correspond one-to-one with points on the original spatial band defined by the X and Y coordinates. Since the amplitudes from spectral analysis are ordered in the units of frequency, the power spectrum computed from them possesses an additional feature of the topography which are only indirectly present in the harmonic vector magnitude. However, the latter is easier to compute and has more flexibility in applications than the former. 27

It should be noted that the power spectrum can be computed

through the fourier transform of the autocorrelation or autocovariance function. The computational procedures are very similar to those of the spectral analyses except that the tapering and pre-whittening functions are different. Also, the resulting values of power spectrum

are different and difficult to interpret although Rayner [41] contends

that they contain the same Information as the power spectrum computed

directly from the fourier transform of the data.

2.3 PARAMETERS FOR DESCRIBING SPATIAL DISTRIBUTION OF SURFACE IRREGULARITIES

The parameter which can be used for describing the nature

of spatial distribution of surface irregularities is known as

autocorrelation. This parameter is a measure of the degree of

internal correlation between one point and another. When related to

a terrain, this parameter may be more appropriately called spatial autocorrelation.

2.31 AUTOCORRELATION IN ONE-DIMENSION

We begin by defining the covariance between and the value

Zfc+k separated by k interval of time or distance as the autocovari­

ance at lag k tfk - covCzc- zt+k D - «&-*) (vs*) (2-23)

The autocorrelation at lag k is given by 28 j k = E (Z - M ) / O z = & (2.24)

where M i s the population mean of Z,

is the variance of Z

The autocrorelations computed for various possible lags in a given data, form what is called the autocorrelation function defined as

R(k) + 1 V Z (t) * Z(t+k) dx (2.25) N Jo

where Z (t) and Z(t+k) represent adjacent elevation

separated at a variable distance k

N = total number of elevation points considered.

The finite discrete approximation for the autocorrelation function in eqn (2.25) may be expressed as, Jenkins and Watts [28]

r(k) = -■ Z(t) • Z (t + k) dt (2.26) “ X=1

Eqn (2.26) may be used to analyze the characteristic of a

terrain with measurements of elevations taken at equal intervals along certain profiles. The graph of the autocorrelation function is called correlogram which may possess any of the following characteristics.

(a) r(t) r (t+k) strictly monotone decresing (2.27) -ve (b) r(t) =’zero (2.28) r * ■,-fve (c) r(t) = L^t (Linear) (2.29) L (d) r (t) = e ^ Exponential (2.30) where k is the lag

L = linear combination of autocorrelations at different lags X = an arbitrary constant 29 2.32 AUTOCORRELATION IN 2-DIMENSIONS

The autocorrelation characteristics of topography may also be considered from a 2-dimensional perspective. For example, the autocorrelation between elevation points separated by distances u and v along x- and y- directions may be expressed as

J^CujV) = E (2.31) (_Zi + u, j + v ^ j C Z ± 3

The sample autocorrelation function may be approximated by

nn-q. n - P TT~ ^ \ (2.32) r

where q = 0, 1, 2,.... T. (Lags in x-direction)

p = 0, 1, 2,.... T (Lags in y-direction)

m, n represent the number of data points

along x- and y- directions respectively

The autocorrelation function in 2-dimensions is extremely difficult to interpret graphically. However, certain terrain features corresponding to those of one-dimension may be given as follows:

(a) r(u,v) > r(u+l, v-t-1) , strictly monotone (2.32a) -ve izero (2.32b) +ve k-v L-u (c) r(u,v) = , linear correlopiped (2.32c)

(d) r (u,v) = e 1 1 , exponential

correlopiped [DAS (23)] (2.32d)

where k = Linear combinations of autocomlations computed along x-axis and L is the corresponding linear combination , _ along y-axis ______30 .

A, , M, are arbitrary constants.

Since the autocorrelation properties are related to the spatial distribution of surface irregularities the characteristics given in eqns (2.JZa-d) will be applied to the problem of optimum sampling pattern in Chapter III. These characteristics are not related to roughness parameters and, therefore, they will not be used for terrain classification.

2.4 COMPUTER PROGRAMS FOR TERRAIN ANALYSIS

The following computer programs were written in Fortran

IV for Ohio State University IBM 370/168, to compute the parameters of quantitative terrain description discussed in the preceding sections.

TERAIN is a program for simulating terrains (see

Appendix A, Figures Al-All).

TERANA consists of eight subroutines which are briefly described below:

1. DIRCOS computes the direction cosines of a terrain in

X, Y, Z directions their skewness and Kurtosis as well

as their associated eigen values and elgen vectors.

2. DERFOR computes the gradients, curvatures, their means

and variances at certain points of the terrain

3. VECTOR

4. HUMP were originally written by Hubson (26] for the

3400 CDC computer but were modified by the author for the

Ohio State University IBM 370/168 Computer. VECTOR computes the Vector Strength, Vector dispersion, and

Mean Dip while HUMP computes Bump Frequency parameters.

5. BERKLN determines the total number of breaklines per

unit area using eqn (2.5).

6. AREVOL computes the ratio between planar area and

surface area.

7. HVM determines the harmonic vector magnitude and

8. AUTOCO computes the spatial autocorrelation from

eqn (2.32).

SPETRM was originally written by Rayner and McCalder [42] for comparisons of surfaces but it was modified to compute the power spectrum after the trend has been removed. The major outputs of these programs are displayed in Table 2.1. TABLE 2.1 QUANTITATIVE CHARACTERISTICS OF FIFTEEN TERRAINS Mean Variance Harmonic of Vector Power Vector Magnitude Spectrum Gradient Curvature Comparea Strength See Vl V2 Xl X2 Appei x 3 Dl A 1 8.003 .00005 1.414 0.0 1.9998 •75.775E06 A 2 0.135 .0065 • 0.6646 1.1012 1.3196 50491 A 3 1.730 .000001 0.099 .094 1.0003 .43756E06 A 4 4.672 32.93 23.897 23.922 213.447 .1034E05 A 5 34.533 697.1 227.247 31.316 1087.09 .77162E05 A 6 277.811 5464.0 207.129 247.358 19730.746 •16682E07 A 7 432.60 9350000.0 2499.869 344.405 3273296 .26714E05 A 8 8.820 13.73 25.680 14.56 132.068 .44055E05 A 9 37.431 242.34 219.78 27.34 6683.6 .57225E04 A10 22.402 34.038 65.361 36.111 841.371 15.721 All 51.931 1342.769 228.345 37.679 12863.476 77.137 A12 30.456 57.336 64.327 31.569 2416.00 97.988 A13 2.134 lO.348 2.4619 1.8249 6.240 388.81 A14 6.789 9.785 21.7424 9.7789 267.544 125.60 A15 10.789 15.672 25.056 14.535 738.534 205.28

w N> TABLE 2.1 (Cont'd) QUANTITATIVE CHARACTERISTICS OF FIFTEEN TERRAINS No. of Variance Resultant Breakline Bump Frequency of Kurtosis Per Sq. Vector Direction of Unit of Surface Dispersion_____ Dip A______B______C______Cosine______Dir. Cosine______Area_____ oee 02 Xi* d 3 xs x 6 x 7 x 8 Ad d end lx A

A 1 -.00106 27.506 10.0 .010 .0081 .276 1.3374 0.0 A 2 2.7076 42.453 .0606 .3999 .0013 .40724 1.1082 0.0 A 3 -.00183 69.973 9.162 7.723 .0411 .4246 2.2615 0.0 A 4 -.12515 1.7948 -104.48 68.576 -14.376 .99051 6.750 .111 A 5 -.01514 10.915 -1691.5 606.361 -630.737 .8409 8.303 .029 A 6 -.00069 0.6341 -55245 945.158 15505.9 .99758 12.2968 .151 A 7 -.04503 -.17586 -1110.9 15505.973 605.701 .99975 180.317 .049 A 8 -.0268 4.880 -445.4 48.48 53.608 .983 4.169 .0243 A 9 -.2518 10.54 -51.854 1519.99 567. .974 7.814 .0313 A10 1.0391 1.948 -175.526 123.393 -87.150 .99498 7.4976 .04 All 1.152 9.007 •-2763.127 2118.172 2118.859 .5560 9.1414 .0144 A12 1.203 4.804 694.446 219.289 217.798 .9797 10.7919 0.0 A13 3.072 47.881 881.927 14.962 13.301 .6827 .8431 .0000 A14 1.2766 9.473 6264.2 106.931 -105.977 .97965 2.3346 0.0 A15 1.551 30.024 1073.004 213.209 207.211 .8483 1.993 0.0 2.5 TERRAIN CLASSIFICATION

Classification is one of the fundamental methodologies in science. It involves the grouping of variables according to their common characteristics thereby introducing simplicity and orderliness

into what might have been a complex multiplicity of intertwined and

interwoven facts. When the scientist is confronted with such complex­

ity, he must hold tenaciously to the belief expressed by Gould [67],

"Order there is if only we can unscramble the chaos that lies all around." The scientific theory of classification attempts to make

Gould’s conviction a reality to scientists at large irrespective of

their disciplines. Before discussing the parameters and methodology used for terrain classification in this research it is important to

identify the qualities of a good classification system with special

reference to terrain classification.

(a) A good classification must be logical, simple, and

orderly

(b) It must be mutually exclusive. Classes must not be

overlapping, and classes must also be finite.

(c) It must be collectively exhaustive or all inclusive.

No category of the population of items being considered

must be left out.

(d) The terminologies used in classification must be

concise and intelligible and should have some physical

bearing to terrain characteristics

(e) The classification must be objective as far as it is

humanly possible. This implies that a classification based on quantitative characteristics of the terrain

is to be preferred to that based on qualitative

characteristics. Also the techniques for classification

must be based on sound principles of statistical

analysis and cluster theory

(f) The classification must be adaptable for Digital

Terrain Model applications. For example, such a

classification should be useful for proper sampling of

a terrain.

The only known serious attempt on the terrain classification for DTM is credited to Silar [15] who presented his paper (wtitten in Germany) to the ISP Working Group IV/l in Czechoslovakia in 1969.

The four parameters used for his classifications are: slope of the surface, shape of the surface, the number of local peaks and depres­ sions and the roughness of the terrain. El-Ghazali's [2] comments on Silar's work is that the classification is based on qualitative— quantitative measurements but comparatively simple for practical application.

The classifications performed in this research are based on purely quantitative measurements which have their bearing in qualitative properties of the terrain. Also the methodology of , classification performed here is based on the theory of Multivariate statistical analysis which has been successfully applied in other dis­ ciplines such as Geology, Botany, Biology, Zoology, and Taxonomy. 36

2.51 STATISTICAL METHOD USED FOR CLASSIFICATION

Only a brief outline of the statistical techniques used for terrain classification is given below. The three criteria for evaluating a classification are based on the familiar quantities used in analysis of variance technique; namely, '’within" "between" and "total" sums of squares which are defined by the following equations.

wi j - V j ) <2-3*>

k - Nh( xh.i - x..i)Cxh .3 - x ..j) (2-35)

M N. T ij S t ^ C2-36)

1 Nh where v _ A s r — n y V l ‘ Nn ^ l ^kl

•-1 Mh

N = no of items to be classified

M ° no of groups to be identified in the classification

p = no of variables used for classification

It can be shown, Anderson’ [49], that

T = W + B

The three criteria used to assess and compare classification in an iterative solution are:

(i) The pooled within— groups sum of squares 37

(ii) Wilks’ Lambda and

(ill) The sum of the eigen values associated with discriminent

functions (see-Griffiths and'Demirmen [50] for details).

The trace of pooled "within-groups" sum of squares may be expressed as: M N. r x 2 Trace of W - Tr(W) = ~ \ i ] (2.37) h=l k=l V

Which is the total within-groups sum of squares in respect of all the p variables used in all the m groups. According to this criterion a small value of Tr(W) shows that the total variability with m groups about the respective means is small and, therefore, the classification is in this sense "compact" or "better" than a classification in which

Tr(W) is bigger.

Wilks’ .Lambda criterion relates to the well-known Wilks’

Ratio (see Anderson [61]), given by

* = j f j - (2.38) which is the ratio between the generalized variances of W and T.

This assumes that W and T must be nonsingular matrix and, therefore, places the following restriction on M, N, and p

p £ N-M (2.39)

values of X Indicates a compact classification

The third criterion has been shown to be equivalent to the —1 trace of W B, Demirmen [50]

/ — I \ i M Tr

(2.40) 38

ii *“1 where W J the (i,j)th element of W according to Demirmen [50] 2 represents the generalization of Mahalanobis D to more than two

groups. Unlike the former two criteria, larger values of Tr (W ^B)

correspond to more "compact" or "better" classification.

The use of these criteria assumes that any two classifica­

tions being compared must contain the same number of items, variables, and groups.- Friedman and Rubin [68] have successfully

used the above criteria to improve a partition in cluster analysis and

Demirmen [50] report a satisfactory experience with Tr(W) and Wilks'

Lamder (A) criteria.

2.52 COMPUTER PROGRAM FOR EVALUATION AND IMPROVEMENT OF CLASSIFICATION (TECLAS)

The computer program TECLAS used for the classification of

15 terrains was originally written by Ferruh Demirmen [50]. This program was modified for the O.S.U. IBM 370/168, without changing

the essential features of the original program. Although the program

computes all the parameters of the three criteria discussed above only the Tr(W) criteria was used for the evaluation and improvement of classifications. The nearest neighbor algorithm, in which an item is placed into a group to which it is nearest using Euclidean dis­

tances, operates in the discriminant space. All items are examined and

retained in their existing groups else they are displaced into other

groups, thus creating a new classification from which the Tr(W) is

computed. 39

The procedure is repeated iteratively to yield a smaller Tr(W), until either an improvement of the nearest neighbor algorithm is not feasible or when the maximum number of iterations required by the user is attained.

The variables chosen for the classification were chosen for two reasons: first they meet all the four conditions set forth in

Section 2.1. They also appear to be consistent parameters which are capable of distinguishing one terrain from another. A total of 15 terrains were classified using these five variables. (see Figs. A1 to A15, Appendix A). Eleven of these terrains were simulated on the computer so as to give a wide range of terrain types. The remaining four terrains were chosen from Tobler’s [39] Digital Terrain Library which consists of terrains digitized from topographical sheets. The result of the terrain classification is shown in Table 2.2. It is important to realize that the classification system used in this research falls short of only one of the qualities enumerated in

Section 2.5 notably that of mutual exclusiveness of classes. This can be explained by the complexity of the terrains and It is doubtful whether any classificatory system for the terrains can possess this quality. It is, however, more important to note that some of the parameters used for classification will be employed in the next

Chapter to determine optimum sampling for Digital Terrain Models. TABLE 2.2 TERRAIN CLASSIFICATION

Class Surface HVMV1 Description X1 X2 X3 X5

S-l 0.002 1.414 0.0 2.000 .010 Flat or nearly flat terrain S-2 0.135 0.665 1.101 1.320 .399 Av. Grad. = 0.0 - 3.0 I S-3 1.730 0.099 .094 1.0003 7.723 Av. Curv. ■ 0.0 - 5.0 S-13 2.134 2.462 1.8250 6.240 14.962 NOBPUA =0.0 HVM = 0.0 - 3.0 NOPPUA =0.0

S-1S 10.789 25.056 14.535 738.538 213.209 Gently undulating terrain S-14 6.789 21.742 9.779 267.544 106.931 Av. Grad. =10-70 II S-4 4.672 23.897 23.922 213.447 68.576 Av. Curv. =10-35 S-8 8.820 25.680 14.56 132.068 48.48 NOBPUA = 0.0 - 0.02 S-10 22.402 65.361 36.111 841.371 123.393 HVM =5-30 S-12 30.456 64.327 31.569 2416.00 219.289 NOPPUA =0.34

Rough terrain with pronounced S-5 34.533 227.247 31.316 1087.09 606.361 surface irregularities, and many steep slopes III ' S-9 37.431 219.780 37.340 6683.6 1519.990 Av. Grad. = 100 - 200 Av. Curv. =40-50 NOBPUA = .02 - .05 HVM = 35 - 50 NOPPUA =0 .5 2

Many breaklines Extremely rough terrain with Extreme surface irregularities S-6 277.811 207.129 247.358 19730,746 945.118 and precipitous slope and pronounced curvature IV S-7 432.60 2499.869 344.405 3273296.0 15505.973 Av. Grad. =* 225 - 00 S-ll 51.931 228.345 37.679 12863.476 2118.172 NOBPUA = .05 - 1.0+ HVM = 50 - 1000+ NOPPUA = 1.02+ HVM = Harmonic Vector Magnitude = Bump Frequency NOPPUA = No. of points per X^ = Gradient Av. Grad. = Average gradient unit area Av. Curv. = Average curvature X- “ Curvature NOBPUA = No. of breaklines per unit area X, ■ Ratio of planar area to' surface area CHAPTER III

OPTIMUM TERRAIN SAMPLING FOR DIGITAL TERRAIN MODELS

3.1 INTRODUCTION

Many investigations have been made into Digital Terrain

Model technique using regularly spaced data points. Questions which relate to the problem of optimum number of data points and their associated distribution in relation to different types of terrain are usually ignored although the importance of efficient sampling is often acknowledged over interpolation technique. This is borne out by the Blaschke’s [17] definition of D.T.M. cited in Section 1.2 in . which he emphasized the importance of "storing measured coordinates

X, Y, Z of characteristic terrain points in sufficient quantity and significance...." The definition of D.T.M. given by the author in

Section (1.2) also bears testimony to the importance of evaluating the adequate number of data points as well as the appropriate sample distribution of such points that match that terrain. This is what we mean by "Optimum Terrain Sampling" which is the object of focus in this Chapter. The problem of objective classification of terrains has been dealt with in Chapter II. The question that follows Is this: having identified the class to which a given terrain belongs, what is the optimum sample size and sample type adequate for this terrain? The problem of optimum interpolation which is equally

41 42

recognized by the author's definition, of D.T.M. will be discussed in

the next Chapter.

3.2 OPTIMUM SAMPLE SIZE

3.21 THE HARMONIC VECTOR MAGNITUDE

The problem of how the stereoplotter operator could know when he has taken enough sample elevation observations during the process of generating D.T.M. data is not a simple one because it

involves a proper assessment of the terrain roughness in relationship

to the size of the area occupied by the terrain. This implies that

if any of the roughness parameters discussed in the preceding Chapter is to be used such a parameter should incorporate the size of the entire area.

One parameter which seems to satisfy this condition is the Harmonic

Vector Magnitude (HVM) discussed in Section 2.27. It will be recalled

that one important assumption made in Section 2.27 is that a general­

ization of the original surface is adequately represented by the

coefficients of the Double Fourier function (eqn. 2.7) from which the

HVM is computed. It will also be recalled that the size of HVM

increases with increasing terrain roughness (see Table 2.1). It is well to remember that the fundamental wave lengths in X and Y

directions define the limits of size of the area occupied by the

terrain. From the foregoing explanation it is obvious that the HVM

satisfies the qualities necessary for indicating optimum sample

size. The following steps which are suggested for computing optimum

sample size requires a simultaneous evaluation of the terrain type 43

using HVM program in collaboration with the concept of progressive sampling first proposed by Makarovic [3].

Step 1 : Start with a suitable sample interval for taking 2 N elevation measurements (N =4, say) in a

regular grid pattern on the stereo model. This

permits a least squares solution of eqn. (2 .6 )

which has 11 unknown parameters for m=n=l

Step 2: Compute HVM using HVM program (see Section 3.4) 2 Step 3: Increase the number of elevation points to (N + 1)

using a new interval for regular grid pattern.

Step 4: Repeat Steps 2 and 3 until HVM satisfies the

following criteria:

HVM± - H V M (1+1) ^ CC (3.1) where the quantity CC can be determined in a number of ways as illustrated in the following considerations:

(i) CC - HVM1+1* fHVM (3.2)

f ^ H V M may be taken as the measure of accuracy for

photogrammetric representation of relief given by

Markarovic [3] as

0-2 - 0.5% of elevation for minimum (TZ. H

0-4 - 1.0% of elevation for maximum crH

(ii) CC = HVM x

The CC criterion used in this research is that of

eqn • (3.2)

and ft,™ = *5% of HVM..- was found to be suitable , HVM ±Tl

The interpretation of eqn. (3.1) is that as the sample size is increased iteratively the addition of new elevation measurements do not make any significant contribution to the size of

HVM which in turn remains relatively constant as the sample size approaches the optimum density for a given terrain. The steps outlined above were used to determine the optimum sample size of nine simulated terrains using the computer program TERRAIN and HVM (see

Appendix A, Figs. A1-A9). The results are shown in Table 3.1 The procedure analyzed above for determining the optimum sample size of a given terrain assumes that the stereoplotting instrument is on- lined with a digital computer or that a versatile programmable desk calculator is readily available to the stereo-plotter operator. Its merit lies in the fact that it is flexible and could be used either for sampling the whole model or for sampling a model which is divided into patches. 45 TABLE 3.1 OPTIMUM SAMPLE SIZE USING MEAN HARMONIC VECTOR MAGNITUDE CRITERION _____ (Units in Meters)____ SAMPLE MEAN HARMONIC SAMPLE MEAN HARMONIC SURFACE SIZE VECTOR MAGNITUDE SURFACE SIZE VECTOR MAGNITUDE

Surface 16 .0026 81 35.119 with 25* .0026 D . Fourier 256 266.603 Line 36 .00263 Surface 289 23.241 Trend (1) 49 .00256 with 324 270.834 Synthetic 361 272.760 16 0.449 Coeffs 400 26.5841 Exponential 25 0.137 441 276.741 Surface 36* 0.135 484* 277.811 49 0.131 529 278.275 (2) 441 0.154 (6)

16 2.060 289 4847.969 Loga­ 25 1.906 Poly­ 361 4683.835 rithmic 36 1.736 nomial 400 4511.977 Surface 49* 1.730 Surface 441 4553.312 64 1.764 No- 2 484 4497.551 (3) 441 1.978 529 4447.608 576 4400.941 100 6.97B 625 4358.812 Double 121 4.848 676* 4349.053 Fourier 144 4.672 (7) 729 4340.406 Surface 169* 4.679 with 225 4.674 36 14.726 Random 256 4.680 47 10.136 Coeffs 289 4.687 D . Fourier 64 13.901 (4) 400 4.882 Test 81 9.717 Surface 100 10.569 144 39.592 121 .8.834 Poly­ 109 38.214 144* 8.820 nomial 196 37.004 169 8.811 Surface 1 225 36.092 196 8.801 256* 35.257 (8) 225 8.802 289 34.533 (5) 324 33.897 121 50.157 Test 144 46.782 Poly­ 169 44.034 nomial 196 41.759 Surface 225 39.8654 256 38.336 289 37.439 324* 37.431 361 37.677 * Optimum Sample Size (9) 400 37.939 46

3.22 MULTIPLE LINEAR REGRESSION EQUATION FOR OPTIMUM SAMPLE SIZE

The purpose of this Section is to investigate alternative method of determining optimum sample size which will Involve less computation then the H.V.M Method. Multiple regression provides a method by which a "dependent" variable can be explained by two or more independent (predictor) variables. In this particular application an equation of the form is sort

fS = bo + b x X1 + b 2 X2 + b 3 X 3 + .... + \ \ (3.5) where N ~ dependent (response) variable = optimum

sample size:

^1 ' ^2 > are Pre^^ct^on variables which

correspond to the roughness parameters in Table 3.2

bQ, b^, are t*ie coefficients of the regression

equation which correspond to the relative contribution of

prediction variables in "explaining" or "causing" the

dependent variable ft

bQ represents a joint contribution of all the prediction

variables 2 R, the multiple correlation coefficients is defined as

(3.5a) TABLE 3.2 DEPENDENT AND INDEPENDENT VARIABLES FOR REGRESSION EQUATIONS Resultant Kurtosis No. of No. of Bump Var. of of Breakline Points Optimum Mean Mean Surface Mean Freq. Direction Direction Per Unit Per Unit Sample Gradient Curvature Area/Planar Dip IB Cosine Cosine Area Area Surface Size (N)

A 1 25 1.141 0.0 1.998 27.506 .010 .276 1.3374 0.0 .00025 A 2 36 .6646 1.1012 1.3196 42.453 • .39990 .40724 1.1682 0.0 .000036 A 3 49 .099 .094 1.0003 69.973 •7.723 .4246 2.2615 0.0 .000049 A 4 169 23.897 23.922 213.447 1.7948 68.576 .99051 6.750 .111 .2934 A 5 289 227.247 31.316 1087.07 10.915 606.361 .84091 8.303 .029 .5017 A 6 484 207.129 247.358 19730.746 0.6341 945.158 .9975812.2968 .151 .8403 A 7 676 2499.869 344.40532732.96 -.17586133505.873 .99975 180.317 .049 1.1736 A 8 196 25.680 14.56 132.068 4.880 48.48 .983 4.169 .0243 .391

-vl 48 where N n

N is an estimate of N from eqn (3.5)

The dependent variables X^, > X^, X ^ , X^, Xg X^» Xg, Xg,

used for computing the regression equations were taken from Table 2.1

except Xg. The addition of this variable is justified in that

it is a measure of optimum number of points per unit of area.

Table 3.2 shows that we can have a multiple linear regression

equation of nine predictor variables. Such an equation will defeat

the purpose of this Section. Before we appeal to various methods of

selecting the "best" regression equation out of the possible regression

equations which can be formulated from various combinations of N with

Xf, X£> Xg, the following principles and assumptions should be

stated:

1. It is hypothesized that optimum sample size is a

function of one or more of the predictor variables X^,

^2 ’---- ^9 * are t*ie roughness parameters of a

terrain.

2. For reasons of economy and ease of computation we wish

to include only a minimum number of variables possible.

3. Both objective and subjective methods will be used in

monitoring the variables for selecting the "best"

regression equation. "Best" refers to the one which

will give the most accurate prediction with the

minimum of variables which -dlso require the least •

amount of computational effort. 49

2 4. The multiple correlation coefficients R provides an

estimate of the variation of N "explained" or accounted

for by prediction variables.

The following procedure was used for selecting the

variables which could be included in the "best" regression equation.

(See Drapper and Smith [69] and ASA Manual [70] for details.)

3.22 SUBJECTIVE SELECTION

This procedure involves the inspection of the variables in

Table 3.2 and deciding which of their contributions will yield

the least computational effort. This procedure gives rise to the

following "Economic Models"

"ECONOMIC MODELS

"Best" 1 variable Xg

(3.6) '9 "Best" 2 variables 3 - x X 1 . * 9 ;3 + 525 .653Xg (3.7)

482.495Xg (3.8)

’Best" 3 variables XX , X2 , Xg

XX , X3 , Xg

X2* X3* X9

.1742X2 + 480.0075Xg (3.9)

N = 32.0078 + .OOO^Xj^ + .0000082X3 + 525.618Xg (3.10) 50

N = 34.88 + .17298X2 + .00000446X3 + 483.066X9 (3.11)

"Best" 4 variables X-J » X„,3^2 I X~,X^ ) XXg

x 3 , x5 , Xfi, X y

xr x2, x6, x 7

N = 36.412 + .1511X1 + .2703X2 - .0000984X3 + 418.146Xg

(3.12)

N = -16.146 -.00195X- + .3457XC + 102.256X, + 8.9464X-, 3 5 6 7

(3.13)

N * -28.481 + 0.5210X. + 1.11430X- + 203.6779X, 1 Z b

~6-6295X? (3.14)

"Best" 6 variables X ^ , X 2 , X 3 , , X g , X g ,

N = 7.477 -2.747X1 -1.538X2 -.0003X3 + 1.167X5

+ 77.5103Xg + 367.743Xg (3.15)

3.23 "OBJECTIVE" METHODS OF SELECTION

The "objective" methods refer to any of the five techniques given below which can be used to find which of the variables X^, X2 ,----

------Xg are most likely to be included in the regression model under certain specified criteria. The term "F-Statistic to include" is used

to mean the F Statistic which miikes a variable significant.

3.231 MAXIMUM R2 IMPROVEMENT

This technique produces the best 1 variable, 2 variable, etc. from a given set of prediction variables by first finding the 2 "best" one-variable model which yields the highest R . Then the

"best" two-variable model is determined by finding which of the remaining variables when combined with the "best" one-variable gives 2 the maximum R increase. The best "three-variable model" is 2 determined by the maximum R criteria. The result of this procedure is given as eqns (3.6) and (3.16) - (3.20)

MAXIMUM R-SQUARED IMPROVEMENT

Best 1 variable Xg

N = 29.3868 + 54l.9068Xg same as eqn. (3.6)

Best 2 variables X,, X„ ------4 9

N = 1.2772 + 0.7215X + 572.6318X. (3.16) 4 ” Best 3 variables X^> X^, Xg

N = 8.1918 + 0.0968X2 + 0.6026X4 + 539.5534Xg (3.17)

Best 4 variables X„» X.. X„. X„ ------2 4 8 9

N = 10.9173 + 0.1032X2 + 0.551X4 -29.4883X0

+ 537.647Xg (3.18)

Best 5 variables X2, X^, X0, X0, Xg

N - 9.2308 + 0.1165X- + .5424X. + 5.531X, Z h b -44.73O4X0 + 531.373Xg (3.19)

Best 6 variables X^, X2, X^, Xg, X^, X^

N = 4.1653 -0.00627X1 + .2032X2 + .4608X4 + 29.547Xg 3.232 MINIMUM R-SQUARED IMPROVEMENT

Best 1 variable same as (3.6) X9 Best 2 variables (3.16) V X9 same as

Best 3 variables v same as (3.17) v V 9

Best 4 variables (3.18) X2* X4* > ^g GQine 3S

Best 5 variables x2, x3, X*. ^ 7 » ^9

N == 11.2005 + .1116X2 + .0000256X3 + .5614X4

-.478X ? + 535.804Xg (3.21)

Best 6 variables x3 , x6 , xy, x5 V X4 *

N == 7.511 + .1987X2 + .0000944X3 0.5398 X^

+ 18.294X. ~1.8103X7 + 510.7175Xg (3.22) 6 2 This procedure is similar to maximum R improvement except 2 that the smallest decrease in R criterion is used. The result of this procedure is given in eqns. (3.6, 3.16-3.18, 3.21-3.22).

3.233 BACKWARD ELIMINATION PROCEDURE

A multiple regression equation is performed by using all the given variables. The "partial" F-statistics is computed for each variable and compared with a specified f-statistic to remove at a given significance level. If F

Variables which do not make any significant contribution to sample

size according to this procedure are

X3> Xg , and X?

Best 6 variables

N = -3.7261 +4.4361X^ +2.157X2 +.8174X4 -.6561X5

+3073.5957Xg -962.9166X9 (3.23)

3.234 FORWARD SELECTION PROCEDURE

2 The single variable with the maximum R is selected and

tested for its significance according to a specified F-statistic to y include . Then compute the partial correlation coefficient Fx and

choose the variable with the largest Fx to be included in the model.

Use "partial" F-statistic to test whether the new variable should be

included according to F to include and continue until all the

significant variables have been included.

Variables considered to make significant contribution

X2 * X 4 * X8 ’ X9

are the same as in eqn. (3.18).

3.235 STEPWISE REGRESSION PROCEDURE

Stepwise procedure is a combination of Forward and Backward

procedures in that although it adds or drops a variable according to

specified F-statistic to include or to exclude, at each step all the variables which have been previously included are reexamined. The variables considered significant according to this procedure are the same as in Section (3.18). The outline of steps involved will be given below and detailed discussion is reserved for

Section 4.31 (see Draper and Smith [69]).

Step 1: Compute the simple correlation coefficients between

the dependent and the independent variables and

select the variable with the highest coefficient

X^ say, for regression equation.

Step 2: Compute the partial correlation coefficients and

select the variable with the highest partial

coefficient as the next variable X^, say.

a Step 3: Compute regression equation N = f(X^, X^) and

using the criteria -F* to exclude and the partial

F > decision is made whether to retain X, in the x 4 light of including X^. The partial correlation

coefficients for the remaining variables are

computed and the next variable X 2 say, is selected

as in Step 2.

A Step 4: The regression equation N = (X^,X^, X2 ) is then

computed and X^ and X2 are examined as to their

significance. The decision is then made as to

whether they should be retained before an additional

variable to be included is determined as in Step 3.

This continues until all the variables are

exhausted. 55

The variable considered significant according

to the Stepwise procedure are the same as those

of eqn. (3.18).

3.24 RESULT OF TESTING MULTIPLE REGRESSION EQUATION FOR OPTIMUM SAMPLE SIZE

All the equations obtained through both "subjective" and

"objective" procedures were tested on terrain No. 9 (see Fig. A9,

Appendix A) to see whether they give similar results as the HVM

criteria. The results are displayed in Table 3.3 from where it can be observed that the optimum sizes from some regression equations ob­

tained through "subjective" procedures differ significantly from that

obtained by HVM criteria. On the other hand, other regression equations

obtained through "objective" procedures seem to produce optimum sample

sizes which compare favorable with those from HVM criteria. It may be noted from Table 3.4 that the independent variables excepting Xg,

appear to be fairly constant after the optimum sample size is exceeded. This shows that the optimum sample size from HVM is enough

to describe the terrain roughness features. The merit of some of the

linear regression equations especially those from "subjective" methods

is that predictor variables can be computed by a stereoplotter operator using an inexpensive desk calculator. A very simple criteria

for determining optimum size may be given by

\ — Nk+1 | < TR (3.24) where TR = ^k+i x ^

f = .5% 56

ft = last sample size computed from

regression equations.

It should be noted that eqns. (3.12-3.15) and (3.21) are not recommended for use because of their gross errors. 57

, TABLE 3.3 DIFFERENCE BETWEEN GIVEN SAMPLE SIZE AND COMPUTED SAMPLE SIZE USING EQNS (3.6-3.23) ON SURFACE 9

Eqn. N=169 N=196 N=225 N=256 N=289 N=324* N=361

(3.6)** 19 18 16 14 12 10 8 (3.7)** 17 15 12 10 7 4 1 (3.8)** 7 2 -2 -7 -13 -18 -24 (3.9)** 7 2 -2 -8 -13 -19 -25 (3.10)** 17 15 12 10 7 4 0 (3.11)** 18 3 -1 -6 -12 -17 -23 (3.12)** -11 -18 -26 -34 -43 -53 -63 (3.13)** 436 412 390 362 335 303 271 (3.16) 8 8 7 7 7 7 6 (3.17) 4 2 0 -2 -4 -6 -8 (3.18) 5 4 2 -1 -3 -6 -8 (3.19) 5 3 1 -3 -6 -8.334 -11 (3.20) 7 4 0 -9 -13 -18.591 -22 (3.21) -118 -146 -182 -214 -253 -289.805 -•331 (3.22) 2 0 -2 -4 -7 -9.008 -12 (3.23) -7 -10 -14 -18 -23 -27.911 -32 (3.15 -1419 -1499 -1583 -1671 -1764 -1862.17 -•1965 (3.14)** 1701 1705 1705 1704 1699 -1695.566 1687

** Economic Model

* Optimum sample size using harmonic vector magnitude criteria TABLE 3.4 PREDICTOR OF VARIABLES: TERRAIN 9 (Parameters of Terrain Roughness in Relation to ______increasing sample size)______

N X 1 X2 X3 X4 X5 X6 X7 X 8 X9

169 30.69 200.96 6677.570 10.171 1481.451 0.6074 6.7102 .0263 0.2934

196 38.86 206.98 6678.910 10.354 1493.369 0.6034 6.640 .0243 0.3403

225 36.65 281.69 6681.734 10.397 1503.543 0.5883 7.270 .0261 0.3906 256 36.99 219.64 6680.66 10.55 1512.30 0.5840 7.271 .0310 0.444 289 37.89 219.87 6683.6 10.54 1519.99 0.5730 7.814 .0313 0.5018 *324 37.69 219.89 6683.29 10.6569 1526.71 0.5690 7.877 .0313 0.563

361 37.58 219.67 6684.13 10.4785 1532.66 0.5596 7.3475 .0313 0.6267

* Optimum sample size •

Ln 00 59

3.3 OPTIMUM SAMPLE TYPE

It will be observed from the definition of Digital Terrain

Model given in Section 3.1 that emphasis is placed on "making use of adequate elevation measurements compatible in number and distribution with that terrain." The last Section has been devoted to finding the elevation measurements compatible in number but not in distribution with a given terrain. It will be recalled that the progressive sampling method proposed in Section 3.2 is performed using a regular point grid distribution of elevation measurements. While such a sampling plan is suitable for the analysis of terrain roughness and the determination of optimum sample size, it may not be the optimum plan for interpolation of points on the surface because of the influence of sampling error. Miesch and Connor [62] have pointed out that in trend analysis the residual, "includes all variation having resulted from local geologic factors and from data errors of various kinds. The local geologic factors give rise to sampling errors which in many instances may form the major problem of the total noise (residual) in the data."

The problem of finding relative efficiency of various sample types for certain categories of populations has been the subject of many published papers in statistics namely, Das [23],

Quenoulle [24], Cochran [6 6 ], and others. Cochran [6 6 ] working with one dimensional autocorrelation has developed some theorems to show when a particular sample type (Random, Stratified, or Systematic) is more efficient than the other. Das [23] has demonstrated a 2- dimensional extension of Cochran's theorems. 60

Before stating these theorems we need to know what types

of sampling plans (sample type) can be generated in a 2-dimensional

surface.

3.31 MAJOR SAMPLING PATTERNS

There are three major sample types well known in statistics.

These are:

Random sampling Stratified sampling Systematic sampling

Quenouille 1.24] has rightly pointed out that there are many ways

in which we can sample a two-dimensional space since there is flexibility

in employing random, stratified, or systematic sampling in either

direction. However, Morrison [63] has demonstrated that six

sampling plans can theoretically represent nearly all the possible

sample point scatters in a plane. This author feels intuitively that

a seventh sampling plan which combines all the features of the three

major sampling plans has some merits which deserve investigation.

These seven sample types are now briefly described.

UNALIGNED RANDOM SAMPLING

The Unaligned Random Sampling is a type of sampling in which each point is chosen randomly. In two-dimensional plane this

means that the coordinates X and Y of a given point is selected at

random either by using a table of random numbers or by generating

random numbers through a computer program. This gives rise to an 61 uneven areal coverage.. It is believed by statisticans that this type of sampling is efficient if there is periodic variation or any type of trend in the population since it gives rise to an uneven areal coverage.

ALIGNED RANDOM SAMPLING

This type of sampling is similar to the previous one except that the random number in one direction, X or Y is fixed and the other is chosen randomly. This type of sampling does not have a good areal coverage as its counterpart without alignment.

UNALIGNED STRATIFIED SAMPLING

This is a type of sampling in which the area concerned is subdivided into strata within which sampling points are chosen randomly in a manner similar to unaligned random sampling. The advantage of stratified sampling is that it tends to increase the precision of the estimate of a population parameter without increasing the number of points since the areal coverage of points seem to be more representative of the population.

ALIGNED STRATIFIED SAMPLING

This is the same as its unaligned counterpart except that one of the coordinates is aligned within each stratum. 62

UNALIGNED SYSTEMATIC SAMPLING

Unaligned systematic sampling is generated by dividing the area into sections (rectangulars or squares) and points are sampled randomly in each Section.

ALIGNED SYSTEMATIC SAMPLING

This is by far the most popular type of sampling used in

Digital Terrain Model studies because it is the easiest to generate.

The initial point is selected randomly or purposefully and all others determined by a fixed interval.

UNALIGNED STRATIFIED SYSTEMATIC RANDOM SAMPLING

As the name implies this sampling type is a combination of all the three major sampling types. The area concerned is covered with squares or rectangular grids and the first point is selected at random in the first square. The X coordinate of this first point is then used with a new random Y coordinate to locate the new point in the second square.

A new point is similarly treated in the subsequent squares in the first row. The second and subsequent rows of squares are treated like the first row to generate the points required. Fig. 3.1 shows the examples of sample point scatters resulting from each of the seven sampling plans discussed above. 63

58a

t • # • * Unaligned Aligned t - rt Random « t Random • * * 4 (2) (1) • 1 H # » 1 «• * . . • * * * # • • 4 44 • • * 4

I • ft I Unaligned Aligned Stratified Stratified k 9 f► « ft '

(3) (4) 1 « • • 4

t * • * ” > 1 ' * 1

■------

9 • ft *

» 1 r Unaligned Aligned Systematic Systematic ► k k » r (5) (6) > 1 i

► 1 I 1 r

m 9 ► V k • 9 »

• « » 4 •

1 9 • r 9 Unaligned Random • Stratified Systematic • * • 4 « (7) • l * * ♦

Fig. 3.1 SEVEN SAMPLING PATTERNS 64

3.32 DAS' THEOREM FOR RELATIVE EFFICIENCY OF SAMPLING PATTERNS

Before stating the conditions under which one type of sample type is more efficient than the other for estimating population parameters it is important to discuss the following notations and properties of autocorrelation which are essential for understanding

Das' [231 theorems.

A^ r(u,v) = r(u +l»v) -r(u,v) (3.25a)

&2 r(u,v) = r(u,v+l) -r(u,v) (3.25b)

r(u,v) = r(u+l,v) + r(u-l,v) (3.26a)

Ai A 2 r(u,v) = Air(u,v+1) -Air(u,v) (3.26b)

r(u,v) - r(-u,-v) (3.27)

r(u,-v) = r(-u,v) (3.28)

^(u.v) = r(u,v) + r(-u,v) (3.29) where r(u,v)sthe spatial autocorrelation between points separated by

t distances u,v in x’ and y directions, respectively

Theorem 1 for all infinite population in which

(i) A x 4* £ 0

(ii) 4 2 f < 0

(iii) 2 y(u,v+l) ^ T (u+1 ,v) + ^(u+l,v+l) and

(iv) 2 t(u+l,v) “j p + V(u+l,v+l)

* 2 2 q (i.e. stratified sampling is more efficient than random St r s-2 sampling) for any size of the sample and <5 - ^ 6 unless equality St r holds in each of the above cases. Corrolary for all infinite populations in which _ 2 _2 ii v = A2 v £ o Ost <0 ^or an^ s^ze t*ie samPle and < 2 J S r 8 Ay unless equality holds in each of the above cases.

Theorem 2 : When stratification is made- by parallel strips along

u-direction if (i) Ai V < o

(ii) A 2 £ 0 aad

(iii) 2Y (u+1 ,V) ^ T (u,v+l) + Y (u+l,v+l)

Then (fet ± g ; 2

Corrolary When stratification is made by parallel U-strips if

A 2 V 1 * 1 V £ 0 then < g r

Theorem 3: for all patterns of stratification if

(i) Ai V <_ 0

(11) A 2 ^ 1. 0 aad

(iii) Ai A 2 V £ 0 then

Theorem 4: for all infinite population in which

£ 1 j* (u»v)‘ £ 0

(ii) £ 2 9 (u.v) ^ 0 2 2 Then (7" £ AT for any sample size and 2 2 6 sy < 6 st un^ess equallt:y holds

i.e., Systematic is better than stratified

Theorem 5 : for all infinite population in which

(1 ) A : V (u,v) £ 0

(ii) A 2 V (u,v) £ 0

(iii) ^ ! 2J ( u , v ) £ 0 66

(iv) y (u,v) >, 0

Then > (T* >. 6 * *^r s vst ' ^sy

i.e., Systematic is better than stratified and stratified

is better than random

It should be noted that the physical interpretation of these conditions are very difficult to perceive except in one- dimension. For example, in Theorem 5 conditions (i) and (ii) are nothing but strictly monotone autocorrelation properties and (iii) and (iv) are properties of second differences of autocorrelation in u and v directions.

3.33 THEORETICAL EXPECTATION AND EMPERICAL RESULT OF THE EFFICIENCY OF SAMPLING PATTERNS

The objective of this Section Is to investigate the validity of these theorems when they are applied to a topographic surface as the infinite population of interest, and when efficiency of sampling pattern relates to interpolation accuracy. An experiment set up to determine the practical efficiency of the various sampling plans, is briefly described below.

Step A ; Determine the theoretical relative efficiency of the six standard sampling patterns according to Das' theorems. A computer program called AUTOCO computes the autocorrelation properties described in Theorems 1-5. From this^theoretical efficiency of a sampling plan can be determined. For example, if all the conditions in Theorem 5 are satisfied then systematic sampling is "better" than stratified which in turn is "better" than random sampling. 67

Step B : Determine the practical (emperical) relative efficiency of the six standard sampling patterns in the following manner:

(i) A surface is generated (or simulated) by a mathematical

function using the program TERRAIN. Let the parent function 2 2 be Z 3 a + bx + cx + dy + ey , for example o (ii) A program called GENSAP was written to generate Z values

for seven sampling patterns discussed above, at a given

or optimum sample size using this parent function.

(iii) A ‘’deficient1’ function is established by dropping one or 2 2 two terms of.the parent function e.g., Z* — BX + CX + EY .

Then by Least Squares technique the Z values so generated

for the seven sample plans are used in fitting the deficient

function to determine the parameters B, C, and E which are

in turn used to compute the Z^ values at some points whose

X, Y, Z coordinates are known on the parent surface.

Step C : The RMS (root mean squares) of differences between the Z’s from parent surface and the Z^’s from "deficient" surface are then computed. The sampling plan with the smallest RMS is then chosen as the most efficient for that terrain. This result is then compared with the most efficient, theoretically speaking, sampling plan obtained from Step 1.

The seven terrains used are shown in Figs. A1-A6, A9, Appendix A. The parent functions and the "deficient" functions these seven terrains are briefly described below.

Surface 1

Parent: Z = a + bX + cY

Deficient: Z* = bX + cY

Surface 2

Parent: Z = a e c^x ^ ^ ^ -( 2 + Z ) Deficient: Z* = a + a_e X ^ 1 o 1

Surface 3

P a r e n t : Z = a- log X + a_ log Y 1 °e 2 e

Deficient: Z* = a + a^ l°ge X + a 2 l°Be Y

Surface 4

Parent: Double Fourier Function as in eqn. (2.7)

faith m = n = 5 and coefficients

chosen randomly

Deficient: Double Fourier function as in eqn- (2.7)

with m = n = 4

Surface 5

Parent: Power series polynomial

Z - [ bjj ( x V / j

m = n = 5

Deficient: Same as above except that

the last term was dropped Surface 6

Parent: Double Fourier function as in eqn. (2.7)

m = n = 5 with synthetic coefficient

Deficient: Double Fourier Function as in eqn. (2.7)

with m = n - A

Surface 9

Parent: Same as Surface 5 but with different coefficents

Deficient: Case 1 - Same as parent but the last three terms were

dropped

Case 2 - Same as parent function but the last terms

were dropped.

From the results of the empirical verification of the relative efficiency of the seven sampling patterns shown in Table

B1-B9, Appendix B, the following observations could be made:

1. The unaligned systematic stratified random sampling

pattern is the most efficient since it produces the

least RMS for almost all sample sizes and terrain types.

2. The relative efficiency of the sample patterns is not

related to the terrain roughness or class of terrain

or to sample size>

3. The sample pattern which produces the most stable

normal coefficient matrix in a least squares solution

gives the best accuracy for interpolation. In other

words, the relative efficiency of sampling patterns

could be determined by computing the determinant or TABLE 3.5 THEORETICAL EXPECTATION AND EMPERICAL RESULT OF THE EFFICIENCY OF SAMPLING PATTERNS

Sample Theoretical Emperical Surface Class Size Expectation Result

1 I 25 rand.< str. rand. < str.; 64 sys. < str. sys. < str. sys. < str.

2 I 36 rand.< str. rand.. < str. ; sys.< str. 100 No decision No decision

49 rand. < str. rand. < str. s y s .< str. 3 I 100 rand. < str.; rand. < str.: rand. < sys. rand. < sys.

100 No decision No decision 4 II 169 sys. < str. sys. < str.

5 III 256 sys. < str. sys.< str. 324 sys. < str. sys < str. 400 sys. < str. sys.< str.

100 sys. < str. rand.< str. sys. < str. 6 IV 196 No decision No decision 484 No decision sys. < str.

256 sys.< str. sys. < str. 9 III 394 sys.< str. sys. < str. Case 1 400 sys.< str. sys. < str.

256 sys.< str. sys.< str. 9 III 324 sys.< str. sys.< str. < r a n d ... Case 2 400 sys.< str. sys. < str. i . . . sys. = systematic sampling , , , . - better than 1 ; ,,,, , , . rand. = random sampling str. = stratified sampling r ° the trace of the dispersion Matrix. According tio

Fedorov [43], if D^, are the Dispersion Matrices for

adjusted parameters from sample pattern 1 and 2

respectives and

if | D]_| < | | (3*30)

or if Trace (D^) < Trace (D2) (3.31)

then experiment 1 is better than 2

This phenomenon can be observed from Table B.5 in which

D? < Og < Dj. < D2 < D3 < Ds < Dl

4. If the result of theoretical expectation of the relative

efficiency of the sampling patterns is compared with

that in Table Bl-9 it will be observed that Das 1 [23]

theorems can be a fairly good a priori indicator of

what the optimum sampling pattern for any given terrain

is expected to be (see Table 3.5). Also from Table 3.5

one can conclude that the optimum sample pattern for

any terrain is dependent on its autocorrelation

properties. It is well to remember that it was shown

in Section 3.2 that the optimum sample size is related

to the degree of terrain roughness.

3.4 AUTOMATIC OPTIMUM SAMPLING FOR D.T.M.

As a result of the investigations performed in this research, a program called "ATOSAP" was developed which could be used inter­ actively by the stereoplotter operator or in a fully automated system to determine the optimum sample size and also to generate the optimum sample pattern for digitizing any given terrain. The essential features of "ATOSAP" are as follows:

1. It uses HVM program to iteratively sample on a

stereo-model to obtain the optimum sample size using

the criteria in eqn. (3.1). As an alternative, the

operator could use any of the linear regression

equations, particulary eqns. (3.6), (3.7), (3.8),

(3.9), (3.10), (3.11), (3.16), (3.17), (3.18), (3.19),

(3.20), (3.22), and (3.23) in combination with criteria

(3.24).

2. ATOSAP then calls "AUTOCO" to determine a priori which

of the sampling pattern is relatively most efficient

according to Das' theorem.

3. GENSAP is called by ATOSAP to generate the X,Y

coordinates for the theoretically most efficient

sampling patterns. If the operator wishes, he might

call any type of sampling pattern of his own choice.

Since the most convenient pattern for interpolation is

the aligned systematic pattern it is recommended that it

be used for sampling unless any of the following

conditions occurs:

(a) The first condition is when there are frequent

occurrence of "Breaklines" which are incompatible

with the class of terrain. According to Table 2.2

this case can be detected from ATOSAP which also computes other parameters of terrain roughness.

This means that more sample data should be purposely taken in areas with unusual occurrence of "Breaklines."

This procedure may be called "purposeful sampling."

Systematic Sampling should not be used if there is no means of determining an optimum interpolation algorithm for a given terrain. The method to deriving such an optimum algorithm and its relationship to sampling error will be discussed in the next Chapter. 74

CHAPTER IV

INTERPOLATION PROCEDURES FOR DIGITAL TERRAIN MODELS

4.1 DEFINITION OF BASIC TERMINOLOGIES

The branch of statistics/mathematics which is related to interpolation and prediction has borrowed some terminologies from other branches of science particularly electrical engineering. Basic definitions of some terms are therefore justifiable.

Data Smoothening is a data-processing method whose main objective is to reduce the effect of observational errors which are usually called "noise.11 Smoothening may be achieved by

Least Squares procedures, Moving Average, etc. Filtering is synonymous with smoothening. Signal is a component of the data which has a meaningful pattern of variation. It is relevant and wanted component of the data whereas "noise" is irrelevant, unwanted, and

"short-term" (i.e., it fluctuates rapidly) and it is a random component of the data. By contrast, signal tends to be "long-term."

In Curve or Surface Filtering, Regression or Trend Analysis a function which includes unknown parameters is assumed to express the relation­ ship between the dependent and the independent variables.

Interpolation consists of estimating some unknown quantities by making use of some given data (for curve fitting) whose sphere of influence is assumed to extend over the unknown quantities. 75

Extrapolation procedure is the same as interpolation except that estimation of quantities are made outside the sphere of influence of the given data. Prediction, strictly speaking, refers to both interpolation and extrapolation although interpolation is sometimes used to indicate prediction. Point Interpolation involves fitting of a curve Z - f(X) to a given data at some points usually arranged in one-dimension and estimating the unknown data from other points.

Patch-wise interpolation is a type of interpolation using locally defined functions Z (x,y), for an area divided into patches^

Whereas in Global Interpolation a single function Z (x,y) is used for interpolation all over the whole area. Trend is that part of data which is systematic and can be predicted. The term "White

Noise'1 refers to residuals which are random and whose autocorrelation is not significant.

4.2 INTERPOLATION FORMULAS

Many interpolation formulas abound for various methods of

interpolation viz, the classical numerical polynomial formulas, the

logarithmic formulas, the exponential formulas, the Least Squares polynomial, and formulas, the Fourier Series formulas, and the

Multiquadric formulas. These formulas which are briefly discussed below,

seek to express elevation Z of a point in terms of one or both

cartesian coordinates.

4.21 NUMERICAL POLYNOMIAL INTERPOLATION FORMULAS

Classical numerical polynomials have not found much favor with DTM researchers for reasons best known to them. One of the objectives of this research is to investigate the efficiency of some of these polynomials as a DTM technique. In general numerical

interpolation formulas are obtained by the use of operators A or V or 6 indicating "Forward Difference," "Backward Difference," and

"Central Difference" respectively (see Watson [54]). In the appli­ cation of numerical interpolation a function Z = f(x) or F(y) is not known and we wish to predict or infer the values of Z for values of x or y intermediate between given data r points (x^, y^), (x^*

(x , y ), by constructing a "Difference Table." (See Table 4.1 for m m ° 4 an example of forward difference table for Z = f(x)=x .

TABLE 4.1 FORWARD DIFFERENCE TABLE

JL f(X) A f A2f A3f A^f x— 2 0 0

1 x— i 1 1 14 15 36 x 2 16 50 24 c 65 60 x 2 3 81 65 24 175 84 x 2 4 256 174 369 x 3 5 625

Note that the A**f of the fourth difference is a constant. Only a brief discussion of formulas will be offered here (see [52-55] for details). 4.211 NEWTON'S FORWARD INTERPOLATION FORMULA

Newton's Forward Interpolation formula is derived by

Taylor's Series expansion as

Z = fo + o- (X-Xo) + A 2 fn (X-X ) (X—X i ) h “ 2h *

+ (X-Xo) (X-XO (X-X2) + ......

... An fn (X-X0) (X-X!)_____(X“Xn 1 ) <4 ‘1 5 ,.n n ! h

where n = degree of polynomial through equally spaced (n+1 ) points

Putting p - (X-X0) /h eqn. (4.1) may be written as

A 2 ir A 3 -F Z = fo + Af 0 P + P(P-l) + P(p-l) (P-2) + 2! 3!

.Dr + ...... =-**>• P(P-l) (P-2) ...... n !

42.2 SPLINE INTERPOLATION FORMULA

Spline interpolation in analogous to estimation of the

value of a function at any new point by using a draftsman's spline

to draw a smooth curve through the given data points. Spline Fit

interpolation consists of determining which two points (X^, Z^) and

(X^_^, x^j_) the given value X lies and finding Z by the formula 2 " Cl,k (\ + l - X)3 + °2,k (X- \ )3 + C3,k

+ C4 k (X-^) + . . . . (4.3)

where

Ci . C , , C 3 , , C 3 . are computed from ,K.j 2 * ^ » R

c',k - 3V 6dk

c^,k = a2W 6dk

c 3,k - V dk - a2zk ' V 6

°4,k = zk+l 1 dk “ d X + l dk /&

dk “ *k+l “ \

From eqn. (4.3) it is clear that spline fit interpolation is accomplished by connecting each pair of adjacent points with a section of a 3rd degree polynomial such that the first and second derivatives are continuous at each point between various sections.

4.213 AITKEN - NEVILLE FORMULA

Aitken-Neville formula is a successive linear interpolation and extrapolation. The formula for linear interpolation for the value of f±+1 (x) between the interval (xi, *1+1) ±g g±ven by

(1) fi+l(K) *“ f(xi+ ph) = pfi+l + (1“ p)fi **• (4.4) where 79

P = X“Xi

h = Xi " Xi+1

A quadratic interpolation f^(x) may also be obtained by linear interpolation from

and in the interval using the formula

(2) . f±(X)= i i-Pd-Df^^n-p] [l+pjf^d-p)^] and this interpolation method can be extended to obtain higher- degree interpolation e.g., nth degree interpolation.

4.214 STIRLING’S INTERPOLATION FORMULA

Stirling's Interpolation formula is one of the "central difference" formula usually written as a polynomial of degree 2n passing through 2n+l points e.g.,

(x__ n» •** (x0» Z0), (xi, Z\) ... txn » yn )

The formula may be written as

Z = fo+ pA f -i + Afp + p 2 A 2f -i 2 2 ) + P(P2- l 2) A 3f -2+ a3f —! + p2(p2 - ]2) 4„f _ 2

31.2 4!

+ ......

+ P2(p2. !*, Ip2.(n-1)2] Aan f _n ...

(2n) ! 80 where

P = (x - xo) /h

4.215 LAGRANGE FORMULA

Lagrange's interpolation formula is an equation expressing nth degree polynominal through (n+1 ) points (xozo) (X 1 > zi) •••

(xn , z^) which are not necessarily equally spaced. The formula can be written in shortened form as

n z = f (x) = Z L (x). f. (4.7) i=0 1 where the coefficients L^(x) for i = 0 ,1 ,2 ,..., n may be written as:

^ ( x ) = (x “x o) (x - xx) ... (x - ...(x - xn )

(x± - x0) (Xj-xi) ... (x.j-x.^ ) (xi”Xn)

Eqn. (4.7) may be written more explicitly as:

Z = L 0 (x)fo + Li(x)fL2 (x)f2+ --- + L^(x)f (4.8)

4.216 DIVIDED DIFFERENCE FORMULA

Divided difference formula like Lagrange's formula does not require equal interval between successive points. It is given by:

f(x) » a 0 + ai(X-Xo) + a2 (X-Xq) (X-X^ + ...

+ a (X-X0) (X-Xi)...(X-X ,) n n— 1

+ (X-X„) (X-X1)....(X-Xn)Rn+1 (4<9) where

~ , Xg]

a 0 = f[X0 ]

= f[Xit X0] Xi-X0

a 2 - ftX^xn-fULXp] = [X2, x„ X„] x 2-x0

f [X . . •X-.l-ffx ...X ] =f[X X 1 t...Xn] a = L n, 1J L n-1 0 a, n - 1 ’ u n v xn-x0

4.22 POLYNOMINAL INTERPOLATION FORMULAS

The polynomial in 1-dimension is given by

Z = ao + ajy + a2y 2 + anyU * * * (4.10) passing through [n+2] points for a least squares solution. The need for a least squares solution arises from the desire to reduce the effects of observational errors commonly called noise. Using Uotila’

[85]notations we have the following formulation

Maths Model: L = f(X ) (4.11)

Observation Equation: V = AX + L (4.12)

Normal Equation: X = (ATA)” ^ATL (4.13) where

a - 3 F A - 3X

L = L^ = observations 82

X = parameters

The purpose of adjustment is to get the best estimate of a parameter with minimum variance and minumum sum of squares of residuals.

An exact solution where the number of parameters is equal to the number of observations was attempted and the results compared with the least squares solution.

Equation (4.10) may be extended in 2-dimensions as

9 3 Z = ag + ajx + a 2x ‘i + x +■*:.. .

+ bjy+ b 2y 2 + b 3y 3+ ...

+ cixy + c 2;c2y + csxy2 + cijX2y 2+ ... (4.14)

4.23 LOGARITHMIC FORMULAS

The Logarithmic Formula, for interpolation may be expressed both in one- or two-dimensions e.g.,

Z = alog x (4.14) and Z - alogx + blog y (4.15)

The logarithm may be Napierian or natural logarithm.

Since the coefficient is linear a least squares solution of the above equations does not require iteration.

4.24 EXPONENTIAL FORMULAS

Exponental formulas may be expressed as: 83

Z = ao£ (A.16) ax2+ b y 2 Z = ao£. (4.17) Elevations are defined as exponential function of x and x,y.

Although the coefficients are not linear a least squares solution with observation equation is feasible.

4.25 FOURIER SERIES FORMULAS

The single fourier series formula may be expressed in the way

CosIIix + b Sinllix] (4.18) -- n ---

The two dimensional version of Fourier series has been given in

Section 2 as eqn (2.7) under which the notations used in eqn. (4.18) have been defined.

4.26 MULTIQUADRIC FORMULAS

Multiquadric equations of topography and other irregular surfaces were first developed by Hardy [22] as a result of the failure of polynomial and Fourier Series to represent any continuous irregular surface to any required degree of exactness and to provide a logical interpolation of the surface between data points.

A type of 3-dimensional multiquadric surface may be expressed as:

n (4.19) where

smoothening factor (see [93] for details) 84

Z = f(x,y) resulting from the summation of single quadric surfaces, (e.g., hyperpoloid, paraboloid, cones, cylinders, spheres, ellipsoids, etc.)

C determines the algebraic sign and flatness of the quadric term and the vertical axis of symmetry of each quadric term is located at a discrete horizontal position x^,y^ for a summation of cones eqn. (4.19) becomes

z = cj^[(xj-x) 2 + (y^-y)2] 1 /2 i-<5j (4.20) J“1

In one-dimension

z = c J k Xj - x )2 ]17 2 -!-/^ (4.21) " j = f

One of the objectives of this Section is to investigate

Hardy’s claim that if control data are located at significant points on the topography, multiquadric should perform better than the conventional interpolation methods.

4.3 OPTIMUM INTERPOLATION AND LEAST SQUARES METHOD

Two basic conceptual backgrounds (assumptions) are very

important to the understanding of what is meant by optimum inter­ polation. The first will be called a priori assumption and it should be made for the D.T.M. data sample.

Value at data Regional Variation + / Noise I (4.22a) point + Local Deviation 85

The above equation simply means that sample data have three components, namely, (i) Regional variation which extends all over the major parts of the area (ii) the local deviations from the regional variation which extends only to a limited part of the area and beyond the mean interval of data points and (iii) the "noise" which includes errors due to localized random variation between interval of data points. This gives rise to sampling errors due to lack of optimum sample size and sample pattern. Optimum sampling for D.T.M. may be defined as sampling that satisfies the relationship in the above equation such that the noise is simply "white noise." This also implies that data blunders or measurement errors which is part of the third component, is minimum.

The second assumption may be called a posteriori assumption given by

Value at Deterministic Data point functional value

This equation which states the post picture of least squares interpolation, asserts that value at data point consists of the deterministic functional value and the residuals. The problem of determining optimum interpolation consists primarily of relating the two conceptual equations above. The deterministic function is essentially the interpolation formula for surface fitting. In least squares solution the residuals are assumed to be white "noise" having zero mean,.common variance and uncorrelated. Under these assumptions the least squares solution of the deterministic function is optimal and accounts for both the regional variation and the local deviation in the apriori equation. The most important assumption here

is that of uncorrelated residuals because if it is not fulfilled the

least squares solution is not optimal. That is, the adjusted parameters

although unbiased do not have minumum variance properties. This

conclusion is admitted by Draper et al [69] and Neter et al [91].

In this case, the residuals contain not only the "noise" but also

part of the local deviations. This situation may arise when (a)

one or more significant terms of the deterministic function is

missing and (b) there is the presence of gross sampling or/and

observational errors in D.T.M. data. The method whereby the sampling

errors can be removed have been discussed In Chapter III even though

the merits of aligned systematic sampling have been admitted. Two

sophisticated approaches to optimal interpolation which seem to handle these two solutions properly are now discussed. The first approach which will be called Stepwise interpolation is an attempt

to construct the "best" deterministic function using some statistical

tools and the second which is an advanced least squares commonly

called least squares collocation, seeks to filter part of the local

deviation (signal) from the residuals so as to make it pure "white noise." In both of these approaches the statistical test on auto­ correlation of residuals will give the ultimate indication as to whether an optimum interpolation is attained. 87

4.31 STEPWISE REGRESSION PROCEDURES FOR OPTIMUM INTERPOLATION

The Stepwise procedure for choosing the most significant independent variables to be included in the "best" multiple linear regression equation of the form

Y = b + b, X. + b. X, + + b X o 1 1 2 2 T n i was briefly discussed in Section 3.3. In this Section application of the Stepwise regression will be made to regression equations with polynomial and non-polynomial terms of the form

i f + (4.23) k+1 k+1

j + rf j + i where

f x 1 1 2

2

f Sin X j+1 1

In other words f^ can assume any function of interest. The procedure enumerated in Section (3.235) will be used to determine the significant terms to be included in the "best" interpolation formula. Criteria are set up to add or delete a term. One of the most important features of the stepwise regression is that a significant, term which has been added at an earlier stage is reexamined along with other terms at a later stage and may then be considered insignificant, and therefore, deleted.

At the end of the search only the most significant variables which make the greatest improvement to the "goodness of fit" are retained in the regression equation model. By "goodness of fit" is meant accounting for a large.proportion of the total variance in the dependent variable. The coefficients corresponding to these significant terms determined by least squares on other similar pro­ cedures represent their "best" estimates. There are many methods for selecting the significant terms in the regression model which include

1. The percentage of the total sums of squares of the

dependent variable accounted for by the terms

Whitten [8 6 ]y

2. The statistical significance of the percentage sums of

squares in 1)*Allen and Krumbein [87].

3. The degree of autocorrelation in the residuals * Connor

and Miesch [8 8 ].

4. The magnitude of the sura of squares or variance of the

residuals. Criteria 2) is by far the most common and

two variations are recognized. In the first, critical

F^ and TF^ values are specified, Efroymson [89] and

used in adding or deleting terms by comparing with the 89

computed F-value according to F-test. In the second

variation a probability level Q Is specified rather

than the critical F values and this is compared with

the probability Q of computed F value occurring by

changes,Miesch and Connor [62].

The merits of stepwise regression to interpolation are:

1. It removes the problem of knowing the number of terms

that are sufficient for interpolation for any given

situation. The same result can only be attained by

trial and error which would be very expensive In terms

of time and computational effort .

2. Since only significant terms are included in the

regression model problems related to "ill-conditioned"

normal coefficient Matrix are removed. This

may be very significant in terms of removing sampling

error in D.T.M.

3. Another advantage of the stepwise procedure is that

"composite" formulas described in Section 4.27 could

be used. This will, therefore, strengthened the

quality of the "best" regression model that is

finally arrived at.

4.311 COMPUTATIONAL PROCEDURE FOR STEPWISE REGRESSION

Efroymson [89] has successfully applied the steps involved in the Gaussian elimination method for solving simultaneous equation 90

to stepwise regression procedure. The procedure is now discussed briefly:

Step 1 : Specify a priori value of Q

Generate the functions of all the terms required

to be considered in the regression equation (4.23)

at each of the data point (X,Y,Z) or (X,Z)t as the

case may be, and store them in a Matrix T (n,n) where n = number of terms to be considered

let NDF = n-1 and VAR" 1 and set Ci = -1, l

NDF = degrees of freedom

Step 2 : Compute the covariance Matrix and correlation

Matrix R from Matrix T

Step 3 : Augment Matrix R to obtain Matrix A which is

partioned as follows:

~ I A = R(kxk) j B1 (kxl) I(kxk)

I B(lxk) S (lxl) O(lxk)

•I(kxk) 0 (kxl) O(kxk) where R(kxk) = correlation Matrix for k independent variables

B(lxk) = correlation vector of k independent variables

with dependent variable Z

B^(kxl)= transpose of B

S(lxl) = correlation of dependent variables with itself. 91

I(kxk) = identity Matrix

-I(kxk) - negative identify Matrix

0 (k,k) = zero Matrix initially

Step 4; Computer Vi = ^+1 iJ * i=l»2,....,k

4 (a) find minimum Vi for = 1 and 0.00001.

If V^ minimum does not exist then go to step 46 otherwise

set Vmin = minimum Vi and compute F = (vmin, NDF)/VAR

Compute QF, probability of F using eqns. ,(4.24-4.26)

QFF - [F1 / 3 (1 ]- 7/a (4.24)

2FZ/32F2/ 3 A>\1/2 V 9 9NDF /

The area under F distribution curve above QFF may be computed

from eqn.(4.25) for large degrees of freedom.

f *0 -t2/ ** J q f f 8 dc <*-25> which is approximately computed from

qrt# e~QFF f 2 (a1 t+a2 t2+a3t3+a4 t4+a5 t5) (4.26) vizir '

where t = 1/(1+02316419 |q Ff Q; +1.782125 5978

a- = 0.319381530 a. = -1.82125 5978 1 4

a2 = -.35656 3782 =+1.33027 4429 (See Miesch and Conner [62]) 92

If QFF Is negative, then QF = (1.0 - QF); (4.27)

If NDF <£ 100 and is even QFF = NDF /(NDF+F) (4.28)

and QF = 1 - 1-QFF (1+ ^ QFF2 + ), (4.29)

If NDF ~ 100 and odd QFF = (1 - 2 (NDF)) (4.30) where d(NDF) = 20TT for NDF = 1

d(NDF) = (0 + sin 0(Cos9jO + 2/3 cos^O + ) for NDF 1

. 1/2 and 9 = arctan (F/NDF)'

If QF > Q set k=i of minimum v^, and increase var by vmin and increase

NDF by 1 and go to 5

If QF < Q go to 4b

4(b) Find maximum Vi and set Vmax = Vi max. , , „ (Vmax. DF) and compute F = (Var_Vmax) M.31)

Determine QF from eqns (4.25) - (4.30)

If QF Q set K=i of maximum Vi decrease var by vmax

and decrease NDF by 1 and go to 5

Step 5 Adjust Matrix A to B for variable entered

First divide B,=(A,/A) (4.32) mj mj mm' *

Bij - A1J - W-33) mm where m=k for variable just entered and change Cm to -Cm; for all Bii=l and go to step 4

Step 6 Compute all the regression coefficients for which

Ci = +1 from AS r \ bj = il * z for all i^where Ci = +l) (4.34) Si The regression constant is computed by bo = Z (b^.X^) for all i, where - +1 (4.35) where Z = mean of dependent variable

means of Independent variables

= standard deviation of Zs dependent variables

= standard deviation of independent variables

The four modes of stepwise regression depicting the composite interpolation formulas used in one- on two-dimensions are depicted in Table 4-2 TABLE 4.2 STEPWISE REGRESSION COMPOSITE MODES

Mode 1: One-dimension Polynomial Dominant Polynomial Terms Linear ------X

Quadratic------X2

Cubic------x3

Quantic------

Quintic------x5

Nonpolynomial terms

Square root - X 5? 2X Exponential------e , e , e

Logarithmic------log X, (logX)2

Reciprocal------X \ X 2

Mode 2 : One-dimensional single Fourier Series Dominant fourier terms i = 1 Cosnx, SinJTX L L

i = 2 ------Cos 2IIX , Sin 2IIX L L i = 3 ------cos 3IIX , Sin 3]1X L L

NonFourier Terms 2 Polynomial------X, X

Square root

Logarithmic------logX

Reciprocal------, X ^ , X 2

i = harmonic in X-direction 95 TABLE 4.2 CONTINUED

Mode 3: 2-dimensions: Polynomial Dominant Polynomial Terms L i n e a r ------X, Y

Quadratic------X2 , XY, Y 2 3 2 2 3 Cubic------X , X Y, XY , Y

4 3 2 2 3 4 Quantic------X,XY, X Y , XY , Y 5 4 3223 45 Quintic------X , X Y, X Y , X YJ , XY*, YJ

Non Polynomial terms

Square root--- J x J x Y p T

Exponential------eX , eY , e^X ,eX+Y, e^Y 2 Logarithmic------logX, logY, (logX.) , log X*log Y , (logY)2

Reciprocal------x"1 , Y- 1 , X_ 2 , (XY) - 1 , Y-2

Mode 4 : 2-dimensional Double Fourier Dominant

Double Fourier Terms

see eqn. (2.7)

Non Double Fourier Terms

Polynomial X, Y, XY

Square roots JXY

Logarithmic (logX) (log Y)

Reciprocal (XY) ^ 96

4.32 LEAST SQUARES COLLOCATION COMPUTATION FORMULAS

The detailed concepts and derivations of collocation are

contained in Geodetic Science Report No. 175, 1972, by Helmut Moritz ![94]

Only a brief discussion of basic formulas and derivations is offered here.

The generalized least-squares collocation formula is given by Moritz [14]

x = AX +s + n (4.36)

where x = vector of observation

X - vector of parameters

A = partial derivative of a function (viz., a polynomial) 3F 2 with respect to X is A = where F = b(j + b^u + b 2 U + * * *

for interpolation

n = "noise" - vector of measuring error

s = "signal" = vector of random quantity.

Statistically

From equation (4.36) above it is clear that x the observation consists of a systematic part (AX) and a random part (s and n). Note that s existvboth at observation and predicted points.

If the collocation model, eqn. (4.36) above, is applied to interpolation using a polynomial we have

X f(u^) + s^ + n^ i 97

ITm m where f(ui> = ai + a2U + a3U3 + uf ' * anrfl T. ak+l k=0

F (U) = f(u) “ s(u)

X = [ai, a 2 ... a^^]'

Linearizing eqn (ii)

3F(u) A = 3 X

x = AX + S 1 + n (4.37)

-,T where x = lx ^> X2 * • .. x ] = observations q

liT 1 _ [si, s * , ... s*] = signals at observation points

n [ni , n 2, ... n ] = measurement errors. q

Fig. 4.1 Below illustrates the quantities which are observed, and

predicted, and the polynomial as well as the signal

Fig. 4.1 LEAST SQUARES COLLOCATION QUANTITIES 98

u i , U 2 ---- U 5 = data points

fcl» t2 ---- t 5 = predicted points (interpolated points)

f(u) = polynomial without the signal

F(u) = f(u) + s(u)

From eqn (4.37)

x — A X + Z

Zl where Z2

Z 1 = S 1 + n

Zq

‘Si ‘ Let s 2 = p + q random nu S = signal for computation point

V I ie Z = signal for observation point -H- Zi V = vector of residual z 2

Zq

From the equations above and according to Uotila [85]

AX + BV - x = 0 (4.38) where

B = [0, I]

The covariance matrix of V is computed using

c c ss sx Q = (4.39) C C X S X X where

C = C i 1 + C (4.40) xx s 1s 1 nn T Minimizing V PV and using Lagrange's Multiplier we have

= VTPV -2 KT (AX + BV - jc )

Solving for X and S we have

X = (ATC“ 1A)“ l A ^ ^ x (A.44)

S = C C“ 1(x-AX) (4.45) sx

The covariance matrix and covariance function is computed as follows

The covariance function is given by the Gaussian function

_ 2 2 C(r) = Co ta r (4.46)

where C0 = var^-ance unit weight

a = coefficient and it is unknown

r = distances between data points

Co and "a" are determined by fitting eqn. (4.46) using the data points

Cgisi = Cr*(s^sl) = cov(r±j) = CoS "a 2 (rij)2 (4.47) where

r ^ = |U^-Uj [ at observation point 100

Similarly 1 C = C ( r . .) (4.48) ss ij 7 where

r ^ = Iv^-V^l at predicted point

Csx = cov(Bij3J‘) = cov(rij) <4'49> where

ril - 'ui -vjl

The Error Estimates are computed as follows:

Ex = £ - x (4.50)

Eg = s - s (4.51)

Exx = (AT"C U )" 1 (4.52)

E = C - C C-^C + HAE ATHT SS SS SX XS XX

E = -E ATHT (4.53) X S X X ' where

H = C C- 1 SX

Interpolation or prediction is obtained using the following formulas: 101

t = A i K + S (4.54) at predicted points

t f (v £) = f(V±) + S(V±) i

Error of interpolation is computed using

(4.55)

4.33 PATCHWISE INTERPOLATION

Patchwise interpolation method could be used to improve the accuracy capability of various 2-dimensional interpolation formulas.

The problems and solutions associated with discontinuity along the boundaries of the patches into which the whole surface is divided are discussed in [44] and [95].

4.4 INTERPOLATION, SAMPLING PATTERN, SAMPLING ERROR, AND ACCURACY

In this Section Interpolation methods will be discussed as they relate to sampling pattern, sampling error, and accuracy considerations.

4.41 INTERPOLATION AND SAMPLING PATTERN

Table 4.3 shows the types of Interpolation formula suitable for certain sampling patterns. 102

TABLE 4.3 INTERPOLATION AND SAMPLING PATTERNS

Random Interpolation Random Stratified Systematic Stratified/ Method UNALALUNAL ALUANLAL Systematic

Numerical Interpolation v /

Least Squares Polunomial v / ✓ 1-D 2-D y ✓ ✓ w / y y Least Squares Fourier Series y' 1-D 2-D ✓ W ✓ ✓ y Least Squares Exponential 1-D 2-D ✓ ✓ ✓ W" V'- Least Squares Logorithm v ' 1-D 2-D y ✓ y

Multiquadric \S n/ 1-D 2-D y ^ v / - ✓ y

Stepwise ✓ 1-D — 2-D ' ------y ✓ y ✓ Collocation ✓ 1-D

2-D ' z ? y ✓ ✓ ✓ y

Patchwise v / / 2-D y

AL = Aligned UNAL = Unaligned 103 Table 4.3 shows the restrictions placed on the choice of sampling

pattern for D.T.M. data generation. It can also be observed from

Table 4.3 that aligned systematic sampling pattern is universal

in the sense that it is suitable for all types of interpolation methods. It is not surprising, therefore, that systematic sampling has been favored in most D.T.M. applications.

4.42 INTERPOLATION AND SAMPLING ERROR

One of the merits of stepwise regression method for interpolation

referred to in Section 4.31 was that it could suppress or reduce sampling errors. Emperical evidence for this merit can be observed from Table 4.4 which shows the result of interpolation using step­ wise regression method on surface 4 (see Appendix A, Figure A.4) Two

dimensional polynomial dominant functions (see Table 4.2) and

"deficient" function, were used.

4.43 INTERPOLATION AND ACCURACY CONSIDERATIONS

The advantages of systematic sampling over other types of sampling methods have been discussed in section 4.41. Also since stepwise regression is insensitive to sampling error the accuracy considerations being explored in this section relates only to systematic sampling.

Durbin/Watson test [96] will be used as an indicator of optimum inter­ polation . In otherwords if autocorrelation of residuals is signi­

ficant then optimum interpolation is attained, otherwise improvement of accuracy can be further made by applying patchwise-method or linear least Squares collocation method? 104

Table 4.5 shows the accuracy attained by various interpolation methods discussed in the foregoing sections. The following conclusions may be reached from the results shown in Table 4.5

(1) Autocorrelation of residuals is a good indication of optimum

interpolation.

(2) The class of terrain is a good indicator of the number and

type of terms to be included in the interpolation formula.

(3) Some of the classical or numerical interpolation formulas

which do not involve "smoothening" have comparable accuracies

with those of optimum interpolation formulas and also patch­

wise-method.

(4) Justification for classifying interpolation formulas accord­

ing to their accuracy standards can be found from Table 4.5.

(5) Multi quadric method of Interpolation seems to be consistently

superior to other methods In two dimensions.

(6) Interpolation methods are classified (see Table 4,6) according

to their accuracy capabilities as judged by their reactions

to four types of terrain indentified in Table 2.2. Multivariate statistical method of classification discussed in Chapter 2

was used for the classification of interpolation methods

into four groups. Group one consists of methods with the

highest accuracy capabilities and group four, those with the

poorest accuracy standards for interpolation. The usefulness

of such a classification is to guide the automatic choice

of interpolation method according to accuracy requirements

and application. For example if the user is interested 105 TABLE 4.4 OPTIMUM INTERPOLATION AND SAMPLING ERROR

Surface and Stepwise Arbitrary Sample Type Optimum Interpolation Interpolation Formula Std. Error Unitless Std. Error Unitless

1. UN-Random 15.768 19.697 2. AL-Random 11.523 23.149 3. UN-Stratified 15.526 16.701 4. AL Stratified 13.911 19.055 5. UN-Systematic ■ 15.268 16.947 6 . AL-Systematic 16.113 19.222 7. UN-Systematic Stratified 14.690 24.045 only In predicting elevation on a surface with high accuracy require­ ments "Lagrange" formula would be an automatic choice because it in­ volves the least computation effort and time among its peers in class

I, Table 4.6. Whereas if the application is for volume computation the user would prefer some powerful global approach to interpretation such as stepwise 2-D fourier function, or 2-D multiquadric method.

The justification for the choice of global function for volume compu­ tation will be discussed in Section 5.3.

4.5 CONCEPT OF AUTOMATED D.T.M.

The term "mechanization" and "automation" are often used to indicate the replacement of a human operator totally or partially, by a machine. Automation as used in this research implies that some aspects of D.T.M. operations-data generation and interpolation - which involve human efforts are partially being mechanized and optimised such that information thus obtained is fast and it is used to control further TABLE 4.5: INTERPOLATION METHODS AND ACCURACIES

INTERPOLATION INTERPOLATION TERRA T N CLASS MODE METHOD I II III IV R.M.S. R.M.S R.M.S. R.M.S. (Units of Ft.) (Units of cm) (Uni (Un-fr1f*<5q) Autocorrelation (0.124) of raw data (0.377) (-.5649) (.678) 1. Lagrange 3.983 4.844 0.0005 2505.099 2. Divided Difference 3.369 4.366 2.5950 4261.709 3. 1-D Spline 3.713 4.690 1.0960 2641.474 Ailken-Neville 3.983 4.844 0.0005 2505.099 POINT-WISE Stirling 4.257 6.123 3.4560 5678.483 e! Newton Forwar 17.226 11.467 102.325 24491.553 7. Exact 1-D Poly- 3.983 4.844 0.0005 82789.348 nominal

2-D Polynomial 11.719 5.479 8.276 9734.332 4-Patches 2-D Polynomial PATCH-WISE g_ 8-Patches 11.691*** 4.328*** 7.056*** 4925.532*** 2-D Polynomial 16-Patches 18.26 4.503 45.611 7353.006 ■ POINT-WISE 9. Stepwise 1-D 4.427 4.814 5.826 5362.798 GLOBAL 10. Polynomial 2-D 27.946 [.1364*] 5.1384 [.0514**] 187.201 [-283*] 22281.745 [.0484*J POINT-WISE 11* Collocation 1-D 3.257 4.344 5.705 5016.391 g l o b a l 12. Polynomial 2-D 9.143 3.748 5.707 2673.047 POINT-WISE 13* Stepwise 1-D 3597.162 8567.345 5678.897 1834678.777 GLOBAL 14. Fourier Fun -2-D 171.289 [t535*] 4.643 [.021**] 23.469 [.068**] 5870.253 [.1489*] ... Cfio CollocationV d 3.561 3.996 11.164 8792.745 POINT-WISE 15* _ with „ GLOBAL 16. Fourier Fun-2-D 10.769 3.459 211.904 15834.426

POINT-WISE Multiquadric 2-D GLOBAL 17* Representa­ 10.200 [.0232] 4.432.[.086] 17.016 [.532] 6660.188 [.0813] tion of Topography

[ *] Significant autocorrelation of residuals at 52 level according to Durbin/Watson [96] [ **] Inconclusive test regarding significance of autocorrelation *** Optimum Patch Size RMS “ Root mean square of differences between true and computed elevations TABLE 4.6 CLASSIFICATION OF INTERPOLATION METHODS ACCORDING TO ACCURACY CAPABILITY

CATEGORY

Lagrange* Divided Difference* Patchwise 2-D Stepwise 2-D Interpolation Polynomial Polynomial

Spline 1-D* Stirling* 1-D Polynomial Newton* Methods cum L.S. Collocation Ailken-Neville* Step-wise 1-D Stepwise 1-D Fourier 2-D Fourier Cum Exact Polynomial Ploynomial L.S. Colloca tion

2-D Polynomial Multiquadric* cum L.S. Collo­ cation Stepwise 2-D Fourier

1-D Fourier cum L.S. Collo­ cation

* Methods which require the least computational effort in each category. 107 108

rapid progress of the overall operation. As a result of this research a computer program ATODTM was developed which can be used in a fully automated system or by a stereoploter operator to obtain the following results automatically.

(1) to determine the optimum sample size and optimum sample

pattern of data point distribution, for generating elevation

information for any given terrain either on a stereo-model

or a topographic map. (See ATOSAP, Appendix C).

(2) to objectively classify the terrain of investigation using

some quantitative features of the terrain (see CLAS Appen­

dix C) .

(3) to determine for any application the optimum interpolation

or surface fitting technique for the terrain using informa­

tion from Table 4.6.

(4) to analyse numerical parameters and to show graphical dis­

play of the terrain, for example, area, volume, center of

gravity, cross-section, contour perspective drawings and

statistical analyses (see figures 4.2 for flow chart of

automated D.T.M.)

Detailed description of ATODTM can be found in Appendix C. FIG. 4.2 ATODTM FLOWCHART

Sterro Model (Application) HVM Optimum" Sample ■Size

YES ATOSAP Terrain Analysis

SAMP Optimum Sample Pattern

CLAS Terrain Classification

y T ^ \ T E P O OTTIMMUM^s. INTERPOLATION NO (Durbin/Watso . T e s t l X ^

YES

GRAFT C Numerical, Statistical Analysis, Graphical - CHAPTER V

DIFFERENTIAL PHOTO MAPPING APPLICATIONS

5.1 INTRODUCTION

The automated D.T.M. as designed in this research has unreserved flexibility and adaptability to a wide range of applications in which the basic source of data is either a photogrammetric stereo- model or a topographic map. The particular application of interest in this research has to do with differential mapping. Differential mapping is defined as the mapping of an object relative to itself with proper considerations to physical dimensional changes in time and space or relative to another object/objects with which it bears some physical, genetic, etc. relationships. In this research the better part of the definition of differential mapping is illustrated and the crucial relationship .being considered in the mapping of cows from the same parents, is essentially genetic. The objective of this

Chapter is to demonstrate the application of automated D.T.M. to the procedure for constructing the map of a cow relative to another and for constructing an average cow for the purpose of making comparisons between the cows involved. The Interest in this research was generated by the OSU Department of Dairy Science through Professor F. Allaire who is investigating the possibility of deriving genetic relationships

110 Ill from the physical characteristics of cows as displayed on their maps.

5.2 DIFFERENTIAL MAPPING PROCEDURES

The procedures described here assume that the instruments used are the Wild Stereometric camera (C-4) and the Wild-A-7 online with the IBM Computer and the IBM 1620 Calcomp Plotter. Also, it is assumed that no absolute ground controls are given. The set up of the photography is depicted in Fig. 5.1. In a previous study,

Ayeni [51] investigated the relative efficiency of three differential mapping procedures illustrated in Fig. 5.2 and found procedure II to be an optimum procedure for the problem of mapping cows.

Procedure II is now described with appropriate modification in the application of automated D.T.M.

5.21 PROCEDURE II FOR MAPPING A COW

Step 1: Interior Orientation. The center of the two fiducials on each of the photographic plates was determined on the

Wild A-7 coordinatograph. Each stereopair of the original negatives were then carefully centered on the non-compensating plate-holders inserted in the Wild A-7 projectors. Due to the limitations of the

Wild A-7, a nominal principal distance of 100 mm was introduced

Instead of the calibrated 88.2 mm determined by Oswald [82] for an object distance of 3 + 0.025 meters. However, the Model Z readings taken after absolute orientation is to be corrected for the incorrect focal length Introduced at the Instrument. Light Beige Cloth

Leveling Staff

Camera

Figure 5.1a Photography Set-up

White Mark; Dia.= 25mm

Black Mark; Dia.= 5mm

Figure 5.1b Markers 113

. INTERIOR Step 1 ORIENTATION at Wild A7

RELATIVE- Step 2 ORIENTATION at Wild A7

I ^ _ i = " v III ABSOLUTEABSOLUTE Step 3 DIGITIZATION ORIENTATIONORIENTATION at Wild A7 at Wild A7 at Wild A7 i f I ABSOLUTE • Step 4 DIGITIZATION DIGITIZATION ORIENTATION at Wild A7 at Wild A7 at Computer ♦ ♦ CONTOUR CONTOURCONTOUR Step 5 PLOTTINGPLOTTING PLOTTING at Wild A7 at Computer at Computer

PROCEDURE I PROCEDURE II PROCEDURE ffl

Figure 5.2 Flow Chart Indicating the Three Procedures 114

Step 2: Relative Orientation. Both "b^ elements were set at 75 mm, or any other suitable value, so as not to exceed the Z range of the instrument. B f element was also set at a desired value

to give a convenient model scale. For example, bx = 4 cm, for C-4 camera photography to achieve a model scale of 1:10. Then empirical relative orientation was performed and the "by" element used to determine the standard error of residual y-parallaxes (not to exceed

+ 0.02 mm at the six standard points).

Step 3: Absolute Orientation. The scale correction was

computed from the image of the leveling staff and applied at the

instrument following the procedure enumerated by Ghosh [75].

Leveling was performed using the three control points Ci, C 2 , and C 3 , as known in Figure 5.1. For the solution of common K, it was intended

to make Y-coordinates of C* and C 2 equal and for the solution of and the heights of Cj, Cg, and C 3 were to be made equal. These constraints were applied to the transformation formula eqn. (5.1)

(see Ghosh [75] and a computer program was written to do a least squares solution. ■ X X x 0 Y = s [MT ] y + Y o (5.1) H h H 0 1_ where [M] = rotation matrix X S = scale Y = Ground Survey Coordinates x 0 H yD = Translation x

z 0 y = Model Coordinates h The correction applied to the elevation of points read on the instrument is given by 115 Hg = 100 + (H-lOO)* (5.2) where Hg = Adjusted spot height

H = Elevation read at the Wild A-7

X = 88.2/100

Step 4: Digitization. Digitization consists of automatic

recording of the X, Y, H absolute oriented model coordinates after

the centroid has been fixed at X - 500 ' mm , Y - 500 mm and H = 100 mm.

As part of ATODTM the subroutines ATOSAP is called to determine the

optimum sample size and to generate the optimum sample pattern in

the manner described in Chapter III.

Step 5: Terrain Analysis. This step also using ATODTM

calls subroutine ATOSAP to compute various roughness parameter

other parameters like volume- Subroutine CLAS is also called to

classify the terrain as illustrated in Chapter II.

Step 6 : Plotting and Contouring. This is the graphical

, m u * _ | „ _ ^ . output section of ATODTM. A computer program was written to do the

following:

(a) Reads the X, Y, H of digitized points in Step 4

(b) Corrects the H readings for the improper focal

length using eqn. (5.2)

(c) Transforms the X, Y coordinates into that of the

Calcomp Plotter which is off-line the O.S.U. IBM 360/65

by a suitable shift of origin.

(d) Plots the boundary of the cow and the contours and

spot height of digitized points at the desired scale

1:5, by calling subroutines PLOT, SYMBOL CNTOUR. I

!T» 1““

& •p *«/•*») 111 L — r. w

* t ■

1 1 1 • IV • cr it' II w It it M H M FIGURE S - 3 : MRP OF COW =50: PROCEDURE II

CQNTOUH !NTERVRL=-2.5CM MRP SCALE* 1:5

FRESH DATE * 7-02-73

CURRENT ORTE =■ 02-01-74

CfiRTCCRflpHlC REPRESENTATION ON THE CRLCOHP PLOTTER

PROJECT SUPERVISOR: OR. SRNJIB GHOSH OPERATOR ON HILO R7: OLU. RTENI 116 117

. The out-put of this program consists of a deck of cards which can be used on the IBM 1620 Calcomp Plotter. The map of cow

#50 in Figure 5.3 illustrate the final product of procedure II using ATODTM.

5.21 PROCEDURE FOR CONSTRUCTING A DIFFERENTIAL MAP OF A COW AND OF AN AVERAGE COW

The main objective of this section is to develop a computer assisted method of computing and mapping the physical dimensions of an average cow from offsprings of the same parent. The following principles were used in computing an average cow viz.

1. The three points Cj, C 2 » and C 3 were to be used to

control the size of the average cow because of their

unique and strategic (always identifiable) locations.

2. An average cow would be computed by transforming each

individual offspring cow from the same parent into the

standard cow chosen from amongst them and then by

taking the simple average of both the standard and the

transformed cow.

Ghosh and Jiwalai [76] have suggested a number of approaches to this problem based on graphical, empirical/graphical and com­ putational techniques. The approach being used in this research is a graphical-computational method of constructing an average cow. 118

5.22 GRAPHICAL/COMPUTATIONAL METHOD FOR CONSTRUCTING AN AVERAGE COW

The steps Involved in the construction of an average cow

is described below.

Step 1: Interior Orientation. The interior orientation

is performed in the same way as that for the standard cow. (See

Step 1, Procedure II)

Step 2: Relative Orientation. This is the same as for

the standard cow. (See Step 2, Procedure II)

Step 3: Transformation for , fi, and Solution. This

step involves the transformation of relatively oriented model

coordinates x, y, z of Ci, C z , and C 3 on any individual cow into the

absolutely oriented model coordinates X, Y, H of Cj, C2 , and C 3

on the standard cow, using eqn. (5.1) without imposing any constraints.

The unknown parameters would be Xo , Yo, Ho, and S, and since we have nine observation equations, there are two degrees of freedom

and a least squares solution is possible. A computer program was written using the mixed model F(X , L ) = 0 d. cl The observation equations are the type (using Uotila [85] notations).

AX + BV + W = 0 (5.3) where A _ F(Xa,La> Partials with Respect to

Ti Parameters L X q , YQ , Z 0) 119

_ ^F(Xa>La) a Partials With Respect to £)( X,Y,Z) Observations

„ _ F(Xa>La) Discrepancy Vector Evaluated at +B(L -L V = ^ servat^ons ant^ Current b a Parameter Estimates

V = Vector of Observational Residuals

X = Solution Vector

The resulting normal equation is given by:

(A1M_ 1A) X + A ^ ^ W = 0 (5.4) where

M = BP 1BT

P = I i.e., equal weights (unity) for all the observations.

The solution vector is given by:

X = - (A1!!-1 A) ”*1 A lM-1W (5.5)

The updated adjusted parameter vector becomes

Xa = X + X q (5.6)

where

Xo = Approximate value of the parameters

The output of this program consisted of ft ,. $» S,

Xjj, Y0, H0. The corrections due to elements ft * and S were then introduced manually at the Wild A-7 to complete the transform­ ation of the individual cow into the standard one. The iteration was performed in accordance with Pope's [83] method.

Step 4: Digitization. The elevation gears fro elevation were set to "Ba" and the desired plotting scale was chosen to be 1:5. 120 The Centroid of the three control points was physically determined on the instrument and an arbitrary coordinate (X = 500 mm, Y = 500 mm, and H = 100 mm) were assigned to this point as was done in Stage 4 of

Procedure II. After this, all the operations described in Stage 4 of

Procedure II follow:

Step 5: Terrain Analysis. This is the same as in

Procedure II.

Step 6 : Plotting and Contouring of Transformed Cow and and the Average Cow. This step is essentially the same as Step 6 of

Procedure II.

A computer program was written to perform the following functions viz.:

(a) Corrects the H readings of transformed cow using

eqn. (5.2).

(b) Computes the average of the corrected H reading for

both the transformed and the standard cows.

(c) Transforms the X,Y coordinates of the digitized points

of both the transformed and the standard cows into the

coordinate system of the Calcomp Plotter.

(d) Plots the boundary outline of the transformed and the

"standard" cow and the average H reading as spot heights

along with their associated contours at the desired scale

1:5.

Step 7: Graphical Interpolation. A graphical interpolation between the boundaries of the transformed and standard cow was made to produce the boundary of the average cow. Fig. 5.4 illustrates 121

the final product of this procedure for cow 51 while Fig. 5.5 cor­

responds to the average cow, constructed from Cows 50 and 51.

• m rm iltt n a i a l n «

lll.ll

~*r »* wriif.jin «

rSul.T^litH . .iH.i - 1 1 * n I*,

7*** -* ' 7 n * [« ''> i* H '*rl + m .» i

in titin n ^iita^pw1 .in.*! ,in in titin MVll m m*y ^11 Aj> t W * i in t* .In +1111* yn » \m +HI ll /lll.ll inn „L ^

W//---l>\ / 0 '1' «S ,!•*.»- _ _ _ .19_ •* + IJ* ii *ir»n +in u 41* n L« ** +ini*_.,ii« * +\n*i Jm.n fdi- Av^rs ^ + ■****• +m li^m n'^'yl* ** +)*«« «i)i ft +l\* ** - * / x ^ y s 1 V' 'lipll .111 «'» ItX+Jlf H •H'a tipH j^l^* .ifur1' .in ]K t'Mu /|iil (t/1 '•ittl- .’'f v /^ T r n ^ > j » > . s . i M . ' fr v ^ ' i T ifarTtxrn—

i «* *a i m «« in ia ia ia m it «c « • « t* n \*m tt n FIGURE 5-4: MRP OF COW *51 CONTOUR lNTEflVAL*2,5CM MAP SCALE* IiS

PROJECT SUPERVISOBi OR. SANJ19 CH05H - OPERATOR ON HILO A7i OLU. ATEN1

CARTOGRAPHIC REPRESENTATION ON THE CALCOHP PLOTTER

FRESH DATE * 7-03-73

CURRENT DATE * 02-01-74 122

irrr ,T £*** ,>n ^ ,<«i n '% *• M \+>t> * ♦■ j ► , J. • m --r'-^v

I 4 ■ M ♦ » • V * rr \ w * v .* t* 11> FIGURE 5-5: MRP OF RVERRGE COW FROM COW «50 AND «51 C C M O O S If« T E RVfiL '-^. OC H HflP SCRLt - 1:5

C0510Gfi

TABLE 5.1 APPLICATION OF GLOBAL FIT FOR PREDICTING ELEVATION ON THE BODY OF THE AVERAGE COW Sample Size = 200 No. of points predicted = 15

Interpolation Type Interpolation Formulas RMS in cm

2-D Polynomial 2.721 " Global Interpolation 2-D Fourier 2.360 " 2-D Multiquadric 2.0136 "

2-D Polynomial 5.239 ” Global Extrapolation 2-D Fourier 3.445 " 2-D Multiquadric 2.567 " 123

5.3 INTERPOLATION AND VOLUME COMPUTATION

From Table 4.5 it is observed that global fit can give comparable accuracy with point and patch-wise interpolation. This possibility offers an application in the global fit of the entire body of a cow especially an average cow (including head). For example if after the construction of an average cow established from progenies of the same parent as des­ cribed in Section 5.2, it is found that there are some random points for which the elevations are required. Such points may not be common to all the progenies hence their elevations could not be computed for an average cow. Such points may also require interpolation or extra­ polation. If an appropriate single function could be found for a global fit, the problem of predicting by interpolation or extrapolation would be solved. An attempt was made for a global fit with 200 data points on the body of a cow, to predict 15 random points on any part of the cow body. Table 5.1 displays the results of the investigation and it shows that the Double Fourier function performs better than a Least Squares power series polynomial for interpolation and extrapolation regardless of the terrain type. Multiquadric appears to be superior' to the other two

•formulas.

Another application of global fit is the computation of volumes of a body bounded by topographical bodies. Various approximate formulas have been used, instead of using the formula.

Volume (5.7)

Method A. The volume can be computed using the area enclosed by a contour (Hou and Veress [79]) n 124

tl where A 3 21rr L **-,i

h = total no. of contours

M = no. of points digitized on the contour

n = total number of contours

Method B . This method also uses the ones enclosed by a contour to compute the volume. k

. cT h ) v = i r 1 (Aj ' <5 -9> where C^H = contour interval

k = number of contour lines

Method C. Cubic formula which is widely known in engineering applica­ tions may be expressed as

v - f - F(M.) (5.10) 4 i-1 where F(M^)are the elevation readings at the four corners of a- 2 square or rectangle whose area is a . This formula gives

accurate results only if the surface can be approximated by

a polynomial f(X,Y) of degree one. This implies that the

region of interest should be divided into sections suffici­

ently small to justify such an approximation. 125

Method P. Simpson’s Method -- Simpson’s 1/3 rule for the integral over a

rectangle 2h by 2k centered at i,j . : is given by the double

integral Is

f(x,y)dxdy

i-1

— — ff + & f + f 1 3 3 ^ i+1^ j+1 i, 1-1, j+l'

+ (5.11)

In this section investigation is made into the accuracies of various formulas for volume computation. These accuracies are compared with those obtained by using gobal functions of varying degrees to compute the volume from the double intergral version of Simpson's Rule with Richardsons' extrapolation improvement [58]. The test surface was surface 5, whose true volume was computed by using eqn. (5.7). From the results displayed in Table 5.2, it is clear that the optimum global function with Simpson's

Rule (Method D) performs better than the approximate formulas (Methods

A, B, and C ) . 126

Table 5.2 Volume Computation on Surface 5

Method Volume in cu. units

A (volume from contours 14478.926

B (volume from contour interval 13264.342

C (cubic methods) 14580.1020

D Optimum Interpolation 17971.609 with Simpson .Rule

True Value by Simpson 139477.250

True Value by Direct Integral 139477.750

5.4 COMPARISON BETWEEN TWO OBJECTS

Another application of Automated D.T.M. developed in this

research is that it provides various parameters of surface which can be used for making comparisons between two objects. Parameters such as

surface area, volume, average gradient, vector strength, vector dis-

pension, Harmonic Vector Magnitude, variance spectrum and others have been discussed in the previous section. One important parameter!' which has not been discussed so far relates to cross spectral analysis - in

2-dimensions. As in the case of variance spectrum^the cross spectra may be derived from the 2-dimensional lag covariance function or directly

from the original data say elevation measurements, from two terrains

(arrays, X(i,j) and Y(i,j)). From Step 5 of Section 2.28^ the amplitudes

ax (k^,k2 >, b^Ck^jk^), ay (k^,k2 ) and b^ (k^,k2) are computed. The co­

spectrum are then computed using (see Rayner[40] for details), The quadrature spectrum is obtained by

XY (5.13) o 2 i

The correlation or coherence between two surfaces may be computed by

which is analogous to the sample correlation coefficient formula. The 2 coherence between two surfaces may be interpreted as correlation

coefficient. The coherence between cow.-50 and 51 is 0.8 and this is considered significant*. In other words the two surfaces are correlated

.or similar. Table 5.3 shows some parameters which could be used as a basis for making comparisions between the two cows and the average cow.

5.41. PLOTS OF MULTIVARIATE DATA

Good as the contour maps of the elevation are, they do not bring out clearly how close together cows from the same parents are. This

type of comparison is multi-dimensional. For example, we may wish to compare the elevations at 215 points for any two or more given cows.

Plots of infinite-dimensional function which has long been one of the most useful tools in data analysis provide an answer. Such a method i of plotting data of more than two dimensions was proposed by D. F.

Andrew [72], For instance, the spot heights h^, ...... ^215 may be mapped into a function of the form.

f(t) = ~ f- + h sin t + h_ cos t + h. sin 2 t + \J 2 1 J 4

+ cosl07t (5.15) TABLE 5.3: PARAMETERS FOR HAKINC COMPARISONS BETWEEN SURFACES i Mean Standard Hean. Hean Surface Vector Vector Buop H.V.M. Voluae Auto Coherence Terrain cow t Elevation Deviation Gradient Curvature Area/ Strength Dlspersln Fteq. Corre­ Class cubic lation of Elevation Plane CO CO CD en Area

55.360 0.384 13.*37 31.831 16752.992 0.531 50 115.421 32.226 3.714 1.B08 9.690 I

13.410 34.007 16993.678 0.607 0.805 51 117.792 47.357 3.850 2.035 3.106 12.7370 0.521 1

Average 116.606 34.143 3.543 1.641 7.789 12.845 32.840 16928.56 0.626 Gov 55.217 0.383 1 128 129

This function is plotted over a range - TT*

A computer program (part of ATODTM) was written to generate 215 terms of this series and the output device for the plot was on the

O.S.U. IBM 1620 Calcomp Plotter. Figure 5.6 shows the spot heights of cow 50, 51, 50/51 plotted as a function of eqn(5.15). Before any interpretation or inference is advanced from these plots it is per­ tinent to consider two of the properties of the plot function, eqn(5.8).

(a) The function preserves the means (X^) of a set

of n multivariate observation X ■. i (b) The function also preserves distances and, therefore,

close points will appear as close functions.

The proofs of these properties are briefly stated in Andrews [72], From the properties outlined above, one may conclude that if some plotted functions are at a distant apart from others for all values of t, then the corresponding points are also at a distant apart in the Euclidean metric and such plots represent outliers. Conversely, if some plots form a bond by remaining close together for all values of t, then such a bond represents a multivariate cluster.

Figure 5.6 shows that the three cows form multivariate clusters.

In other words, no outlier cow exists although at certain values of t * (t^ and for example) the cows show marked differences. It will be recalled that the elevation of the average cow was computed as a simple arithmetic mean from two cows from the same parent. This linear rela­ tionship which implies, for example, that

f(t) = [f(t) + f(t)] 12 (5.16) ave. 50 51 is borne out by the plots which also show that the average cow lies

exactly midway between the two cows. 130

FIGURE 5- 6 HULT J -DIMENSIONAL, PLOT-COH* 50,51*50/51 CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 CONSLUSIONS

Based on the investigations and analysis presented in the foregoing sections, a number of conclusions can be drawn.

1. Conclusion related to Digital Terrain Models in general.

(a) The importance of according proper-recognition to the

various components of D.T.M. must be emphasizes (see

Section 1). The weakness in some definitions of D.T.M.

have been pointed out in Section 1.

(b) From the idealistic definition of D.T.M., four important

components of D.T.M. can be recognized viz (i) the terrain

type, (ii) the sample size and (iii) pattern sampling and

(iv) the interpolation method. The interaction between these

components as influenced by accuracy Is illustrated in Fig. 1 .1 .

(c) The importance of attaining maximum efficiency with minimum

effort through an automated D.T.M. was stressed throughout

the research.

2. Conclusions related to terrain.analysis and classification.

In Chapter 2 attention was focused on the problem of

quantitative description and classification of terrains.

131 132

(a) Based on certain criteria (for describing terrain rough­

ness) only five parameters,viz,average gradient, curva­

ture^: surface area, Harmonic Vector Magnitude and Bump

frequency can be used for a.fairly objective terrain classi­

fication.

(b) Certain characteristics of a good classification

identified and multivariate statistic (the trace of

pooled within - groups sum of squares) can be used as a

criteria for classification improvement.

(c) Two groups of parameters for quantitative description

of the terrain can be identified.

(i) Parameters describing surface roughness itself.

(ii) Parameters describing spatial distribution of surface

irregularities.

15 terrains of contrasting characteristics were

analysed and classified into four major classes of

terrain types (see Table 2.2} Computer programs

TERAIN, TERANA and CLAS were used for this analysis.

3. Conclusions related to optimum sampling for D.T.M.

The problem of determining optimum sample size and sample

pattern were investigated.

(a) It was found that optimum sample size is a function of

terrain roughness while optimum sample pattern is related

to autocorrelation properties. (b) The HVM (Harmonic Vector Magnitude) can be used as a criterion

for determining optimum sample size. Similar results can be

obtained from linear regression equations using terrain

roughness parameters as dependent variables.

(c) The empirical relative efficiency of seven sampling pattern

as investigated, indicate‘that their results compare favorably

with those from theoretical relative efficiency from Das'

theorems.

(d) A program ATOSAP developed as a result of this investigation

can be used in a fully automated system or by a steroplotter

operator to generated data efficiently from a stero-model

or from a topo map.

CONCLUSIONS RELATED TO INTERPOLATION PROCEDURES FROM D.T.M.

Four terrains representing the four major classes of terrain

types were used to test the accuracy capabilities of all the

Interpolation methods discussed in this research. Using

multivariates statistical procedure discussed in Chapter 2

these interpolation methods were grouped into 4‘ classes, } (See Table 4.6.) . Such a classification was used as an aid

to the choice of optimum interpolation method for a given

application in the program ATODTM.

(i) It was observed (Section-4.43) that while the class

of terrain is a good indicator of the number and type of

terms to be included in the interpolation formula the auto­

correlation of residuals is a good indicator of optimum

interpolation. 134

5. CONCLUSIONS RELATED TO APPLICATION OF AUTOMATED D.T.M.

The primary application of interest in this research concerns

only differential photo mapping with close-range photogrammetry.

Examples were shown how the differential mapping of cows can

be done efficiently with high performance in accuracy and time

by using automated D.T.M. It should be noted that other applica­

tions regarding computation of volumes from a topographic body,

interpolation on the body of an average cow and making compari­

sons between two similar objects were also illustrated.

(a) It was noted that a global function obtained from

optimum interpolation in conjunction with Simpson's rule provides

a far more accurate method for volume computation than most

other approximate formulas widely used in engineering applica­

tions .

6 .2 RECOMMENDATIONS

In the light of the results obtained in this research and the conclu­ sions made in the previous section the following recommendations are made for future research.

1. RECOMMENDATIONS IN CONNECTION WITH OPTIMUM SAMPLING.

(a) Only 7 terrains were used for forming the linear regression

equations employed to determine optimum sample size in

Chapter 3. It is recommended that more terrains (both real

and stimulated) be used to generate new regression

equations and their accuracies tested under various terrain

types to ascertain their consistency and reliability. 135

Such equations especially the "economic models" of Section

3.22 should provide a less sophisticated method of determin­

ing optimum sample size.

(b) There is need for research into optimum sampling on a

terrain with unusual occurrence op frequency of breaklines

in order to avoid some of the problems of breaklines discussed

in Section 2,26.

2. RECOMMENDATIONS RELATED TO INTERPOLATION PROCEDURE

There is need for further research into the accuracy capability

of least square collocation for interpolation. The empirical approach

to the computation of the covariance function by fitting residuals

to the "Gaussiam" function seems to limit the accuracy standard of

this advanced least squares method since the Gaussian function and other similar functions (see Rampal [90]) may not be suitable for a given

terrain. It is therefore recommended that future research should be directed towards the application time series analysis in finding the appropriate theoretical covariance functions,three of which are widely used in time-series analysis namely (see [28]) >

i. Simple moving average process (MA).

ii. A purely autoregressive process (AR) and

iii. Autoregressive moving Average Process (ARMA).

It is the author's belief that D.T.M. should be regarded as one of the most effective photogrammetric methods for the extension of elevation control and its accuracy standards should be pursued vigorously. 136 3. RECOMMENDATION RELATED TO AUTOMATION

The approach to automation in this research has been "batch" oriented, due to lack of time and facilities for developing an interactive approach to automation. It is therefore recommended that investigation should be made into developing an interactive approach to automated D.T.M. by using the Cathode Ray Tube (CRT) so that results can be displayed on the CRT screen from time to time. .Also important decisions related to optimum sample size, and pattern, optimum interpolation could be made at any stage by an operator (human correlator) who becomes the crucial link between the "terrain", the computer program, the computer, and the stereoplotter/digitizer.

4. RECOMMENDATION RELATED TO APPLICATIONS

Other possible areas of application not dealt with in this research are listed below for the attention of future researchers.

i. Digitization of existing topomaps and applications to auto­

mated topographic mapping,

ii. Study of physical changes in an ecosystem, for example a lake

resort, dam site deformation, gorge erosion, in time and space.

Thus a 4th and 5th dimension may be introduced to various aspects

of mapping such ecosystems,

iii. Application to highway design and construction,

iv. Applications to microscopic 3-D mapping with the electron micro­

scope .

v. Seismic related studies such as land slides and earthquakes; and

vi. Biomedical applications such as the study of changes in parts of

the body. APPENDIX A

TERRAINS AI - A15 USED FOR

CLASSIFICATION 137 FIG. A-l: LINEAR SURFACE

\

i JUT. \

\ ocv.v \

Z = a + b X + CY 138 FIG. A-2: EXPONENTIAL SURFACE

x+y Z = A e o 139

FIG. A-3: LOGARITHMIC SURFACE

Z = Log X + Log Y 140

FIG. A-4: DOUBLE FOURIER SURFACE (RANDOM COEFS) (A)

7

( i

\)' i, \ ' i I 0 / r

/ 1 /' \ 1 1 >■ ( 7 (\ !< \ \ i v/ A \ \ \

•. \ \ ! I - ' -jA' i A i ■ . ! b'l V v4' l‘ ■ ''V u ’ 1 ' ' ’ 1 )A\n )/v:> v\7!!. !j m - A w,'-,' . • -»r*\ i*'*- ' j ,

r\ \( V f 1 , - V.' • > / j

\ -i

\ .. » \ *- f \ I \ J v

0 \ /

..A i _ 1 \ ;

(See Equation (2.7) in the Text) 141

FIG. A-5: 2-D POWER SERIES POLYNOMIAL SURFACE - I

Z A 0 + A_X + A_X + A„ X + A.X4 + A_X3

+ b^Y + b2Y? + b3Y3 + b^Y4 + bsY5

+ c1x 2y2 + c2x 2y 3 + c3xy2 + c^xy3 + c5xy 5

FIG. A- 6 : DOUBLE FOURIER SURFACE: SYNTHETIC COEFS +15

J .... \

( ] I* r ,!jl )(r' ■ ^ y( * ...... f ' : J L I \ >" \ " o ' "j " . .1 /_ y„ ••••■: ■' \ a ( 15 .L__ A ™ . _ I . . - *J -ip 0 + ];?

(See eqn (2.7) in the text) FIG. A-7: 2-D POWER SERIES POLYNOMIAL SURFACE II ” j';-, I ■ P i W . / / / .

’■ .Ilf

v :11 4 "! It. .'Ill' ■ I''. ;i» - fl'r ’ | IJ"' i'V':i K I". / * V MNbhnp i | ; !|v j l ;. j ; / I ■ i ' ‘ ‘,,.! f ; ; (* I i s \ s ^ ‘;.k; i r? i i / . \

:l virfr I : i . - ■ ' 4

------t - -

Example of Surface Generated From Ploynomlal Reconstructed from Aligned Random Sampel: Size =* 196

(See eqn. for Surface 5) 143

FIG. A-8: DOUBLE FOURIER (RANDOM COEFS) SURFACE (8)

V I <11 l'' - i— A (See Equation (2.7) in the Text) FIG. A-9: 2-D POWER SERIES POLYNOMIAL SURFACE III

■im1

(See Equation for Surface 5)

FIG. A-10: DOUBLE FOURIER SURFACE (RANDOM COEFS)

-171

- (See Equation (2.7) in the Text) FIG. A-11: 2-D POWER SERIES POLYNOMIAL SURFACE IV

(See Equation for Surface 5) 146

FIG. A-12: ALMA, WISCONSIN (from Tobler's DIM Library [39J)

FIG. A-13: EMERADO, N. DAKOTA (from Tobler's DIM Library [39j) 147

FIG. A-14: MAVERICK SPRINGS, WYOMING (from Tobler’s Library [39])

1

FIG. A-15: OGEMAW COUNTY, MICHIGAN (from Tobler's Library [39]) APPENDIX B

RESULTS OF THE

INVESTIGATION INTO THE EMPIRICAL

RELATIVE EFFICIENCIES OF SEVEN

SAMPLING PATTERNS

FOR INTERPOLATION ON 9 TERRAINS Table B-l; LINEAR SURFACE (Fig. A-l) R.M.S of Differences Between True and Com­ Sample Type Sample Size puted Elevations, H.. and H (Unitless) c

Unaligned Random 64 3.985 Sampling 25 3.975 Aligned Random 64 3.996 Sampling 25 3.980 Unaligned Stratified 64 3.952 Sampling 25 4.019 Aligned Stratified 64 3.990 Sampling 25 4.011 Unaligned Systematic 64 3.943 Sampling 25 4.011 Aligned Systematic 64 3.951 Sampling 25 3.970 Unaligned Random 64 3.939 Stratified Systematic 25 3.965

Note: Hfc = True Elevation

H = Computed Elevation Table B-2: SURFACE 2 - EXPONENTIAL SURFACE (Fig. A-2)

Sample Type Sample Size R.M.S. of Differences (unitless)

Unaligned 16 .0000011 Random 36 .000007 100 .000009 Aligned 16 .000002 Random 36 .000004 100 .000005 Unaligned 16 .000002 Stratified 36 .000007 100 .000005 Aligned 16 .000002 Stratified 36 .000007 100 .000007 Unaligned 16 .00001 Systematic 36 .000005 100 .000007

Aligned 16 .000001 Systematic 36 .000006 100 .000006 Unaligned Stratified 16 .000002 Systematic Random 36 .000004 100 .000006 150

Table B-3 SURFACE 3 - LOG SURFACE (Fig. A-3) Sample Type Sample Size R.M.S. of Differences (unitless) Unaligned 64 .000758 Random 49 .000143 100 .001414 Aligned 64 .001613 Random 49 .000275 100 .001402 Unaligned 64 .00204 Stratified 49 .000810 100 .00366 Aligned 64 .00427 Stratified 49 .005342 100 .001926 Unaligned 64 0.003351 Systematic 49 0.000417 100 .002815 Aligned 64 .018346 Systematic 49 .000283 100 .01223 Unaligned Systematic 64 .00435 Stratified Random 49 .000157 Sampling 100 .00434 151

Table B-4: SURFACE 4 - D. FOURIER RANDOM

Sample Type Sample Size R.M.S. of Difference fnn-f tlp.ds'l Unaligned 64 70.613 Random 100 • 19.722 196 15.223 169

Aligned 64 57.719 Random 100 34.965 196 21.333 169

Unaligned 64 26.467 Stratified 100 40.259 196 16.906 169

Aligned 64 89.719 Stratified 100 24.199 196 16.111 169

Unaligned 64 41.159 Systematic 100 22.415 196 15.856 169

Aligned 64 22.2003 Systematic 100 17.244 196 14.959 169

Unaligned 64 283.337 Systematic 100 21.823 Stratified 196 14.598 Random Sampling

I 152

Table B~5: POLYNOMIAL SURFACE (Fig. A-5) R.M.S. of Sample Type Sample Size Determinant Trace Differences of H.& H L C Unaligned 256 16.695 0.6889 .0639 Random 324 15.128 0.4814 .0259 400 14.259 0.4036 0.0217

DI l Aligned 256 15.285 0.6576 0.0414 Random 324 19.055 0.6515 0.0349 400 16.112 0.4140 0.0223

°2 Unaligned 256 15.384 . 6434 0.0346 Stratified 324 14.219 0.5137 0.0276 400 14.0590 0.3953 0.0213 Do J Aligned 256 15.890 0.6701 0.0360 Stratified 324 15.667 0.5197 0.0278 400 14.806 0.4099 0.0220 D* Unaligned 256 14.711 0.6240 0.0345 Systematic 324 13.855 0.4985 0.0678 400 13.808 0.3965 0.0213

DS3 Aligned 256 14.503 0.6480 0.0344 Systematic 324 14.376 0.5112 0.0275 400 14.289 0.4117 0.0222

D6

Unaligned 256 14.114 0.6141 0.0330 Systematic 324 14.162 0.5441 0.0292 Stratified 400 13.675 0.3822 0.0205 Random D 7 153

Table B-6: D FOURIER SYNTHETIC COEFFS. (Fig. A-6)

Sample Type Sample Size R.M.S. of Differences (unitless) Unaligned Random 64 705.244 100 210.605 196 190.063 484 152.311

Aligned Random 64 2181.776 100 342.245 196 258.222 484 230.422

Unaligned Stratified 64 349.781 100 351.204 196 209.926 484 158.638

Aligned Stratified 64 748.791 100 426.223 196 191.694 484 161.40683

Unaligned Systematic 64 384.848 100 304.300 196 196.34 484 151.974

Aligned Systematic 64 349.840 100 237.563 196 100.702 484 151.135

Unaligned 64 2058.96 Systematic 100 243.724 Stratified 196 240.356 Random 484 Table B-7: POLYNOMIAL SURFACE (Fig. A-7) R.M.S. of Sample Type Sample Size Differences (unitless) Unaligned 64 2202.658 Random 100 2064.817 196 1497.041 676 1561.238

Aligned 64 6178.387 Random 100 2384.975 196 1764.842 676 1575.774

Unaligned 64 2936.706 Stratified 100 1587.672 196 1418.532 676 1395.439

Aligned 64 1896.521 Stratified 100 1945.651 196 1538.368 676 1388.846 64 Unaligned 1545.511 100 Systematic 1543.476 196 1419.478 676 1353.447

64 Aligned 1425.774 100 1393.172 Systematic 196 1364.967 676 1347.933

64 Unaligned 1510.599 100 Systematic 1451.349 . 196 1735.807 Stratified 676 Random 1464.214 Table B-8: DOUBLE FOURIER SURFACE (Fig. A-8)

R.M.S. of Sample Type Sample Size Differences (unitless)

Unaligned 256 289.0515 Random 324 217.1567 400 233.238

Aligned 256 353.003906 Random 324 240.1956 400 236.867

Unaligned 256 248.8567 Stratified 324 231.313 Sampling 400 205.493

Aligned 256 307.5413 Stratified 324 278.925 Sampling 400 231.846

Unaligned 256 233.96887 Systematic 324 203.831 400 212.613

256 194.180 Aligned 324 192.103 Systematic 400 191.213

Unaligned 256 207.0443 Systematic 324 207.1082 Stratified 400 198.433 Random 156

Table B-9a: CASE 1; POLYNOMIAL SURFACE (Fig. A-9)

R.M.S. of Sample Type Sample Size Differences fnnitless)

Unaligned 256 109.924 Random 324 106.351 400 105.128

Aligned 256 144.973 Random 324 185.810 400 120.1941

Unaligned 256 110.880 Stratified 324 106.890 400 102.232

256 114.126 Aligned 324 106.323 Stratified 400 109.5793

Unaligned 256 103.410 Systematic 324 106.297 400 101.966

Aligned 256 102.692 Systematic 324 102.385 400 102.183

Unaligned 256 101.227 Stratified 324 109.046 Systematic 400 102.491 Random Table B-9b: CASE 2; POLYNOMIAL SURFACE (Fig. A-9)

R.M.S. of Sample Type Sample Size Differences (unit]r s r )

Unaligned 256 .041436 RAndom 324 .108874 400 .120045

Aligned 256 .172883 Random 324 .365151 400 .203786

Unaligned 256 .039912 Stratified 324 .084591 400 .142462

Aligned 256 .03412 Stratified 324 .090372 400 .146659

Unaligned 256 .028047 Systematic 324 .060178 400 .102222

Aligned 256 .044862 Systematic 324 .044852 400 .110796

Unaligned 256 .035521 Stratified 324 .029863 Systematic 400 .082742 APPENDIX C

COMPUTER PROGRAMS FOR

ATODTM

(Major Subroutines Only) LEVEL ? . l I JAN 75 ) D S/360 FORTRAN H EXTENDED DATE 7 6 •1 5 4 /0 2 * 0 7 * 2 2 REQUESTED OPTIONS: O P T = 2 OPTIONS IN EFFECT: NAME(MAIN) OPTIMIZE!?) LINECOUNT160] SIZE(MAX) AUTOPRL[NONE] SOURCE EBCDIC NOLIST NODFCK OSJECT MAP NOFORMAT NOGOSTMT NDXRFF NOALC NOANSF NCJTERHINAL F ISN 000? IMPLICIT REAL*? (A-H.O-Z) ISN 0003 REAL*'- x x .y y .f ISN U00 4 REAL'S NA ISN 0005 INTIGFR P ISN G0G 6 DIMENSION X 1160 ),Y(160),Z(169),X3(169).ZZ(160,ll) ,ZAt 169,11),AZ(11 l.lbFJ,ZZI11,169),KA111),K0(11 ),XX(1’,13I,YYU3.13 1, 3E(13*13), XDNI169J,YDN1169) , ZD!J( I6<> I ,DI VI169) ,XN0RM<16«], lNal 11,11), SKI 5),CUT I 5),CI10, 10). VC 110, 10 J,T( 10,10) ,R(l<>,?). 2EE(6),B(6),CC(3),SUHD(3),SUMDS(3).:HEAM13),ZPIPt169 ),a R C Z (169), AYfiORMI lt.9) .ZNIKHI 169) ISN 000 7 DIMENSION DAT(65,65),LORR (104,65),DDT (65,65 ),C H 6 5 , 10),CM (10,10) ISN 0C0 6 DIMENSION IA (A) ,10(4),IC(9,2),ID(9,2),IE(9,51,IF[9)

------PRflGRA.M A T O D T M ------AUTOMATIC DIGITAL TERRAIN MODELS (ATODTM) WRITTEN bY ------OLU. AYENI, *------DEPARTMENT nF GEODETIC SCIENCE OHIO STATE UNIV. ------CCLUHPUS OHIO, 1976 ------

“ ~ ” t 5 d TM CONSISTS OF 8 MAIN SUBROUTINE~PROGRAMS DESCRIBED BELOW *— A T O S A P — p e r f o r m s t h e f o l l o w i n g OPFRATIDNS A O LOMPUTFS ROUGHNESS PARAMETERS SUCH AS AVF. GRADIENT(X1),AVE. CURVATURE(X2I.SURFACE AREA/PLANAR AREAIX3).MEAN DIP(X4), BUMP FREQUENCY(X5), VARIANCt OF OIRfCTIONA COSINE(X6),RESULTANT KURTOSIS OF DIRECTION COSINE! X7J.N0 OF SREAK.LINES PFR SOT. UNIT DF ARFA(XE),X NO OF POINTS PER SO. UNIT OF AREA IX9).VECTnR STRENGTH(01).VECTOR PISPERSI0NID2), BUMP FREOUFNCY(A(D3),SUMP FREQUENCY C,(D4) SOME OF THESE PARAMETERS ARE LISTED IN TABLE 2.1 AND THE HARMONIC VECTOR MAGNITUDE(HVM) ----- ASSUMPTION IS THAT YHE SAMPLE SIZE IS OPTIMUM ALSO THE DATA USED IS IS DN SYSTEMATIC PATTFRN (M ). THF ROUGHNESS PARAMETERS CAN EE USED FOR IB). THE ROUGHNESS PARAMETERS CAN BE USFO FOR COMPUTING EXPECTFD SAMPLE SIZE USING REGRESSION EQUATIONS DESCRIBED IN CHAPTER 3,DR USED FQR THfc CLASSIFICATION OF THE TERRAIN INTO ITS CLASS TYPE (C). AUTOCORRELATION PRPPERTIES FOR DETERMINING OPTIMUM SAMPLE PATTCRN ACCORDING TO D a S* THEORFKS ARE ANALYSEO — (2 ) . ---- - C L A S ------:------— SUBROUTINE CLAS USES THE ROUGHNESS PARAMETERS HVH.Xl,X2,X3.X5, COMPUTED FROM ATOSAP TO PERFQRH AN OBJECTIVE CLASSIFICATION DESCRIBED IN LEVEL 2.1 t JAN 75 ) MAIN 0 S /3 6 0 FORTRAN H EXTENDED DATE 7 6 .1 5 4 /0 2 .0 7 .2 2 CHAPTER 2 ( 3 1 . ------NUMERI IT IS AN OPTIONAL 5UBRPUT INF HUMERI IS ONE OF THE 6 OPTIONAL SUBROUTINES CALLED BY THE USER NJf'EKl CONSISTS OF 7 SUB ROUT INE S FOR N U M ERICAL INTCRoaLATITID No^ERl CONSISTS OF 7 SUBROUTINES FOR NUMERICAL INTERPOLATION PORMULAS DtSCRIBCn IN CHAPTER 4

IT IS AN O P T I O N A L S'JFROUTINF PATCH DIVIDES AGIVEN 2-PIMENSIONAL ARRAY INTO A GIVEN NO. OF PATCHES IT USES A SPECIFIED DEGPREE OF POLYNOMIAL FOR INTERPOLATION ( 5 ) . ----- — HQRT IT IS AH OPTIGHAL SUEROUTINE HURT IS MULT I OL'A URIC REPRESENTATION OF TOPOGRAPHY HURT MODELS A TOPOGRAPHY KY USING MULTIOI'ADR IC FORM'JLARS PROPOSED GY PROF. HARDY .HCRT IS ALSO USED FOR VOLUME CnM°UT4TIPN I<= THE USER SPECIFIES TT.

IT IS AN OPTIONAL SUBROUTINE STEPO REFERS TO STEPWISE INTERPOLATION METHOOE DESCRIBED BY THE AUTHOR IN CHAPTLR A STEPO IS A TECHNILEU OF CHOnSING THE OPTOMUM INTERPOLATION FQRHU l A FROM A GIVEN MJL'6FR OF TERMS A COMBINATION OF FUNCTIONS AS DESCRIBED IN CHAPTER 4 TABl E 4.2 IN CHAPTER 4 TATLE 4.2 OPTIONS ARE DSSCRT3ED UNDER THE SUBROUTINE (7). ------COLLOC IT IS AN OPTIONAL SUBROUTINE COLLOC PERFORMS COLLnCATION BY LEAST SQUARFS METHODS ITS PURPOSE IS TO INPROVE INETERP0LATIO.N BY USING THE SIGNIFICANT TERMS FROM I STEPO TO OBTAIN 4 MORE ACCURATE MODELLING OF THE TERRAIN tfi). ------GRAFIC ------IT IS AN OPTIONAL SUBROUTINE GRAFIC PERFORMS TWO MAJOR OPERATIONS I A) IT COMPUTES AND PLOTS MULTIVARIATE GRAPHS FOR MULTIVARIATE COMPARISONS OF DIGITAL INFORMATION BY MAKING USE OF FnURIER TERMS PROPOSED BY ANDREWS .F.ISEE BIOMETRICS ,VDL. 54, 1972. IB). PLOTS CONTOURS OF /FROM A 2-0 ARRAY USING PROF. RAPP'S PRIVATE LIBRARY

ISN 00 0 9 G CAL-3G.G GOAL THE VALUE OF GOAL REFFERS TO OPTION REQUIRED BY THE USER G0AL=1 CALLS n u m e r i G0AL=2 c a l l s p a c t h w i s e interpolation g o a l =3 c a l l s multiouadric interpolation G0AL=4 CALLS STEPWISE INTERPOLATION C0AL=5 CALLS COLLOCATION FOR INTERPOLATION ------ATOSAP ------159 LEVEL 2 .1 t JAN 75 ) MAIN 0 5 /3 6 0 FORTRAN H EXTENDED DATE 76 .154/02. 07.22 ISN 0 0 1 0 S CALE=1.0 SCALE = SCALE FACTOR TO BE USED IN CONVERTING RESULTS TO CORRECT SCALE I5N C 011 NL=6 NL IS FIXED AT CURRENT VALUE ISN C01Z L L L -13 ISN u013 KKK &11 LLL = &cil DIMENSION PF INITIAL DATA KKK =COLUMN DIMENSION OF INITIAL DATA ISN 0 0 1 4 YMAX=11.2 31 ISN C01 5 XMAX=13.362 C XMAX—MAX X COORD. c YMAX=MAX Y COORD. ISN 001 6 C.XX=1.0*SCALE ISN 0017 DYY=1.0*SCALE c DXX = INTERVAL IN X COORDS c DYY= INTERVAL IN Y COORDS ISN (018 Nf'T=3 ISN 001 9 NNN=1 ISN 0C2 G N T = 3 ISN 0021 M* —11 c. M=NO OF PARAMETERS FOR C L.S. FITTING OF SURFACE TO BE USED FOR HVH ISN 0022 NN1 =9 ISN CC23 N K=3 ISN 0 0 2 4 NI = 4 C NNT*Nf)N*NTiNNI.NK»NI,ARE FIXED AT CURRENT VALUES CNK= NO. OF PARAMETEkS OF POLY. USED FDR EREAKLINE 0 TEST C NI = NO. OF POINTS USED FOR FITTING POLY. FOR BREAKLINE 0 TEST ISN 0025 L L = L LL*KKK ISN 0 026 CALL ATDSAP(X'Y,Z'X3tZZ,ZA,AZ.Z?iKA.Ka,XX,YY,E'XnN'YDN.ZnNt lCIV.XNORM.NA,SK ,CUT,C.VC,T,R,CM.SB,B,CC.SUMP,SUMDS,ZHFAN,ZDIPT 2ARCZ.YN0RKtZNDRM,SCALE,LLL,KKK,DXX,DYY,YMAX,XMAX,NNT,M,NT,NNI, C 3NKi *11,XLEN,YBRET, LL.NLtNNN.XXO> XXI,XX2,XX3,XX5) ______ISN 0 0 2 7 I N = 0 : ■ ISN 0028 DD 300 11=1 ,LLL ISN (•029 00 300 JJ=1,KKK ISN 0 030 1H=1N*1 ISN 0C31 X I IN)=XX(IT , JJ) ISN C032 YYII1,JJ)=Y(IN] ISN ’.•033 Z(IN»= E 11 £ ,J J) ISN 1.034 300 CONTINUE ISN 0 0 3 5 LJ=LL C Jl= NO Or ITEMS =9 HAXIMUM C JZ=N0 OF VARIABLES =5 C J3= NO OF CLASSES =4 C J4 =2 ISN 0 036 Jl=9 ISN 0 0 3 7 J2=5 ISN 0038 J3=4 ISN 0 039 J4=J1*J2 ISN CC4 0 J5=J2*J2 ISN CC-41 J6=5 ISN 0042 J7=J2*tJ2+1)/2 C Jl= NO OF TERRAINS TO BE CLASSIFIED INCLUDING CURRENT TERRAIN C J2* NO OF VARIABLES TO USED FOR CLSSIFICATIGN c j3= n o o f c l a s s e s C J6=J2 ISN 0 043 C A L L Cl ASIIA,IB,IC,I0,KA,KB*ZZtZA,AZ,Z2,CM,IE«IF,BB,3,X3,XDN,YPN? 160 LEVEL 2.1 t JAN 75 ) MAIN OS/36Q FORTRAN H EXTENDED DATE 7 6 .1 5 4 /0 2 .0 7 .2 2 IXNORM .YNORM ,SK, CUT,DA T.CORR,DOT,Cl , VC , C,T,NA.ZON,OIV t ZDIP,ARCZ. 2Zt.nRH.XX0, XXI ,XX2,XX3.XXS.J1,J2,J3,J*,J5,J6,J7) C TEST WHFTHER L’ATA IS OBTAINED BY SYSTEMATIC PATTERN C IF CATA IS SYSTEMATIC DO NOT READ X,Y,Z ARRAYS ISN 0 044 lL=n ISN or.<,5 401 I 6=1 L+l ISN 00*6 SYST=l.G ISN G0*7 IF(SYST.EC.1.C1 GO TO *02 ISN ;>u*9 IF(SYST.NE.l.C) READ(5,400) X IIL),Y(IL),ZIILI ISN 0051 IFltL.LE.LL) GO TO *01 ISN lif.53 *00 FORMATI3F10.3) ISN 0064 *02 CONTINUE ISN 0C55 KK=1 ISN 0 056 LN= 13 C LN=LLL ISN 0057 LL=KKK*LLL ISN OGSE L = 6 ISN 0 059 P=8 C L= NO OF DATA POINTS FOR INTERPOLATION C P= NO OF POINTS TO BF INTERPOLATED C --- **♦* VALUE OF TEST =1,2..7 DETERMINES WHICH FORMULAR WILL BF USED C --- ♦*** FOR INTERPOLATION ISN 0 060 IFtGOAL.EC. 1,0 ) CALL NUMERIIXDN,ZDN.X3,ZDIP,ARCZ,ZNnRM, X X ,YON, lZZ,KA,KB,C0RR,DIV,XN0kH,YNnRH,9B,NA,DAT,LN,L,P,M,X,Y,Z,KK,LL,TESTJ ISN G O 62 KK=1 ISN 0063 L=30 ISN 0 0 6 * M = o C L= NO OF POINTS DFS1RED FOR EACH PATCH C H= MO OF COEFFS. OF POLY. USED FOR PATCHWISE INTERPOLATION ISN G O 65 LL=LLL*KKK ISN 0C>66 IFIGPAL.E0.2.0JCALL PATCHIXDN,YON,ZON,ZA,AZfCORR,Z2*0IV,NA,KA,X8, 1X3.5B,X,Y.Z.LL.L.M,KK1______I S N GC68 WRITE(6,5C0) ISN 0064 500 FOKHATt*0*,5X,"RESULT flF H5RT •) ISN CC70 N=65 ISN 0071 NN=1D* C N= NO OF DATA POINTS C NN= NO OF POINTS TO BE INTERPOLATED ISN CC72 It.=0 ISN 0073 DO 2CC 1=1,LLL ISN C 074 00 200 J=1,KKK ISN 0075 IN=IN*1 ISN 0076 XXI I,J)=X {IN) ISN CG7 7 YY11, J)=Y11N) ISN CO70 El I , J) = Z11NI ISN 0079 200 CONTINUE i s n ooeo IF(GQAL.EQ.3.C)CALL MQRTIX* V,Z,XNORM,YNORM,ZNORM,OAT» XDN,CORR,N 1,NN,OOT,XX,YY,E,LLL,KKX,KK) ISN C O 62 LA=39 ISN 008 3 LB=NN ISN C0E 9 NNN = 1 ISN 0085 LC=3 C LA = NO OF TERMS TO BE USED I FOR I STEPWISE INTERPOLATION C LC IS FIXED AT CURRENT VALUE 161 LEVEL 2.1 ( JAN 75 ) MAIN OS/360 FORTRAN H EXTENDED DATE 76.159/02.07.22 ISN C O 86 IF(G0AL.SQ.9.l>I CALL TEPO(DIV ,DAT,XT.ARCZ,ZDIP,XDN,YDN,CDRR,X,Y»Z, 1ZFF AN ,ZDN,L A, L9 ,LC,N ,XNORM,YNDkM,ZNDRH ,NNN t C C M=NO OF PARAMETERS OF FORMULAS Fo r ' INTERPOLATION C LP= NO OF POINTS TO BE INTERPOLATED AT ATIME ISN 0 0 6 B H=IC ISN 008*) LP=10 ISN 0 0 VO IF tGOAL>E0.5.)CALL COLLOC(Z,R,DOT,DAT.VC,AZ,Z2,NA,ZA.ZZ, T.XHORM, 1YNORM*Z.NORM ,DIV , ZDIP, KA,KB» A.RCZ,C,CM ,CORR,X, Y»XDN ,C 1, YON, Z0f!»X3.N, 2N:i,M,K<,LP) ------GRAFJC ------AIM AIM THIS IS AN OPTION AIM=1 CALLS FOR GRAPHICAL OUT PUTS SUCH AS CONTOURS ANO FOURIER PLOTS ISN 0092 A I M = 1 . ISN 0093 LLX=13 ISN •'■099 KK Y=1 3 ISN 0095 IFIAIM.EQ.1.1 CALL GRAFICCXX,YY,E ,LLX,KKY,LJ,Z) ISN 0 0 9 7 STOP ISN 0098 END 163

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rg m *t y 0 f^cGco^gnjrwiiW hkcuokOHNpigio<]M09 0H(Nj(ngiA *r g *vf .j vf sf g*«n*nuiir.snin.n,0X 4 .4 < 0 gj 4 ■gj-4/*O»»fwf*r*|"*fi“f'*4,“f**p“gDU>»ijCE»UJCLiUjiUd*ujo :co* juo^; ocjOOGG £ g C OOg .10C0S ;•.*■ 3tOO J?CCC JJ.'COgUOOQ -1 zz iT Z Z Z z zzzzzzzzzzjtzzzzzzzzzzzzzzzzzz yii/iua i/> in MMM «-« »-« OS/360 FORTRAN H EXTENDED DATE 76.15A/D2.07.26 LEVEL 2.1 ( JAN 75 I ATOSAP ISN CC9 6 DIFF=<'3E-SAP.P ISN W l W«ITE(6 ,FP8 ) BB5.SAHP.DIFF ISN COPS ISN iiOPV SJS!?SK;!»Sf»a6.S<-P.0>FF ISN 01C0 ISN 0101 S in fl” ; '” ' U1.5AW. PI FF ISN G1C2 ISN ■">103 ) t?Bf * SAMP ,DI FF D I 1 F=:>8*»-SA^P ISN C10A WRITE 16 .BFfl) PB9.SAMP.DIFF ISN i i H o ISN 01C6 DI FF= 8310-SAMP I Sr. 010 7 WRITFI6.6PP) BB1C.SAMP.DIFF ISN c u e D1FF=3911-SAMP ISN •1199 WRIT; tfc t6f?E ) PB11 .SAHP.DIFF ISN Cl 10 ISN 0111 ISN o u z FoJIlTlio'JJx.'PBtDICTOR VARIJSl'S'l ISN 0113 800 WKITE(6.601 I XXl.XX2.XX3tXXA.XX5.XX6fXX7.XX9.XX9 ISN Cl 1A ISN •M15 601 F3RMATI2X.9F1A.6) ISN 0116 100 CONTINUE ISN 0117 RETURN ISN o n e END

LEVEL 2. 1 I JAN 7*i ) ttS/360 FORTRAN H EXTENDED “DATi REQUESTED OPTIONS: 0PT=2 ~TrPTIONS~IN'EFFECTTTIAMF {MAIN! "OPT IHlIFr2T~OfiECnUNT7^T^r^lMAxy_ADT0r»?LTN0N?T SOURCE FPCDIC NOLI ST NODECK OBJECT HAP NOFORMAT NOGOSTMT NOXREF NOALC ISN 0002 RDUJ1N |_NVMJL X._y.2.A.G,EXS.£XX.KA.KB.R.V.LL.M.NNN.LLL.KKK.XHAX. ISN C P 03 IMPLICIT RE AL*8 IA-H.0-I1 ISN On-jA 1 S M T . 0 P 5 - ISN 0006 DO 100 11=1,NNN ISN f C*(>7 IC0UNT=2 ISN CC-cF NP=? TfFT77.7C.C- T^Rpzr ISN C-010 IP=NNP ISN 0011 CALL 9HAT(A .X.Y.L.H.NP.IP.XHAX.YHAX.ICOUNTI ISN Q C 12 - . . - ^ - C Al L GL0FI.1[fA.G . g X S . E X X . V . R . K A . K B . L . H . V H . Z l______' T E N T C 13” 2 2 2 WRITEi6,A6) NP 1SN G DIA 66 FORMAT(5XMAX.SINE/COSINE HARMONICS IN X.Y DIRECTION^,151 I S N 0015 ILL=NF*NP ISN 0016 ISN f.017 T flfl 'C c41IN ‘16i.EXX»R i VC .NR t HaJ Q .I I L .XXQ) ISN 00 18 999 RETURN ISN CO 19 END 166

u. -J -t * z rg M • X 00 Of o UJ • e rg c o z "v + u. in tn 1-4 Z • u C u ? h 91 —X «4 -JH a 0 t£lrt UJ CO a a OO o. z z h C z o -t UJ DZ * u H - t ~ a ♦ X H- O X UJ -* < X — XX x c X •I nC » o t o z - l z —IL ► •—> 0) I < u-C > CL Ifi i tf NZ ^ z » I H- ►4 X « UJ or 1TQ * a. C < a z UL — s; Z '- o » u <6h ui U to z o — J a » Uj o \ X>efi •• z r Ul CCJ o ► □c a O u o *■ U ftf ►rv*Z h z z o O 1 —’ * O UJ •—UJ u a » «n uj «< -JO 4 4(Q ► > o ffiX ^ & « z 03 1 « m z o rg *< o . <43 » H <—t- < - z O a z U.U1 < ► w u> — — f, z o *gf-4 <—00(0. • < ^ q . • I ►4-J o* z lA <1- Ifl OC r o U .J — a ^ a — m w *k*« CL < ^ •z L-<-C 0 . 0 . *.a a.o x. U.Q.U * CL cl o! X u. 0.0. n o n x aj< z z h o n z w o z z a »-• eg z z <— 17 * *-<**• o c a o ii* c t ► 4 * » » Q VIH tff* » < C'-rt Z 7 c a c ^ «-tp4H a o x rtf- C D - 5 0 x > z a ►Hh-O II II II II II If-4 II Ui N M • II IT. II UJ II II * Il'OllJ-'/V II |l II # M I/)Uj9“ *l/» OuJ « u v-»-»•-• C^I—r3*-«Tr>»--'*'r^»— ihT m -s— ;? bUt"3 *1Zh ♦ ”»->“> II X X — H w3z rtU V * 0 *►— iTlA * T) »HOHaOIX>UtJ5U.aUXCCUJ 9* 1 3 a. «rO in O ZUI r - m lA** min IA<4><0 CO »• •« 10 - on Art OOfdO* 00 z - < z o T o UJ M u. h LL a UJ o rgng- u> o r^ o3 0* ^ «-i rg tj-in <£> r-ju ^ arA ^ in «o r- ±0 9* O AJ ia ■U'j> o 9* co c •** rg in m rii«iirireii*vnmr*W‘rg / <1 j- v# »f •J’mm m m • *3 rt OCu3 CCCiCUOwJOOOCOOJoO1; OCOUOOCiOOOOOCOO oocoooooooo IN UJ c c o **r 3c^j oo*.*coo z ?‘ ooo jg-rccooc-:ocQfjo1: ccou OOC OO'I j'jOJO -J Ul z z z UJ UJ o ITiUII/j > 3 ►4 HH m irtHHrtt-rHrtHHMHHrtNNrtHN Wh HHM w MH m HH UJ O 1- .J UJ a. CC a LEVEL 2.1 { JAN 75 J O S/360 FORTRAN H EXTENDED DATE 7 6 .1 5 4 /0 2 .0 8 .2 9 REQUESTED OPTIONS: OPT=? OPTIONS IN EFFECT: NAME IN AIN) OPTIMIZED) LTNECOUIlT 160) SIZE(MAX) AllTODBL INON'E) SOURCE EBCDIC NQLIST NOOECX OBJECT HAP NOFORMAT NQGQSTMT NOXREF NOALC NOANSF NOTERMINAL F ISN 0C02 SUBROUTINE AUT7C0IU,Z,LLL,KKK,L,A,S»C,D,£,F,G,H,LAK,HG) i s n 0003 IMPLICIT REAL'S (A-H,IJ-Z1 ISN *iftft4 Kl A l *a it ISN uG(>5 D1MENSI ON A(LAK),b(LAG),C(LAG),D(LAG),E(LAG),F(LAG),GtLAG),HtLAGI ISN CSCfr DI“LliSinN UILLL tKKK) ,ZIL ) ISN vOu7 h=lll ISN oooe N=KKK ISN (1009 K=<> ISN Oft in 00 700 1=1,H ISN 0011 OO 700 J=1,N ISN CC 12 K=K*1 ISN CG 13 7 0 0 U11»J)=Z(K) ISN CC 14 LL® N* M ISN 00 15 PN=LL ISN CO 16 CALL STOVAR(ZtLLtPN,ZMEAN,VAR) ISN 0017 nn tos i i = i ,l a k ISN coie Mh=H-II ISN O'llP NN=N-II ISN 0020 5U M =‘.i» 0 ISN i*02l OC -30C- 1 = 1,KM ISN 0022 on ^00 J=1»NN ISN 0023 8 0 0 SUM=*UM< I'Jl I*II , J+II)—ZMEAN )* (UlI,J)-ZHEAN) ISN 002* AUT0='UM/|M'**NN) ISN OC- 25 a u t ;"i= aijTi?/v a r ISN 0c26 A(II)=AUTO ISN 00 2 7 WRITE16,666 ) II,AUTO ISN 0020 666 EOkJUTl*0*,2X»*AUTOCORRELATION RIU.V) FOR LAG*,15,»«'»F12.6) ISN C0*9 805 CONTINUE ISN KM=M-i ISN GO 31 NN=N-1 ISN 0032 00 705 11=1,LAG ISN 0033 S U N =0.0 ISN 005 4 00 701 I=l,Mrt ISN C035 0 0 7C1 j=!,r:N ISN ‘>(■36 SUM=SUM+(UiI+II ,J)-ZMEAN )*(UtI*J+1 l—ZMEAN) ISN 0037 701 CONTINUE ISN 0039 AUT^SUVIMH^NN) ISN 003', AU10=AUTO/VAR ISN Cu*0 e n i ) = A U T o ISN GC91 NN = NN-1 ISN j"*2 M.p=r.i-l ISN 00 43 WRITE16 ,466) II,AUTO ISN Oft 4 4 466 FOKPATI*0*,JX,'AUTOCORRELATION R(-U,V) FOR LAG',15,•=*,F12.6) ISN GC*»5 705 CONTINUE ISN CC46 H “ =H-1 ISN 00*7 NN=N-2 ISN oO*-8 DO 6N5 11=1,LAG ISN 1H I49 SUM=0.0 ISN ftOSO DO 601 1=1,MM ISN 00 61 DO 6til J=1,NN ISN CC52 Sli.M=cUM*( U(I'll »J+II+1J-ZNEAN}*(U(I»J)— ZKEAN) ISN CO 53 601 CONTINUE ISN C054 AUTO=SUK/(MH*NN) 168

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o O H - < > •— of a t-> *U »»•«*+■z z to Of to CO h- zzcrocinc/oo < « t H J > * to °s HtCUi UfkLI LLIUJUJUJ Of Of 4 to CO > • ••••# 104 O J C J oz O J 0 ( 0 zx ------*zz— ^5* — * < z . i^Z— *z_-_ nlu v oc o arc ^ o cec arc aeo of U CC^-J «J*J JSH JO II «jr> H it «JO h — J 3 ii * J I J xuj r-— Jh-r j»-r-fcK-»HQ f\l - U . U.U. tuU.3«LUf D < i^fJ 3 M H M W H t M i m H h i il n I* 4 I *i U J n »- -J I C i a O f a ISN 0002 SUC(OUTINE AREVOLISUMM.X,Y, Z,LL,NT.XLENtYBRET,LLL»KKK,NNT,XX,YY, 1ZZ.0,A,M,KK,CC.NP,NNP,XKAX,YMaX,XX3,XX9j ISN 0 0 0 3 IMPLICIT REAL+B (A - H ,0 - Z 1 IFN OCGA KEA l *A XX,YY,ZZ I SN 00 j 5 DIMENSION SUMM(NNT),X!LL).YtLLl,Z(LL),XX(LLL,KKK),YYlLLL.KKK), 1Z Z(LL LtKKK) ISN 00 OS N U “.X=l ISN0007 H JMY=1 ISN COLd TOTAL=0.0 ISN0009 TVCL=C.O ISN ••'<10 T ' v n L = 0 .0 ISN 0011 TSAHEAsf.O ISN 0012 PAREA=XLEN*YBRET ISN OC 13 AK= LL ISN (■CIA XX9=AN/P4REA ISN t 0 IS LT=LLL-1 ISN ■■r.it k t =k k k - i I SN CO 17 NC=KT ISN 00 IE n k = l t ISN 0019 1=0 ISN 0020 DC 2C0 JJJ=I,LLL ISN CC21 DO 2L>*. 11 1 = 1 , KICK. ISN .v.?2 1 = 1+1 ISN U023 XX(JJJ,UI)=X(I I ISN 0(|?A YY!JJJ,III)=Y(I> ISN 002 5 200 2 Z (J J J , III )=Z(T ] ISN 00 26 00 ICO j j j = i ,l t ISN C027 00 100 111 = I« KT ISN ■i0 2£ YA=YY(JJJ,III) ISN 0.0 29 XA=XX(JJJ,III1 ISN 0030 XB=XX1JJJ+1,III+1) ISN CC51 YB=YYtJJJ+l,III+l) ISN OC 32 CALL VOL! X4,XfJ»YA,YB, VOLUME,SAREA,SVOL,LLL,KKK,JJJ,III,ZZ) ISN 0C33 TSVOl=TSVOL + SVr>L ISN •Vj 3A TSAKEA=TSAKF4+SAREA ISN 00 3 5 TVC'L=T VOL+VULUM E ISN 00 j 6 100 CONTINUE ISN 0037 Wkl TE !6 ,9AJ PAR FA ISN (.038 AA FORMAT('O',EX.'PLANAR AREA=*,F25.A) ISN 0039 WRITE(A.551TSAREA ISN 00 A b 55 FQR“AT(5X.'SURFACE AREA=•,F25.9) ISN OuAl WRITE (6 ,771 T VOL ISN 0092 WRITE16,8P1 TSVOL ISN 00 A 3 77 FORMAT I'O','VOLUME BY' TRAPEZOIDAL R'JLE=',F25 .A) ISN COAA sa FORMAT 1'f.','VOLUME BY IMPROVED TRAPEZOIDAL RULE = *.F25.A! ISN '■095 AVOL=TflTAL/PARE A ISN CO 96 EVP l =TVOL/P AREA ISN Ch>97 c v o l = t s v o l / p a r e a ISN u O a E PAR l A=T s a r e a / p a r e a ISN 0049 XX3=DARFA ISN CO SC WrtITE!6,511 ISN 0051 51 FORMAT!»C • ,10X, 'VOLUHE/PL.AREA',3X,'VOLUME/PL.AREA',3X,'VOLUKE/PL IAKEA',3X.'SURF.AREA/PL.AREA*) ISN C05 2 WRITE 16,99) AVOL.BVOL.CVOL.DAREA ISN 0053 99 FORMAT!'0',10X,AF19.A) I SN 00 59 RETURN ISN 0055 END 172

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COMPUTE Eir.I'N v a l u e AND EIGEN VECTOR OF COVA/MATRIX ISN G031 CALL COVCOR(C,CM,VC,BB,T,R,L.N,IR,IC,B) TRANSFORM RAW DIRECTION COSINE. INTO PRINCIPAL AXES I S N 0032 CALL LGMPKD(ZZ,T,ZA,NN, IR ,1C) . ISN r.0 33 DO 33 I=1,NN ISN 0034 X«NORM( 11= Z A ( I , U ISN 0035 YM]KM(I»= ZAII.2) ISN 00 3 6 X3(I»=ZA(I,3) ISN 0037 33 CONTINUE COMPUTE VARIANCE AND MEAN OF TRANSFORMED COSINE DIRECTION ISN 0038 CALL COVA (XNORM .YNORM,X3 ,-"N, VC.C ,CH ,NK.TOT) ISN 0 039 CALL COVCOR(C,CM,VCtBB,T.K,L.NrIR,IC,6) ISN GCAC DO 66 1=1,NM ISN 0G41 66 X 3 1 11=0.0 :VCL 2.1 ( JAN DIRCOS QS/B60 FORTRAN H EXTENDED DATE 76.154/02.C7.4#. ISN 0042 IEF.=0 C c COMPUTE SKEWNESS AND KOURTOSIS ISN GO 43 WKITEI6.07) ISN MU 44 971 F PP HAT ( '0' » 7X .' K E A N ',6X,' STD DEV. * «3Xt'SKEWNESS*,3X,'KOURTO ISIS') ISN CC45 HLN=NK»r.K ISN LU CALL PISRINN.Ml .ZA,X3»B,RB,5K.CUT,R,NA,VC,T,C,TER,MLM) ISN C0*-7 DC' 7F I -1 iNK ISN C048 78 WRITS (6.57) B (I ). BB 11 1, SK U) . CUT( I) ISN (.'049 57J rOHHAT(5X,4Fl5.4) ISN Of-50 DIh C0A= ( Pt11**2* B(2)**2♦ Rt 3)**2) ISN or, s i 0IRC?K=DS(3RT(r.I KCDA) UN 0052 D IRCOT= IHF1 ll**2+(JBt2)**2+B9t3)**2) ISN ■>053 DIKCnV=PSCRT(DIKCnT) ISN 1.054 XX6=DI*<,0V ISN <055 XXAA= (CUT{1)**2+CUT[21+*2*CUT 13)**?} ISN 6C5 6 XX7=DS')RTIXXAA) ISN 0057 WPnr.(6,90) D1RC0H,PIRC0V ISN CC58 991 FORMAT[*G■.2X»1RESJuTANT DIRECTION COSINE:•,2X,'MEAN*',F15.5,PX,» lVAhIANCE=»,H5.5) C COMPUTE 1 A*AZ=CM FOM RAW DIRECTION COSINE HATRIX ISN 0059 CAtU DCKTRA12A,A2,NN,HK) ISN OC 60 CALL PGMPRfi [ AZ A. CK tNK t NN.NK) ISN CuEl CALL VFCT(CK.S.N,L) ISN GC62 CALL U.'JVCT(CM,B,N,L) c FINC EI5EN VALVE OF COVARIANCE MATRIX c COMPUTE EIGEN VALUES OF RAW ZA*AZ HATRIX c ISN G063 CALL DEIGENIBiR*Nt0) ISN f->64 HKITE(6•861) ISN C0o5 8811 F O r HATI*1 *FIGfN VALUES*) ISN (■C'66 CALL WSVCT(P,L.N] ISN O u 6 7 CAL l DGMTRA(R,T,IR,IC) ISN 0C-6P RETURN ISN 0069 END SUBROUTINE COVCrjRIC,CM,VCtBBtT.R,L,N,IR.IC.B) .. . ISN 0002 DIMENSICN CCN.HI.CHtN.NI.VC(N.N).83(L).B(L)»T(j.N)*RtN.NI ISN l C l 3 c CONVERT CORRELATinN MATRIX TO A FULL MATRTRIX ****************** ISN 0 0 0 4 CALL VECTIC.BP.N.L) ISN 00t>5 c ****AL1'CONVERTTCOV'MATRIX TO A FULL HATRIX ******************** ISN 0CP6 CALL VcCTtVC.tl.N.Ll CGu7 ISN r FIhDLEIGENTVALU£ 'o^COVARIANC5 HATRIX ISN i>oce CALL- DEIGEN

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T8I LEVEL 2 .1 ( JAN 75 > OS/360 FORTRAN H EXTENDED DATE 7 6 .1 5 4 /0 2 .C9.58 REQUESTED OPTIONS: 0PT=2 OPTIONS IN EFFECT: NAHE(MAIN) OPTIMIZED) LINFCnUNT(60) SIZE(MAX) AUTDDBL(NONE I SOURCr FFCDIC NOLIST NODICK OBJECT HAP NOFORMAT NQGQSTMT NOXREF NOALC NOANSF NOTERHINAL FI ISN 0002 SUBROUTINE NUMFRUFO,FA,fi,FF,U,F,A,V,X,KA,KB,B,D,S,G.Z,C,PY,LN,L, lP,M,XX,YY,ZZ,KK,LL,TESTt ISN IMPLICIT READ9 (A-H.O-Z) ISN COCA INTTGIR P ISN CGu5 DIMENSION FQ{ lN J , FA ( LNI , R (L 1 , FF (L) .U( P ) ,F(P ). A ( L,L) , V IL ),X( L. K K ) , IKA ID,K?(L)»BtP,L),niL),SIL),G(L),Z(L),C(L,Li,OY(L,L) ISN 0006 DIMENSION XX(LLI,YYILL),ZZ(LL) C NUMERICAL INTERPOLATION C SULrlU'JTINE TLA G, STIR, STIR ,SPLICON WRI T T E N BY PENNINGTON C AND MODIFIED DY OLU AYFNI ISN 10C7 LLK=c ISN u008 s u m u :..o o ISN $ UH7=u.D» ISN OrtlO St)MT=.T.CO ISN (.011 S U M 5=0.00 ISN C01 ? SliM5=C.L0 ISN GO 13 SUM6=G.D0 ISN 0G1A SUM7=l.r.O ISN 0015 SUMF=0.0 ISN 0016 SIJMNcfl. O ISN 0017 DO «-F7 11 = 1,LN ISN C01B LP=L-1 ISN CP 19 911 pn 505 1=1,LN ISN UOZC LLH=Ll M+1 MEAD R 11 ) ,FF(I I ISN 0021 F0(I)=YY(LLM) ISN C O 22 FA( I )=ZZ(LLK) ISN 3023 995 CONTINUE ISN C02A 10 FORMAT(FIO.3,1PXtFlO.3) ISN 0025 WRITE(6,30) ISN 6u26 30 FPOMATI>0*,A7X,’DATA SET •/• ' ,AAX»*UtI I’.SXt’F d ) * / * »,<,2X, ******* ISN O P 27 00 ?2 1=1,LN IS;, 3029 WRITE(6.31) F0(I),FA(I) ISN 10 29 32 CONTINUE ISN V<30 J = 1 ISN 0031 00 6fi 1=1.L C -- READ UID.FII I ISN 0032 R(I)=FO(J) I«N 0023 FF(I)=FA(J) ISN C03«. 68 J=J+3 ISN v,035 JJ=1 ISN CC36 JK=i ISN uO37 JJJ=2 ISN M . z B OCT 593 1 = 1, lB ISN 0039 U(JJ)=FO(JK) ISN C G a C F I J J )= F A (J K ) ISN COM JK=JK*1 ISN 00A2 U(JJJ)=FOIJK) ISN 00A3 f{JJJl=FA(JK| ISN 0GA4 JK=JK*2 ISN 00*5 JJ=JJ*2 ISN C0A6 583 JJJ=JJJ*2 182

j LEVEL 2.1 ( JAN 75 I NUMERI □ S /3 60 FORTRAN H EXTENDED DATE 7 6 .1 5 A /O 2 .OB.5 8 UN 00*7 31 FORMAT(• ',<.2X,F9.2,2X,"",2X,F12.3) ; ****** l a g k a n g i a n f o r m u l a h ISN uOAfi WRITE t6 >8) II lit; 00 A9 a FOkMATl'i'.SX.'LINE NO',13) ISN <■>(150 WRIT! 16,81) ISN CO 51 01 FORMAT! 'it*,50X,'RISULT OF PREDICTION BY LAGRAGIAN POLYNOMIAL* ) ISN 0052 IF(TEST.EO. 1.0).CALL TLAGCR,FF,U,F,L.P,STD) ISN l 056 SU“ ls5UMl+STD C *».*** DIVIDED DIFFERENCE ISN C055 w « m ( 6 , p ) i i ISN (•056 WRITE(6,e2) ISN (■067 82 FORMAT!*n*,50X,'RESULT OF PREDICTION BY DIVIDED DIFFERENCE') UN OC 68 DO 516 1=1,L ISN c359 515 AC 1, 1) = FF (11 ISN LG tO IFCIFST.FB. 2.0) CA l L DDIFU.A ,Ut F,L,P,STD) ISN T'n62 Su F * = S U ”.2 + STD C ****** SPLINE ISN 0063 W R I T E (6,B J II ISN Of 65 WK1TE16,G3) ISN OCeS 83 FORMAT!'P',*.0X»'RESULT OF PREDICTION BY SPLINE INTERPOLATION*) ISN 00 66 IFtTfcST.EO. 3.0) CALL SPLINE{«,FF,L,C,XINT,YItlT,U,F,P,STD,D,S,V,A, 1G, I ) ISN 0068 SUM3=SUW3*ST0 C ****** AIT-NEVILLE ISN 006 9 W R i r c tj),8r i i ISM G07C WKllf(6,PA) ISN CC71 85 FORMAT! '(j'.AOX,'RESULT OF PREDICTION BY AIT-NEVILLE FORMULAR'l ISN u072 H=R(?)-H(l) ISN 0073 X B E C = K ( 1) ISN 00 76 IMTEST,C3. 5.0) CALL AITNEVIR,FF,L,XBEG,U,F,P,STD,V,H) ISN UG76 SUK5= S'JMA+S TD C ****** STIRLING ISN CG77 WRITE(6,8) II ISN i*07B WRITE(6,6 5) ISN 0075 85 FORMAT!*i)',50X,'RESULT OF PREDICTION BY STIRLING FORMULAR*) ISN l.iOEO XCE G=R( 1) ISN (10 ei E RR=3.S ISN WCF? D X= R t 2) -R (1) ISN 008 3 ro S9 i = i,l ISN ■ 9t 5 99 fY(1.I)=FF(11 ISN OOPS CALL OIFTtB(R,OY.L) ISN 0 06 6 IF(TFST.E3. 5.01CALL STIR(FF,L,DY,XBEG,DX,XINT»ERR,YINT,R,U,F,P, 1STD.V) ISN 0088 SUM5=?UM5*ST0 c ****** NEWTON FORWARD DIFFERENCE ISN 1'0P9 WRITE 16,8) II . ISN 0u9 0 WRITE(6,86) ISN <•001 86 FORMAT!'O',60X,'RESULT OF PREDICTION BY NEWTON FORMULAR'l ISN uO 52 N=3 ISN CC63 X8EG=R( 1) ISN 00 05 D X = k ( 2 ) - R (1) ISN i” S5 . IFCTEST.EC.^6.0) CALL FNEW(VtL,XBEG,0X,N,XINT,YINTtR,FF,U,F,P,57D) ISN 0097 ISN (.098 WRI t I (6,201) ISN GG99 SUrt6=SUH9*STT I ****** EXACT POLYNOMIAL ISN 0100 WRITE (6 ,B ) II ISN 0101 WRIT £(6,67) l e v e l 2.1 I J A N 75 1 HUMERI 0S/360 FORTRAN H EXTENDED DATE 76.154/02.08.58 ISN CIO? 87 FORMAT!•0*(40Xt ’RESULT OF PREDICTION BY EXACT POLYNOMIAL*1 ISN C103 M=-c ISN > K 4 IF1 TEST.FQ.7.C) CALL EXACTIR,FF,A,B,X.KA,KB,U,F,V,L 1.1 Y v C I ISN (jK'6 S,>.7-SUM7*STD ISN O U T WHIT!. (6.200) ISN (■UP M=4 ISN O K 9 987 CONTINUE I FN Clio 201 Ffl.R .MAT I ’0',2X,' RESULTS FOR SINGLE DIMENSION MULTIQUADRIC * ISN Cl 11 200 FORMAT!*0*.2X »• RESULTS FOR SINGLE FOURIER*> ISN o n ? RMSl=SUMl/LN ISN Ill 15 KMS?=S«JM2/LN ISN Cl 14 RM S 3 = S U M 3 / l N ISN 0115 RMS4=:S:u:'.4/Lf. ISN Cl 16 R“ '5=S!J':S/LN ISN (.•117 R^SfisSUFF /LN ISN 0118 RMS7 = S'JM7/LN ISN 0119 RMF >' = SU'-,S/uN ISN 0120 ISN c m WRITEI6.61) RMS 1 IFN Cl 22 WRITE(6,62) RMS 2 ISN 0123 WKITt(6.6?) R*S3 ISN 0 1 2 4 WKI TE(6,64) RMS4 ISN 0125 KK1 IE (6,651 RMS 5 ISN 0126 HRITE(6,66) RMS6 ISN a 27 W K I T t (6 ,67) RMS 7 ISN 0128 WR1TF16,5 8) RMS 0 ISN •129 W n l T t (6 ,69) RVS4 ISN 0130 61 FORMAT!*C*,5X,* FINAL RMS FOR LAGRANGIAN s*,F19.6) ISN 0131 62 FORMAT! 'O', *>X»* FINAL RMS FOR DIVIDED DIFF =*,F1°.6) ISN 0132 63 FORMATI*0*.5X,‘FINAL PH5 FOR SPLINE =*,F19.6) ISN 0133 64 FORMAT!*0',5X,'FINAL RMS FOR ATT-NEVILLE =*,F19.6) ISN Cl 34 65 FORMAT! 'C-',5X,* FINAL RMS FOR STIRLING =*,F19.6) ISN ul 55 66 FORMAT!*C*,SX,'FINAL RMS FOR NEWTON =',F19.6I ISN 0136 67 FORMAT! ")*,6X,'FINAL KMS FOR EXACT POLY =*,FI9.6) ISN 0 137 58 FORMAT!'0*,5X,‘FINAL RMS FOR S-FOURIER =*,F19.6) ISN 0158 69 FORMAT! *0 * ,5X, 'FINAL RKS FOR 1-D MQRT =*,F19.6) ISN 0139 886 STOP ISN 0140 END 184 LEVEL 2 .1 ( JAN 75 ) 0 S /3 6 0 FORTRAN H EXTS'JOEO DATE 76.154/02.09.38 REQUESTED OPTIONS: 0PT=2 OPTIONS IN EFFECT: NANFI MAIN I (1PTIMIZM?) LINPCDUNT (601 SIZF(MAX) 4UT0DBL(NONF) SOURCE F5CD1C NO Ll 5. T MHDECK CBJECT MAP NOFPRMAT NDCOSTMT NOXREF NQALC HOANSF NOTERHINAL F ISN t.002' SURRD'JTINt PATCH(XtY,Z,A,G,CCC,FXStV,GXX,KAt!

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in m •* »n 4> r* oo o m rMn^^i^r-coff o minm-* tft'OH® J>O4-r;vi«wj‘tOH®0'OfNM*i,.n«Gr-<0i> OMPjrniAHO1 M rvrWm4> f-® 4444444 4^^ lAtnov.T.nuwi.A »► <►•£ O'vJj'4 - 3 ^ ^ r-h-h-r'i'^J’-r-f'-h-f'-ccaJcugjujixou®® 0*0-0 O'J O'? *» O O w O O O'J CQCOCOOCOO ocoo-.i uc ^ a oo iooooocrooac ouoc oouo^ooooo OO&Q i’i CO h m m m m m m m uo;ojoocoj oCw.rf- ‘ jcoc ;:oooo3Quojcjuocoouojocuoc OJOOOOJwOw2OO OO zrezzzzzzz Z^2JS2 .r?*iZ r^Z.!ZZZ27222Z22Z^ZZi2i!2^Zi:z z z zr.zzzz zzzzz zz tm/ttni/iuitAw’ttn i^t/j lQw>fc.rLSltOvlUiVO^I (/J*/) V^VH/IVl t/|M# WOtO MMMmmmMmmm MMMMrHMMMM mmmmmmmmmmmmmmmmmmmmmmmmmmmm M M M M ^ h M M M M M M M M m m LEVEL 2 .1 ( JAN 75 ) 0 S /3 6 0 FflRTRAN H EXTENDED DATF 7 6 .1 5 6 /0 2 .0 9 .6 8 REQUESTED OPTIONS: 0PT=2 OPTIPNS IN EFFECT: NAHFIM a IN) OPTIMIZED) LINECOUNT 160) SIZE(MAX) AUTOPAL(NONE) SOURCE EBCDIC NO s. I ST NODECK OBJECT HAP NOFORMAT NOGOSTHT N7XRSF NOALC NOANSF NOTERHINAL FI ISN CCC? SI.'R knuT INE uCKT(XD>YDtZn,XtYtZ|AiCtBtNfNNTD ,XX ,YY,ZZ,LLL,KKK, KK) ISN •-0J3 IMPLICIT REAL»6 (A-H,C1-Z) ISN 0GC6 KCAL*6 XX,YY,ZZ ISN >»rti'>5 PIMINSION XPIN] ,YD(N),ZD(N).XINN>,YINN),Z(NN»,A(N,N1,C(N),8(NN,H) ISN OCC-6 0 16;N T I Qr; Dl N ,N I, XX (LLL.KKK J , YY (LLL,KKK t , ZZ ILLL.KKK I C -- REAP XXtl,J),YY(IJ,),ZZII,J) ISN CCC7 12 c o n t i n u e ISN coca 13 FPRVATI3FI0.3) C MUT =HUTI3UADRIC REPRESENTATION OF TOPOGRAPHY ISN CCL9 SC'«=0.0 ISN Cj uIO Nrtl=N-l ISN -Oil STKMI N=699V060V9.0 ISN u012 DC 2 J=11NH1 ISN .111 13 E=XDIJ) ISN C 0 1 6 F=Yt'( J) ISN UG13 NI=J+I ISN CCI6 DU 3 I=HI,N ISN iC17 . A(J,I)= (E-XPI II )**2* ISN 0061 CALL VERSOLIA,C,N) L>r ISN 0062 DO 11 I=1,H ISN 0063 C( I )=O.C ISN C066 DO 11 J=1,N ISN .065 n c(U=cin+A(i,j)«zD{j] ISN 0066 VK=G.G ISN 0067 VPV =0.0 ISN Ou6B D O 101 1 = 1, H ISN 0 0 6 9 S U M =0.0 ISN GO 50 DO IOC J=1,N ISN 0051 109 SUM-SUM-»0 (I ,J t*C[ J) 188

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SOME SAMPLE OUTPUTS FOR COW #50 CLAS ATOSAP

GRADIENT = 3.714, CURVATURE = STATISTICAL MULTIVARIATE CLASSIFICATION RESULTANT DIR. COSINE MEAN = .411, VAR = .914 HARMONIC VECTOR MAGNITUDE = 0.318 OF TERRAIN FOR D M T VECTOR STRENGTH = 0.384 SUMMARY OF 2ITERATION ECTOR DISPERSION = 1.613 NO. OF BREAKLINES = 0.0 1 2 BUMP FREQ. = 13.435 SUBS tT 1 I S-l 1 1 2 S-9 1 1 3 S —2 I 1 SUBSET 2 4 S-3 2 2 5 S-4 2 ? SUBSET 3 6 S-5 3 3 7 S-6 3 3 SUBS tT A e s - i 4 4 o s-6 4 4 THIS TERRAIN BELONGS TO CLASS 1 BIBLIOGRAPHY 193

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