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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.