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By Urho Antti Kalevl Uotila I960

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By Urho Antti Kalevl Uotila I960 Copyright "by Urho Antti Kalevl Uotila i960 INVESTIGATIONS ON THE GRAVITY FIELD AND SHAPE OF THE EARTH DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy In the Graduate School of The Ohio State University By URHO ANTTI KALEVI UOTILA, M.Sc. ****** The Ohio State University 1959 Approved hy: W , R Adviser Department of Geology (Geodetic Sciences) ii PREFACE Geodetic science, as Dr. Welkko A. Heiskanen has long maintained, has had a great need for a world-wide geodetic system which would connect the different regional geodetic systems of the world. In recent years, the need of such a globe-spanning system has also been increasingly recognized by the Armed Services, particularly the United States Air Force, as a basis for obtaining the accurate distances and directions so important to our national defense In this Jet flight era. These needs, the academic and the military, were admirably combined with the establishment In 1950 of the World-Wide Gravity Project by Dr. Heiskanen, under the sponsorship of the United States Air Force, at the Mapping and Charting Research Laboratory of The Ohio State University Research Foundation. Under this significant program, observational gravity data from various agencies all over the world were, and continue to be, channeled into Columbus, adjusted to one reference system (the Potsdam system), translated Into mean free air gravity anomalies, and finally used for computations of the undulations of the geold and the deflection of the vertical. Because of the vast amounts of observational data handled by the Laboratory and the complexity of the computations Involved, the writer Ill had the opportunity to develop a variety of new methods and techniques particularly as pertains to high-speed computational operations. The more important of these, which are covered In some detail in this dissertation, are; 1. A new method for high-speed computations of the undulations of the geold. The writer, insofar as he knows, was the first to develop a practical method, based on mean anomalies, for the high-speed computer determination of the undulation of the geold. This method was used in computing the Internationally known "Columbus Geoid". It was also used in detailed investigation of the undulations of the geoid for a number of regional areas in the northern hemisphere. How­ ever, the results of these investigations have not, as yet been released for publication. 2. A method for high-speed computations of the deflection of the vertical. Again, the writer believes himself to have been the first to develop a high-speed computer method for the numerical Integration of the effect of distant areas on the deflection of the vertical. 3. A template method for manual computations of the deflection of the vertical. With this practical method, the numerical Integration of the effect of distant areas on the computation point is manually obtained by the use of templates of original design in conjunction with mean anomaly stereographic charts. iv 4. System for uniform high-speed handling of gravity data. The writer presents, in detail, a system he has conceived and organized for the high-speed handling of vast amounts of gravity data with IBM machines. 5. A high-speed integration method for the determination of the shape of geopotential surfaces at high elevations. The writer has for the first time determined the shape of geopotential surfaces at elevations greater than 100 nautical miles, using this new, high-speed integration method. 6. Correlations of free air anomalies and elevations on land. The writer shows, by means of sample graphical analyses of free air anomalies from various parts of the world, the existence, within limited areas,of a general relationship between free air anomalies and elevation. Such correlations have been useful in estimating mean free air anomalies for regions in and near areas where only a few gravity observations are available. 7. General technique for determination and estimation of world free air gravity anomalies. The writer describes in this dissertation a general technique he has devised and used, which takes into consideration free air anomaly elevation correlations and various methods of extrapolation and inter- o o polatlon, for the determination and estimation of 1 x 1 mean free air gravity anomalies wherever possible throughout the world. With this technique he has determined and estimated approximately 1 0 ,0 0 0 l°x 1° mean free air anomalies. These anomalies, however, have not V as yet been released for publication. It should be noted that these seme 1°X 1° mean free air anomalies have also served as the base material for recent investigations of the Army Map Service and other agencies in the United States. 8. Relating national reference stations to the same vorld gravity system. As a prerequisite to the many types of geodetic computations carried out under the World-Wide Gravity Project, it was necessary for the writer to relate the various national reference stations of the world to a single world gravity system. There are many individuals and organizations to whom the writer is indebted in the preparation of this dissertation. Foremost among these is Dr. Weikko A. Heiskanen, Director of the Finnish Geodetic Institute (in absentia), Director of the Institute of Geodesy, Fhotogrammetry and Cartography of the Ohio State University, and Supervisor of the World-Wide Gravity Project, who, as adviser to the writer, has not only provided constructive criticism, giving liberally of his time, but has been a source of understanding inspiration as well. The other members of the reading committee, Dr. Paul M. Pepper, Ohio State University, and Dr. C. Tsuboi, Tokyo University, furnished constructive advice for which the writer is grateful. High-speed computational facilities have been made available in the numerical Computation Laboratory of the Ohio State University. The programming of the IBM 650 has been done by Mr. Marvin Hardenbrook whom the writer wishes to thank for his excellent cooperation. Hie Ti writer it very grateful to Mr. Lasti A. Kivioja and to Mr. Clarence R. Johnson for helping the writer in many cosqputatlons, and to the latter again for checking the dissertation in regard to correct English expression. The writer wishes to express his gratitude to Dr. Paul M. Pepper end Arthur S. Cosier, former directors of the Mapping and Charting Research laboratory and to the Geophysics Research Directorate, Air Force Cambridge Research Center, Air Researoh and Development Comand, Bedford, Massachusetts, which is administering Air Force Contract Kb. AF19(6o 4)-1963, under which a part of this dissertation has been executed. The writer is also very grateful to Mr* Lynn Freisner for drawing most of the illustrations, to Mrs. Mary Ann Humphrey for her excellent typing, and to all others who have assisted with this dissertation. TABLE OP CONTENTS Section Page PREFACE INTRODUCTION 1 1 GRAVITY POTENTIAL AND GRAVITY FIELD OF THE EARTH........ 1+ 1.1 Potential........................................ 4 1.11 Potential of A t t r action.................... 5 1.12 Potential of Centrifugal Force ........... 13 1.13 Geopotential............................. 13 1.14 Spheropotential........................... 13 1.2 Gravity......................................... 14 1.3 Gravity Formulas ............................... 16 1.31 Theoretical Gravity Formula............... 16 1.32 Computations of Gravity Formula........... 19 2 GRAVITY MEASUREMENTS..................................... 24 2.1 types of Gravity Measurements. ..........24 2.2 Absolute Gravity Measurements................... 24 2.3 Relative Gravity Measurements....................26 3 GRAVITY REFERENCE SYSTEM ..................................27 3.1 Potsdam System ................................. 27 3*2 Standardization Lines........................... 26 3*3 Gravity Reference Stations .................... 29 3*4 Relating and Recording of Gravity D a t a .......... 3? vii viii TABU OF CONTENTS (cont'd) Section Page 4 GRAVITY ANOMALIES ..................................... 41 4.1 T^pe of Anomalies .......................... 41 4.2 Mean Free Air Anomalies .................... 44 5 GRAVITY MATERIAL....................................... 65 6 COMPUTATIONS OF UNDULATIONS OF THE CBSOID.............. 66 6.1 Basic Formulas.............................. 66 6.2 Numerical Computations...................... 69 6.21 Oeneral Technique.................... 69 6.22 Selection of the Size of Surface Elements............................... TO 6.23 High Speed Computation Method ........ 74 6.231 Computational Technique ........ 74 6.232 Computations of Stokes' Coefficients.................... 7? 6.2321 Coefficients for l°x 1° Squares............ 75 6.2322 Coefficients for 5 X 5 Squares............ 8l 6.233 Summation of the Products: Stokes' Coefficients X Gravity Anomalies .............. 83 6 .2 3 3 1 l°x 1° Squares............ 83 6.2332 5°x 5° Squares............ 85 6.234 Total Undulations of the G e o l d .......................... 88 6.24 Accuracy of Undulation Values ........ 88 ix TABLE OP CONTENTS (cont'd) Section Page 7 COMPUTATIONS OP DEFECTIONS OF THE VERTICAL ........ 92 7.1 Basic Formulas ............................. 92 7.2 Numerical Computations...................... 94 7 • 21 General Technique.................... 94 7.22 Selection of the Size of Surface Elements............................. 95 7.23 Manual Computation Method.......... 97 7*231 Base Map and Templates........ 9? 7.232 Estimation
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