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I MUELLER, Ivan Iatvan. the GRADIENTS of GRAVITY and THEIR APPLICATIONS in GEODESY* the Ohio State University, Ph.D., 1960 Geology

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I MUELLER, Ivan Iatvan. the GRADIENTS of GRAVITY and THEIR APPLICATIONS in GEODESY* the Ohio State University, Ph.D., 1960 Geology This dissertation has been microfilmed exactly as received | Mic 00-4118 X ’ I MUELLER, Ivan Iatvan. THE GRADIENTS OF GRAVITY AND THEIR APPLICATIONS IN GEODESY* The Ohio State University, Ph.D., 1960 Geology University Microfilms, Inc., Ann Arbor, Michigan THE GRADIENTS CP GRAVITY AND THEIR APPLICATIONS IN GEODESY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the, Graduate School of The Ohio State University By IVAN ISTVAN MUELIER, DIPL. ENG. ****** The Ohio State University i960 Approved by Adviser Department of Geology (Division of Geodetic Science) TO THE MEMORY OF ROLAND EOTVOS PREFACE In 1901 Professor Roland Efttvtts, Hungarian physicist, in his opening speech as President of the Hungarian Academy of Sciences, said that scientists have tried to determine the shape and size of the Earth for centuries but ... good results can be obtained only if we concentrate upon the investigation of the Earth's gravity field, since it was gravity which determined the shape of the oceans and’ the surface of the continents when the Earth was formed.... He mentioned that Geodesy is that science which deals with these problems but ...at the present time (1901) Geodesy cannot determine the detailed shape of the Earth. It cannot answer questions such as: What does the detailed shape of the surface, formed by gravity, look like? What is the shape of the surface of a body of water immediately around us? How large is the curvature of this surface, and in which di­ rection is this curvature a maximum? In which direction dees gravity change with a maximum rate, and what is the magnitude of that change? All of these questions are yet unanswered. The geodesist is like the farsighted person who can enjoy the sight of the distant blue moun­ tains but is unable to read a letter bringing good news. This same man, to use another analogy, can determine the curvature of the oceans, but not the curvature of a glass of water. Etitvtis, in order to bring all these problems closer to solution, had designed an instrument, the "torsion balance," which measures the horizontal gradients of gravity. In 1915, only a few years after the first Instruments were in use, Helmert wrote that Geodesy has 1. two miraculous instruments, the level and the torsion balance. Both are very simple instruments^even so they give crucial informa­ tion about the shape and structure of the Earth. In spite of this fact, the torsion balance has been used only in commercial geophysi­ cal explorations and not for geodetic purposes. Even in geophysics it was superseded by the faster and cheaper gravimeters after about thirty years' use. It is interesting to note that with the develop­ ment of the sensitive gravimeters it has become possible to measure the vertical gradient of gravity with a relatively high accuracy. The horizontal gradients, together with the vertical gradient, i.e., the torsion balance together with the gravimeter, can give even more geodetic information than described in the Efltvfls quotation above. Thus we see that the advent of sensitive gravimeters leads to the further development of geodetic applications of the torsion balance. However, not much further has been done as far as geodetic applica­ tions are concerned since Efitvfls' death. This can be partly ex­ plained by the fact that geodesists in the past did not need very detailed information about the gravity field of the Earth or about the shape of the equipotentlal surfaces. At present the situation is different. In the age of space flight and of the developing world geodetic system even the small details of the gravity field may have Increasing significance. Someone might say that we still do not need all of the detailed geodetic data which it is possible to obtain from the gradients of gravity. But with the present rate of development of science the time is near when the best of these data may not be sufficient. When the author first became interested in this topic in 1951*-> his original aim was to emphasize the importance of the horizontal and the vertical gradients of gravity in geodesy. During this work, which continued until the end of 1956 and was started again the be­ ginning of 1959# the writer realized that no publication was avail­ able which covered all of the problems of determining and applying the gradients of gravity in geodesy. One purpose of this paper is to report and Introduce these problems with as many details as are needed for practical applications. (For instance, help tables for computations which are out of publication were also included. See Tables 3> 7 and 8 .) The author collected and used all of the available publications (350, mostly in German, Russian, French, English and Hungarian languages.) Most of these publications are listed in two sections of the Bibliography. In the first section those publications are given to which reference is made. The second section contains the rest. Note the relatively few publications in English, especially in connection with the horizontal gradients of gravity. Another purpose of this paper is to fill this gap in the English coverage of the subject. The third section of the Biblio­ graphy contains those papers which deal with the geophysical appli­ cations of the gradients of gravity. Since the topic of this dis­ sertation is the geodetic applications of the gradients of gravity, the third section is included only for the sake of those who are interested also in the geophysical applications. Vi In preparing this paper several conclusions and suggestions have been made (see chapters 2*21, 2.22, 2*31? 3*5* 3*6? 4.14, 4.33/ and 4.6.) These together with chapters 3*1* 3»7. 3*8# 3*9* 3«10» 4.4 and 4.5 are the contributions of this dissertation to geodesy. During the summers from 1954 to 1956 numerous field measurements were carried out. The computations based upon measurements together with other computations have been also included as examples in the proper chapters (see Tables 1, 2, 5> 6 , 10, 14, 15,and l6 .) In addi­ tion, several formulas have been derived, which are given also in other publications but without derivations (see Formulas 49, 107, 135 136, 181, 184, 193 and 196.) This paper is dedicated to the memory of Professor Roland E&tv8s,and therefore the author feels that it is proper to end this preface with another quotation from his work: "The scientist is not a person who knows all about science, but one who forwards it." Measured against this scale the author considers himself only on the very first rung of the ladder of science. vii ACKNOWLEDGMENT The author wishes to acknowledge his indebtedness for the help he received during the preparation of this dissertation, to Pro­ fessor W. A. Heiskanen, Director of the Institute of Geodesy, Photo- grammetry and Cartography, The Ohio State University, who has been his adviser; to Dr. R. A. Hirvonen, Professor of Geodesy of the Finland Institute of Technology, to Dr. H. J. Pincus, Professor of Geology of The Ohio State University, the members of the reading committee; and to Mr. R. J. Feely for his great work in correcting the manuscript. The first part of this paper in a shorter and somewhat different form was prepared in Hungary, and was circulated among leading Hun­ garian scientists for criticism. The author is especially grateful to Professor I. Redey, who has been his teacher and adviser for more than five years; to Professor A. Tarczy Hornoch; to Dr. J. Renner; and to Dr. L. Homor&Ly, for their constructive criticisms. An Etttvfts torsion balance, has been made available for field measurements by Mr. T. Dombay, Director of the Hungarian Roland EtttviJs Geophysical Institute, whom the writer wishes to thank. Special thanks are due to the author's wife, Mrs. Marianne Mueller, for her patience and help during the preparation of this dissertation; to Mrs. Margery Corrigan for her excellent typing; to Mr. Leslie Cunningham for his help with the figures and the repro­ duction; and to all others who have assisted with this paper. viii TABLE OF CONTENTS ; Chapter Title Page 1. INTRODUCTION . .......... ............... 1 1.1. The Gravitational Force. .......... 1 1.2. The Potential of the Gravitational Force . « ■ . ' 4 1.3* The Gravity. The Gradients of Gravity....... 7 2. THE DETERMINATION OF THE HORIZONTAL GRADIENTS OF GRAVITY . 13 2.1. The Principle of the EdtvBs Torsion Balance . 13 2.11. The Curvature Variometer. .......... 13 2.12. The Horizontal Variometer .................. 24 2.121. The Horizontal Variometer with One Swinging System ............... 24 2; 122. The Horizontal Variometer with Two Swinging Systems............... 30 2.13. The Gradiometer........... 36 2.2. The Torsion Balance Apparatus. ..................39 2.21. The Different Instruments............. 39 2*22. Field Operations......................... 58 2.3. The Reduction and Transformation of the Torsion Balance Measurements ............... 6l 2.31> Definitions............................... 6l 2.32. The Normal Gradients. ............ 64 2.33• The Terrain Effect......... 67 2.331«- Analytical Methods........ ......... 67 2*332. Mechanical M e t h o d s . ........ 86 ix TABLE OF CONTENTS Chapter Title Page 2.34. The Cartographic E f f e c t ...................... 91 2.341. Analytical Methods........... * . 91 2.342. Mechanical Methods. ......... 94 2.343. Graphical Methods .................. 95 2.35* The Transformation of the Torsion Balance ■ Measurements. ..................... 100 2.4. The Determination of the Instrument Constants. 103 2.41. The Instrument Constants..................... 103 2.42. The Determination of the Quantity — ..........104 2 A 3 . The Determination of the Torque Constant, r • 111 2.431. The Swinging Period Method........... Ill 2.432. The Cavendish Method................. 114 3. THE APPLICATIONS OF THE TORSION BALANCE MEASUREMENTS IN G E O D E S Y ..............................................
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