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GEOMETRIC GEODESY Part 2.Tif GEOMETRIC GEODESY PART II by Richard H. Rapp The Ohio State University Department of Geodetic Science and Surveying 1958 Neil Avenue Columbus, Ohio 4321 0 March 1993 @ by Richard H. Rapp, 1993 Foreword Geometric Geodesy, Volume 11, is a continuation of Volume I. While the first volume emphasizes the geometry of the ellipsoid, the second volume emphasizes problems related to geometric geodesy in several diverse ways. The four main topic areas covered in Volume I1 are the following: the solution of the direct and inverse problem for arbitrary length lines; the transformation of geodetic data from one reference frame to another; the definition and determination of geodetic datums (including ellipsoid parameters) with terrestrial and space derived data; the theory and methods of geometric three-dimensional geodesy. These notes represent an evolution of discussions on the relevant topics. Chapter 1 (long lines) was revised in 1987 and retyped for the present version. Chapter 2 (datum transformation) and Chapter 3 (datum determination) have been completely revised from past versions. Chapter 4 (three-dimensional geodesy) remains basically unchanged from previous versions. The original version of the revised notes was printed in September 1990. Slight revisions were made in the 1990 version in January 1992. For this printing, several corrections were made in Table 1.4 (line E and F). The need for such corrections, and several others, was noted by B.K. Meade whose comments are appreciated. Richard H. Rapp March 25, 1993 Table of Contents 1 . Long Geodesics on the Ellipsoid ..................................................................... 1 1. 1 Introduction ...................................................................................... 1 1.2 An Iterative Solution for Long Geodesics .................................................... 1 1.2 1 The Iterative Inverse Problem ...................................................... 16 1.22 The Iterative Direct Problem ........................................................ 19 1.23 The Improved Iteration Procedures .............; .................................. 22 1.24 The Non-Iterative Direct Problem .................................................. 24 1.3 The Non-Iterative Inverse Problem ......................................................... 31 1.4 A Numerical Integration Approach to the Solution of the Direct and Inverse Problem ........................................................................................ 37 1.5 Geodesic Behavior for Near Anti-Podal Lines ............................................ 40 1.5 1 Anti-Podal Behavior for Two Points on the Equator ............................ 41 1.52 Geodesic Behavior for Near Anti-Podal Points .General Case ................ 47 1.52 1 A Convergence Problem ............................................................ 50 1.6 The Behavior of "Backside lines" ........................................................... 50 1.7 TestLines ...................................................................................... 52 1.71 StandardTestLines .................................................................. 52 1.72 Anti-Podal Lines ..................................................................... 52 1.73 Backside Lines ....................................................................... 53 References............................................................................................ 55 2 . Transformation of Geodetic Data Between Reference Datums ................................. 57 2.1 Introduction .................................................................................... 57 2.2 Similarity Transformation .................................................................... 58 2.2 1 The Bursa-Wolf Transformation Model ...........................................60 2.22 The Veis Transformation Model .................................................... 68 2.23 The Molodensky Transformation Model ..........................................72 2.24 The VaniEek-Wells Transformation Models ......................................74 2.3 Geodetic Coordinate Transformation ....................................................... 76 2.3 1 A Differential Projective Transformation Procedure ............................. 76 2.3 1.1 The Molodensky Geodetic Coordinate Transformation ............... 79 2.3 1.2 Geodetic Coordinate Changes Caused by Changes at the Datum Origin Point Due to Shift and Ellipsoid Changes ...................... 80 2.3 1.3 Differential Change Formulas in Terms of Deflections of the Vertical and Geoid Undulations (or Height Anomalies) .............. 82 2.31.4 Special Cases of Transformation Involving Origin Shifts and Ellipsoid Parameter Changes ............................................. 83 2.3 1.5 Geodetic Coordinate Changes Due to the Scale Change .............. 85 2.3 1.6 Geodetic Coordinate Changes Due to the Three Rotation Angles .... 85 2.3 1.7 The Total Change in Geodetic Coordinates From the Sum of the Individual Components ................................................... 85 2.3 1.8 Azimuth Changes Due to Parameter Changes .......................... 86 2.32 A Differential Development Transformation Procedure ......................... 88 2.32.1 Comparison of Certain Projective and Development Change Formulas .................................................................... 90 2.3 3 Non-Conventional Transformation of Geodetic Coordinates ................... 91 References............................................................................................ 92 3 . The Determination of Geodetic Datums and Ellipsoid Parameters ............................. 95 3.1 Introduction .................................................................................... 95 3.2 Horizontal Geodetic Datums .Theory ...................................................... 95 3.3 Datum Definition and Horizontal Networks Through the Use of Positions Derived from Space Observations ........................................................... 99 3.3.1 Introduction ........................................................................... 99 3.3.2 Space Positions to Horizontal Datum System ....................................99 3.3.3 Horizontal Positions to Space Positions ..........................................102 3.4 Local Geodetic Datums of the World ...................................................... 106 3.4.1 The European Datum ............................................................... 109 3.4.2 The Australian Geodetic Datum ................................................... 111 3.4.3 The North American Datum 1983 (NAD83) ..................................... 113 3.5 Space Based Reference Systems ...........................................................113 3.6 The BIH Terrestrial System (BTS) ........................................................ 114 3.7 The IERS Terrestrial System ............................................................... 115 3.8 The World Geodetic System 1984 (WGS84) ............................................. 115 3.9 The Estimation of the Datum Origin Coordinates and Ellipsoid Parameters .......... 118 3.9.1 The Determination of a Best Fitting Datum and Reference Ellipsoid .......... 118 3.9.1.1 A Modified Best Fitting Ellipsoid ...................................... 121 3.9.2 A General Terrestrial Ellipsoid Based on Astro-Geodetic Data ................ 121 3.9.3 Remarks on the Area Method ......................................................123 3.9.4 The Molodensky Correction to Development Computed Astrogeodetic -- Data ...................................................................................121 3.9.5 The Arc Methods for Datum and Ellipsoid Parameter Determination ......... 127 3.10 The Determination of the Parameters of a General Terrestrial Ellipsoid ............... 131 3.10.1 GM and J2 Determinations ........................................................ 132 3.10.2 The Angular Velocity o ........................................................... 132 3.1 0.3 The Equatorial Radius ............................................................. 133 3.10.3.1 The Determination of the Equatorial Radius from Space Derived Station Positions ...................................................... 133 3.10.3.2 The Determination of the Equatorial Radius from Satellite Altimeter Data ......................................................... 134 3.1 1 Other Considerations on Ellipsoid Determination ........................................ 135 3.12 Future Determinations ......................................................................-136 3.13 Vertical Datums ...............................................................................136 3.13.1 The Geoid ...........................................................................136 3.1 3.2 The Mean Sea Level ...............................................................136 3.13.3 Determination of Orthometric Heights ........................................ 138 3.1 3.4 Specific Vertical Datums ..........................................................139 3.14 Future Vertical Datums .....................................................................-140 References ...........................................................................................141 4 . Fundamentals of Three Dimensional Geodesy .................................................. 147 4.1 Introduction.................................................................................. -147 4.2 Coordinate Systems
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