PHYSICAL GEODESY

By MIA JACKSON

INTERNAL GRAVITATIONAL FIELD OF A HOMOGENEOUS ELLIPSOID OF REVOLUTION THE GRAVITATIONAL ATTRACTION COMPONENTS FX AND FY THE GRAVITATIONAL ATTRACTION COMPONENT FZ GRAVITATIONAL POTENTIAL IN THE INTERIOR EXTERNAL GRAVITATIONAL FIELD OF A HOMOGENEOUS ELLIPSOID OF REVOLUTION GRAVITATIONAL POTENTIAL OF A HOMOGENEOUS ELLIPSOIDAL HOMOEOID OF REVOLUTION GRAVITATIONAL POTENTIAL IN THE EXTERIOR SURFACE DENSITY THE EARTH’S NORMAL GRAVITY FIELD MACLAURIN ELLIPSOID NORMAL GRAVITATIONAL POTENTIAL THE REFERENCE EARTH ELLIPSOID NORMAL GRAVITY ON THE REFERENCE EARTH ELLIPSOID NORMAL GRAVITY ABOVE THE REFERENCE EARTH ELLIPSOID GRAPHICAL REPRESENTATION SECOND ORDER APPROXIMATE FORMULAE THE NORMAL POTENTIAL PARAMETERS OF THE REFERENCE EARTH ELLIPSOID REFERENCES STOKES’ THEORY AND BEYOND STOKES’ BOUNDARY VALUE PROBLEM DISTURBING POTENTIAL AND GRAVITY DISTURBANCE GEOIDAL HEIGHT AND DE?ECTION OF THE VERTICAL GRAVITY ANOMALY AND FUNDAMENTAL GRAVIMETRIC EQUATION SOLUTION OF STOKES’ BOUNDARY VALUE PROBLEM SOLUTION OF THE DISTURBING POTENTIAL AND GEOIDAL HEIGHT SOLUTION OF THE GRAVITY DISTURBANCE AND DE?ECTION OF THE VERTICAL GRAVITY ANOMALY OUTSIDE THE GEOID BOUNDARY VALUE THEORIES BEYOND THAT OF STOKES FORMULATION WHEN THE GRAVITY DISTURBANCE IS KNOWN FORMULATION WHEN THE DE?ECTION OF THE VERTICAL IS KNOWN FORMULATION WHEN THE GEOIDAL HEIGHT IS KNOWN INCLUSION OF ERRORS OF THE REFERENCE EARTH ELLIPSOID IN STOKES’ THEORY FUNDAMENTAL RELATIONS SOLUTION OF DISTURBING POTENTIAL, GEOIDAL HEIGHT, AND DE?ECTION OF THE VERTICAL RELATION BETWEEN THE REFERENCE EARTH ELLIPSOID AND THE GEOID SOME CHARACTERISTICS OF THE EARTH’S GRAVITATIONAL FIELD A GLOBAL MODEL OF THE GRAVITATIONAL POTENTIAL DEGREE POWER SPECTRUM OF THE GRAVITATIONAL POTENTIAL AND GEOIDAL HEIGHT DEGREE POWER SPECTRUM OF THE GRAVITATIONAL ATTRACTION AND DE?ECTION OF THE VERTICAL GRAVITY REDUCTION BASIC CORRECTIONS AND GRAVITY ANOMALIES FREE AIR CORRECTION AND FREE AIR ANOMALY PLATE CORRECTION AND INCOMPLETE BOUGUER ANOMALY TERRAIN CORRECTION AND COMPLETE BOUGUER ANOMALY A PRACTICAL FORMULA FOR COMPUTING ORTHOMETRIC HEIGHT USING SPIRIT LEVELING DATA HELMERT CONDENSATION AND HELMERT ANOMALY ISOSTASY, ISOSTATIC CORRECTION, AND ISOSTATIC GRAVITY ANOMALY BACKGROUND FACT AND THE PRATT–HAYFORD MODEL THE AIRY–HEISKANEN MODEL THE VENING MEINESZ MODEL ISOSTATIC CORRECTION AND ISOSTATIC ANOMALY DETERMINATION OF ISOSTATIC MODELS: THE CONCEPT GRAVITATIONAL FIELD OF A LAYER OF MASS AROUND THE EARTH’S SURFACE A LATERALLY HETEROGENEOUS LAYER OF MASS AROUND THE EARTH’S SURFACE A HOMOGENEOUS CAP-SHAPED SHELL AROUND THE EARTH’S SURFACE GRAVITY REDUCTION FOR THE SPHERICAL EARTH MODEL SOME RELATIONS BETWEEN THE FORMULATIONS FOR THE FLAT AND SPHERICAL EARTH MODELS THE COMPLETE BOUGUER ANOMALY THE HELMERT ANOMALY THE ISOSTATIC ANOMALY INDIRECT EFFECT OF GRAVITY REDUCTION ON THE GRAVITY FIELD INDIRECT EFFECT AND COMPUTATIONAL PROCEDURE OF THE EARTH’S GRAVITY FIELD INDIRECT EFFECT OF THE GLOBAL GRAVITATIONAL POTENTIAL, THE TOPOGRAPHIC–ISOSTATIC MODEL INDIRECT EFFECT OF THE GEOIDAL HEIGHT INDIRECT EFFECT OF THE DE?ECTION OF THE VERTICAL FURTHER READING MOLODENSKY’S THEORY AND BEYOND MOLODENSKY’S BOUNDARY VALUE PROBLEM NORMAL HEIGHT, HEIGHT ANOMALY, TELLUROID, AND QUASIGEOID MOLODENSKY’S GRAVITY ANOMALY AND GRAVIMETRIC BOUNDARY VALUE PROBLEM A GENERALIZED DE?NITION OF GRAVITY ANOMALY IN SPACE MOLODENSKY’S SOLUTION MOLODENSKY’S INTEGRAL EQUATION MOLODENSKY’S SHRINKING SOLUTION OF THE HEIGHT ANOMALY SOLUTION OF THE DE?ECTION OF THE VERTICAL REDUCTION OF THE SOLUTION TO A PRACTICAL FORM SOLUTION BY DOWNWARD CONTINUATION A PROPERTY OF MOLODENSKY’S SOLUTION ANALYTICAL DOWNWARD CONTINUATION TO A SURFACE THROUGH THE POINT OF INTEREST A DISCUSSION ON THE POSSIBLE ANALYTICAL DOWNWARD CONTINUATION TO THE QUASIGEOID AN ITERATIVE APPROACH OF HARMONIC DOWNWARD CONTINUATION GENERAL DESCRIPTION INDIRECT EFFECT OF THE HEIGHT ANOMALY DOWNWARD CONTINUATION AFTER GRAVITY REDUCTION: A CONCEPTUAL PROCEDURE THE GNSS-GRAVIMETRIC BOUNDARY VALUE THEORY THE GRAVITY DISTURBANCE AND THE GNSS-GRAVIMETRIC BOUNDARY VALUE PROBLEM SOLUTION OF DISTURBING POTENTIAL BY MOLODENSKY’S SHRINKING SIMPLI?CATION OF THE SOLUTION OF THE DISTURBING POTENTIAL SOLUTIONS OF THE HEIGHT ANOMALY AND THE DE?ECTION OF THE VERTICAL ANALYTICAL DOWNWARD CONTINUATION AND GRAVITY REDUCTION COMBINATION OF DIFFERENT TYPES OF DATA CONVERSION OF DIFFERENT TYPES OF DATA INTO GRAVITY ANOMALY CONVERSION OF DIFFERENT TYPES OF DATA INTO GRAVITY DISTURBANCE APPENDIX: SOME REMARKS ON THE USE OF RUNGE’S THEOREM FUNDAMENTALS OF COMPUTATION AND DETERMINATION PREPARATION OF GRAVITY DATA TRANSFORMATION BETWEEN DIFFERENT GEODETIC REFERENCE FRAMES EVALUATION OF SURFACE INTEGRALS IN GRAVITY REDUCTION INTERPOLATION OF POINT-WISE GRAVITY ANOMALY COMPUTATION OF GRID MEAN GRAVITY ANOMALY COMPUTATION OF THE GLOBAL MODEL OF THE GRAVITATIONAL POTENTIAL COMPUTATION BY EVALUATING INTEGRALS COMPUTATION OF INTEGRALS OF THE ASSOCIATED LEGENDRE FUNCTION PELLINEN AND GAUSSIAN SMOOTHING DE-SMOOTHING OF THE GRID MEAN GRAVITY ANOMALY COMPUTATION OF GRAVITATIONAL POTENTIAL BASED ON LEAST SQUARES FITTING GEOIDAL HEIGHT, HEIGHT ANOMALY, AND DE?ECTION OF THE VERTICAL BASED ON A GLOBAL MODEL OF THE GRAVITATIONAL POTENTIAL COMPUTATION OF THE GEOIDAL HEIGHT, HEIGHT ANOMALY, AND DE?ECTION OF THE VERTICAL USING GRAVITY DATA GEOIDAL HEIGHT AND HEIGHT ANOMALY THE DE?ECTION OF THE VERTICAL ASTRO-GEODETIC DETERMINATION OF THE DE?ECTION OF THE VERTICAL ASTRO-GEODETIC MEASUREMENT OF THE DE?ECTION OF THE VERTICAL INTERPOLATION BASED ON THE GRAVIMETRIC DE?ECTION OF THE VERTICAL INTERPOLATION BASED ON ISOSTATIC DE?ECTION OF THE VERTICAL ASTRO-GEODETIC DETERMINATION OF GEOIDAL HEIGHT AND HEIGHT ANOMALY ASTRONOMICAL LEVELING ASTRO-GRAVIMETRIC LEVELING GNSS AND GNSS-GRAVIMETRIC LEVELING FLATTENING AND GRAVITY INSIDE THE EARTH HYDROSTATIC EQUILIBRIUM AND THE SPHERICAL EARTH MODEL GRAVITATIONAL POTENTIAL OF A HOMOGENEOUS ASPHERICAL SHELL GRAVITATIONAL POTENTIAL OF A HETEROGENEOUS ASPHERICAL BODY CLAIRAUT’S EQUATION OF THE EARTH’S INTERNAL FLATTENING AN ANALYTIC APPROXIMATE SOLUTION OF CLAIRAUT’S EQUATION EARTH’S INTERNAL FLATTENING AND GRAVITY REFERENCE GRAVITATIONAL ATTRACTION GRAVITATIONAL ATTRACTION OF A POINT MASS ACCORDING TO NEWTON’S LAW OF GRAVITATION, ANY TWO OBJECTS ATTRACT EACH OTHER, I E , EACH OBJECT ATTRACTS THE OTHER TOWARD IT THE FORCE OF ATTRACTION IS CUSTOMARILY REFERRED TO AS FORCE OF GRAVITATIONAL ATTRACTION BETWEEN MASSES IF THE SIZES OF THE OBJECTS ARE MUCH SMALLER THAN THE DISTANCE BETWEEN THEM, THE FORCE OF GRAVITATIONAL ATTRACTION BETWEEN THEM IS PROPORTIONAL TO THE MASSES OF BOTH OBJECTS AND IS INVERSELY PROPORTIONAL TO THE SQUARE OF THE DISTANCE BETWEEN THEM AS SHOWN IN FIG 1 1, WE ASSUME THAT TWO POINT MASSES (WHICH COULD BE UNDERSTOOD AS IN?NITELY SMALL PARTICLES), M AND M PRIMEM?, ARE LOCATED AT M AND P, RESPECTIVELY WE USE LEFT PARENTHESIS XI COMMA ETA COMMA ZETA RIGHT PARENTHESIS(?, ?, ?) AND LEFT PARENTHESIS X COMMA Y COMMA Z RIGHT PARENTHESIS(X, Y, Z) TO DENOTE THEIR COORDINATES, AND USE MODIFYINGABOVE L WITH RIGHT ARROW??L AND L TO REPRESENT THE RELATIVE POSITION VECTOR OF P WITH RESPECT TO M AND ITS LENGTH, GRAVITATIONAL ATTRACTION, CENTRIFUGAL ACCELERATION, AND GRAVITY