EARTH TIDE EFFECTS ON GEODETIC OBSERVATIONS by K. BRETREGER A thesis submitted as a part requirement for the degree of Doctor of Philosophy, to the University of New South Wales. January 1978 School of Surveying Kensington, Sydney. This is to certify that this thesis has not been submitted for a higher degree to any other University or Institution. K. Bretreger ( i i i) ABSTRACT The Earth tide formulation is developed in the view of investigating ocean loading effects. The nature of the ocean tide load leads to a proposalfor a combination of quadratures methods and harmonic representation being used in the representation of the loading potential. This concept is developed and extended by the use of truncation functions as a means of representing the stress and deformation potentials, and the radial displacement in the case of both gravity and tilt observations. Tidal gravity measurements were recorded in Australia and Papua New Guinea between 1974 and 1977, and analysed at the International Centre for Earth Tides, Bruxelles. The observations were analysed for the effect of ocean loading on tidal gravity with a •dew to nodelling these effects as a function of space and time. It was found that present global ocean tide models cannot completely account for the observed Earth tide residuals in Australia. Results for a number of models are shown, using truncation function methods and the Longman-Farrell approach. Ocean tide loading effects were computed using a simplified model of the crustal response as an alternative to representation by the set of load deformation coefficients h~, k~. It is shown that a ten parameter representation of the crustal response is adequate for representing the deformation of the Earth tide by ocean loading at any site in Australia with a resolution of ±2 µgal provided extrapolation is not performed over distances greater than 10 3 km. This is assessed as being of sufficient accuracy for all purposes in high precision geodesy. The formulation of sol id Earth and ocean tide perturbations of satellites is reviewed, and these equations used to obtain graphs of the variations in the orbital elements of four satellites Beacon Explorer - C (BEC), GEOS 1, GEOS 2 and GEOS 3, with particular reference to the latter. The effect of Earth tides on satellite altimetry measurements is examined with the view of suitable corrections, if required. A numerical estimate is given on the possibility of determining ocean tide models from satellite altimetry, based on the signal-to-noise ratio. The degradation of the analysis is obtained by progressively adding specified levels of randomly distributed white noise which simulates observational and orbital errors. The conditions for the successful recovery of tidal signals are dependent on the signal-to-noise ratio and this is a function of the particular tidal signal considered. (iv) TABLE OF CONTENTS Page ABSTRACT iii TABLE OF CONTENTS iv NOTATION vi ABBREVIATIONS ix CONVENTIONS X INDEX OF FIGURES AND TABLES xi ACKNOWLEDGEMENTS xiii 1. INTRODUCTION 1.1 Significance of the Project 1. 2 Method of Analysis 3 1.3 Synopsis of Contents 5 2. DEVELOPMENT OF TIDAL THEORY 7 2. 1 Development of the Tidal Theory 7 2.2 Components of the Tidal Potential 12 2.3 The Constituent Potentials 15 2. 3. 1 The Ti de Producing Potential 15 2.3.2 The Deformation Potential 15 2.3.3 The Load Potential 16 2. 3. 4 The Stress Potential 17 2.4 Variations in Potential and its Components 17 2. 4. 1 Change in Potential 17 2.4.2 Change in Gravity 19 2.4.3 Changes in the Deflection of the Vertical 19 3. INSTRUMENTATION 21 3. 1 lntroduct ion 21 3.2 General lnstal lation Aspects of Gravimeters 23 3,3 Instrumental Characteristics of Gravimeters 23 3. 3. 1 Gravimetric Drift 23 3.3.2 Calibration of Gravimeters 25 4. AUSTRALIAN TIDAL GRAVITY RESULTS 28 4. 1 Introduction 28 4.2 Instrumental Properties 30 4.3 Results from Australian Stations 30 4.4 Interpretation of Tidal Records 31 4.5 Discussion of Results 33 5. OCEANIC PERTURBATIONS ON EARTH TIDES 39 5.1 Hodell ing of Ocean Tides 39 5. 1. 1 Ocean Tides 39 5. 1. 2 The Laplace Tidal Equations 39 5. 1.3 Cotidal-Corange Charts 41 5.2 Oceanic Loading 44 5.2. 1 Introduction to Ocean Load Hodell ing 44 5.2.2 Mathematical Formulation of Truncation Function Method 45 5. 2.2. 1 Gravity Variations 45 5.2.2.2 Variations in the Deflection of the Vertical 47 (v) Page 5.2.2.3 The Combination of the Elastic Components of the Potent i a 1 48 5.2.2.4 Representation of the Ocean Loading Effects on the Radial Deformation 50 5.2.3 Dutl ine of Longman-Farrell Method for Calculation of Oceanic Perturbations 52 6. COMPUTATIONAL RESULTS FOR TIDAL LOADING FOR AUSTRALIA 55 6. 1 I nt roduc t ion 55 6.2 Computational Procedures for Truncation Function Method 56 6.3 A Method of Representing the Ocean Tide Deformation with Truncation Functions 61 6.4 Calculation of Outer Zone Effects by Longman-Farrell Method 62 6.5 Inner Zone Calculations Using Longman-Farrell Method 64 6.6 Comparison Between Different Methods 69 6. 7 Hodel 1 ing Ocean Loading Effects in Australia 74 6. 7.1 Introduction 74 6. 7.2 Method of Analysis 77 6.7.3 Analysis of Australian Results 79 6. 7.4 Conclusions on the Modelling Method 86 7. TIDAL PERTURBATION THEORY ON SATELLITES 87 7. 1 Introduction 87 7.2 Tidal Perturbation Theory 87 7. 2. 1 Simplified Tidal Perturbation Theory 88 7.2.2 General Development of Earth Tide Perturbation Theory 94 7-3 Ocean Tide Perturbations 98 8. TIDAL EFFECTS ON VARIOUS SATELLITES 100 8. 1 Introduction 100 8.2 Major Tidal Perturbations in Inclination 100 8.3 Sol id Earth Tidal Perturbations on GEDS 3 104 9. SATELLITE ALTIMETRY STUDIES 119 9. 1 Introduction 119 9.2 Ocean Tide Models from Satellite Altimetry 119 9.2. 1 Introduction 119 9.2.2 Analysis of Altimeter Measurements 120 9.2.3 Error Sources in Satellite Altimetry 122 9.2. 3. 1 Instrumental Errors 123 9.2.3.2 Other Errors 124 9.2.4 Results on the Recovery of Ocean Models 124 9-3 Earth Tide Effects on Satel 1 ite Altimetry 128 9. 3.1 Introduction 128 9.3.2 Determination of Radial Perturbations 129 9.3.3 Orbit Determinations 13D 9.4 Cone 1us ions 132 10. CONCLUSIONS 134 Tidal Gravity and Ocean Loading 134 Satellite Studies 136 REFERENCES 139 APPENDIX A The Harmonic Development of the Tide Generating Potential 143 APPENDIX B Location Sketches of Sites 151 (v i) NOTATION Frequently used symbols are Symbol Meaning Page Azimuth of the ocean load from point of computation 5.2.2.2 47 a Mean radius of the Earth 2. 3,2 15 Semi-major axis of the disturbing body's orbit 2. 1 9 Semi-major axis of the satellite orbit 7 .2. 1 90 A multiplying parameter 2.3.4 17 clnm A parameter defining the ocean tide, related to amplitude 7,3 98 Cn Multiplying parameter c of degree n 2. 3, 4 17 Do Undisturbed ocean depth 5. 1. 2 L10 dm An element of mass 2.3.2 15 dr Radial displacement 2.4.1 18 do An element of sol id angle 2.3.2 15 Es Eccentric anomaly of the satel 1 ite 9, 3.2 129 eb Eccentricity of the Disturbing body's orbit 7.2.2 95 es Eccentricity of the satellite orbit 7.2. 1 90 Fb Disturbing function due to the disturbing body tide at 90 satellite altitude 7. 2. 1 Fnmj (is) Polynomial in terms of is 7.2.2 95 Fnmp ( i b) Polynomial in terms of ib 7.2.2 95 FA Zonal component of the ocean bottom st~ess 5. 1.2 40 F,p Meridional component of the ocean bottom stress 5. 1.2 40 f A Love number fb True anomaly of the disturbing body 7. 2. 1 89 fe Flattening of the Earth 2.3.3 16 fs True anomaly of the satel 1 ite 7. 2. 1 89 G Gravitational constant 2. 1 7 GD Doodson constant 2. 1 9 Gnjg(es) Polynomial in terms of es 7.2.2 95 Gnpq(eb) Polynomial in terms of eb 7.2.2 95 g Gravity at the Earth's surface 2.2 12 g' Gravity at the tidal deformed Earth 2.4.2 19 H Hour angle of the disturbing body 2. 1 10 h Mean longitude of the Sun 2. 1 10 Love number 2. 4. 1 18 Height of a point on the Earth's surface 2.4.2 19 Love number h of degree n 2.4. 1 18 Load deformation coefficient of degree n 2.4.1 18 Inclination of the disturbing body's orbit 7 .2. 1 89 Inclination of the satellite, to the celestial equator 7. 2. 1 89 Second zonal harmonic of the Earth's potential 7 .2. 1 91 Multiplying parameter of degree n 5.2.3 53 Love number 2.3.2 16 Love number k of degree n 2.3.2 16 Load deformation coefficient of degree n 2.3.4 17 Argument of latitude of the disturbing body 7 .2. 1 89 Argument of latitude of the satellite 7 .2. 1 89 Load deformation coefficient of degree n 5.2.3 53 Global mean value of the enclosed quantity (vii) Symbol Meaning Page Hean anomaly of the disturbing body 7 .2.