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Collocation and least squares methods as a tool for handling gravity field dependent data obtained through space research techniques

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Abstract

Least squares adjustment and collocation methods have in the last decade been the tool for extracting gravity field information from data obtained through space research techniques (satellite orbit tracking, altimeter observations, doppler determined positions), and when combining these data with data observed at the surface of the Earth.

The mathematical framework for the two models is described and the models are compared. It is shown that the two methods only become equivalent in cases where the number of parameters are equal to the number of observations.

It is pointed out that several arbitrary choices (of parameters, weights and norms) will have to be made before the methods can be applied, and that further investigations are needed in order to justify the specific choices.

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Bibliography

  • K. ARNOLD: An Attempt to Determine the Unknown Parts of the Earth's Gravity Field by Successive Satellite Passages, Bulletin Géodésique, No. 87, pp 97–101, 1968.

    Article  Google Scholar 

  • G. BALMINO: Representation of the Earth Potential by Buried Masses, In: The Use of Artificial Satellites for Geodesy, AGU, Geophysical Monograph, 15, 1971.

  • G. BALMINO: Contribution à l'Amélioration du Potentiel Terrestre, GRGS, Bull, No. 12, 1974.

  • G. BALMINO: Introduction to Least-Squares Collocation Lectures, Second International Summer School in the Mountains, Ramsau, Austria, 1977 (In print).

  • G. BALMINO, Ch. REIGBER and B. MOYNOT: The GRIM 2 Earth Gravity Field Model, DGK, Reihe A, Heft Nr. 86, 1976.

  • W. BENNING: Detaillierte Geoidstruktur aus Beobachtungen der Satellitenaltimetrie, Zeitschrift für Vermessungswesen, Vol. 101, No. 2, pp. 50–59, 1976.

    Google Scholar 

  • R.D. BROWN: Geoid Determinations from Satellite altimetry using Sample Functions, Proceedings of the International Symposium on Applications of Marine Geodesy, pp. 315–329, 1974.

  • A. DERMANIS: Probabilistic and Deterministic Aspects of Linear Estimation in Geodesy, OSU rep. No. 244 „1976.”-OSU rep.=Reports of the Department of Geodetic Science, The Ohio State University, Columbus, Ohio.

  • E.M. GAPOSCHKIN: 1972 Smithsonian Institution Standard Earth 3, J.G.R., Vol. 79, pp. 5377–5411, 1974.

    Google Scholar 

  • G.E.O. GIACAGLIA, and C.A. LUNDQUIST: Sampling Functions for Geophysics, SAO Special Report No. 344, 1972.

  • S. GOPALAPILLAI: Non-Global Recovery of Gravity Anomalies from a Combination of Terrestrial and Satellite Altimetry Data, OSU rep. No. 210, 1974.

  • E.W. GRAFAREND: Geodetic Applications of Stochastic Processes, Physics of the Earth and Planetary Interiors, Vol. 12, pp. 151–179, 1976.

    Article  Google Scholar 

  • Erik GRAFAREND: Operational Geodesy, Lecture notes, Second International Summer School in the Mountains, Ramsau, Austria, 1977. (In print).

    Google Scholar 

  • E. GROTEN: Combination Solutions in Geoid Computations, Proceedings of the International Symposium on Applications of Marine Geodesy, pp. 357–369, 1974.

  • D. HAJELA: Direct Recovery of Mean Gravity Anomalies from Satellite to Satellite Tracking, OSU rep. No. 218, 1974.

  • W.A. HEISKANEN and H. MORITZ: Physical Geodesy, Freeman & Co., San Francisco, 1967.

    Google Scholar 

  • B.B. HOLLAND, A. EISNER and S.M. YIONOULIS: The Effect of WGS-72 Geopotential in NNSS on Station Surveys, Presented AGU Fall Meeting, 1976.

  • Stanley K. JORDAN: Statistical Model for Gravity, Topography, and Density Contrasts in the Earth, Preprint. J.G.R., Vol. 83, No. 84, pp. 1816–1824, 1978.

    Google Scholar 

  • W.M. KAULA: Test and Combination of Satellite Determinations of the Gravity Field with Gravimetry, J.G.R., Vol. 71, No. 22, pp. 5303–5313, 1966.

    Article  Google Scholar 

  • K.R. KOCH: Alternate Representation of the Earth's Gravitational Field for Satellite Geodesy, Boll. di Geofisica Teorica ed Applicata, Vol, X, No. 40, pp. 318–325, 1968.

    Google Scholar 

  • K.R. KOCH: Earth's Gravity Field and Station Coordinates from Doppler Data, Satellite Triangulation, and Gravity Anomalies, NOAA Technical Report NOS 62, 1974.

  • K.R. KOCH: Least Squares Adjustment and Collocation, Bulletin Géodésique, Vol. 51, No. 2, pp. 127–135, 1977.

    Article  Google Scholar 

  • E.J. KRAKIWSKY: A Synthesis of Recent Advances in the Method of Least Squares, Lecture Notes, No. 42, Dep. of Surv. Eng. Univ. of New Brunswick, 1975.

  • T. KRARUP: A Contribution to the Mathematical Foundation of Physical Geodesy, Geodaetisk Institut, Meddelelse No. 44, 1969.

  • S.L. LAURITZEN: The Probabilistic Background of Some Statistical Methods in Physical Geodesy, Geodaetisk Institut, Meddelelse No. 48, 1973.

  • D. LELGEMANN: On the Recovery of Gravity Anomalies from High Precision Altimeter Data, OSU rep. 239, 1976.

  • D. LELGEMANN: On a Mathematical Model for Remote Sensing Altimetry, Paper presented at the First General Assembly of the European Association of Remote Sensing Laboratories, Strasbourg, 1977.

  • R.S. MATHER: Geoid Definitions for the Study of Sea Surface Topography from Satellite Altimetry, Proceedings of the International Symposium on Applications of Marine Geodesy, pp. 279–289, 1974.

  • R.S. MATHER: On the Evaluation of Stationary Sea Surface Topography using Geodetic Techniques, Bulletin Géodésique, No. 115, pp. 65–82, 1975.

    Article  Google Scholar 

  • R.S. MATHER: On the Realization of a System of Reference in Four Dimensions for Ocean Dynamics, Presented at the Col, on Radio Oceanography, Hamburg, 1976.

  • R.S. MATHER: A Geodetic Basis for Ocean Dynamics Prepared for “Modern Trends in Geodesy”, A Volume dedicated to prof, Marussi, In print, 1977.

  • P. MEISSL. Elements of Functional Analysis, In: Brosowski, B. and E. Martensen (Ed.): Methoden und Verfahren der matheamtischen Physik, Vol. 12, pp. 19–78, 1975.

  • P. MEISSL: Hilberts Spaces and Their Application to Geodetic Least Squares Problems, Bollettino de Geodesia e Scienze Affini, Vol. XXXV, No. 1, pp. 181–210, 1976.

    Google Scholar 

  • H. MORITZ: Least-Squares Estimation in Physical Geodesy, OSU rep. No. 130, 1970.

  • H. MORITZ: Combination of Satellite Harmonics and Gravimetry, OSU rep. No. 146, 1970a.

  • H. MORITZ: Advanced Least-Squares Methods, OSU rep. No. 175, 1972.

  • H. MORITZ: Least-Squares Collocation, DGK, Reihe A, Nr. 75, 1973.

  • H. MORITZ: Stepwise and Sequential Collocation, OSU rep. No. 203, 1973a.

  • H. MORITZ: Some First Accuracy Estimates for Applications of Aerial Gravimetry, OSU rep. No. 209, 1974.

  • H. MORITZ: Precise Gravimetric Geodesy, OSU rep. No. 219, 1974a.

  • H. MORITZ: Combination of Aerial Gravimetry and Gradiometry, OSU rep. No. 223, 1975.

  • H. MORITZ: Covariance Functions in Least-Squares Collocation, OSU rep. No. 240, 1976.

  • H. MORITZ: On the Computation of a Global Covariance Model, OSU rep. No. 255, 1977.

  • H. MORITZ and K.P. SCHWARZ: On the Computation of Spherical Harmonics from Satellite Observations, Bollettino di Geodesia e Scienze Affini, Ann. 32, No. 3, pp. 185–200, 1973.

    Google Scholar 

  • I.I. MUELLER: Global Satellite Triangulation and Trilateration Results, J.G.R., Vol. 79, No. 35, pp. 5333–5347, 1974.

    Article  Google Scholar 

  • R.H. RAPP: Analytical and Numerical Differences between Two Methods for the Combination of Gravimetric and Satellite Data, Bollettino di geofisica teorica ed applicata Vol. XI, No. 41-42, pp. 108–118, 1969.

    Google Scholar 

  • R.H. RAPP: Numerical Results from the Combination of Gravimetric and Satellite Data Using the Principles of Least Squares Collocation, OSU rep. No. 200, 1973.

  • R.H. RAPP: Results from a Combination of Satellite and Gravimetric Data Considering the Collocation Concept, Bollettino di Geodesia e Scienze Affini, Vol. XXXII, No. 3, 1973a.

  • R.H. RAPP: Gravity Anomaly Recovery from Satellite Altimetry Data Using Least Squares Collocation Techniques, OSU rep. No. 220. 1974.

  • R.H. RAPP: The Gravitational Potential of the Earth to Degree 36 from Terrestrial Gravity Data, Paper presented at the IUGG, General Assembly, Grenoble, 1975.

  • R.H. RAPP and R. RUMMEL: Methods for Computation of Detailled Geoids and Their Accuracy, OSU rep. No. 233, 1975a.

  • R.H. RAPP: Anomalies Recovered from GEOS-3 Altimeter Data Using Least Squares Collocation, Paper presented at the AGU Spring Meeting, Washington D.C., April 1976.

  • R.H. RAPP: Potential Coefficient Determinations From Anomalies Given on a Bounding Sphere as Derived by Least Squares Collocation Prepared for “Modern Trends in Geodesy”, A Volume dedicated to prof. Marussi, In Print, 1977.

  • R.H. RAPP: Potential Coefficient Determinations from 5° Terrestrial Gravity Data, OSU rep. 251, 1977a.

  • G.B. REED: Application of Kinematical Geodesy for Determining the Short Wave Length Components of the Gravity Field by Satellite Gradiometry, OSU rep. No. 201, 1973.

  • C. REIGBER and K.H. ILK: Vergleich von Resonansparameterbestimmungen mittels Ausgleichung und Kollokation, Zeitschrift für Vermessungswesen Vol. 101, pp. 59–67, 1976.

    Google Scholar 

  • R. RUMMEL: Downward Continuation of Gravity Information from Satellite to Satellite Tracking or Satellite Gradiometry in Local Areas, OSU rep. no. 221. 1975.

  • R. RUMMEL: GEOS-3 Altimeter Data Processing for Gravity Anomaly Recovery, Abstract, EOS, Trans. Amer. Geophys. Union, Vol. 57, No. 4 p. 234, 1976.

    Google Scholar 

  • R. RUMMEL: A Model Comparison in Least Squares Collocation, Bulletin Géodésique, Vol. 50, pp. 181–192, 1976a.

    Article  Google Scholar 

  • R. RUMMEL, D. HAJELA and R.H. RAPP: Recovery of Mean Gravity Anomalies from Satellite-Satellite Range Rate Data Using Least Squares Collocation, OSU rep. No. 248, 1976.

  • R. RUMMEL and R.H. RAPP: Undulation and Anomaly Estimation Using GEOS-3 Altimeter Data without Precise Satellite Orbits, Bulletin Géodésique, Vol. 51, No. 1, pp. 73–88, 1977.

    Article  Google Scholar 

  • K.P. SCHWARZ: Even Zonal Harmonics from Satellite Observations by Collocation, In G. Veis (Ed): The Use of Artificial Satellites for Geodesy and Geodynamics, Athen, 1974, pp. 225–235.

  • K.P. SCHWARZ: Tesseral Harmonic Coefficients and Station Coordinates from Satellite Observations by Collocation, OSU rep. No. 217, 1974a.

  • K.P. SCHWARZ: Zonal Harmonic Coefficients by Least Squares Collocation Using Satellite and Gravimetric Data, In: Mitteilungen der geodätischen Institute der TU Graz, Folge 20, pp. 241–265, 1975.

  • K.P. SCHWARZ: Zonale Kugelfunktionskoefficienten aus Satellitendaten durch Kollokation, DGK, Reihe C, Nr. 209, 1975a.

  • K.P. SCHWARZ: Least Squares Collocation for Large Systems Bollettino di Geodesia e Scienze Affini, Anno XXXV, No. 3, pp. 309–324, 1976.

    Google Scholar 

  • K.P. SCHWARZ: Geodetic Accuracies Obtainable from Measurements of Frist and Second Order Gravitational Gradients, OSU rep. No. 221, 1976a.

  • K.P. SCHWARZ: Numerische Probleme bei der Bestimmung des globalen Erdschwerefelds durch Kollokation. Zeitschrift für Vermessungswesen, Vol. 101, No. 6, pp. 221–230, 1976b.

    Google Scholar 

  • K.P. SCHWARZ: Simulation Study of Airborne Gradiometry, OSU rep. No. 253, 1977.

  • D.E. SMITH, F.J. LERCH, J.G. MARCH, C.A. WAGNER, R. KOLENKIEWICZ and M.A. KHAN: Contribution to the National Geodetic Satellite Program by Goddard Space Flight Center, J.G.R., Vol. 81, No. 5, pp. 1006–1026, 1976.

    Article  Google Scholar 

  • G. SMITH: Mean Gravity Anomaly Prediction from Terrestrial Gravity Data and Satellite Altimeter Data, OSU rep. No. 214, 1974.

  • C.C. TSCHERNING and R.H. RAPP: Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variance Models, OSU rep. No. 208, 1974.

  • C.C. TSCHERNING: A FORTRAN IV Program for the Determination of the Anomalous Potential Using Stepwise Least Squares Collocation, OSU rep. No. 212, 1974.

  • C.C. TSCHERNING: Application of Collocation for the Planning of Gravity Surveys. Bulletin Géodésique, No. 116, pp. 183–198, 1975.

    Article  Google Scholar 

  • C.C. TSCHERNING: Covariance Expressions for Second and Lower Order Derivatives of the Anomalous Potential, OSU rep. No. 225, 1976.

  • C.C. TSCHERNING: A Note on the Choice of Norm when using Collocation for the Computation of Approximations to the Anomalous Potential, Bulletin Géodésique, Vol. 51, No. 2, pp. 137–147, 1977.

    Article  Google Scholar 

  • C.C. TSCHERNING: Introduction to Functional Analysis with a View to its Applications in Approximation Theory, Lecture Notes, Second International Summer School in the Mountains, Ramsau, Austria, 1977a (in print).

    Google Scholar 

  • C.A. WAGNER, F.J. LERCH, J.E. BROWND and J.A. RICHARDSON: Improvement in the Geopotential Derived from Satellite and Surface Data (GEM 7 and 8), GSFC-Report X-921-76-20, 1976.

  • H. WOLF: Über Verallgemeinerte Kollokation, Zeitschrift für Vermessungswesen, Vol. 99, No. 11, pp. 475–478, 1974.

    Google Scholar 

  • H. WOLF: Zur Grundlegung der Kollokationsmethode, Zeitschrift für Vermessungswesen, Vol. 102, No. 6, pp. 237–239, 1977.

    Google Scholar 

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  1. Geodaetisk Institut, Gamlehave Allée 22, DK-2920, Charlottenlund, Danmark

    C. C. Tscherning

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Tscherning, C.C. Collocation and least squares methods as a tool for handling gravity field dependent data obtained through space research techniques. Bull. Geodesique 52, 199–212 (1978). https://doi.org/10.1007/BF02521773

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