Table Of ContentInternational Association
of Geodesy Symposia
F emando Sanso, Series Editor
Springer-Verlag Berlin Heidelberg GmbH
Internationa I Association
of Geodesy Symposia
Femando Sanso, Series Editor
Symposium JOI: Global and Regional Geodynamics
Symposium 102: Global Positioning System: An Overview
Symposium 103: Gravity, Gradiometry, and Gravimetry
Symposium 104: Sea Surface Topography and the Geoid
Symposium J05: Earth Rotation and Coordinate Reference Frames
Symposium 106: Determination of the Geoid: Present and Future
Symposium 107: Kinematic Systems in Geodesy, Surveying, and Remote Sensing
Symposium J08: Application of Geodesy to Engineering
Symposium J09: Permanent Satellite Tracking Networks for Geodesy and Geodynamics
Symposium 110: From Mars to Greenland: Charting Gravity with Space and Airborne Instruments
Symposium 111: Recent Geodetic and Gravimetric Research in Latin America
Symposium 112: Geodesy and Physics of the Earth: Geodetic Contributions to Geodynamics
Symposium 113: Gravity and Geoid
Symposium 114: Geodetic Theory Today
Symposium 115: GPS Trends in Precise Terrestrial, Airborne, and Spaceborne Applications
Symposium 116: Global Gravity Field and Its Temporal Variations
Symposium 117: Gravity, Geoid and Marine Geodesy
Symposium 118: Advances in Positioning and Reference Frames
Symposium 119: Geodesy on the Move
Symposium 120: Towards an Integrated Global Geodetic Observation System (IGGOS)
Symposium 121: Geodesy Beyond 2000: The Challenges of the First Decade
Symposium 122: IV Hotine-Marussi Symposium on Mathematical Geodesy
IV Hotine-Marussi Symposium
on Mathematical Geodesy
Edited by Battista Benciolini
Convened by Battista Benciolini, Willi Freeden
and Roland Klees
Trento, Italy
September 14-17, 1998
, Springer
Volume Editor Series Editor
Professor Dr. Battista Benciolini Prof. Dr. Fernando Sansa
University of Trento Polytechnic of Milan
Department of Civil and Department of Hydraulic,
Environmental Engineering Environmental and Surveying
Via Mesiano, 77 Engineering
38050 Trento Piazza Leonardo da Vinei, 32
ltaly 20133 Milan
E-mail: [email protected] Italy
ISSN 0939-9585
ISBN 978-3-642-62574-9
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Hotine-Marussi Symposium on Mathematical Geodesy (4, 1998, Trento): IV Hotine-Marussi Symposium on Mathematical Geodesy: Trento,
Italy; September 14-17, 1998 / ed. by Battista Benciolini. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris;
Singapore; Tokyo: Springer, 2001
(International Association of Geodesy symposia; Symposium 122)
ISBN 978-3-642-62574-9 ISBN 978-3-642-56677-6 (eBook)
DOI 10.1007/978-3-642-56677-6
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Contents
Report on the Symposium
Riccardo Barzaghi ................................................................................ 1
The Molodensky Scalar Boundary Value Problem in Spherical Coordinates: a New Result (abstract
only)
Jesus Otero, Fernando Sanso ...................................................................... 3
The Slepian Problem on the Sphere (abstract only)
A.Albertella, F.Sanso, N.Sneeuw .................................................................. 4
White Noise Stochastic BVP's and Cimmino's Theory
F. Sanso, G. Venuti ................................................................................ 5
Simulation of the Goce Gravity Field Mission
Nico Sneeuw, Raul Dorobantu, Christian Gerlach, Jiirgen Miiller, Helmut Oberndorfer, Reiner Rum-
mel, Radboud Koop, Pieter Visser, Peter Hoyng, Avri Selig, Martijn Smit ........................ 14
Quality Improvement of Global Gravity Field Models by Combining Satellite Gradiometry and Air
borne Gravimetry
Johannes Bouman, Radboud Koop ............................................................... 21
On the Determination of Geopotential Differences from Satellite-to-Satellite Tracking
Christopher Jekeli ............................................................................... 33
On the Topographic Effects of Helmert's Method of Condensation
Lars E. Sjoberg .................................................................................. 40
Distance Measurement with Electromagnetic Wave Dispersion
Michele Caputo .................................................................................. 48
A Global Topographic-Isostatic Model Based on a Loading Theory
Wenke Sun, Lars E. Sjoberg ..................................................................... 61
Stochastic Modelling of Non-stationary Smooth Phenomena
V. Tornatore, F. Migliaccio ...................................................................... 77
Deformation Detection According to a Bayesian Approach
B. Betti, F. Sanso, M. Crespi ..................................................................... 83
Block Elimination and Weight Matrices (abstract only)
Kai Borre ........................................................................................ 89
Construction of An-isotropic Covariance-Functions Using Sums of Riesz-Representers (abstract only)
C.C. Tscherning ................................................................................... 90
New Covariance Models for Local Applications of Collocation
R. Barzaghi, A. Borghi, G. Sona ................................................................. 91
Approximation of Harmonic Covariance Functions on the Sphere by non Harmonic Locally Supported
Ones (abstract only)
G. Moreaux, C.C. Tscherning, F. Sanso ......................................................... 102
Integration of a Priori Gravity Field Models in Boundary Element Formulations to Geodetic Boundary
Value Problems
Roland Klees, Riidiger Lehmann ................................................................ 103
v
On the Solution of the Inverse Stokes Problem Including Ellipsoidal Effects (abstract only)
Bernhard Heck .................................................................................. 110
The Spheroidal Stokes Boundary Value Problem (extended abstract)
A. Ardalan, E. W. Grafarend, M.G. Sideris ..................................................... 111
The Hotine Rectified Skew Orthomorphic Projection (Oblique Mercator Projection) Revisited (abstract
only)
E. W. Grafarend, J. Engels ..................................................................... 122
A Solution of the Korn-Lichtenstein Equations of Conformal Mapping which Directly Generates El
lipsoidal GauB-Kriiger Conformal Coordinates or the Transverse Mercator Projection (abstract only)
R. Syffus ........................................................................................ 123
An Iterative Solution of the Scalar Free Boundary Value Problem and the Choice of the Reference
Surface (abstract only)
Kurt Seitz ...................................................................................... 124
On the Use of Galerkin's Method in the Solution of the Geodetic Boundary Value Problem (abstract
only)
Petr Holota ..................................................................................... 125
Numerical Aspects of Altimetry-Gravimetry Problems
Riidiger Lehmann .............................................................................. 126
Fast Numerical Solution of the Vector Molodensky Problem
Roland Klees, Christian Lage, Christoph Schwab ................................................ 137
Softly Unbiased Prediction
Part 2: The Random Effects Model
Burkhard Schaffrin ............................................................................. 145
Fractale, Brownian Motion, Wiener Process, Krige Variogram, Kolmogorov Structure Function:
Geodetic Examples (abstract only)
E. W. Grafarend ................................................................................ 146
The General Linear Model - a Solution by Means of the Condition Adjustment
Gabriel Nkuite, Jan van Mierlo ................................................................. 147
Iteration Procedure for Evaluating High Degree Potential Coefficients from Gravity Data
M.S. Petrovskaya ............................................................................... 158
On the Adjustment of Combined GPS/Levelling/Geoid Networks (abstract only)
Christopher Kotsakis, Michael G. Sideris ........................................................ 165
Alternative Formulas for Terrain Reduction and Comparison with Existing Formulas
Hussein A. Abd-Elmotaal ....................................................................... 166
Error Detection in GPS Observations by Means of Multi-Process Models
Enrik F. Thomsen .............................................................................. 172
A Concept for Satellite Orbit Sensitivity Analysis (abstract only)
Erik W. Grafarend, Christof Schafer ............................................................ 181
Index of Authors ................................................................................ 182
List of Participants .............................................................................. 183
vi
Report on the Symposium
The IV Hotine-Marussi Symposium on Mathematical Geodesy was held in Trento
from September 14th to September 17th, 1998. It continues the long-standing tradition
of symposia that was initiated by Martin Hotine and Antonio Marussi in Venice in 1959.
It is the 12th symposium of this series and the fourth one associated with the names of
Hotine and Marussi. The arguments treated were connected with the theoretical and
methodological aspects of geodesy. Infact, these kind of symposia are usually devoted
to the development of the founding aspects of geodesy. The principal themes that were
discussed during this last meeting have been related to the boundary value problems,
to the satellite geodesy and to the stochastic methods in geodesy. The boundary value
problems were tackled both from the theoretical and the numerical points of view. New
advancements were presented in the Molodensky scalar boundary value problem as well
as in the application of the Slepian theory for the sphere. Furthermore, a theoretical
scheme for handling white noise stochastic boundary value problem was illustrated, the
importance of ellipsoidal effects were investigated in inverse Stokes problem and the
solution of the spheroidal Stokes problem was presented. Methods for computing the
solutions of the boundary value problems were also illustrated. The boundary element
formulations were introduced and numerically tested in geodetic boundary value prob
lems; the use of Galerkins method has been proposed as a tool for solving geodetic
boundary value problems as well as iterative solution applied to the scalar boundary
value problem. Numerical investigations were also presented, with particular atten
tion to the methodological aspects involved in the computations. Satellite geodesy also
entered in the framework of this Hotinee-Marussi Symposium. The main arguments
treated were the simulation of the GOCE mission, the integration of satellite gradiom
etry and airborne gravity for gravity filed recovery, satellite-to-satellite tracking and
orbit sensitivity analysis. A relevant part of the Symposium was then devoted to the
stochastic methods applied to geodesy. The covariance modelling problem both on
a global and on a local scale was presented in three different papers. Specific argu
ments related to the stochastic modelling of non stationary fields, to Bayesian inference
applied to deformation analysis and to the joint adjustment of GPS/levelling/geoid
observations were also discussed. Furthermore, more theoretical papers on stochas
tic process theory were presented on unbiased estimators, on general stochastic and
non stochastic estimation models applied to geodesy and on least squares estimation
in general linear model. These three main parts of the symposium were completed
by other contributions related to important geodetic topics. Topographic reduction
methods were revisited; particularly, the Helmert condensation method and the global
topographic-isostatic model were discussed. ~~umerical tests with alternative formulas
for terrain reduction were also presented. Two papers on the reduction of gravity ob
servations and on the asymptotic behaviour of high degree potential coefficients gave
1
important contributions to these two basic arguments. Error detection in GPS oserva
tions, a discussion on some topics of map projection and matrix algebra complete the
list of the subjects that were presented at the meeting. From this short summary it
is possible to assume the relevance of the IV Hotine-Marussi Symposium; all the main
topics of geodesy were touched and important steps forward have been done on many
theoretical aspects. The deep and interesting discussions which were carried out during
the meeting proved the interest and the relevance of the presented items.
Riccardo Barzaghi
Previous symposia on Mathematical Geodesy
1. 1st Symposiwn on Three Dimensional Geodesy, Venezia 1959
2. 2nd Symposium on Three Dimensional Geodesy, Cortina d'Ampezzo 1962
3. 3rd Symposiwn on Mathematical Geodesy, Torino 1965
4. 1st Hotine Symposium on Mathematical Geodesy, Trieste 1969
5. 2nd Hotine Symposium on Mathematical Geodesy, Firenze 1972
6. 3rd Hotine Symposium on Mathematical Geodesy, Siena 1976
7. 4th Hotine Symposium on Mathematical Geodesy, Assisi 1978
8. 5th Hotine Symposium on Mathematical Geodesy, Como 1981
9. 1st Hotine-Marussi Symposiwn on Mathematical Geodesy, Roma 1985
10. 2nd Hotine-Marussi Symposiwn on Mathematical Geodesy, Pisa 1990
11. 3rd Hotine-Marussi Symposium on Mathematical Geodesy, L'Aquila 1994
2
The Molodensky Scalar Boundary Value Problem
in Spherical Coordinates: a New Result
Je Slis Otero
Istituto de Astronomia y Geodesia, Facultad de Ciencias Matematicas, Universidad Complutense -
Madrid
Fernando Sanso
DIIAR -Politecnico di Milano - Piazza L. da Vinci 32 - Milano - Italy
Abstract
The theory and analysis of free boundary value One of the basic tools to get this result
problems has attained a certain popularity in has been the intermediate Schauder estimates
Geodesy in the last 20 years, being considered for oblique derivative problems obtained by
as a sort of reference theory providing a sound G.M.Lieberman. The drawback of what we have
scientific background to the problem of determin proved is as always the use of existence theorems
ing the figure of the earth and its gravity field. in the small, which do not say much about how
In this framework an effort has been done to "close" should be the data to the reference field
come to a solution of the most relevant prob in order to guarantee existence and uniqueness of
lem, namely the so-called scalar geodetic bound solution. Furthermore, as in many other works
ary value problem, under the most general condi in this field, the result is obtained here in the
tions of regularity of the boundary values. In fact so-called spherical approximation for the refer
the basic theorem proven here guarantees a solu ence field and one could think that a better result
tion for the (modified) Molodensky scalar bound could be achieved by working with an ellipsoidal
ary value problem under the conditions that (on approximation for the reference field; this could
the boundary)the gravity belongs to He" and the be a line of research worth some effort in future.
gravitational potential to Hl+cr•
3
The Slepian Problem on the Sphere
A.Albertella, F .Sanso
DIIAR - Politecnico di Milano - Piazza L. da Vinci 32 - Milano - Italy
N.Sneeuw
Institut fuer Astronomische und Physikalische Geodaesie, Technische Universitaet Muenchen -
Arcistrasse 21 - Muenchen - Germany
Abstract
The Slepian problem consists in determining The relation between the new basis and the or
a sequence of functions that constitute an or dinary spherical harmonic basis can beexplicitely
thonormal basis of a subset of]R (or ]R2) con written and numerically studied.
centrating the maximum information in the sub What turns out is that this tool is a natural so
space of square integrable function with band lution to capture the maximum amount of infor
finite spectrum. The same problem can be stated mation from a non-polar gradiometric mission.
and solved on the sphere.
4
