Mid-term Report of IAG Special Commission 1
MATHEMATICAL AND PHYSICAL FOUNDATIONS OF GEODESY
for the period 1999-2001
by
Petr Holota
Research Institute of Geodesy, Topography and Cartography
250 66 Zdiby 98, Praha-vychod, Czech Republic
e-mail: holota@pecny.asu.cas.cz

1. Introduction

The Special Commission on Mathematical and Physical Foundations of Geodesy (CMPFG) was established by the International Association of Geodesy on the occasion of the 20th General Assembly of the International Union of Geodesy and Geophysics in Vienna in 1991. It expresses the need for a permanent structure working on the foundations of geodesy. The establishment of the special commission is essentially associated with the preparatory work done by K.-P. Schwarz (the president of Section IV at that time) and the Section IV Steering Committee.

The main objectives  of the special commission are the following:

• to encourage and promote research on the foundations of geodesy in any way possible;
• to publish, at least once every four years, comprehensive reviews of specific areas of active research in a form suitable for use in teaching as well as research reference;
• to actively promote interaction with other sciences;
• to closely cooperate with the special study groups in Section IV - General Theory and Methodology. (As an IAG structure the CMPFG belongs particularly to this section).

This formulation is short in its form but in reality it represents a challenging program that may be also found in the 2000 issue of The Geodesist's Handbook [Journal of Geodesy (2000), Volume 74, No. 1]. In addition one can read information and details concerning the CMPFG (including the bibliography) on the website of this special commission at the address: http://pecny.asu.cas.cz/IAG_SC1/

It is natural that the research program of the CMPFG represents a continuation of the activities developed already in the period of the last 8 years when E.W. Grafarend successfully chaired the special commission. The research program of the CMPFG mainly focuses on statistical problems in geodesy, numerical and approximation methods, geodetic boundary value problems, on problems in geometry and differential geodesy, relativity, cartography, on equilibrium reference models and also on the theory of orbits and dynamics of systems. 

In this field the CMPFG derives important driving impulses especially from the work of the IAG itself. As a minimum let us mention two problems that were discussed at a special plenary session held in Birmingham on the occasion of the 22nd General Assembly of the International Union of Geodesy and  Geophysics in 1999: 1) "Are our contemporary theoretical and computer models sufficient to handle the 1:10⁹ accuracy in frame realization, Earth rotation, positioning etc. consistently?"; - 2) "Can we be sure that sensor and/or model deficiencies do not enter into geophysical interpretation?"

The broad spectrum of research objectives is connected with a subdivision of the research program into specific tasks. In 1999 immediately upon approval of the CMPFG program by the IAG the following subcommissions were established: 

• Subcommission 1 "Statistic and Optimization"
  Chair: P. Xu (Japan)
• Working Group "Spatial statistics for geodetic science"
  Chair: B. Schaffrin (USA)
• Subcommission 2 "Numerical and Approximation Methods"
  Chair: W. Freeden (Germany)
• Subcommission 3 "Boundary Value Problems"
  Chair: R. Lehmann (Germany)
• Subcommission 4 "Geometry, Relativity, Cartography and GIS"
  Chair: V. Schwarze (Germany)
• Subcommission 5 "Hydrostatic/isostatic Earth's Reference Models"
  Chair: A.N. Marchenko (Ukraine)

The theory of orbits and dynamics of systems is an exception. In general problems that by nature have a tie to this topic are given a considerable attention in many branches of science. Here the topic was left within the framework of the special commission itself. It focuses on the interplay between mathematics (especially analysis) and applications that together with problems related to methods of integration, modelling, analysis of perturbations and qualitative aspects in the evolution of trajectories reach the field of space geodetic methods and inertial systems. After two years the original intention is associated with visible achievements. The work of the CMPFG members resulted in a number of very valuable contributions. They concern e.g. dynamic satellite geodesy on the torus; the relation between analytical and numerical integration in satellite geodesy; energy relations for the motion of satellites within the gravity field; asymptotic series in mathematics, celestial mechanics and physical geodesy; satellite geodesy on curved space-time manifolds; differential equations in inertial navigation systems etc. Considerable activities of members develop also in the filed of dedicated satellite mission and in a contact with IAG Special Commission 7.

2. Subcommissions

Also the subcommissions are very productive. It can be immediately seen from the bibliography of the CMPFG that is directly accessible on the website of the special commission (at the address given above and in the References). It contains a rich list of entries that document the research done by the members of the special commission in the reported period. (On the website the bibliography is an open material that is under a process of a permanent completion.)  

For page limit let us mention some highlights only. Subcommission 1 placed emphasis on areas such as the theory of (geo)inverse problems, nonconventional models for space applications, spatial information theory, global optimization methods and also the advancement of traditional topics. The IAG Executive Committee at its meeting in Nice, 2000 have decided to give Dr. Peiliang Xu (the chair of the subcommission) the IAG young authors award for his paper "Biases and the accuracy of, and an alternative to, discrete nonlinear filters", published in the Journal of Geodesy, Vol. 73(2000), pp. 35-46. Within the general discipline of statistics, methods which relate each "data point" to a location allow for an analysis with a spatial resolution that might otherwise be lost. The data do not need to be point data themselves, but could have been derived from a certain area by averaging. Key is that we have to deal with both probabilistic and spatial distributions. This in short is the field of research of the Working Group that develops its activities in a close cooperation with Subcommission 1. Members of the subcommission and the working group brought significant contributions to the IAG 1st International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS in  Zurich, March, 2001.

An intensive mathematical research oriented to problems in the representation and approximation of the Earth's gravitational potential, to problems in physical geodesy and in the treatment of modern space geodetic data was in the focus of Subcommission 2. The season is that one has to think of the geopotential as a "signal" in which the spectrum evolves over space in significant way. This space-evolution of the frequencies is not reflected in the Fourier transform in terms of non-space localizing spherical harmonics. Wavelet transforms are a counterpart. Therefore, aspects of constructive approximation, decorrelation, data compression etc. were treated within the wavelet theory. Moreover, an uncertainty principle was formulated and used as it gives an appropriate bound for the quantification of space and frequency properties of trial functions in geodesy. In the focus there were also combined models, where expansions in terms of spherical harmonics are combined with local methods, e.g. radial base function techniques as splines, wavelets, mass-points, finite elements etc. In the limited time span of the first two years the subcommission also significantly progressed in the methodology of the treatment of spaceborn observations. In addition to a number of presentations and entries in the bibliography an important contribution on "Multiscale modelling of GOCE data products" was prepared for the ESA International GOCE User Workshop held in Noordwijk in April, 2001.

Subcommission 3 focused on boundary value problems (BVP) in physical geodesy. They are essentially connected with the use of potential theory and the theory of partial differential equations in the determination of the gravity field and figure of the Earth. In the reported period the research carried out by the subcommission concentrated on the refinement of the solution of the standard problems and new mathematical models, on free-datum and multi-datum BVPs, as they arise from unknown height datums; on mixed BVPs and especially various types of altimetry-gravimetry problems with their capability to give a mathematical model for a combined use of different data on the boundary; on stochastic BVPs; overdetermined and constraint BVPs; BVPs on special surfaces and also on pseudo BVPs. The research covered also non-classical methods in the solution of BVPs, as variational methods  with their close tie to the concept of the so-called weak solution, boundary element techniques, various aspects in the use of ellipsoidal harmonics and other function bases. Within traditional concepts the role of the BVPs is rather well-known in physical geodesy, but nowadays the work of the subcommission is strongly influenced by new striking impulses. Among others they reflect the progress in the data collection, data accuracy, higher requirements on the accuracy of the solution and also a need for mathematical modelling associated with the use of modern technologies, as e.g. airborn gravimetry and dedicated satellite missions (a spacewise approach, Slepian's problem etc.). The results of the subcommission were clearly visible at the IAG International Symposium on Gravity, Geoid and Geodynamics 2000 in Banff, July/August, 2000 and also at the 26th General Assembly of the EGS in Nice, March, 2001.

Geometry oriented problems, relativity aspects, cartography and GIS define the field of interest of Subcommission 4. Here under geometry one understands the Marussi-Hotine approach to differential geodesy, foundations of Gaussian differential geodesy, geometry of plumblines as geodesics in conformal 3-manifould, Fermi's coordinates etc. Nevertheless the main progress was achieved in the use of the theory of relativity, in particular in the reformulation of geodetic measurement processes within the framework of general relativity. Here the metric tensor plays an important role and it was represented with respect to a set of appropriate charts. Using the words of the chairman, we knew that almost every quantity of interest in geodetic and geophysical applications refers to a geocentric, Earth-fixed coordinate system (chart). Therefore, the space-time metric with respect to an Earth-fixed chart was derived at first post Newtonian order. The field equations determining the terrestrial gravitational field were derived and its explicit representation was outlined. On this basis the impact of the results on the modelling of geodetic measurement process including space-time positioning scenarios as well as the high-precision gravitational filed estimation was discussed. Finally, results achieved in cartography and GIS were presented at the IAG 1st International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS in  Zurich, March, 2001.

Subcommission 5 is a completely new substructure of the CMPFG. Nevertheless it proved to be very active. In the reported period it attacked the construction of piecewise radial density models, stable determination of parameters of radial density models, variational problems and the interpretation of some reproducing kernels, it focused on the low-frequency Earth's gravity field and the evolution of the Earth's principal axes and moments of inertia completed with a canonical form of the solution. Some research was also oriented to incompressible fluid Earth, compressibility and vicoelestic perturbations. For the density recovery from seismic velocities the solution was based on three differential equations and the density function was separated into a hydrostatic (main) part and an additional small part due to chemical/phase inhomogenieties or superadiabatic temperatures. Some famous laws (Legendre-Laplace, Roche, Darwin, Gauss) were considered for radial density distribution in connection with the solution of the famous Clairaut, Poisson and Williamson-Adams differential equations. In the interpretation of reproducing kernels it was shown that the set of all suitable kernel functions may be interpreted as a finite sum of two point singularities (pole and dipole) and also straight line singularities. In addition an optimum point mass model of the global gravitational filed was compiled. 

3. Business Meeting in Banff

The CMPFG is an important discussion forum. This was evident from  the business meeting of the special commission organized in Banff on the occasion of the IAG International symposium "Gravity, Geoid and Geodynamics 2000". A circular letter distributed by the special commission chairman well before the meeting proved to be a stimulus that met with a good response. "What you think is the most urgent problem to be solved related to the foundations of geodesy" this was a key question formulated by C.C. Tscherning and circulated with the letter. It turned out that what is natural. The response reflects the impact of the future or up-coming satellite missions. In particular the following urgent problems were mentioned (in the formulations by R. Rummel):

• How to deal in a proper way with the actual Earth boundary when bringing down the high resolution gravity information from GOCE from satellite altitude to the Earth's surface? Should one simply apply some kind of topographic correction or are there better and/or more correct ways?
• A mission like GRACE drifts (in the course of the entire mission length) down from, say 500 km to 300 km. While going down the gravity field is sampled in a changing manner and at the same time it drifts through several regimes of resonance. At the same time one tries to recover the temporal variations of the gravity field. Thus one is faced with a very complicated sampling/aliasing problem, which to my best knowledge is not sorted out so far.
• A similar problem in altimetry which certainly affects, for example, estimates of sea level change. Again the satellite samples time variable effects such as tides in a very complicated time and space pattern. What is the aliasing situation there, what could be done to improve it?
• In real world satellite sensors such as accelerometers or gradiometers do not show a normally distributed noise behavior but drifts and a similar effects cause systematic distortions which result in e.g. 1/f  (f = frequency) behavior. We have taken into account these things but in an ad hoc manner. Could one work on stochastic models on the sphere for processes with less favorable error behavior?
• Currently several groups try to determine center-of-mass changes of the Earth from the analysis of global tracking data. What is the proper formulation of datum definition and S-transformation for a real deformable earth with moving plates, how should a center-of-mass change determination be correctly formulated from the theoretical point of view?

K.-H. Ilk expressed another view. He pointed out three problem areas related to the satellite missions CHAMP, GRACE and COCE: - analysis of the observation system; - modelling and data analysis aspects; - applications in geosciences, oceanography, climate change studies and other interdisciplinary research topics. 

In addition M. Vermeer suggested, loosely speaking "best practices" and the use of common sense in connection with the use of modern techniques in geodesy. Using his words, we know that in traditional geodesy there were these common sense rules such as "working from the large to the small" and many many more. With new techniques, and the availability of fast computers and complex theories, sometimes it seems that common sense has been a bit forgotten.

The subsequent discussion at the business meeting concerned some reflections on the running process towards the new structure of the IAG. B. Heck, the president of IAG Section IV outlined the key aspects that motivate this initiative. His information were then amplified by F. Sanso, the IAG president who first paid a considerable attention to the work of the CMPFG itself and than focused on a detailed explanation of the principles and actions that are most frequently discussed within the IAG executive in preparing the concepts for the new IAG structure. The business meeting of the CMPFG was well attended, not only by the members, but also by a number of participants of the Banff symposium on Gravity, Geoid and Geodynamics 2000. The CMPFG will hold its future business meeting in Budapest, concurrently with the IAG Scientific Assembly, 2-8 September 2001. For 2002 the CMPFG prepares an active participation in the Hotine-Marussi symposium on mathematical geodesy which by tradition will be held in Italy under the sponsorship of the IAG.

References:  see please  http://pecny.asu.cas.cz/IAG_SC1/

Acknowledgements. Concluding this brief report, I wish to express my sincere thanks to all my colleagues from IAG Special Commission SC1 for excellent cooperation and all the results achieved that often mean months or years of a great endeavor and devoted work. Much success in your further work!