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I - Objectives

The Special Commission on Mathematical and Physical Foundations of Geodesy (CMPFG) was established by the International Association of Geodesy during the XXth General Assembly of the International Union of Geodesy and Geophysics in Vienna in 1991. It is Special Commission 1 in Section IV of the IAG and 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 decision taken by the Section IV Steering Committee. For two periods the special commission was then successfully chaired by E.W. Grafarend.

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.

The research program of the CMPFG envisaged for the next four years will mainly focus on statistical problems in geodesy, numerical and approximation methods, geodetic boundary value problems, on problems in geometry and differential geodesy, relativity and cartography, on equilibrium reference models and also on the theory of orbits and dynamics of systems.

This broad spectrum of research objectives is connected with a subdivision of the research program into specific tasks. The majority of them will be assigned to subcommissions within the CMPFG, which in a certain measure will continue that structure of the CMPFG that proved to be efficient in the last period.

The theory of orbits and dynamics of systems is an exception. The experience from the last period supports our opinion that the topic should be treated within the special commission itself, rather than in a subcommission. The intention is to embody problems related to integration methods, modelling, to the analysis of perturbations and their causality, but also to qualitative aspects in the temporal evolution of trajectories.

Workshops of the CMPFG will be organized at least once between general assemblies and specialists from other disciplines will be invited to contribute to these workshops. Representation on scientific bodies which can contribute to the work of the CMPFG or which should be aware of the research results will be sought on mutual basis.
   
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II - Structure, Ex-officio members and Individuals

Membership of the CMPFG is restricted by the IAG by-laws. One third of members is replaced every four years. Chairmen of special study groups within Section IV are automatically members of the CMPFG. The Section IV President as well as Section IV Secretaries are ex officio members. Other members are proposed by the Special Commission President and approved by the Section President.

Subcommissions and working groups will be formed by the CMPFG as deemed appropriate to study defined aspects in its field. In 1999 an updated structure and program of the CMPFG were approved by the IAG. This includes also an approval of the following subcommissions and the working group together with the respective leadership. The first four subcommissions will continue and further develop the work already done in the last period. The last subcommission is new:

Subcommission 1 "Statistics and Optimization"
Chairman: P. Xu (Japan)

Working Group "Spatial statistics for geodetic science"
Chairman: B. Schaffrin (USA)

Subcommission 2 "Numerical and Approximation Methods"
Chairman: W. Freeden (Germany),  http://www.mathematik.uni-kl.de/~wwwgeo

Subcommission 3 "Boundary Value Problems"
Chairman: R. Lehmann (Germany)

Subcommission 4 "Geometry, Relativity, Cartography and GIS"
Chairman: V. Schwarze (Germany)

Subcommission 5 "Hydrostatic/isostatic Earth's Reference Models"
Chairman: A.N. Marchenko (Ukraine),  http://people.polynet.lviv.ua/sc5/

The following distinguished scientists have been invited to work in the CMPFG, in its individual Subcommissions and in its Working Group:

Ex officio Members

B. Heck (Germany)
President of Section IV

C. Jekeli (USA),  http://www-ceg.eng.ohio-state.edu/~cjekeli/ssg4-191.htm
Secretary of Section IV,
Chairman SSG 4.191: Theory of fundamental height systems

Yuanxi Yang (China)
Secretary of Section IV

W. Keller (Germany),  http://www.uni-stuttgart.de/iag
Chairman SSG 4.187: Wavelets in geodesy and geodynamics

G. Strykowski (Denmark),  http://research.kms.dk/~ssg4188/study_group/index.html
Chairman SSG 4.188: Mass density from integrated inverse gravity modelling

D. Wolf (Germany)
Chairman SSG 4.189: Dynamic theories of deformation and gravity field

H. Kutterer (Germany),  http://www.dgfi.badw.de/ssg4.190/welcome.html
Chairman SSG 4.190: Non-probabilistic assessment in geodetic data analysis

E.W. Grafarend (Germany),  http://www.uni-stuttgart.de/gi/
Chairman SSG 4.195: Fractal geometry in geodesy
(!! New  - The proposal for this SSG was adopted at the Executive Committee meeting in Budapest, 
September 4, 2001)

Individuals

M. Bougeard (France)
C. Cui (Germany)
A. Dermanis (Greece)
E.W. Grafarend (Germany),  http://www.uni-stuttgart.de/gi/
E. Groten (Germany)
K.H. Ilk (Germany),  http://www.geod.uni-bonn.de/SC7/index.html
J. Janak (Slovak Republic)
R. Klees (The Netherlands)
L. Kubacek (Czech Republic)
Z. Martinec (Czech Republic)
G. Moreaux (Denmark)
J. Otero (Spain)
M. Petrovskaya (Russia)
R. Rummel (Germany)
F. Sacerdote (Italy)
K.P. Schwarz (Canada)
M. Sideris (Canada)
N. Sneeuw (Germany)
H. Sünkel (Austria)
L. Svensson (Sweden)
P. Teunissen (The Netherlands)
C.C. Tscherning (Denmark)
P. Vanicek (Canada)
G. Venuti (Italy)
M. Vermeer (Finland)
R.J. You (Taiwan)

Maintenance of liaisons with related activities: 
IUGG Committee on Mathematical Geophysics: M. Vermeer (Finland)
   
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III - Subcommissions, Additional and Corresponding Members

Subcommission 1 "Statistics and Optimization"
Chairman: Peiliang Xu (Japan)

Research Program

After discussing the goals of this Subcommission for the next four years with various (old and new) members, we felt that the scope should be extended somewhat to also include optimization issues, and that particular emphasis should be placed on areas such as the Theory of Inverse Problems, Nonconventional Models for Space Applications, Spatial Information Theory, Global Optimization Methods, and the Advancement of Traditional Topics. The extended scope should be documented in the augmented name of the Subcommission.

We anticipate particularly fruitful, original and fundamental contributions to the aforementioned topics by our regular as well as corresponding members. By no means, however, do we want to restrict our creativity to the areas as described; on the contrary, we welcome all participants who distinguish themselves by a certain degree of curiosity, novelty, and inventiveness while keeping a sense for systematic issues. In this spirit we offer the following agenda:

(i) Inverse Problem Theory. We strongly encourage all novel theoretical work on improved estimation, comparison, testing, from all kinds of points of view. However, our final goal is to find an estimate that is the closest to the true but unknown field, to check or to confirm whether a certain (geo)physical hypothesis can be tolerated by the collected data, and to provide a guide for further improvement and theory development.

(ii) Models for Space Applications. We are used to a linear or nonlinear model, with all the model parameters being real-valued and with all the noise being assumed to be additive. These assumptions are not acceptable in the time of space techniques. For GPS, we have to deal with integer unknowns; for SAR, we have to deal with multiplicative noise. Thus we strongly recommend all kinds of statistical work, for instance, model parameter estimation, variance-component estimation, and testing, on these new types of observation models.

(iii) Spatial Information Theory. In this regard, we will encourage any research on probabilistic models for manifolds that are of special interest in the Earth Sciences, for instance, directions, referentials and tensors. The error model to assess spatial informatic data is also far from satisfactory. Hypothesis testing techniques should be developed correspondingly. Thus all novel ideas to set spatial data on a solid foundation are most welcome.

(iv) Global Optimization Method. Global optimization should also be an important area to work, since almost all practically important models are nonlinear. On the other hand, this fascinating area is still waiting for great development. Again, we would be pleased to see the contributions from the Earth scientists.

(v) Traditional Topics. In addition to the above mentioned areas, let us not forget those traditional topics, stochastic boundary value problem; linear and/or nonlinear model; linear/nonlinear dynamical models; observability and invariance, to name a few, but keeping in mind that we are working on the foundation, or the scientific side of geodesy.

Beside the ex officio and individual members of the CMPFG the following distinguished scientists have been invited to work in Subcommission 1 as

Additional Members

J.A.R. Blais (Canada)
Y. Kagan (USA)
M. Sambridge (Australia)
Y.X. Yang (China)
S.D. Pagiatakis (Canada)

Corresponding Members

O. Abrikosov (Ukraine)
M.Y. Markuze (Russia)
C. Shi (China)
   
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Working Group "Spatial Statistics for Geodetic Science"
Chairman: B. Schaffrin (USA)

Research Program

Within the general discipline of statistics, methods which relate each "data point" to a location allow for an analysis with 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, however, that we have to deal with both probabilistic and spatial distributions.

In many cases, even for nonergodic or nonisotropic processes, the spatial connection supports the statistical analysis and vice versa. This has been known in geodetic science for a long time, and various techniques have been developed over the years to take this fact into account. Now is the time to find a unifying framework with respect to which every single procedure can be interpreted and compared to others as far as their advantages and shortcomings are concerned.

There is also ample room for new developments, particularly in view of increased use of Geographical Information Systems for geodetic purposes. Among the topics of interest are:

- the role of point, line and block data;
- the data quality (accuracy, reliability, correlations, etc.) and their spatial distribution;
- geostatistical methods for TINs (Triangular Irregular Networks) and raster data;
- the use of homeograms versus variogram and covariance function, respectively "reproducing kernel";
- hypothesis testing for spatial regions;
- spatial support for ill-posed problems;
- applications to laser scanning and INSAR satellite data.

Beside the ex officio and individual members of the CMPFG the following distinguished scientists have been invited to work in this Working Group as

Additional Members

Shaofeng Bian (China)
W. Caspary (Germany)
K. Kraus (Austria)
S. Meier (Germany)

Corresponding Members

N. Cressie (USA)
J. Pilz (Austria)
   
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Subcommission 2 "Numerical and Approximation Methods"
Chairman: W. Freeden (Germany),  http://www.mathematik.uni-kl.de/~wwwgeo

Research Program

- The spherical representation of the Earth's external gravitational potential by means of spherical harmonics is essential to solving many problems in today's physical geodesy. In future research, however, spherical harmonic expansions may not be the most natural or useful way. The season is that one has to think of the potential as a "signal" in which the spectrum evolves over space in significant way. In other words, at each point the potential refers to a certain combination of frequencies; and in dependence of the mass distribution inside the Earth, the contribution to the frequencies and, therefore, the frequencies themselves are spatially changing. This space-evolution of the frequencies is not reflected in the Fourier transform in terms of non-space localizing spherical harmonics, at least not directly. In conclusion, specific methods of achieving a space-dependent frequency analysis in geopotential determination have to be proposed. Known from recent publications are windowed Fourier transform and wavelet transforms. The results of the wavelet theory include aspects of constructive approximation (i.e. basis property), decorrelation, fast computation, data compression, etc. In future, biorthogonalization processes, lifting scheme, regularization by multiresolution, fast evaluation techniques, etc. should be tested numerically.

- The uncertainty principle gives appropriate bounds for the quantification of space and frequency properties of trial functions in geodesy. The main statement is that sharp localization in space and in frequency are mutually exclusive. Extremal members in the space/frequency relation are spherical harmonics and the Dirac function(al)s. Members "in between" are bandlimited and non-bandlimited kernel functions (in the jargon of constructive approximation: radial basis functions). Although all these trial functions may be used to approximate the Earth's gravitational potential (in a suitable topology), their different numerical properties must be investigated in more detail. In this respect the uncertainty principle should be consulted as an appropriate tool for quantitative decision. In addition, new types of trial functions like the Shannon kernel, smoothed Shannon kernels, the Abel-Poisson kernel, the Gauss-Weierstrass kernel, the Tikhonov kernel and locally supported kernels have to be discussed in several aspects.

- Seen from numerical point of view, the use of spherical harmonic expansions of higher and higher degree for the determination of the gravitational filed and the figure of the Earth meets with essential problems for several reasons (e.g. Nyquist rate, uncertainty principle). It is not appropriate to model local behaviour by non-localizing functions. The polynomial nature of these functions causes severe numerical difficulties due to their oscillatory character. The evaluation of high order spherical harmonics may be unstable. Therefore one should concentrate on combined models, where expansions in terms of spherical harmonics are combined with local methods, e.g. radial basis function techniques as splines, wavelets, mass-points, finite elements, etc.

- Isotropy preserving approximation methods must be compared with non-preserving techniques for geodetic obsevables of scalar, vectorial and tensorial nature.

- Future spaceborne observations combined with terrestrial and airborne activities will provide huge datasets of the order of millions of data. Standard mathematical theory and numerical methods are not at all adequate for the solution of data systems with a structure such as this, because these methods are simply not adapted to the specific character of the spaceborne problems. They quickly reach their capacity limit even on very powerful computers. A reconstruction of the gravitational field from future data material requires much more: Economical and efficient combination of large data sets of different types and from different heights. In particular, the vectorial and tensorial nature of satellite data in satellite-to-satellite tracking and satellite gravity gradiometry, respectively, is a challenging problem in this connection. Future numerical procedures should be able to handle such problems automatically.

- The demanded high accuracy of future models has to take into account the true surface of the Earth. Therefore, numerical methods usable for non-spherical boundaries should be an important goal for future developments. This includes finite difference methods, finite element methods, all boundary element techniques as well as sphere-oriented methods (like harmonic splines or harmonic wavelets). For the numerical efficiency, the use of multilevel or multiresolution techniques is indispensable.

Beside the ex officio and individual members of the CMPFG the following distinguished scientists will be invited to work in Subcommission 2 as

Additional Members

C. Jekeli (USA)
W. Keller (Germany)
J. Mason (England)
V. Michel (Germany)
F.J. Narcowich (USA)
Z. Nashed (USA)
E. Schock (Germany)
L. Schumaker (USA)
J. Sloan (Austria)
G. Steidl (Germany)
L. Sjöberg (Sweden)
L. Svensson (Sweden)
C.C. Tscherning (Denmark)
J. Ward (USA)
   
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Subcommission 3 "Boundary Value Problems"
Chairman: R. Lehmann (Germany)

Research Program

The following geodetic boundary value problems (BVPs) will be considered:

- Pseudo-BVPs with boundary conditions relating to different  boundaries
- Free-datum and multi-datum BVPs, as they arise from unknown height  datums and/or unknown local datums
- Mixed BVPs, e.g. various types of altimetry-gravimetry problems
- Stochastic BVPs with stochastic boundary conditions and/or  stochastic boundary surface
- Overdetermined and constraint BVPs, particularly with a truncated  global spherical harmonics expansion as constraint for the solution
- Gravitational BVPs, where the centrifugal part of the Earth's  gravity field has been reduced from the boundary conditions
- BVPs on special surfaces (ellipsoids, etc.)

Program of activities

- Further development of the hierarchy of BVPs and adaptation to the changing and increasingly complex world of geodetic measurements
- Theoretical properties of geodetic BVPs in advanced formulations
- Analytical versus numerical solutions of geodetic BVPs
- Application of advanced mathematical tools for the solution of  BVPs (e.g. the theory of wavelets)
- Numerical case studies for testing and comparison purposes

Beside the ex officio and individual members of the CMPFG the following distinguished scientists have been invited to work in Subcommission 3 as

Additional Members

M. Günter (Germany)
Yu.M. Neyman (Russia)
Yu.A. Rozanov (Russia)
N. Weck (Germany)
W. Wendland (Germany)
K.J. Witsch (Germany)
A.I. Yanushauskas (Lithuania)
   
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Subcommission 4 "Geometry, Relativity, Cartography, GIS"
Chairman: V. Schwarze (Germany)

Research Program

Geometry

- the generalized Marussi-Hotine approach to differential geodesy, including schemes for integrating the Hotine-Marussi equations
- conceptual foundations of Gaussian differential geodesy
- further investigations of the geometry of plumblines (orthogonal trajectories of a family of equipotential surfaces), plumblines as  geodesics in a conformally flat 3-manifold
- Fermi coordinates on a biaxial ellipsoid of revolution as a generalization of the Soldner coordinates, the deviation equation

Relativity

- definition of Earth-fixed and slowly non-rotating reference systems within the framework of General relativity
- consistent modelling of geodetic measurement techniques within the  first post-Newtonian approximation
- analysis of the gravitational field of the Earth within the first post-Newtonian approximation (orbit perturbations and gravity  gradients in a local pseudo-orthogonal frame)
- propagation of electro-magnetic signals travelling through curved  space-time from a satellite to the Earth's surface in geometric-optical approximation for a refractive, dispersive and  magnetized medium

Cartography

- curvilinear datum transformation
- the Earth's topographic surface as a 2-manifold and its embedding in  an Euclidean 3-space, geodesics on the Earth's topographic surface and its Delaunay triangulation, map projections of the Earth's  topographic surface
- map projections of the geoid (Law of the Sea) in  spheroidal-spherical harmonic series
- map projections of an ellipsoid of revolution: the Hotine oblique Mercator projection, pseudo-cylindrical/equiareal projections of an ellipsoid of revolution, the triple map projection: The Earth's topographic surface, ellipsoid of revolution, plane; map projections  based on the second fundamental form of a surface
- map projections of a space-time manifold; analysis of various space-time charts from the viewpoint of cartography

Geodetic Information Systems (GIS)

- representation of the terrestrial gravitational field and the  Earth's surface
- Geodetic coordinate systems as a basis of GIS; quality control

Beside the ex officio and individual members of the CMPFG the following distinguished scientists have been invited to work in Subcommission 4 as

Additional Members

S.A. Klioner (Germany)
D. Milbert (USA)
M. Schmidt (Germany)
   
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Subcommission 5 "Hydrostatic/isostatic Earth's Reference Models"
Chairman: A.N. Marchenko (Ukraine),  http://people.polynet.lviv.ua/sc5/

Research Program

The problem of the standard Earth's density model was formulated by the IAG in 1971 and some models (as e.g. PREM, 1981) were produced by geophysicists. Later also temporal variations of low-degree harmonic coefficients were observed.

At present a logical step is to collect and integrate available data of geodetic, geophysical and astronomical nature (as e.g. fundamental geodetic constants, seismic velocities, temporal variations, etc.) for the construction of (1D and 3D) global density models that can also serve as a reference for subsequent interpretations of the gravity field.

Obviously, an immediate way of constructing Earth's reference models can be essentially associated with an analysis of solutions of the famous Clairaut, Poisson, Williamson-Adams and other differential equations which corresponds to the so-called hydrostatic/adiabatic Earth (time-independent version). In this step the work should result in a suitable parametrization of the continuous and also piecewise continuous density distribution.

The second step then consists of a transformation of an hydrostatic/adiabatic model into a hydrostatic/adiabatic/isostatic Earth's model in a global scale. Obviously, the time-dependent Earth's model is the most difficult case. It requires investigations of existence, uniqueness and stability problems of solutions to an inverse time-dependent geodetic/geophysical problem.

The main task of the Subcommission is the evolution of a hydrostatic/isostatic Earth's density model for the interpretation of the global gravity field. In particular the research will focus on:

- the integration of available data (including fundamental geodetic constants and seismic data, as supporting information) and the stratification of the Earth;
- solutions of differential equations governing hydrostatic/adiabatic/isostatic models and their parameterization.
- piecewise hydrostatic/adiabatic Earth's models;
- the estimation of isostatic radial deviations from the hydrostatic/adiabatic Earth's model;
- deviations of the real Earth from a hydrostatic/isostatic reference model;
- temporal variations and a space-time mass density distribution.

The first three items have a dual character. The solution of the respective inverse problem, which essentially is of non-linear nature, yields a hydrostatic/adiabatic Earth's model. This may then be used for further linearization and investigations of occurring deviations, thus also in the solution of the remaining tasks. Special attention will be given to the last item.

One of the first opportunities for the Subcommission to meet are the International Conferences which will be organized by the State University "Lviv Polytechnic" in Lviv (Ukraine), April, 2000 and in Alushta (Crimea, Ukraine), September, 2000.

Beside the ex officio and individual members of the CMPFG the following distinguished scientists have been invited to work in Subcommission 5 as

Additional Members

O.A. Abrikosov (Ukraine)
S. Kostyanev (Bulgaria)
D. Lelgemann (Germany)
H. Moritz (Austria)
G. Papp (Hungary)

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