**President:** Bernhard Heck (Germany)

**Secretary:** Christopher Jekeli (USA), Yuanxi Yang (China)

##### Terms of Reference

As stated in the by-laws, Section IV has primarily a methodological character. Its scope is not confined to one particular topic in Geodesy which would be peculiar to this Section only, but rather all topics are shared in one way or another with other IAG Sections, with the accent of the research pointing towards the systematic mathematical treatment of geodetic problems.

The Section keeps its basic structure as in the last period which originated from the preparatory work done by K.P. Schwarz and the decision of the Section IV Steering Committee to adopt a new organization, by establishing a novel structure in the core of the Section at that time, i.e. the Special Commission on Mathematical and Physical Foundations of Geodesy.

This Special Commission (even this name was born in Section IV) follows the original and standing concern to collect real specialists on the mathematical treatment of various geodetic problems, e.g. geodetic boundary-value problems, statistical problems in geodesy or problems in geometry, relativity, cartography, theory of orbits and dynamics of systems, and put them work on the assessment of difficult questions, open ever since many 4-years periods.

In this concept the new S.S.G.'s are on the contrary in duty to treat a much smaller range of problems, focussing on some very specific open questions to be solved as a rule in one 4-year period. Collective numerical experiments in the framework of S.S.G's are

encouraged, when possible.

##### Structure

__Special Commissions__
**SC1: Mathematical and Physical Foundations of Geodesy**
**Chair:** P. Holota (Czech Republic)

__Special Study Groups:__
**SSG 4.187: Wavelets in Geodesy and Geodynamics**
**Chair:** W. Keller (Germany)

**SSG 4.188: Mass Density from Joint Inverse Gravity Modelling**
**Chair:** G. Strykowski (Denmark)

**SSG 4.189: Dynamic theories of deformation and gravity fields**
**Chair:** D. Wolf (Germany)

**SSG 4.190: Non-probabilistic assessment in geodetic data analysis**
**Chair:** H. Kutterer (Germany)

**SSG 4.191: Theory of Fundamental Height Systems**
**Chair:** C.Jekeli (USA)

##### Special Commission SC1: Mathematical and Physical Foundations of Geodesy

**Chair:** P. Holota (Czech Republic)

**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.

__Structure__
Membership of the CMPFG is restricted to 30 members, one third of which 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 immediately upon approval of the CMPFG program by IAG the following subcommissions will be established, the first four of them will continue and further develop the work already done in the last period:

**Subcommission 1 "Statistics 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 following distinguished scientist have been invited to work in the CMPFG and its Subcommissions:

__Ex officio members__
B. Heck (Germany), President of Section IV

C. Jekeli (USA), Secretary of Section IV, Chair SSG 4.191

Yuanxi Yang (China), Secretary of Section IV

W. Keller (Germany), Chair SSG 4.187

G. Strykowski (Denmark), Chair SSG 4.188

D. Wolf (Germany), Chair SSG 4.189

H. Kutterer (Germany), Chair SSG 4.190

__Individuals__
M. Bougeard (France)

C. Cui (Germany)

A. Dermanis (Greece)

E.W. Grafarend (Germany)

E. Groten (Germany)

K.H. Ilk (Germany)

J. Janak (Slovak Republic)

R. Klees (The Netherland)

L. Kubacek (Czech Republic)

Z. Martinec (Czech Republic)

G. Moreaux (Denmark)

J. Otero (Spain)

M. Petrovskaya (Russia)

R. Rummel (Germany)

F. Sacerdore (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:

##### Subcommissions:

**Subcommission 1 - Statistics and Optimization**
**Chair:** 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. Tn this spirit we offer the following agenda:

- 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 developement.

- 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.

- 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.

- 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.

- 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 also the following 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)

**Working Group - Spatial Statistics for Geodetic Science**
**Chair:** 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, correl ations, etc.) and their spatial distribution;

- geostatistical methods for TINs (Triangular regular 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.

__Additional Members__
Shaofeng Bian (Germany)

W. Caspary (Germany)

K. Kraus (Austria)

S. Meier (Germany)

__Corresponding Members__
N. Cressie (USA)

J. Pilz (Austria)

**Subcommission 2 - Numerical and Approximation Methods**
**Chair:** W. Freeden (Germany)

__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 also the following scientists have been invited to work in Subcommission 2 as

__Additional Members__
S. Beth (Germany)

M. Belikov (Russia)

J. Engels (Germany)

O. Glockner (Germany)

V. Michel (Germany)

F.J. Narcowich (USA)

L. Söberg (Sweden)

**Subcommission 3 - Boundary Value Problems**
**Chair:** 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.)

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

__Additional Members__
M. Günter (Germany)

Yu.M. Neyman (Russia)

N. Weck (Germany)

W. Wendland (Germany)

K.J. Witsch (Germany)

A.I. Yanushauskas (Lituania)

**Subcommission 4 - Geometry, Relativity, Cartography, GIS**
**Chair:** 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 also the following scientists have been invited to work in Subcommission 4 as

__Additional Members__
C. Boucher (France)

G. Hein (Germany)

D. Milbert (USA)

J. Mueller (Germany)

M. Schmidt (Germany)

**Subcommission 5 - Hydrostatic/isostatic Earth Reference Models**
**Chair:** A.N. Marchenko (Ukraine)

__Research Program__
The problem of the standard Earth's density model was formulated by the IAG in 19971 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 also the following scientists have been invited to work in Subcommission 5 as

__Additional Members__
O.A. Abrikosov (Ukraine)

D. Lelgemann (Germany)

G. Papp (Hungary)

**Special Study Group 4.187 - Wavelets in Geodesy and Geodynamics**
**Chair:** W. Keller (Germany)

__Program of Activities__
- Application of wavelets as base functions for the numerical solution of Geodetic Boundary Value Problems:

- Investigation of the convergence rate of the corresponding Galerkin method

- Development of error estimates

- Application of wavelets for pattern recognition:

- Test of the suitability of standard wavelets for the detection of typical geophysical signatures

- Modification of standard wavelets for the improvement of their performance in pattern recognition

- Modification of wavelet analysis techniques

- Construction of smooth orthogonal wavelets in Euclidean spaces as well as on the sphere

- Construction of efficient quadrature formulas for the computation of the initial wavelet coefficients of a given data set

- Develpment of error estimates for the wavelet approximation- and aliasing error

__Members__
B Schaffrin (U.S.A.)

P. Maass (Germany)

R. Lehmann (Germany)

J. Zavoti (Hungary)

L. Liu (China)

K. Arfa-Kaboodvand (Germany)

P. Zatelli (Italy)

R. Haagmans (Netherlands)

C. Kotsakis (Canada)

A. Bruton (Canada)

M. Schmidt (Germany)

__Associate member__
**Special Study Group 4.188 - Mass Density from Joint Inverse Gravity Modelling**
**Chair:** G. Strykowski (Denmark)

__Objectives__
The objective of this study group is to go more directly and explicitly into a problem of mass density modelling from surface gravity data and from other types of geophysical data (seismic, magnetic, heat flow,... etc.). Thus, the work of the group will focus on adequate modelling of the source of the primary observables in physical geodesy, the various quantities associated with the surface gravity signal. There are many reasons for undertaking this task: (i) the adequate mass density model yields adequate models both of the external and of the internal gravity field; (ii) the adequate mass density model makes it possible to construct an accurate gravimetric geoid; (iii) terrain corrections of the surface gravity data can be viewed as a reduction for the gravitational attraction of the mass density anomaly associated with the transition from the terrain to the free air; (iv) a number of geophysical statements about the Earth's interior are based on the surface gravity data and, thus, a construction of the adequate model of the mass density distribution for the shallow parts of the subsurface will help to separate the surface gravity signal generated by these sources. For example, a correct separation is vital for studies of isostasy.

The overall task of this study group will be to bridge the existing gap between solid earth's geophysics and physical geodesy in a similar way as it is done in satellite altimetry, where geodesists and oceanographers work together. Present days methods of physical geodesy are based on mathematical tradition of gravity reductions, which often circumvents the problem of explicit and direct addressing of the local mass density distribution. In this way, and because of the inherent ill-posedness of the inverse gravimetric problem, a gravi-equivalent mass density distribution is implicitly/explicitly used. By using these equivalent models it is still possible to reproduce the existing surface data, but the extrapolation of the statements outside the scope of the data is doubtful. This will not be the case if an adequate mass density model is constructed.

In geophysics, we now have gotten beyond the need to restrict density models to two-dimensional cross sections. There is clearly a need to develop three-dimensional models of mass distributions. Furthermore, what need to be studied are the mathematical limitations inherent in potential-field modeling (especially nonuniqueness and existence) and how these limitations are relaxed with the addition of ancillary datasets (e.g., seismic tomography and heat flow).

__Program of Activities__
- Mathematical description based on physics. Studies of the ambiguity domain of the inverse gravimetric problem especially in connection with the limitations of the domain imposed by the physics, e.g. a physical mass density contrast is bounded. Definitions: What do we understand by a mass density model? (discrete/continuous description, spatial resolution, ... etc.) How does mass density information manifest itself in surface gravity data and in other types of geophysical data? Such consideration are important in order to understand how to couple different types of geophysical data (strong/weak coupling). Thus, the goal of this study will be to formulate some general statements about how to use different types of geophysical data in order to extract information about local mass density distribution.

- Modelling and case stories. Both the methodological and the practical aspects of modelling will be studied, especially in connection with the above. Is mass density adequately parametrized in the methods used? Is the mathematical coupling between different types of geophysical data adequate? How to design observables or combination of them to expose certain aspects of mass density distribution? Thus, the focus of this activity will be on clarifying a somehow difficult problem of the appropriate choice of coupling and parametrization in practical modelling. In the existing methods these choices are often taken out of mathematical convenience. However, are these choices also adequate as compared to the physics of the problem?

- Mass density and physical geodesy. The goal of this activity is to study (often implicit) assumptions on the mass density distribution in standard methods used in physical geodesy. In these way the relation between e.g. different methods of geoid modelling and the assumptions will be exposed. Hopefully, this gives an insight into why the results obtained by different methods do not agree. Furthermore, one can ask whether the assumptions are adequate to describe the local mass density distribution? If not, the group will try to come with some ideas of what to do about it?

__Members__
L. Ballani (Germany)

R. Barzhaghi (Italy)

F. Boschetti (Australia)

G. Gerstbach (Austria)

P. Holota (Czech Republic)

B. Meurers (Austria)

O.C.D. Omang (Norway)

G. Papp (Hungary)

I.L. Prutkin (Russia)

T. Romanyuk (Russia)

S. Schmidt (Germany)

D. Sharni (Israel)

V. Starostenko (The Ukraine)

G. Strykowski (Denmark)

I. Tziavos (Greece)

M. Veronneau (Canada)

__Associate members__
W. Featherstone (Australia)

O. Legostayeva (The Ukraine)

I. Makarenko (The Ukraine)

D. Rossikopoulos (Greece)

T. Yegorova (The Ukraine)

**Special Study Group 4.189 - Dynamic theories of deformation and gravity fields**
**Chair:** D. Wolf (Germany)

__Terms of References__
Recent advances in ground-, satellite and space-geodetic techniques have revealed deformation and gravity changes covering a wide range of periods. These changes have been related to different types of processes acting near the earth’s surface or in its interior. Predictions of the deformation and gravity changes or their inversion require the development of dynamic theories.

__Program of Activities__
- Use of conventional Love-number formalism for studying elementary sources (normal and tangential surface forces, volume forces)

- Development of generalized Love-number formalism for studying complex sources (dislocations)

- Development of generalized Love-number formalism for studying periodic processes (Fourier-transformed Love numbers) and aperiodic processes (Laplace-transformed Love numbers)

- Solutions for generalized viscoelastic earth models to study geophysical processes responsible for deformation and gravity changes

- Study of effects due to density stratification, compressibility, phase boundaries, rheology and lateral heterogeneity

- Prediction of deformation and gravity changes caused by atmospheric, cryospheric, hydrospheric and solid-earth forcing functions

- Inversion of observed deformation and gravity changes in terms of mantle rheology or forcing function

__Members__
H. Abd-Elmotaal (Egypt)

J.-P. Boy (France)

V. Dehant (Belgium)

R. Eanes (USA)

J. Engels (Germany)

J. Fernandez (Spain)

G. Kaufmann (Australia)

Z. Martinec (Czech Republic)

G.A. Milne (United Kingdom)

H.-P. Plag (Norway)

W. Sun (Sweden)

P. Varga (Hungary)

K. Wieczerkowski (Germany)

D. Wolf (Chair) (Germany)

P. Wu (Canada)

__Associate Members__
P. Gegout (France)

E.W. Grafarend (Germany)

J. Hinderer (France)

L.E. Sjöberg (Sweden)

**Special Study Group 4.190 - Non-probabilistic assessment in geodetic data analysis**
**Chair:** H. Kutterer (Germany)

__Terms of Reference__
Geometrical and physical models are more or less approximations of reality. The remaining differences between the data and the chosen model are not exclusively of the type "random". Thus, there are cases when probability theory and statistics are not the only nor even the proper mathematical theories to handle all kinds of problem-immanent uncertainties. The common empirics-based formulation of the stochastic model in adjustment calculus may serve as an example. As a starting point it is necessary to identify and to classify the occuring uncertainties. Within the SSG this shall be done regarding some typical geodetic fields of application. This topic can be seen as the qualification of the different kinds of uncertainty within the data analysis. A very promising approach to deal with non-probabilistic imprecision is given by fuzzy-theory and the related possibility theory. The work shall focus on these fuzzy-theoretical concepts which are still new in geodesy. Their comparison with the well-known theory of statistics shall be given. This includes both differences and common aspects. The consideration of statistical concepts in a fuzzy-theoretical framework is planned. At least three different fields will be considered: GPS data processing, deformation analysis, and GIS (primarily topological aspects). In addition, there will be input from local geoid computation and multi sensor mobile mapping systems. The data-analytical points of interest are the data handling in the preprocessing steps and the corresponding setup of models. This item represents the link between the qualification of the uncertainty and its quantification. As an example the uncertainty of GPS results introduced by different operators and different software packages is mentioned. In addition to the analytical studies the numerical quantification of the uncertainties of the data, the model, and their impact on the quantities of interest shall be done. This will be mostly in an exemplary and instructive way to understand the interaction between statistical and non-statistical viewpoints and methods.

__Objectives__
- Identification and classification of different kinds of uncertainties in geodetic models and inferences; supply of a proper terminology

- Collection of approaches to deal with uncertainty, to infer and to decide under uncertainty like fuzzy-theory, possibility theory, evidential reasoning, etc, in addition to the well-known concepts of statistics and probability; compilation of a bibliography

- Discussion of the applicability of the different approaches to the data analysis in selected fields of geodetic interest

- Comparison of the different methods by studying selected problems numerically

__Members__
H. Kutterer (Germany)

M. Brovelli (Italy)

Brunn (Germany)

Carosio (Switzerland)

B. Crippa (Italy)

G. Joos (Germany)

K. Heine (Germany)

S. Leinen (Germany)

Merminod (Switzerland)

F. Neitzel (Germany)

W. Niemeier (Germany)

J. Ou (China)

Rossikopoulos (Greece)

Schaffrin (USA)

T. Schenk (USA)

T. Shyllon (Nigeria)

J. Wang (Australia)

Y. Yang (China)

J. Zavoti (Hungary)

__Corresponding Members__
M. Molenaar (The Netherlands)

R. Viertl (Austria)

A. Wieser (Austria)

**Special Study Group 4.191 Theory of Fundamental Height Systems **
**Chair:** C.Jekeli (USA)

__Terms of reference__
Height systems (datums) around the world invariably are based on the geoid (the geopotential), although, the realization of the geoid is far from uniformly precise enough to enable the definition of a world vertical datum for all applications. Moreover, geodetic vertical datums are based on different height systems, such as the North American Vertical Datum (NAVD88) which uses orthometric heights, the European heights that are mostly normal heights, and the South American Geocentric Reference Systems (SIRGAS) that is now considering to propose the use of geopotential numbers as fundamental height units. Other datums, such as the International Great Lakes Datum, already are based on dynamic heights. Against this background strictly geometric heights are becoming increasingly and dominantly prevalent Ð consider the GPS heights and topographic heights from SAR, INSAR, as well as laser altimetry, as obtained on space-borne and airborne platforms. Accuracy in vertical positioning with GPS is approaching subcentimeter levels and global topographic (land) data from the other sources are being obtained with accuracies on the order of a few meters, or better. Of course, satellite radar altimetry (over oceans) continues to improve with centimeter and better accuracy foreseen in the near future. Each database refers to a possibly different reference system. Geometric and geopotential-based heights must be consolidated within the context of accurate geoid determination. And, height reference systems must be defined within the broader scope of three-dimensional terrestrial reference systems, as well as time systems. Time frequency standards are now accurate enough to be sensitive to general relativistic effects due to the geopotential. To achieve a uniform base for definition and transformation between different height systems, it is important to consider the fundamental definition and realization of the geoid and its relation to international terrestrial reference systems. Equally important in assessing achievable and desirable accuracies is the identification of the future needs of users of heights in many disciplines, ranging from farming and flooding control to global concerns of rising sea level and tectonic processes.

__Objectives__
- To consider the fundamental definition and realization of the geoid, with full recognition of temporal and general relativistic effects, for the purpose of establishing a global reference for geopotential heights and time systems.

- To investigate height systems, including the role of the geopotential, and their utility for different geodetic, geophysical, and engineering applications that are current and that are projected in the year 2020 (and beyond). What accuracies will be needed, what height systems will be useful, how will heights from different measurement systems be consolidated into a uniform frame? How should the vertical dimension be treated in terms of reference systems within the context of three-dimensional geodesy?

- To determine if the IAG should establish recommendations for the future definitions of height systems for geodetic control and geophysical and engineering applications.

- To determine (and recommend) if existing IAG-affiliated services should be expanded to provide information specific to vertical data.

__Members__
V. Andritsanos (Greece)

C. Boucher (France)

M. Bursa (Czech Republic)

H. Drewes (Germany)

M. Higgins (Australia)

C. Jekeli (USA)

S. Kenyon (USA)

U. Marti (Switzerland)

M. Poutanen (Finland)

F. Sacerdote (Italy)

M. Sideris (Canada)

D. Zilkoski (USA)

__Corresponding Members__
M. Bevis (USA)

R. Forsberg (Denmark)

E. Groten (Germany)

C. Hwang (Taiwan)

B. Kearsley (Australia)

R. Luz (Brazil)

A. Mainville (Canada)

I. Tziavos (Greece)

M. Vermeer (Finland)

P. Woodworth (U.K.)