IC SG4

IC-SG4: Inverse theory and global optimization Chair:C. Kotsakis (Greece) Affiliation:Comm. 2

Introduction

At the Sapporo IUGG General Assembly (June 30 - July 11, 2003), the International Association of Geodesy (IAG) has approved the establishment of an 'inter-commission' working group (WG) on Inverse Problems and Global Optimization, with the aim of supporting and promoting theoretical and applied research work in various areas of modern geodetic data analysis and inversion. This WG has successfully operated during the last four years under the umbrella of the Intercommission Committee on Theory (ICCT) and the chairmanship of Dr. Juergen Kusche. During the IAG-EC meeting at the Perugia IUGG General Assembly (July 2 - 13, 2007) the new structure of the ICCT and its associated WGs was discussed, and a decision was made that the ICCT/WG on Inverse Problems and Global Optimization will continue its operation for another 4-year period. The purpose of this document is to give an (updated) description of the WG's potential study areas and research objectives, and its associated terms of reference for the upcoming research period 2007 - 2011.

Terms of Reference

It is well recognized that many, if not most, geodetic problems are in fact inverse problems: we know to a certain level of approximation the mathematical and physical models that project an Earth-related parameter space and/or signal onto some data space of finite discrete vectors; given discrete noisy data we then want to recover the governing parameter set or the continuous field (signal) of the underlying model that describes certain geometrical and/or physical characteristics of the Earth. The sitŹuation is further complicated by the fact that these problems are often ill-posed in the sense that only generalized solutions can be retrieved (due to the existence of non-trivial nullspaces) and/or that the solutions do not depend continuously on the given data thus giving rise to dangerous unstable solution algorithms. In order to deal successfully with geodetic data inversion and parameter/signal estimation problems, it is natural that we have to keep track with ongoing developments in inverse problem theory, global optimization theory, multi-parameter regularization techniques, stochastic modeling, Bayesian inversion methods, statistical estimation theory, data assimilation, and other related fields of applied mathematics. In modern geodesy we also have to develop special inversion techniques that can be used for large-scale problems, involving high degree and order gravity field models from space gravity missions and high-resolution discretizations of the density field or the dynamic ocean topography.

Earth's gravity field modeling from space gravity missions has been (and will surely continue to be in the future) a key study area where existing and newly developed tools from Inverse Problem Theory need to be implemented (including the study of regularization methods and smoothing techniques and the quality assessment of Earth Gravity Models, EGMs). With the cutting-edge applications of the latest and upcoming gravity missions (recovery of monthly surface mass variations from GRACE, constraining viscosity/ lithospheric/postglacial rebound models from GRACE time-variable gravity and from GOCE static geoid pattern analysis), it can be expected that Inverse Problem Theory will increase its importance for the space gravity community.

Furthermore, there still exist other, more classical geodetic problems that have been identified as inverse and ill-posed and have traditionally attracted the interest of many researchers: the inverse gravitational problem where we are interested in modeling the earth's interior density from gravity observations, various types of downward continuation problems in airborne/satellite gravimetry and geoid determination, certain problems in the context of satellite altimetry and marine gravity modeling, the problem of separating geoid and dynamic ocean topography, the problem of inferring excitations/earth structure parameters from observed polar motion, the determination of stress/strain tensors from observational surface monitoring data, or certain datum definition problems in the realization of global geodetic reference systems. Another, relatively recent, geodetic problem of ill-posed type is the orbit differentiation problem: non-conventional gravity recovery methods like the energy conservation approach and the acceleration approach require GPS-derived kinematic satellite orbits to be differentiated in time, while counteracting noise amplification at the same time. The above nonexhaustive list of inverse problems provides a rich collection of study topics with attractive theoretical/practical aspects, which (in conjunction with the increasing data accuracy, coverage and resolution level) contain several open issues that remain to be resolved.

Objectives

The aim of the WG is to bring together people working on inverse problem theory and its applications in geodetic problems. Besides a thorough theoretical understanding of inverse problems in geodesy, the WG's central research issue is the extraction of maximum information from noisy data by properly developŹing mathematical/statistical methods in a well defined sense of optimality, and applying them to specific geodetic problems. In particular, the following key objectives are identified:

• Identification and theoretical understanding of inverse and/or ill-posed problems in modern geodesy
• Development and comparison of mathematical and statistical methods for the proper treatment of geodetic inverse problems
• Recommendations and communication of new inversion strategies

More specific research will focus, for example, on global optimization methods and theory, on the mathematical structure of nullspaces, on the treatment of prior information, on nonlinear inversion in geodetic problems and on the use of techniques for treating inverse problems locally. It is also necessary to investigate the quality assessment and numerical implementation of existing regularization methods in practical geodetic problems (e.g. dealing with coloured noise and/or heterogeneous data, using partially over- and underdetermined models, dealing with different causes of ill-posedness like data gaps and downward continuation, coping with data sets that have entirely unknown noise characteristics, etc.).

Program of Activities

The WG's activities will include the launching of a webpage for dissemination of information, for presentation, communication and monitoring of research results and related activities, and for providing an updated bibliographic list of references for relevant papers and reports in the general area of geodetic inverse problems. This would also provide WG's members (and other interested individuals) with a common platform to communicate individual views and results, and stimulate discussions. Although the discussion will be in general based on email, it is planned to have splinter meetings during international conferences and, if possible, a workshop or a special conference session.

Membership

The following is a proposed (tentative) membership list for the IAG/ICCT WG on Inverse Problems and Global Optimization. The final list will be confirmed within 2007.

C. Kotsakis (Greece, chair)
J. Kusche (Germany)
S. Baselga Moreno (Spain)
J. Bouman (The Netherlands)
P. Ditmar (The Netherlands)
B. Gundlich (Germany)
P. Holota (Czech Republic)
M. Kern (The Netherlands)
T. Mayer-Guerr (Germany)
V. Michel (Germany)
P. Novak (Czech Republic)
S. Pereverzev (Austria)
B. Schaffrin (USA)
M. Schmidt (Germany)
Y. Shen (China)
N. Sneeuw (Germany)
S. Tikhotsky (Germany)
C. Xu (Russia)