The main instrument for information and steering the group were the Circular Letters
(No. 0 up to No. 4). The first letter (No. 0) at the beginning was a questioning to find out the very different setups, fields, data and strategies in inversion used. Two meetings to improve the personal contact within the group could be organized. The first meeting
(28-30 Oct 1996 Walferdange/Luxembourg) was supported by the fund of the “ Open Partial Agreement on Major Hazards” offered by the European Center for Geodynamics and Seismology (E.C.G.S.) in Luxembourg and Belgium (Prof. Bernard Ducarme), by the Executive Board of the GeoForschungsZentrum Potsdam (GFZ) and the International Association of Geodesy (IAG). The participants of the meeting were 20 members of the SSG and 3 guests. 21 papers were presented followed by intensive discussions. A second shorter meeting took place during the EGS General Assembly in Vienna/Austria 1997. The mainly discussed points referred to the structure and the contents of a possible monograph covering all the topics of the group as well as some open problems to give an orientation for the future work. A monograph and comparative tests with data between the algorithms used in the group, as were intended at the beginning of the time period and were prepared to a certain stage, could not be finally realized. A home page showing the basic facts of the group at the internet site of the IAG was created. Some of the members of the group were engaged in the organization of symposia and partly also in the edition of proceedings afterwards. Subsequently some significant meetings, symposia and sub-symposia are listed:

	Geodätische Woche (Geodetic Week), Session G12 “ Geodätische Inverse Probleme/Geodetic Inverse Problems” , Stuttgart, Germany, October 7-12, 1996.
	First Meeting of IAG-SSG 4.170 “ Integrated Inverse Gravity Modelling” , Walferdange, Luxembourg, October 28-30, 1996.
	EGS XXII General Assembly, Symposium EGS1 “ oint Inversion as a General Problem in Earth Sciences” , Vienna, Austria, 21-25 April 1997.
	EGS XXIII General Assembly, Symposium EGS3 “ Modelling techniques and joint inversion in Earth sciences” , Nice, France, 20-24, April, 1998.
	2nd Joint Meeting of the International Gravity Commission and the International Geoid Commision, Session VII - Inverse gravimetric and related problems. Trieste, Italy, September 7-12, 1998,
	EGS XXIV General Assembly, Symposium G15 “ Topography and bathymetry in geodetic and geophysical applications” , Den Haag, The Netherlands, April 19-23, 1999.
	EGS XXIV General Assembly, Symposium SE46-02 “ Inversion and interpretation of gravity and magnetic anomalies on all scales, case histories” , Den Haag, The Netherlands, April 19-23, 1999.
	IUGG General Assembly, Inter-Association Symposium JSA40, “ Solid-Earth Geophysical Data Fusion and Analysis Methods” , Birmingham, UK, July 28, 1999

 

3. Results and Problems

As can be seen from the references (see below) a broad spectrum in inverse modelling is covered by the work of the group which can be, of course, summarized in different way. From the aspect of the aims and orientation of the SSG the following lines are important:

	ordinary (“ single” ) inversion of gravity, magnetics, seismics and stress
	joint inversion of gravimetry and seismics or magnetics
	theoretical and computational aspects of joint inversion
	geodynamical inverse problems
	geophysical and geological applications

Only a few special points shall be stressed here: The inherent nonuniquenss and instability in the single inversion was investigated under different views (the structure of the null space and of parameter spaces, the evaluation of the a-priori and a-posteriori information, the application of the maximum entropy principle of the information theory). New algorithms reflecting diverse backgrounds and also non-linearity (simulated annealing, genetic algorithms, neural networks, Bayesian inference) were studied and applied. Optimization and regularization strategies (hybrid norms, sequential processing) were directly adapted to the inversion problem structure. There is a convergency between geodesy and geophysics in studying subjects as the downward continuation of the Earth’s gravity field from the topography to the geoid. As an important aspect appears the study of mass density modelling and its different role in geodesy and geophysics. To overcome the known difficulties with the varying relationship between density and seismic velocity this relationship was included as unknown parameter in the joint gravito-seismic inversion procedure. Gravity information is more and more coupled in the inversion with numerous physical fields and geo-data. Then it often acts as a control to validate the geophysical model. The modelling situations comprehend not only regional and global layer models but also subducting plates and tectonic phenomena, e.g. collision zones. Dynamic inverse modelling relates e.g. the geoid with mantle viscosity and convection information or can give indications on rheological properties.

Future investigations in the field of integrated inverse gravity modelling should study

	the specific assumptions in the different seismic models and codes together with the investigated geophysical situation and the non-uniqueness of single inverse problems
	the different forms of coupling and correlation in the joint inversion, especially the role of geometry as a common quantity
	the connection of the mass density with the different geophysical fields and other geo-data, its role in topography, especially the study of the inverse gravimetric problem together with the geoid determination (boundary-value problem) making use of all types of geodetic measurements

 

4. References 1995 - 1999

A selection only of printed articles, with a few exceptions, can be given here (in alphabetical order related to the SSG members). Sobolev, S.V.; Zeyen, H.; Granet, M.; Achauer, U.; Bauer, C.; Werling, F.; Altherr, R.; Fuchs, K.: Upper mantle temperatures and lithosphere-asthenosphere system beneath the French Massif Central constrained by seismic, gravity, petrologic and thermal observations. Tectonophysics, vol. 275 (1997) 1-3, 143-164.

Ballani, L.: Solving the Inverse Gravimetric Problem: On the Benefit of Wavelets. in: Sanso, F. (ed.), Geodetic Theory Today. IAG Symposia 114, Springer -Verlag Berlin etc., 1995, pp. 151-161.

Ballani, L.; Greiner-Mai, H.; Stromeyer, D.: Determining the Magnetic Field in the Earth's Deep Mantle by an Inverse Boundary Value Problem. Scientific Technical Report STR99/12, GeoForschungsZentrum Potsdam, Potsdam, Germany, 1999, 80 pages.

Barzaghi, R.: SSG 4.141 Integrated Inverse Gravity Modelling. in: Section IV Bulletin, General Theory and Methodology, IAG, IUGG, No.1, Delft, The Netherlands: The Office of the Netherlands Geodetic Commission, 1995, pp. 65-67.

Tondi, R.; De Franco, R.; Barzaghi, R.: Sequential integrated inversion of traveltimes and gravity data for 2D structures. Paper 2nd Joint Meeting of the International Gravity Commission and the International Geoid Commision, Session VII - Inverse gravimetric and related problems. September, 7 - 12, 1998, Trieste, Italy. (submitted to IGJ)

Buyanov, A.F.; Glaznev, V.N.; Mitrofanov, F.P.; Raevsky, A.B.: Three - dimensional modelling of the Lapland Granulite Belt and adjacent structures of the Baltic Shield from geophysical data. in: Nor. geol. unders. Special Publ. vol. 7, 1995, pp. 167-178.

?adek, O.; ?i?kova, H.; Yuen,D.A.: Mantle viscosity inferred by genetic algorithms from the geoid and seismic information. Phys. Chem. Earth (A), vol. 23 (1998) No. 9/10, 865-872.

Kido, M.; ?adek, O.: Inferences of viscosity from the oceanic geoid: indication of a low viscosity zone below the 660-km discontinuity. Earth Planet. Sci. Lett., vol. 151 (1997), 125-137.

?i?kova, H.; ?adek, O.; Yuen, D.A.; Hua-wei Zhou: Slope of the geoid spectrum and constraints on mantle viscosity stratification. Geophys. Res. Lett., vol. 23 (1996) 21, 3063-3066.

Kido, M.; Yuen, D.A.; ?adek, O.; Nakakuki, T.:Mantle viscosity derived by genetic algorithm using oceanic geoid and seismic tomography for whole-mantle versus blocked-flow situations. Phys. Earth Planet. Int., vol. 107 (1998) 4, 307-326.

Campos-Enriquez, J.O.; Delgado-Rodriguez, O.; Chavez-Segura, R.;Gomez-Contreras, P.; Flores-Marquez, E.L.; Birch, F.S.: The subsurface structure of the Chalco sub-basin (Mexico City) inferred from geophysical data. Geophysics, vol. 62 (1997) 1, 23-35.

Everaerts, M.; Poitevin, C.; De Vos, W.; Sterpin, M.: Integrated geophysical/geological modelling of the western Brabant Massif and structural implications. Bulletin de la Societe Belge de Geologie, vol.105 (1996) 1-2, 41-59.

Kahle, H.-G.; Geiger, A.; Buerki, B.; Gubler, E.; Marti, U.; Wirth, B.; Rothacher, M.; Gurtner, W.; Beutler, G.; Bauersima, I.; Pfiffner, O.A.: Recent crustal movements, geoid and density distribution; contribution from integrated satellite and terrestrial measurements. In: Pfiffner, O.A. et al (eds.), Results of NRP 20; Deep Structure of the Swiss Alps. Birkhaeuser, Basel, Switzerland. 1997, pp. 251-259.

Galichanina, L.D.; Glaznev, V.N.; Mitrofanov, F.P.; Olesen, O.; Henkel, H.: Surface density characteristics of the Baltic shield and adjacent territories. in: Nor. geol. unders. Special Publ. vol. 7, 1995, pp. 349-354.

Glaznev, V.N.; Raevsky, A.B.; Skopenko, G.B.: A three-dimensional integrated density and thermal model of the Fennoscandian lithosphere.Tectonophysics, vol. 258 (1996) 15-33.

Mints, M.V.; Glaznev, V.N.; Raevsky, A.B.: Three-dimensional geological model of the Earth's upper crust, the Kola ultradeep borehole and adjacent areas of the Kola Peninsula. Geotectonics, vol. 28 (1995) 6, 457-475.

Ilk, K.H.: Inverse Geodetic Problems. in: Travaux de l’Association Internationale de Géodésie, Paris, vol. 29 (1992) pp. 356 – 391.

Gret, A,; Klingele, E.: Application of Artifitial Neural Networks for Gravity Interpretation in Two Dimension. Geodäsie und Photogrammetrie ETH-Z. Bericht Nr.279. 1998. ISBN 3-906467-10-4. 88 pages.

Klingele, E.E.; Kahle, H.; Seismic Velocity-Density Relationship: A Critical Review. Paper Symposium EGS1 "Joint inversion as a general problem in Earth sciences", EGS XXII General Assembly, Vienna, Austria, April 21-25, 1997.

Lehmann, R.: The Integrated Inverse Gravimetric Problem - a Separated Minimum-Norm-Solution. Paper presented at the First Meeting of IAG - SSG 4.170 "Integrated Inverse Gravity Modelling", Walferdange, Luxembourg, October 28-30, 1996.

Lehmann, R.: Maximum entropy density determination for the earth and terrestrial planets. Studia geoghys. et geodaet., vol. 43 (1999) 1, 35-60.

Martinec, Z.: Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. J. Geod., Vol. 70 (1996) 805-828.

Martinec, Z.: Boundary-Value Problems for Gravimetric Determination of a Precise Geoid. Lecture Notes in Earth Sciences. Springer, Heidelberg, 1998, 223 pages.

Smolyaninova, E.I.; Mikhailov, V.O.; Lyakhovsky, V.A.: Numerical modelling of regional neotectonic movements in the northern Black Sea. Tectonophysics, vol. 266 (1996) 1-4, 221-231.

Mosegaard, K.; Tarantola,A.: Monte Carlo sampling of solutions to inverse problems. J. Geophys. Res. vol. 100 (1995) B7, 12431-12447.

Vasin, V.V.; Prutkin, I.L; Timerkhanova, L.Yu. : Retrieval of a Three-Dimensional Relief of Geological Boundary from Gravity Data. Izvestiya, Physics of the Solid Earth, vol. 32(1996) 901 - 905. (Russ.: Fizika Zemli, 1996, No.11, 58-62.)

Prutkin, I.L.: Determination of the 3D Topography for a Conductive Half-Space Boundary from Electromagnetic Data. Izvestiya, Physics of the Solid Earth (Russ.: Fizika Zemli, 1998, No.4, 62-66.).

Romanyuk, T.V.: Seismic and density modeling of the crust and upper mantle along the Quartz traverse. Izvestiya, Physics of the Solid Earth, vol. 31 (1996) 9, 735-747.

Chunduru, R.K.; Sen,M.K.; Stoffa, P.L.: Hybrid optimization methods for geophysical inversion. Geophysics, vol.62 (1997) 4, 1196-1207.

Zhao Lian She; Sen, M.K.; Stoffa, P.L.; Frohlich, C.: Application of very fast simulated annealing to the determination of the crustal structure beneath Tibet. Geophys. J. Int., vol.125 (1996) 2, 355-370.

Sen, M.K.; Stoffa, P.L.: Bayesian inference, Gibbs' sampler and uncertainty estimation in geophysical inversion. Geophys. Prosp., vol. 44 (1996) 2, 313-350.

Smilde, P.: Verwendung und Bewertung von a-priori Information bei potentiell singulären Inversionsproblemen am Beispiel der gravimetrischen Bestimmung von Dichteverteilungen (Application and evaluation of a-priori information in potentially singular inverse problems demonstrated by the gravimetric determination of density distributions). DGK: Series C, No. 490, München: Beck, 1998, 90 pages.

Starostenko, V.I.; Kozlenko, V.G.; Kostyukevich, A.S.; Baranova, E.P.; Korenevich, K.A.; Koyfman, L.I.: Structure of the Earth's Crust in Ukraine at the II and III Geotraverses Intersection according to Seismogravimetric Modelling Data. Krystalinikum, vol. 22 (1995) 137 - 147.

Starostenko, V.I.; Legostaeva, O.V.: Calculation of the Gravity Field from an Inhomogeneous, Arbitrary Truncated Vertical Rectangular Prism. Izvestiya, Physics of the Solid Earth (Russ.: Fizika Zemli, 1998, No.12, 31-44).

Buryanov, V.B.; Makarenko, I.B.; Starostenko V.I.: Thickness and structure of the crust of the Black Sea Basin according to gravity modeling data. Geophys. J., vol. 16 (1997) 5, 603-623.

Stepanova, I.E.: The Integral Equation of the Reverse Potential Problem in Three-Dimensional Case. Dokl. Akad. Nauk, vol. 348 (1996) 2, 252-254. (Russ.)

Stepanova, I.E.: On the Numerical Algorithm of the Solution of the Three-Dimensional Reverse Problem of Potential. Dokl. Akad. Nauk, vol. 354 (1997) 2, 249-251. (Russ.)

Stepanova, I.E.: The Integral Equation of the Three-Dimensional Structural Inverse Problem. Dokl. Akad. Nauk, vol. 353 (1997) 1, 108-110. (Russ.)

Stepanova, I.E.: Reconstruction of a Gravity Source in Ore-Type Problems. Izvestiya, Physics of the Solid Earth (Russ.: Fizika Zemli, 1998, No.11, 86-89).

Strakhov, V.N.; Schäfer, U.; Strakhov, A.V.: Neue Lineare Approximationen Linearer Elemente des Gravitationsfeldes der Erde - Probleme und Perspektiven (New linear approximations of linear elements of the Earth gravitational field - problems and perspectives). in: Freeden, W. (ed.), Progress in Geodetic Science, Aachen: Shaker, 1998. pp. 315-322.

Strakhov, V.N.; Strakhov A.V.: On the Regularization Method of the Least Squares Method (Russ.), Geophysical Journal, vol. 20 (1998) 6, 1-24.

Strakhov, V.N.; Strakhov A.V.: On the Construction of Stable Approximating Solutions of Great and High-dimensional Systems of Linear Algebraic Equations Originating in Problems of Gravimetry and Magnetics.(Russ.) Geophysical Journal, vol. 21 (1999) 1, 1-21.

Stromeyer, D: Plate boundary forces acting on the South America Plate derived from stress field inversion. Annales Geophysicae, vol. 14, Suppl. I, C256, 1996.

Strykowski, G.: Dealing with the non-uniqueness of the solution to the inverse geophysical problems. in: Hansen, P. Ch. (ed.) Proc. Interdisciplinary Inversion Workshop 4, Danish Technical University, Lyngby, Denmark, Oct 22, 1997. pp. 67-74.

Strykowski, G.: Experiences with a Detailed Estimation of the Mass Density Contrasts and of the Regional Gravity Field using Geometrical Information from Seismograms. Phys. Chem. Earth (A) vol. 23 (1998) 9/10, 845-856.

Strykowski, G.: Elements of the Joint Gravimetric-Seismic Modelling - Coupling the Available Information. in: Bo Holm Jacobsen (ed.), Proc. Interdisciplinary Inversion Workshop 5, Dept. Earth Sciences, Univ. Aarhus, Denmark, Sept. 24, 1997. pp. 24-30.

Strykowski, G.: Some Technical Details Concerning a New Method of Gravimetric-Seismic Inversion. Phys. Chem. Earth (A)., vol.24 (1999) 3, 207-214.

Strykowski, G.: Formulation of the mathematical frame of the joint gravimetric-seismic modelling problem based on analysis of regionally distributed borehole data. In: Segawa, J.; Fujimoto, H.; Okubo, S. (eds.), Gravity, Geoid and Marine Geodesy. IAG Symposia 117, Springer-Verlag. New York, 1997. pp. 360-367.

Strykowski, G.: Borehole data and stochastic gravimetric inversion. PhD-thesis, Univ. Copenhagen, Kort & Matrikelstyrelsen, Skrifter 4. rk, bd. 3 (Public. National Survey and Cadastre, Publications 4. series, vol. 3), Copenhagen, Denmark, 1996, 110 pages.

Tziavos, I.N.; Sideris, M.G.; Suenkel, H.: The effect of surface density variations on terrain modeling; a case study in Austria. In: Tziavos, I.N.; Vermeer, M. (eds.), Techniques for local geoid determination. Reports Finn. Geod. Inst., vol.96 (1996) 2, 99-110.

Vasco, D.W.: Regularization and trade-off associated with nonlinear geophysical inverse problems: penalty homotopies. Inverse Problems, vol. 14 (1998) 1033-1055.

Vasco, D.W.: Groups, algebras and the non-linearity of geophysical inverse problems. Geophys. J. Int., vol. 131 (1997) 9 - 23.

Vasco, D.W.; Peterson, J.E.; Majer, E.L.: Nonuniqueness in traveltime tomography: Ensemble inference and cluster analysis. Geophysics, vol.61 (1996) 1209-1227.

Vasco, D.W.: A transformation approach to geophysical inverse problems. Geophys. J. Int., vol. 123 (1995) 1, 183-212.

Wang Qianshen; Yang Xinshe: Stratified Gravity Image and Its Application - The Test and Combined Application of Seismic Tomography. Paper submitted to the First Meeting of IAG-SSG 4.170 “ Integrated Inverse Gravity Modelling” , Walferdange, Luxembourg, October 28-30, 1996, (booklet of the meeting, 6 pages).

Wang Qianshen; Yang Xinshe: Stratified Gravity Image and Its Application Acta Geophysica Sinica, vol.40 (1997) No.5, 649-659.

Wang Qianshen; Wu Chuanzhen; Jiang Weiwei: Gravity field and crustal structure of Gyirong-Lugu Region in western Xizang (Tibet) Plateau. Chinese Sci. Bull., vol.42 (1997) 8, 669-673.

Kong Xiangru; Wang Qianshen; Xiong Shaobai: Comprehensive geophysics and lithospheric structure in the western Xizang (Tibet) Plateau. Science in China. Series D, Earth Sciences., vol. 39 (1996) 4, 348-358. 1996.

Yegorova, T.P.; Kozlenko, V.G.; Starostenko, V.I.; Shen, E.L.; Botev, E.A.: Density inhomogeneities of the upper mantle of the Central Balkans. Geophys. J. Int., vol. 132 (1998) 283-294.

Yegorova, T.P.; Starostenko, V.I.; Kozlenko, V.G.; Pavlenkova, N.I.: Three-dimensional gravity modelling of the European Mediterranean lithosphere. Geophys. J. Int., vol. 129 (1997) 2, 355-367.

Yegorova, T.P.; Starostenko, V.I.: Gravity Modeling Constraints on the Structure of the Transition Zone between Western Europe and East European Platform. (Russ.: Fizika Zemli, 1998, No. 6, 19 - 36.) Izvestiya, Physics of the Solid Earth.

Zeyen, H.; Achauer, U.: Joint Inversion of Teleseismic Delay Times and Gravity Anomaly Data for Regional Structures. in: Fuchs, K. (ed.), Upper Mantle Heterogeneities from Active and Passive Seismology, NATO-ASI series 1, vol.17, Kluwer Academic Publishers, Dordrecht, Netherlands, 1997, pp. 155-168.

Shaorong Zhao: Detection of the active segment at the Red River fault zone by inversion of observed gravity changes. J. Geodyn., vol. 20 (1995) 1, 41-62.

Shaorong Zhao: Joint inversion of observed gravity and GPS baseline changes for the detection of the active fault segment at the Red River fault zone. Geophys. J. Int., vol.122 (1995) 1, 70-88.

 

The cooperation of my colleagues R.Barzaghi, Z.Martinec, V.O. Mikhailov and G. Strykowski in preparing this report is gratefully acknowledged.