**Physics of the Earth
results achieved at Department of Geophysics in 2000 (reported in December, 2000)**

We have studied viscoelastic response of elastically compresible Earth's models in the time domain (Hanyk et al., 2000; Hanyk et al., submitted). By means of numerical simulations of a magnetohydrodynamic system we have demonstrated that the adiabatic heating/cooling stabilizes convection pattern in the Earth's core (Velimsky and Matyska, 2000). The effects of adiabatic heating/cooling and viscous dissipation on 3-D rapidly rotating thermal convection was also studied by Mistr et al. (submitted). We have employed the extended-Boussinesq convection models to extract also the Bullen parameter and showed that the profiles of the Bullen parameter have a definite potential of being useful in constraining the physical parameters and flow structures associated with the Earth's mantle convection (Matyska and Yuen, 2000; Matyska and Yuen, submitted). Classical Boussinesq approximation of thermal convection was employed by Vecsey and Matyska (in press) to demonstrate the ability of wavelet spectra to reveal the multiscale temporal dynamics of convecting systems.

We have developed a spectral-finite-element approach to the forward modelling of the viscoelastic response of a spherical earth with a 3-D viscosity structure to a surface mass load. It represents an alternative to a variety of numerical methods for 2-D and 3-D postglacial rebound modelling used recently (the finite-element method, the perturbation method, the semi-analytical approach and the spectral--finite-difference method). We constructed an analytical form of the layer propagator matrix for the response of a locally incompressible, layered, linear-viscoelastic sphere to an external load assuming that the initial density stratification within each layer is parametrized by Darwin's law. We showed that a locally incompressible earth model relaxes faster than a materially incompressible model. This is a consequence of the fact that the perturbations of the initial density are zero during viscoelastic relaxation of a locally incompressible medium, so that there are no internal buoyancy forces associated with the continuous radial density gradients, only the buoyancy forces generated by internal density discontinuities. On the other hand, slowly decaying internal buoyancy forces in a materially incompressible earth model cause it to reach the hydrostatic equilibrium after a considerably longer time than a locally incompressible model. We solved the satellite gradiometric boundary-value problem in terms of Green's function as a series of tensor spherical harmonics. This form can be applied to develop higher order multipole moments of the Earth's external gravitational potential and the shape of the geoid, one of the main objective of the GOCE mission. Alternatively, we converted the spectral form to closed spatial form. This form was used to construct the upward and downward continuation operators that can be applied to transform the measured gradiometric data from a (non-spherical) satellite orbit to a mean orbit sphere (Martinec, 2000; Ardestani and Martinec, 2000; Martinec et al., in press; Vanicek et al., in press; Novak et al., submitted; Ardestani and Martinec, submitted; Martinec, submitted; Martinec and Grafarend, submitted).

Development of the algorithms for two-point ray tracing and travel-time interpolation in 3-D continued. The first quantitative results have been achieved in the description of the ray chaos due to heterogeneities in the velocity model by means of the average Lyapunov exponents. Various kinds of the coupling ray theory for weakly anisotropic models have been studied and compared with the exact solution derived for the "twisted crystal" model. Particular attention is devoted to the resolution of seismic inversion techniques, model fitting, conversion and smoothing with the application of medium correlation functions and Sobolev scalar products. (Klimes, 2000 a,b; Bulant et al. 2000; Klimes submitted, a-f; Bulant and Klimes, submitted; Bulant, submitted; Zacek, submitted).

Inhomogeneous waves in dissipative media have been studied in detail; extensive calculations of reflection/transmission coefficients at plane interfaces in various dissipative models have been performed and compared with the corresponding coefficients in elastic non-dissipative media (Brokesova, in press). Ray and finite-difference methods have been used to simulate a measured CDP seismic section for the shallow lignite deposit at the Domenico lignite site, Greece (Brokesova et al., 2000).

Records of several Turkey earthquakes of 1999 obtained at broad-band seismic stations in the Corinth Gulf, Greece, have been used to study the dispersion of Love and Rayleigh waves. As compared with previous models for the territory of Greece, the dispersion data require significantly lower velocities in the uppermost crust, and smaller crustal thickness (Novotny et al., submitted).

A hybrid method has been developed and applied to earthquakes at EUROSEISTEST near Thessaloniki where it allowed a more realistic modeling than the conventional methods with plane waves (Riepl et al., 2000). The hybrid method has been generalized to 3D media (Oprsal and Zahradnik, submitted). A new ASPO method for the focal-mechanism retrieval from earthquake amplitude spectra and polarities has been developed and applied in Corinth Gulf (Zahradník et al., in press). Seismic stations of the Charles University, operating in Greece, have contributed to the explanation of the damaging Athens 1999 earthquake (Tselentis and Zahradnik, 2000 a,b).

**References **

Ardestani, V. E., Martinec, Z., 2000.
Ellipsoidal Stokes boundary-value problem with ellipsoidal
corrections in boundary condition.
*Studia Geoph. et Geod.*, submitted.

Ardestani, V. E., Martinec, Z., 2000. Geoid determination through ellipsoidal Stokes boundary-value problem.* Studia Geoph. et Geod.* **44**, 353-63.

Brokesova, J., Zahradník, J., Paraskevopoulos, P., 2000. Ray and finite-difference modelling of CDP seismic sections for shallow lignite deposits. *J. Appl. Geophys.* ** 45**, No. 4, 261 – 272.

Brokesova, J. Reflection/transmission coefficients at a plane interface in dissipative and nondissipative isotropic media: a comparison, *J. Comp.Acoustics*, in press.

Bulant, P., Klimes, L. Numerical algorithm of the coupling ray theory in weakly anisotropic media.
*Pure and appl. Geophys.*, submitted.

Bulant, P. Sobolev scalar products in the construction of velocity models application to model Hess and to SEG/EAGE Salt Model. *Pure and appl. Geophys.*, submitted.

Bulant, P., Klimes, L. & Psencik, I., 2000. Comparison of ray methods with the exact solution
in the 1-D anisotropic "twisted crystal" model. *Expanded Abstracts with Author's Biographies*,
**70**, 2289-2292.

Hanyk, L., Matyska, C. and Yuen, D.A., 2000. Determination of viscoelastic
spectra by eigenvalue analysis.* Glacial Isostatic Adjustment and the
Earth System: Sea-Level, Crustal Deformation, Gravity and Rotation*,
eds. J. X. Mitrovica and L. L. A. Vermeersen, AGU, submitted.

Hanyk, L., Matyska, C., Yuen, D.A.,2000. The problem of viscoelastic relaxation of the Earth solved by a matrix eigenvalue approach based on discretization in grid space.
*Electronic Geosciences*,** 5**.

Klimes, L., 2000a. Lyapunov exponents for 2-D ray tracing without interfaces.
*SEG Expanded Abstracts*,** 70**, 2293-2296.

Klimes, L., 2000 b. Comparison of ray-matrix and finite-difference methods
in a simple 1-D model. *Expanded Abstracts with Author's Biographies*,** 70**, 2325-2328.

Klimes, L. Calculation of the third and higher travel-time derivatives in isotropic and anisotropic media, *Geophys. J. int.*, submitted.

Klimes, L., Perturbation of the polarization vectors in the isotropic ray theory.
*Pure and appl. Geophys.*, submitted.

Klimes, L. Lyapunov exponents for 2-D ray tracing without interfaces.
*Pure and appl. Geophys.*, submitted.

Klimes, L. Application of the medium covariance functions to travel-time tomography.
*Pure and appl. Geophys.*, submitted.

Klimes, L. Estimating the correlation function of a self-affine random medium.
*Pure and appl. Geophys.*, submitted.

Klimes, L. Correlation functions of random media.
*Pure and appl. Geophys.*, submitted.

Martinec, Z., 2000. Spectral-finite element approach to three-dimensional viscoelastic
relaxation in a spherical earth. *Geophys. J. Int.*,** 142**, 117-141.

Martinec, Z., Thoma, M., Wolf, D., 2000.
Material versus local incompressibility and its influence on
glacial-isostatic adjustment.
*Geophys. J. Int.*, in press.

Martinec, Z., 2000.
Viscoelastic relaxation in a three-dimensional spherical earth.
*Glacial Isostatic Adjustment and the Earth System:
Sea-Level, Crustal Deformation, Gravity and Rotation*,
AGU Monograph Series, eds. J.X. Mitrovica and B. Vermeersen, submitted.

Martinec, Z., Grafarend, E. W., 2000.
Green's function solution to gradiometric boundary-value problems.
*phys. J. Int.*, submitted.

Matyska, C. and Yuen, D.A., 2000. Are mantle plumes adiabatic?
*Earth Planet. Sci. Lett.*, submitted.

Matyska, C., Yuen, D.A., 2000. Profiles of the Bullen parameter from mantle
convection modelling,* Earth Planet. Sci. Lett.*, **178**, 39-46.

Mistr, Z., Matyska, C. and Yuen, D.A., 2000. Modeling of rapidly rotating thermal convection at finite Prandtl number with vorticity, vector potential and temperature.
*Studia Geophys. et Geod.*, submitted.

Novak, P., Vanicek, P., Martinec, Z.,Veronneau, M., 2000.
Effects of the spherical terrain on gravity and the geoid.
*J. Geod.*, submitted.

Novotny, O., Zahradnik, J., and Tselentis, G-A. North-western Turkey earthquakes and the crustal structure inferred from surface waves observed in the Corinth Gulf, Greece. *Bull. Seism. Soc. Am*, submitted.

Oprsal, I., Zahradnik, J. 3D finite-difference method and hybrid modeling of earthquake ground motion, *J. Geophys. Res.*, submitted.

Riepl, J., Zahradník, J., Plicka, V., Bard, P.-Y., 2000. About the efficiency of numerical 1D and 2D modeling of site effects in basin structures. *Pure and Appl. Geophys.*, **157**, 319-342.

Tselentis, G.-A., Zahradnik, J., 2000a. The Athens earthquake of September 7, 1999. *Bull. Seism. Soc. Am.*,** 90**, No. 5, 1143-1160.

Tselentis, G.-A., Zahradnik, J., 2000b. Aftershock monitoring of the Athens earthquake of September 7, 1999. *Seism. Res. Letters *,**71**, No. 3, 330-337.

Vanicek, P., Novak, P., Martinec, Z., 2000.
Geoid, topography, and the Bouguer plate or shell.
* J. Geod.*, in press.

Vecsey, L., Matyska, C. Wavelet spectra and chaos in thermal convection
modelling. *Geophys. Res. Lett.*, in press.

Velimsky, J., C. Matyska, 2000. The influence of adiabatic heating/cooling
on magnetohydrodynamic systems, *Phys. Earth Planet. Inter.*,** 117**,
197-207.

Zacek, K. Smoothing the Marmousi model.
*Pure and appl. Geophys.*, submitted.

Zahradnik, J., Jansky, J., Papatsimpa, N., 2000. Focal mechanisms of weak earthquakes from amplitude spectra and polarities. *Pure and Appl. Geophys *, in press.