Charles University in Prague

This research had been carried out at:

Charles University, Faculty of Mathematics and Physics, Department of Geophysics, V Hole sovi ckách 2, 180 00 Praha 8, Czech Republic

tel.: 42-2-8576 2546

fax.: 42-2-8576 2555

Email: io@karel.troja.mff.cuni.cz

jz@karel.troja.mff.cuni.cz

1 INTRODUCTION

2 BASIC EQUATIONS

3 PSi-2 SCHEME FOR IRREGULAR GRID

3.1 Irregular grid

3.2 A non-mixed derivative

3.3 A mixed derivative

3.3.1 Short form

3.3.2 Full form

3.3.3 The finite-difference approximation to the

3.3.4 The finite-difference approximation to the

3.3.5 Consistency with free-surface boundary conditions

3.3.6 Stability and accuracy

4 BOUNDARY CONDITIONS

4.1 Non-reflecting boundaries

4.1.1 Equations

Edges

Corners

4.1.2 Finite difference formulations

Edges

Corners

4.2 Tapers

4.3 Reflecting boundaries

4.3.1 Finite difference formulations

5 NUMERICAL EXPERIMENTS

5.1 Plane free surface

5.1.1 Homogeneous half-space

5.1.2 Two quarter-spaces

Model with regular grid

Model with irregular grid

5.1.3 Low-velocity layer

Model with regular grid

Model with irregular grid

5.2 Topographic features

5.2.1 Step-like surface of homogeneous half-space

5.2.2 Ridge-like surface of homogeneous half-space

Coarse grid model M1 (regular grid)

Fine grid model M2 (regular grid)

Fine grid model M3 (irregular grid)

Model M4 (regular grid)

5.2.3 Ramp-like surface of homogeneous half-space

5.2.4 Step-like layer of low velocities underlain by homogeneous half-space

Model S1 (regular grid)

Model S2 (irregular grid)

6 CONCLUSION

REFERENCES

INTRODUCTION

The models containing material discontinuities have been solved by two approaches: homogeneous and heterogeneous. The homogeneous approach is based on the use of different FD formulas for the internal grid points, and the surface grid points, corners and interfaces (the latter provided by the traction-continuity condition). The heterogeneous approach is using a single FD formula for all the points of the model. The interface conditions for the heterogeneous formulation are fulfilled through the discontinuous material parameters that are entering the discretized equation of motion.

The heterogeneous formulation used for the free surface is called "vacuum formalism", and it is represented by the elastic parameters, and the displacement equal to 0 above the free surface. The effective parameters used in the "vacuum formalism" (Zahradní k, Moczo & Hron, 1993); Zahradní k, O'Leary & Sochacki, 1994) are evaluated by geometrical averaging that converts the prescribed topography of the free surface into a step-like approximation. After that, the approximated free surface is composed of elementary steps of minimum height and width given by the vertical and the horizontal grid step, respectively.

To avoid the diffraction caused by the "step-like" surface, we have to employ a finer grid in the vicinity of the topographic free surface. The other reason for the fine meshing is the presence of the low velocity zones in which the fine gridding (with respect to wavelength) is necessary for keeping the accuracy of the computation. The fine grid is combined (via the stability condition) with a small time step, hence more computer time and memory is required.

One of the ways how to reduce the computer memory and time is to use an irregular grid. This grid is supposed to be dense in the places with surface topography and/or the low velocity zones.

The FD homogeneous-formulation modelling on the irregular grids [I], and topography [T] have recently been studied by Falk, Tessmer & Grajevski (1995) [I, irregular time step]; Jastram & Tessmer(1994) [I,T]; Hestholm & Ruud (19..)[I,T; Transformation from curved to rectangular grid]; Jih, McLaughlin & Der (1988) [Polygonal topography]; Hong & Bond (1986) [T]; Illan (1977) [T, arbitrary polygonal surface]. Heterogeneous FD formulation was studied by Moczo & Kristek (1996) [I]; Zahradní k & Priolo (1995) [regular grid, free surface]; Zahradní k & Hron (1992) [regular grid].

The topography models investigated by other methods were recently published by Seriani, Priolo, Carcione & Padovani, (1993)[Spectral Element Method]; Gaffet & Bouchon [Boundary Integral Equation Method]; Nielsen (1994) [Elimination of grid artifacts]; Tessmer, Kosloff & Behle (1992) [Surface topography achieved by mapping a rectangular grid onto a curved grid]; Kawase (1990) [Discrete wavenumber method, hybrid method].

The aim of this thesis is to derive a new FD scheme for the irregular grid (heterogeneous formulation), and to apply it to topography modelS on the irregular grid. The scheme is called PSi-2. The derivation of the PSi-2 IS based on the PS-2 scheme for the regular grid (Zahradní k, 1995).

The PSi-2 is numerically tested and the results are compared with other
methods. The purpose of the tests is to show, whether modelling on the
irregular grid can improve the accuracy with respect to the regular grids,
what is the efficiency of the irregular grids with respect the regular
fine grids, whether the irregular gridding produces numerical artifacts,
and how the irregular grids handle the topography models.

**For the whole thesis, see the top of this page.**