# Atle 2007 May 2

### From NA-Wiki

**Speaker:** Andreas Atle, Dept. of Earth Sciences, Memorial University of Newfoundland

**Title:** Seismic modeling and dispersion minimizing finite difference stencils

**When:** The NA seminar meets om Wed May 2 at 15:15 in D4523

**Abstract**

The seismic modeling with acoustic, acoustic variable density and elastic wave equations is extensively used within geophysical research community as the basis for inverse methods. These model driven inverse methods range from reverse time migration to complex adjoint schemes. Reverse time migration is an old idea; it uses the time reversal nature of the wave equation. The adjoint schemes compute the model error forward in time and back propagate the error using reverse time migration. These numerical methods can be implemented using traditional finite differences. However, the dispersion error of wave propagation limits their effectiveness. Typical earth models are large with a scale of 500 to 1000 wave lengths. With large models and poor difference approximations new methods are needed for effective seismic modeling. With these motivations we have undertaken a research project into high quality seismic modeling.

A new 29-point diamond-shaped stencil for the acoustic wave equation in 2D has been developed. The stencil minimizes the error in a discrete dispersion relation. A similar technique has been used by Shin et al for Helmholtz equation in frequency domain for 9 and 25 points. In time domain we find it necessary to extend the grid to 29 points. We use symmetries that reduce the number of parameters to 8. Each parameter is a polynomial in the cfl-number cdt/h, so the total number of unknowns are 24. The stencil is designed to be good for problems with few points per wavelength. We chose to minimize the error in the dispersion relation down to 4 points per wavelength. The stencil is stable for cfl-numbers up to $0.708$ using von Neumann analysis. Computations indicates very good results up to 600-1000 wavelengths. Generalizations to 3D has also been developed. In 3D we have a 129-point stencil with 10 parameters with 30 unknowns.