A Kirchhoff Approach to Seismic Modeling and Prestack Depth Migration

The Kirchhoff integral provides a robust method for implementing seismic modeling and prestack depth migration, which can handle lateral velocity variation and turning waves. With a little extra computation cost, the Kirchoff- type migration can obtain multiple outputs that have the same phase but different amplitudes, compared with that of other migration methods. The ratio of these amplitudes is helpful in computing some quantities such as reflection angle. Here, I develop a seismic modeling and prestack depth migration method based on the Kirchhoff integral, that handles both laterally variant velocity and a dip beyond 90 degrees. The method uses a finite-difference algorithm to calculate travel times and WKBJ amplitudes for the Kirchhoff integral. Compared to ray-tracing algorithms, the finite-difference algorithm gives an efficient implementation and single-valued quantities first arrivals on output. In my finite difference algorithm, the upwind scheme is used to calculate travel times, and the Crank-Nicolson scheme is used to calculate amplitudes. Moreover, interpolation is applied to save computation cost. The modeling and migration algorithms here require a smooth velocity function. I develop a velocity- smoothing technique based on damped least-squares to aid in obtaining a successful migration.