Extrapolation of vector (elastic) displacements by displacement potential field continuation

Abstract

Extrapolating recorded elastic displacement data from the surface downwards is the first step of a two-step elastic migration scheme. The second step is the image formation condition. Although an elastic (vector) reverse time migration can extrapolate elastic displacement data downwards, the subsurface reflections and the loss of energy due to the use of the two-way wave equations are unavoidable. The elastic Kirchhoff-Helmholtz type integrals can not be used to extrapolate elastic displacement data downwards in media in which the velocity changes in both the x-direction and the z-direction. The present method attempts to first obtain the P- and S-wave displacement potentials near the surface. Then the P- and S-wave displacement potentials are each extrapolated (in a scalar fashion) with the split-step Fourier method, subject to modest velocity gradients. Finally, the elastic displacement fields within the subsurface can be obtained from the extrapolated potential fields. In order to obtain the P- and S-wave displacement potentials near the surface, the finite difference elastic reverse time migration method is used to extrapolate the recorded elastic surface displacement data downwards only a few depth intervals, assuming a constant velocity over this small depth region, so that the P- and S-displacement potentials can be computed from the calculated elastic displacement fields. Therefore the method can be used to extrapolate recorded elastic displacement fields from the surface downwards with almost no velocity limitation except near the surface (only a few depth intervals). It was tested on the numerical model data produced by a fourth order finite difference elastic forward modelling program with good results. The finite difference elastic reverse time migration method was used because a finite difference elastic forward modelling program had already been developed and could be run inverse. In addition, the use of the Kirchhoff-Helmholtz type integrals is preferable for the initial extrapolation of elastic surface displacement data downwards a few depth intervals to calculate potentials because of its high accuracy, no velocity limitation in the z-direction and no spurious boundary reflections. However such a scheme is computationally demanding and was not used in the present study.