Explicit 3-D migration using equiripple polynomial expansion and Laplacian synthesis

Abstract

In f-x explicit finite‐difference depth migration schemes, wavefield downward extrapolation is carried out through spatial convolution using finite‐length filters. Existing methods for computing these filters are based on nonlinear least‐squares, with a high computational cost, or on Taylor series expansion, which is suboptimal. In the 3-D case, the physics of wavefield extrapolation requires 2-D extrapolation filters with circular symmetry. Recently, McClellan transformation has been used to design circularly symmetric extrapolation operators. But this approach exhibits artifacts when the data are not spatially oversampled. We describe an alternative method to take advantage of the circular symmetry: the radial response of the filter is expanded as a polynomial in the Laplacian, which is synthesized as the sum of two 1-D second‐derivative filters. Using the Laplacian rather than the McClellan transform yields an artifact‐free impulse response for wavenumbers much closer to the Nyquist wavenumber at the same computational cost. Other advantages of the proposed method are the easy extension to a rectangular grid and the possibility of time‐migration implementation. The coefficients of the polynomials are optimized in the [Formula: see text] norm, because the stability condition is expressed more easily with this norm. The Remez exchange algorithm, which is a fast [Formula: see text] norm spectral synthesis algorithm, is adapted to obtain these [Formula: see text] coefficients of the polynomials, as well as the coefficients of the second derivative filters.