Inverse extrapolation of primary seismic waves

Abstract

Forward wave‐field extrapolation operators simulate propagation effects from one depth level to another. Inverse wave‐field extrapolation operators eliminate those propagation effects. Since forward wave‐field extrapolation can be described in terms of spatial convolution, inverse wave‐field extrapolation can be described in terms of spatial deconvolution. A simple approximation to a stable, spatially band‐limited deconvolution operator is obtained by taking the complex conjugate of the convolution operator. A one‐way version of the Kirchhoff integral that contains the conjugate complex Green’s function is derived. Unlike the situation with respect to the forward problem, the modification of the closed surface integral into an open boundary integral involves an approximation that is identical to the approximation in the conjugate complex deconvolution approach. This approximation neglects the evanescent field and causes a second‐order amplitude error. For a plane acquisition surface, the one‐way Kirchhoff integral is transformed into a one‐way Rayleigh integral. For media with small to moderate contrasts, the one‐way Rayleigh integral with the conjugate complex Green’s function describes true amplitude inverse extrapolation of primary waves. This is illustrated with an example, in which the Green’s function has been modeled with the Gaussian beam method.