Double Plane Wave Kirchhoff Depth Migration

Abstract

We present a migration method in the coupled rayparameter domain that is fast and efficient for seismic data that are densely sampled in the source-receiver configuration space. The method is based on slant stacking over both shot positions and receiver positions (or offsets) for all the recorded data. If the data acquisition geometry permits, both in-line and cross-line source positions and receiver positions (or offsets) can be incorporated into a multidimensional phase velocity space which is regular even for randomly positioned input data. By noting the maximum time dips that are present in the shot and receiver gathers and constant offset sections, the number of plane waves required can be estimated and this generally results in a data reduction of at least one and possibly two orders of magnitude. The required travel time computations for depth imaging are independent for each particular plane wave component and thus can be used for either the source or the receiver plane waves during extrapolation in phase space, reducing considerably the computational burden. Even so, each source and receiver plane wave component must be combined with all other receiver and source components for a complete diffraction summation. Since only vertical delay times are required, many travel time techniques can be employed and the problems with multipathing and first arrivals are either reduced or eliminated. Further, the shot plane wave integral can be pruned to concentrate the image on selected targets. In this way the computation time can be further reduced and the technique lends itself naturally to a velocity modeling scheme where for example, horizontal and then steeply dipping events are gradually introduced into the analysis. Of course this imaging scheme can be implemented in parallel using a distributed architecture like a PC cluster to compute various plane wave sections since they are independent of each other. The common ray-parameter image gathers can be used exactly like common angle image gathers for residual migration velocity analysis. The migration method lends itself to imaging in anisotropic media since phase space is the natural domain for such an analysis. Introduction Typically the τ − p transformation is performed for each recorded seismic trace relative to its fixed source position (that is, with respect to the receiver’s offset relative to the source position, see Schultz and Claerbout, 1978; Stoffa et al., 1981). Given modern multi-coverage data , where s is the source location and r is the receiver location, there is no practical reason or obstacle not to apply the ( , , ) P s r t τ − p transformation with respect to r, or s, or even both (Fokkema and van den Berg, 1993). In the frequency domain the decomposition of the recorded data into source and receiver planewave components is accomplished using a variant of slant stacking which uses a linear phase delay of each trace with respect to its shot or receiver position. For receiver plane waves, we have the following forward and inverse stacking formulas: ( , , ) P s r t