Depth migration by the Gaussian beam summation method

Abstract

Seismic depth migration aims to produce an image of seismic reflection interfaces. Ray methods are suitable for subsurface target-oriented imaging and are less costly compared to two-way wave-equation-based migration, but break down in cases when a complex velocity structure gives rise to the appearance of caustics. Ray methods also have difficulties in correctly handling the different branches of the wavefront that result from wave propagation through a caustic. On the other hand, migration methods based on the two-way wave equation, referred to as reverse-time migration, are known to be capable of dealing with these problems. However, they are very expensive, especially in the 3D case. It can be prohibitive if many iterations are needed, such as for velocity-model building. Our method relies on the calculation of the Green functions for the classical wave equation by per-forming a summation of Gaussian beams for the direct and back-propagated wavefields. The subsurface image is obtained by cal-culating the coherence between the direct and backpropagated wavefields. To a large extent, our method combines the advantages of the high computational speed of ray-based migration with the high accuracy of reverse-time wave-equation migration because it can overcome problems with caustics, handle all arrivals, yield good images of steep flanks, and is readily extendible to target-oriented implementation. We have demonstrated the quality of our method with several state-of-the-art benchmark subsurface models, which have velocity variations up to a high degree of complexity. Our algorithm is especially suited for efficient imaging of selected subsurface subdomains, which is a large advantage particularly for 3D imaging and velocity-model refinement applications such as subsalt velocity-model improvement. Because our method is also capable of providing highly accurate migration results in structurally complex subsurface settings, we have also included the concept of true-amplitude imaging in our migration technique.

A fast algorithm for prestack Gaussian beam migration adopting the steepest descent approximation

“…Gray and Bleistein (2009) derived true-amplitude prestack Gaussian beam migration formulae based on the cross-correlation imaging condition and the deconvolution imaging condition, respectively, and proposed the method to solve the formulae by the steepest descent approximation. Popov et al (2010) expressed the Green function in the Kirchhoff integral formula by the Gaussian beam summation method to extrapolate the recorded wavefields and proposed a prestack depth migration method which was similar to the reverse time migration. Bleistein and Gray (2010) proposed the method for computing the Hessian matrix in the formula of 3D prestack Gaussian beam migration by the steepest descent approximation.…”

Dynamically Focused Gaussian Beams for Seismic Imaging

“…Nonetheless, this type of imaging can be important with certain true amplitude formulations where planar and localized beams are required at the imaging points in the subsurface [20,21]. These formulations originally involved launching beams directly from the scattering points in the subsurface up to the surface [18]. In the faster alternative proposed here, dynamically focused Gaussian beams are launched from the surface down to the imaging points at International Journal of Geophysics depth instead.…”

“…This reduces the number of beams required for the summation at the receiver, and also planar beam fronts at the receiver provide more stable beam summations. Recent true amplitude migration formulations using Gaussian beams have used beams launched directly from the scattering points up to the surface with the beam waists specified at the scattering points [18,20,21]). However, it is more economical to launch beams from the source and receiver positions down into the subsurface since there are fewer source and receiver locations than subsurface scattering points, and this minimizes the amount of beam tracing required.…”

“…Here curved initial beams are used to decompose the data and these are then propagated into the subsurface using dynamically focused Gaussian beams. An alternative is to shoot beams from the scattering locations upwards to the source and receiver plane as was done by Popov et al [18], but this will be computationally slower than shooting beams from the surface.…”

An amplitude-preserved adaptive focused beam seismic migration method

“…Because the smoothed velocity is required in ray-based migration for numerical stability in ray tracing, here we use a damped least squares algorithm to smooth the velocity (see Fig. 8b), which permits us to specify the degree of smoothing of the firstand second-order derivatives of the velocity (Popov et al 2010). The depth images migrated with the Kirchhoff method and GBM with different initial widths l 0 , one-way wave equation migration and the proposed method are shown in Fig.…”