Multifrequency beam-based migration in inhomogeneous media using windowed Fourier transform frames

Abstract

Summary We consider the problem of inhomogeneous subsurface imaging using beam waves. The formulation is based on the ultra-wide-band phase space beam summation (UWB-PS-BS) method, which is structured upon windowed Fourier transform (WFT) expansions of surface fields and sources. In this approach, the radiated field is given as a superposition of beam propagators. Here we use the beams first for expanding the surface sources and the scattered data, and then for imaging where we use the backpropagation and cross-correlation of beams. This formulation enables a target oriented imaging approach, where we take into account only pairs of source and receiver beams that pass near a region of interest, and thus extract only the relevant data arriving from this region. It also leads to a-priori sparse representation of both the beam domain data and the beam propagators. A physical cogent for the beam domain data is obtained under the Born approximation. The beam domain data can be approximated as the local interaction between the beam propagators and the medium reflectivity. Thus, one may interpret the beam domain data as a local Snell’s law reflection in the direction defined by the vector summation of the incident beam and backpropagated beam ray parameters. We demonstrate a physical model for the beam domain data and the salient features of the proposed imaging algorithm using numerical examples.

Introduction to a Two‐Way Beam Wave Method and Its Applications in Seismic Imaging

“…From physical analysis, Hill (1990) proposed a dual‐parameter GBSM to represent seismic data recorded in an acquisition aperture at surface, which provides a self‐consistent beam parameter selection strategy. Currently, the frame theory has been introduced to optimize the coefficients of Gaussian beam summation for representing seismic data (Geng et al., 2014; Tuvi et al., 2020; Zacek, 2006), which provides rigorous mathematical basis for the beam‐forming strategy using an overdetermined frame in Hill (1990); Hill (2001)'s early work.…”

“…From physical analysis for plane waves at surface, Hill (1990) proposed the basic framework of wavefield decomposition using Gaussian beam summation. Later, the Gabor frame expansion is introduced to optimize the coefficients of Gaussian beams in wavefield decomposition, which mainly include the beam center position and initial propagation direction (Geng et al., 2014; Tuvi et al., 2020; Zacek, 2006). The windowed slant‐stacks used in Hill (1990) constitute an overdetermined collection or frame of elements representing observed seismic data (Nowack et al., 2007).…”

“…The windowed slant‐stacks used in Hill (1990) constitute an overdetermined collection or frame of elements representing observed seismic data (Nowack et al., 2007). The analysis based on the frame theory provides a rigorous mathematical foundation for the GBSM (Daubechies, 1990; Shlivinski et al., 2004; Tuvi et al., 2020). In addition, with a finite paraxial‐ray width, GBSM calculates the seismic wavefield at an observation point by summing the contributions of adjacent Gaussian beams, which does not need iterative adjustment of rays to exactly pass through the observation point as in classical KRT methods.…”