Least‐squares migration of incomplete reflection data

Abstract

A least‐squares migration algorithm is presented that reduces the migration artifacts (i.e., recording footprint noise) arising from incomplete data. Instead of migrating data with the adjoint of the forward modeling operator, the normal equations are inverted by using a preconditioned linear conjugate gradient scheme that employs regularization. The modeling operator is constructed from an asymptotic acoustic integral equation, and its adjoint is the Kirchhoff migration operator. We tested the performance of the least‐squares migration on synthetic and field data in the cases of limited recording aperture, coarse sampling, and acquisition gaps in the data. Numerical results show that the least‐squares migrated sections are typically more focused than are the corresponding Kirchhoff migrated sections and their reflectivity frequency distributions are closer to those of the true model frequency distribution. Regularization helps attenuate migration artifacts and provides a sharper, better frequency distribution of estimated reflectivity. The least‐squares migrated sections can be used to predict the missing data traces and interpolate and extrapolate them according to the governing modeling equations. Several field data examples are presented. A ground‐penetrating radar data example demonstrates the suppression of the recording footprint noise due to a limited aperture, a large gap, and an undersampled receiver line. In addition, better fault resolution was achieved after applying least‐squares migration to a poststack marine data set. And a reverse vertical seismic profiling example shows that the recording footprint noise due to a coarse receiver interval can be suppressed by least‐squares migration.

Multi‐source least‐squares reverse time migration

“…The least-squares migration method (Lailly, 1984;Cole and Karrenbach, 1992;Schuster, 1993;Nemeth et al, 1999;Duquet et al, 2000) has been shown to sometimes produce migration images with better quality than those computed by conventional migration. When least-squares migration is implemented with the RTM method (Tang and Biondi, 2009;Dai and Schuster, 2010a;Dai et al, 2010;Wong et al, 2011;Dai et al, 2012), it can reduce not only the acquisition footprint but also the artifacts in the RTM image, while enhancing the image resolution.…”

“…Least-squares migration was originally implemented with Kirchhoff migration (KM) method (Nemeth et al, 1999;Duquet et al, 2000), but was later developed for phase shift migration algorithms (Kaplan et al, 2010;Huang and Schuster, 2012a). In this dissertation, I propose to implement LSM with reverse time migration (RTM) (Baysal et al, 1983;Whitmore, 1983;McMechan, 1983), where the Green's functions are calculated by finite-difference solution to the full wave equation instead of asymtotic Green's function in Kirchhoff migration (Bleistein, 1987), or one-way solution in one-way wave equation migration (Claerbout, 1985).…”

“…In reality, the conventional migration image usually contains migration artifacts caused by a limited recording aperture and/or coarse source and receiver sampling. To improve the image quality, Nemeth et al (1999) proposed to solve the problem by an iterative method and named the method least-squares migration (LSM), and it has been shown to have the following advantages: (1) it can reduce migration artifacts; (2) it can balance the amplitudes of the reflectors; and (3) it can improve the resolution of the migration images.…”

“…In this way, the effects of the source wavelet, limited recording aperture, geometric spreading, etc, are taken into account to produce images with reduced acquisition footprints, balanced amplitudes and improved resolution. Following the work of Lailly (1984) and Beydoun and Mendes (1989), Nemeth et al (1999) proposed an iterative linear inversion method they denoted as least-squares migration (LSM).…”

“…To indirectly account for the inverse Hessian matrix, an iterative conjugate gradient method can be used to solve either the linear (Lailly, 1984;Nemeth et al, 1999;Duquet et al, 2000) or non-linear (Tarantola, 1984;Mora, 1987) optimization problem. In this way, the effects of the source wavelet, limited recording aperture, geometric spreading, etc, are taken into account to produce images with reduced acquisition footprints, balanced amplitudes and improved resolution.…”