The Theory of True Amplitude One‐Way Wave Equation Migrations

Abstract

In this technical report, we review the recent development of true amplitude one-way wave equation migration and summarize the migrated amplitude performance from different one-way wave equations. To produce correct amplitude from postack phase-shift migration, we have to apply accurate geometrical spreading compensation in preprocessing. Although Dubrulle's common-offset phase-shift migration gives correct imaging position, it is not a true amplitude migration except for zero offset or flat reflectors. We have shown that the conventional common-shot one-way wave equation migration does not produce correct migration amplitude. Based on the true amplitude one-way wave equations and the surface boundary condition corrections, we have developed true amplitude common-shot migration and have proved that it produces an inversion output that agrees asymptotically to Kirchhoff inversion. In the next step, we propose a way to obtain true amplitude common-angle image gathers from both single-square-root and double-square-root wave equation migrations. Our analysis indicates that the pUp * D imaging condition is a good candidate to produce true amplitude angle domain common imaging gathers. Compared to the pU/pD imaging condition which is required for true amplitude common-shot migration, the new approach suggests a more stable way of doing inversion and is more attractive to the real seismic data processing.

Preserved amplitude migration based on the one way wave equation in the angle domain

“…One way preserved amplitude pre-stack depth migration Based on the one way split equation proposed by Zhang Guanquan, Zhang Yu (Zhang et al, 2005;Zhang, 2006) defi ned a new pressure wave fi eld and derived the following preserved amplitude common shot migration algorithm which satisfies the one way equation and boundary condition:…”

“…using the wave fi eld splitting method proposed by Zhang (1993;2000). Zhang et al (2005) and Zhang (2006) use a split one-way wave equation operator for true amplitude and an initial condition for the correcting surface to reconstruct conventional migration into a true amplitude migration algorithm which has been shown to be equivalent to true Kirchhoff migration. Cui et al (2004) and Liu et al (2004) have done some work on mixing preserved amplitude migration with 3-D heterogeneous media.…”

Amplitude preserved VSP reverse time migration for angle-domain CIGs extraction

“…In order to solve these problems, the advanced angle domain CIGs is introduced which can naturally deal with the artifacts in ODCIGs and can be directly used for AVO/AVA analysis (Zhang et al, 2007a). ADCIGs can also be output by both Kirchhoff (Brandsberg-Dahl et al, 1999;Xu et al, 2001) and wave-equation migration (de Bruin et al, 1990;Prucha et al,1999;Mosher et al, 2001;Fomel, 2003 and2006;Fomel, 2004). If the velocity model is accurate, the ADCIGs generated by these two methods have similar characteristics, which are fl at-event gathers.…”

“…However, when lateral velocity varies signifi cantly, Kirchhoff migration based on the high-frequency approximation assumption suffers from diffi culties in imaging and severe artifacts in ODCIGs because the multiple paths can cause ODCIGs artifacts (Xu and Gilles, 2006;Zhang et al, 2006 and. In order to solve these problems, the advanced angle domain CIGs is introduced which can naturally deal with the artifacts in ODCIGs and can be directly used for AVO/AVA analysis (Zhang et al, 2007a).…”

Anisotropic converted wave amplitude-preserving prestack time migration by the pseudo-offset method

“…Zhang et al (2000; also provided amplitudepreserving migration based on the wave equation and estimated that the amplitude-preserving migration evolution for the wave equation is equivalent to the Kirchhoff equation. Zhang (2006) developed amplitudepreserving migration based on travel time and derived an amplitude-preserving migration equation.…”