Yu Zhang

Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration

One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By “true-amplitude” one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in the standard wave-equation migration imaging condition. The boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation. When the full wave-form solutions are replaced by their ray-theoretic approximations, the imaging formula reduces to the common-shot Kirchhoff inversion formula. In this sense...

Practical Issues of Reverse Time Migration - True-amplitude Gathers, Noise Removal and Harmonic-source Encoding

Summary We analyze the amplitude behavior of reverse-time migration and show that modifying the initial-value problem into a boundary-value problem for the source wavefield, plus implementing an appropriate imaging condition, yields a true-amplitude version of RTM. We also discuss different ways to suppress the migration artifacts. Finally, we introduce a “harmonic-source” phase-encoding method to allow a relatively efficient delayed-shot or plane-wave RTM. Taken together, these yield a powerful true-amplitude migration method that uses the complete two-way acoustic wave equation to image complex structures.

True amplitude wave equation migration arising from true amplitude one-way wave equations

One-way wave operators are powerful tools for use in forward modelling and inversion. Their implementation, however, involves introduction of the square root of an operator as a pseudo-differential operator. Furthermore, a simple factoring of the wave operator produces one-way wave equations that yield the same travel times as the full wave equation, but do not yield accurate amplitudes except for homogeneous media and for almost all points in heterogeneous media. Here, we present augmented one-way wave equations. We show that these equations yield solutions for which the leading order asymptotic amplitude as well as the travel time satisfy the same differential equations as the corresponding functions for the full wave equation. Exact representations of the square-root operator appearing in these differential equations are elusive, except in cases in which the heterogeneity of the medium is independent of the transverse spatial variables. Here, we address the fully heterogeneous case. Singling out depth as the preferred direction of propagation, we introduce a representation of the square-root operator as an integral in which a rational function of the transverse Laplacian appears in the integrand. This allows us to carry out explicit asymptotic analysis of the resulting one-way wave equations. To do this, we introduce an auxiliary function that satisfies a lower dimensional wave equation in transverse spatial variables only. We prove that ray theory for these one-way wave equations leads to one-way eikonal equations and the correct leading order transport equation for the full wave equation. We then introduce appropriate boundary conditions at z = 0 to generate waves at depth whose quotient leads to a reflector map and an estimate of the ray theoretical reflection coefficient on the reflector. Thus, these true amplitude one-way wave equations lead to a true amplitude wave equation migration (WEM) method. In fact, we prove that applying the WEM imaging condition to these newly defined wavefields in heterogeneous media leads to the Kirchhoff inversion formula for common-shot data when the one-way wavefields are replaced by their ray theoretic approximations. This extension enhances the original WEM method. The objective of that technique was a reflector map, only. The underlying theory did not address amplitude issues. Computer output obtained using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data over the entire survey area must be collected from a single common-shot experiment. Multi-experiment data, such as common-offset data, cannot be used with this method as currently formulated. Research on extending the method is ongoing at this time.

Delayed-shot 3D depth migration

For 3D seismic imaging in structurally complex areas, the use of migration by wavefield extrapolation has become widespread. By its very nature, this family of migration methods operates on data sets that satisfy a wave equation in the context of a single, physically realizable field experiment, such as a common-shot record. However, common-shot migration of data recorded over dipping structures requires a migration aperture much larger than the recording aperture, resulting in extra computations. A different type of wave-equation record, the response to a linear or planar source, can be synthesized from all the common-shot records. Synthesizing these records from common-shot records involves slant-stack processing, or applying delays to the various shots; we call these records delayed-shot records. Delayed-shot records dont suffer from the aperture problems of common-shot records since their recording aperture is the length of the seismic survey. Consequently, delayed-shot records hold potential for eff...

True-amplitude, angle-domain, common-image gathers from one-way wave-equation migrations

True-amplitude wave-equation migration provides a quality migrated image of the earth’s interior. In addition, the amplitude of the output provides an estimate of the angular-dependent reflection coefficient, similar to the output of Kirchhoff inversion. Recently, true-amplitude wave-equation migration for common-shot data has been proposed to generate amplitude-reliable, shot-domain, common-image gathers in heterogeneous media. We present a method to directly produce angle-domain common-image gathers from both common-shot and shot-receiver wave-equation migration. Generating true-amplitude, shot-domain, common-image gathers requires a deconvolution-type imaging condition using the ratio of the upgoing and downgoing wavefield, each downward-projected to the image point. Producing true-amplitude, angle-domain, common-image gathers requires, instead, the product of the upgoing wavefield and the complexconjugate of the downgoing wavefield in the imaging condition. Since multiplication is a more stable comput...

Reverse-time Migration: Amplitude And Implementation Issues

We formulate reverse-time migration (RTM) based on the theory of true-amplitude migration, and we give formulations for true-amplitude RTM angle-domain common-image gathers. Then we address some implementation issues for RTM. Specifically, we compare RTM’s efficiency using different orders of finite differencing along the time direction. Finally, we propose “harmonic-source migration”, a phase-encoding technique that allows increased efficiency in a delayed-shot implementation of RTM.

Imaging complex salt bodies with turning‐wave one‐way wave equation

We propose a modified version of one-way wave equation migration which incorporates true amplitude corrections to enhance steep dips and propagates wavefields to any possible directions. With this new method, turning waves can be properly imaged and the imaging capability of one-way wave equation is greatly improved. Introduction One-way wave equation migration has been widely used in 3-D seismic processing for imaging complex structures. To image steeply dipping reflectors, sometimes we have to depend on the reflections conveyed by turning waves. However, conventional one-way wave equation can only compute the wavefield with propagation angles less than 90o relative to the vertical direction. Therefore steeply dipping salt flanks and the underside of domes, which require turning waves to generate an image, are often absent from seismic images. To properly delineate complex salt bodies, we need a wave equation based migration that can image all dips, even beyond 90o. In this paper we propose a modified version of the one-way wave equation migration that can propagate wavefields to any possible direction. Also, we incorporate true amplitude corrections in the migration to enhance steep dips. With this new method, turning waves are properly imaged by a prestack depth migration. Theory and Algorithm According to Zhang (1993), the two-way wave equation can be approximately split into the following coupled one-way wave equation system

Migration/inversion: think image point coordinates, process in acquisition surface coordinates

We state a general principle for seismic migration/inversion processes: think image point coordinates; compute in surface coordinates. This principle allows the natural separation of multiple travel paths of energy from a source to a reflector to a receiver. Further, the Beylkin determinant (Jacobian of transformation between processing parameters and acquisition surface coordinates) is particularly simple in stark contrast to the common-offset Beylkin determinant in standard single arrival Kirchhoff .A feature of this type of processing is that it changes the deconvolution structure of Kirchhoff operators or the deconvolution imaging operator of wave equation migration into convolution operators; that is, division by Greens functions is replaced by multiplications by adjoint Greens functions.This transformation from image point coordinates to surface coordinates is also applied to a recently developed extension of the standard Kirchhoff inversion method. The standard method uses Greens functions in the integration process and tends to produce more imaging artefacts than alternatives, such as methods using Gaussian beam representations of Greens functions in the inversion formula. These methods point to the need for a true-amplitude Kirchhoff technique that uses more general Greens functions: Gaussian beams, true-amplitude one-way Greens functions, or Greens functions from the two-way wave equation. Here, we present a derivation of a true-amplitude Kirchhoff that uses these more general Greens functions. When this inversion is recast as an integral over all sources and receivers, the formula is surprisingly simple.

Delayed-shot 3D depth migration, GEOPHYSICS, 70, E21–E28

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How to Obtain True Amplitude Common-angle Gathers From One-way Wave Equation Migration?

True amplitude wave equation migration (WEM) provides the quality image of wave equation migration along with proper weighting of the output for estimation of an angularly dependent reflection coefficient, similar to the output of Kirchhoff inversion. In an earlier paper, Zhang et al. (2003) presented a true amplitude WEM for common-shot data in heterogeneous media. Here we present two additional true amplitude heterogeneous media WEM techniques to produce common-angle gathers. The first is based on the same common-shot image gathers, while the second is based on a modified double-square-root one-way wave equation. True amplitude common-shot migration was based on an imaging condition using the ratio, pU /pD . In contrast, common-angle gathers require the product pU p ∗ D in the imaging condition. We demonstrate this new method with a simple synthetic example.