Guanquan 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...

An effective imaging condition for reverse-time migration using wavefield decomposition

Reverse-time migration (RTM) exhibits great superiority over other imaging algorithms in handling steeply dipping structures and complicated velocity models. However, low-frequency, high-amplitude noises commonly seen in a typical RTM image have been one of the major concerns because they can seriously contaminate the signals in the image if they are not handled properly. We propose a new imaging condition to effectively and efficiently eliminate these specific noises from the image. The method works by first decomposing the source and receiver wavefields to their one-way propagation components, followed by applying a correlation-based imaging condition to the appropriate combinations of the decomposed wavefields. We first give the physical explanation of the principle of such noises in the conventional RTM image. Then we provide the detailed mathematical theory for the new imaging condition. Finally, we propose an efficient scheme for its numerical implementation. It replaces the computationally intensiv...

A stable TTI reverse time migration and its implementation

Modeling and reverse time migration based on the tilted transverse isotropic (TTI) acoustic wave equation suffers from instability in media of general inhomogeniety, especially in areas where the tilt abruptly changes. We develop a stable TTI acoustic wave equation implementation based on the original elastic anisotropic wave equation. We, specifically, derive a vertical transversely isotropic wave system of equations that is equivalent to their elastic counterpart and introduce the self-adjoint differential operators in rotated coordinates to stabilize the TTI acoustic wave equations. Compared to the conventional formulations, the new system of equations does not add numerical complexity; a stable solution can be found by either a pseudospectral method or a high-order explicit finite difference scheme. We demonstrate by examples that our method provides stable and high-quality TTI reverse time migration images.

One-step extrapolation method for reverse time migration

We have proposed a new method, a one-step extrapolation algorithm, to solve the acoustic wave equation. By introducing a square-root operator, the two-way wave equation can be formulated as a first-order partial differential equation in time, which is similar to the one-way wave equation. To solve the new wave equation, we used a stable explicit extrapolation method in the time direction and handled lateral velocity variations in the space and wavenumber domains. Unlike the conventional explicit finite-difference schemes, the new method does not suffer from numerical instability or numerical dispersion problems. It can be used to design cost-effective and high-quality reverse time migration or modeling code.

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...

An anisotropic acoustic wave equation for modeling and migration in 2D TTI media

In this abstract, we propose a new anisotropic acoustic wave equation for 2D TTI media. This is the extension of our previous work on VTI media to TTI media. Similar to VTI case, we follow the same procedures as Zhou et al. (2006), by the introduction of an auxiliary function, the original fourth-order differential equation becomes a coupled system of lower-order differential equations. However, unlike VTI case, a cross-derivative term has been added to each of the coupled system of the hyperbolic differential equations because of its TTI characteristics. Of these two equations, one equation plays a key role in governing the propagation of the wavefront, with the other equation compensating for the loss of anisotropy for TTI media not only in the lateral but also in the depth directions. The new anisotropic acoustic wave equation has the obvious physical meaning and is much easier to implement. Impulse responses for both modeling and migration have been tested to show the validation of the proposed anisotropic acoustic wave equation.

Reverse-time Migration Using One-way Wavefield Imaging Condition

Reverse-time migration has the capability to image all dips including overturned structures. However, the conventional imaging condition produces high-amplitude noises in the image, which often seriously mask the shallow structures. In this paper, we propose a new imaging condition to eliminate these noises which works by decomposing the full wavefields to their one-way components, and applying the imaging condition to the appropriate combinations of the wavefield components. Numerical tests verify that this new imaging condition successfully removes the undesired noises.

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

Removing S-wave noise in TTI reverse time migration

Summary Anisotropy is intrinsically an elastic phenomenon. The widely-used acoustic anisotropic wave equation results in significant shear wave presence in both modeling data and reverse time migration images. In this abstract, we discuss two approaches to reduce the S-wave noise in migration. The first is based on an elastic anisotropic wave equation with non-zero S-wave velocity to attenuate S-wave noise. The second approach uses a supplementary routine to remove the S-waves after wavefield modeling in each time step. Impulse responses and a simple 2D TTI example show that both methods reduce S-wave energy significantly and lead to a cleaner image than those based on the conventional acoustic anisotropic wave equation.

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.