True-amplitude One-way Wave Equation Research Articles

True-amplitude one-way wave equation migration in the mixed domain

One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

Including lateral velocity variations into true-amplitude common-shot wave-equation migration

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

Migração PSPI baseada nas equações da onda unidirecionais com amplitude verdadeira

As equações da onda unidirecionais com amplitude verdadeira, propostas nos trabalhos mais recentes, levam em conta de forma correta a parte cinemática e dinâmica da equação completa da onda equivalentemente às obtidas a partir da aproximação de altas freqüências. Isto é, as equações iconal e de transporte obtidas para estes operadores se correspondem com as obtidas com a equação completa da onda. Num meio com variação lateral de velocidade, a implementação das equações unidirecionais com amplitude verdadeira envolve o cálculo numérico de operadores integro-diferenciais que podem ser aproximados por somatórios e resolvidos por técnicas de diferenças finitas. Para um meio que varia somente com a profundidade, as novas equações unidirecionais com amplitude verdadeira podem ser solucionadas através do tradicional método phase-shift , e as novas condições de fronteira e de imagem se reduzem a multiplicações no domínio de Fourier. Neste trabalho estamos propondo estender a migração phase-shift com amplitude verdadeira para meios com variação lateral v(x,z), através da metodologia Phase shift plus interpolation (PSPI). Experimentos numéricos no domínio de tiro comum com o novo método PSPI com amplitude verdadeira e sua extensão split-step (PSPI-SS), produzem imagens com amplitudes mais corretas em relação aos operadores unidirecionais convencionais, enquanto as características cinemáticas da migração PSPI e PSPI-SS são preservadas.

The Theory of True Amplitude One‐Way Wave Equation Migrations

AbstractIn 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 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.

Analysis of the neighborhood of a smooth caustic for true-amplitude one-way wave equations

In earlier papers it was shown that true-amplitude one-way wave equations provide “true amplitude” in the sense that the WKBJ or ray-theoretic solutions agree with the corresponding solutions of the full (two-way) wave equation. In the neighborhood of smooth caustics – smooth envelopes of rays – these ray-theoretic solutions break down with the predicted amplitudes of the solutions becoming infinite. For the two-way wave equation, that breakdown is well understood. The exact solution of the wave equation remains finite and it is only the WKBJ formalism that fails. Asymptotic solutions have been developed using higher functions (Airy functions). These solutions remain finite in a neighborhood of smooth caustics, adequately describing the exact solution, even in the limit where the observation point is on the caustic. A corresponding theory for the one-way wave equations is not yet available. However, numerical examples indicate that the solutions of the one-way wave equations remain finite and smooth near caustics. Furthermore, the mechanism for deriving the uniformly valid WKBJ solutions near smooth caustics strongly suggests that the uniform theory as derived for the two-way wave equation should be extendable to the one-way wave equation. Absent that extension, we present here a specific example in which we can analytically derive a uniformly valid asymptotic expansion near a smooth caustic via a standard extension of the method of stationary phase designed to deal with the neighborhood of smooth caustics.

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, the migration is true amplitude as well. On the other hand, this new method retains all of the fidelity features of wave equation migration. Computer output 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 must be collected from a single common-shot experiment. Multiexperiment data, such as common-offset data, cannot be used with this method as presently formulated.

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

One-way wave operators are powerful tools for use in forward modelling andinversion. Their implementation, however, involves introduction of the square rootof an operator as a pseudo-differential operator. Furthermore, a simple factoringof the wave operator produces one-way wave equations that yield the same traveltimes as the full wave equation, but do not yield accurate amplitudes except forhomogeneous media and for almost all points in heterogeneous media. Here, wepresent augmented one-way wave equations. We show that these equations yieldsolutions for which the leading order asymptotic amplitude as well as thetravel time satisfy the same differential equations as the correspondingfunctions for the full wave equation. Exact representations of the square-rootoperator appearing in these differential equations are elusive, except incases in which the heterogeneity of the medium is independent of thetransverse spatial variables. Here, we address the fully heterogeneous case.Singling out depth as the preferred direction of propagation, we introduce arepresentation of the square-root operator as an integral in which a rationalfunction of the transverse Laplacian appears in the integrand. This allows usto carry out explicit asymptotic analysis of the resulting one-way waveequations. To do this, we introduce an auxiliary function that satisfies a lowerdimensional wave equation in transverse spatial variables only. We prove thatray theory for these one-way wave equations leads to one-way eikonalequations and the correct leading order transport equation for the full waveequation. We then introduce appropriate boundary conditions at z = 0 togenerate waves at depth whose quotient leads to a reflector map and an estimateof the ray theoretical reflection coefficient on the reflector. Thus, these trueamplitude one-way wave equations lead to a ‘true amplitude wave equationmigration’ (WEM) method. In fact, we prove that applying the WEMimaging condition to these newly defined wavefields in heterogeneous medialeads to the Kirchhoff inversion formula for common-shot data when theone-way wavefields are replaced by their ray theoretic approximations.This extension enhances the original WEM method. The objective ofthat technique was a reflector map, only. The underlying theory did notaddress amplitude issues. Computer output obtained using numericallygenerated data confirms the accuracy of this inversion method. However, thereare practical limitations. The observed data must be a solution of thewave equation. Therefore, the data over the entire survey area must becollected from a single common-shot experiment. Multi-experiment data, suchas common-offset data, cannot be used with this method as currentlyformulated. Research on extending the method is ongoing at this time.