Maarten V. de Hoop

Focusing in dip and AVA compensation on scattering‐angle/azimuth common image gathers

Common image gathers (CIGs) in the offset and surface azimuth domain are used extensively in migration velocity analysis and amplitude variation with offset (AVO) studies. If the geology is complex and the ray field becomes multipathed, the quality of the CIGs deteriorates. To overcome these problems, the CIGs are generated as a function of scattering angle and azimuth at the image point. The CIGs are generated using an algorithm based on the inverse generalized Radon transform (GRT), stacking only over migration dip angles. Including only dips in the vicinity of the geological dip, or focusing in dip, suppresses artifacts in and results in improved signal‐to‐noise ratio on the CIGs.Migration velocity analysis can be based upon the differential semblance criterion. The analysis~is~then carried out by minimizing a functional of the derivative of the CIGs with respect to horizontal coordinates (offset/azimuth or scattering‐angle/azimuth), but AVO/amplitude variation with angle (AVA) effects will degrade the...

Generalization of the Bremmer coupling series

An operator formalism is developed to expand the acoustic wave field in a multi‐dimensionally smoothly varying medium, generated by a source localized in space and time, into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times. This expansion is a generalization of the Bremmer coupling series. The condition of smoothness of the medium relates to the width of the signature of the source in the configuration. Both the existence and the convergence (in the weak sense) of the expansion are discussed. The operator calculus involved leads to a natural generalization of the concept of slowness surface to multi‐dimensionally smoothly varying media. The operator associated with the corresponding generalized vertical slowness induces the full one‐way wave operator in the type of media under consideration. In addition, a wavefield decomposition operator as well as an interaction operator that couples the d...

Modeling and imaging with the scalar generalized‐screen algorithms in isotropic media

The phase‐screen and the split‐step Fourier methods, which allow modeling and migration in laterally heterogeneous media, are generalized here so as to increase their accuracies for media with large and rapid lateral variations. The medium is defined in terms of a background medium and a perturbation. Such a contrast formulation induces a series expansion of the vertical slowness in which we recognize the first term as the split‐step Fourier approximation and the addition of higher‐order terms of the expansion increases the accuracy. Employing this expansion in the one‐way scalar propagator yields the scalar one‐way generalized‐screen propagator. We also introduce a generalized‐screen representation of the reflection operator. The interaction between the upgoing and downgoing fields is taken into account by a Bremmer series. These results are then cast into numerical algorithms. We analyze the accuracy of the generalized‐screen method in complex structures using synthetic models that exhibit significant m...

Leading-order seismic imaging using curvelets

Curvelets are plausible candidates for simultaneous compressionofseismicdata,theirimages,andtheimagingoperator itself. We show that with curvelets, the leading-order approximation in angular frequency, horizontal wavenumber, and migrated location to common-offset CO Kirchhoff depth migration becomes a simple transformation of coordinates of curvelets in the data, combined with amplitude scaling. This transformation is calculated using map migration, whichemploysthelocalslopesfromthecurveletdecomposition of the data. Because the data can be compressed using curvelets, the transformation needs to be calculated for relatively few curvelets only. Numerical examples for homogeneous media show that using the leading-order approximationonlyprovidesagoodapproximationtoCOmigrationfor moderate propagation times.As the traveltime increases and raysdivergebeyondthespatialsupportofacurvelet;however, the leading-order approximation is no longer accurate enough. This shows the need for correction beyond leading order,evenforhomogeneousmedia.

GENERALIZED RADON TRANSFORM INVERSIONS FOR REFLECTIVITY IN ANISOTROPIC ELASTIC MEDIA

The resolution analysis of generalized Radon transform (GRT) inversion of seismic data is carried out in general anisotropic media. The GRT inversion formula is derived from the ray-Born approximation of the wave field for volume scatterers. However, by considering scattering surfaces in the resolution analysis, rather than parameter perturbations, we show that the inversion provides a reflectivity map and reflection/transmission coefficients as functions of scattering angles and azimuths. Those coefficients can be subjected to any type of amplitude versus angle (AVA) or amplitude versus offset (AVO) analysis. By applying the inversion to Kirchhoff approximate data rather than Born approximate data, we show that the output is actually linear in the reflection coefficients and, hence, a nonlinear function of the change in medium parameters across discontinuity surfaces - the reflectors of the medium.

Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients

We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmanders definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.

The resolving power of seismic amplitude data : An anisotropic inversion/migration approach

A description of the theory and numerical implementation of a 3-D linearized asymptotic anisotropic inversion method based on the generalized Radon transform is given. We discuss implementation aspects, including (1) the use of various coordinate systems, (2) regularization by both spectral and Bayesian statistical techniques, and (3) the effects of limited acquisition apertures on inversion. We give applications of the theory in which well-resolved parameter combinations are determined for particular experimental geometries and illustrate the interdependence of parameter and spatial resolutions. Procedures for evaluating uncertainties in the parameter estimates that result from the inversion are derived and demonstrated.

A Novel Application of Time Reversed Acoustics: Salt Dome Flank Imaging Using Walkaway VSP surveys

In this paper we present initial results of applying Time-Reversed Acoustics (TRA) technology to saltdome flank, seismic imaging. We created a set of synthetic traces representing a multilevel, walkaway VSP for a model composed of a simplified Gulf of Mexico vertical-velocity gradient and an embedded salt dome. We first applied the concepts of TRA to the synthetic traces to create a set of redatummed traces without having to perform velocity analysis, moveout corrections, or complicated processing. Each redatummed trace approximates the output of a zero-offset, downhole source and receiver pair. To produce the final salt-dome flank image, we then applied conventional, poststack, depth migration to the zero-offset section. Our results show a very good image of the salt when compared to an image derived using data from a downhole, zero-offset source and receiver pairs. The simplicity of our TRA implementation provides a virtually automated method to estimate a zero-offset, seismic section as if it had been collected from the reference frame of the borehole containing the VSP survey.

Modeling of seismic data in the downward continuation approach

Seismic data are commonly modeled by a high-frequency single scattering approximation. This amounts to a linearization in the medium coefficient about a smooth background. The discontinuities are contained in the medium perturbation. The high-frequency part of the wavefield in the background medium is described by a geometrical optics representation. It can also be described by a one-way wave equation. Based on this we derive a downward continuation operator for seismic data. This operator solves a pseudodifferential evolution equation in depth, the so-called double-square-root equation. We consider the modeling operator based on this equation. If the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal, the singular part of the data is described by the solution to an inhomogeneous double-square-root equation.

Kinematics of shot-geophone migration

Recent analysis and synthetic examples have shown that many prestack depth migration methods produce nonflat image gathers containing spurious events, even when provided with a kinematically correct migration velocity field, if this velocityfieldishighlyrefractive.Thispathologyoccursinall migration methods that produce partial images as independent migrations of data bins. Shot-geophone prestack depth migration is an exception to this pattern: each point in the prestack image volume depends explicitly on all traces within the migration aperture. Using a ray-theoretical analysis, we have found that shot-geophone migration produces focusedsubsurface-offsetdomainorflatscattering-angledomain image gathers, provided there is a curvilinear coordinate system defining pseudodepth with respect to which the rays carrying significant energy do not turn, and that the acquisitioncoverageissufficienttodetermineallsuchrays.Although the analysis is theoretical and idealized, a synthetic example suggests that its implications remain valid for practical implementations, and that shot-geophone prestack depth migration could be a particularly appropriate tool for velocityanalysisinacomplexstructure.