Samuel H. Gray

An overview of depth imaging in exploration geophysics

Prestack depth migration is the most glamorous step of seismic processing because it transforms mere data into an image, and that image is considered to be an accurate structuraldescriptionoftheearth.Thus,ourexpectationsofitsaccuracy, robustness, and reliability are high.Amazingly, seismic migration usually delivers. The past few decades have seen migration move from its heuristic roots to mathematically sound techniques that, using relatively few assumptions, render accurate pictures of the interior of the earth. Interestingly,theearthandthesubjectswewanttoimageinside itarevariedenoughthat,sofar,nosinglemigrationtechnique has dominated practical application.All techniques continually improve and borrow from each other, so one technique mayneverdominate.Despitetheprogressinstructuralimaging,wehavenotreachedthepointwhereseismicimagesprovide quantitatively accurate descriptions of rocks and fluids. Nor have we attained the goal of using migration as part of a purelycomputationalprocesstodeterminesubsurfacevelocity. In areas where images have the highest quality, we might be nearing those goals, collectively called inversion. Where data are more challenging, the goals seem elusive. We describe the progress made in depth migration to the present and the most significant barriers to attaining its inversion goalsinthefuture.Wealsoconjectureonprogresslikelytobe made in the years ahead and on challenges that migration mightnotbeabletomeet.

Prestack Gaussian-beam depth migration in anisotropic media

Gaussian-beam depth migration is a useful alternative to Kirchhoff and wave-equation migrations. It overcomes the limitations of Kirchhoff migration in imaging multipathing arrivals, while retaining its efficiency and its capability of imaging steep dips with turning waves. Extension of this migration method to anisotropic media has, however, been hampered by the difficulties in traditional kinematic and dynamic ray-tracing systems in inhomogeneous, anisotropic media. Formulated in terms of elastic parameters, the traditional anisotropic ray-tracing systems aredifficult to implement and inefficient for computation, especially for the dynamic ray-tracing system. They may also result inambiguity in specifying elastic parameters for a given medium.To overcome these difficulties, we have reformulated the ray-tracing systems in terms of phase velocity.These reformulated systems are simple and especially useful for general transversely isotropic and weak orthorhombic media, because the phase velocities for thes...

Taking apart beam migration

The years 2000–2001 sparked a flurry of activity on various flavors of beam migration. GEOPHYSICS papers by Yonghe Sun et al. on slant-stack Kirchhoff migration and Ross Hill on Gaussian-beam migration showed the potential of migration methods that combine aspects of Kirchhoff migration with some novel preprocessing. As a result, a number of variant beam-migration methods have arisen in the last few years, some promising great efficiency and some promising great imaging fidelity. On the other hand, because of beam migrations extra preprocessing, a simple interpretation of beam migration, analogous to that of Kirchhoff migration, has been hard to pin down. In this article, we try to add some intuition to the discussion of beam-migration methods. Our task is challenging since, for the most part, we will describe Gaussian-beam migration, which is possibly the most complicated of the slant-stack migrations. Of course, a successful understanding—even a partial understanding—of this important method will make ...

Amplitude calculations for 3D Gaussian beam migration using complex-valued traveltimes

Gaussian beams are often used to represent Greens functions in three-dimensional Kirchhoff-type true-amplitude migrations because such migrations made using Gaussian beams yield superior images to similar migrations using classical ray-theoretic Greens functions. Typically, the integrand of a migration formula consists of two Greens functions?each describing propagation to the image point?one from the source and the other from the receiver position. The use of Gaussian beams to represent each of these Greens functions in 3D introduces two additional double integrals when compared to a Kirchhoff migration using ray-theoretic Greens functions, thereby adding a significant computational burden. Hill (2001 Geophysics 66 1240?50) proposed a method for reducing those four integrals to two, compromising slightly on the full potential quality of the Gaussian beam representations for the sake of more efficient computation. That approach requires a two-dimensional steepest descent analysis for the asymptotic evaluation of a double integral. The method requires evaluation of the complex traveltimes of the Gaussian beams as well as the amplitudes of the integrands at the determined saddle points. In addition, it is necessary to evaluate the determinant of a certain (Hessian) matrix of second derivatives. Hill (2001 Geophysics 66 1240?50) did not report on this last part; thus, his proposed migration formula is kinematically correct but lacks correct amplitude behavior. In this paper, we derive a formula for that Hessian matrix in terms of dynamic ray tracing quantities. We also show in a simple example how the integral that we analyze here arises in a true amplitude migration formula.

Converted-wave beam migration with sparse sources or receivers

Gaussian beam depth migration overcomes the single-wavefront limitation of most implementations of Kirchhoff migration and provides a cost-effective alternative to full-wavefield imaging methods such as reverse-time migration. Common-offset beam migration was originally derived to exploit symmetries available in marine towedstreamer acquisition. However, sparse acquisition geometries, such as cross-spread and ocean bottom, do not easily accommodate requirements for common-offset, common-azimuth (or common-offset-vector) migration. Seismic data interpolation or regularization can be used to mitigate this problem by forming well-populated common-offset-vector volumes. This procedure is computationally intensive and can, in the case of converted-wave imaging with sparse receivers, compromise the final image resolution. As an alternative, we introduce a common-shot (or common-receiver) beam migration implementation, which allows migration of datasets rich in azimuth, without any regularization pre-processing required. Using analytic, synthetic, and field data examples, we demonstrate that converted-wave imaging of ocean-bottomnode data benefits from this formulation, particularly in the shallow subsurface where regularization for common-offset-vector migration is both necessary and difficult.

Kirchhoff Inversion In Image Point Coordinates Recast As Source/Receiver Point Processing

Kirchhoff inversion theory tells us that reflection data provides information about the Fourier transform of the reflectivity function at each point in the illuminated subsurface. Thus, inversion formulas expressed as integrals in image point coordinates that closely characterize that Fourier domain are attractive for their relative simplicity. On the other hand, integrals over source/receiver coordinates are more natural to implement on seismic data. We propose a general principle for seismic migration/inversion (MI) 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. Furthermore, the Beylkin determinant is particularly simple in this formalism, and transforming to surface coordinates transforms deconvolution-type imaging and inversion operators into convolution-type operators with the promise of better numerical stability.

Computational inverse scattering in multi-dimensions

Many authors have shown that multidimensional inverse scattering using the Born approximation is asymptotically equivalent to the procedure for imaging seismic data known as Kirchhoff migration. The author shows how to carry out the imaging or inversion as accurately and efficiently as possible. Kirchhoff migration consists of two parts: computing travel-time curves linking every output point (at which an image is desired) with every source or receiver location (at which scattering data is collected), and for each output point, summing data over a travel-time curve to obtain a diffractivity value. If the input and output volumes are two-dimensional, then the Kirchhoff summation requires O(N3) multiply/add operations, but computing travel-time curves for all the output points by standard ray-tracing techniques requires O(N4) operations. It is shown how to reduce the operation count for the travel-time calculations to O(N3), and also how to do this without violating the inverse-scattering requirement that the background velocity structure vary slowly.

Amplitude Calculations For 3-D Gaussian Beam Migration Using Complex-valued Traveltimes

Gaussian beams are often used to represent Green’s functions in three-dimensional Kirchhoff-type true-amplitude migrations because such migrations yield superior images to similar migrations using classical ray-theoretic Green’s functions. Typically, the integrand of a migration formula consists of two Green’s functions, each describing propagation to the image point—one from the source position and the other from the receiver position.