Jack K. Cohen

Mathematics of multidimensional seismic imaging, migration, and inversion

Preface.- Multi-Dimensional Seismic Inversion.- The One- Dimensional Inverse Problem.- Inversion in Higher Dimensions.- Large-Wavenumber Fourier Imaging.- Inversion in Heterogeneous Media.- Two-and-One Half Dimensional Inversion.- The General Theory of Data Mapping.- Distribution Theory.- The Fourier Transform of Causal Functions.- Dimensional vs. Dimensionless Variables.- An Example of Ill-Posedness.-An Elementary Introduction to Ray Theory.- Author Index.- Subject Index.

Nonuniqueness in the inverse source problem in acoustics and electromagnetics

A recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.

Velocity inversion procedure for acoustic waves

An approximate solution is presented to the seismic inverse problem for two‐dimensional (2-D) velocity variations. The solution is given as a multiple integral over the reflection data observed at the upper surface. An acoustic model is used, and the reflections are assumed to be sufficiently weak to allow a “linearization” procedure in the otherwise nonlinear inverse problem. Synthetic examples are presented demonstrating the accuracy of the method with dipping planes at angles up to 45 degrees and with velocity variations up to 20 percent. The method was also tested under automatic gain control, in which case velocity estimates were lost but the method nonetheless successfully migrated the data.

Two and one-half dimensional Born inversion with an arbitrary reference

Multidimensional inversion algorithms are presented for both prestack and poststack data gathered on a single line. These algorithms both image the subsurface (i.e., give a migrated section) and, given relative true amplitude data, estimate reflection strength or impedance on each reflector. The algorithms are “two and one‐half dimensional” (2.5-D) in that they incorporate three‐dimensional (3-D) wave propagation in a medium which varies in only two dimensions. The use of 3-D sources does not entail any computational penalty, and it avoids the serious degradation of amplitude incurred by using the 2-D wave equation. Our methods are based on the linearized inversion theory associated with the “Born inversion.” Thus, we assume that the sound speed profile is well approximated by a given background velocity, plus a perturbation. It is this perturbation that we seek to reconstruct. We are able to treat the case of an arbitrary continuous background profile. However, the cost of implementation increases as one...

Three‐dimensional Born inversion with an arbitrary reference

Recent work of G. Beylkin helped set the stage for very general seismic inversions. We have combined these broad concepts for inversion with classical high‐frequency asymptotics and perturbation methods to bring them closer to practically implementable algorithms. Applications include inversion schemes for both stacked and unstacked seismic data. Basic assumptions are that the data have relative true amplitude, and that a reasonably accurate background velocity c(x, y, z) is available. The perturbation from this background is then sought. Since high‐frequency approximations are used throughout, the resulting algorithms essentially locate discontinuities in velocity. An expression for a full 3-D velocity inversion can be derived for a general data surface. In this degree of generality the formula does not represent a computationally feasible algorithm, primarily because a key Jacobian determinant is not expressed in practical terms. In several important cases, however, this shortcoming can be overcome and ...

An Inverse Method for Determining Small Variations in Propagation Speed

We consider the inverse problem of determining small variations in propagation speed from observations of signals which pass through the medium of interest and are then observed remotely. We show that the variation satisfies an integral equation of integral transform type with an atypical kernel. In a variety of examples for the scalar and vector wave equation (Maxwell’s equations) and the equations of linear elasticity, we solve this integral equation by elementary means.

The singular function of a surface and physical optics inverse scattering

Abstract It is shown how to recover both the location and the reflection coefficient of a scatterer using only high frequency backscattered data. The result is obtained without use of the far field approximation although a separate identity is derived when this approximation is introduced. This latter result improves upon previously derived physical optics far field inverse scattering identities.

Computational and asymptotic aspects of velocity inversion

We discuss computational and asymptotic aspects of the Born inversion method and show how asymptotic analysis is exploited to reduce the number of integrations in an f-k like solution formula for the velocity variation. The output of this alternative algorithm produces the reflectivity function of the surface. This is an array of singular functions—Dirac delta functions which peak on the reflecting surfaces—each scaled by the normal reflection strength at the surface. Thus, imaging of a reflector is achieved by construction of its singular function and estimation of the reflection strength is deduced from the peak value of that function. By asymptotic analysis of the application of the algorithm to the Kirchhoff representation of the backscattered field, we show that the peak value of the output estimates the reflection strength even when the condition of small variation in velocity (an assumption of the original derivation) is violated. Furthermore, this analysis demonstrates that the method provides a m...

Prestack Kirchhoff inversion of common‐offset data

In trying to resolve complex geologic structures, the pitfalls in employing the CDP method become evident. Additionally, stacking multioffset traces corrupts the amplitudes necessary for stratigraphic analysis. In order to preserve whatever structural and amplitude information is in the data, prestack processing should be performed.Given common-offset data and the velocity above a reflector, prestack acoustic Kirchhoff inversion resolves the location of the interface. When amplitude information has been preserved in the data, the method additionally calculates the reflection coefficient at each interface point. For band-limited seismic data, the inversion operator produces a sinc-like picture of the reflector, with the peak amplitude of this band-limited singular function equal to the angularly dependent reflection coefficient. The inversion development is based upon high-frequency Kirchhoff data which are inserted into a general 3-D inversion operator. Asymptotically evaluating the four resulting integrals by the method of four-dimensional stationary phase permits an inversion amplitude function to be chosen so that the inversion operator produces a singular function of support on the reflector, weighted by the reflection coefficient. Specializing the three-dimensional inversion operator to two and one-half dimensions allows for processing of single lines of common-offset data. Synthetic examples illustrate the accuracy of the method for constant-velocity Kirchhoff data, as well as the problems in applying constant-velocity data to multivelocity models.

Velocity inversion using a stratified reference

We present an inversion algorithm for backscattered (“stacked”) seismic data which will reconstruct the velocity profile in realistic earth conditions. The basic approach follows that of the original (constant reference speed) Cohen and Bleistein paper (1979a) in that hig‐hfrequency asymptotics and perturbation methods are used. However, we use a reference speed which may vary with depth, and this greatly enhances the validity of the perturbation assumption and hence the inversion results. The new algorithm enjoys the same economies and stability properties of the original algorithm, making it very competitive with current migration schemes. Four major assumptions are made: (1) the acoustic wave equation is an adequate model; (2) stacked data have amplitude information worth preserving fairly accurately; (3) the actual reflectivity coefficients can be adequately modeled as perturbations from a continuous reference velocity which depends only on the depth variable; and (4) the subsurface can be adequately ...