Inverse scattering on the line

We present a multidimensional multiple‐attenuation method that does not require any subsurface information for either surface or internal multiples. To derive these algorithms, we start with a scattering theory description of seismic data. We then introduce and develop several new theoretical concepts concerning the fundamental nature of and the relationship between forward and inverse scattering. These include (1) the idea that the inversion process can be viewed as a series of steps, each with a specific task; (2) the realization that the inverse‐scattering series provides an opportunity for separating out subseries with specific and useful tasks; (3) the recognition that these task‐specific subseries can have different (and more favorable) data requirements, convergence, and stability conditions than does the original complete inverse series; and, most importantly, (4) the development of the first method for physically interpreting the contribution that individual terms (and pieces of terms) in the inv...

An inverse-scattering series method for attenuating multiples in seismic reflection data

This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task, and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four task-specific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks, i.e., no terms corresponding to coupled tasks are ever computed. This inversion in stages provides a tremendous practical advantage. The achievement of a task is a form of useful information exploited in the redefined and restarted problem; and the latter represents a critically important step in the logic and overall strategy. The individual subseries are analysed and their strengths, limitations and prerequisites exemplified with analytic, numerical and field data examples.

https://iopscience.iop.org/article/10.1088/0266-5611/19/6/R01/pdf

Inverse scattering series and seismic exploration

Density and bulk modulus variations in an acoustic earth are separately recoverable from standard reflection surveys by utilizing the amplitude-versus-offset information present in the observed wave fields. Both earth structure and a variable background velocity can be accounted for by combining the Born and WKBJ approximations, in a "before stack" migration with two output sections, one for density variations and the other for bulk modulus variations. For the inversion, the medium is considered to be composed of a known low-spatial frequency variation (the background) plus an unknown high-spatial frequency variation in bulk modulus and density (the reflectivity). The division between the background and the reflectivity depends upon the frequency content of the source. For constant background parameters, computations are done in the Fourier domain, where the first part of the algorithm includes a frequency shift identical to that in an F-K migration. The modulus and density variations are then determined by observing in a least-squares sense amplitude versus offset wavenumber. For a spatially variable background, WKBJ Green's operators that model the direct wave in a medium with a smoothly varying background are used. A downward continuation with these operators removes the effects of variable velocity from the problem, and, consequently, the remainder of the inversion essentially proceeds as if the background were constant. If the background is strictly depth dependent, the inversion can be expressed in closed form. The method neglects multiples and surface waves and it is restricted to precritical reflections. Density is distinguishable from bulk modulus only if a sufficient range of precritical incident angles is present in the data.

/pdf/a-born-wkbj-inversion-method-for-acoustic-reflection-data-47x253in30.pdf

A Born-WKBJ inversion method for acoustic reflection data

Seismic migration and inversion describe a class of closely related processes sharing common objectives and underlying physical principles. These processes range in complexity from the simple NMO‐stack to the complex, iterative, multidimensional, prestack, nonlinear inversion used in the elastic seismic case. By making use of amplitudes versus offset, it is, in principle, possible to determine the three elastic parameters from compressional data. NMO‐stack can be modified to solve for these parameters, as can prestack migration. Linearized, wave‐equation inversion does not inordinately increase the complexity of data processing. The principal part of a migration‐inversion algorithm is the migration. Practical difficulties are considerable, including both correctable and intrinsic limitations in data quality, limitations in current algorithms (which we hope are correctable), and correctable (or perhaps intrinsic) limitations in computer power.

Migration and inversion of seismic data

This paper investigates the problem of obtaining an analytic solution and practical computational procedures for recovering the properties of an unknown elastic medium from waves that have been reflected by or transmitted through the medium. The medium consists of two homogeneous half‐spaces in contact with a heterogeneous region. The analytic solution is obtained by transforming the equation of motion for the propagation of plane waves at normal incidence in a stratified elastic medium into a one‐dimensional Schrodinger equation for which the inverse‐scattering problem has already been solved. The practical computational procedures are obtained by solving the corresponding discrete inverse‐scattering problem resulting from approximating the heterogeneous region with a sequence of homogeneous layers such that the travel time through each layer is the same. In both the continuous and discrete inverse scattering problems, the impedance of the medium as a function of travel time is recovered from the impulse response of the medium. A discrete analogy of the continuous solution is also developed. Similar results are obtained for a stratified elastic half space bounded by a free surface.

Jerry A. Ware

Papers

Continuous and Discrete Inverse‐Scattering Problems in a Stratified Elastic Medium. I. Plane Waves at Normal Incidence