Robert H. Stolt

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

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...

Inverse scattering series and seismic exploration

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.

Migration and inversion of seismic 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.

Toward a unified analysis for source plane-wave migration

In complex areas with large lateral velocity variations, wave-equation-based source plane-wave migration can produce images comparable to those from shot-profile migration, with less computational cost. Image quality can be better than in ray-theory-based Kirchhoff-type methods. This method requires the composition of plane-wave sections from all shot gathers. We provide a general framework to evaluate plane-wave composition in prestack source plane-wave migration. Our analysis shows that a plane-wave section can be treated as encoded shot gathers. This study provides the theoretical justification for applying plane-wave migration algorithms to sparsely sampled shot gathers with irregularly distributed receivers and limited offset. In addition, we discuss cylindrical-wave migration, which is 3D migration of 2D-constructed plane waves along the inline direction. We mathematically prove the equivalence of shot and plane-wave migration, and their equivalence to cylindrical wave migration in 3D cases when the sail lines are straight. Examples (including the Sigsbee 2A model) demonstrate the theory.

Inverse scattering series for multiple attenuation: An example with surface and internal multiples

A multiple attenuation method derived from an inverse scattering series is described. The inversion series approach allows a separation of multiple attenuation subseries from the full series. The surface multiple attenuation subseries was described and illustrated in Carvalho et al. (1991, 1992). The internal multiple attenuation method consists of selecting the parts of the odd terms that are associated with removing only multiply reflected energy. The method, for both types of multiples, is multidimensional and does not rely on periodicity or differential moveout, nor does it require a model of the reflectors generating the multiples. An example with internal and surface multiples will be presented.

Plane Wave Source Composition: An Accurate Phase Encoding Scheme For Prestack Migration

Wave equation based shot profile migration can produce higher quality images than Kirchhoff type methods. However, the high costs largely limit their production use. The cost of a complete wave equation shot profile migration equals the cost of migrating a single shot times the number of shots migrated. Thus, methods to reduce the number of migrations required for a complete experiment while preserving the image quality are as significant as those that increase the efficiency for each migration. One way to increase the efficiency of shot migration is to extrapolate the linearly combined wavefields from multiple shots together. However, special treatment is needed to take care of the cross-term artifacts, which are generated in applying the imaging condition. This technique is generally called phase encoding. Plane-wave source migration is one such method to reduce the number of migrations. In this paper, we demonstrate the concept of plane-wave source composition from the phase encoding point of view. We will show that plane wave source composition is a specific phase encoding technique in shot profile migration. This study demonstrates a systematic error analysis for the plane wave source approximation in migration. This offers the possibility of applying different composing schemes in each migration. We can show that this approach can generate results nearly equivalent to those of shot profile migration. Numerical tests on the Marmousi model and others prove the efficiency and accuracy of this method.

Migration‐inversion revisited (1999)

Seismic reflection events are typically classified as primary or multiple. The standard view considers primary as signal and multiple as a form of coherent noise. Methods for separating primaries from multiples were reviewed in TLE’s January 1999 issue.

A prestack residual time migration operator

Larner and Beasley (1987) present cascaded migration as a way to increase the power and effectiveness of relatively simple migration methods. In particular, f-k migration (Stolt, 1978) can be made to accommodate a depth‐dependent velocity as a cascade of constant‐velocity migrations. The core concept is that data which have been migrated with an approximate velocity can be effectively migrated to their true velocity by migrating with a velocity that is equal to the square root of the difference between the squares of the true and approximate velocities.

Multiple attenuation Recent advances and the road ahead (2011)

Multiple removal is a longstanding problem in exploration seismology. Although methods for removing multiples have advanced and have become more effective, the concomitant industry trend toward more complex exploration areas and difficult plays has often outpaced advances in multiple-attenuation technology. The topic of multiples, and developing ever more effective methods for their removal, remains high in terms of industry interest, priority and research investment. The question as to whether today, in 2011, multiples or multiple removal is winning is a way of describing what we are about to discuss. This paper focuses on recent advances, progress and strengths and limitations of current capability and a prioritized list of open issues that need to be addressed.

Altering offsets and azimuths—in a computer

For years, seismic processing geophysicists used dip moveout (DMO) to alter the acquisition geometry. Our industry has found many uses for DMO because it converts the timing of nonzero-offset data to the timing of zero-offset data. More recently, theoreticians created data mapping as a generalization of the principle behind DMO. Data mapping converts data obtained at an observed offset and azimuth to data at a new offset and/or azimuth. The precise derivation of the data mapping resides in an arduous solution of the wave equation. Thus, data mapping transformations may appear magical. This article provides a simple, geometric understanding of the data mapping transformation. Before turning to the more general 3-D case, we first present the 2-D case. For the 2-D case, data mapping transforms data obtained at one offset distance into data “observed” at a second offset distance. To understand this procedure, we will use one principle, one requirement, and two assumptions. The principle is linear superposition and the requirement is physical invariance. For conceptual convenience, we assume reflection coefficients do not change with offset, and we also assume a constant velocity earth. The principle of linear superposition simplifies our task. As shown in Figure 1, it allows us to construct any input data from a linear superposition of spikes or impulses. In addition, data-mapped output is a linear superposition of data-mapped spike responses. Consequently, we need to understand only the data-mapping operation on a single spike at an arbitrary location. Figure 1. Linear superposition simplifies data mapping. The physical invariance restriction is obvious—the earth does not change as a result of altering the acquisition offset. Likewise, our image of that earth should not change. Depth migration should image the same subsurface using either the original data or the offset-transformed data. The constant velocity assumption …