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 inverse series make toward these tasks in the inversion process, which realizes the selection of task‐specific subseries. To date, two task‐specific subseries have been identified: a series for eliminating free‐surface multiples and a series for attenuating internal multiples. These series result in distinct algorithms for free‐surface and internal multiples, and neither requires a model of the subsurface reflectors that generate the multiples. The method attenuates multiples while preserving primaries at all offsets; hence, these methods are equally well suited for subsequent poststack structural mapping or prestack amplitude analysis. The method has demonstrated its usefulness and added value for free‐surface multiples when (1) the overburden has significant lateral variation, (2) reflectors are curved or dipping, (3) events are interfering, (4) multiples are difficult to identify, and (5) the geology is complex. The internal‐multiple algorithm has been tested with good results on band‐limited synthetic data; field data tests are planned. This procedure provides an approach for attenuating a significant class of heretofore inaccessible and troublesome multiples. There has been a recent rejuvenation of interest in multiple attenuation technology resulting from current exploration challenges, e.g., in deep water with a variable water bottom or in subsalt plays. These cases are representative of circumstances where 1-D assumptions are often violated and reliable detailed subsurface information is not available typically. The inverse scattering multiple attenuation methods are specifically designed to address these challenging problems. To date it is the only multidimensional multiple attenuation method that does not require 1-D assumptions, moveout differences, or ocean‐bottom or other subsurface velocity or structural information for either free‐surface or internal multiples. These algorithms require knowledge of the source signature and near‐source traces. We describe several current approaches, e.g., energy minimization and trace extrapolation, for satisfying these prerequisites in a stable and reliable manner.

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 taskspecific 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.,

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 pres tack migrl).tion.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.

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Almost every link in the chain of exploration seismology methods used to process recorded data has been affected by Green’s theorem. Among the seismic processes that can be related to, and/or have benefited from, Green’s theorem are wavelet estimation, multiple elimination, regularization, redatuming, imaging, deghosting, and interferometry. This tutorial on various seismic exploration methods derived from Green’s theorem emphasizes seismic data reconstruction (including regularization and redatuming) and its relationship to interferometry as well as to wavelet estimation and wavefield separation. The last decade has witnessed ever-increasing attention within the energy industry and its concomitant representation in the published literature to methods dealing with wavefield reconstruction through in-terferometry or virtual-source techniques. The attention has re- newed interest in Green’s theorem because all different ap-proaches to interferometry can be derived from it. This tutorial provides a derivation and explication of the limitations of interferometric techniques (when interferometry is used to process measured data from marine surface seismic experiments with controlled sources) as approximations to Green’s theorem. This tutorial provides a definite statement of the comprehensive framework given by Green’s theorem to wavefield reconstruction and shows how different techniques are directly understood as specific mathematical forms and/or approximations to the theorem. The use of approximations can have shortcomings and create artifacts. These artifacts and errors are also analyzed and explained. All methods discussed in this tutorial recognize their foundation on Green’s theorem and have a secure mathematical-physics cornerstone to recognize the assumptions behind distinct approximate solutions and to guide the search for more accurate, effective techniques.

A procedure is derived which incorporates a generalized linear inverse viewpoint within a multidimensional Born inversion method. The method we present is a more general Born theory which can accommodate insufficient and inaccurate data. This general method reduces to the ordinary Born procedure when the data requirements of the latter technique are satisfied.

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