Arthur B. Weglein

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.

Multiple attenuation; an overview of recent advances and the road ahead (1999)

This paper is an overview of the current state of multiple attenuation and developments that we might anticipate in the near future. The basic model in seismic processing assumes that reflection data consist of primaries only. If multiples are not removed, they can be misinterpreted as, or interfere with, primaries. This is a longstanding and only partially solved problem in exploration seismology. Many methods exist to remove multiples, and they are useful when their assumptions and prerequisites are satisfied. However, there are also many instances when these assumptions are violated or where the prerequisites are difficult or impossible to attain; hence, multiples remain a problem. This motivates the search for new demultiple concepts, algorithms, and acquisition techniques to add to, and enhance, our toolbox of methods.

Wavelet estimation for a multidimensional acoustic or elastic earth

A new and general wave theoretical wavelet estimation method is derived. Knowing the seismic wavelet is important both for processing seismic data and for modeling the seismic response. To obtain the wavelet, both statistical (e.g., Wiener-Levinson) and deterministic (matching surface seismic to well-log data) methods are generally used. In the marine case, a far-field signature is often obtained with a deep-towed hydrophone. The statistical methods do not allow obtaining the phase of the wavelet, whereas the deterministic method obviously requires data from a well. The deep-towed hydrophone requires that the water be deep enough for the hydrophone to be in the far field and in addition that the reflections from the water bottom and structure do not corrupt the measured wavelet. None of the methods address the source array pattern, which is important for amplitude-versus-offset (AVO) studies.This paper presents a method of calculating the total wavelet, including the phase and source-array pattern. When the source locations are specified, the method predicts the source spectrum. When the source is completely unknown (discrete and/or continuously distributed) the method predicts the wavefield due to this source. The method is in principle exact and yet no information about the properties of the earth is required. In addition, the theory allows either an acoustic wavelet (marine) or an elastic wavelet (land), so the wavelet is consistent with the earth model to be used in processing the data. To accomplish this, the method requires a new data collection procedure. It requires that the field and its normal derivative be measured on a surface. The procedure allows the multidimensional earth properties to be arbitrary and acts like a filter to eliminate the scattered energy from the wavelet calculation. The elastic wavelet estimation theory applied in this method may allow a true land wavelet to be obtained. Along with the derivation of the procedure, we present analytic and synthetic examples.

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.

Source signature estimation based on the removal of first‐order multiples

The estimation of the source signature is often one of the necessary first steps in the processing of seismic reflection data, especially if the processing chain includes prestack multiple removal. However, most methods for source estimation are based on poststack data or assume that the earth is 1-D. In this work, a new source estimation method for prestack data is presented. It consists of finding the source signature that permits the removal of events attributable to the first‐order free‐surface reflections (i.e., first‐order multiples). The method exploits the formulation of the relationship between the free‐surface reflections and the source signature as a scattering Born series. In this formulation, the order of the scattering series coincides with that of the free‐surface reflections, and the series is constructed exclusively with seismic data and the source signature without any knowledge of the subsurface other than the velocity of sea water. By restricting the problem to first‐order free‐surface...

Green’s theorem as a comprehensive framework for data reconstruction, regularization, wavefield separation, seismic interferometry, and wavelet estimation: A tutorial

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 derivatio...

On the construction of an absorptive–dispersive medium model via direct linear inversion of reflected seismic primaries

Arbitrary multi-dimensional distributions of two absorptive–dispersive acoustic medium parameters (P-wave velocity and Q) may be determined to first order from reflected seismic primary data with an inverse scattering formulation. The problem can be considered a form of Q-estimation, but one that distinguishes itself by using the dispersive reflection coefficient as its primary source of information. If the overburden above a target to be characterized is known, synthetic Q estimates are successfully derived, albeit using subtle data variations. If the overburden is unknown, the linear estimate is instead best interpreted as the starting point of a nonlinear inverse scattering procedure. Simple analytic and numerical examples may be used to characterize associated issues of conditioning, leakage, detectability and transmission error.

The impact of migration on AVO

Amplitude variation with offset (AVO) analysis is often limited to areas where multidimensional propagation effects such as reflector dip and diffractions from faults can be ignored. Migration-inversion provides a framework for extending the use of seismic amplitudes to areas where structural or stratigraphic effects are important. In this procedure, sources and receivers are downward continued into the earth using uncollapsed prestack migration. Instead of stacking the data as in normal migration, the prestack migrated data are used in AVO analysis or other inversion techniques to infer local earth properties. The prestack migration can take many forms. In particular, prestack time migration of common-angle sections provides a convenient tool for improving the lateral resolution and spatial positioning of AVO anomalies. In this approach, a plane-wave decomposition is first applied in the offset direction, separating the wavefield into different propagating angles. The data are then gathered into common-angle sections and migrated one angle at a time. The common-angle migrations have a simple form and are shown to adequately preserve amplitude as a function of angle. Normal AVO analysis is then applied to the prestack migrated data. Examples using seismic lines from the Gulf of Mexico show how migration improves AVO analysis. In the first set of examples, migration is shown to improve imaging of subtle spatial variations in bright spots. Subsequent AVO analysis reveals dim spots associated with dry-hole locations that were not resolvable using traditional processing techniques, including both conventional AVO and poststack migration. A second set of examples shows improvements in AVO response after migration is used to reduce interference from coherent noise and diffractions. A final example shows the impact of migration on the spatial location of dipping AVO anomalies. In all cases, migration improves both the signal-to-noise ratio and spatial resolution of AVO anomalies.