Simon A. Shaw

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.

Comparisons of adaptive subtraction methods for multiple attenuation

Coherent noise may be removed from seismic data by first making an approximate model of the noise, then producing an even better estimation of the noise by adaptively matching the modeled noise to the data. This modified model of the noise may then be subtracted from the data, eliminating most coherent noise. The success of this approach depends both on how well the initial model matches the true noise and the success of the adaptive matching in modifying the initial noise prediction to match the true noise. The adaptive matching step is complicated by the presence of the signal and other noise in the data. In this article, the noise of interest is surface-related multiples, although other types of coherent noise may be removed with this approach.

New approaches to deghosting towed-streamer and ocean-bottom pressure measurements

Summary A variant of the extinction theorem is combined with a recent wavefield prediction technique (Tan, 1999, Weglein et al., 2000) to produce a direct method for deghosting towed streamer data through an integral of the pressure measurements along the cable. Only those pressure measurements are required. Neither finite difference, nor Taylor Series type approximations, nor dual streamer measurements are required. Single sensor towed streamer acquisition is a practical prerequisite. We also present a separate method for deghosting oceanbottom pressure measurements (deeper than 10 meters) that requires the source signature in water but completely avoids the need to measure the vertical particle velocity. Background

An inverse‐scattering sub‐series for predicting the spatial location of reflectors without the precise reference medium and wave velocity

The accurate location, resolution and identification of targets beneath complex media (e.g., salt, basalt and karsted sediments) are high priority and essentially unsolved problems today. Imaging methods that are tested and compared using synthetic data with precise model velocity as input are not addressing the real-world problem. At the very least, imaging methods need to be tested using synthetic data and a velocity model that corresponds to what would be estimated from the data using current best velocity analysis techniques. This bit of realism would help focus on the relevant issue: how do we achieve accurate imaging at depth given our current ability to estimate the velocity, especially under complex geologic circumstances? There are two responses to this challenge: (1) to significantly improve velocity estimation capability, and (2) to develop methods that can produce accurate images at depth without precise velocity. We support both approaches and this paper represents an effort in the second category.

Using the inverse scattering series to predict the wavefield at depth and the transmitted wavefield without an assumption about the phase of the measured reflection data or back propagation in the overburden

The starting point for the derivation of a new set of approaches for predicting both the wavefield at depth in an unknown medium and transmission data from measured reflection data is the inverse scattering series. We present a selection of these maps that differ in order (i.e., linear or nonlinear), capability, and data requirements. They have their roots in the consideration of a data format known as the T-matrix and have direct applicability to the data construction techniques motivating this special issue. Of particular note, one of these, a construction of the wavefield at any depth (including the transmitted wavefield), order-by-order in the measured reflected wavefield, has an unusual set of capabilities (e.g., it does not involve an assumption regarding the minimum-phase nature of the data and is accomplished with processing in the simple reference medium only) and requirements (e.g., a suite of frequencies from surface data are required to compute a single frequency of the wavefield at depth when the subsurface is unknown). An alternative reflection-to-transmission data mapping (which does not require a knowledge of the wavelet, and in which the component of the unknown medium that is linear in the reflection data is used as a proxy for the component of the unknown medium that is linear in the transmission data) is also derivable from the inverse scattering series framework.

Convergence properties of a leading order depth imaging series

The objective of seismic depth imaging is to produce a spatially accurate map of the re ectivity below the Earths surface. Current methods for depth imaging require an accurate velocity model in order to place re ectors at their correct locations. Techniques to derive the velocity model can fail to provide this information with the necessary degree of accuracy, especially in areas that are geologically complex.

A Decade of Field Trials in the Panhandle of Oklahoma and the Development of ZenSeis

Summary ConocoPhillips has spent the last decade in the field developing a high production Vibroseis acquisition technique based upon the concepts of point source, point receiver acquisition combined with high fold and simultaneous sourcing. This technology is the result of many field experiments and production 3D’s. The methodology has now been used to acquire over one thousand square miles of very high fold 3D seismic data. The purpose of our paper is to document some of the field experiments and touch upon the lessons and conclusions learned in the field. This paper will also set the stage for additional papers that will deal with the analysis of the test results and the production 3D’s that were acquired with this new method.

Vibroseis Source Signature Uncertainty and its Impact on Simultaneous Sourcing

Knowledge of the source wavelet in a controlled source seismic experiment significantly improves our ability to extract information from the resulting seismic data. The radiated source signature in Vibroseis field experiments is found to deviate from the pilot sweep and the ground force estimate (GFE) signal put out by the controller, especially at higher frequencies. The Vibroseis source signature uncertainty is a problem for simultaneous sweeping techniques that require reliable phase control and an accurate GFE in order to separate simultaneous sweeps.

Prediction of the wavefield anywhere above an ordinary towed streamer: Application to source waveform estimation, demultiple, deghosting, data reconstruction and imaging

In principle, it is not possible to compute the total two-way propagating pressure field above a cable from measurements of only the pressure field on a single typical towed streamer. It might appear that knowing the pressure field on the measurement surface together with the fact that the total field vanishes at the air-water “free-surface”, would be sufficient information to compute the two-way field at all points between. However, the latter argument assumes knowledge of all medium properties and sources between the two levels where the pressure is known. The fact that the energy source lies between these two surfaces and that the source and its waveform are generally unknown, precludes computation of the two-way field between the cable and the free-surface. Weglein and Secrest (1990) describe how to compute the scattered field between the measurement surface and the free surface, and the source waveform below the measurement surface, given a cable (or in 3D, a surface) where both the pressure and its normal derivative are measured. Osen et al. (1998) and Tan (1992) show how the wavelet due to an isotropic source can be determined from pressure measured on a typical cable plus one extra phone between the cable and the free surface. While in principle it is not possible to determine the field above the single towed streamer, it has recently been observed by Tan (1999) that this is possible in practice, for the frequencies and geometry corresponding to the typical marine seismic experiment. A typical depth of the towed streamer below the free-surface is ~10 m and the dominant seismic frequencies are less than ~125 Hz. It turns out that the term in the equation that blocks the ability to predict the field above the towed streamer is negligible due to the confluence of these depth and frequency factors. Hence, the typical depth of streamers and seismic frequencies conspire to make practice more accommodating than theory. Tan (1999) exploits this fact and then introduces a mathematically complex Wiener-Hopf Green’s function to provide a stable wavelet estimation scheme from a single cable. In this paper we review and further clarify these recent developments by placing them within the context of the general inverse-source problem. We also show that the ability to predict the field above the cable opens up a plethora of new seismic processing opportunities (in addition to the important application described by Tan, 1999). The new opportunities for progress include: the calculation of full source waveform both below and above the cable from single cable pressure measurements only; calculation of the scattered field between the cable and the free-surface, again with a single cable pressure measurements only; demultiple techniques based on updown separation; creation of a vertical cable above the towed streamer; deghosting; data reconstruction; and twoway wave migration.

Cooperation of the leading order terms in an inverse-scattering subseries for imaging: 1-D analysis and evaluation

Summary Our ability to image targets beneath complex overburdens is compromised by our ability, under these circumstances, to find the precise velocity model with which to downward continue the measured wavefield. As part of a research effort responding to this challenge, a subseries of the inverse scattering series has been identified that is responsible for locating seismic events in depth using measured reflection data and only the wave velocity of a reference medium – not the actual medium. Analytical and numerical studies of this subseries for a 1-D constant density acoustic medium have revealed a welcome property. The leading order contributions to this imaging subseries cooperate to locate reflectors at the correct depth. In the context of the convergence of the imaging subseries, this cooperative behavior suggests that the leading order terms in the subseries will not have the deleterious property that successive terms overshoot/undershoot the correct depth. This cooperation also serves to unambiguously indicate whether the reference velocity is slower or faster than the actual medium velocity. The information comes from the amplitudes and kinematics of the data at and above the reflector being imaged.