Introduction  

Prospecting by the seismic reflection method has revolutionized hydrocarbon exploration. Accurate 3-D reflection seismic imaging of complex structures (along with horizontal drilling) has increased success rates to the point where exploration and production in thousands of feet of water is now often economically feasible. This success, however, appears to fly in the face of common sense. Despite an inherently noisy earth, weak reflected signal, deep reservoirs, and a complex wavefield, seismic images constructed with singly-reflected P-waves (henceforth, ``primary reflections'' or ``primaries'') alone often suffice to plan drilling activities.

Modern marine seismic acquisition generally yields higher quality recorded data than terrestrial acquisition. Marine towed-streamer surveys sample the wavefield densely, regularly, and at a relatively low cost. Marine data is immune from a variety of factors which combine to degrade terrestrial data quality: a non-flat acquisition datum, near-surface inhomogeneity, and strong surface waves. However, the water column's relative homogeneity and the near-perfect reflectivity at the water's surface almost always produce observable multiply-reflected P-waves (henceforth, ``multiple reflections'' or ``multiples''). Multiples often erect the most significant impediment to the successful construction and interpretation of an image of the primaries, especially in regions with anomalously strong reflectors (e.g., ``hard'' water bottom or salt bodies). Multiple suppression techniques have, by necessity, advanced contemporaneously with reflection imaging for fifty years.

Despite its nuisance, however, energy from multiples penetrates deeply enough into the earth to illuminate the prospect zone. In this sense, the multiples can be viewed as perfectly viable signal, rather than as noise. Moreover, since they illuminate different angular ranges and reflection points, a primary and its multiples are more than simply redundant. In theory and in practice, multiples provide subsurface information not found in the primaries.

To actually exploit the information provided by multiples, the multiples and primaries must first be mapped into a domain where they are directly comparable, and then combined in some fashion. Imaging algorithms like migration reduce the signal to a compact form by removing the effects of wave propagation through a the overburden. Additionally, if the prestack images are arranged in angle-domain common-image gathers (see () for a review), the events can be analyzed for angle-dependent phenomenon. We conclude, therefore that the prestack image domain, and in particular, the angle domain, is the best one in which to integrate the information contained in the multiples and primaries.

An important class of multiple suppression techniques create from the data a ``model'' of the multiples, which may then be adaptively subtracted from the data. Many of these algorithms use wavefield extrapolation to ``add a multiple bounce'' to recorded data, and thus transform primaries into an estimate of the multiples (, , , , , , ). The imaging of multiples can be viewed roughly as the reverse process of modeling. Prestack imaging of multiples ``removes a multiple bounce'' from the data and transforms multiples into pseudo-primary events (, ) which can then be imaged using conventional imaging techniques.

Existing migration techniques for multiples perform the reverse modeling process either implicitly or explicitly. () imaged pegleg multiples with Kirchhoff prestack depth migration. () present a least-squares joint imaging scheme for multiples that uses poststack Kirchhoff depth migration. () and () migrate peglegs with shot-profile depth migration, while () used a similar crosscorrelation technique. () uses source-geophone migration after crosscorrelation at the surface. In many ways, however, these techniques fail to fully leverage the valuable information contained in the multiples.

In effect, primaries and each mode of multiple constitute semi-independent measurements of the earth's reflectivity at depth. Unfortunately, these independent measurements are embedded in a single data record. We would like to improve signal-to-noise ratio or fill illumination gaps by averaging the images. However, simple averaging of the raw images (, , ) encounters two problems, illustrated by Figures 1 and 2. First, unless the multiple images have undergone an appropriate amplitude correction, the signal events are incommensurable. Secondly, just as multiples constitute noise on the primary image, primaries and higher order multiples constitute noise on the first-order multiple image. The unmodeled events on each image are called ``crosstalk'' (). Because corresponding crosstalk events on the primary and multiple images are kinematically quite consistent, especially at near offsets, averaging the images may not increase the signal-to-noise ratio or improve signal fidelity.

 

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Figure 1 Common-offset section and common-midpoint (CMP) gather from 2-D field data example after normal moveout (NMO) for primaries (left panels) and a particular prestack, true relative amplitude imaging method for pegleg multiples (Section ). Signal events are consistent between all panels, both kinematically and in terms of amplitudes. The multiples provide near offset information not found in the primaries. However, the multiple image contains crosstalk events - overcorrected primaries (``P'') and multiples from other reflectors (``R1M'') - that inhibit simple averaging of the multiple and primary images. The crosstalk events shown here are, however, inconsistent between images, and to some extent curved with respect to offset, and can thus be distinguished from signal events, which are both flat and consistent between images.

 

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Figure 2 Same format as Figure 1, but taken from deeper in the section, after the arrival of the first seabed multiple. Weak signal events are visible on both images, but corresponding crosstalk events (e.g., ``R1M'') are generally consistent between images and would greatly inhibit the effectiveness of simple image averaging. However, they are curved with respect to offset, while signal events are flat.

The previous paragraph underscores the main obstacle facing algorithms which attempt to jointly image multiples and primaries: while multiples provide additional information about the earth's reflectivity, we cannot exploit it unless we separate the individual modes. Cleanly separating a variety of different multiple modes from prestack data is both expensive and difficult. Moreover, by casting mode separation as a preprocessing step, as is the norm, we may bias the amplitudes in the separated modes and thus inhibit the integration of primaries and multiples.

In this thesis I introduce the LSJIMP (Least-squares Joint Imaging of Multiples and Primaries) method, which solves the separation and integration problems simultaneously, as a global least-squares inversion problem. The model space of the inverse problem, as illustrated in Figure 3, contains a collection of images, with the energy from each mode partitioned into one, and only one image. Moreover, each image has a special form: because the forward modeling operator contains appropriate amplitude correction operators, the signal events in multiple and primary images are directly comparable, in terms of both kinematics and amplitudes.

Minimization of the modeling error ($\bold d - \bold d_{\rm mod}$ in Figure 3) alone is an ill-posed problem. Forward-modeled crosstalk is indistinguishable from forward-modeled signal. I devise three model regularization operators which discriminate between crosstalk and signal and thereby properly segregate energy from each modeled wave mode into its respective image. Figure 4 illustrates these discriminants on a field data example. The model regularization operators serve a higher purpose than crosstalk suppression alone, however. By applying differential operators along reflection angle and between images, we can ``spread'' signal from other angles or images to fill illumination gaps and increase signal fidelity. Furthermore, by exploiting an additional, and hitherto ignored dimension of data redundancy - that between primaries and multiples - we can, with a degree of rigor, solve the integration problem and rightly claim to have solved a ``joint imaging'' problem.

 

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Figure 3 LSJIMP schematic. Assume that the recorded data consist of primaries and pegleg multiples. Prestack imaging alone (applying adjoint of modeling operator $\bold L_{i,k}$) focuses signal events in zero-offset traveltime (or depth) and offset (or reflection angle), but leaves behind crosstalk events. If the $\bold m_{i,k}$ images contain only signal, then we can model all the events in the data that we desire. The LSJIMP inversion suppresses crosstalk and endeavors to fit the recorded data in a least-squares sense. The model regularization operators used to suppress crosstalk simultaneously enable LSJIMP to exploit the intrinsic redundancy between and within the images to increase signal fidelity.

 

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Figure 4 Illustration of intrinsic redundancy within and between prestack images of primaries (panels (a) and (c)) and peglegs (panels (b) and (d)), and the regularization schemes used by LSJIMP to exploit this redundancy to suppress crosstalk, increase signal fidelity, and fill illumination gaps. All panels are CMP gathers from the same midpoint, but the upper and lower panels zoom into small portions of the time axis for viewing purposes. Between panels (a) and (b), notice how signal events are consistent (kinematics and amplitudes) between images, while crosstalk events are not. A differencing operator between images increases signal consistency and penalizes crosstalk. On panel (c), notice that signal events are flat with offset, while crosstalk events are generally curved. Differencing between adjacent offsets increases signal fidelity and suppresses crosstalk. Lastly, from panels (a) and (c), notice how we can use signal to predict crosstalk events. The predicted crosstalk can be used as a model penalty weight which penalizes crosstalk.