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# LSJIMP Review

LSJIMP models the recorded data as the sum of primary reflections and p orders of pegleg multiples from multiple generators. An -order pegleg splits into i+1 legs. Denoting the primaries and the leg of the order pegleg from the multiple generator , the modeled data takes the following form:
 (1)
If we have designed imaging operators that map primaries and multiples to comparable signal events (kinematics and angle-dependent amplitudes), we can write the as linear functions of prestack images. We can similarly denote the modeling operators (adjoint to imaging) for primaries and peglegs as and , respectively, and the images of the primaries and peglegs as and , respectively. Rewriting equation (1), we have:
 (2)
The LSJIMP method optimizes the primary and multiple images, , by minimizing the norm of the difference between the recorded data, , and the modeled data, :
 (3)
Minimization (3) is under-determined for most choices of and , implying infinitely many solutions. Crosstalk leakage is a symptom of the problem. For instance, maps residual first-order multiple energy in to the position of a first-order multiple in data space. Minimization (3) alone cannot distinguish between crosstalk and signal. Without model regularization, the basic LSJIMP problem is intractable.

Previously Brown (2003b), I devised discriminants between crosstalk and signal, and used them to derive three model regularization operators which choose the set of primary and multiple images which are optimally free of crosstalk. Moreover, these operators exploit signal multiplicity-within and between images-to increase signal fidelity, fill coverage gaps, and combine multiple and primary information. These model regularization operators are the key to the LSJIMP method's novelty.

To solve the regularized LSJIMP problem, we supplement minimization (3) with the three model regularization operators:
 (4)
, , and are the model residuals corresponding to differencing across images, differencing across offset, and crosstalk penalty weighting, respectively. Scalars and balance the relative weight of the three model residuals with the data residual. I use the conjugate gradient method for minimization (4).

Previously Brown (2003a,c), I outlined an efficient prestack, true relative amplitude imaging scheme for pegleg multiples. We can rewrite as the cascade of a differential geometric spreading correction for multiples (), Snell Resampling to normalize a multiple's AVO to its primary (), HEMNO (Heterogeneous Earth Multiple NMO Operator) to kinematically image split peglegs (), and finally, application of the multiple generator's spatially-variant reflection coefficient ():
 (5)
The imaging operator for primaries, , is simply NMO. The imaged multiples and primaries on the in equation (2) are directly comparable in terms of kinematics and amplitudes. We can exploit this important fact (with model regularization) to discriminate crosstalk from signal, and also to spread information between the images. A true relative amplitude imaging scheme like equation (5) is crucial to fully leverage the multiples in this joint imaging algorithm. Since we apply the operator and its adjoint many times in iterative optimization, computational efficiency is crucial. Because HEMNO images (like NMO) by vertical stretch, equation (5) is fast, memory-conserving, and robust to poorly sampled wavefields, which are the norm with 3-D acquisition. Brown and Guitton (2004) demonstrated that this modeling/imaging approach can accurately model multiples, even in a moderately complex earth.

Next: 3-D Theory Up: Brown: 3-D LSJIMP Previous: Introduction
Stanford Exploration Project
5/23/2004