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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