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LSJIMP models the recorded data as the superposition of primary reflections and
*p* orders of pegleg multiples from multiple-generating surfaces. A schematic for LSJIMP is given in Figure .
An order pegleg splits into *i*+1 legs. If we denote the primaries
as and the leg of the order pegleg from the
multiple generator as , the modeled data takes the
following form:
**schem-LSJIMP-seg
**

Figure 1 LSJIMP schematic. Assume that the recorded
data consist of primaries and pegleg multiples. Prestack imaging alone
(applying adjoint of modeling operator ) focuses signal events
in zero-offset traveltime (or depth) and offset (or reflection angle), but
leaves behind crosstalk events. If the 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.

| |
(1) |

If we have designed an imaging operator that produces primary and multiple images
with consistent signal (kinematics and angle-dependent amplitudes), then we
assume that we can model the important events in the data. We can rewrite equation
(1) as
| |
(2) |

| (3) |

where is the modeling operator of the primaries, and is the image of primaries. Similarly, for the leg of the
order pegleg from the multiple generator, and are modeling operator and image respectively.
Brown (2004) derived appropriate imaging and amplitude correction operators to map the primary image, , into data-space primary events using the Normal Move-Out (NMO) operator, . Similarly, a given pegleg image, , is mapped into data space by sequentially applying the differential geometric spreading correction (), Snell resampling (), the Hetrogeneous-Earth Multiple NMO operator (HEMNO) (), and finally, a reflection coefficient (). HEMNO is a slight improvement over NMO that takes into account kinematics of mildly dipping reflectors. Using these operators, we can rewrite equation (2) as follows:

| |
(4) |

The LSJIMP seeks to optimize the primary and multiple images, ,by minimizing the norm of the data residual, defined as the difference
between the recorded data, , and the modeled data, [equation (3)]:
| |
(5) |

Minimization (5) is under-determined for many choices of
prestack imaging operator, which implies an infinite number of
least-squares-optimal solutions. The problem of infinite optimal solutions manifests
itself as crosstalk leakage. Of this infinity of possible 's, we seek the one which not only fits the
recorded data, but which also has minimum crosstalk leakage and maximum
consistency between signal events on different images. In general we compensate for a
correlated or poorly scaled data residual by adding a residual weighting
operator, :
| |
(6) |

where strictly speaking,
| |
(7) |

To effect the final step of LSJIMP and penalize crosstalk, we use three regularization operators. As discussed in detail by Brown (2004), the operators penalize roughness in reflection angle and between images, and also penalize the model after weighting with a prior model of the crosstalk on each . For estimation of the optimal set of , we minimize a quadratic objective function which consists of the sum of the weighted norms of the data residual [equation(6)] and of the three model residuals:
| |
(8) |

where and are scalars that balance the relative weight of the three model residuals (damping factors) with the data residual. These three residuals are calculated by differencing across images, differencing across offset, generating a crosstalk model, and calculating their penalty weights.

** Next:** Implementation of LSJIMP
** Up:** LSJIMP: Vyas and Brown
** Previous:** Introduction
Stanford Exploration Project

4/5/2006