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Nemeth et al. (1999) introduced an under-determined least-squares
formulation to jointly image compressional waves and various (non-multiple)
coherent noise modes. Guitton et al. (2001) extended this technique to
multiple suppression, using a prior noise model and prediction-error filters to
model the noise and signal.
We can model the recorded data, here a CMP gather, as the superposition of the
primary reflections and *p* orders of pegleg multiples from a single interface
(e.g., the seabed). The *i*^{th} order pegleg has *i*+1 ``legs'', or independent
arrivals Brown (2003a). If we denote the recorded data as
, the primaries as , and the *k*^{th} leg of the *i*^{th} order
pegleg as , the model of the data takes the following form.

| |
(1) |

The main goal of LSJIMP is to use the multiples as a constraint on the primaries.
Thus the model space of the LSJIMP process will be a collection of images, one
corresponding to the primaries, and one for each leg of each order of pegleg.
When the process is finished, the events in each image should have the same
timing and AVO signature as the primaries; they should be ``copies of the
primaries''. The forward modeling equation is the physical mapping which
transforms these copies of the primaries into the multiples recorded in the data.
Brown (2003b) derived ``HEMNO'' (Heterogeneous Earth Multiple
NMO Operator), an operator which independently images each leg of a pegleg
(flattens and shifts to the zero-offset traveltime of the primary).
Brown (2003a) derived a series of linear operators which
modify the amplitude of peglegs to account the effects of the multiple leg of the
raypath (reflection and transmission). Taken together, these operators transform
a single-CMP image that resembles the primaries, into ``data'' that resembles a
pegleg multiple. Let us rewrite equation (1):

| |
(2) |

is the primary image, flattened by the adjoint of the NMO operator
. is image of the *k*^{th} leg of the *i*^{th}-order
pegleg, flattened by , the adjoint of the HEMNO equation. To all
*i*^{th}-order peglegs, applies a differential geometric spreading
correction, applies Snell resampling, and applies the
appropriate reflection coefficient. The motivation and implementation of
, , and are discussed in detail by
Brown (2003a).
For clarity, we can rewrite equation (2) in matrix notation:

| |
(3) |

and define the data residual as the difference between modeled and recorded data:
| |
(4) |

Viewed as a standard least-squares inversion problem, where the model is adjusted
to minimize the *L*_{2} norm of the data residual, equation (4)
is under-determined. The addition of model regularization operators, defined in
later sections, forces the problem to be over-determined.

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Stanford Exploration Project

7/8/2003