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Nemeth Forward Model

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 ith order pegleg has i+1 ``legs'', or independent arrivals Brown (2003a). If we denote the recorded data as $\bf d$, the primaries as $\bold d_0$, and the kth leg of the ith order pegleg as $\bold d_{i,k}$, the model of the data takes the following form.  
 \begin{displaymath}
\bold d = \bold d_0 + \sum_{i=1}^p \sum_{k=0}^i \bold d_{i,k},\end{displaymath} (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):  
 \begin{displaymath}
\bold d = \bold N_0 \bold m_0 
 + \sum_{i=1}^p \sum_{k=0}^i \bold R_i \bold N_{i,k} \bold S_i \bold G_i \bold m_{i,k}\end{displaymath} (2)
$\bold m_0$ is the primary image, flattened by the adjoint of the NMO operator $\bold N_0$. $\bold m_{i,k}$ is image of the kth leg of the ith-order pegleg, flattened by $\bold N_{i,k}$, the adjoint of the HEMNO equation. To all ith-order peglegs, $\bold G_i$ applies a differential geometric spreading correction, $\bold S_i$ applies Snell resampling, and $\bold R_i$ applies the appropriate reflection coefficient. The motivation and implementation of $\bold G_i$, $\bold S_i$, and $\bold R_i$ are discussed in detail by Brown (2003a).

For clarity, we can rewrite equation (2) in matrix notation:  
 \begin{displaymath}
\left[ \bold N_0 \hspace{0.1in} (\bold R_1 \bold N_{1,0} \bo...
 ...ots \  \bold m_{p,p}
 \end{array} \right] = \bold L_n \bold m,\end{displaymath} (3)
and define the data residual as the difference between modeled and recorded data:  
 \begin{displaymath}
\bold r_d = \bold d - \bold L_n \bold m.\end{displaymath} (4)
Viewed as a standard least-squares inversion problem, where the model is adjusted to minimize the L2 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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Next: Consistency of the Data Up: Least-Squares Joint Imaging of Previous: Least-Squares Joint Imaging of
Stanford Exploration Project
7/8/2003