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BACKGROUND

A PEF is estimated by minimizing the output of the known data convolved with the PEF, where the first coefficient of the PEF is constrained to be 1 and the other PEF coefficients are unknown. This can be written as  
 \begin{displaymath}
\bold{W(DKf} + \bold{d}) \approx \bold 0
,\end{displaymath} (1)
where K is a mask that constrains the first filter coefficient to 1, W is a diagonal weighting operator that is equal to 1 only when all filter coefficients lie on known data (and is otherwise), D represents convolution with the data, d is simply a copy of the data, and the unknown PEF is denoted as f.

Fitting goal (1) works well for estimating the PEF if there are contiguous data. If the data are sparse enough so that there are an inadequate number of fitting equations, such as case (a) in Figure [*], a multi-scale method is used Curry (2002). With this method, more fitting equations are generated by rescaling the data (d) to multiple different grid sizes (d_i), and then convolving the filter with all of the different scales of data. An example of rescaled data is case (d) of Figure [*]. This is done by first performing linear interpolation on the original data in (a), followed by adjoint linear interpolation onto a coarser grid in (c). That is: for each bin, create a data point located at the center of that bin, and then use those data points for linear interpolation onto a coarser grid.

 
scale
scale
Figure 1
A valid fitting equation requires that the PEF (thick line) falls on completely known data (gray cells). If any cells are missing (white), a fitting equation is no longer possible. Starting from sparse data in (a) with no valid fitting equations, perform linear interpolation from the grid in (a) to the points shown in case (b), and then perform adjoint linear interpolation from the points in (b) onto a coarser grid in (c), so that valid fitting equations are possible in (d).
view

All of these different scales of data (d_i) can then be convolved (D_i) with the single unknown PEF (f), which leads to a better-determined system of equations,  
 \begin{displaymath}
\bf W \left( 
 \left[ 
 \begin{array}
{c} 
 \bf D_0 \\  \bf ...
 ... 
 \bf d_n \\  
 \end{array} \right] 
 \right) 
 \approx 0 
.
 \end{displaymath} (2)
The weight (W) now includes a weight for missing data in all of the scaled versions of the data.

In the case of a non-stationary prediction error filter, the PEF in fitting goal (1) changes from a vector f(ia) to a a much longer vector f(ia,id), but the fitting goal looks the same. A full description of the changes in K and D in fitting goal (1) are explained elsewhere Guitton (2003). To extend fitting goal (2) for non-stationary PEFs, more changes need to be made, since the different scales of data (d_i and D_i) have different sizes, and we have a PEF that varies with position. We now need to subsample the PEF to match the different data sizes, so that  
 \begin{displaymath}
\bf W 
 \left( 
 \left[ 
 \begin{array}
{c} 
 \bf D_0 \\  
 ...
 ...\  
 \bf d_n \\  \end{array} 
 \right] 
 \right) 
 \approx 0
. \end{displaymath} (3)
A sub-sampler P_i has been introduced that reduces the size of the non-stationary PEF from $na \times nd$ to $na \times nd_{i}$. Since this non-stationary filter is now likely under-determined due to a large increase in the number of unknown filter coefficients, a regularization fitting goal,  
 \begin{displaymath}
\epsilon \bf A f \approx 0
, \end{displaymath} (4)
must also be added, where $\bf{A}$ is a regularization operator that roughens common filter coefficients spatially. This improves the stability of the PEF, as well as insures that it will vary smoothly in space.

Once the PEF has been determined, a second minimization problem is solved, where the output of a convolution of the newly found stationary or non-stationary PEF (F representing convolution with f) with the partially unknown model m is minimized. The known points of the model are constrained to their actual values m_k, via a missing data mask, K_data. The output of this final step is the interpolated model, m, where  
 \begin{displaymath}
\bold{K_{data}m} = \bold{m_{k}}\end{displaymath} (5)
 
 \begin{displaymath}
\bold{Fm} \approx \bold 0
.\end{displaymath} (6)

All of the above fitting goals are minimized in a least-squares sense, by using a conjugate-gradient solver. Iteratively re-weighted least-squares (IRLS) is a method where a weighting function that varies with iteration is applied to a conjugate-gradient solver, so that different norms can be approximated. I will not go into the details of IRLS here, as they are covered in-depth elsewhere Claerbout (1999); Darche (1989); Fomel and Claerbout (1995); Guitton (2000); Scales and Gersztenkorn (1988), except to mention that several parameters need to be set, including the quantile of the residual used, and the frequency of recalculation of the weights.


next up previous print clean
Next: IRLS-based interpolation Up: Curry: Iteratively re-weighted least-squares Previous: INTRODUCTION
Stanford Exploration Project
7/8/2003