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Size of the parameter space

In the multi-scale method of PEF estimation there are a multitude of parameters that must be selected. They can be put into two general categories: those that determine the structure of the PEF, and those that determine how the PEF is estimated.

The first group of parameters includes the size of the PEF; the shape of the PEF; the size of the gap; and in the case of non-stationary PEFs, the size and shape of the micropatches. The size of the PEF presents a trade-off between an improved ability to capture the inverse data covariance, and the number of fitting equations available to estimate the PEF. As the size of the PEF increases, the inverse data covariance is more accurately captured, but the number of fitting equations decreases.

The gap and center parameters both involve the shape of the PEF, and both have a nonintuitive effect on the result. They have a very light impact on the number of fitting equations, and a more pronounced effect on the ability of the PEF to capture the inverse covariance of the data.

The size and shape of the micropatches determine the non-stationarity of the filter. There is another trade-off here, where smaller micropatches allow for greater non-stationarity in the PEF, but also increase the size of the model space, i.e. the number of PEF coefficients.

This next set of parameters is used in the PEF estimation process. In the case of a stationary PEF on regularly-sampled data, no extra parameters are required for the regression except for the number of conjugate-gradient iterations. However, once the data are sparse enough that a PEF cannot be estimated in the traditional manner, rescaling of the data can provide more fitting equations. The choice of these scales is present in the $\bf{D_{i}}$ and $\bf{d_i}$ in fitting goals (2) and (4).

When estimating a non-stationary PEF, quite often the problem will be under-determined; that is, that there will be more unknowns in model space than fitting equations. This is overcome by adding a regularization term to the optimization which introduces more parameters, namely the choice of regularization ($\bf{A}$ in fitting goal 3) and $\epsilon$. The choice of regularization is data-dependent, but it normally operates over common filter coefficients to insure a smoothly varying filter. In the case of common midpoint gathers, the regularization is a radial roughener, since in a CMP gather dips are approximately constant over radial lines Crawley (2000). The choice of $\epsilon$ can be handled by either balancing the model residual with the data residual, or by balancing the size of the model and data gradients within the conjugate-direction solver Claerbout (1999).

Most of the parameters in this problem are independent. The two exceptions to this are micropatch size and scale choice. Since non-stationary filters are linked to data, if that data is regridded, the non-stationary filter must also be regridded. So, the choice of scales must maintain the aspect ratio of the data and also maintain the spatial location of the micropatch boundaries, such that the same filter coefficients are in the same place regardless of the scale. This means that the choice of scale must cleanly divide all of the data axes as well as all of the micropatch axes. Figure [*] has an illustration of both a valid and invalid scales, given existing data and micropatch sizes.

Figure 1
Plot of data space (grid) with the non-stationary PEF micropatches (overlay). Above: the rescaling doesn't cleanly re-sample the PEF, so the filter coefficients could be in different places depending on the scale. Below: The appropriate choice of scale maintains both the aspect ratio of the data and of the PEF.

There is also a trade-off when optimizing micropatch size and scale choice. When the size of the micropatches is increased, the size of the model space decreases, and the number of possible scales increases. However, the PEF is then able to capture less non-stationarity in the data.

When choosing the micropatch and scale parameters, the goal is to have as many fitting equations as possible. In the non-stationary case, the need to have these fitting equations spatially distributed over a wide area is also important so that there are as few unconstrained micropatches (those with no fitting equations) as possible. In addition to varying scale and micropatch size, we can also vary the origin of the micropatch grid by shifting the positions of the micropatches so that they are more evenly constrained.

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