(1) |

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.

Figure 1

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,

(2) |

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

(3) |

(4) |

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

(5) |

(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.

7/8/2003