BACKGROUND

A PEF can be estimated by minimizing (in a least-squares sense) the output of the known data (d) convolved (D) with a filter f that is unknown except for the first coefficient, that is constrained to 1 by K. This is expressed below, where W is a diagonal weighting operator that is equal to 1 when all filter coefficients lie on known data, and is otherwise. This is written as:  

$$\bold{W(DKf + d)} \approx \bold 0$$ (1)

When the data are sparsely sampled, W will be zero everywhere, since there are not enough contiguous data to estimate a PEF. An example of this is shown elsewhere in this report Curry (2003). More fitting equations can be added to this regression by convolving a single filter on multiple copies of the data that have been rescaled to different grid sizes Curry and Brown (2001). This can be written as  

$$\bf W \left( \left[ \begin{array} {c} \bf D_0 \\ \bf D_1 \... ...f ... \\ \bf d_n \\ \end{array} \right] \right) \approx 0 .$$ (2)

where W is now a much larger diagonal weight for all of the copies of data (d_i), D_i represents convolution with d_i, and K and f are the same as in fitting goal (1).

Often the covariance of the data is not stationary, so it cannot be adequately described by a single filter. Non-stationary PEFs may be used to overcome this limitation. These filters vary with position, so that a filter that looked like f(ia) now looks like f(ia,id). These filters can be estimated in an fitting goal that looks identical to fitting goal (1), except that W, K, D, and f are all now non-stationary counterparts to those in fitting goal (1). The details of these changes are documented elsewhere Guitton (2003).

Since this non-stationary filter is now likely under-determined due to a large increase in the number of filter coefficients, a regularization fitting goal,  

$$\epsilon \bf A f \approx 0 ,$$ (3)

must also be added, where $\bf{A}$ is a regularization operator that roughens common filter coefficients in space. This improves the stability of the PEF, and insures that it will vary smoothly in space. In order to reduce the number of filter coefficients from $na \times nd$ to something more manageable, the filter coefficients are taken to be constant over a small spatial area, so that the number of coefficients reduces to $na \times nd/np$,where np is the size of this small area, known as a micropatch Crawley (2000). Non-stationary PEF estimation can also be performed on sparse data Curry (2002), in a multi-scale fashion similar to that used in fitting goal (2). The only changes to fitting goals (1) and (3) are that fitting goal (1) changes to  

$$\bf W \left( \left[ \begin{array} {c} \bf D_0 \\ ... ...\ \bf d_n \\ \end{array} \right] \right) \approx 0 ,$$ (4)

where W, K, D, and f are all now nonstationary versions of those in fitting goal (2). The only new term in this equation is P_i, that sub-samples the non-stationary PEF from $na \times nd$ to $na \times nd_{i}$.