Background

A PEF can be estimated by solving the minimization problem where known data ($\bf{d}$) is convolved ($\bf{D}$) with an unknown PEF ($\bf{f}$), so that

 

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

where $\bf{W}$ is a weight for missing data and $\bf{K}$ constrains the first PEF coefficient to be 1.

When all of the equations contain missing data, $\bf{W}$ is everywhere, and the problem cannot be solved. In this case, rescaled copies of the data can be substituted for the original data in equation (1), resulting in

 

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

In this case, $\bf{d_i}$ represents the various different rescaled copies of the data, $\bf{D_i}$ is convolution with that rescaled data, and $\bf{W}$ is now a weight for all scales of data. The data is rescaled by taking the original finely gridded data, and transforming it to a series of points located at the center of cells with data. Adjoint linear interpolation is then performed to move the data points onto the new grid.

For the case of a non-stationary PEF, the equations remain largely the same, except that the PEF varies with position. When convolving different sizes of data with a non-stationary PEF, the PEF must be sub-sampled so that the spatial dimensions of the non-stationary PEF and the data match. This is accomplished by the introduction of a sub-sampling operator $\bf{P_i}$, so

 

$$\bf W \left( \left[ \begin{array} {c} \bf D_0 \\ ... ...\ \bf d_n \\ \end{array} \right] \right) \approx 0 .$$ (3)

In addition, since the model space has now increased substantially, a second fitting goal must be added,

 

$$\epsilon \bf A f \approx 0 ,$$ (4)

that ensures that the PEF will vary smoothly over space. In equation (4), $\bf{A}$ is a regularization operator (in this paper, a spatial Laplacian), and $\epsilon$ is a scale factor.