A PEF can be estimated by solving the minimization problem where known data () is convolved () with an unknown PEF (), so that
where is a weight for missing data and constrains the first PEF coefficient to be 1.
When all of the equations contain missing data, 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
In this case, represents the various different rescaled copies of the data, is convolution with that rescaled data, and 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 , so
In addition, since the model space has now increased substantially, a second fitting goal must be added,
that ensures that the PEF will vary smoothly over space. In equation (4), is a regularization operator (in this paper, a spatial Laplacian), and is a scale factor.