Let us consider theory to construct a map that fits dense seismic data and the well data .The first goal says that when we linearly interpolate from the map, we should get the well data. The second goal (where is a roughening operator like or )says that the map should match the seismic data at high frequencies but need not do so at low frequencies.
Although () is the way I originally formulated
the well-fitting problem, I abandoned it for several reasons:
First, the map had ample pixel resolution compared to other sources of error,
so I switched from linear interpolation to binning.
Once I was using binning,
I had available the simpler empty-bin approaches.
These have the further advantage that it is not necessary
to experiment with the relative weighting between
the two goals in ().
A formulation like () is more likely
to be helpful where we need to handle rapidly changing functions
where binning is inferior to linear interpolation,
perhaps in reflection seismology where high resolution is meaningful.