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.
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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.