Regularization is model styling

Abandoned theory for matching wells and seismograms

Let us consider theory to construct a map $\bold m$ that fits dense seismic data $\bold s$ and the well data $\bold w$. The first goal $\bold 0 \approx \bold L \bold m - \bold w$ says that when we linearly interpolate from the map, we should get the well data. The second goal $\bold 0 \approx \bold A (\bold m - \bold s)$ (where $\bold A$ is a roughening operator like $\nabla$ or $\nabla^2$) says that the map $\bold m$ should match the seismic data $\bold s$ at high frequencies but need not do so at low frequencies.

$$\begin{array}{lll} \bold 0 &\approx & \bold L \bold m - \b... ... \bold 0 &\approx & \bold A (\bold m - \bold s) \end{array}$$ (18)

Although (18) is the way I originally formulated the well-fitting application, 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 approaches have the advantage that it is not necessary to experiment with the relative weighting between the two goals in (18). A formulation like (18) 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.

Regularization is model styling