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.  

$$\begin{array} {lll} \bold 0 &\approx & \bold L \bold m - \b... ...\\ \bold 0 &\approx & \bold A (\bold m - \bold s) \end{array}$$ (19)

Although (19) 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 (19). A formulation like (19) is more likely to be helpful where we need to handle rapidly changing functions; for example, in reflection seismology where high resolution is meaningful.

EXERCISES:

1. It is desired to find a compromise between the Laplacian roughener and the gradient roughener. What is the size of the residual space?