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.  

$$\begin{array} {lll} \bold 0 &\approx & \bold L \bold m - \b... ...\\ \bold 0 &\approx & \bold A (\bold m - \bold s) \end{array}$$ (17)

Although (17) 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 (17). A formulation like (17) 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.

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?

2. Like the seismic prospecting industry, you have solved a huge problem using binning. You have computer power left over to do a few iterations with linear interpolation. How much does the cost per iteration increase? Should you refine your model mesh, or can you use the same model mesh that you used when binning?