Spatial interpolation with $\partial_x-p\partial_t$

Given data on a mesh and the dip filter form $\partial_x-p\partial_t$we can find the best fitting value of p by a univariate linear-least squares procedure in chapter 4. In summary, take the filter output $v(t,x)=\partial_x u(t,x) - p\partial_t u(t,x)$and minimize it, $v(t,x)\approx 0$,in the least-squares sense by variation of the scalar variable p. The resulting p is simply a ratio of dot products (summation over t and x) with a numerator of u_x dotted into u_t and a denominator of u_t dotted into itself.

The difference star representing $\partial_x-p\partial_t$is unchanged when the mesh is interleaved simultaneously on t and x. Finding the missing values on the t-axis is straightforward by any interpolation method. Finding the missing values on the x-axis seems straightforward by the CG method. Those with wave-equation solving experience might suggest a systematic procedure to have v(t,x)=0 vanish exactly on the new traces. I'll outline the procedure to give confidence that the least-squares task need not be a horrific one, despite the huge number of unknowns. The differencing star is $2\times 2$.Suppose the right side is placed on a known trace, the left side multiplies an unknown trace. The unknown trace can be found by solving a bi-diagonal set of simultaneous equations. This will involve recursion up or down the time axis depending on the sign of p. Since p will typically vary with t and take both signs in the range, well-known methods are needed. Hopefully the CG method takes care of this in a fairly automatic way. This deserves investigation. Potentially the CG method requires as many iterations as there are data points, so this could be prohibitive when the task is set up in a huge window of data. If the CG method is slow here, we'll gain some valuable insights on how to speed it up because we think we understand the mechanisms so well in this example. On the other hand, the CG method does not try for zero error when extrapolating from right to left, but for minimum averaged error when extrapolating both ways. Unfortunately, I have burned myself out on programming, so I have no examples to show you.