Integrating time differences

A reason I particularly like the Vesuvius exercise is that slight variations on the theme occur in many other fields. For example, in 2-D and 3-D seismology we can take the cross-correlation of neighboring seismograms and determine the time lag $\tau$ of the maximum correlation. Thus, analogous with Vesuvius, we pack a vector $\bold d$ with measurements of $d\tau/dx$ and $d\tau/dy$.Now we hypothesize that there exists a lag $\tau(x,y)$ whose gradient matches $\bold d$.Instead of solving for phase $\phi$, our regression says $\nabla \tau(x,y) \approx \bold d$,and we can approach it as we did Vesuvius. Actually, I'd like to rewrite the book with just such an example because for many people time lag $\tau(x,y)$ is more concrete than phase $\phi(x,y)$.Unfortunately, the real problem requires visualizing the raw data (voltage as a function of (t,x,y) which requires learning to use 3-D volume data visualization tools. Likewise the raw data back shifted by $\tau(x,y)$ is 3-D. Additionally, the codes would be more cluttered because the raw data would be a cube of numbers instead of a plane, and we'd need to fumble around doing the crosscorrelations. That crosscorrelation business is a little tricky because we need to measure time shifts less than one mesh point.

Old-time reflection seismic interpreters would track a strong event along a seismic line going off into the 3-D world when they would jump from one line to a crossline. Eventually they would work their way back to the starting line where they would hope they were on the same event. They would say, ``The lines should tie.'' The mathematician (or physicist) is saying something similar with the statement that "The curl should vanish everywhere." If the sum around all possible little loops vanishes, logically it means that the sum around all big loops also vanishes.

Here is a real-world problem you could think about: You have earthquake seismograms recorded at i=1,2,...,N locations. You would like to shift them into alignment. Assume a cartesian geometry. You have measured all possible time lags $\tau_{i,j}$between station i and station j. What operator would you be giving to the solver?