Straight Ray Tomography Example


OBJECTIVE: Solve Traveltime Tomography Ls=t by s=[LTL]-1LTt.

BACKGROUND: Straight ray tomography does not take into account ray bending but can provide a good quick first velocity model. Code below is quite simple and does a transmission tomography experiment where sources are along top boundary and receivers along bottom boundary.

PROCEDURE:

  1. Download the programs timetomo.m and name it timetomo.m. Also download the following files ray.m, coord.m, raymat.m, merge.m, and raypath.m.

  2. There is a rule of thumb in traveltime tomography and the slowness resolution: the best resolved features in the reconstructed tomogram are velocity gradients perpendicular to the rays. That is, horizontal rays are good for resolving velocity distributions vfor horizontal layers, and vertical rays are best for resolving veryically oriented layers.
  3. Type timetomo, and it will first give you a movie of rays, where shot is along top surface and receivers is along bottom boundary. The model is a vertical layer model. Extract the diagonal components of the [LTL]-1 matrix by typing "di=diag(inv(SLL'*SLL));". Plot them up by typing imagesc(reshape(di,nx-1,nx-1)');colorbar. Make sure that you add a diagnoal matrix with damping parameter of .1 to LTL matrix before you invert. This will lessen singularity of matrix so covariance values are better displayed. This is the plot of the variance terms of the slowness model. Do the high variance values correspond to poorly resolved parts of model? Why? Which cells have rays with widest variability of ray angles? Are they best resolved according to the covariance matrix?

  4. Adjust calculations by changing model to a horizontal layer model. Model can be changed to a 2-layer model by uncommenting the 2-layer model line in timetomo.m.

    Does the new covariance matrix vlaues predict areas of poor resolution? Make sure that you add a diagnoal matrix with damping parameter of .1 to LTL matrix before you invert. This will lessen singularity of matrix so covariance values are better displayed.

  5. Adjust calculations for an anomaly model. An anomaly model can be included by uncommenting the 2 anomaly lines in timetomo.m.

  6. Explain why vertical layer model is well resolved but horizontal layer model is not. Comment about resolvability of anomaly model.

    (Extra Credit) Try different damping parameters and constraints to see if the result improves.

  7. (Extra Credit) Determine optimal Marquardt damping parameter by creating a plot similar to that in Figure 2.4 in here.