Relinearization

As mentioned above tomography is actually a non-linear problem. We can potentially achieve a better result by relinearization around our updated slowness model. We can write our updated travel times in terms of non-linear iteration i,

$$\bf \Delta t_i = \bf \Delta t- \bf T_{i} ({\bf s_i} - {\bf s_0}).$$ (5)

We performed two more non-linear iterations of tomography. The right panel of Figure  shows our updated ray field. The ray field shows moderate changes from the initial field (left panel of Figure ). After two additional non-linear iterations we get a moderately changed velocity model, shown in the left panel of Figure . The overall reduction in the RMS travel-time error is about 80%, with the majority coming on the first iteration.

 

final
Figure 6 The final set of rays (right) and the final velocity model (left) after three non-linear iterations of tomography. Note how many of the feature seen in Figure  have become more defined.