Tomography

Tomography, or at least the definition of tomography I will be using in thesis, starts from the idea that there is an operator $\bf T_{nl}$ that relates slowness $\bf s$to travel times ,

$$\bf t = \bf T_{nl} \bf s$$ (1)

One way to think of this operator is in terms of rays. Figure 1 shows a simple two layered model with sources along top and left edges and receivers along all four edges. The simplest way to think of the operator is that it simply integrates the slowness along the raypath.

 

cor.overlay Figure 1 A simple example of tomography with the sources and receivers every .2 km in depth and every .5 km in x. Overlaid are the raypaths connecting the sources to the receivers.

This simplistic view of tomography has a number of inconsistencies with the reflection tomography problem. The first is that we wouldn't be doing tomography if we actually knew the slowness field beforehand. What we actually have is some initial guess, $\bf s_0$, of the slowness field. If for our initial model we assume a constant slowness field and find the raypaths connecting the same source and receivers, Figure 2, we encounter another problem: the raypaths are significantly different. An alternate way to state this observation is that the operator is model dependent and therefore non-linear.

 

init.overlay Figure 2 The same recording geometry as Figure 1, with a constant velocity field. Note that the ray paths are significantly different.