THEORY

Following the method described in Clapp 1998, we began by linearizing the tomography problem around an initial guess at our slowness model $\bf s_{0}$. We assumed ray stationarity and described the change in travel time ($\bf \Delta t$) as being linearly related to our change in slowness ($\bf \Delta s$):

$$\bf \Delta t\approx \bf T_{z} \bf \Delta s.$$ (1)

$\bf T_{z}$ is composed of two portions. The first, $\bf T_{z,ray}$ simply applies  

$$\delta t = \tilde{l} \delta s ,$$ (2)

or that the change in the travel time is ($\delta t$) is equal to change in slowness ($\delta s$) times length of the ray of the ray segment ($\tilde{l}$) of the ray connecting the source, reflector, and the receiver. The second component, $\bf T_{z,ref}$, can be thought of as a chain of two operators: the first maps our change in slowness ($\bf \Delta s$) into reflector movement, the second maps the reflector movement into our change in travel times ($\bf \Delta t$) van Trier (1990). This second term amounts to performing residual migration and can be done by back projecting a ray located at the reflection point perpendicular to the reflector Stork (1994), Figure 1. Taking both components into account our tomography fitting goal becomes

$$\bf \Delta t\approx ( \bf T_{z,ray} + \bf T_{z,ref} ) \bf \Delta s.$$ (3)

 

schematic
Figure 1 The two portions of the back projection operator. $\bf T_{z,ray}$ is the pair of rays through $\bf s_{0}$ from the source to the receiver that obey Snell's law at the reflector. $\bf T_{z,ref}$ is the raypath from this reflection point to the surface.