Endpoints, edge effects, and errors

To set up our tomography problem we need to cover some final details. We can convert our semblance picks back into a $\Delta z$ shift by applying

$$\Delta z = \gamma \theta^2 .$$ (8)

We then note that our tomography problem is set up for time rather than depth errors, To convert our depth error to a time error we multiplying by RMS slowness s_rms of. If we use a straight ray geometric assumption, we can approximate the time error at a given offset by multiplying by $\cos \theta$ and $\cos \phi$ (where $\phi$ is the geologic dip) obtaining as our final relationship  

$$\Delta t = \frac{ 2 \gamma \theta^2 \cos \theta\cos \phi } {v(z,{\bf x}) } .$$ (9)

In constructing our raypaths we benefit from having our CRP gathers in terms of angle. If errors were in terms of offset we would have to either

• shoot rays from every source and receiver location, and find ray pairs that obey Snell's law at the position on the reflector imaged at the offset dictated by the ray pair
• or shoot rays from our CRP locations along where our reflector is imaged at every offset and then interpolate the ray field to our source and receiver locations.

Both options require significant additional ray-tracing in 2-D, and even more in 3-D. In addition, we are always faced with the tradeoff of how much should we interpolate our rays versus how many additional rays should we shoot.

With our moveout errors in terms of angle we only need to shoot a single ray-pair up from our imaging point at the angle $\alpha$ and $\beta$,

$$\begin{eqnarray} \alpha &=& \phi + \theta \\ \nonumber \alpha &=& \phi - \theta\end{eqnarray}$$ (10)

where $\theta$ is one-half the aperture angle, $\phi$ is the geologic dip (Figure 1). If the rays emerge at surface locations corresponding to an offset and CMP location inside our acquisition geometry we have a valid ray pair.

 

sketch Figure 1 How the takeoff angle for a ray-pair are defined.