Tomography

The way I formulate my tomography fitting goals requires some deviation from the generic multi-realization form. My tomography fitting goals are fully described in Clapp (2001a). Generally, I relate change in slowness $\bf \Delta s$,to change in travel time $\bf \Delta t$ by a linear operator $\bf T_{}$The tomography operator is constructed by linearizing around an initial slowness model $\bf s_{0}$. I regularize the slowness $\bf s_{}$rather than change in slowness and obtain the fitting goals,

$$\begin{eqnarray} \bf \Delta t&\approx&\bf T_{} \bf \Delta s \\ \nonumber \epsilon \bf A\bf s_{0} &\approx&\epsilon \bf A\bf \Delta s.\end{eqnarray}$$ (4)

The calculation of $\sigma_{d}$ is the same procedure as shown in equation (3). The only difference is now we initiate $\bf r_{m}$ with both our random noise component $\sigma_{m} \bf \eta$ and $\epsilon \bf A\bf s_{0}$.A cororarly approach for data uncertainty is discussed in Appendix A.

Results To test the methodology I decided to start with a structurally simple 2-D line from a land dataset from Columbia provided by Ecopetrol. Figure 1 shows the estimated velocity for the data. Note how it is generally v(z) with some deviation, especially in the lower portion of the image. Figure 2 shows the result of performing split-step phase shift migration and Figure 3 shows the resulting angle gathers Sava (2000). Note how the image is generally well focused and the gathers with some slight variation below three kilometers at x=3.5. Figure 4 shows the moveout of the gathers in Figure 3. Note the traditional `W' pattern associated with the velocity anomaly can be seen in cross-section at depth.

 

vel-init Figure 1 Initial velocity model.

 

image-init Figure 2 Initial migration using the velocity shown in Figure 1.

 

mig-init Figure 3 Every 10th migrated gather using the velocity shown in Figure 1.

 

semb-init Figure 4 Moveout of the gathers shown in Figure 3.

To start we need to solve the problem without accounting for model variance. If we solve for $\bf \Delta s$ using fitting goals (4) our updated velocity is shown in Figure 5. The change of the velocity is generally minor, with an increase in the high velocity structure at x=3.5, z=3.2. The resulting image and migration gathers are shown in Figures 6 and 7. The resulting image is slightly better focused below the anomaly and the migration gathers are, as expected, a little flatter.

 

vel-none Figure 5 New velocity obtained by inverting for $\bf \Delta s$ using fitting goals (4).

 

image-none Figure 6 New image obtained by inverting for $\bf \Delta s$ using fitting goals (4) using the velocity shown in Figure 5.

 

mig-none Figure 7 New gathers obtained by inverting for $\bf \Delta s$ using fitting goals (4) using the velocity shown in Figure 5.

If we apply equation (3) using the $\bf r_{n}$ when estimating our improved velocity model we can find the right amount of noise to add to our fitting goals. We can now resolve for $\bf \Delta s$ accounting for the model variability. Figure 8 shows four such realizations. Note that they have the same general structure as seen in Figure 5 but within additional texture that is accounted for by covariance description. If we migrate with these new velocity models we get the images and migrated gathers shown in Figures 9 and 10. In printed form these images appear identical, or close to identical. If watched as a movie, amplitude differences can be observed.

 

vel-multi
Figure 8 Four different realizations of the velocity accounting for model variability.

 

image-multi
Figure 9 Four different realizations of the migration accounting for model variability. Note how the reflector position is nearly identical in each realization and with the image without variability (Figure 6), but the amplitudes vary slightly.

 

mig-multi
Figure 10 Four different realizations of the migration accounting for model variability. Note how the reflector position is nearly identical in each realization and with the image without variability (Figure 7).