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 (). Generally, I relate change in slowness 141#141,to change in travel time 224#224 by a linear operator 223#223The tomography operator is constructed by linearizing around an initial slowness model 499#499. I regularize the slowness 500#500rather than change in slowness and obtain the fitting goals, The calculation of 502#502 is the same procedure as shown in equation (). The only difference is now we initiate 490#490 with both our random noise component 503#503 and 504#504.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  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  shows the result of performing split-step phase shift migration and Figure  shows the resulting angle gathers (). Note how the image is generally well focused and the gathers with some slight variation below three kilometers at x=3.5. Figure  shows the moveout of the gathers in Figure . 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 .

 

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

 

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

To start we need to solve the problem without accounting for model variance. If we solve for 141#141 using fitting goals () our updated velocity is shown in Figure . 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  and . 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 141#141 using fitting goals ().

 

image-none Figure 6 New image obtained by inverting for 141#141 using fitting goals () using the velocity shown in Figure .

 

mig-none Figure 7 New gathers obtained by inverting for 141#141 using fitting goals () using the velocity shown in Figure .

If we apply equation () using the 505#505 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 141#141 accounting for the model variability. Figure  shows four such realizations. Note that they have the same general structure as seen in Figure  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  and . 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 ), 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 ).