Tau tomography

In depth tomography we must constantly deal with the depth-velocity ambiguity problem. Put another way, we are simultaneously trying to estimate both a focusing (S_f) and a mapping (S_m) slowness. Biondi et al.1997 showed that by mapping (z,x) into ($\tau,x$) through

$$\tau(z,x) = \int_0^z 2 S(z',x) \delta z'$$ (8)

we can write a focusing eikonal equation which only indirectly depends on the mapping velocity

$$4\left( {\partial t (\tau,x) \over \partial \tau }\right) + ... ...m(\tau,x) {\partial t (\tau,x) \over \partial \tau}\right)^2 =1$$ (9)

where $\sigma_m$ is the differential mapping operator defined as

$$\sigma_m(\tau,x) = \int_0^\tau S^{-1}_m (\tau',x) {\partial \over \partial x}S_f(\tau',x) \delta \tau' .$$ (10)

From this eikonal equation we can derive a new relation for the change in travel time due to a change in the focusing velocity:  

$$\delta t = \left( {\tilde{\delta x}^2 \over \tilde{l}} \righ... ...ilde{\delta x} \tilde{\sigma_0}(x,\tau)) \right) \delta \sigma$$ (11)

where $\tilde{\delta x}$ and $\tilde{\delta \tau}$ are the change in x and $\tau$ position of the ray segment, $\tilde{\sigma_0}(x,\tau)$ is the differential mapping factor of our initial slowness model at the ray location, and $\delta \sigma$ is defined as

$$\delta \sigma(\tau,x) = {\sigma_0(\tau,x) \over s_0(\tau,x)}... ...,x) - 2 s_0(\tau,x) {\delta (\delta z(\tau,z)) \over \delta x}$$ (12)

where

$$\delta z(\tau,z) = \int_0^\tau{ \delta s(\tau',x) \over 2 s_0^2(\tau',x)} \delta \tau .$$ (13)

We now have a way to back project travel time errors and can write a new set of fitting goals,

$$\begin{eqnarray} \bf \Delta t&\approx&(\bf T_{\tau,ray} + \bf T_{\tau,ref}) \bf A^{-1}\bf p\nonumber \\ - \bf A\bf s_{0} &\approx&\epsilon \bf p\end{eqnarray}$$ (14)

where $\bf T_{\tau,ray}$ and $\bf T_{\tau,ref}$ use (11) rather than (2) to back project.

SYNTHETIC TESTS

To test the effectiveness of the method we create a simple anticline synthetic velocity model, Figure 2. We simulated six reflectors, one on top, four in the anticline, and a basement reflector. We calculated travel times to all the reflectors for an offset range of 4 kms. These travel times represent our `recorded travel times'. For our initial model we created a v(z) model by taking the lateral average of the velocity field, Figure 2.

 

model
Figure 2 Left panel is our synthetic model superimposed by the six reflectors. The right panel is our starting guess for our velocity function and the map migrated reflector position using this initial velocity estimate.

From this initial model we attempted to invert the velocity function by three progressively advanced methods:

Depth-Standard

: Inverted for a depth model, using an inverse Laplacian preconditioner Claerbout (1998b) for $\bf A^{-1}$ in our depth fitting goals (7)

Depth-Steering

: Inverting for a depth model, using a steering filter operator for our preconditioner in our depth fitting goals (7)

Tau-Steering

: Inverting for a tau model, using a steering filter operator for our preconditioner in tau fitting goals (14)

Figure 3 shows the result of one non-linear iteration for all three inversions schemes. All three methods were able to recover the dome shape after one iteration. When using a Laplacian smoother the velocity increase is spread too far both laterally and vertically. As a result, the bottom reflector is located too deep throughout the model. When using steering filters we still have a significant velocity-depth ambiguity problem, but we have done a little better job position the bottom reflector. In the case of tau tomography with steering filters we have done almost a perfect job after a single iteration. We have not perfectly recovered the lower portion of the anticline structure but we have almost completely flattened the bottom reflector.

 

model1-iter1
Figure 3 Left panel is Depth-Standard, middle is Depth-Steering, right is Tau-Steering. All after 1 non-linear iteration.

To see if, and how fast, we could converge to the correct solution in depth we performed several more non-linear iterations. As Figure 4 shows we did a decent job recovering the anticline with all three methods. In this case, where the model is fairly simple and we have good travel time coverage, the big advantage seems to be speed. We got a high quality result with steering filter tau tomography in a single iteration, while it took three with steering filter depth tomography, and four when using the Laplacian and depth tomography.

 

model1-best
Figure 4 Top-left: Depth-Standard, after 4 non-linear iterations; top-right: Depth-Steering after 3 iterations; bottom-left: Tau-steering after 1 iteration; and bottom-right: a comparison of the reflector positions using all 3 methods. The solid, white line is correct reflector position, the small dashes represent Tau-Steering: large-dashes:Depth-Steering; and the solid black line is Depth-Standard.

The smoothness of the anticline was well suited for the Laplacian so we decided on a slightly more difficult challenge that could better differentiate between a Laplacian and steering filter regularization. Our new model keeps the same basic shape for the model but adds a low velocity layer within the anticline. Figure 5 shows the correct, initial, and the result of 4 iterations using both the Laplacian and steering filters to precondition the problem. After 4 iterations the steering filters have done a much better job recovering the low velocity layer.

 

model2
Figure 5 Top, left Our new model with a low velocity layer within the anticline; top-right, our starting model; bottom-left, Depth-Standard after 4 iterations; bottom-right Depth-Steering after 4 iterations.

TEST ON REAL DATA We next decided to test the method on real data. For this initial test we decided to work with a relatively clean data which still had some residual move-out in the common reflection point (CRP) gathers. The data is from the Blake Outer Ridge as was used by Ecker1998 to characterize methane hydrate structures. For our initial velocity model we used Ecker's Dix 1955 derived model, Figure 6.

Our general philosophy was to limit human time as much as possible. Therefore we chose to do tau migration Alkhalifah (1998) using a generic Kirchhoff packageBiondi (1998). By using tau rather than depth migration, we were quickly able to compare CRPs from iteration to iteration and it allowed us to pick reflector positions only once.

 

christine-vel0
Figure 6 Initial velocity model in depth. Note the low velocity zone caused by the gas hydrate starting at approximately 32000 kms and extending to the end of the section.

After performing the migration we picked six reflectors, Figure 7. We picked the sea floor, a strong reflector above the bottom simulating reflector (BSR), the BSR itself, the flat reflector below the BSR, and two deeper reflectors.

 

stack
Figure 7 Initial stack overlaid by reflectors picked for tomography.

Rather than pick move-out differences we decided to create residual semblance panels at each reflector location, Figure 8. The panels indicate that there is significant residual curvature, especially where the BSR meets the lower reflector. From these semblance panels we picked smooth curves at approximately the maximum semblance at each reflector. To check to see if a single parameter adequately described the move-outs we back projected the picked semblance into our CRP gathers. Figure 9 shows that the semblance picks did a fairly good job describing the move-out.

 

sem-vel0
Figure 8 Residual semblance panels for the bottom 5 reflectors. The black line in each panel represents the picked maximum.

 

overlay.vel0
Figure 9 Common reflection point gathers from every 2000 meters starting from 28000. The lines are the result of mapping back the picked residual slowness values. Note how the curves do an excellent job matching the actual reflector move-out.

We used our picked reflectors to construct our steering filters and then applied our tau tomography fitting goals (14). Generally, we have increased velocity, Figure 10, but the changes still keep velocity following reflector dip.

 

christine-vel1
Figure 10 The velocity after 1 iteration of tau-steering tomography

The next step is to see if our new velocity model flattens our CRP gathers and improves the focusing of the data. Figures 11 and 12 indicate that we have accomplished both of these goals. Figure 9 shows that all of our reflectors are significantly flatter, with only significant curvature left along the BSR. Figure 12 shows a much more continuous BSR reflection along with overall improved focusing of the section above and below.

 

overlay.vel1
Figure 11 CRP gather from 24000-48000 meters. Note how they are considerably flatter than Figure 9.

 

stack-comp
Figure 12 The stack using our initial velocity and the velocity after 1 iteration of tau steering tomography. Note how the reflectors are generally better focused at a, b, c, and d.