TOMOGRAPHY IN TIME VS. TOMOGRAPHY IN DEPTH

Because of the velocity-depth coupling, non-linear tomographic velocity estimation in the depth coordinates often does not converge effectively. To avoid some of the drawbacks of traditional tomography Biondi et al. 1997 proposed a tomographic velocity estimation in the two-ways traveltime coordinates ($\tau$,$\xi$). The method is based on a transformation of the eikonal equation from the depth coordinates (z,x) into the time coordinates ($\tau$,$\xi$), according the following transformation ov variables:

$$\begin{eqnarray} \tau(z,x) &=& \int_0^z {2\over V(z',x)} dz' \nonumber \\ \xi(z,x) &=& x .\end{eqnarray}$$ (7)

 

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Figure 8 Velocity model after one, two, three, and four non-linear iterations.

Starting from the eikonal equation in the time coordinates, we can remap the tomography goals of equation (6) into the time coordinates. The operators $\bold{L_s^i}$, $\bold{H}$, $\bold{G}$ are different in the time domain from the equivalent depth-domain operators; and also $\bold A^{-1}$ has a slightly different orientation, because it now operates on a velocity function defined in ($\tau$,$\xi$).

Some of the advantages of formulating the tomography problem in ($\tau$,$\xi$) domain can be seen in Figure 9. In this new domain reflector movement is minimal, significantly decreasing the velocity-depth connectivity problem that we saw in (z,x) space. The initial reflector position is closer to the correct reflector position. As a result, the orientation of the steering filters more accurately follows true reflector dip.

 

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Figure 9 Left, depth velocity model with the correct reflector position (dashed) and the estimated reflector position using the initial guess at the velocity model (solid). Right, ($\tau$,$\xi$) velocity model with the same reflectors superimposed. Note how reflector movement is significantly less than in the ($\tau$,$\xi$) case.

To analyze the advantages of working in ($\tau$,$\xi$) space we performed two non-linear iterations of tomography and remapped the resulting velocity model back into (z,x) space (Figure 10). Comparing the results of tomography in the time domain (Figure 10), with the results of tomography in the depth domain (Figure 8), we can notice how the doublet, seen in the depth tomography case, has significantly decreased and how the dome shape is much more prominent.