# Dix conversion

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The horizon-consistent stacking velocity profiles (Figure 9.1-3) at each of the layer boundaries are used to perform Dix conversion to derive the interval velocity profiles for each of the layers. Dix conversion is based on the formula

 ${\displaystyle v_{n}={\sqrt {\frac {V_{n}^{2}\tau _{n}-V_{n-1}^{2}\tau _{n-1}}{\tau _{n}-\tau _{n-1}}}},}$ (1)

where vn is the interval velocity within the layer bounded by the (n − 1)st layer boundary above and the nth layer boundary below, τn and τn−1 are the corresponding two-way zero-offset times, and Vn and Vn−1 are the corresponding rms velocities. Derivation of equation (1) is provided in Section J.4.

Equation (1) is based on the assumptions that the layer boundaries are flat and the offset range used in estimating the rms velocities Vn and Vn−1 corresponds to a small spread. Additionally, keep in mind that the rms velocities used in equation (1) are based on a straight-ray assumption; thus, ray bending at layer boundaries are not accounted for in Dix conversion.

The procedure for estimating the layer velocities and reflector depths using Dix conversion of stacking velocities includes the following steps:

1. For each of the layers in the model, pick the time of horizon on the unmigrated CMP-stacked data that corresponds to the base-layer boundary (Figure 9.1-2a). These times are used in lieu of the two-way zero-offset times in equation (1).
2. Extract the rms velocities at horizon times (Figure 9.1-3).
3. Use equation (1) to compute the interval velocities for each of the layers from the known quantities — rms velocities and times at top- and base-layer boundaries.
4. Use interval velocities and times at layer boundaries to compute depths at layer boundaries. If the input times are from an unmigrated stacked section as in Figure 9.1-2a, use normal-incidence rays for depth conversion. If the input times are from a migrated stacked section, use image rays for depth conversion.

Interval velocity profiles derived from Dix conversion are shown in Figure 9.1-4a. The earth model can be constructed by combining the estimated interval velocity profiles and depth horizons (Figure 9.1-4b). Comparison with the true model shown in Figure 9.1-1b clearly demonstrates that the interval velocity estimation based on Dix conversion is not completely accurate. The interval velocity profiles derived from Dix conversion (Figure 9.1-4a) exhibit the sinusoidal oscillations caused by the swings in the stacking velocity profiles themselves (Figure 9.1-3).

The fundamental problem is that the stacking velocity estimation is based on fitting a hyperbola to CMP traveltimes associated with a laterally homogeneous earth model. If there are lateral velocity variations in layers above the layer under consideration, and if these variations are within a cable length, then stacking velocities would oscillate in a physically implausable manner [1] [2] [3]. As a consequence, the resulting interval velocity estimation based on Dix conversion is adversely affected. In the present case, Dix conversion has produced fairly accurate estimates for the interval velocities of the top three layers — H1, H2, and H3 as shown in Figure 9.1-4a. But the interval velocity estimates for layers H4 and H5 have been adversely affected by the laterally varying velocities within the layer above, H3.

The pragmatic approach would be to smooth out the oscillations in the stacking velocities before Dix conversion and smooth out the oscillations in the velocity profiles after Dix conversion (Figure 9.1-5a). Then, the resulting earth model is expected to be free of the adverse effects of stacking velocity anomalies (Figure 9.1-5b). A closer look at the central portions of the estimated models using Dix conversion is shown in Figure 9.1-6. Note that the model derived from the smoothed interval velocities is closer to the true model. We shall make an attempt in model updating to update this result by using tomography.

## References

1. Lynn and Claerbout, 1982, Lynn, W. S. and Claerbout, J. F., 1982, Velocity estimation in laterally varying media: Geophysics, 47, 884–897.
2. Loinger, 1983, Loinger, E., 1983, A linear model for velocity anomalies: Geophys. Prosp., 31, 98–118.
3. Rocca and Toldi, 1983, Rocca, F. and Toldi, J., 1983, Lateral velocity anomalies: 53rd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 572–574.