Velocity-independent time-domain seismic imaging using local event slopes

Dix inversion

Not only moveout velocities but also interval velocities can be estimated by using the local slope information. Employing the Dix inversion approach (Dix, 1955) and defining $p_t = \partial p/\partial t$, one can deduce from equations 3 and 4 an expression for the interval velocity $v_i$, which becomes another attribute directly mappable from the data, as follows:

$$v_i^2 = %\frac{d}{d\,t_0}\,\left[t_0\,v^2(t_0)\right] = ... ...rac{p\,l (p + t\,p_t) - 2 p_t\,t^2}{2\,t - l\,(p + t\,p_t)}\;.$$ (15)

Equation 15 shows that then the interval velocity can be regarded as another attribute mappable directly from the data. The derivation is detailed in Appendix A.

Figure 5 shows the interval velocity mapping according to equation 15 for the synthetic data example shown in Figure 1a. The exact interval velocity profile (red curve) is recovered perfectly. An analogous example for the field dataset from Figure 3 is shown in Figure 6. The field data result is noisy because of the instability of numerical differentiation but clearly shows the overall range of interval velocities.

Equation 15 enables direct mapping from data slope attributes into interval velocities. Thus, it provides an analytical solution to the stereotomography problem (Billette and Lambaré, 1998) for the special case of horizontal reflection layers and vertically variable velocities.

svin
Figure 5. Oriented mapping to interval velocity for the synthetic data set shown in Figure 1a. The exact interval velocity profile is shown by a red curve.

bvint
Figure 6. Oriented mapping to interval velocity for the field data set shown in Figure 3a.

Velocity-independent time-domain seismic imaging using local event slopes