Summary

We have discussed the mathematical and physical development of our theoretical Kirchhoff migration/inversion solution from an intuitive perspective (we hope!).

The Kirchhoff reflectivity estimation is an elastic prestack depth migration. It is elastic because it uses the elastic properties V_p, V_s, $\rho$, and anelastic Q factors in the background migration (velocity) model, and backpropagates the recorded wavefield in accordance with the elastodynamic wave equation, as opposed to the standard but more approximate acoustic wave equation. It is a depth migration because it uses interval properties (velocities) specified as a function of spatial coordinates, as opposed to time migration and rms velocities specified as a function of a temporal coordinate. This implies that the elastic depth migration will tend to be more correct than acoustic time migration if the elastic interval velocity model is known.

The migration is true amplitude in the sense that it can recover the correct relative amplitude reflectivity information, without any special preprocessing of the data, provided the data amplitudes are reliable and the elastic migration model is sufficiently accurate. reliable. The migration can estimate the elastic Zoeppritz reflectivity coefficients $R_{ij}(\theta)$ as a function of specular reflection angle. The subscripts (i, j) indicate that any of the coefficients $\grave{P}\!\acute{P}$, $\grave{P}\!\acute{S}$, $\grave{S}\!\acute{S}$, etc., can be estimated, although at present only the $\grave{P}\!\acute{P}$ implementation is available. The $R_{pp}(\theta)$estimates are theoretically correct for heterogeneous media (including lateral velocity variations) that are ray valid (i.e., smoothly varying velocities within the model and each layer, if any). The present implementation is for a 2-D geometry, but the theory is valid for 3-D implementation, which will be considered in the near future. Finally, the technique is well suited to flexible acquisition geometries, and should be adaptable to VSP, crosshole, and other non-standard acquisition geometries.

The elastic parameter inversion is fast and robust. Several (six at present) parameterization choices are available, as well as confidence estimator maps which give quantitative measures of the relative confidence in the elastic parameter inversion depth images.