METHOD

Plane-layer reflectivity or Zoeppritz based modeling techniques cannot capture the effects of heterogeneities in the subsurface. On the other hand, full finite difference methods are too computationally intensive for modeling large 3-D multi-offset surveys. This led to the need to use a scattering method such as the first order Born approximation as developed by Wu and Aki 1985.

For a model that consists of weak scatterers embedded in a smoothly varying background medium, the first Born approximation is that the scatterers act independently. The total wavefield is then the linear sum of the scattered field from each diffractor. A more complete discussion of the method, which includes its applicability to both forward modeling and inverse problems, as well as a review of the literature is given by Beydoun and Mendes 1989.

The model was parameterized in terms of the Lamé parameters and density. Small perturbations $\delta\rho$, $\delta \lambda$ and $\delta \mu$ were considered in a slowly varying background medium described by $\rho_0$, $\lambda_0$ and $\mu_0$. The scattered wavefield from each point diffractor is then proportional to a linear combination of relative perturbations,

$$\frac{\delta\rho}{\rho_0}, \; \frac{\delta \lambda}{\lambda_0+2\mu_0} \; {\rm and} \; \frac{\delta \mu}{\lambda_0+2\mu_0}.$$

Care was taken to use the correct scattering characteristics, so that the amplitudes of the modeled events would be meaningful as long as the initial assumptions were valid.

The amplitudes of these relative perturbations in $\rho$ and $\mu$ had r.m.s. values <0.05, which fits into the $\ll 1$ regime. However, the perturbations in $\lambda$ were larger, having an r.m.s. value about 0.2, with individual scatterers larger than 0.5.