Born elastic modeling

Each point within the reservoir volume is considered as an independent point diffractor of much smaller size than the wavelength of the incident wave. Formulae for Rayleigh scattering are then used to describe the scattering characteristics of each single diffractor, and a modeled trace can then be constructed as the linear sum of contributions from all the elements within the reservoir.

The important assumptions that have to be made for this technique to be applicable are:

1.

The Earth model consists of two parts: firstly, a smoothly varying background medium through which ray-paths can be traced; and secondly, within this there are high-frequency perturbations or point diffractors which act to scatter the wavefield.

2.

The amplitude of the perturbations are small relative to the background medium. This allows each scatterer to be treated independently (first order Born approximation). It also means that the scattered wavefield can be modeled as being linearly related to the amplitude of the perturbations.

3.

The physical size of scatterers is small with respect to the wavelength of the incident wave, or $ka \ll 1$, where k is the wavenumber of the incident wave and a is the length scale of the heterogeneity.

Following this approach, the field recorded at the surface can be approximated by the following integral taken over the whole reservoir:

$$U(t) = \int \; A_{s} \; A_{r} \; R \; D \; \frac{d}{dt} \, \psi(t - \tau_r - \tau_s) \; dV$$ (5)

where A_s and A_r are the geometric spreading factors from the source to the diffractor and from the diffractor to the receiver, D is a correction for the directionality of the receiver, $\tau_r$ and $\tau_s$ are ray-traced travel-times from the source to the diffractor and from the diffractor to the receiver, $\psi(t)$ is the source signature, and R is amplitude from Rayleigh scattering which can be calculated from the average perturbations $\bar{\delta \rho}$, $\bar{\delta \lambda}$ and $\bar{\delta \mu}$ from the background Lamé parameters $\rho_0$, $\lambda_0$ and $\mu_0$ and the total scattering angle, $\theta$

$$R = \frac{\bar{\delta \rho}}{\rho_0} \cos \theta - \frac{\b... ...- \frac{2 \bar{\delta \mu}}{\lambda_0 + 2\mu_0} \cos^2 \theta$$

Calculation of R, $\tau_r$ and $\tau_s$ can be facilitated by precalculating and tabulating travel-times, geometric spreading factors and propagation angles so that they can be quickly referenced from the inner-loop of the modeling program and do not have to be calculated each time on the fly.