Elastic impedance

Elastic impedance is a pseudo-property or seismic attribute developed by Connolly in 1998 and 1999. It is a generalization of acoustic impedance for non-normal angles of incidence. To invert a vertical-offset (normal incidence) seismic trace, acoustic impedance is used directly from well log bulk density and P-velocity. However, for offset traces, an equivalent of the acoustic impedance can be used to calibrate the non-zero-offset seismic reflectivity.

Theory
Offset-dependent elastic impedance is given by


 * $$ EI(\theta_\mathrm{P}) = \alpha^a \beta^b \rho_\mathrm{B}^c $$

where &alpha; is P-wave velocity, and &beta; is S-wave velocity, &rho;B is bulk density, and


 * $$ a = 1 + \tan^2\theta_\mathrm{P}, \ \ \ b = -8 K \sin^2\theta_\mathrm{P} , \ \ \ c = 1 - 4K \sin^2\theta_\mathrm{P} $$

where K is usually set to the average value of (&beta;/&alpha;)2 over the log interval of interest. Elastic impedance aids in the inversion of non-zero offset data because it provides a log trace derived from a set of P-wave velocity, S-wave velocity,and density logs consistent with the reflectivity of a far-offset-angle stack in the same way that acoustic impedance logs are used to calibrate zero-offset seismic data.

Derivation
It comes from the Aki–Richards equation, sort of like taking an integral of it.

Normalized elastic impedance
An undesirable feature of the elastic impedance function is that its dimensions (units) varies with incidence angle. Strictly speaking, absolute numerical values of EI are rather meaningless and change significantly as a function of $$\theta_{P}$$. Whitcombe (2002) modified the EI function using reference constants $$\alpha_{0}$$, $$\beta_{0}$$, $$\rho_{0}$$. This eliminates the variable dimensions of the elastic impedance equation and returns normalized impedance values. using units of density times velocity (kgm2s-1) for all angles $$\theta_\mathrm{P}$$.

The normalizing constants $$\alpha_{0}$$, $$\beta_{0}$$, $$\rho_{0}$$ are often arbitrarily chosen, but if they are chosen to be averages over the log interval of interest then EI(&theta;) will vary around unity. If the function is further scaled by a factor $$\alpha_{0}\rho_{0}$$ the dimensions of EI become the same as acoustic impedance.


 * $$\mathrm{normalized}\ EI(\theta_\mathrm{P})=\alpha_{0}\beta_{0}\left(\frac{\alpha}{\alpha _{0}}\right)^a\left (\frac{\beta}{\beta_{0}}\right )^b\left(\frac{\rho }{\rho_{0} }\right)^{c}$$

To achieve the new normalized form, we have effectively scaled the original definition by $$\alpha_{0}^{1-a}\beta_{0}^{-b}\rho_{0}^{1-c}$$

This modification allow for direct comparison between elastic impedance values across a range of angles in a manner that was not possible with the EI formulation.