# Elastic impedance

Elastic impedance, ${\displaystyle EI(\theta )}$, is an effective impedance dependent on the incident angle, ${\displaystyle \theta }$. It is therefore the impedance equivalent of angle dependent reflectivity ${\displaystyle R(\theta )}$; if ${\displaystyle EI(\theta )_{1}}$ is the elastic impedance above an interface and ${\displaystyle EI(\theta )_{2}}$ the elastic impedance below then the standard impedance/reflectivity relationship applies;

${\displaystyle R(\theta )={\frac {EI(\theta )_{2}-EI(\theta )_{1}}{EI(\theta )_{2}+EI(\theta )_{1}}}}$

It follows that EI(0) =AI. Just as AI can be used as the basis for well-ties, inversion and rock property analysis of zero-offset seismic data, EI does the same for theta angle stacks.

The most commonly used version of elastic impedance [1] is derived from the above reflectivity equation and the Wiggins linearized AVO equation;

${\displaystyle R(\theta )\approx A+B\sin ^{2}\theta +C\sin ^{2}\theta \tan ^{2}\theta }$

where

${\displaystyle A={\frac {1}{2}}({\frac {\Delta V_{P}}{V_{P}}}+{\frac {\Delta \rho }{\rho }})\,\,\,\,\,\,\,\,B={\frac {\Delta V_{P}}{2V_{P}}}-4k({\frac {\Delta V_{S}}{V_{S}}})-2k({\frac {\Delta \rho }{\rho }})\,\,\,\,\,\,\,\,C={\frac {\Delta V_{P}}{2V_{P}}}}$

This results in;

${\displaystyle EI(\theta )=V_{P0}\rho _{0}[{\frac {V_{P}^{1+\tan ^{2}\theta }}{V_{P0}}}{\frac {V_{S}^{-8k\sin ^{2}\theta }}{V_{S0}}}{\frac {\rho ^{1-4k\sin ^{2}\theta }}{\rho _{0}}}]}$

Where ${\displaystyle k={\frac {V_{S}^{2}}{V_{P}^{2}}}}$ and ${\displaystyle V_{P0}}$, ${\displaystyle V_{S0}}$, and ${\displaystyle \rho _{0}}$ are normalisation constants.[2] If only the first two terms of the Wiggins equation are used for the derivation the resultant EI expression is the same except that the ${\displaystyle \tan ^{2}\theta }$ in the first exponential becomes ${\displaystyle \sin ^{2}\theta }$.

The derivation of EI requires that the background velocity ratio k be kept constant for an EI series. If the value of k used in a calculation is close to the correct average value then the EI model is usually reasonably accurate but an incorrect value of k will cause inaccuracies. The values of the normalisation constants do not affect the accuracy of the model but ensure that EI values are in a similar range to AI values.

Similar impedance expressions can be derived for converted waves. Anisotropy terms can also be included.

The advantages of EI over reflectivity based AVO expressions follow from its main characteristic that it is an intrinsic property of the medium with values for each sampled measurement rather than an outcome of property contrasts across an interface. This allows numerical and visual correlation. EI curves can be displayed as part of a suite of petrophysical curves to indicate the visual correlation with reservoir properties or EI values can be cross-plotted against other properties for quantitative analysis.

Elastic impedance provides a convenient way to tie well logs to seismic angle stacks. A ${\displaystyle V_{P}}$ term can be factored from the EI equation and the remaining term used in place of the density in a standard well tie procedure;

${\displaystyle EI(\theta )=V_{P}[V_{P}^{(\tan ^{2}\theta )}V_{S}^{(-8k\sin ^{2}\theta )}\rho ^{(1-4k\sin ^{2}\theta )}]}$

(normalization terms not shown).