# Extended elastic impedance

Extended elastic impedance ${\displaystyle EEI(\chi )}$, is an effective impedance dependent on the coordinate rotation angle ${\displaystyle \chi }$ in the AVO intercept/gradient domain. It is therefore the impedance equivalent of ${\displaystyle R(\chi )}$;

${\displaystyle R(\chi )=A\cos(\chi )+B\sin(\chi )}$

where ${\displaystyle A}$ is the intercept, ${\displaystyle B}$ the gradient. If ${\displaystyle EEI(\chi )_{1}}$ is the value above an interface and ${\displaystyle EEI(\chi )_{2}}$ the value below then the standard impedance/reflectivity relationship applies;

${\displaystyle R(\chi )={\frac {EEI(\chi )_{2}-EEI(\chi )_{1}}{EEI(\chi )_{2}+EEI(\chi )_{1}}}}$

It also follows that EEI(0) = AI. EEI(90) is referred to as Gradient Impedance, GI, and is the impedance equivalent of the AVO gradient term ${\displaystyle B}$ (or ${\displaystyle G}$). EEI can be used as the basis for well-ties, inversion and rock property analysis of chi angle stacks in the same way as EI does for theta angle stacks.

The formula for EEI can be derived from the above two equations;[1]

${\displaystyle EEI(\chi )=V_{P0}\rho _{0}[({\frac {V_{P}}{V_{P0}}})^{(\cos \chi +\sin \chi )}({\frac {V_{S}}{V_{S0}}})^{(-8k\sin \chi )}({\frac {\rho }{\rho _{0}}})^{(\cos \chi -4k\sin \chi )}]}$

where ${\displaystyle k={\frac {V_{S}^{2}}{V_{p}^{2}}}}$ and ${\displaystyle V_{P0}}$, ${\displaystyle V_{S0}}$, and ${\displaystyle \rho _{0}}$ are normalisation constants. As with elastic impedance, the value of the background velocity ratio k must be kept constant for an EEI series.

EEI Can also be expressed as a combination of AI and GI;

${\displaystyle EEI(\chi )=AI^{\cos(\chi )}GI^{\sin(\chi )}}$

EEI curves at various chi angles typically exhibit high correlation with a number of elastic properties. Ball et al (2014) show that the optimum chi angles depend only on the value of k (see table).[2]

Extended elastic impedance has similar advantages to elastic impedance but applied to the intercept/gradient crossplot domain. AIGI crossplots can form the basis of a seismic rock property study by showing what facies can be separated in 2-term AVO analysis and at what chi angles.

The table shows the relative weightings of the intercept and gradient in terms of k to optimize correlation with a range of elastic properties. The right hand column shows the equivalent chi angle for a k of 0.25 (after Ball et al, 2014).[2]
${\displaystyle W_{(int)}}$ ${\displaystyle W_{(grad)}}$ Chi (k=0.25)
P-impedance 1 0 0
S-impedance 0.5 ${\displaystyle {\frac {-1}{8k}}}$ -45
Mu-rho 1 ${\displaystyle {\frac {-1}{4k}}}$ -45
K-rho ${\displaystyle {\frac {2k-6}{4k-3}}}$ ${\displaystyle {\frac {-1}{4k-3}}}$ 10
Lambda-rho ${\displaystyle {\frac {2k-2}{2k-1}}}$ ${\displaystyle {\frac {-1}{4k-2}}}$ 18
E-rho ${\displaystyle {\frac {4k^{2}-6k+6}{4k^{2}-7k+3}}}$ ${\displaystyle {\frac {-8k^{2}+16k-6}{8k(4k^{2}-7k+3)}}}$ -24
Poisson's ratio ${\displaystyle {\frac {k}{2k^{2}-3k+1}}}$ ${\displaystyle {\frac {1}{8k^{2}-12k+4}}}$ 45