Elastic Impedance Calculator: Connolly EI for Angle Stack Inversion

Acoustic impedance inversion is one of the most trusted workflows in seismic interpretation: invert the near-offset stack, tie it to impedance logs at the wells, and map rock quality between them. But the near stack is only part of the data. Far-angle stacks carry the fluid information that AVO analysis chases, and for years there was no impedance-like quantity to invert them against. Patrick Connolly closed that gap in 1999 with elastic impedance, a pseudo-impedance defined so that reflectivity between layers at a chosen angle works exactly the way acoustic impedance reflectivity works at zero:

EI(theta) = Vp^(1 + tan^2 theta) x Vs^(-8K sin^2 theta) x rho^(1 – 4K sin^2 theta)

where K is a background (Vs/Vp)^2 held constant, conventionally around 0.25. Set theta to zero and every exponent collapses to give Vp x rho, plain acoustic impedance, which is the built-in sanity check.

What it buys you in practice

• Far-angle stacks become invertible with the same post-stack machinery used for AI, no prestack inversion required
• An EI log computed from dipole sonic and density at the well ties the far-stack inversion just like an AI log ties the near
• The difference between near (AI) and far (EI) inversions is a fluid-sensitive volume, effectively AVO in impedance form
• Crossplotting AI against EI(30) separates fluids that either volume alone leaves ambiguous

Worked example

Take a shale (Vp 3000 m/s, Vs 1500 m/s, 2.30 g/cc) over a gas sand (Vp 2900, Vs 1750, 2.15). Their acoustic impedances are 6900 and 6235, a modest contrast that could be missed in noisy data. But at 30 degrees the elastic impedances are about 2090 and 1760, and the EI reflectivity of roughly -0.086 is several times stronger than the normal-incidence response. That is a class 2 to 3 AVO anomaly translated into impedance language, and it is precisely the kind of target the far-stack inversion lights up.

The fine print

EI has famously awkward units: because the exponents change with angle, the absolute numbers at 30 degrees are not comparable to those at 0 or to anything physical. The Whitcombe 2002 normalization fixes the cosmetics by referencing everything to constants, and most commercial software applies it silently. Two more real limitations: the derivation inherits the small-contrast assumptions of the linearized AVO equations, so at very strong contrasts or beyond about 35 to 40 degrees it drifts from exact Zoeppritz behavior, and the constant-K assumption smears somewhat when the true Vs/Vp varies strongly through the section. None of this has stopped EI from becoming a standard workflow; it just means the exact solver is the right cross-check before trusting far-angle quantitative work.

Related Calculators and Reference

The zero-angle limit of this calculation is the acoustic impedance calculator, the exact-angle benchmark is the Zoeppritz equations calculator, and missing Vs inputs can be estimated with the Castagna mudrock line calculator. The SEG Wiki page on elastic impedance covers the Connolly derivation and the Whitcombe normalization.