geophysics · impedance
Elastic Impedance (Connolly)
EI(θ) = Vp^(1+tan²θ) · Vs^(−8k·sin²θ) · ρ^(1−4k·sin²θ)
click formula to derive ↑
Inputs
ft/s
ft/s
g/cc
°
—
ft/s
ft/s
g/cc
Description
Computes Connolly's (1999) Elastic Impedance, the angle-dependent generalization of acoustic impedance built from the Aki-Richards linearization, plus the Whitcombe (2002) normalized form that keeps EI on a fixed, physically comparable (g/cc·m/s) scale at every angle.
Variables
| Symbol | Unit | Description |
|---|---|---|
| EI(θ) | — | Elastic Impedance (Raw, Connolly) |
| EIN(θ) | (g/cc)·(m/s) | Elastic Impedance (Normalized, Whitcombe) |
| Vp | ft/s | Compressional velocity of the formation, from sonic log or check shot/VSP. |
| Vs | ft/s | Shear velocity of the formation, from a dipole/multipole sonic log. |
| ρ | g/cc | Bulk density of the formation, typically from the density log. |
| θ | ° | Incidence angle matching the target angle-stack or angle-gather. |
| k | — | Average (Vs/Vp)² over the log interval or zone of interest, typically computed from the interval's own sonic logs — locally regressed, not a universal constant. |
| Vp₀ | ft/s | Reference (typically well/zone-average) Vp used to normalize EI onto a fixed impedance-like scale (Whitcombe, 2002). |
| Vs₀ | ft/s | Reference (typically well/zone-average) Vs used in the Whitcombe normalization. |
| ρ₀ | g/cc | Reference (typically well/zone-average) density used in the Whitcombe normalization. |
Assumptions
- Same small-contrast linearized-Zoeppritz assumptions underlying Aki-Richards — EI is derived from, not independent of, that approximation
- k represents a locally-appropriate average (Vs/Vp)² for the interval being analyzed, not a universal constant
- Normalization references (Vp0, Vs0, rho0) are held constant across the well/project for comparability
Limitations
- Raw (unnormalized) EI's units and numeric scale vary with θ — do not compare raw EI values computed at different angles; use the normalized output for that
- Inherits Aki-Richards' ~30-40° validity ceiling
- A single k value is an approximation; genuinely angle-dependent or spatially-varying Vs/Vp is not captured
Use Cases
- → Angle-stack seismic inversion calibration: Calibrate an angle-stack inversion volume to well-log EI computed at the matching stack angle.
- → Rock-physics template extension: Extend acoustic-impedance-based rock-physics templates to non-zero-offset/angle-stack data.
- → Multi-angle-stack comparison: Compare normalized EI traces across several angle stacks on a common, physically meaningful scale.
Related Calculations
Region Notes
Permian Basin
Wolfcamp benches typically show k≈0.25-0.30 (Vp/Vs≈1.8-2.0); use interval-specific k rather than a single field-wide value given the facies variability within the Wolfcamp.
Gulf of Mexico
Unconsolidated Miocene-Pliocene sands often show higher k (softer, lower Vp/Vs) — recompute k per interval rather than reusing a deeper, more consolidated section's value.
Global
Always hold Vp0/Vs0/rho0 fixed across an entire project once chosen — changing the normalization reference between wells defeats the purpose of using normalized EI for comparison.
References
Primary source
Primary source
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