AVO from PINN-augmented FWI

§10.1-§10.3 inverted velocity (acoustic). §10.4 takes the next step: invert ELASTIC parameters — V_p, V_s, and density ρ — from the AMPLITUDE variation of seismic reflections with source-receiver offset. This is AVO (Amplitude Versus Offset), the workhorse of oil-exploration "bright-spot" interpretation.

Why AVO matters

An interface between two layers reflects seismic energy with amplitude that DEPENDS ON THE INCIDENCE ANGLE θ. At normal incidence (θ=0), only the acoustic impedance contrast matters: $R(0) = (Z_2 - Z_1) / (Z_2 + Z_1)$ where $Z = \rho V_p$. At oblique incidence (θ>0), the shear-wave velocity V_s also contributes via mode conversion at the interface. Plotting R(θ) vs θ — the AVO CURVE — encodes the contrasts in V_p, V_s, AND ρ separately.

Why does anyone care? Because GAS-SAND reservoirs have a characteristic AVO signature. Below shale, gas-saturated sand has:

• LOW V_p (gas slows down P-waves in pores)
• SLIGHTLY HIGHER V_s (V_s depends on rock matrix, not fluid)
• LOW ρ (gas is much less dense than water/oil/rock)

The combination produces "Class III AVO": LARGE NEGATIVE $R(0)$ (a "bright spot" on near-offset stack) AND amplitude that GROWS in magnitude with offset (a brightening AVO trend). When an interpreter sees this, the prior probability of gas is high. Drill there.

The Aki-Richards 3-term linearised formula

For a single horizontal interface and small-perturbation contrasts, the reflectivity is

where $dV_{pR} = (V_{p,2} - V_{p,1}) / V_{p,\mathrm{avg}}$ is the FRACTIONAL contrast in V_p (similarly for V_s, ρ), and $k = V_s / V_p \approx 0.5$ for typical sediments. This is LINEAR in the three unknown contrasts; given R(θ) at $\ge 3$ angles, classical AVO inversion solves a 3×3 least-squares problem.

The AVO 3-term ambiguity

Classical 3-term AVO is ill-posed at SMALL ANGLES because the V_s coefficient $-4 k^2 \sin^2 \theta \to 0$ as $\theta \to 0$. At normal incidence only V_p and ρ contrasts matter, AND they enter the same way: $R(0) = \frac{1}{2}(dV_{pR} + d\rho_R)$. Any V_p increase can be compensated by a ρ decrease. As angles grow, V_s starts to register, and the ambiguity shrinks — but only PARTIALLY, because real surveys rarely sample beyond 30-40° usable incidence.

Gardner-relation PINN prior

The fix: use a known empirical relationship between density and V_p in sedimentary rocks — the Gardner relation:

$\rho = 0.31 , V_p^{0.25}$ (in cgs units; Gardner et al 1974)

Linearised: $d\rho_R = 0.25 , dV_{pR}$. This breaks the (V_p, ρ) ambiguity by COUPLING the two contrasts. We add it as a §9.2-style PINN regulariser:

Total loss: $\mathcal{L}${\mathrm{tot}} = \frac{1}{2} \sum_i (R(\theta_i; d) - R{\mathrm{obs}}^{(i)})^2 + \lambda , \mathcal{L}{\mathrm{pinn}}(d)Ltot​=21​∑i​(R(θi​;d)−Robs(i)​)2+λLpinn​(d). Gradient descent on $d = (dV${pR}, dV_{sR}, d\rho_R).

Try it

The widget simulates a shale-over-tight-limestone interface (typical of Permian Basin and North Sea Chalk targets):

• Above (shale): V_p = 2.4 km/s, V_s = 1.0 km/s, ρ = 2.30 g/cc
• Below (tight limestone): V_p = 3.6 km/s, V_s = 1.8 km/s, ρ = 2.50 g/cc
• Truth fractional contrasts: dV_pR ≈ +0.40, dV_sR ≈ +0.57, dρ_R ≈ +0.083
• R(θ) sampled at 10 angles from 2° to 38°, σ_pick = 0.005 noise

Two inversions race: (1) classical (λ=0, pure data misfit) and (2) PINN-augmented (λ=1.5, with Gardner-prior penalty). Two panels: R(θ) curves with data + truth + both fits, and a bar chart comparing recovered vs truth fractional contrasts.

Expected behaviour: classical fits the R(θ) curve well (10 data points, 3 unknowns) but the recovered (V_p, ρ) split has the small-angle Vp-ρ ambiguity bias. PINN-augmented locks dρ_R ≈ 0.25 · dV_pR through the Gardner prior, producing physically-consistent contrasts that match truth more accurately. Truth here SATISFIES Gardner (intra-lithology, no fluid change) so the prior is informative.

IMPORTANT CAVEAT for the wider literature: Gardner applies to INTRA-LITHOLOGY contrasts only. At fluid-substitution boundaries — Class-III gas-sand AVO is the canonical example — the gas drops density much more than Gardner predicts (truth Δρ/ΔVp ≈ 0.9, not 0.25). Production AVO calibrates λ PER-TARGET using local well logs: drop the Gardner prior near known fluid contacts; raise it elsewhere. Mudrock (Castagna et al 1985) and modified-Gardner alternatives exist for fluid-substitution scenarios.

Production AVO

Real AVO inversion happens AT EVERY CDP gather in a 3-D seismic survey — millions of inversions per processing run. Tools like Hampson-Russell's STRATA and JASON Geosystems' RockMOD apply this exact recipe at production scale, with additional refinements:

• Pre-stack waveform inversion. Instead of picking R(θ) and inverting, fit the full pre-stack seismic wavelet via convolution + AVO + spike train — full waveform AVO, the elastic-FWI cousin.
• Multi-prior framework. Combine Gardner with rock-physics templates (Han et al 1986, Castagna et al 1985 mudrock relation V_s = 0.86·V_p − 1.17), porosity-cement models, and well-log calibration.
• Bayesian posterior. §9.6 bootstrap-posterior over the 3-term inversion produces uncertainty maps on V_p, V_s, ρ — what the geomodellers actually need to compute oil-in-place ranges.
• Anisotropy. Elastic AVO is intrinsically anisotropic. Real AVO uses the Rüger 1997 anisotropic Aki-Richards extension with Thomsen parameters $\delta$ and $\epsilon$.

What §10.5 will do

§10.5 puts AVO inversion + PINN-augmented FWI to work on the hardest target in exploration seismology: SUB-SALT IMAGING in a Gulf-of-Mexico class scenario. Salt is fast (V_p ~ 4.5 km/s) and irregular; the resulting wavefield distortion makes classical FWI fail catastrophically without strong priors.

References

• Aki, K., Richards, P.G. (1980; 2nd ed. 2002). Quantitative Seismology. University Science Books. The 3-term reflectivity formula derivation.
• Gardner, G.H.F., Gardner, L.W., Gregory, A.R. (1974). Formation velocity and density — the diagnostic basics for stratigraphic traps. Geophysics 39(6), 770–780. The Gardner relation paper.
• Avseth, P., Mukerji, T., Mavko, G. (2005). Quantitative Seismic Interpretation. Cambridge University Press. The standard reference for AVO + rock physics in production.
• Hampson, D.P., Russell, B.H., Bankhead, B. (2005). Simultaneous inversion of pre-stack seismic data. SEG Annual Meeting Expanded Abstracts. The Hampson-Russell STRATA methodology.
• Castagna, J.P., Batzle, M.L., Eastwood, R.L. (1985). Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks. Geophysics 50(4), 571–581. The mudrock V_s = 0.86 V_p − 1.17 relation.
• Rüger, A. (1997). P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry. Geophysics 62(3), 713–722. Anisotropic AVO.