AVO basics: Zoeppritz, Aki-Richards, intercept and gradient

The Zoeppritz equations

The exact theory was worked out by Karl Zoeppritz in 1919: at a planar boundary between two elastic half-spaces, an incoming P-wave produces FOUR outgoing waves — a reflected P, a reflected S, a transmitted P, and a transmitted S — whose amplitudes are determined by 4×4 boundary-condition equations. The Zoeppritz equations give the exact reflection and transmission coefficients as functions of incidence angle and the elastic properties of both layers.

They are exact, but they are also opaque — a 4×4 system that does not yield clean intuition. For routine AVO interpretation, almost no one solves Zoeppritz directly. Instead they use approximations.

The Aki-Richards approximation

Aki & Richards (1980, "Quantitative Seismology") published a linearized approximation that has become the workhorse of AVO interpretation:

**

**

where:

• R₀ (intercept) $= \frac{1}{2}\left( \frac{\Delta V_p}{V_p} + \frac{\Delta \rho}{\rho} \right)$ — the normal-incidence reflection. Identical to the impedance formula of §5.1.
• G (gradient) $= \frac{1}{2}\frac{\Delta V_p}{V_p} - 4 \left(\frac{V_s}{V_p}\right)^{2} \frac{\Delta V_s}{V_s} - 2 \left(\frac{V_s}{V_p}\right)^{2} \frac{\Delta \rho}{\rho}$ — the slope of R vs $\sin^{2}\theta$. This is the term that captures the fluid sensitivity, because it depends on the Vs contrast that fluid substitution affects through density only.
• F (curvature) $= \frac{1}{2}\frac{\Delta V_p}{V_p}$ — a small correction at large angles. Negligible below 30°; matters past 35–40°. Often dropped in the "two-term Aki-Richards" form.

The notation: Δ means (bottom − top); the unsubscripted V or ρ means the average across the two layers. The angle θ is the incidence angle of the P-wave at the boundary.

Why this approximation matters: it is LINEAR in the elastic contrasts. That linearity means we can fit (R₀, G) from real seismic data using simple regression of amplitude vs sin²θ, and we can interpret the fitted (R₀, G) values directly using rock-physics intuition. Most production AVO software does exactly this.

The intercept-gradient parameterization

The two-term Aki-Richards form $R(\theta) \approx R_0 + G \sin^{2}\theta$ is a straight line in (R, sin²θ) space. Real seismic AVO analysis fits this straight line to the angle-dependent amplitudes at each (i, x, t) voxel and extracts two volumes:

• Intercept volume R₀ — the zero-offset reflection. Looks like a stacked seismic; the data at zero angle.
• Gradient volume G — the slope. Captures the fluid/lithology contrast information that the stack alone cannot reach.

These two volumes together constitute the AVO output. Section §5.5 shows how to crossplot them to classify reservoirs (Class I/II/III/IV) and to identify hydrocarbons.

Exercise — see fluid AVO in action

• The widget defaults to comparing two scenarios: shale on brine sand (blue) vs shale on gas sand (red). Look at the two curves. The brine sand starts as a small positive reflection (R₀ ≈ +0.03) and crosses zero into a small negative at far offsets — a polarity reversal (Class IIp). The gas sand starts at a large negative reflection (R₀ ≈ −0.11) and grows even MORE negative with angle (Class III) — the reservoir BRIGHTENS at far offsets.
• This dramatic difference is why AVO is the gas-detection workhorse. Stacked amplitude alone shows a fairly visible reservoir for both cases, but it cannot distinguish them. AVO separates them with one look at the gradient.
• Try other comparisons. Switch Scenario A to oil-saturated sandstone. Notice how the oil sand sits between brine and gas — R₀ is small negative, G is large negative — a Class II response. Real oil reservoirs often show this subtle pattern; they are easier to miss on stack than gas reservoirs.
• Compare two scenarios within the same lithology, like shale on hard shale (Class I). Notice both R₀ and G are positive in some regimes, giving a peak that grows with angle — the unusual Class I+ pattern.
• Try the carbonate response: shale on limestone. Strong positive R₀ (carbonate-on-shale impedance jump) with negative G (because limestone has high Vp/Vs relative to shale, contributing a strongly negative gradient term). Class I behaviour: bright on stack, dimming at far offsets.
• Look at R(30°) in the summary panels. This is the "far-offset amplitude" — typically what AVO interpretation uses to compare against R₀. The fluid-sensitive ratio R(30°)/R₀ is the simplest single AVO discriminator.

The four AVO classes (Rutherford-Williams 1989, Castagna-Swan 1997)

Real AVO behaviours fall into a small number of canonical patterns, named for their typical reservoir geology:

• Class I: R₀ large positive (high-impedance reservoir), G negative. Bright peak on stack that DIMS at far offsets. Common at the top of carbonates, chalks, tight gas sands. The reservoir is acoustically harder than its overburden, so at zero offset you see a peak, but the negative gradient pulls the amplitude toward zero at far offsets.
• Class IIp: R₀ small positive, G negative. Weak peak on stack that flips to a trough with angle (polarity reversal). Common for moderate-impedance brine sandstones against shale.
• Class II: R₀ near zero, G strongly negative. Very subtle stacked response that becomes a strong negative reflection at far offsets. Often missed by amplitude-only screening; AVO catches it. Many oil sands show this pattern.
• Class III: R₀ large negative (bright trough on stack), G also strongly negative. The reflection BRIGHTENS at far offsets — classic gas-sand DHI signature. The textbook AVO anomaly that built the technology’s reputation in the 1980s.
• Class IV: R₀ negative, G positive. Trough on stack that DIMS at far offsets. Characteristic of low-impedance, low-Vp/Vs soft sands or shaley reservoirs. Produces the awkward "reverse AVO" pattern that confuses simple screening tools.

The classification is not just descriptive. Each class corresponds to a distinct region of the (R₀, G) crossplot, and that crossplot is the canonical AVO interpretation tool — covered in detail in §5.5.

Why the gradient G is the fluid discriminator

Look at G’s formula: it has three terms, but the dominant one is $-4 (V_s/V_p)^{2} (\Delta V_s / V_s)$. This term sums the Vs contrast across the boundary, with a coefficient that is squared in the average Vs/Vp ratio.

Now recall Gassmann: gas substitution drops Vp substantially but RAISES Vs slightly. So in (top: shale, bottom: gas-substituted sand) the ΔVs is LARGER than in (top: shale, bottom: brine-substituted sand). The gradient G — which is multiplied by −4(Vs/Vp)²(ΔVs/Vs) — becomes more strongly negative for the gas case.

This is why gas substitution shifts a reservoir’s AVO response toward more negative gradient: because Gassmann’s μ-invariance combined with the density drop means ΔVs (and therefore G) carries the fluid-substitution information. The combined effect on R₀ (which depends on Vp and ρ, both of which drop with gas) and G gives the (R₀, G) crossplot its diagnostic power.

Common AVO pitfalls

• Far-offset amplitude is hard to measure. Real seismic at 35–40° incidence has lower fold (fewer traces stacked together), more noise, and processing-related amplitude distortions. The G volume is always noisier than the R₀ volume.
• Aki-Richards breaks down at large angles. Above 35–40°, the linear approximation fails; the F term and higher-order corrections matter. Don’t extrapolate AVO trends past your data’s usable angle range.
• Anisotropy distorts AVO. Real shales (especially) are vertically transversely isotropic; their Vp differs in the horizontal vs vertical direction. The Aki-Richards form assumes isotropy; real interpretation needs the anisotropic extension (Thomsen 1986, Tsvankin 1997) for shale-dominated overburden.
• Tuning effects masquerade as AVO. Thin-bed tuning produces amplitude variations with offset that mimic AVO. Always check the bed thickness; sub-tuning beds produce ambiguous AVO.
• Incidence angle vs offset. AVO is properly defined as Amplitude vs ANGLE, not vs OFFSET. Real seismic gathers record amplitude vs offset; converting offset to angle requires a velocity model. Errors in the velocity model become errors in the AVO interpretation.
• "AVO from amplitude maps" is NOT AVO. True AVO requires pre-stack data; you cannot recover the gradient from a stacked amplitude map alone. Workflows that claim "post-stack AVO" via amplitude analysis are using a different (and weaker) technique.