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

The Zoeppritz equations

A plane P-wave meeting a planar, welded boundary between two elastic rocks makes four waves: a reflected P, a reflected S, a transmitted P and a transmitted S. Continuity of displacement and of traction across the boundary gives four equations in their four amplitudes (Zoeppritz 1919). Their solution is the exact plane-wave reflection coefficient $R(\theta)$ at every incidence angle $\theta$.

Exact, but opaque: a four-by-four system does not say which rock property does what. Two of its features no simple formula reproduces. At normal incidence it reduces exactly to the impedance formula of Section 5.1, $(Z_2 - Z_1)/(Z_2 + Z_1)$. And when the lower rock is faster in $V_P$, there is a critical angle, $\theta_c = \arcsin(V_{P1}/V_{P2})$: approaching it the reflection climbs steeply to a sharp peak, and beyond it the transmitted P-wave no longer travels and $R$ is complex, its phase shifted. The peak is total reflection, $|R| = 1$, only where no shear waves exist, as between two fluids; between rocks, converted shear waves still carry energy away, and over limestone under medium shale the peak is 0.94.

Aki-Richards and Shuey: the linear forms

For small contrasts, Aki and Richards (1980) linearised the exact coefficient. Shuey (1985) rearranged their result into three terms, each a little harder to see in data than the one before:

• Intercept $R_0 = \frac{1}{2}\left(\frac{\Delta V_P}{V_P} + \frac{\Delta\rho}{\rho}\right)$, the linearised normal-incidence coefficient. For small contrasts it is close to the exact impedance formula: medium shale over brine-saturated sandstone gives +0.033 both ways. For large ones it is not: over coal it gives −0.388 against the exact −0.376.
• Gradient $G = \frac{1}{2}\frac{\Delta V_P}{V_P} - 2\left(\frac{V_S}{V_P}\right)^{2}\left(\frac{\Delta\rho}{\rho} + 2\frac{\Delta V_S}{V_S}\right)$, the slope of $R$ against $\sin^{2}\theta$. Its shear part, $-4(V_S/V_P)^{2}\,\Delta V_S/V_S$, is usually the largest, and it is what makes the gradient respond to the shear contrast the stack cannot see.
• Curvature $F = \frac{1}{2}\frac{\Delta V_P}{V_P}$, the third term. Its multiplier $\tan^{2}\theta - \sin^{2}\theta$ is only 0.08 at 30° but 2.25 at 60°, so it matters only at far angles.

Here $\Delta$ means the lower rock's value minus the upper's, and an unsubscripted $V_P$, $V_S$ or $\rho$ is the average of the two rocks. Aki and Richards define $\theta$ as the mean of the incidence and transmission angles; the usual practice, followed in the figure, is to use the incidence angle, and then Shuey's three terms are an exact rewrite of the Aki-Richards expression.

Dropping the third term leaves Shuey's two-term form, $R(\theta) \approx R_0 + G\sin^{2}\theta$, a straight line against $\sin^{2}\theta$. Fitting that line to the amplitudes of each gather gives an intercept and a gradient at every sample: an intercept volume, which looks like a stack at normal incidence, and a gradient volume, which carries what the stack leaves out. Section 5.5 crossplots the two.

Where the linear forms hold

The approximations are only as good as the contrast is small, and the figure measures how good. With a tolerance of 0.01 in $R$, under medium shale:

• Over gas-saturated sandstone Shuey's two terms stay within 0.01 of the exact curve until 55°, the three-term form only until 35°. The two terms do better here by a cancellation: their gradient, −0.275, is steeper than the exact curve's own slope at normal incidence, −0.237, and the exact curve steepens with angle until the two meet again.
• Over hard shale the two terms leave the band at 26° and the three terms at 32°; a critical angle follows at 49.7°.
• Over limestone, almost twice as fast as the shale, the critical angle is 31.8°. Approaching it the exact reflection climbs steeply to 0.94 while both linear forms keep falling, and both have left the 0.01 band by about 20°.

Agreement can also come back by accident. Over coal, both forms start more than 0.01 off the exact curve, and Shuey's two terms pass back through the band between 16° and 37° only because their error changes sign there; the figure reports such a window but does not count it as a range where the form holds.

The familiar rule that the linear forms are good to 30 or 35° is a rule of thumb for modest contrasts, not a law. Even the gradient is a linearisation: a two-term line fitted to exact amplitudes reads a slope that depends on how far out the angles go, and that differs from $G$ wherever the shear contrast is large.

Angle and offset

A gather records amplitude against offset; AVO is read against angle, and converting one to the other needs a velocity model. Over a uniform overburden the rays are straight and $\tan\theta = x/2z$ for offset $x$ and reflector depth $z$: a 3000 m spread reaches 37° at a reflector 2000 m deep, 56° at 1000 m and only 23° at 3500 m. Where velocity grows with depth the rays bend and a given offset reaches a larger angle at the reflector. Errors in the velocity model become errors in angle, and so in the gradient.

Exercise, read the figure

1. The figure opens on brine-saturated sandstone (A) and gas-saturated sandstone (B) under medium shale. The brine sand starts as a faint peak, +0.033, that fades and turns into a faint trough: at 28.5° on the two-term line, at 32.5° on the exact curve. The Part reads the class off the two-term line, a sign change before 30°, so the brine sand is Class IIp. The gas sand starts as a trough, −0.106, that deepens to −0.168 at 30°: Class III. The rock frame is the same; only the pore fluid differs.
2. Pick out A and tighten the tolerance to 0.005 (the second setup). Shuey's line crosses zero at 28.5°, four degrees before the exact curve. A polarity reversal read from the two-term form is placed at the wrong angle, and so at the wrong offset.
3. Put limestone under the shale and lengthen the spread (the third setup). The critical angle at 31.8° appears in (a), and plate (b) shows both approximations leaving the tolerance near 20°.
4. Raise and lower the reflector with the spread fixed, and watch (d) and the hatching: the same cable records much wider angles from a shallow target than from a deep one.
5. Lengthen the spread over the gas sand and watch the fitted gradient in (c) and in the table: −0.25 out to 37°, −0.28 out to 56°, against the linear theory's −0.275.

The AVO classes

AVO responses fall into a few patterns, named after the reservoirs that show them (Rutherford and Williams 1989; Ross and Kinman 1995; Castagna and Swan 1997). The figure classifies on $R_0$ and $G$ with the rule used throughout this Part, including Section 5.5:

• Class I: $R_0$ positive and at least 0.04, $G$ negative. A peak on the stack that dims with angle: the top of a reservoir harder than its seal, such as a tight sandstone or a carbonate. Hard shale and limestone under medium shale are Class I.
• Class IIp: $R_0$ positive but small, with $R_0 + G\sin^{2}30^\circ$ negative, so on the two-term line the peak turns into a trough before 30°; the exact curve can turn a little later, as the brine sand's does at 32.5°. Brine-saturated sandstone under medium shale is Class IIp.
• Class II: $R_0$ near zero (under 0.04 in size), $G$ negative. Faint on the stack, a trough that grows with angle, easily missed by amplitude screening. Oil-saturated sandstone under medium shale gives $R_0 = -0.020$, $G = -0.195$.
• Class III: $R_0$ and $G$ both negative. A trough on the stack that brightens with angle, the classic gas-sand response: gas-saturated sandstone under medium shale.
• Class IV: $R_0$ negative, $G$ positive. A trough that dims with angle. It arises when the cap is stiffer than the sand in shear, so that $\Delta V_S$ is negative, for example a soft sand under a hard shale or a carbonate; coal under medium shale behaves this way too.

The 0.04 that separates near-zero from large intercepts is this book's convention; in practice the boundaries are drawn against the local background trend. Each class occupies its own region of the $(R_0, G)$ crossplot, the subject of Section 5.5.

Why the gradient responds to fluid

Gassmann's relation (Section 5.3) says that replacing brine with gas lowers the bulk modulus and the density but leaves the shear modulus unchanged. $V_P$ therefore falls, while $V_S = \sqrt{\mu/\rho}$ rises a little as the density falls.

Under medium shale, the brine sand has $\Delta V_S/V_S = 0.221$ and $\Delta\rho/\rho = -0.063$; the gas sand has 0.274 and −0.178. Both changes drive the gradient negative, from −0.145 to −0.275, while the drop in $V_P$ and density takes the intercept from +0.033 to −0.106. The fluid moves the interface diagonally across the $(R_0, G)$ plane, which is what gives the crossplot its power.

Common AVO pitfalls

• Far-offset amplitude is hard to measure. At 35 to 40° the fold is lower, the noise higher and processing distorts amplitudes more, so the gradient volume is always noisier than the intercept volume.
• The linear forms fail sooner for stronger contrasts. Check them against the exact Zoeppritz curve for the rocks in question before trusting a gradient fitted to far angles; past a critical angle no linear form applies at all.
• Anisotropy distorts AVO. Shales are commonly vertically transversely isotropic, and the forms above assume isotropy; a shale-dominated overburden needs the anisotropic extension (Thomsen 1986; Rüger 2002).
• Tuning masquerades as AVO. In a thin bed the interference of top and base reflections changes with offset and can mimic an AVO trend; check the bed thickness against the tuning thickness.
• Angle is not offset. The conversion needs a velocity model, and its errors pass straight into the gradient.
• AVO needs pre-stack data. A gradient cannot be recovered from a stacked amplitude map; so-called post-stack AVO is a different and weaker technique.