# AVO equations

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Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Consider the two elastic half-space layers in Figure 11.2-3e. The Zoeppritz equations (14) can be solved for the reflected and refracted P- and S-wave amplitudes, A1, B1, A2, and B2. However, our interest in exploration seismology is largely the angle-dependency of the P-to-P reflections given by the coefficient A1. Specifically, we wish to infer or possibly estimate elastic parameters of reservoir rocks from reflection amplitudes and relate these parameters to reservoir fluids.

The exact expression for A1 derived from the solution of the Zoeppritz equations (13) is complicated and not intuitive in terms of its practical use for inferring petrophysical properties of reservoir rocks. The first approximation to the Zoeppritz equation for P-to-P reflection amplitude is given by Bortfeld [1] as

 ${\displaystyle R(\theta _{1})={\frac {1}{2}}\ln \left({\frac {\alpha _{2}\rho _{2}\cos \theta _{1}}{\alpha _{1}\rho _{1}\cos \theta _{2}}}\right)+\left[2+{\frac {\ln \left({\frac {\rho _{2}}{\rho _{1}}}\right)}{\ln \left({\frac {\alpha _{2}}{\alpha _{1}}}\right)-\ln \left({\frac {\alpha _{2}\beta _{1}}{\alpha _{1}\beta _{2}}}\right)}}\right]{\frac {\beta _{1}^{2}-\beta _{2}^{2}}{\alpha _{1}^{2}}}\sin ^{2}\theta _{1}.}$ (14)

 ${\displaystyle CE={\frac {1}{\sqrt {1+A^{-1}z}}},}$ (13a)
 ${\displaystyle CE(\theta )={\frac {1}{\sqrt {1+A^{-1}z/\cos ^{2}\theta }}},}$ (13b)

The arrangement of the two terms on the right-hand side of equation (14) is based on separating the acoustic (the first term) and the elastic (the second term) effects on reflection amplitudes. As such, equation (14) does not explicitly indicate angle- or offset-dependence of reflection amplitudes; therefore, its practical implementation for AVO analysis has not been considered.

Instead, we shall use the approximation provided by Aki and Richards [2] as the starting point for deriving a series of practical AVO equations. Now that we only need to deal with the P-to-P reflection amplitude A1, we shall switch to the conventional notation by replacing A1 with R(θ) as the angle-dependent reflection amplitude for AVO analysis.

By assuming that changes in elastic properties of rocks across the layer boundary are small and propagation angles are within the subscritical range, the exact expression for R(θ) given by the Zoeppritz equation can be approximated by [2]

 ${\displaystyle R(\theta )=\left[{\frac {1}{2}}{\Big (}1+\tan ^{2}\theta {\Big )}\right]{\frac {\Delta \alpha }{\alpha }}-\left[4{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta \right]{\frac {\Delta \beta }{\beta }}+\left[{\frac {1}{2}}{\Big (}1-4{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta {\Big )}\right]{\frac {\Delta \rho }{\rho }},}$ (15)

where α = (α1 + α2)/2, average P-wave velocity and Δα = (α2α1), β = (β1 + β2)/2, average S-wave velocity and Δβ = β2β1, ρ = (ρ1 + ρ2)/2, average density and Δρ = ρ2ρ1, and θ = (φ1 + φ2)/2, average of the incidence and transmission angles for the P-wave (Figure 11.0-2e).

Figure 11.2-8 shows the angle-dependent reflection amplitude associated with an interface with contrast in P- and S-wave velocities and densities based on the exact Zoeppritz equation, the Bortfeld approximation described by equation (14), and the Aki-Richards approximation described by equation (15) [3]. Note that these approximations closely follow the exact solution within the range of angles of incidence that are achievable by the recording of seismic data used in exploration. For very shallow reflectors, however, the approximate solutions would deviate from the exact solution significantly at very wide angles of incidence.

Note that the Aki-Richards approximation to the angle-dependent reflection amplitude R(θ) given by equation (15) has three parts in terms of Δα/α which describes the fractional change in P-wave velocity across the layer boundary and hence may be referred to as the P-wave reflectivity, Δβ/β which describes the fractional change in the S-wave velocity across the layer boundary and hence may be referred to as the S-wave reflectivity, and Δρ/ρ which describes the fractional change in density across the layer boundary.

In practice, we do not observe the separate effects of P-wave reflectivity Δα/α, S-wave reflectivity Δβ/β and fractional change in density Δρ/ρ on the reflection amplitudes R(θ). Instead, we observe changes in reflection amplitudes as a function of angle of incidence. In fact, it is the elastic parameters such as the P-wave reflectivity Δα/α, S-wave reflectivity Δβ/β and fractional change in density Δρ/ρ that we wish to estimate from the observed angle-dependent reflection amplitudes. To use the Aki-Richards equation (15) in the inversion of reflection amplitudes for these elastic parameters, we first need to recast it in successive ranges of angle of incidence. This change of philosophy in arranging the terms in the Aki-Richards equation (15) was first introduced by Shuey [4] and led to practical developments in AVO analysis. The new arrangement in terms of successive ranges of angle of incidence is given by

 ${\displaystyle R(\theta )=\left[{\frac {1}{2}}\left({\frac {\Delta \alpha }{\alpha }}+{\frac {\Delta \rho }{\rho }}\right)\right]+\left[{\frac {1}{2}}{\frac {\Delta \alpha }{\alpha }}-4{\frac {\beta ^{2}}{\alpha ^{2}}}{\frac {\Delta \beta }{\beta }}-2{\frac {\beta ^{2}}{\alpha ^{2}}}{\frac {\Delta \rho }{\rho }}\right]\sin ^{2}\theta +\left[{\frac {1}{2}}{\frac {\Delta \alpha }{\alpha }}\right]{\big (}\tan ^{2}\theta -\sin ^{2}\theta {\big )}.}$ (16)

Another practical matter of concern is that the Aki-Richards equation (15) or any of its modifications that we shall derive in this section describe the modeled reflection amplitudes as a function of angle of incidence. However, the observed reflection amplitudes are available from CMP data as a function of offset. A need then arises either to transform the model equation for the reflection amplitudes from angle to offset coordinates [6] or to actually transform the CMP data from offset to angle coordinates. While the first approach is theoretically appealing, the practical schemes are based on the latter approach. We have already discussed such a transformation in the radon transform — the Radon transform using the linear moveout equation or its robust variation in the form of slant stacking. Figure 11.2-9 shows Zoeppritz amplitude curves as a function of angle of incidence and offset [6].

Based on the theoretical conjecture made earlier by Koefoed [7] that the elastic property that is most directly related to angular dependence of reflection coefficient R(θ) is Poisson’s ratio σ, Shuey [4] introduced a variable transformation from S-wave velocity β to σ. The relationship between the two variables is given by equation (L-49) which we rewrite below as

 ${\displaystyle \beta ^{2}={\frac {1}{2}}\left({\frac {1-2\sigma }{1-\sigma }}\right)\alpha ^{2}}$ (17a)

to perform the necessary differentiation

 ${\displaystyle {\frac {\Delta \beta }{\beta }}={\frac {\Delta \alpha }{\alpha }}-{\frac {1}{2}}{\frac {\Delta \sigma }{(1-\sigma )(1-2\sigma )}}.}$ (17b)

The compressional-wave velocity α is given by equation (3b) and the shear-wave velocity β is given by equation (L-47) which is rewritten below as

 ${\displaystyle \beta ={\sqrt {\frac {\mu }{\rho }}},}$ (17c)
 ${\displaystyle \alpha ={\sqrt {\frac {\lambda +2\mu }{\rho }}}.}$ (3b)

where μ is Lamé’s constant.

We also define the P-wave reflection amplitude RP at normal incidence as

 ${\displaystyle R_{P}={\frac {1}{2}}\left({\frac {\Delta \alpha }{\alpha }}+{\frac {\Delta \rho }{\rho }}\right).}$ (18)

Substitute equations (17a), (17b), and (18) into the Aki-Richards equation (16) and perform some algebraic simplification to obtain

 ${\displaystyle R(\theta )=R_{P}+\left[{\frac {1}{2}}{\frac {\Delta \alpha }{\alpha }}-\left(2{\frac {\Delta \alpha }{\alpha }}+{\frac {\Delta \rho }{\rho }}\right)\left({\frac {1-2\sigma }{1-\sigma }}\right)+{\frac {\Delta \sigma }{(1-\sigma )^{2}}}\right]\sin ^{2}\theta +\left[{\frac {1}{2}}{\frac {\Delta \alpha }{\alpha }}\right](\tan ^{2}\theta -\sin ^{2}\theta ).}$ (19)

Define a new term H

 ${\displaystyle H={\frac {\Delta \alpha /\alpha }{\Delta \alpha /\alpha +\Delta \rho /\rho }},}$ (20)

and by way of equation (18) note that

 ${\displaystyle {\frac {\Delta \alpha }{\alpha }}=2\,R_{P}H.}$ (21a)

Next combine equations (20) and (21a) to obtain

 ${\displaystyle {\frac {\Delta \rho }{\rho }}=2\,R_{P}(1-H).}$ (21b)

Finally, substitute equations (21a) and (21b) into the second term on the right-hand side of equation (19) to obtain

 ${\displaystyle R(\theta )=R_{P}+\left[R_{P}H_{0}+{\frac {\Delta \sigma }{(1-\sigma )^{2}}}\right]\sin ^{2}\theta +\left[{\frac {1}{2}}{\frac {\Delta \alpha }{\alpha }}\right](\tan ^{2}\theta -\sin ^{2}\theta ),}$ (22)

where

 ${\displaystyle H_{0}=H-2(1+H){\frac {1-2\sigma }{1-\sigma }}.}$ (23)

Equation (22) is known as Shuey’s three-term AVO equation. The first term RP is the reflection amplitude at normal incidence. At intermediate angles (0 < θ < 30 degrees), the third term may be dropped, thus leading to a two-term approximation

 ${\displaystyle R(\theta )=R_{P}+G\sin ^{2}\theta ,}$ (24)

where

 ${\displaystyle G=R_{P}H_{0}+{\frac {\Delta \sigma }{(1-\sigma )^{2}}}.}$ (25)
Figure 11.2-9  Reflection and refraction amplitude curves as a function of offset and angle of incidence. P1, S1: reflected P- and S-waves, P2, S2: transmitted P- and S-waves. [6].

Equation (24) is known as Shuey’s two-term AVO equation. In practice, amplitudes picked along a moveout-corrected event on a CMP gather plotted against sin2 θ can be fitted to a straight line. The slope of the line gives the AVO gradient attribute and the ordinate at zero angle gives the AVO intercept attribute. The AVO gradient given by equation (25) is directly related to change in Poisson’s ratio Δσ, which in turn, is directly related to fluid saturation in reservoir rocks. The AVO intercept attribute represents the reflectivity RP at normal incidence. Therefore, the AVO intercept attribute, in lieu of conventional stack, can be used as input to derive the acoustic impedance attribute (acoustic impedance estimation), which is indirectly related to porosity in reservoir rocks.

Shown in Figure 11.2-10a is a portion of a section derived from 2-D prestack time migration. The objective is to identify fluid-saturated reservoir zones at the apex and the flanks of the structural closure. This image section is derived from the stacking of common-reflection-point (CRP) gathers associated with the prestack time-migrated data. The CRP gather in Figure 11.2-10b shows three events with amplitude variations with offset which are plotted in Figure 11.2-10c. By using Shuey’s equation (24), the AVO gradient and intercept sections are computed from the CRP gathers as shown in Figures 11.2-11 and 11.2-12, respectively. Note that the gradient section exhibits a group of AVO anomalies in the vicinity of the structural apex, possibly indicating fluid-saturated reservoir rocks.

At large angles of incidence beyond 30 degrees, the third term in equation (22) gradually becomes dominant. Note that this term is related directly to fractional change in P-wave velocity, Δα/α. So, not only the reflection traveltimes at far offsets (normal moveout) corresponding to large angles of incidence, but also the reflection amplitudes at large angles of incidence make the biggest contribution to the resolution needed to estimate the changes in P-wave velocities.

The two-term equation (24) can be specialized for a specific value of Poisson’s ratio, σ = 1/3 and H0 = −1 so that equation (25) takes the form

 ${\displaystyle G=-R_{P}+{\frac {9}{4}}\Delta \sigma ,}$ (26)

which can be solved for the change in Poisson’s ratio across a layer boundary

 ${\displaystyle \Delta \sigma ={\frac {4}{9}}(R_{P}+G).}$ (27)

This is the AVO attribute equation for estimating changes in Poisson’s ratio [8]. Actually, as described by equation (27), this attribute is the scaled sum of the AVO intercept RP and AVO gradient G attributes.

By recasting the first-order approximation to the Zoeppritz equation, Wiggins [9] and Spratt [10] derived a practical expression for S-wave reflectivity. First, drop the term with tan2 θ in equation (16) and rearrange the remaining terms to obtain

 ${\displaystyle R(\theta )=\left[{\frac {1}{2}}\left({\frac {\Delta \alpha }{\alpha }}+{\frac {\Delta \rho }{\rho }}\right)\right]+\left\{\left[{\frac {1}{2}}\left({\frac {\Delta \alpha }{\alpha }}+{\frac {\Delta \rho }{\rho }}\right)\right]-8{\frac {\beta ^{2}}{\alpha ^{2}}}\left[{\frac {1}{2}}\left({\frac {\Delta \beta }{\beta }}+{\frac {\Delta \rho }{\rho }}\right)\right]\right\}\sin ^{2}\theta +\left[2{\frac {\beta ^{2}}{\alpha ^{2}}}{\frac {\Delta \rho }{\rho }}-{\frac {1}{2}}{\frac {\Delta \rho }{\rho }}\right]\sin ^{2}\theta }$ (28)

such that, much like the definition for the P-wave reflectivity RP given by equation (18), an expression for the S-wave reflectivity RS

 ${\displaystyle R_{S}={\frac {1}{2}}\left({\frac {\Delta \beta }{\beta }}+{\frac {\Delta \rho }{\rho }}\right)}$ (29)

can be explicitly inserted back into equation (28) to get

 ${\displaystyle R(\theta )=R_{P}+\left(R_{P}-8{\frac {\beta ^{2}}{\alpha ^{2}}}R_{S}\right)\sin ^{2}\theta +\left(2{\frac {\beta ^{2}}{\alpha ^{2}}}-{\frac {1}{2}}\right){\frac {\Delta \rho }{\rho }}\sin ^{2}\theta .}$ (30)

Set β/α = 0.5 to make the last term on the right-hand side vanish and obtain [10]

 ${\displaystyle R(\theta )=R_{P}+(R_{P}-2\,R_{S})\sin ^{2}\theta .}$ (31)

This equation is of the form given by equation (24) where

 ${\displaystyle G=R_{P}-2\,R_{S},}$ (32)

which can be rewritten explicitly in terms of the shear-wave reflectivity RS

 ${\displaystyle R_{S}={\frac {1}{2}}(R_{P}-G).}$ (33)

This is the AVO attribute equation for estimating the shear-wave reflectivity. Given the AVO intercept RP and AVO gradient G attributes, simply take half of the difference between the two attributes to derive the shear-wave reflectivity RS as described by equation (33).

Figure 11.2-13 shows the reflection amplitudes as a function of angle as predicted by equation (31) for three combinations of RP and RS. Note that equation (31) is a good approximation to the P-to-P reflection amplitudes as predicted by the exact Zoeppritz equation up to nearly 30 degrees of angles of incidence.

Return to the Aki-Richards equation (15) and consider the case of N-fold CMP data represented in the domain of angle of incidence. Note that the reflection amplitude R(θ) is a linear combination of three elastic parameters — P-wave reflectivity Δα/α, S-wave reflectivity Δβ/β and fractional change in density Δρ/ρ.

Smith and Gidlow [3] argue in favor of solving for only two of the three parameters by making use of the empirical relation between density ρ and P-wave velocity α [5]:

 ${\displaystyle \rho =k\alpha ^{1/4},}$ (34a)

where k is a scalar. This relation holds for most water-saturated rocks. Differentiate to get

 ${\displaystyle {\frac {\Delta \rho }{\rho }}={\frac {1}{4}}{\frac {\Delta \alpha }{\alpha }}.}$ (34b)

Now substitute equation (34b) into the original Aki-Richards equation (15) and combine the terms with Δα/α

 ${\displaystyle R(\theta )=\left[{\frac {1}{2}}\left(1+\tan ^{2}\theta \right)+{\frac {1}{8}}\left(1-4{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta \right)\right]{\frac {\Delta \alpha }{\alpha }}-\left[4{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta \right]{\frac {\Delta \beta }{\beta }}.}$ (35)

Simplify the algebra to obtain the desired two-parameter model equation for prestack amplitude inversion (Section L.6)

 ${\displaystyle R(\theta )=\left[{\frac {5}{8}}-{\frac {1}{2}}{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta +{\frac {1}{2}}\tan ^{2}\theta \right]{\frac {\Delta \alpha }{\alpha }}-\left[4{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta \right]{\frac {\Delta \beta }{\beta }}.}$ (36)

Redefine the coefficients ai and bi and write the discrete form of equation (36)

 ${\displaystyle R_{i}=a_{i}{\frac {\Delta \alpha }{\alpha }}+b_{i}{\frac {\Delta \beta }{\beta }},}$ (37)

where

 ${\displaystyle a_{i}={\frac {5}{8}}-{\frac {1}{2}}{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta _{i}+{\frac {1}{2}}\tan ^{2}\theta _{i}}$ (38a)

and

 ${\displaystyle b_{i}=-4{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta _{i}.}$ (38b)

Consider the case of N-fold CMP data represented in the domain of angle of incidence. We have, for each CMP location and for a specific reflection event associated with a layer boundary, N equations of the form as equation (37) and two unknowns — Δα/α and Δβ/β. We have more equations than unknowns; hence, we encounter yet another example of a generalized linear inversion (GLI) problem (Section J.1). The objective is to determine the two parameters such that the difference between the modeled reflection amplitudes Ri represented by equation (37) and the observed reflection amplitudes Xi is minimum in the least-squares sense [3].

The GLI solution in matrix form is given by (Section L.6)

 ${\displaystyle {\begin{pmatrix}\sum \nolimits _{i}^{N}\,a_{i}^{2}&\sum \nolimits _{i}^{N}\,a_{i}b_{i}\\\sum \nolimits _{i}^{N}\,a_{i}b_{i}&\sum \nolimits _{i}^{N}\,b_{i}^{2}\end{pmatrix}}{\begin{pmatrix}{\frac {\Delta \alpha }{\alpha }}\\{\frac {\Delta \beta }{\beta }}\end{pmatrix}}={\begin{pmatrix}\sum \nolimits _{i}^{N}\,a_{i}X_{i}\\\sum \nolimits _{i}^{N}\,b_{i}X_{i}\end{pmatrix}}.}$ (39)

This matrix equation is solved for the two parameters — P-wave reflectivity Δα/α and S-wave reflectivity Δβ/β. Note from the definitions of the coefficients ai and bi given by equations (38a,38b) that you have to choose a value for the ratio β/α to compute the two reflectivities.

Shown in Figure 11.2-14 are the P- and S-wave reflectivity sections that were derived from prestack amplitude inversion applied to the CRP gathers associated with the data shown in Figure 11.2-10a. Note the AVO anomalies along the flanks of the structural closure. It is apparent that some of the AVO anomalies in the P-wave reflectivity section stand out more distinctively as compared with those in the S-wave reflectivity section. This may be related to the fact that changing from brine to gas causes a small change in S-wave reflectivity, while it causes a significant change in P-wave reflectivity. Such inference is supported by the laboratory measurements of P- and S-wave reflectivities at a gas-brine interface in sandstones [10]. Figure 11.2-15 shows P-wave reflectivities for a group of reservoir sandstones in which the pore fluid changes from brine to gas. For rocks with low induration (weak rocks), there is a large change in P-wave reflectivity, while for rocks with high induration the change becomes less significant (Figure 11.2-15a). The change in S-wave reflectivity is negligible for rocks with low and high induration (Figure 11.2-15b). A demonstrative field data example of P- and S-wave reflectivity contrasts that arise from a change in lithology and fluid saturation is shown in Figure 11.2-16.

Figure 11.2-18  Crossplot of P- to S-wave velocity measured down a borehole [12].

Refer to equation (17b) and note that the difference between the P-wave and S-wave reflectivities is related to change in Poisson’s ratio Δσ — a direct hydrocarbon indicator. In fact, Smith and Gidlow [3] have coined the term pseudo-Poisson reflectivity to describe the difference between the P-wave and S-wave reflectivities

 ${\displaystyle {\frac {\Delta {\tilde {\sigma }}}{\tilde {\sigma }}}={\frac {\Delta \alpha }{\alpha }}-{\frac {\Delta \beta }{\beta }}.}$ (40)

Note from equation (17b) that the pseudo-Poisson reflectivity ${\displaystyle {\Delta {\tilde {\sigma }}}/{\tilde {\sigma }}}$ is not exactly the same as what may be referred to as the proper Poisson reflectivity Δσ/σ. If we define the ratio

 ${\displaystyle {\tilde {\sigma }}={\frac {\alpha }{\beta }}}$ (41a)

by differentiation, we can derive the difference relation given by equation (40) and thus show that

 ${\displaystyle {\frac {\Delta {\tilde {\sigma }}}{\tilde {\sigma }}}={\frac {\Delta (\alpha /\beta )}{\alpha /\beta }}.}$ (41b)

Castagna [11] defined a straight line in the plane of S-wave velocity versus P-wave velocity as shown in Figure 11.2-17. This is called the mudrock line and is represented by the equation

 ${\displaystyle \alpha =c_{0}+c_{1}\beta ,}$ (42)

where the scalar coefficients c0 and c1 are empirically determined for various types of rocks. Suggested values for these scalars are c0 = 1360 and c1 = 1.16 for water-saturated clastics [11]. Gas-bearing sandstones lie above the mudrock line, and carbonates lie below the mudrock line as shown on a crossplot of P- and S-wave velocities sketched in Figure 11.2-17. The crossplot shown in Figure 11.2-18 is based on log measurements at a gas-producing well [12]. Note the separation of gas-sandstone cluster from the water-sandstone and shale clusters in the manner as sketched in Figure 11.2-17.

A way to quantify the prospectivity of the reservoir rock of interest is by defining a fluid factor attribute that indicates the position of the rock property with respect to the mudrock line. First, apply differentiation to both sides of equation (42) and note that

 ${\displaystyle {\frac {\Delta \alpha }{\alpha }}=c_{1}{\frac {\beta }{\alpha }}{\frac {\Delta \beta }{\beta }}.}$ (43)

Then, define the fluid factor ΔF

 ${\displaystyle \Delta F={\frac {\Delta \alpha }{\alpha }}-c_{1}{\frac {\beta }{\alpha }}{\frac {\Delta \beta }{\beta }}.}$ (44)

If ΔF is close to zero, it means that you have water-saturated rock. If you have a gas-saturated sandstone, ΔF will be negative at the top and positive at the base of the reservoir unit [3].

Figure 11.2-19 shows the pseudo-Poisson and fluid-factor sections that were derived from prestack amplitude inversion applied to the CRP gathers associated with the data shown in Figure 11.2-10a. Note the distinctive AVO anomalies along the flanks of the structural closure.

The two parameters Δα/α and Δβ/β, estimated by using the least-squares solution given by equation (39), represent fractional changes in P- and S-wave velocities. As such, they are related to P- and S-wave reflectivities, ΔIP/IP and ΔIS/IS, respectively, where IP and IS are the P- and S-wave impedances given by

 ${\displaystyle I_{P}=\alpha \rho }$ (45a)

and

 ${\displaystyle I_{S}=\beta \rho .}$ (45b)

From Section L.6, the P- and S-wave reflectivities are given by

 ${\displaystyle {\frac {\Delta I_{P}}{I_{P}}}={\frac {\Delta \alpha }{\alpha }}+{\frac {\Delta \rho }{\rho }}}$ (46a)

and

 ${\displaystyle {\frac {\Delta I_{S}}{I_{S}}}={\frac {\Delta \beta }{\beta }}+{\frac {\Delta \rho }{\rho }}.}$ (46b)

Assuming that the density obeys Gardner’s relation given by equation (34a), the P-wave reflectivity ΔIP/IP is simply the fractional change of the P-wave velocity Δα/α scaled by a constant, whereas the S-wave reflectivity ΔIS/IS is a linear combination of the fractional changes of the P- and S-wave velocities, Δα/α and Δβ/β, respectively.

Figure 11.2-22  Framework for derivation of the various AVO equations in analysis of amplitude variation with offset.

A direct estimation of the P- and S-wave reflectivities given by equations (L-46a,L-46b) can be made by using an alternative formulation of prestack amplitude inversion which is based on transforming the Aki-Richards equation (15) to the new variables ΔIP/IP and ΔIS/IS [13]. Solve equation (46a) for Δα/α and equation (46b) for Δβ/β, and substitute into the Aki-Richards equation (15). Following the algebraic simplification, we obtain (Section L.6)

 ${\displaystyle R(\theta )=\left[{\frac {1}{2}}\left(1+\tan ^{2}\theta \right)\right]{\frac {\Delta I_{P}}{I_{P}}}-\left[4{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta \right]{\frac {\Delta I_{S}}{I_{S}}}-\left[{\frac {1}{2}}\tan ^{2}\theta -2{\frac {\beta ^{2}}{\alpha ^{2}}}\sin ^{2}\theta \right]{\frac {\Delta \rho }{\rho }}.}$ (47)

Goodway [13] have implemented a specific form of equation (47) to derive the AVO attributes ΔIP/IP and ΔIS/IS. For a specific value of α/β = 2 and small angles of incidence for which tan θ ≈ sin θ, the third term in equation (47) vanishes. Compare equations (18) and (46a), and equations (29) and (46b), and note that equation (47) with the remaining terms takes the form

 ${\displaystyle R(\theta )=(1+\tan ^{2}\theta )R_{P}-(2\sin ^{2}\theta )R_{S}.}$ (48)

The resulting equation then is solved for the P- and S-wave reflectivities, ΔIP/IP = 2RP and ΔIS/IS = 2RS, respectively. Again, consider the case of N-fold CMP data represented in the domain of angle of incidence. In discrete form, equation (48) can be rewritten as

 ${\displaystyle R_{i}=a_{i}R_{P}+b_{i}R_{S},}$ (49)

where i is the trace index and the coefficients ai and bi are given by

 ${\displaystyle a_{i}={\frac {1}{2}}{\big (}1+\tan ^{2}\theta _{i}{\big )}}$ (50a)

and

 ${\displaystyle b_{i}=-2\sin ^{2}\theta _{i}.}$ (50b)

The reflection amplitude Ri in equation (49) is a linear combination of the two parameters, RP and RS. As for the Smith-Gidlow equation (37), the two parameters, RP and RS, can be estimated for a specific event from CMP data using the least-squares solution given by

 ${\displaystyle {\begin{pmatrix}\sum \nolimits _{i}^{N}\,a_{i}^{2}&\sum \nolimits _{i}^{N}\,a_{i}b_{i}\\\sum \nolimits _{i}^{N}\,a_{i}b_{i}&\sum \nolimits _{i}^{N}\,b_{i}^{2}\end{pmatrix}}{\begin{pmatrix}R_{P}\\R_{S}\end{pmatrix}}={\begin{pmatrix}\sum \nolimits _{i}^{N}\,a_{i}X_{i}\\\sum \nolimits _{i}^{N}\,b_{i}X_{i}\end{pmatrix}}.}$ (51)

Following the estimation of the P-wave reflectivity RP and the S-wave reflectivity RS using the least-squares solution given by equation (51), the P-wave impedance IP and the S-wave impedance IS can be computed by integration.

By using the impedance attributes IP and IS, Goodway [13] compute two additional AVO attributes in terms of Lamé’s constants scaled by density — λρ and μρ. Substitute equation (3b) into (45a) to get the relation

 ${\displaystyle (\lambda +2\mu )\rho =I_{P}^{2},}$ (52a)

and substitute equation (17c) into equation (45b) to get the relation

 ${\displaystyle \mu \rho =I_{S}^{2}.}$ (52b)

Note from equations (52a,52b) that

 ${\displaystyle \lambda \rho =I_{P}^{2}-2\,I_{S}^{2}.}$ (52c)

Figure 11.2-20 shows the μρ and λρ AVO attribute sections which were derived from prestack amplitude inversion applied to the CRP gathers associated with the data shown in Figure 11.2-10a. As in the case of the pseudo-Poisson and fluid-factor AVO attribute sections shown in Figure 11.2-19, note the distinctive AVO anomalies along the flanks of the structural closure. Goodway [13] convincingly demonstrates that separation of gas-bearing sands from tight sands and carbonates is much better with the crossplot of the Lamé attributes (μρ, λρ) in contrast with the crossplot of the P- and S-wave reflectivities or impedances (Figure 11.2-21).

Figure 11.2-22 outlines a summary of the AVO equations that are described in this section based on the various approximations to the Aki-Richards equation (15). Start with Shuey’s arrangement of the terms in the Aki-Richards equation, which itself is an approximation to the exact expression for the P-to-P reflection amplitudes given by the Zoeppritz equations. Then, apply a transformation from S-wave velocity to Poisson’s ratio, and drop the third term to get a simple expression for the P-to-P reflection amplitude that is a linear function of sin2 θ. This linear approximation yields the AVO intercept and gradient attributes.

Alternatively, you may drop the third term in the original Shuey arrangement given by equation (16) and apply Wiggins’ rearrangement of the terms. Then, assume the specific case of the P-wave velocity to be twice the S-wave velocity to obtain the equation that yields, once again, the AVO intercept and gradient attributes. The Wiggins AVO intercept and gradient attributes, unlike the Shuey counterparts, directly lead to deriving the P-wave and S-wave reflectivity attributes, albeit only under the assumption that the P-wave velocity is twice the S-wave velocity.

Returning to the three-term Aki-Richards equation (15), you may reduce it to the two-term Smith-Gidlow equation (36) by using Gardner’s relation of density and P-wave velocity. Then, for a specified ratio of P- to S-wave velocity, perform inversion of prestack amplitudes associated with target events on CMP or CRP gathers to obtain the P- and S-wave reflectivity attributes. In addition, using these attributes to derive two more AVO attributes — psuedo-Poisson and fluid factor.

Finally, you may recast the Aki-Richards equation (15) in terms of P- and S-wave reflectivities (equations 46a,46b) and assume that α/β = 2 to obtain the two-term Goodway et al. equation (48). Again, perform inversion of prestack amplitudes associated with target events on CMP or CRP gathers to obtain the P- and S-wave impedance attributes. Subsequently, by using the relations between impedances and Lamé’s constants, you can compute two additional AVO attributes — λρ and μρ.

The AVO equations derived in this section are all expressed in terms of angle of incidence (equations 24, 31, and 36). In practice, the mapping of amplitudes associated with a reflection event on a CMP gather from offset to angle of incidence needs to be performed. Assume that the CMP gather is associated with a horizontally layered earth model so that the event moveout on the CMP gather is given by the hyperbolic equation

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {x^{2}}{v_{rms}^{2}}},}$ (53)

where t is the two-way traveltime from the source to the flat reflecting interface back to the receiver, t0 is the two-way zero-offset time, x is the offset, and vrms is the rms velocity down to the reflector. The ray parameter p is given by the stepout dt/dx measured along the hyperbolic moveout trajectory (the slant-stack transform). Apply differentiation to equation (53) to get

 ${\displaystyle {\frac {dt}{dx}}={\frac {1}{v_{rms}^{2}}}{\frac {x}{t}}.}$ (54)

Since the ray parameter p is also expressed as the ratio of sin θ/vint, where vint is the interval velocity above the reflecting interface, it follows that

 ${\displaystyle \sin \theta ={\frac {v_{int}}{v_{rms}^{2}}}{\frac {x}{t}}.}$ (55)

By using equation (55), reflection amplitudes on a CMP gather can be transformed from offset to angle domain, and subsequently used in the AVO equations (24), (31), (36), and (48).

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