The Pseudo-Shear Reflection Coefficient

Most practitioners are by now familiar with $A\!-B\!$ plots for AVA analysis. The SNA attribute operates on a permutation of the $A\!-\!B$ plane described below. The Zoeppritz equation approximation for small physical contrasts over intermediate angles presented by Shuey (1985) is:  

$$R(\theta) \; \approx \; A + B\sin^2\theta \;,$$ (1)

where
 

$$A = \;\frac{\Delta \rho}{2 \rho + \Delta \rho}\;+\;\frac{\Delta V_p}{2V_p + \Delta V_p}$$ (2)

and

$$B =\; A - 2 C\;,$$

with C having the form  

$$C = \;\frac{1+\eta}{2}\;\frac{\Delta \rho}{2\rho+\Delta\rho}\;+\;\eta \frac{\Delta V_s}{2V_s+\Delta V_s}$$ (3)

where

$$\eta =\;4 \frac{V_s^2}{V_p^2}\;.$$

$\Delta$ quantities are the differences layer 1 minus layer 2. Density and velocities (without $\Delta$) are those of layer 2. Also, I have reordered the usual relationship to lead the reader toward the goal of orthogonalizing the AVA plane between compressional and shear axes.

With the above development of the Shuey equation, we see A is the normal incidence compressional reflection amplitude and B contains both normal incidence and angular dependence. Other authors, such as Castagna et al. (1998), have attempted to simplify this relationship by expressing B as a complicated function of A multiplied by new (empirical) fitting coefficients. While facilitating ever more cross-plotting possibilities, axes remain mixtures of compressional and shear quantities and yield little more insight.

However, with the above formulation of Shuey equation (1) we can separate B into the compressional, A, and pseudo-shear, C, reflection amplitude coefficients. Notice the very parallel structure of the pseudo-shear, equation 3, to the normal incidence term, equation 2, and its independence from compressional velocity contrast. Encouragingly, the pseudo-shear expression contracts to the normal incidence shear reflection coefficient when the compressional to shear velocity ratio $\gamma$ equals 2.

This separation (effectively between $A(\Delta V_p)$ and $C(\Delta V_s)$) will help guide our intuition by isolating the seismic reflection amplitude into two components that have meaning in an AVA sense. Rather than plotting on the $A\!-\!B$ plane, we can utilize the $A\!-\!C$ plane and avoid an ordinate with codependency of compressional and shear velocity boundary contrasts.

 

cartoon Figure 1 With the clarity of structure of equation (3) for C, simple test case plots can be readily manufactured by applying it to mental scenarios within this cartoon.

With the insight gained from the definitions of A and C, we can now intuitively understand that trends in this AVA plane are due to the variation of real rock properties. Figure 1 indicates the relative position of simple targets in this space. Because I have defined density and velocity as the properties of the lower layer, the negative values of A show a change from harder to softer intervals and the opposite is true for the positive values. Therefore we can understand something of the nature of the bounding layers of an interval as harder bounding lithologies will trend to more negative values of A.

We also know that the shear velocity of a porous medium increases as we lower the density of the included fluid. This tells us that the $\Delta V_s$will be negative as we consider a water filled medium versus a gas filled one and this will increasingly drive the value of C more negative. The interplay between these compressional and shear forces results in the normal NW-SE trend of the data cloud in Figure 1 indicating hard bounding rocks upward and soft ones downward.

Gratwick (2001b) outlined the promise and difficulty of prospecting AVA anomalies on an $A\!-\!B$ plane as explained by Castagna and Swan (1997). Both authors stress the importance of the distance away from the background trend for the analysis of a prospective event. Attempting to quantify this, Gratwick (2001b) calculates the product of A with B, then masks the center mass of reflection amplitudes that are assumed to be background values (non-prospective shale-wet sand or shale-shale reflections). This process is shown in Figure 2. The flaw in this method is the dull spoon that differentiates anomalies from background. Not only is the scalpel dull, but this methodology only appreciates a single model type. More practically, the clumsy transfer in and out of SEP architecture for graphical definition of the mute zone can dissuade all but the most committed from utilizing this tool.

 

plot2 Figure 2 A vs. B scatter-plot with mute fairway defined. Gratwick (2001b)

 

planes
Figure 3 The ordinate axis is transformed from B, Shuey's gradient, to compressional the pseudo-shear reflection coefficient C.

Figure 3 (i) shows the standard Slope-Intercept AVA plot, while (ii) shows the transform to the Compressional-PseudoShear plane. The data are generated from a synthetic provided by BP and explained in detail by Gratwick (2001b). While immediately displeasing, these two plots will highlight the power of the $A\!-\!C$ plane when inspected. First note the strong zero presence on the intercept-axis of the $A\!-\!B$ panel. This is modeled data, boring, and makes our unit vector for the shale trend very simple ($\theta = 0$). In an attempt to provide a small measure of believable scatter (make this plot less boring), a bandpass filter was run over the AVA attributes. This contributes to a few bothersome artifacts, but are easy to neglect. These include: data present to the left of shale trend (bandpassing returns negative values), diagonal sub-trends of events, and incomplete orthogonalization. We see that due to the presence of both shear and compressional velocity contrasts in the formulation of B, the transition of reflections on the $A\!-\!B$ plot from water to oil to gas takes place along a line with an acute angle to the background trend. This leads to one of the paramount problems with interpreting AVA anomalies as explained in Castagna 1997. The $A\!-\!C$ plane, enjoying an ordinate quantity that is a function only of a change across the boundary of the shear velocity, shows nice perpendicular departure from the axis of the compressional reflection coefficient.