In the previous example, we inverted for the elastic impedance parameter set (Ip, Is, ). However, we could have elected to invert for any of several other parameter sets: (Vp, Vs, ) for example. We have been investigating six parameter sets so far: Impedance (Ip, Is, ), Velocity (Vp, Vs, ), Elastic Modulus (, , ), Lamé Parameter (, , ), Vp/Vs Ratio (Ip, Vp/Vs, ), and the so-called ``AVO'' Parameters (A, B, C). In practice, when we are faced with noisy data given over a limited aperture of specular reflection angles, the issue of parameter choice can become critical, as we will show.
Figures and are plots of ``radiation'' curves for the Impedance and Velocity parameterizations respectively. A single curve on one plot (Ip in Figure for example) is obtained by making a unit perturbation in that parameter , and setting the other two parameters in that set to zero .This yields that parameter's coefficient as a function of specular angle from the Bortfeld approximation in the Theory section. By plotting all three curves for a parameter set we get Figures or , for example. These curves contain a lot of intuition about how robust a given parameterization will be.
Consider the Impedance radiation curves of Figure . The first thing to notice is that, in terms of amplitude sensitivity to the data , Ip is the most sensitive, followed by Is and lastly . Next, at near offsets Rpp is due only to Ip, which tends to make it a robust parameter (no ambiguity or non-uniqueness). As offsets increase to about , any remaining AVO response in Rpp not explained by Ip has to be due to Is, since has zero sensitivity. Hence, the intuition from the radiation curves tells us that there is a clear order in the robustness of the three parameters, ranging from Ip the most robust, to the least robust, that for - specular coverage Ip and Is are well resolved, and that Rpp contains no information about density variations for .
In contrast, and to make a point, consider the Velocity radiation curves of Figure . At near offsets it is completely ambiguous whether Rpp is due to changes in Vp or or both. Worse still, as offsets increase, the Vs and curves are almost parallel. This means that if a change in Rpp is not explained by Vp with offset, any inversion scheme will have a difficult time trying to decide if that change is due to Vs, or both, since they both have the same change with offset. In noisy, aperture-limited data, the Vp and parameters will be closely coupled at near offsets, and the Vs and curves will be closely coupled at far offsets , giving rise to what is called parameter leakage.
To reinforce this point numerically, we have inverted a noisy version of our previous synthetic example. Figure is the noisy version of Figure , obtained by adding random noise, normally distributed with zero mean and standard deviation equal to one half the near offset peak gas amplitude at 2960 m (s/n = 2/1 defined), filtered to match the wavelet frequency band, and with Markovian type lateral coherency. Table gives the inverted elastic parameter estimates for the Impedance, Velocity, and AVO parameterizations at the shale/gas contact. The error estimates in Table are normalized with respect to the standard deviation of the noise in the Rpp gather. It is readily evident that our intuitive conclusions from the radiation curves are justified by the impedance and velocity inversion results. For comparison, we have also shown the inversion results with the standard AVO parameterization, which demonstrates a similar order of accuracy in the A and Ip terms, but an inferior accuracy in the B term as compared to Is. This conclusion is intuitively supported by the AVO radiation curves, which are not shown here.
We are actively developing methods to quantify the optimality of certain parameter sets in terms of inversion stability, parameter sensitivity, and parameter accuracy. This work is in preparation for journal publication. At this point, we recommend the Impedance parameterization for robust elastic parameter inversion, given reflection data acquired with a specular angle range of about 0-35.