Choice of parameterization

In the previous example, we inverted for the elastic impedance parameter set (I_p, I_s, $\rho$). However, we could have elected to invert for any of several other parameter sets: (V_p, V_s, $\rho$) for example. We have been investigating six parameter sets so far: Impedance (I_p, I_s, $\rho$), Velocity (V_p, V_s, $\rho$), Elastic Modulus ($\kappa$, $\mu$, $\rho$), Lamé Parameter ($\lambda$, $\mu$, $\rho$), V_p/V_s Ratio (I_p, V_p/V_s, $\rho$), 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 (I_p in Figure for example) is obtained by making a unit perturbation in that parameter $(\delta I_p/I_p = 1)$, and setting the other two parameters in that set to zero $(\delta I_s/I_s = \delta \rho/\rho = 0)$.This yields that parameter's coefficient as a function of specular angle $c_i(\theta)$ 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 $(R_{pp}(\theta)$, I_p is the most sensitive, followed by I_s and lastly $\rho$. Next, at near offsets R_pp is due only to I_p, which tends to make it a robust parameter (no ambiguity or non-uniqueness). As offsets increase to about $30^{\circ}$, any remaining AVO response in R_pp not explained by I_p has to be due to I_s, since $\rho$ 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 I_p the most robust, to $\rho$ the least robust, that for $0^{\circ}$-$30^{\circ}$ specular coverage I_p and I_s are well resolved, and that R_pp contains no information about density variations for $\theta \le 30^{\circ}$.

 

imprad
Figure 7 Radiation curves for the Impedance parameterization.

 

velrad
Figure 8 Radiation curves for the Velocity parameterization.

In contrast, and to make a point, consider the Velocity radiation curves of Figure . At near offsets it is completely ambiguous whether R_pp is due to changes in V_p or $\rho$ or both. Worse still, as offsets increase, the V_s and $\rho$ curves are almost parallel. This means that if a change in R_pp is not explained by V_p with offset, any inversion scheme will have a difficult time trying to decide if that change is due to V_s, $\rho$ or both, since they both have the same change with offset. In noisy, aperture-limited data, the V_p and $\rho$ parameters will be closely coupled at near offsets, and the V_s and $\rho$ curves will be closely coupled at far offsets $(30^{\circ})$, 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 R_pp 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 I_p terms, but an inferior accuracy in the B term as compared to I_s. This conclusion is intuitively supported by the AVO radiation curves, which are not shown here.

 

Rnoise
Figure 9 Noisy synthetics with 50% filtered Gaussian spatially coherent noise.

 

table2
Figure 10 Elastic parameter inversion results for noisy synthetics.

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$^{\circ}$-35$^{\circ}$.