GNGTS 2014 - Atti del 33° Convegno Nazionale

32 GNGTS 2014 S essione 3.1 higher angles. The second eigenvector points, approximately, in the direction of S-impedance perturbations, whereas the third eigenvector does not have any particular physical meaning. In the Vp / Vs >>2 case (Fig. 2a, right column), both the first and second eigenvectors, associated with the first and second singular values, point toward the P-impedance. Conversely, only the third eigenvector, associated with the smallest singular value, points entirely in the R s direction. This fact indicates that this component spans the null-space of the G matrix. This result illustrates that for high Vp / Vs ratios, the S-wave velocity plays a very minor role in determining the AVA response. Moreover, by comparing the first and second eigenvectors for Vp / Vs =2 and Vp / Vs >>2, we can see that an increased Vp / Vs ratio increases the cross-talk between R p and R d : a smaller distance is observed between the R p and R d components as the Vp / Vs ratio increases. This trend makes an independent estimation of these two parameters more problematic in the case of high Vp / Vs values. These observations allow us to draw some important conclusions. First, the difficulty of achieving a reliable R s estimation with increasing Vp / Vs values; second, the cross-talk between R p and R d also increases as the Vp / Vs ratio increases. Now, we briefly consider the eigenvectors in model space for the two-term approximation (Fig. 2b). In the case of Vp / Vs =2 (Fig. 2b, left column), the first eigenvector points toward the P- impedance for small angles, whereas the R J component is not-null only if large incidence angles (grater than 20 degrees) are considered. Conversely, if we increase the Vp / Vs ratio (Fig. 2b, right column), the first eigenvector points toward the P-impedance regardless of the considered angle range. In this case, the R J parameter spans the null space of the G matrix, indicating that, for a reliable estimation of the R J term, a sufficiently high Vp / Vs ratio is needed. Note that the two-term inversion is stable only for sufficiently low Vp / Vs values (see Fig. 1), and in these cases, use of the second eigenvector allows the inversion to extract the R J parameter. In the Vp / Vs =2 case, this eigenvector can be used in the inversion and the R J information can be recovered with a good degree of accuracy. Instead, in cases of Vp / Vs >>2, a regularization method (for example, the T-SVD method) is needed to stabilize the inversion. In these cases, the truncation of the second singular value (and the associated eigenvector) renders a reliable estimation of the R J value impossible. Model resolution and unit covariance matrices. Here, we describe the model covariance and resolution matrices. The former describes how the error in the data space propagates in the model space. The latter describes how the true parameters influence the estimated ones. We start by analyzing the unit covariance matrix (computed by assuming an identity data covariance matrix) for the least-squares inversion, for which the model resolution matrix is equal to an identity matrix (see Aster et al., 2005 for a rigorous mathematical demonstration). Fig. 3a shows the unit covariance matrices computed for both Vp / Vs =2 and Vp / Vs >>2 and for both the three- and two-term approximations. From this figure, we can see that the order of magnitude of the errors decreases from the three- to two-term inversion (independent of the Vp / Vs value) and from the Vp / Vs >>2 to the Vp / Vs =2 case (independent of the parametrization). It is worth noting that the Vp / Vs ratio determines the error propagation from the data to the model space. In fact, for high Vp / Vs values, the parameters most contaminated by noise are those associated with the Vs values ( R s and R J , respectively). Instead, if the Vp / Vs is equal to two, the error is more homogeneously distributed among the parameters although, even in this case, the error most strongly affects R s and R J . Now we eliminate the smallest singular value of the G matrix (applying the T-SVD method) and recompute the unit covariance and the model resolution matrices. Let us first consider the model resolution matrices (Fig. 3b). For the three-term inversion, it is clear that for the Vp / Vs >>2 case, we obtain a null resolution for the R s parameter and a good resolution for both R p and R d (note that the resolution is expressed by the diagonal terms). In the case of Vp / Vs =2, the three parameters can be recovered with almost the same resolution, even if the lowest resolution capability is always related to R s . If we reduce the dimension of the model space considering the two-term equation, we can see that for both cases, the R I parameter is characterized by the