GNGTS 2014 - Atti del 33° Convegno Nazionale

30 GNGTS 2014 S essione 3.1 where the subscripts 1 and 2 refer to the overlying and underlying media, respectively. The Aki and Richards equation is inverted to retrieve the relative contrasts at the reflecting interface that can be conveniently written as In this form R p , R s and R d indicate the P-wave, S-wave and density reflectivity, respectively. To reduce the physical ambiguity inherent to the AVAmethod (Drufuca and Mazzotti, 1995) and to stabilize the inversion process the number of unknowns can be reduced. To this end two- term approximations of the Zoeppritz equations are frequently used. In particular in this work we consider the Ursenbach and Stewart equation (Ursenbach and Stewart, 2008): where the density term is incorporated into the P and S impedance contrasts expressed by R I and R J , respectively. These linear approximations of the Zoeppritz equations enable the description of the relationship between the observed AVA response ( R pp ) and the model parameters ( m ) in a linear, compact, matrix form: R pp ( ϑ ) = Gm In this form the G matrix contains the three- or the two-term equation, whereas the vector m contains the inverted parameters (elastic or impedance contrast at the reflecting interface) . The singular value decomposition of the G matrix splits the reflectivity R pp ( ϑ ) into three orthogonal components in both data space and model space. It is interesting to consider the physical meaning of the decomposition. The eigenvectors V are a basis in the model space. The eigenvalues represent the reflected energy due to medium perturbations along the eigenvectors in model space. The amplitude versus angle effects of the reflections are described by the eigenvectors in data space, which are three orthogonal functions (De Nicolao et al. , 1993). Condition number and eigenvectors in model space. We now compare the condition number for the three- and the two-term inversions by varying the background Vp / Vs ratio. We remind that high condition numbers indicate an ill-conditioned problem. Based on previous experience with linear AVA inversion, we can fix the threshold of stability for the linear AVA inversion between 200 and 500. Therefore, we can determine how the Vp / Vs ratio influences the stability of the inverse problem (Fig. 1). Specifically, if Vp / Vs is equal to 2 (or Vs / Vp =0.5), as is often assumed in deep Fig. 1 – Condition number for the three-term Aki and Richards equation (red line) and the two-termUrsenbach and Stewart equation (blue line) for varying background Vs/Vp ratios. The dotted line represents the assumed threshold of stability for the linear AVA inversion.