GNGTS 2014 - Atti del 33° Convegno Nazionale

GNGTS 2014 S essione 3.1 29 where the superscript T indicates the transpose. In a compact form the solution of a linear inverse problem can be written as follow: m = G − g d where G -g is called the generalized inverse. For a common overdetermined least-squares problem, this matrix is equal to: G − g = ( G T G ) − 1 G T However, to resolve an inversion problem, one must not only find a solution that best fits the observed data but should also investigate the relation between the estimated model and the true model or, in other words, analyze which properties of the true model are resolved in the estimated model. This problem can be approached with the sensitivity analysis method. For linear inverse problems, this analysis essentially consists in computing the model covariance and model resolution matrices. On one hand the model resolution matrix ( R ) describes how well the predicted model matches the true one. It can be demonstrated (Aster et al., 2005) that the resolution matrix for a linear inverse problem can be computed as follows: R = G − g G On the other hand, to understand how an error in the data propagates as an error in the estimated model, it is useful to define the model covariance matrix C m. . If the data are assumed to be uncorrelated and all have equal variance, the covariance matrix (unit covariance matrix) is given by: C m = G − g G − g T Another useful tool in approaching inverse problems is the Singular Value Decomposition (SVD). According to this method the matrix G be broken down into the product of three matrices: G = USV T where S is a diagonal matrix of singular values, V is the matrix of eigenvectors in model space and U contains the eigenvectors in data space. The SVD decomposition is essential in sensitivity analysis because it permits to get a better understanding of the physical meaning of the G matrix. Moreover, the SVD method is also a powerful tool for solving ill-conditioned least-squares problems. In these problems, the process of computing an inverse solution is extremely unstable and a small change in the measurements can lead to a large change in the estimated model. In these cases the G matrix is characterized by a high condition number, which is the ratio between the highest and the smallest singular values of the G matrix. Therefore, in order to stabilize the inversion, the truncated SVD method (T-SVD) can be applied. This method is aimed at eliminating the smallest singular values of the G matrix and at reducing the condition number. We pay a price for this stability in that the regularized solution has a decreased resolution. Very detailed information about geophysical inverse problems can be found inAster et al. (2005) and Tarantola (2005). The Aki and Richards and Ursenbach and Stewart approximations. Starting from the Zoeppritz equations, Aki and Richard (1980) provided approximation for P-P wave reflection coefficients that is valid for small physical contrasts and small incidence angles (generally less than 30-35 degrees). This equation can be written as where the P-wave reflection coefficient is R pp , ϑ is the average of incidence and P-wave transmission angles across the interface, and the P-wave velocity, S-wave velocity and density are indicated by α, β, and ρ, respectively. In this equation Δ x / x denotes the relative contrast for a particular property between the overlying and underlying media, whereas γ is inversely correlated with the background Vp/Vs ratio: