Open AccessExplicit, approximate expressions for the resolution and a posteriori covariance of massive tomographic systems
Author(s) -
Nolet Guust,
Montelli Raffaella,
Virieux Jean
Publication year - 1999
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1046/j.1365-246x.1999.00858.x
Subject(s) - singular value decomposition , covariance , covariance matrix , computation , singular value , mathematics , a priori and a posteriori , inverse , algorithm , matrix (chemical analysis) , resolution (logic) , computer science , eigenvalues and eigenvectors , statistics , physics , geometry , philosophy , materials science , epistemology , quantum mechanics , composite material , artificial intelligence
We present an approximate method to estimate the resolution, covariance and correlation matrix for linear tomographic systems Ax = b that are too large to be solved by singular value decomposition. An explicit expression for the approximate inverse matrix A − is found using one‐step backprojections on the Penrose condition AA − ≈ I , from which we calculate the statistical properties of the solution. The computation of A − can easily be parallelized, each column being constructed independently. The method is validated on small systems for which the exact covariance can still be computed with singular value decomposition. Though A − is not accurate enough to actually compute the solution x , the qualitative agreement obtained for resolution and covariance is sufficient for many purposes, such as rough assessment of model precision or the reparametrization of the model by the grouping of correlating parameters. We present an example for the computation of the complete covariance matrix of a very large (69 043 × 9610) system with 5.9 × 10 6 non‐zero elements in A . Computation time is proportional to the number of non‐zero elements in A . If the correlation matrix is computed for the purpose of reparametrization by combining highly correlating unknowns x i , a further gain in efficiency can be obtained by neglecting the small elements in A , but a more accurate estimation of the correlation requires a full treatment of even the smaller A ij . We finally develop a formalism to compute a damped version of A − .