GNGTS 2024 - Atti del 42° Convegno Nazionale

Session 3.3 GNGTS 2024 respectvely, and is the data associated to the -th model . The matrix represents the so called Kalman gain, given by: . In the previous equaton is the cross-covariance matrix between the model and the associated data , whilst is the covariance matrix of the predicted data. Fig. 1 – Schematc representaton of the EB-FWI algorithm. This scheme also shows the possibility to apply a local FWI using as startng model the result of the global one. This step allows to improve the resoluton of the result. The velocity models forming the startng ensemble are drawn from a Gaussian prior distributon in which a Gaussian variogram has been included to impose the desired spatal variability on the velocity model. The workload of the procedure can be alleviated by adoptng a reparameterizaton technique able to considerably reduce the computatonal complexity of the problem. Among the possible methods, we choose the DCT for its compression ability, its linearity, the possibility to easily extend it to more than one dimension and because its applicaton does not overload the inversion procedure with additonal computatonal tme. The compression power of this method relies on the fact that it is able to concentrate most of the informaton of the signal into the low order coefcients. As a consequence, the majority of these coefcients are very close to zero, and retaining only the low order ones is sufcient to approximate the original signal without losing relevant informaton. Furthermore, compressing data and model space, we considerably reduce the size of matrices and vectors involved in the computatons. The DCT compression also mitgated the ensemble collapse issue, consistng in the fast convergence of the ensemble towards the mean and in the consequent underestmaton of the posterior variance. The most common soluton to d p k k m p k K ~ K ~ = C p md ( C p dd + αC d ) −1 C p md m p d p C p dd