Currently the common approach to perform FWI is nonlinear least square minimization of the standard data misfit functional which characterizes L2 residual between observed data and synthetized one for a current velocity model. As was aforementioned this approach has been developed and studied in a great number of publications. But up to now there are problems with reliable reconstruction of macro velocity component via FWI for realistic frequency bandwidth and offsets. The reason is the structure of the forward map, which transforms the velocity model to the data – it is almost quadratic with a well-conditioned matrix with respect to perturbations of reflectors, but has very complicated nonlinear behavior with respect to propagator perturbation . Intuitively this can be explained by the so-called “cycle- skip” effect when phase shifts of the observed and synthetic data may result in local minima. To mitigate the problem earlier was introduced a multiscale inversion strategy in time and frequency domain when the frequency of the input data is increased progressively and the inversion result for lower frequency becomes an initial guess for the higher frequency. However, such sequential inversion approach also fails due to lack of low frequencies in the data.


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