Seismic travel time inversion for 3D structures regularized with Sobolev norms

Abstract

Asymptotic ray methods are commonly used for forward modelling in seismology and seismic exploration, but rays propagating through heterogeneous media exhibit chaotic behaviour. This means that we can observe a sensitive dependence on initial ray conditions and thus, for long times, the exponential amplification of very small perturbations. The degree of chaos in the model, which may be quantified in terms of the average Lyapunov exponent for all the rays in the model, can be approximated by the square root of the Sobolev norm of order 2 of the corresponding model parameters. In this paper, we present a new inversion technique which considers the Sobolev norm as the regularization term of the ill-conditioned seismic inverse problem. We show that in this way we can control the divergence of rays in our model during the iterations and obtain stable solutions with a negligible dependence on the chosen discretization. The robustness and limitations of the algorithm are illustrated by numerical examples produced with a real heterogeneous 3D structure. The sensitivity of the algorithm to non-Gaussian distribution of noise is also tested.

Unveiling seismic and density structure beneath the Vrancea seismogenic zone, Romania

“…The two wavefields are modelled separately with a two‐point ray shooting forward procedure that searches the range of takeoff angles within which two‐point rays may occur [ Ĉerveny et al , 1988; Bulant , 1996]. Considering the uneven distribution of rays, especially at depths greater than 40 km, the reduction of traveltimes residuals is performed using an algorithm which regularizes the solution by minimizing the Sobolev norm of order two (first and second partial derivatives) of the functionals describing the velocity parameters [ Tondi and de Franco , 2005]. We assume that data have independent, normal distributed errors and so the data misfit is addressed through evaluation of the quantity where Δ t is the travel time residual vector of dimension J, the number of observations, and C d is the data covariance matrix of dimension J × J .…”

Refractor velocity analysis: a signal processing procedure

“…In this paper the velocity analysis refers to the step in which the refractor velocities are determined directly (direct data inversion) from the data without model inversion or the tomographic procedure. The seismic tomography or first arrival inversions are powerful techniques to obtain the velocity distribution for 2D and 3D non‐homogeneous models (Podvin and Lecomte 1991; Schuster and Quintus‐Bosz 1993; Pullammanappallil and Louie 1994; Leung 1997; Zhang, Brink and Toksöz 1998; Watanabe, Matsuoka and Ashida 1999; Osypov 1999; Tondi and de Franco 2005; Mendes 2009). In general, the first arrival inversion procedures are sensitive to the starting model and they often work integrating a priori information or other data‐types derived from scalar‐direct inversion or from processing techniques, which can constrain the starting model.…”