An exact inversion formula from determining a planar fault from boundary measurements

Abstract

This paper considers the inverse problem of determining a timedependent slip fault which releases shear stress elastically. The input data are the accelerations measured on the free external surface. A new formulae for determining explicitly the geometry of the planar fault is proposed. Potential applications to real earthquakes are discussed.

A numerical method for reconstructing the coefficient in a wave equation

“…It may also be viewed as a simplified model for the shear waves in a two‐dimensional isotropic elastic medium when the scalar displacement travels along the direction transversal to the medium. In recent years, some rapid identification techniques have been developed for solving the elastodynamic inverse problem, for instance, crack/fault identification techniques are developed for cracks having free boundary condition using a reciprocity gap function , and linear sampling techniques are designed to locate inclusions in the isotropic elastic medium . In this work, we shall mainly focus on the inverse problem of reconstructing the coefficient μ in the model equation , using some Dirichlet boundary data from a single measurement.…”

Numerical Studies of the Lagrangian Approach for Reconstruction of the Conductivity in a Waveguide

“…In addition the mathematical model governed by the hyperbolic equation studied in this work can also be considered as a special case of a time-dependent transverse magnetic polarized wave scattering problem or as a simplified acoustic wave model for fluids with variable density and a constant bulk modulus. In recent years, some rapid identification techniques have been developed for solving the elastodynamic inverse problem, for instance, crack/fault identification techniques are developed for cracks having free boundary condition using a reciprocity gap function [12,13], and linear sampling techniques are designed to locate inclusions in the isotropic elastic medium [10,19]. To compare performance of the algorithm of this paper with different algorithms of [10,12,13,14,19] can be the subject of a future work.…”

“…In recent years, some rapid identification techniques have been developed for solving the elastodynamic inverse problem, for instance, crack/fault identification techniques are developed for cracks having free boundary condition using a reciprocity gap function [12,13], and linear sampling techniques are designed to locate inclusions in the isotropic elastic medium [10,19]. To compare performance of the algorithm of this paper with different algorithms of [10,12,13,14,19] can be the subject of a future work.…”

Duality and symmetry lost in solid mechanics

“…6, pp. 553-565, 2005, [27]). As shown above, the reciprocity gap R is the external boundary functional which is known from the data u 1 = u d and T (u 1 ) = T d and from the chosen adjoint functions {u 2 , T (u 2 )}.…”

“…The above non-linear variational equation can be solved analytically, step by step, by determining the normal to the crack plane, the plane position and then the crack geometry. In what follows, we only determine the crack geometry, after finding the crack plane by suitable adjoint fields (for more details, see Bui et al, [27,12], Bui [28,34]).…”