This paper is concerned with the increasing stability of the inverse source problem for the elastic wave equation with attenuation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the attenuation coefficient. The ingredients of the analysis include Carleman estimates and time decay estimates for the elastic wave equation to obtain an exact observability bound, and the study of the resonance-free region and an upper bound of the resolvent in this region for the elliptic operator with respect to the complex frequency. The advantage of the method developed in this work is that it can be used to study the case of variable attenuation coefficient.

The inverse problem considered in this paper is to determine the shape and the impedance of crack from a knowledge of the time-harmonic incident field and the corresponding far field pattern of the scattered waves in two-dimension. The combined single- and double-layer potential is used to approach the scattered waves. As an important feature, this method does not require the solution of u and ∂u/∂v at each iteration. An approximate method is presented and the convergence of this method is proven. Numerical examples are given to show that this method is both accurate and simple to use.