An Integral Equation Method for Seismic Modelling and Inversion

Abstract

In this paper a Fredholm integral equation of the second kind for the Green’s function associated to a heterogenous medium is derived. This approach is based on the idea to interpret the space dependent propagation speed of the wave equation as a perturbation of a constant reference velocity. The integral equation can be solved by standard quadrature methods. Once the Green’s function is calculated, the seismic response to an arbitrary source function can be calculated by a simple convolution. Also, since the calculations are performed in the frequency domain, an incorporation of attenuation mechanisms can be applied easily. Some numerical examples for one-dimensional acoustic and viscoacoustic media are presented. The derived integral equation can also be interpreted as first kind integral equation for the perturbation potential. Thus it can be applied to the inverse problem as well. In order to treat the ill-posedness of this problem it is necessary to apply certain regularization methods such as truncated singular value decomposition or Tikhonov regularization. Some numerical results for Born inversion applied to synthetic data obtained by the integral equation modeling are presented.