3-D seismic tomography for velocity and interface structure of the crust and upper mantle (theoretical part)

A method of three dimensional (3-D) model parameterization is presented that makes forward and inverse problems become easy. The velocity and interface structure of crust and upper mantle are described by a set of highly smoothed functions. Shooting ray tracing method is chosen to calculate the ray paths for both forward and inverse problems. The partial derivatives of traveltime with respect to parameters of the model grids are calculated analytically while rays are being traced. Because velocity and interface functions have second-order continuous partial derivatives, the geometrical shadow zones at the surface caused by scattering and focusing of ray paths can be prevented. After ray tracing, an equation consisting of matrix and vectors for inverse problem is obtained. We use singular value decomposition method with damped factor to solve the equation. A synthetic data set which consists of several in-line profiles is used to test the methods. The results show that the methods are robust. Compared with the two dimensional method, the 3-D inversion method can give the right position of interfaces and the velocity structure when the crustal model is complicated.