On The Implementation Of Gaussian Beam-born Modeling And The Computation Of Complex Traveltime Based On Complex Eikonal Equation

Gaussian beam based Born theory is the basis of Gaussian beam migration,tomography,full waveform inversion and diffraction stacking theory,which are based on the representations of Green's function of Gaussian beam summation.Its mathematical expression is the form based on volume integral.This implies that we need to stack all the scattering fields when we implement the Gaussian beam based Born modeling.To address the problem of discrepancy between the volume integral and surface integral,we transform the volume integral in the Born equation and surface integral and modify the implementation of the isochronous surface.In addition,the basis of the Gaussian beam-Born method is the computation of Gaussian beam,which includes two parts: One is the amplitude,the other is the phase.However,recent method for computing the complex traveltime is dynamic ray tracing,which is based on paraxial ray approximation and the two orders Taylor expansion.This approximation only uses the velocity on the central ray.Thus,when the beam propagates in the region of strong variation,especially in the region of asymmetric velocity on the both sides of the central sides,the accuracy of the complex traveltime computation can be influenced.A new method based on the perturbation theory and the GN-FMM by solving the complex eikonal equation directly can be used to solve this problem.However,for practical application,there are two problems:(1),this method can only be used for computing the complex traveltime of vertical ray;(2),this method is time consuming.To solve the problems above,this paper presents the non-uniform finite difference method and modified FMM method.Furthermore,I use the L-BFGS method to compute the imaginary slowness to reduce the computational cost.According to the definition,the complex traveltime used to describe the decay is a kind of local complex traveltime.Thus,if we don't limit the domain for computation,we need to compute the complex traveltime in the whole model.Thus,I present a local algorithm to compute the local complex traveltime.Unlike solving the complex eikonal equation in isotropic media,the complex eikonal in TTI media involves several parameters.Thus,the method above cannot be used to solve the TTI complex eikonal equation.Using the perturbation theory,I derive the linearized form of the complex eikonal equation and develop the analytic formulas.Furthermore,this paper present the numerical examples of Gaussian beam wavefields with the complex traveltime based on the complex eikonal equation.