High-frequency Asymptotic Theories And Numerical Computation Methods For The Gradient Scattering Of Seismic Waves

The smooth velocity models are a kind of velocity models with first-order or higher-order partitioned continuous derivative,which are widely used in seismic migration imaging and full waveform inversion.Compared with the actual underground velocity distribution,there are no velocity discontinuities(first-order discontinuities)in the smooth velocity models,but only velocity derivative discontinuities(higher-order discontinuities).Different from the scattering caused by the discontinuities of the medium parameters in the actual velocity distribution,when seismic waves propagate into a medium with a velocity gradient,the incident waves will interact with the heterogeneities of velocity gradient to produce gradient scattering.Although the energy of the gradient scattering field is very weak,it contributes to improving the quality of seismic migration imaging.However,up to now,people have mainly focused on the scattering caused by the discontinuities of the medium parameters in the actual velocity models while the gradient scattering caused by the change of the velocity gradient has not received widespread attention.In order to compute the gradient scattering field caused by the change of velocity gradient in the smooth velocity models,this paper studies two methods for computing the gradient scattering field in the smooth velocity models: one is the generalized Born approximation(the scattering term is represented by the error between the exact Green’s function and the approximate Green’s function,which is mainly related to the change of velocity gradient)and the other is the FK domain integration method.By the conventional FK domain integration method,we can get the scattering field generated by the perturbation of the medium.In order to obtain the gradient scattering field caused by the change of the velocity gradient,this paper applies Taylor expansion to the perturbation term to obtain the integral formula suitable for the computation of the gradient scattering field in the smooth velocity models.Theoretical research and numerical implementation of the above two methods show that both the generalized Born approximation and the FK domain integration method can be used to compute the gradient scattering field in the smooth velocity models with relatively high accuracy.In addition,in order to solve the problem that the 2-D numerical modeling cannot correctly describe the geometric spreading of the seismic waves,the 2.5-D generalized Born approximation is derived using the stationary phase method based on the 2.5-D ray theories(this method uses the ray family information obtained from 2-D ray tracing to describe the propagation of 3-D seismic waves in the observation plane)in this paper.Besides,by the numerical modeling of 2.5-D gradient scattering field,it’s proved that the 2.5-D generalized Born approximation is superior to the 2-D numerical modeling in describing the geometric spreading of the gradient scattering field.