Computation Of Reflected Waves In Smooth And Smoothed Velocity Models

The smooth and smoothed velocity models are a kind of velocity models with only the velocity derivative(soft interface)and no velocity interfacial(hard interface).It is the most common used velocity model for seismic migration imaging and full waveform inversion.Previous studies have shown that although there is no zero-order interface(intersection between physical parameters)in this model,there are partial reflections caused by high-order inter-section and velocity gradient changes.Although the energy of these reflections is relatively weak,in theory its contribution to the reflection seismic migration imaging is not negligible.However,so far,these reflected waves have not been described and calculated by the ray method(Kirchhoff type migration)and beam method(beam migration)commonly used in migration imaging,resulting in the underground useful information and seismic wave energy carried by this part of the reflected wave are lost,which in turn affects the imaging quality of the migration imaging.In order to solve the above-mentioned calculation of the partial reflection,this paper systematically analyzes and studies two kinds of calculation methods for solving the weak reflection in the smooth and smoothed velocity models: One is the WKBJ method(a kind of classical high-frequency approximation method for solving partial differentials introduced from the field of quantum mechanics).The other is the soft interface hardening method proposed by Sun Jianguo(a model transformation method based on model parameterization and the relationship between velocity derivatives and media properties).At the same time,this paper also analyzes and discusses two methods for calculating high-dimensional partial reflections,Bremmer coupling series method and scattering theory method.Through the programming of the first two methods and the model calculation based on this,the following conclusions can be drawn: Both the WKBJ method and the soft interface hardening method can effectively calculate the partial reflection,and the WKBJ method is more suitable for calculating the reflection and transmission in the medium with slowly varying velocity,and its calculation efficiency is very high;the soft interface hardening method transforms the soft interface in the smooth and smoothed velocity model into a hard interface,so that the calculation method is no longer limited by the slow change of the medium velocity.However,in order to ensure the correctness of the soft interface hardening,it is necessary to accurately calculate the various derivatives of the velocity field and its extreme points and inflection points.For this problem,the related research in this paper is only a preliminary stage,so further theoretical improvement and in-depth study of this soft interface hardening method will be done in the future.