Fractional Viscous Acoustic Wave Equation And Its Numerical Simulation

The conventional seismic wave field numerical simulations are generally assume that the medium is completely elastic,however the underground terrane is closer to the viscoelastic medium,and the viscosity of the medium results in energy loss and affects its amplitude and phase.If the viscosity of the medium is neglected,the seismic record of numerical simulation is different from the real seismic record,which leads to misadvise oil and gas exploration.The fractional calculus provides a new method for describing the viscoelasticity of the medium,which effectively compensates for the defects of the traditional viscoelastic medium model.Therefore,the fractional order viscous acoustic wave equation was used to simulate the seismic wave field in viscoelastic media.In this thesis,the theories related to viscoelastic medium were introduced firstly.And then the four kinds of common definitions of fractional derivatives and numerical approximation methods of Riemann-Liouville fractional derivatives and Riesz fractional derivatives were given.As for the fractional order viscous acoustic wave equation,the second-order central difference algorithms were used to approximate its temporal derivatives,and the fractional order center difference algorithms were used to approximate its spatial fractional derivatives.Two difference schemes for fractional viscous acoustic wave equations were obtained,and convolution perfectly matched layer(CPML)was used as the absorbing boundary to suppress the artificial boundary reflection.On this basis,the stability analysis and dispersion analysis of the difference scheme were also carried out.Finally,the numerical simulations of fractional order viscous acoustic wave equation were carried out.Through numerical simulations in homogeneous medium,layered medium and Hess/Salt medium model,it was verified that the algorithms were stable and feasible.This thesis provided a new method for numerical simulation of fractional viscous acoustic wave equations,which has certain theoretical significance and research value.