Research On Forward Modeling Method Of Scalar Wave In Frequency Space Domain

Frequency-domain wave equation forward modeling plays an important role in studying the propagation of seismic waves in complex underground media.Compared with solving wave equation in time domain,the frequency domain wave equation has the advantages of efficient calculation,flexible data selection,and easy parallelization.Based on the analysis of the forward modeling of the scalar wave equation in frequency domain,this paper studies the 125-point finite difference scheme of the three-dimensional scalar wave equation in frequency domain,which improves the calculation accuracy,then through waveform inversion proves the accuracy of the forward modeling and applies it to actual data.The accuracy of forward modeling determines the quality of waveform inversion.In this paper,based on the generalized Hookes law,Cauchy equation and Navier equation establish the scalar wave equation.Then,analysis the Greens function analytical solution of the scalar wave equation in the time domain and frequency domain based on homogeneous medium,and the numerical solution of scalar wave equations in time domain and frequency domain.Then analysis the characteristics of the forward modling waveform by comparing the difference between the finite difference numerical solution and the analytical solution in the time domain and frequency domain.It has proved that the forward modeling of the scalar wave equation in the frequency domain has high accuracy.In order to improve the accuracy of the forward modeling of the scalar wave equation in three-dimensional frequency domain,this paper proposes an optimized 125-point finite difference scheme based on the optimized 27-point difference scheme.Firstly,analysis the inevitable boundary problems in numerical simulation and introduce the perfectly matched layer to the scalar wave equation.Then,the optimized difference coefficients suitable for different spatial sampling interval ratios are calculated through the normalized phase velocity,and compare the dispersion curve with the 27-point scheme.After that study the large-scale sparse matrix solution involved in the frequency domain forward modeling and compare the accuracy of different solutions.Finally,the numerical simulation of equal sampling intervals proves that the accuracy of the 125-point difference format is higher and the numerical dispersion is weaker and it also fit to the rectangular grid with high simulation accuracy.Waveform inversion makes full use of the kinematics and dynamics information of seismic data to reconstruct complex geological structures and underground model parameters.This paper develops the objective function of the waveform inversion based on the least squares theory,and then introduces the commonly waveform inversion algorithm.The gradient of waveform inversion is obtained by the adjoint state method,the step length of iteration is determined by parabolic interpolation method and the frequency selection strategy for waveform inversion is given.The numerical experiments of waveform inversion of different finite-difference schemes and different inversion methods prove the stability of the inversion algorithm.Then apply the waveform inversion to the actual data,pre-process seismic data to deal with the surface wave interference and energy imbalance in the actual data.For the lack of inverted seismic wavelet and low-frequency information in the actual data,using the hybrid domain inversion method,which means forward modeling in time domain and inversion in frequency domain.Numerical experiments prove that the accuracy of the time domain waveform after Fourier transform is similar to the frequency waveform,which means the method is effective.Finally,the efficiency of the waveform inversion has proved by the actual shot collection,which further proves the accuracy of the forward modeling.