Finite Difference Frequency Domain Simulation Of Acoustic Wave Field

Full waveform inversion is a challenging data fitting method,which extracts relevant information from seismic full-wave field simulation,and frequency domain waveform inversion is one of the important branches.The core content of the frequency domain waveform inversion is an effective frequency domain forward modeling method,which largely determines the accuracy and efficiency of inversion.The current finite difference frequency domain algorithm is implemented in Cartesian coordinate system,and it is difficult to calculate the wellbore field problem efficiently.The best algorithm for the wellbore wave field simulation problem should be implemented in the cylindrical coordinate system,so this article implement the finite difference frequency domain algorithm in the cylindrical coordinate system.Based on the basic elastic dynamic equation,the elastic wave equation of the frequency domain in Cartesian coordinate system and cylindrical coordinate system is deduced independently.And discuss the error of different difference schemes,the forward difference and the backward difference formula has the first order accuracy,the central difference formula has the second order accuracy.With the same condition,the error introduced by the central difference formula is much smaller than the other two formulas.Finally,the specific finite difference scheme of the partial differential operator is proposed by using the optimized nine-point difference grid in the cylindrical coordinate system.The PML boundary condition is used to simulate the propagation of elastic waves in the unboundary medium.Then,the PML boundary condition is introduced into the wave equation of the elastic wave in the frequency domain,and the detailed derivation process is given.The numerical solution of the coupled equations is optimized by the optimal nine-point difference grid.And this paper adopts the parameter average method for fluidsolid interface.Based on the Matlab language,the cosine envelope function is selected as the source function.The frequency value of a certain step is taken to solve the amplitude response of the point on the well axis,and get the frequency domain amplitude curve by connecting the discrete point to a smooth curve.And the amplitude curve of time domain can be obtained by the Fourier transform.The results show that the amplitude response increases firstly and then decreases with the increase of the calculated frequency,showing the properties of the parabola.And the amplitude reaches the maximum value near the center frequency,which is slightly different from the source center frequency,and the reason may be the approximation of finite difference and matrix equation.We can see three kinds of waveforms tn the fullwave waveform,namely primary wave,shear wave and Stoneley wave,and the amplitude of the primary wave is the smallest,and the amplitude of the Stoneley wave is the largest.There is a slight dispersion of Stoneley waves in the low frequency,and nearly none dispersion in the high frequency.From the full-wave array diagram,three distinct wave groups can be clearly seen.Through the velocity extraction,the three wave groups are primary wave,shear wave and Stoneley wave,and the velocity of primary wave,shear wave and Stoneley wave decreased in turn,which meets the basic law.