| Wave equation numerical simulation is the foundation of seismic imaging and inversion,and Finite-difference(FD)methods are the most widely used numerical moldelling methods for seismic wave.However,numerical dispersion(grid dispersion)intrinsically existed in FD methods seriously affect their moldelling accuracy,so suppressing numerical dispersion to increase the moldelling accuracy is an important researching content of FD numerical moldelling.The current FD methods for wave equation numerical moldelling in time domain and frequency domain,have their own technologies,characteristics,advantages and disadvantages in suppressing the numerical dispersion to increase the moldelling accuracy.In the article,based on numerous study on the current FD methods in time domain and frequency domain,by learn from each other's advantage,we propose mixed-grid and mixed-staggered-grid FD methods in time domain,and we improve the existed mixed-grid FD methods in frequency domain,and then study them systematically.When studying wave equation numerical simulation in time domain,in view of that the mixed-grid FD methods in frequency domain can effectively obviously decrease the numerical dispersion to increase the moldelling accuracy,we think that reasonable FD stencils should take full use of the grid points near to the center point to differentially approximate the Laplace operator in the 2nd-order acoustic wave equation or the 1st-order spatial derivative.In this article,by introducing the idea of the mixed-grid FD methods from frequency domain to time domain:(1)For two-dimension 2nd-order acoustic wave equation in time domain,expressing the Laplace operator as the weighted average of M and N Laplace operators in general and rotated rectangular coordinate system,we first propose a new kind of generalized time-space domain mixed 2M+N style FD methods(Two dimension M2M+N-FD).(2)For three-dimension 2nd-order acoustic wave equation in time domain,having no way to introduce triaxial rotated rectangular coordinate system makes it is hard to construct three dimension mixed 2M+N style FD methods(Three dimension M2M+N-FD),so two methods to use the mixed-grid points in three dimension to differentially approximate the Laplace operator are raised in this article.The First method,decomposing the three-dimensional Laplace operator into three two-dimensional Laplace operators and borrowing the concept of rotated rectangular coordinate system in two dimension,we derive the method to differentially approximate the two-dimensional Laplace operator using the mixed-grid points in the coordinate planes,and then summing up the three two-dimensional Laplace operator,we will get the difference expression of the three-dimensional Laplace operator.The second method,utilizing the Taylor expansion of function with three variables,the method to differentially approximate the three-dimensional Laplace operator using the mixed-grid points(The grid points are not in the coordinate axes or in planes)in the three dimension space is raised.With the help of these two methods,we successful put forward the two dimension M2M+N-FD to three dimension and construct three dimension M2M+N-FD.(3)For two-dimension and three-dimension 1st-order stress-velocity acoustic wave equation,using 2M grid points in the axis and 4N(Two-dimension)or 8N(Three-dimension)grid points parallel to but off the axis to differentially approximate the 1st-order spatial derivative of the wavefield,we propose a new kind of generalized time-space domain two-dimension mixed 2M+4N style staggered-grid FD methods(Two dimension M2M+4N-SGFD)and three-dimension mixed 2M+8N style staggered-grid FD methods(Three dimension M2M+8N-SGFD).(4)We derive the approach of calculating the FD coefficients based on the time-space domain dispersion relationship and plane wave theory for two-dimension and three-dimension M2M+4N-FD,two-dimension M2M+4N-SGFD and three-dimension M2M+8N-SGFD,at the same time,propose the concept of fractional order difference accuracy,which can better describe the accuracy of the mixed-grid FD methods.Then,we conduct difference accuracy analysis,dispersion analysis and numerical moldelling experiments,and comparing the corresponding results with the counterpart of the existed FD methods,we can see that,these four mixed-grid FD methods can more effectively suppress the numerical dispersion to achieve higher moldelling accuracy without increasing the computational cost,and they also have better stability.In this article,by implementing NPML Absorbing Boundary Condition(ABC)in two dimension M2M+N-FD and PML ABC in two dimension M2M+4N-SGFD,the reflected energy from the artificial boundary is effectively eliminated,and then we put forward this two mixed-grid FD methods to reverse time migration on complex model,the high quality of migration results further demonstrate the superiority,validity and general applicability of the new methods.When studying wave equation simulation in frequency domain,except for the existed mixed 9-point and square mixed 25-point FD stencils,three new mixed-grid FD stencils,which are mixed 13-point,mixed-21-point and rhombus mixed 25-point ones,are constructed,at the same time,considering the method for calculating the FD coefficients based on time-space domain dispersion relationship in time domain can remarkably decrease the numerical dispersion and increase the moldelling accuracy,we introduce the idea of this FD coefficients calculating method from time domain to frequency domain,and propose two new kind of FD coefficients calculating methods for the mixed-grid FD methods in the frequency domain,which are the method based on frequency-space domain dispersion relation and the adaptively optimized method.And then,we conduct dispersion analysis and numerical moldelling experiments,comparing the results obtained from the two new FD coefficients calculating methods and the traditional least-square optimized ones,which show that,the FD coefficients calculating method based on the frequency-space domain dispersion relationship has some advantage on increasing the moldelling accuracy,and the adaptively optimized one has more obvious advantage.In the end,allowing for wave equation numerical moldelling in frequency domain is carried out for single frequency,we also put forward the variable FD strategy,meaning using different mixed-grid FD stencils for different moldelling frequency arrange,which can gain high moldelling accuracy and computational efficiency at the same time.In this article,from seven aspects: FD stencil construction,FD coefficients calculation,FD accuracy analysis,dispersion analysis,stability analysis,boundary condition and numerical moldelling methods implementation,we build the theoretical framework and the foundation of mixed-grid FD methods for wave equation numerical simulation in time domain,and we further replenish and consummate the relevant theory and implementation strategy of mixed-grid FD methods in frequency domain.Studies have shown that,the innovative and improved work in this article can effectively increasing the moldelling accuracy of FD methods for wave equation numerical simulation in time and space domain. |
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