Frequency-domain Element-free Reverse-time Migration

Seismic modeling and migration imaging,as an important part of exploration geo-physics,has become a hot topic at present with the development of seismic exploration.The most frequently-used methods of numerical simulation are the finite difference method and the finite element method,however,both of them have shotcomings more or less.In this paper,a frequency-domain element-free method(EFM)for seismic mod-eling and reverse-time migration is presented.The application of time-domain EFM has been demonstrated successfully in seismic data processing and its advantages are shown.The absence of elements makes the EFM more flexible than the finite element method(FEM)and it can be applied to more complex problems such as for irregular surfaces.Because of the utilization of the moving-least-squares(MLS)fitting method,the dependent variable and its derivative are both continuous and precise in EFM.Even the low order base vector is used,the element-free method has high precision and reso-lution.However,the large computation time of time-domain EFM limits its usefulness.We have developed an algorithm for frequency-domain EFM that reduces its computa-tion time.Unlike time-domain propagators,the developed algorithm solves array equa-tions to calculate a single-frequency wavefield.Without time iteration,there is very little accumulated error and,because of the spectrum of the source,we only need to calculate parts of the frequencies.The independence of each frequency makes it con-venient to manipulate single-frequency wavefields and easy to accelerate computation with parallelization.For the frequency-domain algorithm feature,we can solve multiple shots at the same time.In this paper,we present the theory of element-free method in detail,including the concept and commonly used formaton of base vector,influence domain and weight function,and the construction of approximate solution.Variational method and Galerkin method are compared and used to do formula derivation,respectiovely,when the element-free method is applied to the acoustic wave equation.The frequency-domain wave modeling algorithm is different from time-domain algorithm because of the feature of frequency domain.The boundary condition has an important impact on wave modeling in frequnecy domain.To eliminate the impact of artificial boundary,several freqency-domain boundary conditons are tested and the PML boundary conditon is chosen,con-sidering the effect of eliminate artificial boundary and the convenience of programming.Large sparse matrix equation has to be solved to obtained single-frequency wavefield from frequency-domain governing equation.The direct solver and the iterative solver are introduced and,their advantages and disadvantages are discussed.The reason that why the direct solver is used to solve the governing equation is also presented.The time-domain wavefield can be obtained by Inverse Fourier Transform and frequency-domain imaging can be obtained by cross-correlation imaging condition.Based on the theory of frequency-domain element-free method,several tests has been done.There are some test examples inlcuding the modeling result of layer structure,the imaging result of 2D SEG/EAGE model and Marmousi model.All these examples show that the frequency-domain element-free method can be a good choise in wave modeling and migration imaging.Several measures are adopted to increse the memory and the time efficiency.The compression storage of sparse matrix is applied to save memory and acceleration computation.The forward-propagating wavefield and the backward-propagating wave-field are calculated at the same time,based on the feature of frequency domain and the feature of direct solver.The multiple shots imaging results are also calculated at the same time.The paralleration can be used to acceleration computation between differ-ent frequencies.The test result shows that the frequency-domain element-free method is much more efficent than time-domain element-free method with the same memory requirement.