Full Waveform Inversion Based On The Ensemble Kalman Filter Method Using Uniform Sampling Without Replacement

Full waveform inversion(FWI)can help us better understand the underground geological structure.By taking advantage of the full kinematics and dynamics information of the waveform,FWI can detect the underground velocity structure and the source information with high precision and high resolution.However,FWI has the disadvantages of large computational cost and non-uniqueness of the solution.It may get completely wrong inversion results when there is large error between the initial model and the true model,Therefore,it is of great significance to develop a FWI method that has a large convergence domain and can effectively reduce the strong dependence of the inversion results on the initial model.To this end,we combine the idea of uniform design in statistics with the ensemble Kalman filter method which is widely used in atmospheric and oceanography,and develope a new FWI method(GEKUS).The method optimizes the inversion model parameters and the theoretical wavefield simultaneously,and can reduce the strong dependence of the inversion results on the initial model.Due to the introduction of uniform design,the GEKUS method can reduce the influence of the uncertainty of random samples on the inversion results.The numerical results show that the GEKUS method effectively expands the convergence domain and reduces the strong dependence of the inversion results on the initial model compared with the FWI method based on the adjoint method and the Kalman filter method with random samples.It can save about 30% of the computational cost compared with the random method.In addition,the GEKUS method only needs to solve the original wave equation in the iterative process,and does not need to solve the adjoint equation.As the samples are irrelevant,the method has the characteristics of natural parallelism and has high parallel efficiency.Forward modeling is the basis of inversion.In order to apply the GEKUS method to 3D full waveform inversion,an effective 3D forward modeling algorithm needs to be developed.For that reasom,we combine the optimized finite difference operator with the modified symplectic partitioned Runge-Kutta scheme and extend it to 3D to develop a modified time-spae optimized symplectic method(MTSOS)for solving the elastic wave equation in 3D inhomogeneous media.The method uses a second-order scheme to achieve third-order accuracy,and is more suitable for solving the inhomogeneous medium model.The numerical dispersion error is about 57.5% of the SPRK method.The numerical results show that the MTSOS method can give numerical simulation results accurately.The various phases in the inhomogeneous medium can be clearly seen,and there is no visible numerical dispersion,which shows the effectiveness of the new method.Finally,we combine the 3D MTSOS method with the GEKUS method to achieve full waveform inversion based on the 3D elastic wave equation.The numerical results show that the proposed method obtains high-resolution inversion results which verifies the effectiveness of the GEKUS method for the full waveform inversion of the 3D elastic wave equation.