Reverse Time Migration Method Preserving Symplectic Geometry Structure And Its Applications

Developing rapid and effective methods to solve wave equations is very important for improving the computational efficiency of reverse time migration. This dissertation proposes a symplectic finite difference method of high precision and high efficiency, conducts theoretical analysis and numerical simulation, applies the method to reverse time migration, uses the gradient calculated by the method to remove noise in reverse time migration.Firstly, we develop the sixth-order nearly analytic discrete operator based on the original analytic discrete operator and the traditional difference operator, propose the sixth-order nearly analytic symplectic partitioned Runge-Kutta(NSPRK) method for solving wave equations by combining the sixth-order nearly analytic discrete operator with a symplectic scheme. Theoretical analysis and numerical simulation show that compared to the eighth-order NSPRK, the stability of the sixth-order NSPRK is increased by about 5% for the 1D case and about 12.3% for the 2D case. The numerical dispersion error of the sixth-order NSPRK is smaller compared to the fourth-order NSPRK, the sixth-order conditional finite difference(CFD) method, the fourth-order Lax-Wendroff correction(LWC) sheme, and almost the same with that of the eighth-order NSPRK, in particular smaller than that of the eighth-order NSPRK in large courant number case. Theory and numerical results show that the sixth-order NSPRK has second-order accuracy in time and sixth-order accuracy in space. Its calculation speed is about 2.3 times of the fourth-order NSPRK, 4 times of the sixth-order CFD and 24.7 times of the fourth-order LWC, while its storage capacity is about 44.6%, 62%, 23.2% of the three methods, respectively. We apply the sixth-order NSPRK to acoustic wave prestack reverse time migration and give the impulse response test and reverse time migration results of the Sigsbee2 B model. Results show that on coarse grids, compared to the fourth-order LWC, the sixth-order CFD and the fourth-order NSPRK, the sixth-order NSPRK can get better imaging result.On the other hand, we apply the original fourth-order NSPRK to acoustic equation in VTI media and conduct theoretical analysis and numerical simulation. The results show that the fourth-order NSPRK can suppress numerical dispersion effectively. Its calculation speed is about 14.2 times of the fourth-order LWC and its storage capacity is about 44.6% of the fourth-order LWC. Then the fourth-order NSPRK method is applied to reverse time migration in VTI media. We show the impulse response test and the reverse time migation results of the Hess VTI model. The results show that on coarse grids, compared to the fourth-order LWC, the fourth-order NSPRK can get better imaging result. It also shows that the result of VTI model is more accurate than isotropic model.Finally, we take the fourth-order NSPRK method as an example, combine the gradient field of NSPRK with the inverse scattering imaging condition, apply it to reverse time migration denoising, and give denoising examples of an impulse response test, Sigsbee2 B model and Hess VTI model. The numerical results show that combining the gradient field of NSPRK methods with the inverse scattering imaging condition can get better denoising results.