探地雷达正演的间断伽辽金法影响因素分析

Analysis of influencing factors of the discontinuous Galerkin algorithm for ground penetrating radar

时域间断伽辽金(discontinuous Galerkin time-domain, DGTD)算法具有守恒性、稳定性、高精度性和间断性等优点,现已成为一种有效的探地雷达(Ground penetrating radar, GPR)正演方法.为了提高DGTD算法的计算效率和精度,作者详细分析了数值通量、时间离散格式、单元大小与局部基函数阶次、网格剖分方式等影响因素.数值实验表明,局部Lax-Friedrichs中τ=1/2的补偿数值通量既可以消除伪解,又可以提高计算精度;在精度相同的情况下,低存储显式Runge-Kutta方案(low-storage explicit Runge-Kutta, LSERK)的稳定性条件和低存储优势要明显优于其它两种时间离散格式,尤其是在大型复杂模型和三维正演模拟中更有优势.而提高基函数的阶次或增大网格数,均可以提高其误差的收敛性,局部基函数阶次N和单元大小d与电磁波波长λ的适用关系为d/N约等于λ/15;当单元数目大致相等时,网格剖分方式对于高阶DGTD算法的影响较小,说明DGTD算法对网格具有较好的适应性.最后,采用DGTD算法对火星乌托邦平原模型进行正演,验证了基于最优参数的DGTD算法模拟精度高,可为火星乌托邦平原GPR实测数据的解译奠定理论基础.

The discontinuous Galerkin time-domain (DGTD) algorithm has become an effective method for forward numerical simulation of Ground Penetrating Radar (GPR), due to its conservation, stability, high precision and discontinuity. To improve the computational efficiency and accuracy of DGTD, we analyzed its related influencing factors in detail, including numerical flux, temporal integration scheme, grid size and basis function order, and mesh generation methods. Through the numerical case, we verified that the partially penalized numerical flux of τ=1/2 in local Lax-Friedrichs can not only eliminate the spurious solution, but also improve the computational accuracy. Under the same precision, the low-storage explicit Runge-Kutta scheme (LSERK) has better stability and lower storage advantages than the other two temporal integration schemes, especially in large complex models and 3D forward simulation. The convergence of the error can be improved by increasing the order of the basis function or the number of meshes. The experimental results reveal the order of basic function N and the size of the grid d are closely concerned with the wavelength λ, for example, an appropriate definition is d/N≈λ/15. When the number of cells is roughly the same, the mesh generation methods has little influence on the high-order DGTD algorithm, indicating that DGTD has good adaptability to the grid. Finally, we used the DGTD algorithm to simulate the Martian Utopian Plain model for GPR, which verifies that the DGTD algorithm based on the optimal parameters has high simulation accuracy and can lay a theoretical foundation for the interpretation of the GPR measured data of the Martian Utopian Plain.