基于优化型UCPML和三维旋转交错网格的GPR快速正演研究

Focus on forward simulation with traditional high order finite difference method (FDM) algorithm for complex 3D GPR model, the approach of Standard Staggered Grid was strictly limited by the CFL condition, in order to reduce the numerical dispersion, a smaller mesh size and time step is needed, which restricted the forward efficiency of 3D GPR simulation. In order to improve the efficiency and stability of the 3D GPR forward modeling, the applicant proposed a novel 3D simulation FDM algorithm based on Rotated Staggered Grid, the algorithm introduces a newly staggered grid strategy, which weaken the CFL stability condition, and a larger space and time step can be choosen, which significantly improved the efficiency of 3D simulation for GPR. In order to further perfect the theory of RSG algorithm and improve low frequency reflections with evanescent mode on the truncated boundary, the optimized Unsplit Convolution Perfectly Matched Layer (UCPML) was loaded around the border of calculation area, of which combined the characteristics of RSG algorithm, optimized the linear distribution of absorption factor and the exponential function of electrical conductivity, which expected to improve overall absorption effect of low frequency waves with evanescent mode, aimed at more truly simulating the open 3D problem space. Predictably, through the research of efficient 3D simulation algorithm based on stable RSG and optimized UCPML theory in complex medium, it can more accurately extract the electromagnetic characteristic information, which will promotes the development of 3D migration and full waveform multi-parameter inversion techniques for GPR.

针对传统高阶差分算法进行复杂GPR模型三维正演时，采用的标准交错网格受CFL条件的限制，为减少数值色散，需采用较小的网格尺寸和时间步长，制约三维GPR正演的计算效率。为提升三维GPR正演模拟效率与稳定性，申请者提出三维旋转交错网格(RSG)算法，该算法引入全新的空间网格交错策略，弱化CFL稳定性条件限制，可以选取更大的网格尺寸与时间步长，显著提高了三维正演效率和稳定性。为进一步完善RSG算法的理论并改善截断边界处的低频反射，将优化型非分裂递归卷积完全匹配层(UCPML)加载到计算区域的边界，融合RSG算法的特点，优化吸收因子的线性分布及电导率的指数分布函数，可望改善对低频凋落模式的整体吸收效果，更真实地模拟开放的三维问题空间。可以预见，通过对复杂介质中高效稳定的GPR三维正演与优化型UCPML理论研究，能更准确地提取电磁特征信息，促进复杂介质中GPR三维偏移及全波形多参数反演技术的发展。

本项目针对探地雷达 (Ground Penetrating Radar, GPR) 正演数值模拟与成像，建立了基于旋转交错网格 (Rotated Staggered Grid, RSG) 差分技术和优化非分裂卷积递归完全匹配层 (Un-splitting Convolution Recursive Perfect Matching Layer, UCPML) 吸收边界的GPR快速正演理论，并发展了差分法+有限元的GPR正演混合算法及GPR反演算法。系统地研究了复杂地质条件下UCPML吸收边界的优化策略及优化型UCPML与RSG算法的有效融合，提出了基于优化型UCPML吸收边界的旋转交错网格的GPR正演算法。首次分析并界定了高频电磁波稳态场的低频杂波中低频掠角波、低频隐失模态和低频传播模态的频谱特征和场值分布等，优化了基于m阶多项式参数分布的UCPML，解决了优化项对阶次的高敏感性难题，拓宽了参数组合的优选范围，解决了低频传播模态难以高效吸收的关键技术瓶颈。同时，在主计算区域采用基于非结构化网格的有限元对异常体进行精细剖面，在边界区域采用高阶时域有限差分法及UCPML，进一步提高了复杂介质模型病害体(不密实、空洞、裂隙等)的GPR正演精度和正演效率，并针对On-ground GPR数据，开展了介电常数和电导率的GPR双参数反演研究。项目主要研究成果在高效精确地刻画高频电磁波在复杂介质中的传播特征及On-ground GPR数据的多参数反演等方面，提供了全新的理论基础和解决方案。