衰减夹层GPR模拟的时频域全局反射误差

摘要

全局反射误差分析是深入研究探地雷达(GPR)吸收边界条件吸收效率的有力工具.基于常规完全匹配层(PML)的标准交错网格有限差分算法必须满足严格的CFL条件限制,即在单位时间步长内,不容许电磁波传播的距离超过单元网格尺寸.为了提高主区域所有网格节点的计算效率,并有效地吸收传播后期出现的低频隐失波,提出基于非分裂递归卷积完全匹配层(UCPML)的旋转交错网格(RSG)GPR正演算法.该算法采用不同的网格交错策略,并在边界条件中引入了吸收低频隐失波的自由可变因子,使得该算法允许选取较大的时间步长,提高了计算效率,并且实现了对低频隐失波的高效吸收.本文首先给出了RSG差分格式,推导了基于UCPML的RSG差分更新方程,探讨了数值色散的稳定性条件,然后以绕射现象严重的衰减夹层数值模拟为例,分别从波场快照、单道波记录、时间域反射误差(TDRE)、频率域反射误差(FDRE)四个方面分析了UCPML与常规PML的全局反射误差,说明了基于UCPML和RSG的GPR正演算法能更有效地吸收低频隐失波.%Global reflection error analysis is an effective way to compare absorbing availability among different absorbing boundary conditions (ABC) in Ground Penetrating Radar (GPR),in particular the reflection errors in time-frequency domain.When solving the partial differential equations of Maxwell numerically with the method of finite differences (FD),the standard staggered grid based on the regular perfectly matched layer (PML) must meet the strict limit of Courant-Friedrichs-Lewy (CFL) conditions.In order to improve the computation efficiency and absorb efficiency in the later stage,this work studies how the un-split convolution perfectly matched layer (UCPML) and the rotated staggered grid (RSG) are combined to accomplish the efficient absorption of evanescent modes,as well as to improve the computational efficiency without numerical dispersion.With different strategies of the space staggered grid,the RSG scheme using the rotated staggered finite-difference operators can be used to calculate the difference of field components and dielectric parameters with the linear combination value across the diagonal coordinate axes,with a looser limit of CFL conditions.In this method,a larger time step can be set,which eventually leads to a higher computational efficiency.And then the UCPML is implemented in the boundary region,and in addition to the decay parameter,the other two real variable factors are introduced,which is exclusively for evanescent modes at grazing incidence.Then a numerical modeling example with an attenuation interlayer model embedded is presented,which contains a serious diffractions.Finally,a detailed comparison of UCPML and regular PML is made in the aspects of full-time snapshots,single scan signal,and reflection errors in the time-frequency domain.Both the full-time snapshots and single scan signal show that the amplitude and energy of reflection waves with regular PML are larger than the UCPML,which indicates that UCPML is more efficient in absorbing evanescent modes.The Time Domain Reflection Errors (TDRE) results demonstrate that the reflection errors of UCPML are much less than the regular PML at most of the electromagnetic wave full-time propagation,and the FDRE results show that UCPML has an obvious advantage over the regular PML in absorbing low frequency components of evanescent modes.The combination of RSG and UCPML proposed in this work can easily to be programmed and performed,which can improve the computational efficiency and absorb efficiency on long-term evanescent modes at grazing incidence.