基于Pade逼近的Cole-Cole频散介质GPR有限元正演

针对Cole-Cole频散介质中的复介电常数是jω的分数次幂函数，传统的时域有限元法难以离散及计算时间域分数阶导数，本文采用Pade逼近算法将含有时间分数阶导数的Cole-Cole频散介质电磁波方程推导为一组整数阶辅助微分方程，提出了一种适用于Cole-Cole频散介质的GPR有限元正演模拟算法.在复数伸展坐标系下，通过在频率域Cole-Cole频散介质电磁波方程中引入2个中间变量，并将其变换到时间域，从而以变分形式将PML边界条件加载到Cole-Cole频散介质GPR有限元方程组中，并给出了详细的求解公式.在此基础上，编制了基于Pade逼近的Cole-Cole频散介质GPR有限元正演程序，利用该程序对均匀模型进行计算，并与解析解进行对比，验证了本文构建的GPR有限元正演算法的正确性和有效性.设计了一个复杂Cole-Cole频散介质GPR模型，利用本文构建的GPR有限元正演算法进行模拟并与非频散介质模型的模拟结果进行对比，分析了电磁波在Cole-Cole频散介质中传播衰减增强、子波延伸，分辨率降低等传播规律，有助于实测雷达资料更可靠、更准确的解释.模拟结果表明，基于Pade逼近的GPR有限元正演算法可用于复杂Cole-Cole频散介质结构模拟，且具有较高的计算精度.

Simulation of GPR in Cole-Cole dispersive media by finite element method based on the Pade approximations

Since the complex permittivity of Cole-Cole dispersive media is a fractional power function of jω, it is difficult to calculate the time domain fractional derivative by using the finite element time domain (FETD) method. In this paper, a set of auxiliary integer order differential equations are derived from the fractional derivative Maxwell equations in the Cole-Cole dispersive media by the Pade approximation algorithm and a FETD algorithm for simulating ground penetrating radar (GPR) in the Cole-Cole dispersive media is proposed. A perfect matched layer (PML) boundary condition in the variational form is applied to the Cole-Cole dispersive media through introducing two intermediate variables to the Maxwell equations in the frequency domain, which is then transformed into the time domain. We firstly validate the proposed FETD algorithm through simulating the GPR waves propagating in a homogeneous dispersive medium and comparing the results with the analytical solution. Then we design a complex subsurface model of the Cole-Cole dispersive media, compare the simulated GPR profile with that from the counterpart model of non-dispersive media. We find that the medium dispersion can cause larger attenuation of GPR waves and elongate the GPR wavelet and degrade the resolution. We conclude that the proposed FETD algorithm can simulate GPR waves in the Cole-Cole dispersive media with a high accuracy and can aid in the reliable interpretation of the field GPR data.