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 j omega, 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.