Scalar Wave Equation for GPR Imaging

Modeling a GPR experiment is a complicated process that, for complete accuracy, requires taking into account such effects as antenna radiation patterns and the vectorial nature of electromagnetic (EM) wave propagation and scattering van der Kruk et al. (2003). It is well known, however, that seismic processing techniques based on a scalar wave equation can often be applied very successfully to GPR data Fisher et al. (1992b). This latter point is not mere coincidence. In many situations, isotropic scattering and scalar wave propagation effectively model the kinematics of a GPR experiment. Here, in applying an imaging algorithm based on a scalar wave equation to GPR data, we argue that radar propagation kinematics are well represented. In addition, we suggest that with further development, effects such as antenna radiation patterns and realistic scattering could be accounted for in the source and receiver wavefields and imaging condition.

Considering a situation where Maxwell's equations can be represented by a 2-D scalar wave equation involves making two approximations. First, we implicitly assume that the subsurface geology and sources are strictly 2-D. This results in the decoupled transverse electric (TE) and transverse magnetic (TM) propagation modes Jackson (1975). Choosing the TE-mode, we next assume that heterogeneities within the earth are small such that the gradients of EM constitutive parameters can be neglected Sena et al. (2003). The result is a scalar wave equation for transverse electric field, E, which in the frequency ($\omega$)domain is given by,  

$$\left[ \frac{\partial^2}{\partial z^2}+\frac{\partial^2}{\partial x^2}\right] E + \omega^2 s^2 E = 0,$$ (1)

where the slowness of wave propagation (i.e. inverse of velocity), s, is dependent on the medium's dielectric permittivity, $\epsilon$,and conductivity, $\sigma$, through,  

$$s = \sqrt{ \mu(\epsilon - \frac{i \sigma}{\omega})} \approx \sqrt{ \mu \epsilon}.$$ (2)

The magnetic permeability of the medium, $\mu$, is roughly constant for most material likely to be encountered in a routine radar application. Hence, in low conductivity media (i.e. $\sigma \ll \omega$), the slowness of wavefield propagation is directly proportional to the dielectric permittivity.