A theoretical study of the boundary conditions for parabolic equation

For many years, parabolic equation simulations have been performed for 2D propagation of electromagnetic waves with a great accuracy. However, considering precisely the ground reflection is still an open problem. A classical solution is to consider at each step a constant impedance that depends on the grazing angle calculated by MUSIC or a ray-launching algorithm. For non-constant impedance conditions, this question is not solved. Numerical instabilities may occur when the reflection coefficient is exactly taken into account. This paper theoretically studies this instability. First, the Green function of the vertical operator used for the split-step computation of the propagation is expressed, taking into account the sampling induced by numerical simulations. From this, the observed numerical instability is related to a radiation condition in the upper medium.