A hybrid (finite difference-surface Green´s function) method for computing transmission losses in an inhomogeneous atmosphere over irregular terrain

A standard parabolic equation (SPE) is used to approximate the Helmholtz equation for electromagnetic propagation in an inhomogeneous atmosphere. An implicit finite difference (IFD) scheme to solve the SPE is applied between the irregularly shaped ground and an altitude z =z_h, below which all inhomogeneities of the medium are assumed localized. The boundary condition at z=z _h is obtained by matching the IFD solution to a surface Green´s function (SGF) solution within the uniform region above z =z_h. For ground slopes above about 1°, the IFD implementation of the impedance boundary condition at the ground is shown to maintain the validity of the approximation only for vertically polarized waves. Predictions using this hybrid finite difference (FD)-SGF method agree well with results obtained using other computational methods