## 1. Introduction

[2] Parabolic wave equation (PWE) techniques have proven to be very powerful and accurate wave propagation modeling tools for many applications. Restricting the wave spectrum to forward or backward traveling waves with respect to one of the coordinate directions, the complexity of the originally elliptic (time harmonic) wave problem is reduced considerably without compromising the rigor of the formulation. Until now, the greatest successes of PWE techniques have been in the field of radar or acoustical wave propagation, where especially the influence of material variations in the propagating medium such as the atmosphere is of importance and the ground or ocean interface can often be assumed locally flat and ideally conducting. However, in the case of mobile terrestrial communications and terrestrial radio broadcasting, the local terrain shape (including vegetation and buildings) is of essential importance since the receiver antennas are usually very close to the ground. In this respect, the performance of PWE techniques still needs to be improved.

[3] In this paper, we restrict ourselves to two-dimensional (2-D) PWE techniques based on the robust and efficient Fourier split-step (FSS) algorithm [e.g., see *Levy*, 2000; *Kuttler and Dockery*, 1991]. Those techniques are the method of choice for macrocellular (transmitter antenna above roof levels) situations whereas three-dimensional finite difference PWE approaches such as those discussed by *Levy* [2000] and *Zaporozhets* [1999] may favorably be applied to microcellular (transmitter antenna below roofs) propagation situations.

[4] Two-dimensional techniques assume the great circle path profile from the transmitter to the receiver to be rotationally symmetric around the transmitter and often even consider the Earth sphericity in the formulation of the PWE [*Fock*, 1965; *Kuttler and Dockery*, 1991]. The simplest PWE-FSS algorithms approximate terrain by a series of knife-edges (terrain masking) and propagate the plane wave spectrum of the source fields in the region above the individual knife-edges by assuming free-space propagation [*Levy*, 2000]. At the next knife-edge the fields below the terrain interface are zeroed, and again, only waves due to the (source) field distribution above the edge are propagated to the next edge. It is quite obvious that this approach can only give a very crude characterization of the ground reflective properties and that it is strongly dependent on the chosen distance between the individual knife-edges. More accurate characterizations of irregular terrain profiles usually perform a transformation of the irregular terrain interface into a plane interface [*Beilis and Tappert*, 1979; *Barrios*, 1994]. Consequently, Dirichlet and Neumann boundary conditions can efficiently be applied in the Fourier domain, and imaging the solution domain may be used to obtain additional degrees of freedom in formulating boundary conditions [*Tappert and Nghiem-Phu*, 1985]. *Kuttler and Dockery* [1991] introduced a mixed Fourier transform technique to handle mixed or impedance boundary conditions on flat or slightly spherical terrain. Later it has been extended to the wide-angle shift-map approach for irregular terrain presented by *Donohue and Kuttler* [2000]. A shortcoming of this impedance boundary condition approach is that it requires knowledge of the varying grazing angle along the profile that must be determined beforehand. Another disadvantage of transformation-based approaches is that they introduce additional approximations in order to obtain a useful PWE. Also, it is observed that all available PWE techniques try to handle the irregular terrain interface in the spatial domain.

[5] In this paper, we aim at an exact treatment of the irregular terrain interface in the Fourier or spectral domain. To this end, we utilize and extend the bridged knife-edge model proposed by *Whitteker* [1990]. There, the well-known terrain masking or knife-edge model was extended by introducing metallic or dielectric (straight) bridges between the individual edges, and the Kirchhoff integrals were asymptotically evaluated in the spatial domain to propagate the fields from one knife-edge to the other. Such a terrain model can obviously be applied within the FSS solution of the PWE to formulate an exact boundary condition for the individual plane wave contributions (in the spectral domain). The resulting transformed spatial domain boundary condition will be exact for any piecewise planar multilayered terrain model composed of an arbitrary number of homogenous layers with arbitrary material parameters as long as the dominant portion of the wave field propagates in the actual solution domain above the terrain and certain restrictions with respect to the angular spectrum are fulfilled.

[6] In what follows, we will first put some basic equations and describe the modeling of arbitrary possibly built-up terrain profiles by the piecewise linear and multilayered terrain model. Then, the FSS technique as applied to this terrain model is formulated before numerical implementation issues are addressed. Finally, validation and application results will be presented, and the essential achievements of the paper will be discussed in the conclusions.