A novel Fourier split-step algorithm for the solution of the parabolic wave equation is proposed. The derivations are based on a two-dimensional linearly bridged knife-edge terrain model that considers ideally conducting as well as dielectric lossy ground together with a dielectric lossy layer as macroscopic representation of vegetation and buildings. The boundary condition at the terrain interface is formulated in the spectral domain, and it is shown that waves with a very wide angular spectrum with respect to the horizontal can accurately be modeled as long as the dominant portion of energy propagates above the terrain. Validation results are given for an ideally conducting as well as lossy dielectric wedge and an ideally conducting rounded obstacle. Also, real-world propagation curves show good agreement with measured data.
 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.
 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  and Zaporozhets  may favorably be applied to microcellular (transmitter antenna below roofs) propagation situations.
 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  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 . 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.
 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 . 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.
 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.
2.1. Geometric Model and Basic Equations
 In this paper, we restrict ourselves to two-dimensional (2-D) wave propagation. That is, we consider the terrain profile along the great circle path (see Figure 1) from the transmitter site to the receiver position and assume it to be rotationally symmetric around the transmitter. Also, we assume transverse electric (TE) or transverse magnetic (TM) wave propagation only, so that it is sufficient to calculate solutions of the scalar Helmholtz equation (ejωt time convention assumed and suppressed throughout)
representative for the full system of Maxwell's equations (including appropriate constitutive relations). Here, is the wave number in a medium with permittivity ε and permeability μ and Φ stands for the Eφ or Hφ components in a cylindrical coordinate system (r, φ, z) centered at the transmitter site (see Figure 1) in the TE or TM cases, respectively. Using the Δ operator in cylindrical coordinates and setting
 Since we are interested in field predictions relatively far away from the transmitter, the assumption kr ≫ 1 is valid in the frequency range of interest and 1/(4k2r2) in (3) can be neglected. Even in the very first step of the FSS solution to be realized (where kr will have the smallest values), there errors will be very small. Detailed investigations of various wave propagators and their behaviors in this situation are given by Kuttler .
 In what follows, we therefore continue with the Cartesian form
of the 2-D Helmholtz equation, and the field solution in the cylindrical coordinate system is calculated by using (2). To find a solution to the Helmholtz equation, it is usually necessary to specify boundary conditions on a closed boundary around the solution domain, and potential numerical solution techniques require that the entire solution domain be handled at the same time or that appropriate iterative procedures be adopted. Since this is often numerically cumbersome, we seek a strategy that is accurate enough for the wave propagation problem at hand and has the potential to be efficiently implemented in a numerical solution.
 Observing that dominant wave propagation occurs from the transmitter to the receiver and neglecting often insignificant backscattering effects, the back-propagation part of the Helmholtz operator can be removed, if it is possible to isolate it from the corresponding forward propagation part. Such a separation can be achieved if the commutation condition
Both of these operators are parabolic, and the forward propagating waves (with respect to r) are found by solving
which is the desired PWE. To formulate an integral solution of this equation, we utilize Huygens' principle and choose a boundary r′ = const ranging from -∞ to +∞ (in parallel with the real characteristics of this equation) and assume a prescribed excitation field distribution Ψ(r′, z′) in this boundary. Because of the special features of the PWE [Morse and Feshbach, 1953; Sommerfeld, 1949], this equivalent source distribution is sufficient to compute Ψ(r, z) for all r >r′ according to
where g(r, z|r′, z′) is the appropriate Green's function of the considered solution domain. In analogy to the Green's function of the 2-D Helmholtz equation and assuming a homogenous medium can be g(r, z|r′, z′) written as [Kong, 1990]
where H0(2) is the zeroth-order Hankel function of the second kind and With the identity
substituted into (8), rearranging the resulting expression results in
where we have formulated the basis of the FSS algorithm since the inner integral of (12) is basically the Fourier transform of the source distribution Ψ(r′, z′) and the outer integral is the corresponding inverse Fourier integral given that it is evaluated on a line r = const. It should be noted that the branch of the square root in (11) must be chosen to select the waves propagating (or attenuating in the evanescent case) in the +r direction to be in compliance with the considered PWE.
 In section 2.2, it will be shown how this equation must be modified for the computation of forward wave propagation over a piecewise linear terrain model as illustrated in Figure 2. Such a piecewise linear terrain model allows for sufficiently accurate representation of real-world terrain profiles as derived from digital terrain data bases, and it has the great advantage that an exact wide-angle FSS technique can be formulated for it. It is even possible to handle arbitrary dielectric and lossy piecewise multilayered material compositions that can be used to model lossy ground and built-up terrain such as forests and cities, as long as the dominant portions of the waves propagate in the air layer above the built-up terrain. The formulation will be given for a model consisting of linear sections as displayed in Figure 3. The actual section ranges from r′ = const to r = const and consists of three layers: the earth or ocean layer of infinite extent in downward direction (layer 2 in the figure), one layer of finite thickness to consider vegetation or buildings (layer 1 in the figure), and the air layer of infinite extent in the upward direction (layer 0 in the figure). Of course, the model could be extended to an arbitrary number of layers; however, this is usually not necessary for practical terrain propagation problems.
2.2. FSS Technique for Bridged Knife-Edge Model
 In homogenous space, (12) can be used for arbitrary choices of r and r′. Consequently, the unknown field on a line r = const can be found from a given source distribution on a line r′ = const by evaluating the integrals once. However, to adapt the equation to a piecewise linear terrain model as shown in Figure 2, each linearly varying terrain section must be handled separately in order to fulfill the required boundary conditions at the layer interfaces. Thus we consider a situation as illustrated in Figure 3 and compute the fields on the right-hand side boundary line of the section from the known source distribution on the left-hand side boundary by applying Huygens' principle in the form of a Fourier domain integral representation as given by (12). Observing that (12) is equivalent to a plane wave expansion of the field distribution in the solution domain, it is clear that the boundary conditions can be fulfilled for each individual plane wave provided that the multilayered structure is of infinite extent. Because of the use of Huygens' principle the fields outside the solution domain of interest are zero or cannot influence the fields inside the solution domain, and consequently, the multilayered configurations of the individual linear sections can be assumed to be of infinite extent during the evaluation of the field integrals pertinent to the corresponding sections (see Figure 3). To adapt the integral representation (12) to the multilayered situation, each plane wave in (12) is supplemented by an image wave according to
where this expression is valid only in the half-space above the multilayered structure and Ψtilde;inc is given by
Here, it was assumed that the incident wave source distribution Ψinc is zero for z′ < 0. This assumption is only exact for wave propagation over an ideally reflecting interface; however, in the frequency range of interest and for the material compositions present in real-world scenarios, it is almost exactly fulfilled. The image wave numbers in (13) are found from Snellius' law for the oblique layer interfaces (θinc = θim, see Figure 3) and are given by
where α is the oblique layer interface slope angle with the horizontal (see Figure 3). At this point, we may run into problems since we may find image wave contributions propagating in the −r direction and thus being in contradiction to the PWE (7). This problem is due to violation of the commutation condition (5) since it can be shown that this condition does not hold for r dependent material variations as encountered for oblique layer interfaces. Obviously, such a violation of (5) is inherent to all attempts aiming at a realistic representation of irregular terrain. However, in contrast to other techniques, such as those based on transformations of the terrain interface [e.g., see Barrios, 1994; Beilis and Tappert, 1979; Donohue and Kuttler, 2000], we have a clear quantification of this effect, and with an appropriate restriction of the spectral mask of the incident field Ψinc no further approximations are needed. In a first step, the considered source spectrum is confined to propagating waves only, since the applied range step (the length of one linear section) is typically many times larger than one wavelength. By this, the spectral integrals in (13) are restricted from −k to +k. In a second step, the integration range is further reduced by eliminating wave contributrions that would cause back-propagating waves when reflected at an oblique interface with extreme slope angles α = αmax or α = −αmax. By simple geometric considerations the allowed integration range is found to be −k cos (2αmax) to +k cos (2αmax), and thus (13) is written in the form
Finally, the image wave amplitudes a(kz) in (17) need to be determined. Here, two different cases must be distinguished. If the incident wave is propagating toward the layer interface, the amplitude of the image wave is found by multiplying the amplitude of the incident wave with the reflection coefficient at the upper layer interface. Otherwise, the incident wave is propagating away from the layer interface, and thus the amplitude of the image wave is obtained by dividing the amplitude of the incident wave by the reflection coefficient at the upper layer interface. The second case will, of course, not be applicable for vanishing reflection coefficients. However, the occurence of such situations would be in contradiction to our assumption of predominant wave propagation in the air layer above the terrain. Projecting the two different cases onto the kz integration range, the following result is obtained:
in the TM or TE cases, respectively, and Yi = (ωεi)/kni, Zi = (ωμi)/kni. Here, εi is assumed to be complex quantity given by εi = ε0 [εri − jσi/(ωε0)], where ε0 is the permittivity of free space and εri and σi are the relative permittivity and conductivity of layer i, respectively. Also, kni is the wave number component in layer i normal to the layer interface pertaining to an upward propagating or decaying wave.
3. Numerical Implementation
 For the numerical solution of wave propagation over a piecewise linear terrain profile as shown in Figure 2, the integral representations (17) and (14) are recursively used to compute the field distributions in the right-hand side boundaries r = const of the individual linear sections from the corresponding field distributions in the left-hand side boundaries r′ = const. Both of these integrals are of Fourier type and are efficiently evaluated by using fast Fourier transform (FFT) algorithms. An appropriate spatial domain source distribution for the first linear section next to the transmitter is computed such that the intended vertical far-field radiation pattern of the transmitter antenna is obtained [see Balanis, 1982; Barrios, 1994]. On evaluating (14), the spectral domain source distribution of the first section is found, which can be used to compute the right-hand side observation field of the first section by evaluating (17). To use this observation field distribution as source distribution of the next section, the z origin of the coordinate system is shifted by Δz (see Figure 3) to the new location of the upper layer interface at the left-hand side of the second section. Now, (14) and (17) can again be used to compute the right-hand side observation field of the second section, where the origin of the coordinate system is again shifted to the new location of the upper layer interface, and so on. Since the integral in (17) contains image wave contributions, the domain of the regular grid for the discrete evaluation of the Fourier integrals must be chosen twice as big as the considered solution domain above the upper layer interface in order to avoid aliasing errors. By computing the integral in (14), all spatial domain source contributions in z < 0 (mostly due to the image waves) are set to zero, which is similar to the simple knife-edge FSS techniques. At the upper side of the spatial Fourier integtral window, an appropriately designed smooth window function is applied during the evaluation of (14) to minimize parasitic wave reflections. Because of the restriction of the integration range in (17), the proposed technique can only deliver accurate results if a certain maximum slope angle αmax is not exceeded along the profile. Allowing a maximum angle αmax = 22.5° can certainly be considered as an upper limit since it would require to restrict the angular spectrum of propagating waves to be within ±45° with respect to the horizontal. Compared to other wide-angle techniques [e.g., see Donohue and Kuttler, 2000], the width of such an angular spectrum can be considered as very broad. Also, it is noted again that within the allowed spectral range of our technique no approximations were introduced during the derivations (except for the discretization and windowing in the numerical implementation of course), whereas other wide-angle techniques are usually based on expansions of the square-root function.
 If there are only small variations of α along the profile, the lengths of the individual linear sections can be increased as compared to a profile with strong α variations. Thus the proposed bridged knife-edge technique is implemented with adaptive step size regulation, where the largest possible section lengths are restricted by the need to avoid aliasing errors in the considered discrete solution domain window.
 The transmitter-wedge configuration as displayed in Figure 4 was analyzed as a validation problem for the proposed FSS-PWE technique since uniform theory of diffraction (UTD) reference results are available for such a configuration. Also, the same problem was studied by an alternative transformation-based PWE algorithm, as presented by Donohue and Kuttler . For the computations the wedge was assumed to be straight and of infinite extent such that the transformation (2) did not need to be applied. The geometric dimensions are given in Figure 4 and agree with those used by Donohue and Kuttler  except that we performed our computations for a larger wedge height. In Figure 5 the propagation results obtained with our FSS-PWE technique are given for a wedge height of 1000 m equivalent to a slope angle of about 14°. Displayed is the propagation factor along the observation line indicated in Figure 4, which is the field intensity in relation to the corresponding free-space values. The curves in Figure 5 show excellent agreement of the results of our new FSS-PWE technique and the corresponding UTD reference results in the ideally conducting case. In contrast to this, the PWE results presented by Donohue and Kuttler  show already slight deviations from the UTD results for a wedge height of 760 m equivalent to a slope angle of about 11°. The lossy dielectric wedge propagation curves obtained with our PWE algorithm exhibit the expected smaller path losses as compared to the ideally conducting situation. The lossy dielectric TM propagation curves show some numerical noise due to the reduced reflection coefficient close to Brewster's angle. As also given by Donohue and Kuttler , the curves of the simple terrain masking approach are quite off the UTD reference values. In Figure 6, further results are presented for the same transmitter-wedge configuration above an ideally conducting ground plane at height 0 m. Those computations were carried out for an ideally conducting wedge, and the results are compared to UTD propagation data as well as the corresponding propagation factors without ground plane. The UTD propagation factors were obtained by collecting four ray paths from the transmitter to the receiver (directly diffracted, diffracted-reflected, reflected-diffracted, and reflected-diffracted-reflected), and the results are certainly not very accurate near the junction of the wedge and the ground plane. However, slightly away from this junction, the agreement between PWE results and UTD results is excellent, and the curves exhibit the expected interference pattern due to the superposition of the different ray contributions.
 A more challenging problem is diffraction by a rounded obstacle as illustrated in Figure 7. Reference results for this problem obtained by a contour integration technique are given by Vogler . In Figure 8, propagation curves for an ideally conducting obstacle obtained with our FSS-PWE technique are compared to the corresponding reference results from Vogler . The displayed modified attenuation as well as the parameters η and ζ are defined by Vogler . Here η basically stands for the curvature of the obstacle, and ζ is mostly dependent on the observation height. Also, the modified attenuation is almost identical to the propagation factor discussed above. Our PWE results show almost perfect agreement with the reference values for modified attenuations up to about 60 dB. For even larger modified attenuations the accuracy deteriorates from numerical noise generated in the PWE algorithm.
 The third investigated configuration is the Hjorring path profile as illustrated in Figures 9 and 10. For this profile, propagation measurements were performed by the University of Aalborg, Denmark, and provided within the Euroean Cooperation in the Field of Scientific and Technical Research (COST) initiative. A detailed description of the measurements and the profile is given by Hviid et al.  together with reference simulation results obtained by an integral equation (IE) technique. The transmitter antenna was located 10.4 m above ground, and the receiver antenna height was 2.4 m. In Figures 9 and 10, path loss simulation results (not including any antenna characteristics) of our PWE technique for bridged knife-edges are compared with the corresponding IE simulation results as well as the measured data. The central processing unit (CPU) time for computing these results with a Pentium II 400 MHz computer was 2.5 s and 14 s for 144 MHz and 1900 MHz, respectively. The larger CPU time for higher frequencies is mainly due to the larger number of FFT samples for smaller wavelengths. The sample distance of the terrain profile used within the PWE simulations was 50 m, and simulations were performed for ideally conducting ground as well as ground material parameters εr = 6 and σ = 6 mS. In the latter case, the simulations were performed for TM waves according to the vertical polarization of the transmitting and receiving antennas used for the measurements. For the lowest frequency (f = 144 MHz, see Figure 9) the agreement between simulation (both IE according to Hviid et al.  and PWE) and measurements is excellent, whereas the prediction errors increase with frequency (see also the statistics of the prediction errors in Figure 11). However, this behavior is to be expected because of the neglected influence of buildings and trees on the profile (see description given by Hviid et al. ). Interesting to note is that our PWE simulation results are slightly closer to the measured results than those of the IE simulations. Also, the statistical evaluation of the prediction errors displayed in Figure 11 reveals slightly smaller standard deviations of the PWE simulation results carried out with real-world ground material parameters. However, the improvements compared to metallic ground computations are rather small, which is due to the dominant near-grazing propagation, where the reflection coeffecients according to (19) are close to −1. Further results obtained with our PWE-FSS algorithm are given in Figure 12. The results show that considerable improvement of the predictions can be achieved by considering vegetation and buildings by a lossy dielectric layer above the ground along parts of the profile. Unfortunately, no exact data about vegetation and buildings along the profile were available, and thus the results may be viewed as being somewhat manipulated. Nevertheless, the mean prediction error was reduced to about 0.8 dB with a standard deviation of about 5.5 dB.
 In this paper, a Fourier split-step (FSS) algorithm was formulated to solve the parabolic wave equation (PWE) for a two-dimensional (2-D) linearly bridged knife-edge terrain model. The boundary condition in the spectral domain was derived for the individual linear terrain model sections by applying Snell's law and including the reflection coefficients at the possibly multilayered planar terrain sections. In contrast to other PWE techniques, the formulation of the irregular terrain boundary condition does not require further approximations once the terrain model is defined and a clear criterion for the allowed angular spectrum range indicates when it starts to fail. The validation results show that the algorithm is very robust for a wide range of application problems. In the dielectric TM case the algorithm will fail when the wave propagation direction is close to Brewster's angle since the reflection coefficient vanishes and the dominant portion of the wave energy is no longer propagating above the terrain as assumed in the derivations. Also, numerical limitations were encountered in the deep shadow region when the diffraction attenuation was larger than about 60 dB.
 At the present stage, the discussed wave propagation model can be used to study propagation phenomena over arbitrary but open terrain and, for instance, evaluate the results in order to improve the quality of so-called semiempirical propagation models widely employed in coverage predictions. The proposed vegetation and buildings model including a homogeneous lossy dielectric layer above the terrain can only provide a macroscopic representation of the corresponding propagation effects since essential details are neglected. However, ways can be thought of to combine the FSS field representation with more detailed propagation models of built-up terrain.