## 1. Introduction

[2] The simulation of radio wave propagation near the Earth's surface is of practical importance for assessing the performance of wireless systems. In order to create a comprehensive channel model, wave diffraction models for different obstacles and discontinuities must be developed. One scenario to consider is that of a coastline, where a transmitting antenna sits atop or near a cliff or bluff overlooking a shoreline or land/sea transition. The effects of both the cliff and the transition on the radio signal, including their mutual interaction, is modeled in this paper. By application of a perturbation technique, the problem of diffraction from a shoreline or land/sea transition is first addressed, including the effect of a gradual impedance transition as opposed to a more abrupt one on the radio signal. The model will then be extended to include the effects of a nearby cliff or bluff by applying standard Uniform Theory of Diffraction (UTD) techniques.

[3] For the problem of diffraction from a land/sea transition, asymptotic techniques such as the Geometrical Theory of Diffraction (GTD) and its extension the Uniform Theory of Diffraction (UTD) have usually been applied. They are only valid however, for abrupt transitions and do not address the more general problem of a gradual land-sea transition. The problem of coastal diffraction was originally examined by *Clemmow* [1953]; however, he did not consider the case of impedance junctions and was not able to derive closed form expressions for the relevant split functions. *Maliuzhinets* [1958] was the first to consider two impedance junctions as a special case of wedge diffraction, and the simpler dual-integral equation method can also be employed [*Senior and Volakis*, 1995]. *Bazer and Karp* [1962] addressed the problem of plane wave diffraction from shorelines in planar land-sea boundaries, using the Wiener-Hopf technique, while *Wait* [1974], *de Jong* [1975], and *Ott* [1992] addressed the diffraction effects caused by an inhomogeneous surface, using an integral equation technique to solve for an attenuation factor. This solution can be shown to be equivalent to a Physical Optics (PO) solution, and is formulated in terms of integrals which must be solved numerically.

[4] When the actual solution of a problem varies only slightly (is perturbed) from a known exact solution, perturbation theory is a viable approach to solve these general problems. In order to analyze a land/sea transition, an analytic solution is applied, in which a one-dimensional (1-D) impedance variation in an infinite impedance surface is characterized as a perturbation from a homogeneous impedance surface, representing the ground plane. The method provides a solution for calculating the diffraction from a surface impedance transition of arbitrary profile, such as a river, shoreline, or trough, when excited by a small dipole of arbitrary orientation. This technique is an extension of a two-dimensional (2-D) model presented by *Sarabandi* [1990] for a resistive sheet when excited by a plane wave. It was shown by *Sarabandi* [1990] that the method could be extended to that of an impedance sheet simply by replacing the resistivity with the complex impedance divided by a factor of two. The technique was extended to include the case of oblique incidence and dipole excitation by *Sarabandi and Casciato* [1999], and this is the formulation applied here.

[5] In order to solve the perturbation problem, an integral equation is first defined on the impedance surface in the Fourier domain, and from this, recursive expressions for the induced surface current of any order are derived. The resulting expressions are analytic and valid for any general one-dimensional impedance transition for which the Fourier transform exists. The limits of the method are the radius of convergence of the perturbation series, and of course the limit of the first-order impedance boundary condition applied [*Sarabandi and Casciato*, 1999; *Senior and Volakis*, 1995]. Asymptotic techniques are then used to solve the field integrals and the resulting expressions are algebraic to first order in the perturbation series.

[6] When a base station, transmitter, or receiver is located on a cliff or bluff near the seashore, the radio signal interacts with both the cliff and seashore. In addition, and depending on the direction of transmission, the surface currents excited by the incident wave in the transition (or cliff) reradiate and excite additional currents in the cliff (or transition) which in turn reradiate. These higher-order effects can have a significant influence on the received signal. In addition line of sight (LOS) transmission may be blocked and the dominant contributor to the received signal is from the fields diffracted off the cliff edge. In order to include the effects of the cliff in the overall model, standard UTD techniques for diffraction from an impedance wedge are applied. These techniques provide a uniform solution through all shadow and reflection boundaries [*Senior and Volakis*, 1995].

[7] While the analytic solution for the impedance transition is a 3-D solution for that of a small dipole radiating in the presence of a 1-D impedance transition, there are currently no asymptotic solutions for the general case of an obliquely incident wave on an impedance wedge with arbitrary wedge angle. Because of this, and for the purpose of this analysis a 2-D UTD solution for an impedance wedge will be applied, and it will be assumed that the source is distant (including secondary sources) from the transition and cliff so that plane wave excitation is assumed, locally.

[8] In the next section the perturbation technique presented by *Sarabandi and Casciato* [1999] is reviewed, and relevant equations are given. The method is then applied to the problem of diffraction from a land/sea transition. An actual land/sea transition is a gradual transition from the impedance of the water to that of the land, and the effects of varying this transition width are analyzed. It is shown that while the effects of the transition on the total fields is significant for observation distances far from the transition, these effects are independent of the gradient of the transition, when both source and observation are near the impedance surface. Following this a coastline will be analyzed by integrating the combined effects of a land/sea transition in proximity to a cliff. A UTD solution for an impedance wedge will be applied to model the cliff edge. Result are then calculated for the case of an aerial vehicle (helicopter, unmanned aerial vehicle (UAV), etc.) flying from the sea, up and over a cliff.

[9] Note that throughout this paper the time convention *e*^{−iωt} is assumed and suppressed. Also the term “scattered fields” implies that all electric field components except the direct are included, while “diffracted fields” does not include either the direct or reflected (Geometrical Optics (GO)) field components.