## 1. Introduction

[2] Wave propagation over a flat conducting surface excited by a dipole is a classic electromagnetic problem and has been studied by *Wait* [1998, 1962] and many others. In this paper, we extend the problem to include a finitely conducting medium bounded by a rough surface of small RMS height (*k*σ < 1.0). Over the years, many investigators have considered this specific problem. Radio wave propagation over a rough surface was first studied by *Feinberg* [1944], who obtained an effective impedance at the interface. Barrick conducted extensive studies on HF/VHF propagation over rough seas [*Barrick*, 1971a, 1971b] and showed that the spherical Earth residue series model should be used for MF-VHF propagation over a rough sea. This was also shown rigorously by *Wait* [1971]. The effective impedance of a rough surface has been extensively studied by *Bryukhovetskii and Fuks* [1985], *Bass and Fuks* [1979], and *Bryukhovetskii et al.* [1984] using an extension of the small perturbation theory and the diagram method [*Tatarskii*, 1967; *Rytov et al.*, 1987]. The diagram method has been applied to the rough surface scattering problem, and the basic equations have been developed and solved for Dirichlet and Neumann surfaces and irregular waveguides [*Freilikher and Fuks*, 1970; *Bass et al.*, 1974]. Scattering by random impedance and the backscattering enhancement have been discussed using a different approach including the pole and the grazing angle considerations [*Freilikher and Yurkevich*, 1993; *Freilikher and Fuks*, 1976]. Multiple scattering theories for rough surface scattering have also been proposed by *Watson and Keller* [1983, 1984], *Ito* [1985], and *Ishimaru et al.* [2000a]. Further studies have been conducted recently for low-grazing-angle (LGA) scattering [*Brown*, 1998; *Barrick*, 1998, 1995; *Fuks et al.*, 1999]. This paper follows and extends the multiple-scattering theories developed by *Bass and Fuks* [1979], *Watson and Keller* [1983, 1984], *Ito* [1985], and *Ishimaru et al.* [2000a]. We make use of the diagram and the first-order modified perturbation method [*Tatarskii*, 1967; *Rytov et al.*, 1987; *Frisch*, 1968] to obtain an expression for the Green's function for both vertical and horizontal polarizations.

[3] To properly consider this problem, an electric or magnetic current source excites the rough surface and can be located very near or far away from the rough surface. These two current sources will provide both vertical and horizontal polarization for wave propagation over the rough surface. When the source is located far above the surface, it is sufficient to consider a plane wave or beam wave incident on the surface resulting in coherent (average) and incoherent (fluctuating) scattered fields radiating away from the surface.

where *G*_{total}, 〈*G*〉, and *G*_{f} are the total, average, and incoherent Green's functions and *r*_{o} and *r* are the source and observation points. If the surface is a flat conducting surface, the incoherent field will be absent. However, for increasing roughness, the coherent field diminishes, and the incoherent field will be generated and become dominant. For a source and/or observation point located near the surface it can no longer be assumed that a real, propagating field is incident upon the surface. Instead, complex wave propagation involving both coherent and incoherent fields will excite the surface. The resulting scattered field includes not only coherent and incoherent radiation but also possibilities of complex wave propagation along the surface. To properly consider the Green's function near the surface, we allow the source and observation point to be near the surface and consider the wave propagation along the surface in a similar manner as the Sommerfeld-Zenneck wave solution [*Ishimaru*, 1991].

[4] The coherent field 〈*G*〉 for both vertical and horizontal polarization is obtained from Dyson's equations. The expression for the coherent Green's function is obtained in the spatial Fourier representation similar to the Zenneck-Sommerfeld solution. For transverse magnetic (TM) propagation the modified reflection coefficient produces a Sommerfeld pole whose location is perturbed by the roughness. The Zenneck wave pole, effective surface impedance, and attenuation function for a rough conducting surface are also obtained. The effective surface impedance is consistent with those obtained by *Feinberg* [1944], *Bass and Fuks* [1979], and *Barrick* [1971a, 1971b] in appropriate limits.

[5] To obtain the fluctuating Green's function, we must consider the second moment of the field, called the mutual coherence function.

Noting equation (1), the mutual coherence function is given by

where the coherent mutual coherence function is given by

which is determined from the coherent Green's function. The fluctuating or incoherent Green's function is given by

To obtain an expression for the second moment Γ, we solve the first-order Bethe-Salpeter equation under the smoothing approximation [*Wait*, 1971]. The incoherent Green's function is excited by the propagating coherent field and accumulates fluctuations from all scattering points over the surface. If we evaluate the incoherent field in the far field, the scattering cross sections are shown to be similar to those of *Watson and Keller* [1983, 1984] and consistent with those of *Fuks et al.* [1999] in the Neumann surface limit. Numerical Monte Carlo simulations were conducted to compare with the bistatic cross section for both vertical and horizontal polarizations. For source and/or field points near the surface the complex wave propagation near the surface cannot be ignored. To compensate for the field near the surface, the Bethe-Salpeter's equation is evaluated along the surface similar to the Sommerfeld solution. A corresponding cross section near the surface is obtained, includes the Sommerfeld attenuation function, and is shown to be dependent on the source location and incident grazing angle. The paper is divided as follows. In section 2 we consider the coherent Green's function and obtain expressions for both vertical and horizontal polarizations. The coherent Sommerfeld-Zenneck propagation along the surface is described for vertical polarization. In section 3 the incoherent Green's function is described. First, the far-field cross sections are obtained from Bethe-Salpeter's equation. Second, incoherent propagation along the surface is identified, and a corrected cross section near the surface is obtained. Finally, in section 4, numerical Monte Carlo simulations are conducted and compared to the total intensity of the field.