This paper presents wave propagation over a finitely conducting half-space whose surface is bounded by a rough surface of small RMS height. An electric or magnetic current source is used to excite the rough surface, and the Green's function for transverse electric (TE) and transverse magnetic wave propagation is obtained. The appearance of roughness at the boundary produces both a coherent (mean) and incoherent (fluctuating) field distribution, which is obtained from Dyson's equation and Bethe-Salpeter's equation, respectively. The coherent Green's function for vertical polarization exhibits similar characteristics to the Sommerfeld dipole problem where the Zenneck wave pole is modified by roughness. The incoherent field generated by rough surfaces is obtained for both vertical and horizontal polarization, and the conventional cross section per unit length of the rough surface is modified to include the effects of surface roughness. For angles near grazing, a low-grazing-angle cross section is obtained by evaluating the Bethe-Salpeter's equation with the Sommerfeld solution. Finally, the coherent and incoherent intensity for the TE rough surface Green's function is obtained and compared to Monte Carlo simulations.
 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 Gtotal, 〈G〉, and Gf are the total, average, and incoherent Green's functions and ro 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].
 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 , Bass and Fuks , and Barrick [1971a, 1971b] in appropriate limits.
 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.  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.
2. Coherent Green's Function
 In sections 2.1 and 2.2 we develop the formulation for transverse electric (TE) and transverse magnetic (TM) wave propagation over finitely conducting rough surfaces. At the surface the field satisfies the impedance boundary condition where the ratio of the tangential magnetic field to the tangential electric field is the surface impedance. This effectively reduces the two-medium problem to a one-medium problem with the impedance boundary condition. Next, an equivalent boundary condition at z = 0 is constructed by perturbing the Green's function about the rough surface height z = h(x). For both TE and TM propagation the random height contribution to the boundary condition, which is called the random surface potential V, is obtained. Next, by making use of Green's theorem, a random surface integral equation is formulated for the rough surface Green's function. Finally, the coherent field is obtained from Dyson's equation by averaging the random surface integral equation.
2.1. Equivalent Boundary Condition for TE and TM
 Let us consider a line source located at xo, zo in free space above a finite conducting half-space with permittivity ϵ and conductivity σ. This half-space is bounded by a rough surface at z = h(x), where h(x) is a random function of the surface height (Figure 1).
 The Green's function satisfies
at z = h(x), where ko is the free space wave number, and satisfies the impedance boundary condition
where βo = −iZs/(koZo), for TE and
where αo = ikoZs/Zo, for TM.
 The constants Zo = are the free space characteristic impedance, ∂/∂n is the normal derivative, and the surface impedance Zs is approximated by that of the flat conducting surface [Wait, 1998; Bass and Fuks, 1979] given by
where n2 = ϵ + iσ/(wϵo) is the refractive index of the conducting medium. Notice, if near low grazing angle, kx → ko, we can then approximate the surface impedance as Zs ≈ (Zo/n), which for large |n| ≫ 1 reduces to Zs ≈ Zo/n. These expressions describe the total radiation field for a line source in the presence of a half-space rough surface. If the line source is an electric current Ie, then the y component of the electric field (TE) is given by
and the impedance boundary condition (7) holds. However, if the line source is a magnetic current Im, the y component of the magnetic field is
and then the impedance boundary condition (8) holds. The fields must satisfy the boundary conditions (7) and (8) at the rough surface z = h(x). However, we can write an equivalent expression for the boundary condition at z = 0 by writing the Green's function as a perturbation expansion about z = 0 and including only the first-order powers of h(x). We note that
Therefore the equivalent surface impedance boundary condition at z = 0 for the Green's function is now
for TE and
for TM. Note that the surface potential V(x) is a random function of the surface height and, for TE and TM, is given by
From Green's theorem we can now derive the random integral equation for the rough surface Green's function.
 By making use of the surface potentials (equations (14) or (15)) and the flat, half-space Green's function given in spectral domain as
where the reflection coefficients for TE and TM propagation are given by
where r = r(x, z), ro = ro(xo, zo), and r1 = r1(x1, z1 = 0). Go is the flat, half-space Green's function and is a deterministic function, while the surface potential V(r1) and the rough surface Green's function G(r, ro) are random functions. From equation (21) we can generate the higher-order moments describing the propagation, such as the coherent field and incoherent intensity. By using the diagram method we can obtain Dyson's equation for the coherent Green's function [Bass and Fuks, 1979]
where the Mass operator under the first-order smoothing approximation [Frisch, 1968] is given by (r1, r2) = 〈V(r1)Go(r1, r2)V(r2)〉 = (r1 − r2). We note that the correlation function of the random surface potential V(x) is related to the correlation function of the rough surface height h(x). We can express the height correlation as
where we assumed that h(x) is a homogeneous random function and W(κ) is the power spectral density function. In this paper, we use the Gaussian correlation function for h(x) with RMS height ho and correlation distance l.
The Gaussian spectrum is used to verify our analytical results by comparing with numerical Monte Carlo simulations based on the Gaussian spectrum. It should be noted, however, that our results can be used for any spectrum which would be used to represent an actual problem.
2.2. Coherent Field, Sommerfeld Pole, and Zenneck Wave for Conducting Rough Surfaces
 In this section, we solve Dyson's equation [Ishimaru et al., 2000b] to obtain the coherent Green's function 〈G〉. For TM propagation there exists a pole contribution which is used to calculate the Zenneck wave. However, for TE propagation there is no pole. To solve Dyson's equation, we write the coherent Green's function in the spectral domain as
By making use of the flat surface Green's function (16), the coherent Green's function (25), and the height correlation (23), we can obtain from Dyson's equation (22) the TE reflection coefficient
and the TM reflection coefficient,
The effective surface impedances obtained from Q in equations (26)–(29) are special cases of the more general cases discussed by Bass and Fuks . The first thing to note is that the coherent field 〈G〉 behaves in exactly the same manner as the deterministic flat surface Green's function Go. The difference lies in the description of the reflection coefficient. In fact, the coherent Green's function can reduce to the flat surface Green's function by allowing the surface height to go to zero, z → 0, causing Q(κ) → Qo, which reduces to the flat surface. Second, the impedance boundary condition reduces to Dirichlet's condition by allowing βo → 0, Qo = 0, and RoTE = −1
and to Neumanns condition by allowing αo → 0, Qo = 0, and RoTM = 1.
These two cases have also been obtained by Watson and Keller [1983, 1984]. For the TM case we can calculate the effective surface impedance for the coherent field
Noting that Δ = (kz/ko)Qo(κ), and in the limit as αo → 0,
 The coherent Green's function for vertical polarization in the far field may be evaluated using the saddle point asymptotic technique and is given by
However, of interest is when the source and observation points near the surface for vertical polarization. If we consider complex propagation along the surface and evaluate the coherent Green's function using the modified saddle point technique which takes into account the Zenneck pole, we arrive at
represents the direct and image source and
represents scattering from along the surface. Now the numerical distance p is the difference between the total phase for the Zenneck wave and free space
where the propagation constant for the Zenneck wave must be determined from the pole.
By solving Dyson's equation for the rough surface field we achieve a new reflection coefficient, a new Sommerfeld pole, and, finally, the new Zenneck wave different from the flat surface. The final form of the solution is identical to that for the deterministic case, but the Sommerfeld pole has been repositioned because of roughness. In order to calculate the propagation constant for the Zenneck wave we first calculate kz from equations (29) and (37), which we can express as the following:
Notice that for the flat surface case, kz = −koΔ. Therefore the integral in equation (38) represents the rough surface effects. The propagation constant for the Zenneck wave is then obtained by
For source and field points located near the surface, zo ≈ z ≈ 0.0, the rough surface Green's function reduces to
where F(p) is the attenuation function of the field along the surface and is given by
In Figure 2 we plot the attenuation function for varying heights of the source above the rough surface with RMS height (σ = 0.1λ). The insert in the figure shows the behavior of the pole, and the effective impedance in the complex λ plane as the RMS height increases. In general, the behavior of the pole is to become more attenuated.
3. Incoherent Intensity
 Let us now consider the second moment of the field, or the incoherent intensity. For small surface roughness the coherent field will dominate. However, as the roughness increases, or at larger distances from the surface, the coherent field diminishes, and the incoherent intensity becomes dominant. The incoherent intensity is obtained from the first-order Bethe-Salpeter equation, which describes the mutual coherence function (MCF), or the correlation of fields at r and r′ due to the sources located at ro and r′o (Figure 3). The MCF may be written as the sum of coherent and incoherent intensity
where the coherent intensity was determined from section 2.2
and the fluctuating intensity is given by the first iteration of the Bethe-Salpeter equation.
Since the coherent Green's function 〈G〉 has been determined (equation (25)), we can then evaluate the Bethe-Salpeter equation in the far field [Ishimaru et al., 2000a] to determine the incoherent intensity.
where σo is the scattering cross section per unit length of the finitely conducting rough surface and κ and κ1 are the wave numbers corresponding to the vectors r–r1 and r1–ro, respectively. The cross section for the TM case is given by
and for the TE case by
where W = W(κ − κ1) is the power spectral density function and κs = κ − κ1. For the Neumann surface, αo = 0, the cross section reduces to a similar expression of Fuks et al. .
 Of particular interest is the ratio of cross section HH/VV, where HH and VV are horizontal transmit and receive polarization and vertical transmit and receive polarization, respectively, in the back-direction, as the angle of incidence approaches grazing angle. We first consider the perfectly conducting rough surface and Dirchlet's (HH) and Neumann's (VV) boundary conditions. It should be noted that for the first-order small perturbative method (SPM) the ratio of HH/VV predicts very little backscattering.
which as the grazing angle approaches zero θg → 0, kz → 0 and therefore
Also of note is that the ratio HH/VV for SPM is independent of any rough surface parameters (i.e., RMS height σ and correlation length CL) and is only dependent on the incident angle. If we now consider the ratio of HH/VV given in equations (46) and (47), for Dirchlet's and Neumann's condition (βo = 0, αo = 0), the corresponding ratio becomes
where the reflection coefficients are given by equation (30) with
Because of the surface spectrum W(κ − κ′) the ratio of HH/VV will have some rough surface dependence. In Figure 4 we plot the ratio of HH/VV in the back-direction for a perfectly conducting surface and compare SPM to equation (48), for grazing angles ranging from approximately 0.001° to 10°. Shown in the figure, the SPM result is independent of rough surface parameters and goes to zero as grazing angle approaches zero. However, our results predict a finite intensity that is dependent on the RMS height of the rough surface. The results are all based upon the first-order scattering theory.
 We now consider the impedance boundary condition or finite ground. If we evaluate the ratio of HH/VV in the back-direction near grazing angles for SPM [Ishimaru, 1997], then SPM predicts a finite return due to the presence of the finite ground. Also, if we go ahead and calculate the ratio of HH/VV given by equations (46) and (47), near grazing angle, in a straightforward manner, we get
However, let us reconsider the evaluation of the vertical cross section (46), as the angle of incidence approaches grazing angle and the source and observation points near the surface. Normally, we would try to evaluate the vertical cross section for grazing angles, by restricting kx → ko as θi → π/2. However, after careful analysis of equation (46) and generating incoherent MCF (44), two noticeable conflicts can be seen as we approach field points near the surface. First, the evaluation of the cross section from equation (44) was conducted through a far-field approximation. This approximation can only be valid for moderate angles of incidence and becomes inappropriate as θ → π/2. Second, in the evaluation of the coherent Green's function, we restricted the scattering to real propagating modes. However, from the analysis of the coherent Green's function in section 2.2, we see that complex wave propagation becomes unavoidable as we near the surface. Thus, to overcome these obstacles, we evaluate the coherent Green's function for vertical polarization in equation (44), in a similar manner to the surface wave propagation where 〈G〉 = Gp(R1) + Gp(R2) − 2〈P〉. In doing so, we restrict our source and receiver to being near the surface (Figure 1), and thus the modified incoherent Green's function for low grazing angle becomes
and Fa is the attenuation function for the TM case (equation (41)). Noticeably, the only difference in the cross-section expressions lies in the difference between 1 + Q(κ) in the far field and F(R) near the surface. However, closer examination of equation (51) reveals the position dependence of the source and scattering center xc of the rough surface. The scattering center is the location at which the incoherent scattering is localized. By varying it, we can vary the amount of attenuation of the field along the surface. Thus the relationship between the low grazing angle θg (= π/2 − θi), the source height, and the scattering center xc is given by
 In Figure 5 the low-grazing-angle backscattering cross section for vertical polarization is calculated from the attenuation function of Figure 2, for varying source heights as a function of grazing angle θg from 0.5° to 10°. As the source point and observation points near the surface, the grazing angle goes to zero, and the corresponding LGA cross section increases. The LGA cross section is shown for three different source heights above the surface (zo = 0.1λ, 0.25λ, and 0.5λ). Included within Figure 5 are the far-field vertical cross section (46) and the SPM cross section. Beyond a certain source height (zo ≈ 0.75λ) the LGA cross section is invalid, and the far-field cross section (46) should be used instead. In Figure 6 we plot the ratio of HH/VV in the back-direction for SPM compared to our far-field cross-section approximation (49). As can be seen, the SPM produces a finite intensity and once again is independent of the rough surface parameters. Our results show much more deviation because of the rough surface height dependence. In Figure 7 we compare the SPM results to the far-field cross section (49) and the ratio of HH/VV which makes use of the vertical near-field LGA cross section (51) for two rough surface heights σ = 0.5/ko and σ = 1./ko. The source heights for the LGA cross section are chosen to be zo = 0.1λ, 0.25λ, and 0.5λ. The arrows point from the LGA cross section to the far-field cross section of similiar rough surface height cases.
4. Rough Surface Green's Function
 In this section, we add together both coherent and fluctuating fields to construct a picture of the scattering process occuring when a line source excites the rough surface. The rough surface Green's function must be constructed in a second-order sense, because of the second-order nature of the incoherent Green's function. If we consider the intensity, then the rough surface Green's function is given by
where the coherent Green's function intensity was given in section 2 and the fluctuating Green's function is from section 3. Having noticed the complex propagation near the surface and the real propagating modes away from the surface, the Green's function must be constructed in such a manner that the approximation must be determined best suited for a region of interest. For instance, if the source and observation points are near the surface, then the surface wave and LGA cross section must be used for the coherent and incoherent fields, respectively. If the observation and field points are away from the source, then we can simply consider the far-field approximations for both coherent and incoherent fields. Finally, if mixed propagation occurs, where the source is near and the observation point is far, what is the scattering process? Since the incident source is near the surface, the surface wave Green's function must be used to excite the surface. The corresponding cross section, however, is a mixture of complex wave and real propagation. From the earlier analysis the cross section was modified by the attenuation function Fa for field points near the surface and the factor 1 + Q(κ) in the far field. Therefore, since the incident field is near the surface and the scattered field is away from the surface, the corresponding cross section makes use of a |Fa(R1)| for the incident wave and a 1 + Q(κ) for the scattered wave. Therefore this mixed propagation cross section can be given as
In a similar manner for the other propagation path, where the source is far from the surface and the observation is near the surface, the corresponding cross section can be computed. In Figures 8 and 9 we construct the intensity of the Green's function and compare to Monte Carlo simulations for the TE propagation case. The source and observation points were chosen as xo = 0.0λ, zo = 3λ, and x = 10λ, and the observation points z were allowed to vary from 0.0 to 50λ. The rough surface heights chosen were for RMS height σ = 0.25/ko, 0.5/ko, and 0.75/ko with correlation length 2.24/ko. Notice that as the RMS height increases, the incoherent Green's function begins to increase. Thus for a given source and observation position, the rough surface Green's function for small RMS heights may be constructed and compares well with the Monte Carlo simulations. Note that because of a limited number of realizations the simulations have variations, which should diminish as the number of realizations is increased.
 In this paper, we discussed the effects of surface roughness on the Sommerfeld propagation problem for a conducting surface. With a rough surface the field consists of the coherent field and the incoherent field. The technique is based on the modified perturbation method and Dyson's equation. The expressions for the new Sommerfeld pole, Zenneck wave, numerical distance, and propagation factors are obtained, numerical examples are conducted, and the analytical results are compared with Monte Carlo simulations. We then considered the incoherent mutual coherent function and gave a general expression. We also obtained the expression for the scattering cross section per unit length of the rough conducting surface. For large grazing angles we use the far-field cross sections (46) and (47). However, as we approach LGA, we make use of the low-grazing-angle cross section (51) for vertical polarization, in which we can incorporate source position and grazing angle.
 The work reported in this paper is supported by ONR (N00014-97-1-0590 and N00014-98-1-0614) and NSF (ECS 9908849). This paper was presented at NATO/RTO Meeting in April 2000.