[16] Here, we perform statistical averaging of backscattered field intensities (12) for the particular case of a Gaussian random rough surface. The integrand in (12), being expanded in accordance with (13)–(15), contains three random functions: *Z*_{1}(*x*), *Z*_{2}(*x*) and *Z*_{3}(*x*). Note that for a statistically uniform (homogeneous) rough surface, *Z*_{2}(*x*) does not correlate either with *Z*_{1}(*x*) or *Z*_{3}(*x*) at the same spatial point *x*. For Gaussian statistics, it results in the independence of *Z*_{2} from *Z*_{1} and *Z*_{3}, and therefore averaging over *Z*_{2} can be easily performed. While averaging, only the two first statistical moments of *Z*_{2} appear: 〈*Z*_{2}〉 = 0 and 〈*Z*_{2}^{2}〉 = *σ*_{2}^{2}. Then, it is possible to perform averaging of *Z*_{3} under the condition that *Z*_{1} is given:

where *σ*_{1}^{2} = 〈*Z*_{1}^{2}〉 is the surface slope variance. The next step is integration over *dξ*, which can be performed in the infinite limits if the inequality holds:

It results in the appearance of the *δ*-function and its derivatives *δ*^{(n)}:

The final averaging over the random slopes *Z*_{1} by integration of the obtained equations with a probability density function (PDF)

is possible to perform in the explicit form, using the *δ*^{(n)} -function property:

The last integration over *dx* results simply in multiplying by factor *L*, because the final result of statistical averaging does not depend on the spatial coordinate *x* due to the statistical homogeneity of surface roughness. The specific (per unit length in the *Ox* direction) backscattering cross sections are introduced by the following definitions:

In problems related to radio wave propagation and scattering, *σ*_{HH}^{0} corresponds to a horizontal-horizontal backscattering cross section, and *σ*_{VV}^{0} – to a “vertical-vertical” one. They can be written in the form:

where *σ*_{0} is the backscattering cross section in the GO limit:

It is worth mentioning that (25) is valid for an arbitrary PDF of slopes *w*_{1}, while the diffraction corrections in the forms (23) and (24) were obtained only under the assumption regarding the Gaussian statistics for the surface roughness. It this case, Δ_{HH} and Δ_{VV} are proportional to the second derivative variance *σ*_{2}^{2} and depend, additionally, on the surface roughness slope variance *σ*_{1}^{2}. Note that diffraction corrections (of the same order ∼1/*k*^{2}) to the backscattering cross sections from the solitary specular point at the 2-D smooth surface depend on the first six (!) derivatives of the directrix *Z*(*x*) [*Fuks*, 2004, 2005a, 2005b, 2006].

[17] In the GO limit, there is no difference between backscattering cross sections at *HH* and *VV* polarizations, so the polarization ratio *R*_{HV} = *σ*_{HH}/*σ*_{VV} is equal to unity. From (22)–(26), we obtain the diffraction correction ∼1/*k*^{2} to the *HH*/*VV* polarization ratio:

The last equation differs from the analogous equation (63) in *Fuks* [2004], where the polarization ratio was introduced as averaged ratio (48) of backscattering cross sections from the solitary specular point, whereas here, by the polarization ratio we mean the ratio of statistically averaged backscattering cross sections for *HH* and *VV* polarizations. As was mentioned above, there is no direct and simple relation between the scattering cross sections from the solitary specular point and the specific backscattering cross section from the extended statistically rough surface. Besides, in *Fuks* [2004] the statistical averaging of polarization ratio (48) for a solitary specular point was not performed in full, because of the divergency of the corresponding integrals, and only the result of the partial averaging of (48) was produced there (equation (67) in *Fuks* [2004]).