What is the meaning of the random surface realization , $\zeta (x,y)$, in optics? When reading Terahertz Spectroscopy and Imaging, I came across the Kirchhoff Approximation on Rough Surface Scattering Approximations. It is stated that:
"The scattered field from the Kirchhoff Approximation may be solved numerically
for a random surface realization, $\zeta(x, y)$, in (5.17)."
$${\vec{E}}_s^h\left(\vec{r}\right)=\hat{y}\left[\frac{ike^{ikr}}{4\pi r}\right]E_0\int_{A_0}{\left[a\frac{\delta}{\delta x}\zeta\left(x,y\right)-b\right]e^{i\left(\vec{k_i}-\vec{k_s}\right)\cdot\vec{r^\prime}}dx^\prime d y^\prime} \tag{5.17}$$
$A_0$ is the total surface area of dimensions $2L_x\times2L_y$, and the coefficient $a=\left(1-R_{01}^h\right)\sin{\theta_s}$ and $b=\left(1+R_{01}^h\right)\cos{\theta_s-\left(1-R_{01}^h\right)\cos{\theta_i}}$ .
${\vec{E}}_s^h$ is the horizontal component of the scattered field.
A further explanation is given in the paragraph below but it gives no explanation as to what the random surface realisation, $\zeta (x,y)$ is conceptually, or what it means.
"(...) in (5.17) the emitter and detector are
assumed to be in the xz plane. (...) only tangent planes of the surface which are perpendicular to the xz plane are relevant in the integration. Although these
tangent planes have no slope in the y-direction, a two-dimensional integration is still
necessary to account for the width of the planes. Since the local normal of each of
these planes will be parallel to the xz plane, the local polarization of the incident
wave is still in the y-direction and the horizontal reflection coefficient can be used
to compute the $a$ and $b$ coefficients."
 A: The term realization is there used in the sense of the theory of probability. In that context a random surface is a two-dimensional random process $Z$ so that $Z(x,y)$ is a real random variable (a random height) at each point $(x,y)$. A realization is then a real function $\zeta:\mathbb{R}^2\rightarrow \mathbb{R}$ representing a possibly actual surface, one among the infinitely many modelled by the random surface (note that I used a capital letter to denote the random process, the lower case letter to denote the realization).
To make things more concrete, consider the toss of a die. You can model the outcome of this experiment with a random variable $X$ taking values in the set $\{1,2,3,4,5,6\}$. The number $x=3$ is then a realization of that random variable.
For instance, if you need to solve that integral numerically, you can generate samples from a random surface by using a pseudo-random number generator, typically available in any numeric computing software. If you generate many different realizations and evaluate that integral for each of them, you can then estimate average quantities, e.g. the average scattered field or the correlation at different points.
