We study the statistical properties of the scattering matrix S(q\k) for the problem of the scattering of light of frequency omega from a randomly rough one-dimensional surface, defined by the equation x(3) = zeta(x(1)), where the surface profile function zeta(x(1)) constitutes a zero-mean, stationary, Gaussian random process. This is done by studying the effects of S(q\k) on the angular intensity correlation function C(q,k\q',k') = (I(q\k)I(q'\k')) -(I(q\k))(I(q'\k')), where the intensity I(q\k) is defined in terms of S(q\k) by I(q\k) = Li-1(-1))(omega/c)\S(q\k)\(2), with L-1 the length of the x(1) axis covered by the random surface. We focus our attention on the C^(1) and C^(10) correlation functions, which are the contributions to C (q, k\q', k') proportional to delta(q - k - q' + k') and delta(q - k + q' - k'), respectively. The existence of both of these correlation functions is consistent with the amplitude of the scattered field obeying complex Gaussian statistics in the limit of a long surface and in the presence of weak surface roughness. We show that the deviation of the statistics of the scattering matrix from complex circular Gaussian statistics and the C-(10) correlation function are determined by exactly the same statistical moment of S(q\k). As the random surface becomes rougher, the amplitude of the scattered field no longer obeys complex Gaussian statistics but obeys complex circular Gaussian statistics instead. In this case the C^(10) correlation function should therefore vanish. This result is confirmed by numerical simulation calculations.