The scattering of light from a weakly rough random silver surface characterized by a double rectangular power spectrum is studied by numerical simulations. This power spectrum can support both the enhanced back- and forwardscattering phenomena, which for weakly rough surfaces, are both related to the excitation of surface plasmon polaritons. Here we review these phenomena and present new results from a numerical study of the dependence of the diffuse scattering on the amplitudes ($\gamma_i$, $i=1,2$) of the two rectangular portions of the power spectrum. It is found that there exist an optimal range of ratios, $\gamma_2/\gamma_1$, over which forwardscattering peaks can be observed. By just changing the correlations along the interface, while keeping all other parameters like roughness, polarization and angle of incidence unchanged, the fraction of the incident light that is scattered diffusely can be as large as $16\%$, while for other parameters as small as $1\%$. Moreover, a change in the correlation function only, can result in a $3.5$ times increase in the amount of light that is absorbed at the weakly rough metal interface ($\sigma=10\mbox{nm}$).