Statistics of two dimensional wave groups, of steep wave events, and of a cascade pattern manifested in the surface geometry in a developed sea state are derived. However, mathematical theories used to parameterize these as well as many other features of random surfaces have very limited ranges of validity. For example, high-order moments of wave spectra appearing in the calculations of wave slope statistics cannot be evaluated because of divergence of the corresponding intergrals. In the present paper the restrictions are reviewed and the difficulties are shown to be due to a pseudo- fractal geometry of the sea surface whose spectrum is known only within a limited range of frequency (characterized by either the resolution of a measuring technique or the constraints of a theoretical model). An approach is presented that solves the problem: treating the surface elevation field as specified on a spatial (temporal) running grid, an averaging procedure is developed employing the Taylor microscale as the mesh size. The technique is illustrated by first exposing errors in direct calculations of the effective surface impedance for a coherently reflected L band radio wave. The errors arise from the use of wave spectra whose high-frequency tail is identified with the Phillips saturation range. The technique is then employed in the study of wave groups and steep waves for a Gaussian, two-dimensional, time-varying surface. In particular, it is found that wave groups are not observable in a developed sea. Finally, the theory is applied to estimating breaking wave statistics. A comparison with field observations is presented.