The statistical analysis of a random, moving surface

The following statistical properties are derived for a random, moving, Gaussian surface: (1) the probability distribution of the surface elevation and of the magnitude and orientation of the gradient; (2) the average number of zero-crossings per unit distance along a line in an arbitrary direction; (3) the average length of the contours per unit area, and the distribution of their direction; (4) the average density of maxima and minima per unit area of the surface, and the average density of specular points (i.e, points where the two components of gradient take given values); (5) the probability distribution of the velocities of zero-crossings along a given line; (6) the probability distribution of the velocities of contours and of specular points; (7) the probability distribution of the envelope and phase angle, and hence (8) when the spectrum is narrow, the probability distribution of the heights of maxima and minima and the distribution of the intervals between successive zero-crossings along an arbitrary line. All the results are expressed in terms of the two-dimensional energy spectrum of the surface, and are found to involve the moments of the spectrum up to a finite order only. (1), (3), (4), (5) and (6) are discussed in detail for the special case of a narrow spectrum. The converse problem is also studied and solved: given certain statistical properties of the surface, to find a convergent sequence of approximations to the energy spectrum. The problems arise in connexion with the statistical analysis of the sea surface. (More detailed summaries are given at the beginning of each part of the paper.)