Invariants of Spectral Moments and their Degeneracy for a Gaussian Surface

The transformation and symmetry properties of surface derivatives and spectral moments under rotation of the coordinate axes are examined. It is shown that the moments as well as the derivatives are generally represented in terms of their rotational invariants. For a Gaussian surface, which is characterized by additional symmetry of correlation matrices, the even moments higher than the second order are degenerate; i.e., only three of them are independent, and only two invariants are nonzero. Specific properties of spectrum symmetry and the joint statistical distribution of the mean and differential curvatures are found in this case. As an application of these results to observation of the sea surface, a simple optical method is suggested for simultaneous remote-sensing measurement of the second- and fourth-order moments. This method is based on the count of reversely reflected light impulses, arising from sea-surface scanning by the continuous laser radiation.