Analytical and Physical Aspects of Two-Dimensional Spectra Associated with Stationary Randc

The use of spectral analysis techniques for the study of stochastic processes that occur over multidimensional spaces, such as ocean wind-waves or ocean bottom roughness, leads to a large number of possible spectral quantities one can use in a description of the process. The multidimensional case presents difficulties since this variety does not arise in the study of processes that depend on one dimension. We discuss the specific case of a process taking place over two dimensions that applies to the nature of the ocean surface at a particular instant in time or to the nature of the ocean bottom. The several possible spectral quantities of interest are described and related to the basic two-dimensional wavenumber spectrum. A physical interpretation of the spectral quantities is developed by relating them to the distribution of spectral energy in the two-dimensional wavenumber plane. Some specific analytical results are obtained for the important special case of isotropy, particularly with respect to power-law models for the two-dimensional spectrum. By use of the ocean surface wind-wave case as an example, the investigation of anisotropy for stochastic processes must be considered as an issue of equal importance to stationarity of the process.