The joint statistics of averaged sample auto- and cross-spectra from a heave-pitch-roll time series is studied. It is found that the corresponding joint probability distribution belongs to the class of Wishart distributions W3. Using this result, a maximum likelihood scheme is proposed for the estimation of target heavepitch- roll auto- and cross-spectra, from given averaged sample spectra, taking into account the statistical variability of the latter as well as the energy constraint between auto-spectra. These maximum likelihood estimates are then used for retrieving the directional wave spectrum; a parametric model is used for the latter, permitting to represent unimodal, bimodal or skewed spectral distributions, per frequency band. It is numerically confirmed that the use of the maximum likelihood estimates, instead of the corresponding averaged sample spectra, improves the estimate of the directional wave spectrum.
The estimation of directional wave spectra from a heave-pitchroll (h-p-r) time series, first proposed by Svesnikov (l959) and Longuet-Higgins et al. (1963), and extensively used since then (Allender et aI., 1989), presents essential difficulties because of the limited information available. Accordingly, it is of fundamental importance both to take into account the underlying physics of the phenomenon and to use estimation techniques that exploit in an optimal way all information contained in the time series recorded. The auto-spectra and cross-spectra, associated to the heave, pitch, and roll stochastic process (called in the sequel h-p-r target spectra) are deterministic functions related to the directional wave spectrum through well-known integral relations. Further, the three h-p-r target auto-spectra are linearly dependent (they satisfy the energy constraint (19)). On the other hand, the commonly used heave-pitch-roll sample auto- and cross-spectra (called in the sequel h-p-r sample spectra), even after averaging and/or smoothing, suffer from statistical variability and, in general, they do not fulfil the energy constraint.
