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In this paper we present a statistical analysis of wave characteristics in oceanographic data, using a transformed Gaussian random process for modelling, and to compare theoretical distributions of wave period and amplitude with observations. A natural transform (estimable from the data) is used throughout, the model compared with that based on purely (untransformed) Gaussian assumptions. The data are measurements of a sea state in deep and shallow water, at different geographical locations. One of the purposes of the paper is to use an appropriately sophisticated method to test the correctness of a Gaussian hypothesis in modelling wave data for evaluation of extreme values and for fatigue analysis.