In this work we look at the spectral evolution of waves from the point of view of the total variation (TV) distance. There are several methods for determining changes in the variance of a random process, which correspond to changes in the total energy of the waves. We look instead at changes in the distribution of the energy as given by the energy spectra, after they have been normalized to correspond to a process with unit variance, using the total variation distance. This corresponds to looking at changes in the distribution of energy instead of changes of the total energy present. The TV distance has been successfully used for the comparison of random samples and probability distributions and measures the difference between two probability distributions determining how much one of them has to be modified to coincide with the other. We consider several sets of waves measured at fixed locations over periods of several hours or days and calculate the wave spectrum for periods of 30 minutes. After all the spectra have been normalized the TV distance between all spectra is calculated and used to determine change points in the sequence of normalized energy spectra. We show examples in which although there is considerable change in the significant wave height for the different time intervals, the energy distribution remains essentially unchanged. The method could be used to determine periods of stationarity for sea waves.


The problem of the evolution in time of wave spectra, and in non-stationary stochastic processes in general, is an important and challenging one. In this work we introduce the total variation distance as a new tool that can help the detection and analysis of changes in the energy distribution of the process. This distance has been used for the comparison of probability densities and quantifies the differences between two densities in a scale ranging from 0 to 1.

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