On the estimation of persistence in geophysical time series

The estimation of persistence (or: self-correlation) is necessary to evaluate the effective degree of freedom (or: the number of statistically independent samples) of a geophysical time series. The application of textbook definitions may result in problems when the time series contains periodic signals. This is demonstrated with analytical solutions for a given auto-correlation function. On physical grounds, in this paper we estimate the persistence time as the integral over the absolute value of the auto-correlation function. This procedure has been proposed by Stratonovich and is shown to work for slow and fast oscillations involved. In the practical part of the paper, a 42-year long time series of phase height measurements at Kühlungsborn (54 ^∘N, 12 ^∘E) is studied. In the period band between 2.7 and 27 years, it shows a persistence time of about 12 years and an effective degree of freedom of 3.5. Hence, the negative trend of 310 meters per decade was found to be significant at a level of 86 %. This level could be raised if the time series were longer.