EOF analysis of a time series with application to tsunami detection

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Fragments of deep-ocean tidal records up to 3 days long belong to the same functional sub-space, regardless of the record's origin. The tidal sub-space basis can be derived via Empirical Orthogonal Function (EOF) analysis of a tidal record of a single buoy. Decomposition of a tsunami buoy record in a functional space of tidal EOFs presents an efficient tool for a short-term tidal forecast, as well as for an accurate tidal removal (Tolkova, E. 2009. Principal Component Analysis of Tsunami Buoy Record: Tide Prediction and Removal. Dyn. Atmos. Oceans, 46 (1-4): 62-82.) EOF analysis of a time series, however, assumes that the time series represents a stationary (in the weak sense) process. In the present work, a modification of one-dimensional EOF formalism not restricted to stationary processes is introduced. With this modification, the EOF-based de-tiding/forecasting technique can be interpreted in terms of a signal passage through a filter bank, which is unique for the sub-space spanned by the EOFs. This interpretation helps to identify a harmonic content of a continuous process whose fragments are decomposed by given EOFs. In particular, seven EOFs and a constant function are proved to decompose 1-day-long tidal fragments at any location. Filtering by projection into a reduced sub-space of the above EOFs is capable of separating a tsunami wave from a tidal component within a few millimeter accuracy from the first minutes of the tsunami appearance on a tsunami buoy record, and is reliable in the presence of data gaps. EOFs with 3-day duration (a reciprocal of either tidal band width) allow short-term (24.75 hour in advance) tidal predictions using the inherent structure of a tidal signal. The predictions do not require any a priori knowledge of tidal processes at a particular location, except for recent 49.5 hour long recordings at the location.