There are four data-based prediction methods which have been implemented in Prognocean+. They all have a common deterministic model, which is known as the polynomial-harmonic model (PH). The PH technique is based on fitting a simple function – which is a sum of a linear term and a series of harmonic components with periods of 62, 90, 120, 182, 365 days – to Sea Level Anomaly (SLA) data in a gridwise fashion. The PH models are fitted every day to 1461 SLA data extracted from a forward-moving window of length 4 years, and such an approach accounts for possible uneven amplitudes and phases of the sea level variability. Subsequently, after a successful fit, the current PH model is utilized for: (1) extrapolation which produces the simplest prediction of Prognocean+ known as DET_PH, (2) calculation of residuals which are used for stochastic modelling.

The residuals from the PH model become inputs to AutoRegressive (AR), Treshold AutoRegressive (TAR) and Mutivariate AutoRegressive (MAR) time series models. The stochastic models are fitted every day to the most irregular part of the SLA residual signal, corresponding to so called Near-Real Time (NRT), is analyzed. The AR and TAR models are fitted to such residuals in a gridwise fashion, however the MAR model uses a spatial window of size 0.75° × 1.25° to account for cross-correlations between the SLA time series at the adjacent grids. The models are described in the paper by Niedzielski and Miziński (2013), and the detailed discussion on skills of AR- and MAR-based approaches to predict gridded SLA data can be found in the article by Niedzielski and Kosek (2009). The fitted AR, TAR and MAR models are subsequently used to produce predictions of irregular SLA terms for every grid, and such prognoses are arithmetically added to the DET_PH forecast so that the following Prognocean+ prediction approaches are computed:

- DET_PH combined with the AR-based prediction of residuals (DET_PH_STOCH_AR),
- DET_PH combined with the TAR-based prediction of residuals (DET_PH_STOCH_TAR),
- DET_PH combined with the MAR-based prediction of residuals (DET_PH_STOCH_MAR).

The above-mentioned four prediction approaches lead to the real-time computation of main products of Prognocean+, namely forecasted 0.25° × 0.25° SLA maps for lead times ranging from 1 to 14 days. Along with these prognoses, their Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) are calculated in real time for every grid so that the user may analyze the spatial distribution of prediction uncertainty. Associated with error analysis are graphical illustrations of prediction skills which are available, along with all aforementioned products, in this Web Map Service (WMS) which is equipped with several interactive tools. If graphical presentation is insufficient, the raw data (predictions and their statistics) are available in the Web Coverage Service (WCS) and can be downloaded using the Internet browser.

**References**

Niedzielski T., Kosek W., 2009. Forecasting sea level anomalies from TOPEX/Poseidon and Jason-1 satellite altimetry. Journal of Geodesy 83, 469–476.

Niedzielski T., Miziński B., 2013. Automated system for near-real time modelling and prediction of altimeter-derived sea level anomalies. Computers & Geosciences 58, 29–39.

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