Case Study I: Caspian Sea Level

Abstract

The case of the Caspian Sea level time series demonstrates that both the long range dependence and some secular long term trend may exist together in geophysical phenomena. Even after removing the long term trend from the Caspian Sea level time series, the residual time series still demonstrate long range dependent behavior. Sample autocorrelation functions (ACFs) and periodograms of the sea level data are investigated and the Hurst coefficients are estimated for various time intervals. Forecasting performance of linear stationary (autoregressive moving average, ARMA), linear nonstationary (autoregressive integrated moving average, ARIMA), long-range memory (autoregressive fractionally integrated moving average, ARFIMA) and Trend Line-ARFIMA (TL-ARFIMA) combination models are investigated by comparing the forecasts with the observed Caspian Sea levels. Forecasts and their confidence bands, estimated by the ARFIMA and TL-ARFIMA models, are compared with the forecasts of the AOGCMs reported in the literature. In this study, the forecast confidence bands and the forecast updating methodology, provided for ARIMA models in the literature, are modified for the ARFIMA models. Sample ACFs are utilized to estimate the differencing lengths of the ARFIMA models. The confidence bands of the forecasts are estimated using the probability density functions of the residuals without assuming a known distribution.