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Fractional Brownian Motions and their multifractal analysis applied to Parana river flow

A number of different analysis techniques have been used to analyze long-term time series data from different rivers, starting with the determination of the Hurst coefficient. We summarize the concept of fractals, multifractals and Fractional Brownian Motion (FBM), and apply some such techniques to daily stream flow data from the Parana River recorded at Corrientes, Argentina, for 106 years. After determining the Hurst coefficient for the entire data set (H = 0.76), we analyze the data for each of four seasons and draw the corresponding FBM graphs and their multifractal spectra (MFS). Three of the seasons are similar, but autumn is very different for both FBM and MFS. Based on the MFS results, we propose a number of indices for measuring variations in stream flow, and determine the values of the indices for the three similar seasons. The indices are based on important parameters of the multifractal spectra. The geometry of the spectra as well as the indices all indicate that Winter is the most stable season. This is in contrast to the Boxplot of seasonal stream flow data where Winter shows the largest variation. Thus, these indices provide insight into river flow stability, not detected in, and indeed contradictory to, that from basic statistical analysis.