A novel multi-fractal quantifying method with wider scaling range: Extended self-similarity based multi-fractal detrended fluctuation analysis

Here, a novel method, the extended self-similarity multi-fractal detrended fluctuation analysis (ESS-MF-DFA), is proposed. The method is based on MF-DFA method and modified by incorporating extended self-similarity (ESS). We illustrate it through two numerical tests and observed boundary-layer wind speed data: Compared with traditional multi-fractal detrended fluctuation analysis (MF-DFA) method, ESS-MF-DFA method can significantly extend scaling range (defined in detrended fluctuation analysis, DFA) and reduce uncertainties when estimating the exponents. ESS-MF-DFA can still works well even when DFA fails due to no scaling range between fluctuation function and scale. Furthermore, a criterion is developed based on ESS-MF-DFA to distinguish multi-fractal from mono-fractal behavior and to quantify multi-fractal strength. We believe that ESS-MF-DFA outperforms MF-DFA in reliably handling multi-fractal quantifications in for much wider fields.