Analysis the Scale and Multi-fractal Behavior of Flow Daily Time Series in the Kashkan River

River flow contain fluctuations with very sharp ups and downs that act as unstable and chaos processes. The complexity of the behavior of such flow can be discovered and identified through the standard Hurst exponent and the generalized Hurst exponent or scale exponent. In this study, in order to identify the multi-fractal characteristics of large-scale and complex dynamics of Kashkan river flow, which has very strong fluctuations, multi-fractal detrended fluctuation analysis (MF-DFA) was used. The results of the detrended fluctuation analysis (DFA) show that there is one crossover with a time scale of 348-405 days in the daily signal of Kashkan river flow, which indicates the existence of fractal structure and different behavior of the time series of the river flow in different time scales. The Hurst exponent (h = 2) of small scales was 1.12, which indicates short-term memory and unstable structure of these time scales, while time series with a scale higher than this scale have a relatively stable structure. The strong dependence and gradual decrease of the generalized Hurst exponent on the degrees of fluctuation (q-order RMS) in the range of -5 to 5, on the one hand, shows the complex multi-fractal nature and dynamics, and on the other hand, the nonlinear memory of the Kashkan river flow signal. In addition, the multi-fractal nature and multiple scales of the river flow were confirmed in terms of the nonlinear relationship between mass exponent (tq) and q-order. The large width and asymmetry of the singularity spectrum, while expressing the intensity of the multi-fractal structure and different dynamics in the river flow signal, indicate the weight imbalance of the effect of large and small fluctuations on the river flow signal. The right tail elongation of this spectrum indicates the predominant effect of local fluctuations with small values on the structure of the Kashkan river flow time series.