Multifractal analysis of daily precipitation of selected stations in the west - southwest of Iran

The area of this study, which has covered large parts of the western-southwestern of Iran, has a special topographic and climatic variety. As this area is exposed to geomorphological features such as mountain and plain. In this regard, western and southwestern rainfall systems entering the area, show different reactions to these mid-scale phenomenon (Jahanbakhsh et al; 2020) that such a process has caused the scale behavior and more complex dynamic structure of the rainfall signal in the area. Therefore, on one hand to cover the whole area and on the other hand in order to have long-term daily rainfall statistics, six synotic stations including Khorramabad, Kermanshah, Sanandaj, Dezful, Ahvaz and Abadan stations were selected that have long-term statistics with 1961-2018 as representatives of this area. Also, in order to identify the scale behavior and the dynamics of the structure of the temporal series of rainfall in the western-southwestern of Iran, the fractal and multifractal changed fluctuation analysis method was used (DAF2, MF-DFA2).
 By using fractal-multifractal analysis of receding fluctuations on daily rainfall signal, it was shown that the rain of all the stations has a scale behavior. In this regard, three different scale periods were identified for records. So that, the fitting of the fluctuation function of DFA2 against different scales show that there are two cross over points that separate three different rainy regimes in the fluctuation function of the stations. These two crossover points are based on a temporal scale of 180 (6 months) and 550 days (approximately 2 years); Therefore, there are three different scale periods including small-scale (less than 6 months), mid-scale (from 6 months to 2 years) and large-scale (more than 2 years) in the rainy temporal series of the stations with different stability and dynamic rainy structure at these three temporal periods. Lovejoy and Mandelbrot, 1985; Matsoukas et al., 2000; Gan et al., 2007; Tan and Gan, (2017) claimed that the existence of cross over points in rainy temporal series, are different mechanisms of raining because temporal scales different. The values of scale exponent in these three periods showed that large-scale rainfalls do not follow a specific spatial pattern and show relatively homogeneous behavior. Although, small-scale raining period has a spatial behavior, in the way that the rain of southwestern stations shows more instability and short-term memory than western stations. Also the results of MF-DFA2 showed that these two cross over points are present in all fluctuations, so that different scale periods are also shown in small to large fluctuations and are not limited to medium period fluctuations. The results of MF-DFA2 showed that the generalized Hurst exponent (hq) has been converged with increasing rainy temporal scale, as the difference between the small fluctuations  and large fluctuations , the small-scale temporal series is larger than the large-scale temporal series; Thus, on a small scale, periods with large fluctuations can be clearly distinguished from periods with small fluctuations. Other multifractal properties, including a decreasing hq with increasing the rank of fluctuation (q), nonlinearity of mass signal  in relation to q indicate the multifractal nature and multiple scale behavior and nonlinear memory of the rainy signal of the studied stations (Adresh et al. 2020; Shimizu et al., 2002 ; Bunde et al., 2012; Tan and Gan, 2017).
On one hand, the comparison of the parameters of the singularity spectrum of the stations shows that all the singularity parameters are similar in the area, but have different intensities. In this regard, the singularity spectrum of all stations in the area is asymmetric and has long left tails. Such a tendency in the singularity spectrum indicates the predominant role of large fluctuations in the multifractal structure of the rainy signal (Telesca and Lovallo, 2011). Thus, the shape of the singularity spectrum reveals that the rainy temporal series in the area has such a multifractal structure which is sensitive to local fluctuations with large values (Kalamaras et al., 2017). In this regard, the rainy temporal series in Khorramabad, Kermanshah and Dezful stations were more complex than other temporal series and Abadan and Ahvaz stations showed a very unstable and noisy structure. On the other hand, the extreme rainfall of southwestern stations including Abadan, Ahvaz and Dezful are much more unstable than the western stations and show heavy rainfall. In this regard, although the structure of Sanandaj station rainfall series is highly sensitive to extreme rainfall, but the intensity of its instability rainfall is lower than the limit rainfall of southwestern stations such as Dezful, which are less sensitive to that of Sanandaj. Its scale exponent is equal to 0.67 with the scale exponent of Khorramabad and Kermanshah stations. In general, such results indicate complexities of temporal series s of rainfall that have very strong local fluctuations.