A fractal is a natural phenomenon or a mathematical set that has a repeating pattern at every scale, featured with self-similarity and scale-invariance. Fractal theory was established by B. Mandelbrot in the 1970s. It has been applied to many areas, including philosophy, mathematics, chemistry, physics, economics, geology, seismology, geography, music, and art (Liu et al., 2006). Fractal theory has been applied to hydrology and water resources, such as the fractal of morphological characteristics of watershed systems, the longitudinal channel profile, and flood forecasting and flood disaster prediction (S. Zhang et al., 2009). J. Zhang et al. (2009) also have applied fractal theory to developing flood sub-seasons.

The current study of fractal is based on the qualitative understanding of the examined object's self-similarity. Whether the shapes measured by ε belong to the same fractal depends on whether the fractal dimension is fixed.

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. As for ordinary geometric shapes, points are 0-dimensional sets, lines are 1-dimensional sets which only have length, surfaces are 2-dimensional sets which have length and width, and cubes are 3-dimensional sets which have length, width and height. For complicated geometric forms whose details seem more important than the gross picture, fractal dimensions are applied as an index describing their complexity while the conventional Euclidean or topological dimension shows its limitation. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry (Mandelbrot, 2004). Unlike topological dimensions, the fractal index can take non-integer values (Sharifi-Viand et al., 2012). Multiple algorithms for calculating fractal dimension exist in fractal theory. The Hausdorff dimension, also called gauge dimension, is the most basic. Others include information dimension, correlation dimension, spectral dimension, distribution dimension and Lyapunov dimension, etc. The box-counting dimension (or Minkowski dimension) is used in this paper.

Using a ruler of length ε to measure a line segment of length L, N(ε) as the ratio of L to ε can be obtained. Similarly, using cubes with side length ε to fill an object, N(ε) is the number of cubes required to cover the object. The fractal dimension obtained in this way is called box-counting dimension Dc (Zhu et al., 2011), and is defined as: Dc=lim⁡n→∞ln⁡N(ε)/ln⁡(1/ε) When ε approaches 0, it becomes: ln⁡N(ε)≈-Dcln⁡ε=Dcln⁡(1/ε) Where ε – the scale at which the fractal is measured, Dc – the box-counting dimension, and N(ε) – the covering number.

If there is a straight part (clear correlation) on the ln⁡NN(ε)-ln⁡(ε) graph with linear fitting, the sequence can be conceived as a fractal. The slope of the straight part Dc is the fractal dimension. Smally (1987) introduced a new variable (NN) when computing the fractal dimension of the earthquake spectrum series of New Hebrides, namely the relative measurement: NN(ε)=N(ε)/NT Where N(ε) – absolute measurement, NT – total number of time intervals, T – total time length, ε – step length.

A fractal problem depends on the existence of a straight part (scale-invariant area) on the ln⁡NN(ε)-ln⁡(ε) curve (Dong et al., 2007; Mandelbrot, 1983; Song et al., 2002; Ding et al., 1999). If the slope of the straight part in the scale-invariant area is b, the capacity dimension can be given by the following equation: Dc=d-b Where d – topological dimension. Points of flood peaks distribute on a Q∼t two-dimensional surface, so d equals to 2, and then: Dc=2-b

Compared with single normal distribution, the Von Mises distribution is a continuous probability distribution on a circle. This model primarily describes directional statistics. It is important in areas like astronomy, biology, geography, medicine, etc. For example, He et al. (2011) applied the Von Mises yield criterion in the study of materials in plastic state in physics; Zheng et al. (2011) applied the Von Mises distribution model of monthly premium to analyze the seasonal fluctuation of the premium in medical science. The Von Mises distribution is also applied in hydrologic events. Fang et al. (2008) employed the Von Mises function to fit the time distribution of annual maximum flood and have established a two-variable joint distribution of annual maximum flood.

The probability density curve of the Von Mises distribution is unimodal. However, the probability density curve of the time of the occurrence of floods in flood season also can be multimodal in practical calculation (Yue et al., 1999). Therefore, fitting result and actual measurement may differ when the Von Mises function is used to fit the probability distribution of flood timing. Replacing the Von Mises distribution with mixed Von Mises distribution achieved well-fitted result (Chen et al., 2010).

Assuming flood date t is normally distributed, its probability density function is (Fang et al., 2008): ft=12πI0(k)exp⁡kcos⁡t-u0≤t≤2π,0≤u≤2π,k>0 Where u – measure of location, k – measure of concentration, I0(k) – modified Bessel function of order 0. Assuming there are L days during flood season, N – number of flood samples, Di – time of the occurrence of sample i, and: a=∑i=1Ncos⁡xi/Nb=∑i=1Nsin⁡xi/N where xi=Di2πL is the time of the occurrence of sample i (in radians), 0≺xi≺2π.

Then u and k can be given by: u=arctan⁡b/aa>0,b>02π+arctan⁡b/aa>0,b<0π+arctan⁡b/aa<0π/2a=0,b>03π/2a=0,b<0indeterminatea=0,b=0u=r=a2+b20≤r≤1 In this study, the probability density function of t is given by: ftt=∑i=1npi2πI0(ki)exp[kicos(t-ui)]0≤t≤2π,0≤ui≤2π,k>0 Where n is the number of Von Mises distributions, pi is the mixing percentage, and their optimal values that produce the best fitting result can be obtained with the Quasi-Newton method (Li et al., 1997).

The flood season of Hongfeng reservoir is from 1 May to 30 September (lasting for 153 days). This study uses the historical hydrology record lasting 43 years.

Table 1 shows that the largest flood within a year appears in the first ten day period of August, until the frequency of the largest inflow is 90.698 %, while the second and the third largest flood occur in the last ten day period of August and the first ten day period of September, until which the frequencies of the second and the third largest inflow are 93.023 and 90.698 % respectively during the whole flood record. In terms of the multi-year average and largest inflow in a ten day period, the late July and the early August were at a low point as well as late August and early September. Therefore, the flood season of Hongfeng reservoir can be separated into three sub-seasons based on the analysis of its changing flood pattern and safety requirement. The pre-rainy season is from 1 May to 31 July, the middle flood season is from 1 August to 31 August, and the late flood season is from 1 September to 30 September.

Earlier researches only sampled the sequence of the largest daily inflows, while this paper also accounts for the second and the third largest daily inflows. Distributions of the three largest daily inflows are shown in Fig. 2.

Figure 2 shows large gaps between the ten day period inflows of May and June, July and August, and August and September. So the flood season can be divided into four sub-seasons. Time scale ε is 1, 2, 3 ... 7, or 8 d. By setting a fixed value Y1=235 m3s-1 (slightly larger than the sample average inflow), N(ε) can be obtained under different time scales by counting the number of time intervals in which the average inflows exceed Y1 (Fig. 3(1)). The ln⁡NN(ε)-ln⁡(ε) graph can be plotted to determine slope b of the straight part and then obtain the box-counting dimension Dc (Dc=2-b). Different ln⁡NN(ε)-ln⁡(ε) graphs can be plotted based on different values of T when changing the ending date of the first sub-season. Calculation of the latter three sub-seasons is similar to the first sub-season, and the average inflows are as Y2=540 m3s-1 (Fig. 3(2)), Y3=265 m3s-1 (Fig. 3(3)), Y4=235 m3s-1 (Fig. 3(4)) respectively. The ln⁡NN(ε)-ln⁡(ε) graphs under different values of T of the four sub-seasons are shown in Fig. 3.

Shi et al. (2010) suggest that the significant linear relation between ln⁡NN(ε) and ln⁡(ε) is inversely proportional to the length of the time scale ε and thus should not exceed 6. This case achieves the best result when ε is 8. The calculated box-counting dimensions of the four sub-seasons are shown in Table 2. From Table 2, the box-counting dimensions of situation A and situation B have a slight difference of 0.01 in the pre-rainy season, while situation C largely differs. According to the principle that the box-counting dimensions in the same sub-season should have similar magnitudes while successive sub-seasons do not, A and B should belong to the same sub-season. So it can be concluded that the pre-rainy season is from 1 May to 31 May. Similarly, the box-counting dimensions of situation D, E and F are close with a relative difference less than 4 % in the main flood season, while G is rather different. So the main flood season is from 1 June to 31 July. In the late-flood season I, there is a discontinuous part due to the comparatively large difference between the box-counting dimensions of situation I and situations H and J. So situation H is regarded as one sub-season and the late-flood season I is from 1 August to 31 August. Under such circumstance, the late-flood season in the conventional sense is divided into two sub-seasons, including the late-flood season I and the late-flood season II. In the late-flood season II, situation M is counted out because October is not included in the flood season. It can only be concluded that the late-flood season II is from 1 September to 20 September, and the remaining ten days until 30 September should be regarded as another sub-season if the fractal principle is strictly followed. However, to make it convenient for reservoir management and operation, the late-flood season II should be from 1 September to 30 September.

The above separation was based on the sequence of the largest daily inflows. The separation results based on the sequences of the second and third largest daily inflows are similar, which proves that taking sequence of only the largest daily inflows as research sample is reasonable for separation.

Due to the scarce inflow records of Hongfeng reservoir, more reasonable flood peak records were adopted as samples to accurately trace changes in floods to make the distribution model more relevant. Based on Peaks-Over-Threshold (POT) sampling, this study selected 156 Peaks-Over-Threshold (POT) floods from Hongfeng's 43 year inflow records and two historical catastrophic floods in May 1830 and August 1892 with a threshold of 160 m3s-1. The selected sample floods satisfy the principles of independence and uniformity. A mixed Von Mises distribution with three parts (n=3) was then established. Relevant parameters are u1=0.50, k1=27.53, P1=0.10; u2=2.28, k2=2.82, P2=0.66; u3=0.48, k3=3.05, P3=0.24. Given these parameters, the density function of this mixed Von Mises distribution are: f1t=12π×0.106.89×10-10exp⁡27.53cos⁡t-0.50f2t=12π×0.664.22exp⁡2.82cos⁡t-2.28f3t=12π×0.245.10exp⁡3.05cos⁡t-0.48ftt=f1t+f2t+f3t According to the above formulas, the fitting graph for the mixed Von Mises distribution of the floods occurring time is plotted in Fig. 4.

As shown in Fig. 4, floods in Maotiao River mainly occur in June and July and sometimes in the middle of May, August and September. Floods in May, August and September account for 16, 15 and 8 % respectively of all floods in flood season, while floods in June and July are 61 % of all floods. The Maotiao River flood season is characterized with sub-seasons. In addition, the hydrologic records show that runoff in Maotiao River changes slightly from year to year but largely changes within one year. The largest annual flood generally occurs before August, mostly in June or July. Based on the selected sample sequence, two sub-season definitions were proposed. Both strategies have May as the pre-rainy season, June and July as the main flood season. But one has August as the late-flood season I and September as the late-flood season II, while another combines August and September into one late-flood season. This paper shows that the theoretical curve based on the latter strategy can better fit the sample sequence, and apparently the mixed Von Mises distribution under such circumstance has three parts (n=3).