# Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{q}(s) versus s. The location for the crossover point is approximately one year, implying an unchanged annual periodicity within the streamflow variations. The annual periodical feature of streamflow was removed by using seasonal trend decomposition based on locally weighted regression (STL). All the decomposed streamflow series were characterized by long-term persistence in the study areas. Strong dependence of the generalized Hurst exponent h(q) on q exhibited multifractal behavior in streamflow time series at four stations in the Yellow River basin. The reduction of dependence of h(q) on q for shuffled time series showed that the multifractality of streamflow series was responsible for the correlation properties, as well as the probability density function of the streamflow series.

## 1. Introduction

## 2. Study Area

^{4}km

^{2}, located between 96° ~ 119° E and 32° ~ 42° N. The river originates from the Qinghai−Tibetan Plateau in western China and flows eastward through the Loess Plateau and the North China Plain before entering into the Bohai Sea, for a total length of 5464 km (Figure 1). From the river source to Toudaoguai, its upper reaches, it is 3472 km long with an area of 38.6 × 10

^{4}km

^{2}. The streamflow in the upper reaches accounts for approximately 61% of the whole basin. The upper reaches belongs to an arid climate with an annual average precipitation of 396 mm. Longmen station lies in the middle reaches of the Yellow River. The region between Toudaoguai and Longmen is suffering from severe soil erosion. The climate is semi-arid and arid with an annual average precipitation of 516 mm.

^{2}(Figure 1). The Huangfuchuan River is a first-order tributary in midstream Yellow River with a river length of 137 km and average channel slope of 2.7%. The watershed is located in the transitional belt of warm temperate and mesothermal zones with an average precipitation of 350–450 mm, more than 80% of which occurs between June and September [26]. The Huangfuchuan watershed is one of the most severe soil erosion areas and brings approximately 0.15 billion t sediment into the Yellow River each year. The Yanhe watershed is located to the south of the Huangfuchuan River with a drainage area of 7687 km

^{2}(Figure 1). The Yanhe River is also a first-order tributary in the midstream of the Yellow River. The Yanhe watershed is dominated by a typical warm temperate continental monsoonal climate. Annual precipitation is approximately 500 mm and the average annual temperature ranges from 8.8 °C to 10.2 °C. The Yanhe watershed is covered by forests in the south and steppe grassland in the north, and arable land is mostly distributed in the alluvial plains and gentle slope hills. In the Yanhe watershed, the loess hilly-gully region accounts for 90% of the basin, and the slope in most areas is >15°, where soil erosion is very serious [3].

## 3. Data and Method

#### 3.1. Data

River | Station | Area (km^{2}) | Series Length | Time Interval | Annual Streamflow (10^{8} m^{3}) |
---|---|---|---|---|---|

Yellow River | Toudaoguai | 367,898 | January 1919–December 2009 | Monthly | 228.01 |

Yellow River | Longmen | 497,552 | January 1919–December 2009 | Monthly | 285.69 |

Huangfuchuan | Huangfu | 3,175 | 1 January 1960–31 December 2009 | Daily | 1.20 |

Yanhe | Ganguyi | 5,891 | 1 January 1960–31 December 2009 | Daily | 1.97 |

#### 3.2. Method

#### 3.2.1. Seasonal Trend Decomposition Based on Locally Weighted Regression

_{t}) was regarded as an additive form of three components: Trend (T

_{t}), Seasonality (S

_{t}) and Remainder (R

_{t}).

#### 3.2.2. Multifractal Detrended Fluctuation Analysis

_{1}, x

_{2}… x

_{N}), N is the length of the time series. The detailed computational procedure can be found as follows [28,29,30,31,32]:

_{s}= int(N/s) segments and N

_{s}is the nearest integer part of N/s. Since the length N of the series may not be an integer multiple of the timescale s, some data may remain at the end of the series Y(i). In order not to ignore the rest of the series, the same computation procedure is repeated from the end to the start of the series. Thus, 2N

_{s}segments can be obtained.

_{s}segments, and the variance is determined by:

_{s}, and

_{s}+ 1,…, 2N

_{s}.

_{v}(i) is the fitted piecewise polynomial trend function in segment v, and any order polynomial can be calculated, such as linear, quadratic, or cubic.

_{s}segments, the qth-order fluctuation function is determined by:

_{q}(s) and s:

_{q}(s) versus s. The slope of logF

_{q}(s) and logs is the generalized Hurst exponent h(q). For stationary time series, the exponent h(2) is identical to the well-known Hurst exponent H, and h(2) varies between 0 and 1 [6]. The exponent h(2) can be used to analyze correlations in time series. The scaling exponent H = 0.5 means that the time series are uncorrelated; 0.5 < H < 1 implies long-term persistence and 0 < H < 0.5 implies short-term persistence [28].

## 4. Results

#### 4.1. Statistical Characteristics

**Figure 2.**Temporal variation of streamflow at the Toudaoguai, Longmen, Huangfu and Ganguyi stations of the Yellow River.

^{8}m

^{3}and 23.6 × 10

^{8}m

^{3}, respectively. The standard deviation of streamflow at the Longmen station is higher than that at Toudaoguai, indicating that streamflow fluctuates greatly at the Longmen station. The skewness and kurtosis are greater than 0, suggesting that the streamflow distribution is not subject to the random normal distribution. The daily streamflow at Ganguyi, with more precipitation and extensive vegetation, is relatively higher than that in Huangfuchuan watershed, which has less precipitation and poor vegetation. The standard deviation of streamflow at Huangfu is greater than that at Ganguyi due to the different geomorphological types. The Huangfuchuan watershed is covered by bare weathered rocks, and low permeability and storage capacity improve runoff yield. In addition, the Huangfuchuan watershed is a rainstorm center, leading to changes of large magnitude in streamflow. The streamflow distributions at both Ganguyi and Huangfu stations are non-normal distribution due to skewness and kurtosis greater than 0.

**Table 2.**Characteristics of streamflow for the main river and tributaries of the Yellow River basin.

Hydrological Station | Mean (10^{8} m^{3}) | Max (10^{8} m^{3}) | Min (10^{8} m^{3}) | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|---|

Toudaoguai | 18.9 | 112.0 | 1.31 | 15.3 | 1.92 | 4.37 |

Longmen | 23.6 | 126.1 | 2.81 | 17.7 | 1.76 | 3.33 |

Ganguyi | 0.005 | 1.296 | 0 | 0.019 | 26.5 | 1246.7 |

Huangfu | 0.003 | 1.305 | 0 | 0.024 | 26.9 | 1036.2 |

#### 4.2. Multifractal Detrended Fluctuation Analysis

_{q}(s) versus s of the streamflow series. The crossover points occur at approximately 12 months at the Toudaoguai and Longmen stations, and after approximately 372 days at Ganguyi and Huangfu stations. The location of the crossover point indicates annual periodicity of streamflow, implying a strong relationship between precipitation and streamflow.

**Figure 3.**The fluctuation function F

_{2}(s) versus time scale s in double logarithmic plots for the streamflow time series at the Toudaoguai, Longmen, Ganguyi and Huangfu stations of the Yellow River.

**Figure 4.**STL decomposition for streamflow at Toudaoguai station. The plots show the original measured streamflow, the seasonal component, trend component and the remainder from top to bottom, respectively.

_{q}(s) versus s. The h(2) values of streamflow are respectively 0.8257 and 0.8231 at Toudaoguai and Longmen stations, suggesting long-term persistence of the streamflow time series. It implies that scaling properties of the streamflow series in the upper and middle reaches of the Yellow River are similar. At the Ganguyi and Huangfu stations, streamflow fluctuations are also characterized by long-term persistence, as the values of h(2) are 0.5742 and 0.5882, respectively. This implies that the changing trends may remain stable for the next few years. The scaling properties of streamflow series at the four stations showed similar characteristics over a scale. This may imply universal scaling properties within the Yellow River basin because they have similar climate patterns.

**Figure 5.**The fluctuation function F

_{2}(s) versus time scale s in double logarithmic plots for the streamflow time series removing the periodicity at the Toudaoguai, Longmen, Ganguyi and Huangfu stations of the Yellow River.

**Figure 6.**The relationships of (

**a**) the generalized Hurst exponent h(q) and q; (

**b**) the mass exponent function τ(q) and q; and (

**c**) the singularity spectrum f(α) and singularity exponent α for the streamflow series at the Toudaoguai, Longmen, Ganguyi and Huangfu stations of the Yellow River.

**Table 3.**Slopes of the mass exponent function τ(q) for the streamflow series at Toudaoguai, Longmen, Ganguyi and Huangfu stations of the Yellow River.

Study Area | Slopes | |
---|---|---|

−10 < q < 0 | 0 < q < 10 | |

Toudaoguai | 1.4070 | 0.6501 |

Longmen | 1.3361 | 0.6191 |

Ganguyi | 1.2538 | 0.2738 |

Huangfu | 1.5456 | 0.2358 |

## 5. Discussion

^{2}) and 1.43 (at Huangfu, 3246 km

^{2}), respectively. The results in this study are consistent with Koscielny-Bunde et al. [9] and Özger et al. [37]. Kantelhardt et al. [17] found that the persistence of streamflow was related to storage processes occurring in the soil and the highly intermittent spatial behavior of rainfall. The h(2) values and multifractal spectra f(α) of the streamflow series at the Toudaoguai and Longmen stations are similar, but the values h(2) and multifractal spectra f(α) are different at the Ganguyi and Huangfu stations. For the mainstream of the Yellow River, most of the streamflow comes from the upper reaches of the Yellow River. Fluctuations in streamflow may be similar at Toudaoguai and Longmen stations, resulting in similar scaling behaviors, but the Ganguyi and Huangfu stations are located in different regions which have uneven spatial and temporal distribution of precipitation, various land covers and different soil types. The average annual precipitation are 364.3 mm and 504.1 mm in Huangfuchuan and Yanhe watershed, respectively [38]. There are many rainstorms in the Huangfuchuan basin, which has complex geomorphological types including a weathered sandstone hilly-gully region, a loess hilly-gully region and a sandy loess hilly-gully region. By contrast, the loess hilly-gully region is the dominant geomorphological type, accounting for 90% of the Yanhe basin. Human activities, such as exploitation, utilization of water resources, and the construction of check dams can heavily affect streamflow variation. The number of check dams had reached 567, and the dam-controlled area was 2216.47 km

^{2}in 2010, accounting for 68.3% of the Huangfuchuan basin area [26]. Yang et al. [39] illustrated that hydrological processes downstream of dams were closely associated with the regulating activities of reservoirs, and dam construction had significant influence on hydrological alterations. The interaction of all these complicated factors leads to the different multifractality of the streamflow series at the Huangfu and Ganguyi stations.

**Figure 7.**Generalized Hurst exponent h(q) as a function of q for original and shuffled streamflow series of the Toudaoguai, Longmen, Ganguyi, and Huangfu stations of the Yellow River.

## 6. Conclusions

- (1)
- One crossover point can be found for the four log−log plots of F
_{q}(s) versus s in the streamflow series. The crossover point occurred in approximately 12 months at the Toudaoguai and Longmen stations, and in approximately 372 days at Ganguyi and Huangfu station, which is attributed to a one-year periodicity of streamflow. - (2)
- The scaling properties of the decomposed streamflow series at the four stations showed long-range correlations. This may imply universal scaling properties within Yellow River basin.
- (3)
- The q dependence of h(q) and τ(q) indicated that streamflow time series have multifractal behavior, which is due to the correlation properties, as well as to the PDF of the hydrological series. Comparing with Yanhe, streamflow time series at Huangfuchuan have a multifractal structure that is sensitive to large magnitudes of local fluctuations. Different precipitation−geomorphological types−runoff relationships at Yanhe and Huangfuchuan watershed may be the major effect factors inducing different multifractal behaviors.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Li, E.; Mu, X.; Zhao, G.; Gao, P.
Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China. *Water* **2015**, *7*, 1670-1686.
https://doi.org/10.3390/w7041670

**AMA Style**

Li E, Mu X, Zhao G, Gao P.
Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China. *Water*. 2015; 7(4):1670-1686.
https://doi.org/10.3390/w7041670

**Chicago/Turabian Style**

Li, Erhui, Xingmin Mu, Guangju Zhao, and Peng Gao.
2015. "Multifractal Detrended Fluctuation Analysis of Streamflow in the Yellow River Basin, China" *Water* 7, no. 4: 1670-1686.
https://doi.org/10.3390/w7041670