# Multifractal Analysis of River Networks under the Background of Urbanization in the Yellow River Basin, China

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}and constitutes the fifth longest river in the world and the second longest in China [26]. The Yellow River basin encompasses a vast area with many mountains and has a great difference in elevation from east to west (about 4800 m). The landforms of the various regions associated with the river basin are also very different. Furthermore, the basin is in the middle latitudes, and so it is more affected by atmospheric changes and monsoon circulation. Thus, there are significant differences in climate within the basin. The temperature difference across the Yellow River Basin is very large. In general, as the terrain of the three-tiered elevation changes from west to east, the temperature changes from colder to warmer. The east–west temperature gradient is notably greater than the north–south gradient. The annual temperature difference in the basin is also relatively large. The general trend is that the annual average temperature in the area north of 37° N is between 31 °C and 37 °C, while the temperature in the area south of 37° N is mostly between 21 °C and 31 °C. Precipitation in the Yellow River Basin is concentrated, unevenly distributed, and has significant inter-annual variation. The annual precipitation in most of the regions is between 200 and 650 mm. It is greater than 650 mm south of the middle of the upper and lower reaches, while the precipitation in the deep inland areas of the Ningxia Hui and Inner Mongolia Autonomous Region in northwestern China is less than 150 mm. The basin has a strong evaporative capacity, with an average annual evaporation of 1100 mm. The upper reaches of Gansu Province, Ningxia Hui Autonomous Region, and the central and western regions of the Inner Mongolia Autonomous Region have the largest annual evaporation in China. The maximum annual evaporation in these regions can exceed 2500 mm.

#### 2.2. Data Description

#### 2.3. Method

- (1)
- The study area F is covered with boxes of size $e\times e$, and the total number of non-empty boxes is denoted $N\left(e\right)$. ${p}_{i}\left(e\right)$ is the probability measure of the region contained in each box; that is, the distribution probability of the characteristic information. ${p}_{i}\left(e\right)$ differs for different units. ${p}_{i}\left(e\right)$ and e are related via Equation (1):$${p}_{i}\propto {e}^{\alpha},$$$${p}_{i}\left(e\right)=\frac{{c}_{i}}{{{\displaystyle \sum}}_{i=1}^{N\left(e\right)}{c}_{i}},$$
- (2)
- The partition function $M\left(e,q\right)$ is defined as the weighted sum of the slope distribution probability ${p}_{i}\left(e\right)$ to power q (Equation (3)):$$M\left(e,q\right)={\displaystyle \sum}_{i=1}^{N\left(e\right)}{p}_{i}^{q}\left(e\right),$$
- (3)
- For a given moment q, the relationship between the mass exponential function $\tau \left(q\right)$ and $M\left(e,q\right)$ is given by Equation (4). In the calculation, the size of the box $e$ under the corresponding q value is changed, and the partition function under the corresponding box size is computed. Then, $\tau \left(q\right)$ can be computed through the coefficient of the straight line fit of $\mathrm{ln}M\left(e,q\right)$~$\mathrm{ln}e$ (Equation (5)). With the change in q, the corresponding $\tau \left(q\right)$ can be calculated using the above procedure.$$M\left(e,q\right)\propto {e}^{\tau \left(q\right)},$$$$\tau \left(q\right)=\underset{e\to 0}{\mathrm{lim}}\frac{\mathrm{ln}M\left(e,q\right)}{\mathrm{ln}e},$$
- (4)
- The generalized fractal dimension ${D}_{q}$ is defined by Equation (6) and varies with q. ${D}_{q}$ can reflect the singularity of each subset of the research object from an overall perspective, so there is the relationship between ${D}_{q}$ and α in Equations (7) and (8).$${D}_{q}=\left\{\begin{array}{c}\frac{1}{q-1}\underset{e\to 0}{\mathrm{lim}}\frac{\mathrm{ln}M\left(e,q\right)}{\mathrm{ln}e}=\frac{\tau \left(q\right)}{q-1}q\ne 1\\ \underset{e\to 0}{\mathrm{lim}}\frac{{{\displaystyle \sum}}_{i=1}^{N\left(e\right)}{p}_{i}\mathrm{ln}{p}_{i}}{\mathrm{ln}e}q=1\end{array}\right.,$$$$\underset{q\to +\infty}{\mathrm{lim}}{D}_{q}={\alpha}_{\mathrm{min}},$$$$\underset{q\to -\infty}{\mathrm{lim}}{D}_{q}={\alpha}_{\mathrm{max}},$$

- (5)
- When $\tau \left(q\right)$ is differentiable, the multifractal spectrum$f\left(\alpha \right)$ and singular exponent$\alpha \left(q\right)$ can be obtained by the Legendre transformation of Equation (9).$$\left\{\begin{array}{c}\alpha \left(q\right)=\frac{d\tau \left(q\right)}{dq}\\ f\left(\alpha \right)=q\xb7\alpha \left(q\right)-\tau \left(q\right)\end{array},\right.$$

- The span of the singular exponent $\alpha \left(q\right)$ is the width of the multifractal spectrum, $\Delta \alpha $ (Equation (10)). $\alpha \left(q\right)$ indicates the degree of fluvial inhomogeneity, irregularity, and complexity in each sub-region within the basin. ${\alpha}_{\mathrm{min}}$ and ${\alpha}_{\mathrm{max}}$ (Equations (7) and (8)), respectively, indicate the singular exponent of the distribution probability of the maximum characteristic information ${p}_{i}{\left(e\right)}_{\mathrm{max}}$ and the distribution probability of the minimum characteristic information ${p}_{i}{\left(e\right)}_{\mathrm{min}}$ with the change in e. The smaller the ${\alpha}_{\mathrm{min}}$, the larger is the ${p}_{i}{\left(e\right)}_{\mathrm{max}}$. Therefore, we can use the span of the singular exponent $\Delta \alpha $ to describe the unevenness in the distribution probability of the river network. A larger $\Delta \alpha $ indicates that the distribution of characteristic information in the basin is less uniform, the internal difference in the research object is greater, and the polarization trend of each subset probability is clearer. In contrast, a smaller $\Delta \alpha $ indicates that the difference is smaller inside the fractal body, and the distribution of subsets tends to be concentrated and uniform.
- The difference between the maximum and minimum values of the multifractal spectrum is $\Delta f$ (Equation (11)). $f\left({\alpha}_{\mathrm{min}}\right)$ and$f\left({\alpha}_{\mathrm{max}}\right)$ represent the number of subsets of the maximum and minimum probabilistic characteristic information, respectively. The difference in $\Delta f$ can be used to calculate the difference between the maximum and minimum distribution probability subset numbers of the basin characteristic information. When $\Delta f<0$, the curve $f\left(\alpha \right)$~$\alpha \left(q\right)$ is hooked to the right, and the number of grid points contained in the maximum characteristic information distribution probability subset is less than the minimum probability subset number. The river network is densely distributed. In contrast, when $\Delta f>0$, the curve is hooked to the left. When $\Delta f=0$, the curve $f\left(\alpha \right)$~$\alpha \left(q\right)$ is symmetrical and bell-shaped.
- Symmetry of curve $f\left(\alpha \right)$~$\alpha \left(q\right)$. The multifractal spectrum is more symmetrical, which indicates that the fluvial distribution proportion is more uniform in the study area.$$\Delta \alpha ={\alpha}_{\mathrm{max}}-{\alpha}_{\mathrm{min}},$$$$\Delta f=f\left({\alpha}_{\mathrm{min}}-{\alpha}_{\mathrm{max}}\right),$$

- (6)
- When calculating the generalized fractal dimension ${D}_{q}$ and the multifractal spectrum $f\left(\alpha \right)$, the value of q plays an important role in the accuracy of the calculation results [32,33,34]. Theoretically,$q\in \left(-\infty ,+\infty \right)$, but in the actual calculation, only a limited range can be selected as the value of q. According to the research of [35], when the convergence coefficient $\mathsf{\zeta}<0.2\%$, the resulting changes to $\frac{d{\alpha}_{\mathrm{max}}}{\mathsf{\Delta}\mathsf{\alpha}}$ and $\frac{d{\alpha}_{\mathrm{min}}}{\mathsf{\Delta}\mathsf{\alpha}}$ are very small. The multifractal spectrum calculated within this range can be considered as a multifractal spectrum that reflects the characteristics of the research object. The value range of $\left|q\right|$ can be calculated using (12):$$\mathsf{\zeta}=\frac{\left|{f}_{q}-{f}_{q-1}\right|}{\left|{f}_{q}-f{\left(\alpha \right)}_{\mathrm{max}}\right|},$$

## 3. Results

#### 3.1. Determination of Multifractal Characteristics

#### 3.2. Multifractal Dimension Analysis

#### 3.3. Multifractal Spectrum Analysis

#### 3.4. Correlation Analysis of Multifractal Indicators and the Urbanization Process

## 4. Discussion

## 5. Conclusions

- During the period of 2000–2020, the river network of the Yellow River Basin has clear multifractal properties. It was found that the river network structure of the Yellow River Basin is greatly affected by areas of higher river density. The river network structure (the number and density of the rivers in the network, etc.) has shown a decreasing trend over the past 20 years, and the degree of the impact of dense rivers has also decreased.
- The changes in river networks were significantly affected by urbanization. Changes in river network structure were significantly correlated with the urbanization process. The average Gray correlation values between the changes in river networks and urbanization were greater than 0.7, which was greater than the resolution coefficient of the Gray correlation analysis (0.5). Their order was ${D}_{r}>{D}_{1}>\Delta f>{D}_{0}>\Delta \alpha $. This result indicates that the greater the urbanization rate, the greater the impact on the river network structure.
- To better study the spatiotemporal characteristics of river network changes in the Yellow River Basin in the context of urbanization, we calculated the fluvial characteristic parameters of provinces in the study area during periods of slow urbanization (2000–2010) and rapid urbanization (2010–2020). Moreover, we analyzed the degree of variation and temporal and spatial differences in these parameters. The results show that the changes in the river network structure are more affected by urbanization during the rapid urbanization stage. The multifractal spectrum width $\Delta \alpha $ is more sensitive to changes in the river network structure.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Study Area: (

**a**) Location of the Yellow River Basin in China; (

**b**) the river network of the Yellow River Basin; (

**c**) The provinces included in the Yellow River Basin: Qinghai Province (QH), Sichuan Province (SC), Gansu Province (GS), Ningxia Hui Autonomous Region (NX), Inner Mongolia Autonomous Region (NMG), Shaanxi Province (SA), Shanxi Province (SX), Henan Province (HN), and Shandong Province (SD).

**Figure 5.**The relationship between $\mathrm{ln}e$ and $\mathrm{ln}M\left(e,q\right)$ of the Yellow River Basin in the (

**a**) 2000s, (

**b**) 2010s, and (

**c**) 2020s.

**Figure 6.**The relationship between τ(q) and $q$ of the Yellow River Basin in the (

**a**) 2000s, (

**b**) 2010s, and (

**c**) 2020s.

**Figure 7.**The relationship between generalized multifractal dimension D

_{q}and order moment q in the 2000s, 2010s, and 2020s.

**Figure 8.**The multifractal spectrum of the Yellow River Basin in the (

**a**) 2000s, (

**b**) 2010s, and (

**c**) 2020s.

**Figure 9.**The variation of river characteristic parameters in provinces of the Yellow River Basin during the period 2000–2010 and 2010–2020. The change rate of river density in the period (

**a**) 2000–2010 and (

**b**) 2010–2020; the change rate of capacity dimension ${D}_{0}$ in the period (

**c**) 2000–2010 and (

**d**) 2010–2020; the change rate of information dimension ${D}_{1}$ in the period (

**e**) 2000–2010 and (

**f**) 2010–2020; the change rate of multifractal spectrum width $\Delta \alpha $ in the period (

**g**) 2000–2010 and (

**h**) 2010–2020; the change rate of multifractal spectrum difference $\Delta f$ in the period (

**i**) 2000–2010 and (

**j**) 2010–2020.

Periods | ${\mathit{\alpha}}_{\mathbf{min}}$ | ${\mathit{\alpha}}_{\mathbf{max}}$ | $\mathit{f}\left({\mathit{\alpha}}_{\mathbf{min}}\right)$ | $\mathit{f}\left({\mathit{\alpha}}_{\mathbf{max}}\right)$ | $\mathbf{\Delta}\mathit{\alpha}$ | $\mathbf{\Delta}\mathit{f}$ |
---|---|---|---|---|---|---|

2000s | 1.6720 | 4.6134 | 0.1286 | 0.0262 | 2.9414 | 0.1024 |

2010s | 1.7376 | 4.3076 | 0.1568 | 0.0324 | 2.5700 | 0.1243 |

2020s | 1.6836 | 3.8057 | 0.0746 | 0.0474 | 2.1220 | 0.0272 |

**Table 2.**Gray correlation between the river network changes and the urbanization process of provinces in the Yellow River Basin.

Urbanization Rate | Provinces | Parameters | ||||
---|---|---|---|---|---|---|

${\mathit{D}}_{\mathit{r}}$ | ${\mathit{D}}_{0}$ | ${\mathit{D}}_{1}$ | $\mathbf{\Delta}\mathit{\alpha}$ | $\mathbf{\Delta}\mathit{f}$ | ||

Gray correlation | QH | 0.7900 | 0.7870 | 0.7935 | 0.7317 | 0.7843 |

GS | 0.7384 | 0.7380 | 0.7397 | 0.7355 | 0.7424 | |

NX | 0.7417 | 0.7343 | 0.7391 | 0.7882 | 0.7303 | |

SA | 0.7453 | 0.7417 | 0.7469 | 0.6977 | 0.7472 | |

NMG | 0.7189 | 0.7239 | 0.7217 | 0.7193 | 0.7171 | |

SX | 0.7698 | 0.7653 | 0.7615 | 0.7211 | 0.7634 | |

HN | 0.7494 | 0.7481 | 0.7543 | 0.7055 | 0.7464 | |

SD | 0.7783 | 0.7664 | 0.7701 | 0.7322 | 0.7657 | |

SC | 0.8896 | 0.8563 | 0.8865 | 0.9326 | 0.8778 | |

YRB | 0.7041 | 0.7159 | 0.7169 | 0.7967 | 0.7086 | |

Average | 0.7626 | 0.7577 | 0.7630 | 0.7561 | 0.7583 |

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**MDPI and ACS Style**

Wang, J.; Qin, Z.; Shi, Y.; Yao, J. Multifractal Analysis of River Networks under the Background of Urbanization in the Yellow River Basin, China. *Water* **2021**, *13*, 2347.
https://doi.org/10.3390/w13172347

**AMA Style**

Wang J, Qin Z, Shi Y, Yao J. Multifractal Analysis of River Networks under the Background of Urbanization in the Yellow River Basin, China. *Water*. 2021; 13(17):2347.
https://doi.org/10.3390/w13172347

**Chicago/Turabian Style**

Wang, Jinxin, Zilong Qin, Yan Shi, and Jing Yao. 2021. "Multifractal Analysis of River Networks under the Background of Urbanization in the Yellow River Basin, China" *Water* 13, no. 17: 2347.
https://doi.org/10.3390/w13172347