Multifractal Analysis of River Networks under the Background of Urbanization in the Yellow River Basin, China
2. Materials and Methods
2.1. Study Area
2.2. Data Description
- The study area F is covered with boxes of size , and the total number of non-empty boxes is denoted . is the probability measure of the region contained in each box; that is, the distribution probability of the characteristic information. differs for different units. and e are related via Equation (1):
- The partition function is defined as the weighted sum of the slope distribution probability to power q (Equation (3)):
- For a given moment q, the relationship between the mass exponential function and is given by Equation (4). In the calculation, the size of the box under the corresponding q value is changed, and the partition function under the corresponding box size is computed. Then, can be computed through the coefficient of the straight line fit of ~ (Equation (5)). With the change in q, the corresponding can be calculated using the above procedure.
- The generalized fractal dimension is defined by Equation (6) and varies with q. can reflect the singularity of each subset of the research object from an overall perspective, so there is the relationship between and α in Equations (7) and (8).
- When is differentiable, the multifractal spectrum and singular exponent can be obtained by the Legendre transformation of Equation (9).
- The span of the singular exponent is the width of the multifractal spectrum, (Equation (10)). indicates the degree of fluvial inhomogeneity, irregularity, and complexity in each sub-region within the basin. and (Equations (7) and (8)), respectively, indicate the singular exponent of the distribution probability of the maximum characteristic information and the distribution probability of the minimum characteristic information with the change in e. The smaller the , the larger is the . Therefore, we can use the span of the singular exponent to describe the unevenness in the distribution probability of the river network. A larger 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 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 (Equation (11)). and represent the number of subsets of the maximum and minimum probabilistic characteristic information, respectively. The difference in can be used to calculate the difference between the maximum and minimum distribution probability subset numbers of the basin characteristic information. When , the curve ~ 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 , the curve is hooked to the left. When , the curve ~ is symmetrical and bell-shaped.
- Symmetry of curve ~. The multifractal spectrum is more symmetrical, which indicates that the fluvial distribution proportion is more uniform in the study area.
- When calculating the generalized fractal dimension and the multifractal spectrum , the value of q plays an important role in the accuracy of the calculation results [32,33,34]. Theoretically,, but in the actual calculation, only a limited range can be selected as the value of q. According to the research of , when the convergence coefficient , the resulting changes to and 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 can be calculated using (12):
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
- 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 . 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 is more sensitive to changes in the river network structure.
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Data Availability Statement
Conflicts of Interest
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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
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/w13172347Chicago/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