## 2 Materials and Methods
### Research Methods
#### 2.2.2 Multifractal Method  
To further understand the role of local conditions in the formation of fractal bodies, researchers have proposed the multifractal theory that mainly discusses the probability distribution of a certain parameter. For the multifractal method, previous studies include [PERSON] (1996) and [PERSON] and [PERSON] (1997). The following is the calculation formula of multifractal parameters:  
\[\mu_{i}(\varepsilon)=\frac{M_{i}(\varepsilon)}{\sum\limits_{i=1}^{N( \varepsilon)}M_{i}(\varepsilon)} \tag{4}\]  
\[\left\{\begin{array}{l}D(q)=\underset{\varepsilon\to 0}{\lim}\frac{1}{q-1} \frac{\lg\left[\sum\limits_{i=1}^{N(\varepsilon)}\mu_{i}(\varepsilon)^{q} \right]}{\lg\varepsilon}(q\
eq 1)\\ D(q)=\underset{\varepsilon\to 0}{\lim}\frac{\sum\limits_{i=1}^{N( \varepsilon)}\mu_{i}(\varepsilon)\lg\mu_{i}(\varepsilon)}{\lg\varepsilon}(q=1 )\end{array}\right. \tag{5}\]  
where \(\mu_{i}(\varepsilon)\) is the mass probability, \(M_{i}(\varepsilon)\) is the variable, \(\varepsilon\) is the scale, \(N(\varepsilon)\) is the number of variables under the scale, \(D(q)\) is the generalized fractal dimension and \(q\) is the order of statistical moment.  
\[\mu_{i}(q,\varepsilon)=\frac{\mu_{i}(\varepsilon)^{q}}{\sum\limits_{i=1}^{N( \varepsilon)}\mu_{i}(\varepsilon)^{q}} \tag{6}\]  
\[\left\{\begin{array}{c}\alpha(q)=\underset{\varepsilon\to 0}{\lim}\frac{\sum \limits_{i=1}^{N(\varepsilon)}\mu_{i}(q,\varepsilon)\lg\mu_{i}(\varepsilon)}{ \lg\varepsilon}\\ f[\alpha(q)]=\underset{\varepsilon\to 0}{\lim}\frac{\sum\limits_{i=1}^{N( \varepsilon)}\mu_{i}(q,\varepsilon)\lg\mu_{i}(q,\varepsilon)}{\lg\varepsilon} \end{array}\right. \tag{7}\]  
where \(\mu_{i}(q,\ \varepsilon)\) is the q-order mass probability, \(\alpha(q)\) is the singularity index and \(f(\alpha)\) is the singularity spectrum.  
\(q\)-\(D(q)\) and \(\alpha\)-\(f(\alpha)\) are two sets of parameters for describing multifractals. In practical application, we often use the partition function method and Legendre transformation [37] to establish the relationship between two sets of parameters:  
\[\chi_{q}(\varepsilon)=\sum_{i=1}^{N(\varepsilon)}[\mu_{i}(\varepsilon)]^{q} \propto\varepsilon^{\tau(q)} \tag{8}\]  
where \(\chi_{q}(\varepsilon)\) is the partition function and \(\tau(q)\) is the q-order mass index.  
The quality index corresponding to each \(q\) value can be obtained by calculating the slope of the fitting line between \(log(\varepsilon)\) and \(log[\chi_{q}(\varepsilon)]\) by the least-squares method. Additionally, \(D(q)\) and \(\tau(q)\) have the following relationship:  
\[D(q)=\tau(q)/q-1\
eq 1) \tag{9}\]  
The relationship between \(\alpha(q)\) and \(f[\alpha(q)]\) and \(\tau(q)\) is as follows:  
\[\alpha(q)=d\tau(q)/dq \tag{10}\]  
\[f[\alpha(q)]=q\alpha(q)-\tau(q) \tag{11}\]  
The value of \(q\) can range from negative to positive infinity. When \(q>0\) and \(D(q)\) decreases with \(q\), the variable has multifractal characteristics [12]. When \(q>>1\), the large part of \(\chi_{q}(\varepsilon)\) is dominant, while when \(q<<-1\), the small part of \(\chi_{q}(\varepsilon)\) is dominant. The higher-order statistical moments can extract extreme values and enlarge their contribution. The relationship between \(\alpha\) and \(f(\alpha)\) can be used to analyse the local characteristics of multiple fractal bodies. The value of \(\Delta\alpha\) reflects the overall heterogeneity of variable distribution; the larger the value of \(\Delta\alpha\), the greater the variability of the variable distribution. The asymmetry of the \(\alpha\)-\(f(\alpha)\) curve reflects the dominant power of the uneven variable distribution. If the curve is shaped like a \"left hook\", the spatial variation of the variable is dominated by smaller values; on the contrary, if the curve is shaped like a \"right hook\", it is dominated by the larger values.