# China's Water Intensity Factor Decomposition and Water Usage Decoupling Analysis
## 2 Materials and Methods  
### Logarithmic Mean Divisia Index Model  
In order to analyze the influence of technological progress, regional scale, industrial structure, and other factors on the water usage change of different industries in China, it is beneficial to analyze the driving factors of water usage in China by using the LMDI proposed by [PERSON] [38]. This method has the advantages of zero value and complete decomposition, and can be completely decomposed [39; 40; 41].  
Water usage index can be expressed by absolute quantity and relative quantity. The absolute quantity refers to total water usage, and the relative quantity refers to the water usage per unit of economic output, that is, water intensity. It reflects the utilization efficiency of water resources, which is influenced by economic growth, technological progress, industrial structure, regional scale, and policy factors. According to the definition of water intensity, it can be expressed as:  
\[w=\sum_{i}\sum_{j}\frac{W_{ij}}{G_{ij}} \tag{1}\]where \(w\) is the water intensity (cubic meters/10 thousand CNY); \(W_{ij}\) is the water usage (cubic meters) of the \(j\)th industry in the ith province; \(G_{ij}\) is the gross output value (10 thousand CNY) of the \(j\)th industry in the ith province.  
According to LMDI analysis framework, by analyzing the influence of each effect on water intensity, we can construct Equation (2) as follows:  
\[w=\sum\
olimits_{i}\sum\
olimits_{j}\frac{W_{ij}}{G}=\sum\
olimits_{i}\sum \
olimits_{j}\frac{W_{ij}}{G_{ij}}\times\frac{G_{ij}}{G_{i}}\times\frac{G_{i}} {G}=\sum\
olimits_{i}\sum\
olimits_{j}\mu_{ij}r_{ij}s_{i} \tag{2}\]  
where \(w\) is the water intensity, \(W_{ij}\) is the total water usage of the \(j\)th industry in the \(i\)th province, \(G_{ij}\) is the gross national product of the \(j\)th industry in the \(i\)th province, \(G_{i}\) is the gross product of the \(i\)th province, \(G\) is the gross domestic product, \(q_{ij}\) is the water intensity of the \(j\)th industry in the \(i\)th province, \(r_{ij}\) is the proportion of the gross product of the \(j\)th industry in the gross product of the \(i\)th province, \(s_{i}\) is the ratio of GDP of the \(i\)th province to total GDP.  
Therefore, the total effect formula of water intensity is:  
\[\Delta tot=\Delta t+\Delta u+\Delta v \tag{3}\]  
where \(\Delta tot\) is the total effect, that is, the sum of all effects, indicating the total change of water intensity; \(\Delta t\) refers to the technical effect, indicating the contribution of the change of resource utilization efficiency caused by technological progress to the total change of water intensity; \(\Delta u\) refers to the industrial structure effect, indicating the contribution of industrial structure adjustment to the total change of water intensity; \(\Delta v\) is the regional scale effect, which indicates the contribution of the ratio of regional economic output to GDP to the total change of water intensity.  
The contribution of each effect is expressed as follows:  
\[\Delta t=\sum\limits_{i}\sum\limits_{j}\frac{\frac{W_{ij}^{T}}{G^{T}}-\frac{ W_{ij}^{0}}{G^{0}}}{\ln(\frac{W_{ij}^{T}}{G^{T}})-\ln(\frac{W_{ij}^{0}}{G^{0}})}\ln( \frac{q_{ij}^{T}}{q_{ij}^{0}}) \tag{4}\]  
\[\Delta u=\sum\limits_{i}\sum\limits_{j}\frac{\frac{W_{ij}^{T}}{G^{T}}-\frac{ W_{ij}^{0}}{G^{0}}}{\ln(\frac{W_{ij}^{T}}{G^{T}})-\ln(\frac{W_{ij}^{0}}{G^{0}})}\ln( \frac{r_{ij}^{T}}{q_{ij}^{0}}) \tag{5}\]  
\[\Delta v=\sum\limits_{i}\sum\limits_{j}\frac{\frac{W_{ij}^{T}}{G^{T}}-\frac{ W_{ij}^{0}}{G^{0}}}{\ln(\frac{W_{ij}^{T}}{G^{T}})-\ln(\frac{W_{ij}^{0}}{G^{0}})}\ln( \frac{s_{ij}^{T}}{s_{ij}^{0}}) \tag{6}\]  
The contribution rates of the three effects to the change of water intensity are \(\Delta t/\Delta tot\), \(\Delta u/\Delta tot\), and \(\Delta v/\Delta tot\), respectively. When the positive and negative impacts of each effect are consistent with the total effect, it shows that this effect has a positive impact on the reduction of water intensity, and vice versa.