# China's Water Intensity Factor Decomposition and Water Usage Decoupling Analysis
## 2 Materials and Methods
### Decoupling Model  
The decomposition model can be used to study the contribution of various factors to the change of water usage intensity, but it cannot directly measure the decoupling state between economy and water usage [22]. Therefore, the Tapio decoupling model is adopted [42; 43; 44], and the decomposition model of water usage is as follows:  
\[W=\sum\limits_{i}\sum\limits_{j}W_{ij}=\sum\limits_{i}\sum\limits_{j}\frac{W _{ij}}{G_{ij}}\times\frac{G_{ij}}{G_{i}}\times\frac{G_{i}}{G}\times G \tag{7}\]So as to decompose the changes of water usage into:  
\[\Delta TOT=\sum_{i}\sum_{j}\Delta T+\Delta U+\Delta V+\Delta Q \tag{8}\]  
Among them, \(\Delta TOT\) is the total effect of water usage, \(\Delta T\) is the technical effect, \(\Delta U\) is the effect of industrial structure, \(\Delta V\) is the effect of regional scale, and \(\Delta Q\) is the effect of output scale. The contribution of each effect is as follows:  
\[\Delta T=\sum_{i}\sum_{j}\frac{W_{ij}^{T}-W_{ij}^{0}}{\ln W_{ij}^{T}-\ln W_{ij }^{0}}\ln(\frac{W_{ij}^{T}/G_{ij}^{T}}{W_{ij}^{0}/G_{ij}^{0}}) \tag{9}\]  
\[\Delta U=\sum_{i}\sum_{j}\frac{W_{ij}^{T}-W_{ij}^{0}}{\ln W_{ij}^{T}-\ln W_{ij }^{0}}\ln(\frac{G_{ij}^{T}/G_{ij}^{T}}{G_{ij}^{0}/G_{ij}^{0}}) \tag{10}\]  
\[\Delta V=\sum_{i}\sum_{j}\frac{W_{ij}^{T}-W_{ij}^{0}}{\ln W_{ij}^{T}-\ln W_{ij }^{0}}\ln(\frac{G_{i}^{T}/G^{T}}{G_{i}^{0}/G^{0}}) \tag{11}\]  
\[\Delta Q=\sum_{i}\sum_{j}\frac{W_{ij}^{T}-W_{ij}^{0}}{\ln W_{ij}^{T}-\ln W_{ ij}^{0}}\ln\frac{G^{T}}{G^{0}} \tag{12}\]  
Decoupling elasticity index is used to discuss the decoupling relationship between economic growth and water usage. The elastic coefficient of GDP water usage is calculated as follows:  
\[\omega(W,G)=\frac{\Delta W/W}{\Delta G/G} \tag{13}\]  
The types of decoupling can essentially be divided into coupling, decoupling, and negative decoupling. In addition, according to the elasticity coefficient, the change of water usage and the change of GDP, the decoupling types can be subdivided into eight cases (Table 1) [45]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**State** & **Type** & \(\triangle\textbf{W}\) & \(\triangle\textbf{GDP}\) & \(\boldsymbol{\omega}\) & **Meaning** \\ \hline \multirow{4}{*}{Connection} & Decline connection & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\((0.8,1.2)\)} & Water usage is declining at the same rate as the economy. \\  & (DC) & & & & & \\  & Expansion connection & & & & & \\  & (EC) & & & & & \\ \hline \multirow{4}{*}{Decoupling} & Decline decoupling & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\((1.2,+\infty)\)} & Water usage is declining faster than the economic recession. \\  & (DD) & & & & & Economic growth is \\ \cline{1-1}  & Strong decoupling & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(+\)} & \multirow{2}{*}{\((-\infty,0)\)} & Water usage is declining at the same rate as the economy. \\ \cline{1-1}  & (SD) & & & & & \\ \cline{1-1}  & Weak Decoupling & \multirow{2}{*}{\(+\)} & \multirow{2}{*}{\(+\)} & \multirow{2}{*}{\((0,0.8)\)} & The growth rate of water usage is slower than that of economy. \\ \cline{1-1}  & (WD) & & & & & The rate of water usage \\ \cline{1-1}  & Weak negative decoupling & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\((0,0.8)\)} & & \\ \cline{1-1}  & (WND) & & & & & \\ \cline{1-1} Negative decoupling & Strong negative decoupling & \multirow{2}{*}{\(+\)} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\((-\infty,0)\)} & Water usage is increasing but the economy is declining. \\ \cline{1-1}  & (SND) & & & & The water usage growth rate is lower than the economic growth rate. \\ \cline{1-1}  & Expansion negative decoupling & \multirow{2}{*}{\(+\)} & \multirow{2}{*}{\(+\)} & \multirow{2}{*}{\((1.2,+\infty)\)} & & \\ \cline{1-1}  & (END) & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Types of Tapio models.