## 3 The Objective Function  
Recent theoretical developments have revealed that multiple types of observations can identify the objective function of inversion analysis. In this study, the hydraulic head and leakage measurements are applied to construct the objective function. It is assumed that the permeability coefficient of the rock stratum is isotropic. \(P_{i}^{m}=\left[P_{i1}^{m},P_{i2}^{m},P_{i3}^{m},\ldots P_{i4}^{m}\right](i=1,2,3,\ldots,M)\) denotes the time series data for pressure tube monitoring, while \(Q_{j}^{m}=\left[Q_{j1}^{m},Q_{j2}^{m},Q_{j3}^{m},\ldots Q_{j4}^{m}\right](j= 1,2,3,\ldots,N)\) denotes the time series data for weir monitoring. A scalar K was selected to stand for the permeability coefficient of the impervious curtain.  
The permeability coefficient of the curtain increases with calcium leaching time. The KC equations were used to simulate the evolution of the permeability coefficient under the influence of calcium leaching. The porosity was applied as an intermediate parameter linking the calcium leaching effect to the permeability coefficient. In the calcium leaching model, the lumped term for sound materials \(\Omega_{0}\), the lumped term increased times \(n\), and the rock hydraulic conductivity \(k_{r}\) have significant implications on the permeability coefficient of the curtain. In this paper, the inverse model was developed to obtain the best estimates of these parameters to minimize the value of the objective function F. The mathematical model of the back analysis is established as described in Equation (8):  
\[\begin{array}{l}\min F=\left(\sum\limits_{i=1}^{M}\frac{||P_{i}(K(\Omega_{0},n,k,r,t))-P_{i}^{m}||_{2}^{2}}{||P_{i}^{m}||_{2}^{2}}\right)^{\frac{1}{2}}+w\left( \sum\limits_{j=1}^{N}\frac{||Q_{j}(K(\Omega_{0},n,k,r,t))-Q_{j}^{m}||_{2}^{2}} {||Q_{j}^{m}||_{2}^{2}}\right)^{\frac{1}{2}}\\ \text{s.t.}\ \ \Omega_{0 min}\leq\Omega_{0}\leq\Omega_{0 max}\\ n_{min}\leq n\leq n_{max}\\ k_{rmin}\leq k_{r}\leq k_{rmax}\end{array} \tag{8}\]  
Here \(P_{i}(K(\Omega_{0},n,k,r,t))\) and \(Q_{j}(K(\Omega_{0},n,k_{r},t))\) are numerically calculated hydraulic head and leakage volume, respectively, and \(w\) is the weight for ensuring a relative balance between the head and flow observation errors. [PERSON] et al. [17] analyzed the relative error of hydraulic head and leakage under different weight values. The results show that the errors between hydraulic head and discharge can be well balanced under the condition of \(w=0.02\). The conclusion is adopted in this study. \(\Omega_{0 max}\) and \(\Omega_{0 min}\) respectively represent the upper and lower bounds of the lumped term. \(n_{max}\) and \(n_{min}\) denote the upper and lower bounds of the lumped term increased times. Similarly, \(k_{rmax}\) and \(k_{rmit}\) signify the upper and lower bounds of rock hydraulic conductivity, respectively. The upper and lower bounds of the parameters can be roughly determined by engineering experience and laboratory experiments. The ranges of parameters are indicated in the results section, for verifying the rationality of the results.  
Within the range of the monitoring data, the hydraulic head and leakage measurements at the initial moment were selected for inversion to obtain the target parameters. The combination of the objective parameters was also applied to predict the trend of curtain permeability coefficient in a century.