## 2 Methodology
### Revision of Rating Scores and Weights
#### 2.2.1 Common Weighting Methods  
Currently, popular techniques for the determination of weights include single-parameter sensitivity analysis, correlation analysis [22], and logistic regression analysis [26].  
The SPSA technique was used to evaluate the influence of the rating and weighting values assigned to each parameter and to judge the significance of subjectivity elements [34]. Today, it is being used as a weighting technique [26]. It might be the most commonly used technique for the determination of DRASTIC parameter weights. Based on this approach, the modified weight for parameter j can be calculated with Equation (3):  
\[\overline{w_{j}}=\frac{\sum_{i=1}^{n}\frac{W_{j}\times R_{ij}}{V_{i}}}{n}, \tag{3}\]  
where \(\overline{w_{j}}\) and \(W_{j}\) are parameter j's modified and original weights, respectively. \(R_{ij}\) is the rating score of parameter j in the \(i\)th subarea of the corresponding input parameter layer, \(V_{i}\) is the vulnerability index in the \(i\)th subarea of the output result layer, and \(n\) is the number of subareas.  
Correlation analysis can also be used to revise the weight of each parameter [22]. In this method, the quantitative correlation between the rating scores of every parameter and the nitrate concentration is computed. Because the rating scores of the parameters vary on an interval scale, the quantitative correlation is expressed using [PERSON]'s \(\rho\) and [PERSON]'s \(\tau\) correlation coefficients. The results are assumed to be the modified weights; they are called CA in this paper.  
Logistic regression has been successfully applied to evaluate groundwater's vulnerability to contamination [6; 12; 35; 36]. Unlike the DRASTIC model, this method provides a probability map to depict aquifer vulnerability, which has also been used to determine the weights of the parameters [25]. Two steps are necessary to implement this approach: (1) the nitrate concentration is reclassified into two classes (0 for a concentration that is below a predefined threshold, and 1 for others), (2) the coefficients of the logistic regression equation are calculated. The logistic regression equation can then be written as Equation (4) [25]:  
\[P=\frac{e^{b_{\text{b}}+b_{\text{b}}X_{1}+b_{\text{b}}X_{2}+\cdots+b_{\text{ J}}X_{\text{j}}}}{1+e^{b_{\text{b}}+b_{\text{b}}X_{1}+b_{\text{b}}X_{2}+\cdots+b_{ \text{J}}X_{\text{j}}}}}, \tag{4}\]  
where \(P\) is the probability of a nitrate concentration over the given threshold, \(X_{\text{j}}\) is the rating score of parameter j, and \(b_{\text{j}}\) is the coefficient for the corresponding parameter. These coefficients (\(b_{\text{j}}\)s) can be optimized with Maximum Likelihood Estimation. The likelihood of the regression model is formulated as follows [25]:  
\[l=\sum_{i=1}^{n}[y_{i}\ln(P_{i})+(1-y_{i})\ln(1-P_{i})],\]where \(l\) is the likelihood, \(y_{i}\) is the binary-coded value (0 or 1) for the nitrate concentration of sample \(i\), \(P_{i}\) is the predicted possibility based on Equation (4), and \(n\) is the number of samples. If \(P_{i}\in[0-1]\), \(\ln P_{i}\) and \(\ln(1-P_{i})\) are negative, \(l\) is also negative. Lastly, these coefficients are considered to be the corresponding weights.