## 2 Materials and Methods
### Data Analysis  
We adopted the PLSR model in the SIMCA-P software to explore the relationships between the HSI spectra and physicochemical characteristics (pH or EC) of the alkaline-saline soil. The multidimensional soil spectra contain not only the useful signals about the pH and EC but also noises due to complicated soil characteristics and data redundancy from spectral collinearity. The noisy signals decrease the model performance in regular regressions. The PLSR model could overcome these problems by projecting the spectral data into a low-dimensional space [31; 32] and extracted the orthogonal predictor variables from all of the predictor variables (\(X\)). First a set of orthogonal predictor variables extracted from all the predictor variables is the first component, which is a linear combination of all the predictor variables to the greatest extent. Then, PLSR can extract the second component from predictor variables and improve the performance of the model. In theory, this process is not repeated until the increase of R\({}^{2}\) is below 0.0975 [29]. Although a large number of components increase the R\({}^{2}\) of the PLSR model, these components bring noises that result in decreased performance in validation. Normally, the number of components does not exceed 4 or 5 in practical applications [40]. Finally, the optimal PLSR model can be identified as a linear combination of predictor (\(X\)) and dependent (\(y\)) variables. In this study, the predictor variables (\(X\)) contain the HSI reflectance bands from 21 to 115. The dependent variable (\(y\)) is soil pH or EC. The PLSR regression model can be expressed as:  
\[y=Xa_{j}+F \tag{1}\]  
where \(F\) is a constant, \(a_{j}\) denotes the final regression coefficients of the corresponding HSI bands. Because the predictor variables contain the HSI reflectance bands from 21 to 115, \(j\) (\(j\) = 1, 2,..., \(n\)) indicates the total number of predictor variablesand \(n\) = 95 in this study.  
During the process of PLSR model building, the performance of the PLSR model was evaluated with the R\({}^{2}\), RMSE, and ratio of performance to inter-quartile distance (RPIQ) proposed by [PERSON] et al. [41]. These equations used are:  
\[R^{2}=1-\sum_{i=1}^{p}(Y-Y^{\prime})^{2}/\sum_{i=1}^{p}(Y^{\prime}-\overline{ Y})^{2} \tag{2}\]  
\[RMSE=\sqrt{\sum_{i=1}^{p}(Y-Y^{\prime})^{2}/p} \tag{3}\]  
where \(Y\) and \(Y^{\prime}\) are the measured and predicted values of soil pH or EC, and \(Y\) represents the average of measured values of soil pH or EC. \(p\) denotes the number of validation samples of model building. The optimal PLSR models were then applied to the HSI image to estimate the pH and EC across the study area. As the training soil spectra have been resampled to the same bandwidth of the HSI image, the prediction effectiveness of the models directly applied to images was also tested with RMSE (Equation (3)) by comparing the predicted values from the image pixels and PLSR models with the actual measured values of sample sites. The image pixels had the same location as the sample sites. \(p\) refers to the number of validation samples of model inversion.  
\[RPIQ=IQ/RMSE \tag{4}\]  
\[IQ=Q_{3}-Q_{1} \tag{5}\]  
The RPIQ index, in which the standard deviation is replaced by inter-quartile distance (IQ), accounts considerably better for the spread of the population than the ratio of performance to deviation [41]. \(Q_{3}\) and \(Q_{1}\) are quartiles. \(Q_{1}\) indicates the value below which 25% of the samples can be found; \(Q_{3}\) represents the value below which 75% of the samples can be found. RPIQ can evaluate the performance of the PLSR model by using thresholds defined by [PERSON] et al. [41]. Specifically, RPIQ > 1.89 predicts excellent models, 0.76 < RPIQ < 1.89 predicts fair models, and RPIQ < 0.76 predicts non-reliable models [41].