## 2 Materials and Methods
### Method of Analysis  
In this study, all sample observations were split into a training (70%) and test (30%) dataset of pairwise pixels (Table 4). Scatterplots were computed to visualize the comparison, where the \(x\)-axis stood for the pixel values of Sentinel-2 or Landsat-8 bands, and the \(y\)-axis represented the pixel values of Landsat-7 or Landsat-8 bands. Each scatterplot also displayed the slope of the Orthogonal Distance Regression (ODR) that shows the linear relationship value considering errors in both the independent and dependent variables. The ODR slope measures the change in the dependent variable (e.g., pixel values) for a unit change in the independent variable (e.g., another set of pixel values) in a regression model. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Training Set** & **Test Set** \\ \hline MSI vs. OLI & 24,664 & 10,570 \\ MSI vs. ETM+ & 18,116 & 7764 \\ OLI vs. ETM+ & 17,948 & 7692 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Sample points numbers of each pairwise comparison split into a training and test set. \begin{table}
\begin{tabular}{c c} \hline \hline
**Sensor** & **Sample Points** \\ \hline MSI vs. OLI & 35,234 \\ MSI vs. ETM+ & 25,880 \\ OLI vs. ETM+ & 25,640 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sample points numbers of each pairwise comparison. The ODR slope closer to one suggests that the two sets of pixel values have a direct, proportional relationship, where an increase or decrease in one set corresponds to an equal increase or decrease in the other set. It indicates a stronger and more consistent linear relationship between the pixel values. Two types of regression models were computed to estimate the relationship between the sensor bands: Ordinary Least Squares (OLS) and Major Axis (RMA) regression models. OLS regression models were used to assess the difference between bands for the respective sensors. To fit prediction models, Equation (1) was fitted for each band:  
\[p_{xi}=a+bp_{xj} \tag{1}\]  
where \(p_{xi}\) is the surface reflectance for Landsat 7/8 and Sentinel-2 at a given wavelength, based on the training set pairwise and parameters \(a\) and \(b\) are intercept and slope applicable to each band, respectively. The RMA regression model, used by [PERSON] et al. [39] was also employed to determine the value of differences between each sensor band. Unlike OLS models, RMA assumes that both the dependent and independent variables can be subject to error [62]. This analysis was implemented using the'maregress' function available in MATLAB ([[https://www.mathworks.com/matlabcentral/fileexchange/27916-maregress](https://www.mathworks.com/matlabcentral/fileexchange/27916-maregress)]([https://www.mathworks.com/matlabcentral/fileexchange/27916-maregress](https://www.mathworks.com/matlabcentral/fileexchange/27916-maregress)) (accessed on 5 May 2023)). The regression coefficients (intercept and slope) derived from the two regression models for each band comparison were employed as correction factors for sensor band harmonization, adapting Equation (1), where \(p_{xj}\) was replaced with the SR value of the specific band being corrected for each comparison (MSI vs. OLI, MSI vs. ETM+, OLI vs. ETM+). A \(t\)-test was used with a significance \(p\)-value \(<\) 0.05 to assess the relationship between pairs of band reflectance values and determine the accuracy of regression model coefficients. Additionally, two metrics of difference were utilized: the mean difference (\(\Delta\)) and the root-mean-square deviation (RMSD). The mean difference metric (Equation (2)) provided insights into the average deviation between the predicted and observed reflectance values, indicating systematic bias in the regression model. \[\Delta=\sum_{i}^{n}\frac{\left(v_{i}^{a}-w_{i}^{b}\right)}{n} \tag{2}\]  
The mean difference represents the arithmetic average of the individual differences between paired values from the two distributions \(\left(v_{i}^{a}\right)\) and \(\left(w_{i}^{b}\right)\), capturing the overall directional discrepancy.