## 2 Materials and Methods
### Data Analysis  
The relationship between the dependent variable and one or more independent variables is described using a linear regression model. All of the processes employed in a regression analysis, as well as the results reached, are based on the assumptions of the regression model. The centering predictor variable in the polynomial regression model is useful when there is reason to believe that the connection between two variables is curvilinear, and proposed centering as a technique of reducing multicollinearity.  
The polynomial model is maintained as simple as possible in terms of the order. Arbitrary fitting of higher-order polynomials can be a major regression analysis miscalculation. A model that is consistent with data knowledge and its surroundings should be considered. It is always possible for a polynomial of order (\(n-1\)) to pass through n locations, resulting in the discovery of a polynomial of sufficiently high degree that provides a \"good fit\" to the data. The \(k\)-th order polynomial model in one variable is given by  
\[y=\beta_{0}+\beta_{1}x+\beta_{2}x^{2}+\ldots+\beta_{k}x^{k}+\varepsilon\]  
If \(x_{j}=x^{j}\), \(j=1,2,3,\ldots,k\) then the model is multiple linear regressions model in k explanatory variables \(x_{1}\), \(x_{2}\), \(henx_{k}\). So, the linear regression model \(y=X\beta+\varepsilon\) includes the polynomial regression model. As a result, the same approaches that are used to fit a linear regression model can also be used to fit a polynomial regression model. For example:  
\[y=\beta_{0}+\beta_{1}x+\beta_{2}x^{2}+\varepsilon\]  
or  
\[E(y)=\beta_{0}+\beta_{1}x+\beta_{2}x^{2}+\varepsilon\]This is a polynomial regression model with a single variable and called a second order regression model or quadratic model. The coefficients \(\beta_{1}\) and \(\beta_{2}\) denotes linear and quadratic effect parameters respectively.  
Second-degree polynomial equation for regression analysis was used to establish and quantify the relationship between variables in a data set. Primary BC data and satellite WSA periodically summarized for three mountain ranges are summarized in Table 1. The resulting second-degree polynomial equations for three sampling sites at Karakoram, Himalaya, and Hindukush, along with cumulative data of BC and WSA using regression, are:  
\[\begin{array}{cc}\hline\hline\begin{array}{c}\text{Hindukush}\\ \text{(R^{2})}\end{array}&\text{\bf Karakoram}\\ \text{(R^{2})}&\text{(R^{2})}\end{array}&\text{\bf Himalaya}&\text{\bf Cumulative}\\ \hline\hline 0.5904&0.0082&0.5135&0.1575\\ \hline\end{array}\]  
For Karakoram:  
\[\begin{array}{c}\text{y}=-0.0023\text{x}^{2}+0.0116\text{x}+0.0406\\ \text{For Himalaya:}\\ \text{y}=-0.0054\text{x}^{2}+0.0408\text{x}-0.0219\\ \text{For Hindukush:}\\ \text{y}=-0.0077\text{x}^{2}+0.0541\text{x}-0.0329\\ \text{For Cumulative Data:}\\ \text{y}=-0.0043\text{x}^{2}+0.0295\text{x}+0.0052\\ \end{array} \tag{1}\]  
where \(x\) represents BC and y represents WSA in the above equations as BC has been taken as an independent variable while WSA has been taken as a dependent variable. Coefficient of determination (R\({}^{2}\)) of second order polynomial for regression are shown in the table below.