# Long-Term and Decadal Sea-Level Trends of the Baltic Sea Using Along-Track Satellite Altimetry
## 2 Methodology
Certainly, the outliers of the SA data must first be identified and removed from further computations using a certain method (e.g., see Section 4.1 for further details). Since initial datasets of different SA missions may refer to different reference ellipsoids, then ellipsoidal correction (\(\Delta\text{E}(\varphi)\)) for each SA data point is required. This ellipsoidal correction (associated with reference ellipsoid parameters) refers all participating SA measurements to the same reference ellipsoid [33]. For making the SA-derived DT comparable with \(DT_{TG\in SA\ cycle}\), a certain amount of the \(DT_{SA}\) data points close the TG station for each SA mission and at each pass/cycle need to be spatially averaged to yield the following:  
\[DT_{SA\in TG}\Big{(}\varphi_{TG},\lambda_{TG},t_{cycle}\Big{)}=\frac{1}{m}{ \sum_{j=1}^{m}DT_{SA}\Big{(}\varphi_{j},\lambda_{j},t_{cycle}\Big{)}} \tag{5}\]  
where \(m\) is the number of \(j\)-th data points within the SA track segment up to a certain radius from each TG station. The resulting \(DT_{TG}\), \(DT_{TG\in SA\ cycle}\), and \(DT_{SA\in TG}\) time series will be used for estimating the sea-level trends for each dataset. The most common method to estimate the sea-level trend is the ordinary least-squares (OLS) estimator [53; 54; 55; 56]. The main advantages of using the OLS estimator for climate variables are that (i) it is consistent with previous estimators of sea-level trends and that (ii) the OLS estimate does not depend on the estimated variance-covariance matrix. This also means that the uncertainty estimates only depend on the variance-covariance matrix construction [13]. Hence, the linear regression model can be fitted to both SA and TG data to estimate the sea-level trend from each dataset. A linear model is defined as an equation that is linear in its coefficients:  
\[Y(X)=\beta X+\epsilon \tag{6}\]  
where the model is denoted by \(Y(X)\), \(X\) is a suitable datum and \(\beta\) is an unknown coefficient that can be estimated through OLS solution, and \(\epsilon\) is an unknown error (residual noise). The least-squares estimator of \(\beta\) (denoted by \(\hat{\beta}\)) can be derived as follows:  
\[\hat{\beta}\ \sim\ \big{(}X^{t}X\big{)}^{-1}X^{t}Y \tag{7}\]In most cases, \(\hat{\beta}\) follows a normal distribution (\(N\big{(}\mu,\sigma^{2}\big{)}\)), where \(\mu\) is the sample mean and \(\sigma^{2}\) is the sample variance. The trend estimate uncertainty can be found by using the coefficient confidence. The coefficient confidence intervals (CIs) provide a measure of precision for regression coefficient estimates. A \(100(1-\alpha)\%\) CI gives the range that the corresponding regression coefficient will be within \(100(1-\alpha)\%\) confidence, meaning that \(100(1-\alpha)\%\) of the intervals resulting from repeated experimentation will contain the true value of the coefficient. The \(100(1-\alpha)\%\) CIs for regression coefficients are as follow [57]:  
\[\hat{\beta}\pm t_{(1-\alpha/2,\ n-p)}SE\big{(}\hat{\beta}\big{)} \tag{8}\]  
where \(SE\big{(}\hat{\beta}\big{)}\) is the standard error of the \(\hat{\beta}\) coefficient estimate (minimum error gives the best solution), \(t_{(1-\alpha/2,\ n-p)}\) is the \(100(1-\alpha/2)\) percentile of \(t\)-distribution with \(n\)-\(p\) degrees of freedom, \(n\) is the number of observations, and \(p\) is the number of regression coefficients (e.g., for 95% confidence, \(\alpha=0.05\)). In our case, a linear regression is fitted to the DT time series (for both SA- and TG-derived ones) to determine the sea-level trend estimates (the line slope \(\beta\) shows the trend). The estimated \(\beta\) now represents the sea-level trend (in mm/yr unit) (where \(Y(X)=DT\) and \(X=t\) in Equations (6) and (7)) and the corresponding uncertainties are obtainable using Equation (8).