# Elevation and Mass Changes of the Southern Patagonia Icefield Derived from TanDEM-X and SRTM Data
## 2 Materials and Methods
### Accuracy Assessment  
The mass balance inaccuracy measures result from the following term for error propagation  
\[\varepsilon_{M}=|M|\times\sqrt{\left(\frac{\delta_{dh/dt_{tot}}}{dt/dt_{tot}} \right)^{2}+\left(\frac{\delta_{A}}{A}\right)^{2}+\left(\frac{\delta_{\rho}}{ \rho}\right)^{2}} \tag{1}\]  
where \(\delta\) identifies the uncertainty of the respective variable of elevation change rate (\(dh/dt_{\text{tot}}\)), area (\(A\)) and volume to mass conversion (\(\rho\)). In the following we describe these error terms with their composition and determination in more detail. The error of the elevation change rate (\(\delta_{\text{dh/dt tot}}\)) comprises the error of the relative vertical accuracy of the two DEMs (\(\delta_{\text{dh/dt vert}}\)), the radar penetration into snow and ice (\(\delta_{\text{dh/dt radar}}\)) and an error contribution resulting from the extrapolation during gap filling (\(\delta_{\text{dh/dt exttpl}}\)). This error term can hence be written as  
\[\delta_{dh/dt_{tot}}=\sqrt{\delta_{\text{dh/dt vert}}{}^{2}+\delta_{\text{dh/dt radar}}{}^{2}+\delta_{\text{dh/dt exttpl}}{}^{2}} \tag{2}\]  
We assessed the relative vertical accuracy of our TDX DEMs to SRTM on land (off glaciers, fjords, lakes) for every examined time step. For each mosaic, we provide a map to judge the spatial distribution of the error and a plot displaying the median elevation change between the respective DEMs (cf. Supplementary Materials Figure S1). This error estimation method uses a conservative approach, including more pixels than assumed to be stable in the DEM adjustment (c.f. Section 2.3). Vegetation effects, landscape changes or outliers due to inaccuracies of the signal are not eliminated a priori. Nevertheless, elevation changes exceeding 15 m were not taken into account for the calculation, as we assume this a threshold for a remaining non-stable target. Since the deviation is known to be dependent on the slope of the examined area [21; 34], we provide the median calculated for each bin of 5\({}^{\circ}\) slope (c.f. Supplementary Materials Figure S4.). The increase in the median absolute deviation (MAD) between the compared elevation values with increasing slope remains in a range of 6 m for all mosaics for slopes \(\leq\) 45\({}^{\circ}\) with the median ranging within \(\pm\)1.2 m.  
\[MAD_{\mathit{Aw}}=\frac{\sum MAD_{\mathit{s}\mathit{p}\mathit{i}}\times A_{ \mathit{s}\mathit{p}\mathit{i}}}{A} \tag{3}\]  
Equation (3) shows the calculation for each area weighted MAD (\(MAD_{\mathit{Aw}}\)), where the area proportion of each slope bin of 5\({}^{\circ}\) (\(A_{\mathit{s}\mathit{p}\mathit{i}}\)) is considered (with \(A\) being the entire measured area and \(MAD_{\mathit{s}\mathit{p}\mathit{i}}\) the respective MAD of the slope bin) This resulted in an \(MAD_{\mathit{Aw}}\) of 3.14 for our 2000-2015 elevation change measurements corresponding to a \(\delta_{\mathrm{dh/dt\,vert}}\) of \(\pm\)0.198 m\(\cdot\)a\({}^{-1}\) when divided by the mean time span of 15.86 years. In order to account for differences radar signal penetration of C- and X-band in the snow and glacier surface, we add the term of \(\delta_{\mathrm{dh/dt\,radar}}\). We assume no penetration on melting surfaces up to the average ELA (1050 m cf. Section 2.3) for both frequency bands and a linear increase of up to 5 m in the highest areas. This results in an uncertainty of \(\pm\)0.081 m\(\cdot\)a\({}^{-1}\) after area weighting. For the error contribution of gap filling (\(\delta_{\mathrm{dh/dt\,extpl}}\)) using hypsometry (cf. Supplementary Materials Figure S5b) the uncertainty is given by the standard deviation in each bin. Performing again area weighting we reach a value of \(\pm\)0.191 m\(\cdot\)a\({}^{-1}\). Using Equation (2) the dh/dt uncertainty computes then to \(\pm\)0.287 m\(\cdot\)a\({}^{-1}\) for the entire SPI including the interpolated areas. For the actually measured area (without interpolation) the error is \(\pm\)0.214 m\(\cdot\)a\({}^{-1}\).