## 2 Materials and Methods
### Research Methods
#### 2.3.2 Correction of Multi-Source Systematic Errors  
The glacier surface elevation change data obtained in the previous step still contain systematic errors due to seasonal cycles and different radar bands of the input data.  
Since the TanDEM-X bi-static InSAR data used in this study use X-band radar and the external DEM is based on C-band radar data, it is necessary to estimate the penetration difference between X-band and C-band radar. Differences in radar penetration are the largest source of uncertainty in the geodetic estimation of glacier mass balance [5]. Instead of using a set value for this difference, which is a common practice in the literature, in this study, we used a more rigorous approach:  
Step 1: We select test sites where the X-band SRTM DEM of the ascending and descending tracks overlap, and the DEM is the fused product of both the ascending and descending SRTM DEM without the influence of the three major radar geometric distortions, such as foreshortening, layover, and shadowing.  
Step 2: We examine the C-band and X-band SRTM DEMs for horizontal offsets. The co-registration algorithms of remote sensing images are very mature, but DEM co-registration is still relatively ambiguous. We used the relationship between the DEM elevation difference (_dh_) and the slope (_a_) and slope orientation (_Y_) to generate a two-dimensional scatterplot. The horizontal offset value (_a_), horizontal offset direction (_b_), and elevation deviation (_c_) between the two DEMs were calculated based on the shift vector that are presented in the scatter plot, where \(a\), \(b\), and \(c\) are the magnitude, direction, and offset of this shift vector, respectively (Equation (2)) [41].  
\[\frac{dh}{\tan(\alpha)}=a\cos(b-\psi)+c \tag{2}\]  
Step 3: The C-band and X-band SRTM DEMs for elevation deviation are reconfirmed in the vertical direction. In addition to the elevation deviation obtained in the previous step, we calculate the difference between the two DEMs and reconfirm the elevation deviation.  
Step 4: The radar penetration difference values are calculated and corrected. After the above three steps, we determine whether there is a horizontal offset and an elevation bias between the two DEMs. If there is an offset, we carry out the corresponding offset correction, and if there is no offset, we generated radar penetration differences for each test site by averaging the elevation differences of the glacier regions in the two DEMs. The final radar penetration difference is then averaged by calculating the radar penetration difference for the two test sites.  
Unlike glaciers in other regions of the Tibetan Plateau, most of the glaciers in the SETP are spring-accumulating, with the maximum ablation occurring in summer. A total of 6 of the 12 pairs of TanDEM-X bi-static InSAR images were acquired at around the end of August, but the external DEM was acquired in February 2000. Therefore, six pairs of TanDEM-X bi-static InSAR images contain a seasonal signal that is not present in the reference DEM. To correct for this seasonal bias, we selected two pairs of TanDEM-X bi-static InSAR images that cover the same glacier (Parlung No. 4) in winter (18 February 2014) and summer (14 August 2014). The main reason for selecting this particular glacier is that there are field measurements of this glacier, such as glacimeteorological measurements and mass balance stake records. The glacier mass balance obtained from the field measurements can be used to verify the glacial mass balance in this study. After the comparison of the elevation changes derived from these two image pairs, followed by additional verification using field measurement data, satellite altimetry ICESat, and satellite gravity GRACE, a seasonal correction factor was applied to all of the TanDEM-X-derived elevation change data.