# Assessment of Hongtu-1 Multi-Static X-Band SAR Constellation Interferometry
## 2 Materials and Methods
### Phase Unwrapping and Phase-to-Height Convertion  
Phase unwrapping is a critical step in the processing of interferometric data and a well-established area of research [34; 35; 36; 37]. Unwrapping the differential interferogram becomes simpler after subtracting the topographic phase, simulated using the Copernicus 1\" DEM, and applying the detrending step described above. Having multiple interferometric pairs available over the same area offers additional possibilities. For the unwrapping we used a minimum cost flow algorithm [35], as available in the Gamma Software [31]. For the investigated dataset, the unwrapping was not critical for pairs with shorter baselines. For the longer baselines (>300 m) the solutions obtained for the shorter baselines were used to check the unwrapped phase for ambiguity errors. Before performing the unwrapping, we masked areas with very low coherence (<0.2), i.e., water surfaces and radar shadow areas, which are incoherent because of backscatter levels below the NESZ. Overall, the unwrapping was not problematic in the multi-static HT1 dataset analyzed.  
For each pair, the unwrapped differential interferometric phases were converted into height corrections, \(\Delta h\), using the height-to-phase sensitivity  
\[\Delta h=\Delta\phi\frac{cRsin\theta}{2\pi fB_{\perp}} \tag{1}\]  
with \(\Delta\phi\) representing the unwrapped interferometric phase, \(c\) the speed of light, \(R\) the slant range, \(\theta\) the incidence angle and \(B_{\perp}\) the baseline component perpendicular to the line of sight. In the conversion of the phase-to-height corrections, \(B_{\perp}\) is recalculated for each pixel considering the geometry with the pre-existing terrain height from the Copernicus DEM. The ambiguity height, \(h_{a}\), corresponds to a height difference that corresponds to a \(2\pi\) interferometric phase  
\[h_{a}=\frac{cRsin\theta}{fB_{\perp}} \tag{2}\]  
In the investigated HT1 dataset, the ambiguity height varies from about 245 m to 29 m between the shortest and longest baselines of the multi-static pairs. For each of the six pairs listed in Table 1 a height correction is thus determined; a combined height correction is then calculated based on the six height correction results. How to best combine the six height correction results into one combined height correction is not entirely clear. Using a coherence estimate or a direct estimate of the phase noise from the data together with the phase-to-height sensitivity, reasonable weights can be determined for the different corrections. But this approach does not consider the possibility of remaining unwrapping errors and effects related to spatial filtering in a fully appropriate manner. The weights for the pairs with the longest baselines are the highest--but small areas or individual pixels may be affected by phase unwrapping errors. [PERSON] et al. [32] successfully used an advanced Two-Stage Programming Approach in the unwrapping and then weighted averaging and obtained good results for several HT1 acquisitions. Another approach, using a maximum likelihood algorithm for multi-baseline phase estimation, has been proposed in [38]. In our work the main objective was not really to independently determine a DEM, but to investigate the usefulness of multi-static acquisition for the estimation of DEM changes. We did not have high-quality ground-truth available to validate the result. As a compromise between avoiding outliers caused by ambiguity errors and achieving good quality in areas without ambiguity problems, we used the median over the six height correction estimates as our combined result. This method is simple, but tends to avoid outliers, e.g., by unwrapping errors, and reduces the noise in the result.