the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Apr 2026
Interferometric synthetic aperture radar (InSAR) phase data assimilation for Bayesian snow water equivalent estimation
Abstract. Active microwave measurements, including those using interferometric synthetic aperture radar (InSAR) techniques, have shown promise for characterizing seasonal snowpacks at high spatial resolution. This study demonstrates a Bayesian InSAR phase data assimilation framework for estimating snow water equivalent (SWE) changes (dSWE) and SWE accumulation from dry-snow phase delay measurements at C-, L-, and P-band wavelengths. Two idealized synthetic end-member observing system simulation experiments (OSSEs) were performed in the context of a deep snow year at sites in the Tuolumne watershed. A baseline case with 6-day temporal repeat was used to evaluate the Bayesian framework relative to a deterministic retrieval approach under the two end-member cases where: (i) phase delay data is perfectly unwrapped and (ii) phase delay data is wrapped. In the case of perfectly unwrapped phase measurements, the deterministic retrieval and Bayesian approaches both show good estimation of dSWE across all three wavelengths (< 23 mm RMSE). The Bayesian approach shows reduced RMSE in dSWE (~63-74% of deterministic retrieval RMSE) and SWE (~5-34% of the deterministic retrieval RMSE). The primary source of error in SWE for the retrieval estimates is at the site where a month-long gap in measurements, due to wet snow early in the accumulation season, leads to missing dSWE events that result in SWE underestimation. In the case of fully wrapped phase measurements, ambiguity due to wrapping leads to very large (bias) errors in deterministically retrieved dSWE. The Bayesian framework uses an appropriate likelihood function to account for phase wrapping so that, when combined with the prior information provided by the modeling framework, results show minimal degradation to the perfectly unwrapped case in most test cases (except for the C-band case with a long temporal measurement gap). Tests examining the sensitivity to measurement error standard deviation and temporal repeat highlight the ability of the Bayesian approach to add value to the retrieval of dSWE and SWE across a range of cases. Future work should test the Bayesian framework with real InSAR phase data (e.g. Sentinel-1 C-band and NISAR L-band) across the range of physiographic and snow characteristics and phase retrieval error expected in mountain snow domains.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-1594', Benoit Montpetit, 04 Jun 2026
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RC2: 'Comment on egusphere-2026-1594', Anonymous Referee #2, 10 Sep 2026
Margulis et al. present an OSSE-based analysis that compares deterministic and Bayesian approaches for SWE estimation using synthetic phase delay time series at three radar frequencies (C-band, L-band, P-band) with 6-day repeat for two experimental cases (perfectly unwrapped vs. wrapped). The experiments are carried out with the FSM2 snow model (for generating “truth” and for assimilation experiments with the particle filter) at three stations in the well-studied Tuolumne River Basin (California, USA) in water year 2017. The analysis demonstrates the efficacy of Bayesian assimilation for SWE estimation for all three radar frequencies, while the deterministic approach yields larger errors, especially when phase change is ambiguous (wrapped between -pi and +pi). Additional sensitivity experiments are conducted to assess the impact of phase error and repeat frequency (6-day vs 12-day).
Overall, this is generally a well-designed and executed study, and provides compelling evidence of the use of Bayesian methods for InSAR SWE estimation. These results are timely and should be of high interest to the snow science community, given the recent launch of NISAR and other ongoing efforts to develop InSAR SWE estimates from existing and proposed SAR missions. While the study is not comprehensive, I think it clearly demonstrates the advantages of the data assimilation approach over the deterministic dSWE retrieval approach. I have several suggestions and questions for the authors to consider as they revise the manuscript.
Comments
- Fundamental to InSAR is the problem of phase ambiguity, but I do not think the paper clearly explains the concepts of perfectly unwrapped vs. wrapped phase in Section 2. I think the paper would be more readily understood by those who are not InSAR experts if a conceptual figure was added in Section 2 to explain these central concepts and the problem of phase wrapping to dSWE retrievals and SWE at different SAR frequencies. This new figure might be similar to Fig. S5, but could be simplified to focus on dSWE rather than the individual SWE components (depth and density).
- On the subject of conceptual figures, I find it odd that the paper’s only conceptual figures (Fig. 1 and 2) are somewhat redundant of each other (and with Fig. S3). These figures present the differences in viewing geometry and snow scattering for monostatic and bistatic SAR configurations. Those two configurations do not appear to be relevant to the rest of the paper, which raises the question: why include two conceptual figures to distinguish these two configurations? I think it would be more effective to merge those into a single figure to emphasize the key information or move them to the supplement. Such a change would accommodate the addition of a conceptual figure about phase wrapping (see previous comment).
- The paper’s focus on C-band, L-band, and P-band is logical given existing, new, and proposed SAR missions. However, a common frequency that the paper does not consider is X-band SAR, which is already available from commercial providers (e.g., ICEYE, Capella) at temporal repeat as fine as sub-daily to 3-day. I think it could help to establish (in Section 2.1) why X-band was not considered in the analysis. Like C-band, I assume X-band would have high SNR but even greater phase ambiguity and more limited penetration into snow. However, the phase wrapping issue may be mitigated somewhat by the higher temporal repeat (1-3 days) of current X-band commercial systems.
- The study focuses on a single study basin (3 stations) and a single year (wet 2017 in California), and as a result only assesses the case of very deep snowpack (1300-2000 mm peak SWE). I appreciate that the authors acknowledge these deep conditions throughout the paper. However, I think it would be useful if they expand the discussion about the potential/limitations of the methodology in more moderate or shallow snowpacks, and how/whether the three SAR bands might perform differently.
- The OSSE framework leverages the FSM2 snow model with a maximum of 3 snow layers. What is the impact of the number of snow layers on the model results? The supplement document indicates that too few snow layers may result in an underestimation of wet snow effects, which is important to the InSAR experiments. At a minimum, it would help to include some review and discussion on the influence of snow model vertical complexity (e.g., Webb et al., 2022; Cristea et al., 2022; Decharme et al., 2016). More ideally, a sensitivity experiment of LWC to number of snow layers could be useful (note: FSM2 allows the user to change the maximum number of layers).
- A consistent result from the OSSE experiments was that the Bayesian approach typically yielded estimates that had little or no bias. To what extent is this an artifact of the OSSE design? Namely, the “truth” in the OSSE setup was an FSM2 simulation with a precipitation bias correction factor identified for each site, while time-varying errors (e.g., random errors) in precipitation were not considered (L. 533). In the current setup, missing observations (e.g. wet snow in the early season) may therefore have similar precipitation errors as other times when observations are available. One could argue that temporally varying precipitation errors might be included as a side analysis (similar to Section 4.3.1, analysis on phase error magnitudes).
- The experiment is conducted independently (i.e., point or pixel mode) at the three sites, but does not consider that data availability/uncertainty may vary in space and how spatial relationships may be leveraged when only certain parts of a basin have data gaps. A prime example is Experiment #2 with C-band, where the Bayesian approach yields large overestimates of SWE during an early season gap in measurements at PDS (Fig. 12 top) when there is no such measurement gap at DAN and SWE estimates have low error (Fig. 11 top, L 590-598). I think it would be helpful to include some discussion on the possibility of adapting the Bayesian SWE estimation approach in a spatial framework in future work.
- The determination of wet vs. dry vs. mixed snow from LWC is unclear. The supplement includes some of the background theory on how the complex permittivity is calculated while considering LWC (equation S4). I assume that dry snow is LWC=0%, but I am not sure how the paper classified the wet and mixed states and how this classification is different for each SAR band (Fig. 6). This is important because it determines the availability of phase observations in the experiments. Is it necessary to distinguish between mixed and wet, or could these be merged into a single wet snow class? It seems that for both of those cases, there are no phase observations available.
- In general, I think many figures might be better presented if the panels were rearranged to align common variables (e.g., so errors can be more easily compared horizontally). Currently, some adjacent panels have SWE errors and dSWE errors, which are not directly comparable. I identified a few figures (see below) that may particularly benefit from this type of redesign. Where possible, it would help to match the limits of these common axes so their relative magnitudes can be easily understood by the reader.
Line Comments
- 148-150: Could consider citing the recent Palomaki et al. (2026) paper that attempted to quantify these non-snow effects on phase change for InSAR dSWE measurements.
- 246: Why was this nominal error value selected for phase measurement error? What types of effects might induce a 20 deg error? More generally, is it reasonable to assume that all three radar bands would have a similar uncertainty? I understand the need for the same error in the context of the experiment, but it would help to elaborate whether a similar phase error would be expected in practice for each of the three bands.
- 292-294: Please provide quantitative justification for the assumption of decorrelation after 12 days.
- 321: Suggest adding “(i.e., equal weights)” after “1/N”.
- 399: What is “significant LWC”? This wording seems subjective / inexact.
- 444-446: Similar to previous comments, it is not clear that the LWC sensitivity is fully established.
- 467-469 and Fig. 8: This result may not be obvious as it relies on the reader to notice that the limits of the x-axis vary by panel. I would suggest revising Fig. 8 to flip the panels to align the common dimension (b value), and match the axis limits. This will make it more obvious that the posterior b-value distributions are quite narrow for C-band and progressively wider at L-band and P-band.
Figures and Tables
- Table 1 – suggest adding a row specifying the wavelength, as some readers may think more in wavelength space rather than frequency space.
- Fig. 10 – suggest stacking these panels vertically to align the common variable (retrievals errors vs. Bayesian errors).
- Fig 10 and Fig. 13 – suggest merging into a single figure, where the left panels are the unwrapped case and right panels are the wrapped case (and top is dSWE errors and bottom is SWE errors). This would allow a more direct comparison and make it obvious how the unwrapped and wrapped cases yield different error magnitudes. The current configuration requires the reader to flip back and forth between Fig. 10 and 13 and remember the values.
- Fig. 14 – Suggest flipping to 2 rows (dSWE errors on top, SWE errors on bottom) by 3 columns (bands). This would align their common information (y-axis errors) for ease of comparison.
- Fig. 16 – suggest flipping so dSWE errors are the top two panels and SWE errors are the bottom two panels.
References
- Cristea, N. C., Bennett, A., Nijssen, B., and Lundquist, J. D.: When and Where Are Multiple Snow Layers Important for Simulations of Snow Accumulation and Melt?, Water Resources Research, 58, e2020WR028993, https://doi.org/10.1029/2020WR028993, 2022.
- Decharme, B., Brun, E., Boone, A., Delire, C., Le Moigne, P., and Morin, S.: Impacts of snow and organic soils parameterization on northern Eurasian soil temperature profiles simulated by the ISBA land surface model, The Cryosphere, 10, 853–877, https://doi.org/10.5194/tc-10-853-2016, 2016.
- Palomaki, R., Hoppinen, Z., and Marshall, H.-P.: A spatiotemporal analysis of errors in InSAR SWE measurements caused by non-snow phase changes, The Cryosphere, 20, 2703–2721, https://doi.org/10.5194/tc-20-2703-2026, 2026.
- Webb, R. W., Musselman, K. N., Ciafone, S., Hale, K. E., and Molotch, N. P.: Extending the vadose zone: Characterizing the role of snow for liquid water storage and transmission in streamflow generation, Hydrological Processes, 36, e14541, https://doi.org/10.1002/hyp.14541, 2022.
Citation: https://doi.org/10.5194/egusphere-2026-1594-RC2
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This paper presents an OSSE on assimilating phase difference in repeat pass interferometric SAR data at C- L- and P-band, and its implication in estimating dSWE and SWE compared to direct retrievals. The manuscript is well written, and I really appreciated reading this paper. In my opinion, this paper is of great relevance to the community and highlights the strengths and weaknesses of the different sensors/platforms with respect to snowpack properties on the ground. The authors are very transparent in highlighting that this is not an extensive experiment covering all the possible outcomes of what can be expected in reality, but present the impact of key components such as wavelength, snow state (wet-dry), temporal correlation, signal-to-noise ratio, and phase ambiguity.
This paper also highlights the advantage of Bayesian approaches compared to more direct retrievals described by Eq. 2. This shows that even the Bayesian approaches to retrievals could be more suitable, simply by providing an uncertainty on the retrieved value.
Very minor comments:
1. One thing that is not discussed and could support the discussion is the higher sensitivity to atmospheric phase delays with lower frequencies. This gives more certainty to estimations at C-Band, since atmospheric corrections mostly rely on modelled atmospheric conditions. Maybe adding a line on this in the discussion could help guide future work.
2. I would standardize the phase units across the text to avoid confusion between degrees and radians. I would stick with the units that is used to convert dSWE to phi.
3. This might be outside the scope of this study and could be included in future studies is combining the different frequencies. It's been shown that combining C- and L-band phase information greatly improves dSWE/SWE estimates. In this data assimilation framework, I feel it would makes things very interesting showing that this method might/would benefit from the advantages of each frequency while minimizing the down sides.
4. If I understood correctly, the conversion from dSWE to phi assumes a homogeneous snowpack. Given that FSM2 can provide up to three layers, it would be nice to have an understanding of the stratification of the snowpack for the three sites. If the snowpacks are very heterogeneous, this also has implications in how phi changes between two observations, which can be a source of uncertainty in this analysis. Maybe adding a line in the discussion about this.
5. As for temporal correlation between wet-wet pairs, this assumes there is no major snow surface roughness change between the two acquisitions. A change in surface roughness, especially during melt/refreeze events is not uncommon. This could be added in the discussion, but I don't feel it is possible to quantify this in the current OSSE.