We will reproduce the reviewer's comments in plain type, and give our responses in bold type.

Snow Water Equivalent (SWE) is a key parameter in hydrological, climatological and meteorological applications. New efforts for spaceborne radar-based SWE retrieval algorithms are under development and this paper offers great insight. This paper focuses on the influence of prior information, first guess SWE in this case, on the retrieval. They use previous SWE retrieval in a time series over a winter which reduces the influence of bias from SWE prior coming from an external source. This has benefits in SWE retrieval for future-based satellite missions. This paper is well structure and easy to read. I only have a few comments that would help the understanding of the reader.

We thank the reviewer for these comments.

Specific comments

1. Line 48: the scattering albedo 𝜔 is not well known in the snow community. I suggest defining it a little bit more.

We agree. We will add a short paragraph with a more detailed description of the parameterized forward model, and the scattering albedo, in the introduction.

1. Line 96: “model predictions of the same...” what?

We will reword to prevent confusion.

1. Section 4.1: Is it snow surface scattering or ground surface scattering (background)? I think I know the answer, but it is a bit confusing. Sometimes both terms are used (Line 188-189). I suggest sticking to one and defining it more clearly.

Good point: we are referring to the ground surface scattering throughout, and we do not consider the snow surface scattering. We will fix this. 

1. Line 173: remove when.

This sentence is confusing. We will reword.

1. Line 185: “Surface scattering is assumed to remain constant throughout the entire winter”. Why is that? perhaps cite a paper about constant soil permittivity over the winter. 

We will add a citation of Lemmetyinen et al., Snow density and ground permittivity retrieved from L-band radiometry: Application to experimental data, Remote Sensing of Environment, 180, 377-391, doi:10.1016/j.rse.2016.02.002

1. Line 186: The observation uncertainty symbol is wrong.

We will fix.

1. Figure 2-3-4: It says on the legend that b) and e) show the SWE of ERA5 + bias. Is it a typo or the bias is indeed shown? It seems like the SWE contains no bias from the curve on the graph.

That figure shows ERA5 with no bias added. We will fix the legend.

1. Section 6: One quick takeaway looking a Fig 2 and 3 is that we don't need retrieval, ERA5 is already good. ERA5 SWE prior is close to the true SWE, even before the retrieval. I doubt this is a takeaway you want the reader to leave with. A comment on this was made in section 2.5 on why ERA5 is good at this site but I think commenting again would help. It feels like the method relies on good prior estimation of SWE to predict SWE.

Thank you for raising this very important point. In most places globally, ERA5 does not perform as well as it does here at Sodankyla; thus there is still a need for remote sensing methods to measure SWE. We will add comments to this effect in the introduction, and will add a citation directly addressing the ERA5 accuracy. We will cite Mortimer et al.,  Evaluation of long-term Northern Hemisphere snow water equivalent products, The Cryosphere, 14, 1579-1594, doi: 10.5194/tc-14-1579-2020. Mortimer et al. show RMSE values for ERA5 ranging from 30 to 80 mm across an extensive global evaluation. An accuracy of 30 mm is inadequate for this study, where average true SWE is ~100 mm. Our target accuracy is 20%, or about 20 mm for a seasonally averaged RMSE.

However, the reviewer’s comments seem more focused on clarifying whether or not the method requires a good prior estimate of SWE for the retrieval to work. In fact, our study was designed precisely to answer this question. We found that if you simply use the ERA5 as the prior and do not use the timeseries information, then one needs an accurate prior, as shown in previous studies. But using the previous retrieval as the prior removes the ned for an accurate prior on SWE. We will make the point much more clearly that the method does not require accurate external prior information when using the previous retrieval as the prior. Specifically, we will focus on improving and clarifying the first paragraph of the discussion, and the first paragraph of the conclusion. These are places where we discussed this topic in the submitted manuscript, but were not direct enough in our communication. 

In order to try to make this point visually, we are experimenting with creating a figure showing some of the biased timeseries results. It was not explicitly asked for by the reviewer. If we are able to create a figure able to capture this dynamic, we will include in the revision.

In summary, we believe these changes will address the reviewers’ point, and improve the manuscript.

1. Line 254-255: This information on the scattering albedo could be useful earlier in the intro. Also, this might concern the Zhu et al 2018 retrieval but why not use correlation length or grain size directly as a variable instead of this proxy? I’m not completely sold on the scattering albedo yet!

We will address this earlier in the manuscript, by stating that scattering albedo can be considered a proxy for snow microstructure. 

The reason to use single scattering albedo instead of correlation length or grain size is computational efficiency, and to focus on a well-constrained retrieval problem, at the expense of fidelity in the radiative transfer model. In order to fully invert a model such as the Snow Microwave Radiative Transfer Model, the Microwave Emission Model of Layered Snowpacks, or the Dense Medium Radiative Transfer Model, one must retrieve multiple snow parameters, such as snow depth, snow density, correlation length, etc. Each of these properties varies vertically within the snowpack, complicating the problem. The parameterized model dramatically reduces the parameter space, such that there are only two unknowns for the snowpack. This comes at the cost of working with the proxy single scattering albedo, instead of the microstructure. Thus, using the single scattering albedo is a pragmatic choice that simplifies the retrieval problem.

In our response we noted that we were experimenting with a figure to try to more directly illustrate the sensitivity of our results to bias for each of the three algorithms we explore. The attached shows our attempt at this, along with some explanation. (This document responds to comments made by both reviewers, so will post the identical document to both reviewers).

We will reproduce the reviewer's comments in plain type, and give our responses in bold type.

The paper describes a new approach to tackle problems in the retrieval of SWE from radar measurements. Typically high accuracy of a priori SWE and grain size are needed, but the presented algorithm is demonstrated to work with (only) highly-biased SWE as a priori. The results are promising for future snow satellite missions, and the topic is highly relevant. The paper is well-written and easy to read. I have only a few minor comments and suggestions.

 We thank the reviewer for these comments.

Minor comments

1. P1L13: Remove the duplicate ‘than’

Will do.

2. P1L23: ‘Roughenss’

Will fix.

3. P8L192: These values are used *in* the analysis.

Will fix.

4. Figs 2, 4, 5: The legend says “ERA+bias”, but caption states “do not include any artificially imposed bias”. Please check. Should Figs 4 and 5 show the used a priori (previous SWE and weighted average) rather than ERA, or are the legends wrong? Please check.

This comment is on the same topic as the Reviewer #1. We here reproduce the same response, which describes our response toboth comments.

Thank you for raising this very important point. In most places globally, ERA5 does not perform as well as it does here at Sodankyla; thus there is still a need for remote sensing methods to measure SWE. We will add comments to this effect in the introduction, and will add a citation directly addressing the ERA5 accuracy. We will cite Mortimer et al.,  Evaluation of long-term Northern Hemisphere snow water equivalent products, The Cryosphere, 14, 1579-1594, doi: 10.5194/tc-14-1579-2020. Mortimer et al. show RMSE values for ERA5 ranging from 30 to 80 mm across an extensive global evaluation. An accuracy of 30 mm is inadequate for this study, where average true SWE is ~100 mm. Our target accuracy is 20%, or about 20 mm for a seasonally averaged RMSE.

However, the reviewer’s comments seem more focused on clarifying whether or not the method requires a good prior estimate of SWE for the retrieval to work. In fact, our study was designed precisely to answer this question. We found that if you simply use the ERA5 as the prior and do not use the timeseries information, then one needs an accurate prior, as shown in previous studies. But using the previous retrieval as the prior removes the ned for an accurate prior on SWE. We will make the point much more clearly that the method does not require accurate external prior information when using the previous retrieval as the prior. Specifically, we will focus on improving and clarifying the first paragraph of the discussion, and the first paragraph of the conclusion. These are places where we discussed this topic in the submitted manuscript, but were not direct enough in our communication. 

In order to try to make this point visually, we are experimenting with creating a figure showing some of the biased timeseries results. It was not explicitly asked for by the reviewer, so we are currently still experimenting with these figures.

In summary, we believe these changes will address the reviewers’ point, and improve the manuscript.

5. It also looks like the ERA SWE is already very accurate. Why do we need the retrieval? I would highlight the results of Fig. 3 even more to show that retrieval results are good despite biased a priori data. Perhaps add retrievals using biased data to Fig 2,4,5?

Thank you for raising this point, which is critical. It was also brought up by Reviewer 1, and our response to Reviewer 1 fully addresses this point as it is raised here.  See our response to Reviewer 1, comment 8. 

6. P10L224: “This divergence in February highlights a weakness of using the previous retrieval.” Does this mean that if the estimate is wrong, then this error is carried on in further retrievals? Please elaborate.

The basic idea of using the previous retrieval as a prior for the current day is to carry information from one day to the next. The risk of using ONLY the previous retrieval, and not a combination of the previous retrieval and ERA5, is that an estimate diverges. Our third algorithm which uses a weighted average of the two does not carry a high risk of divergence, and it performs far better. We will clarify this in the manuscript.

7. P13 L278: ‘dependnence’

Will fix.

In our response we noted that we were experimenting with a figure to try to more directly illustrate the sensitivity of our results to bias for each of the three algorithms we explore. The attached shows our attempt at this, along with some explanation. (This document responds to comments made by both reviewers, so will post the identical document to both reviewers).