The manuscript presents two assimilation methods for semi-positive-definite variables, specifically using the FALL3D model to study the deposition of tephra from the Calbuco eruption in Chile. These methods are proposed instead of more traditional methodologies (such as Kalman filter) as they are expected to work better in the case of positive-definite variables such as volcanic aerosols and tephra deposits.

The manuscript is well-written and organised. The two methods discussed here are the Gaussian with Nonnegative Constraints (GNC) and the Gamma, Inverse-Gamma (GIG) Gaussian Ensemble Kalman Filter are presented in detail and show promising results for the case study. Tephra transport and dispersion studies suffer large uncertainties due to the difficulties with sampling as well as poorly-defined initial conditions. As such, the use of data assimilation (especially using methodologies tailored for such application) represents an important way forward.

Overall, the work presented here represents an exciting step forward and I believe merits publication after a few revisions, as follows:

My biggest concern is that I feel that a comparison against a “control” Kalman filter methodology is necessary to highlight the strength of the new methods tested. As the main aim of the paper is to show that the two methods used are better-suited for tephra, I think that it is necessary to show that a “normal” Kalman filter methodology is problematic, or at least that it leads to worse results.

A smaller point is that I believe that the discussion section could be expanded a bit by including a paragraph that discusses uses of these methodologies beyond tephra, to highlight the strength of the methodologies in other settings.

A number of smaller comments can be found in the attached PDF.

One caveat of my review is that my experience with data assimilation techniques is limited. Judging from the presentation of the techniques, the arguments made by the authors and the results presented, both of the suggested methodologies seem to provide good candidates for such applications; however, the manuscript might also benefit from an additional review from a researcher who is more active in data assimilation techniques and modelling.

I hope that the authors find the points raised constructive and I would like to wish the writers the best of luck with the revisions (and a good time over the holidays).

I enjoy reading this interesting work focusing on two new ensemble-based data assimilation techniques for semi-positive-definite variables with highly skewed uncertainty distributions, including aerosol concentrations and tephra deposit mass loading. The results show that both assimilation methods lead to a significant improvement over the first-guess results obtained from the simple ensemble forecast. I acknowledge that the motivation, method, and results of this study are relatively well written, thus only a very Minor revision is needed before this work can be published in GMD.

L45: Did you assimilate the observation in the prior ensemble mean by using the traditional ensemble Kalman filters? Comparison of which can present the advantage of using GNC and GIG more clearly.

L235: I cannot follow the analysis of observation error, which seems to be irrelevant to this work. What are these 7 groups referred to?

L315: In Fig. 6, there are more than 4 (2) data points laying outside the 1:10 ratio band.

L345: In Fig. 8 there are 5 counter lines. What does the first isopach denote? I also suggest to show the prior ensemble mean in Fig. 8, thus we can see how the (a) GNC and (b) GIG (typo in the title) improve the simulation of Tephra fall deposit. 

In the manuscript entitled "Reconstructing tephra fall deposits via ensemble-based data assimilation techniques", the authors propose and evaluate the benefit of recent well-posed data assimilation techniques to deal with the non-Gaussianity of the control variable and the observations, for the analysis of tephra deposits. This is indeed a relevant scientific lock to lift. Two methods are compared on a recent volcanic eruption, from which tephra deposits have been measured in-situ. The presentation of the manuscript and of figures is overall good. To my view, the results that are presented are original enough to be published. However, the manuscript should be improved along the following recommendations before publication.

General comments

The methodology of validation is well-posed in the sense that validation is done against non-assimilated observations. However, the validation stations and the assimilated measurements (Figure 2) are located in quite different areas. Moreover, many validation stations are out of the regions where tephra deposits were significant. How valid are the validation measurements? I would recommand to build a single dataset with all the available data and then to split it homogeneously (using some random procedure for instance) into a assimilation and a validation one, unless there are good arguments not to do so. 

In order to strenghten the results of the article, some plots may be added in the figures: the prior should be added to Figure 6 and 7, and a panel showing the prior should be added to Figure 8, in order to discuss the benefit of assimilation. It may be interesting, also, to plot the error values (any relevant indicator: bias, rmse or wrmse) of the analyses and the of prior at the different validation stations on a map; this could be an addition to Figure 8 for instance. In this way we could see how much and where error reduction occurs after assimilation.

In its present form, the connection of Section 4 with the purpose of the manuscript is not clear. Section 4 addresses the estimation of the source term, and it seems out of the main scope of the article. It is an interesting topic, that would deserve a complete study by itself, if a more clear connection to the present manuscript is not provided.

Specific comments 

lines 168-169: why only considering the GIG equations that assume a Gamma distribution for the prior and Inverse-Gamma for the observations? Arguments should be provided at this point, even though some results later (Figure 4) give some for the prior.

Section 3.4: it is important indeed to evaluate the probability law of the prior ensemble. It would also be important to evaluate the dispersion of the ensemble and the prior error variance, and particularly to assess it compared to the observation error variance, in order to quantify the relative weight of observations and of prior in the assimilation process.

Figure 2 (upper-right panel): are the isotachs from Van Eaton et al (2016) or from Romero et al (2016), as in Table 2?

Figure 6 and 7 may be merged into a single 4-panels figure, unless adding the prior in this figures (see General comments) makes it difficult.

Line 376: please rephrase "based on the Gaussian hypothesis", since Gaussian hypothesis is not formally a condition for applying EnKF. In case of non-Gaussian errors, the EnKF can apply, but it provides a result (analysis) that is not optimal.

Line 405, it is stated that "These reasons make the GIG method a better candidate for implementation in VATD models". However, "The GIG method is a sequential assimilation procedure proposed by Bishop (2016), in which single observations are sequentially assimilated" (line 402), and from this sequential aspect we may assume that paralleization is not possible. So what is the potential impact of this sequential aspect on runtime and on operations? Are some parallelization strategies possible? This deserves to be discussed when addressing operational implementation.