A non-stationary trans-Gaussian model for daily rainfall over complex topography

Benoit, Lionel; Lucas, Matthew P.; Allard, Denis; Kodama, Keri M.; Giambelluca, Thomas W.

doi:10.5194/hess-30-4799-2026

Articles | Volume 30, issue 14

https://doi.org/10.5194/hess-30-4799-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/hess-30-4799-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 30, issue 14

Research article

|

29 Jul 2026

Research article |

| 29 Jul 2026

A non-stationary trans-Gaussian model for daily rainfall over complex topography

Lionel Benoit, Matthew P. Lucas, Denis Allard, Keri M. Kodama, and Thomas W. Giambelluca

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Final revised paper (published on 29 Jul 2026)
Supplement to the final revised paper
Preprint (discussion started on 19 Jun 2025)
Supplement to the preprint

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-2181', Anonymous Referee #1, 15 Sep 2025

See attached file

Citation: https://doi.org/10.5194/egusphere-2025-2181-RC1
- AC1: 'Reply on RC1', Lionel Benoit, 17 Oct 2025
  
  Dear Editor and Reviewers,
  Thank you for your detailed comments and suggestions about our manuscript entitled “A non-stationary trans-Gaussian model for daily rainfall over complex topography”.
  To capitalize on your propositions of improvement, we suggest to:
  (1) Thoroughly revise the introduction in order to better contextualize this study within the field of stochastic rainfall models, better define trans-Gaussian models, and better highlight the novelty of our approach.
  (2) Add a new section entitled “Example dataset: orographic precipitation on the Island of Hawai‘i” after the introduction in order to illustrate the impact of complex topography on daily rainfall fields before introducing our statistical model.
  (3) Improve the description of the construction of the non-stationary covariance function in section “Making the trans-Gaussian model non-stationary” and better explain the estimation of the covariance parameters in section “Model calibration”.
  (4) Add scatter plots of all target statistics in Figures 3 and 4 in order to complement the existing maps and provide a more quantitative assessment of model performance for stochastic rainfall generation.
  More details about the proposed improvements and a detailed response to all questions raised by the reviewers can be found in our point-by-point responses in the supplementary pdf documents. In these responses, the reviewer comments are in normal font and our responses are in italic.
  We hope that our responses will address your concerns, and that our proposed improvements will meet your expectations.
  Best regards,
  Lionel Benoit, on behalf of the authors.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2181-AC1
RC2:
'Comment on egusphere-2025-2181', Anonymous Referee #2, 17 Sep 2025
Review of « A non-stationary trans-Gaussian model for daily rainfall over complex topography »
By Lionel Benoit et al.

Summary :
The article proposes a non-stationary trans-Gaussian geostatistical model for the analysis of daily rainfall fields over complex topography. The model is applied with very good performance to Hawaii island that shows very strong orographic effects.

Main comments :
The article is well written and very clear. The topic is clearly worth for HESS journal. However,

Although the results are clearly convincing on the ability of the model to reproduce strong spatial variability, it’s not completely clear to me what part of the model makes it suitable for modeling such orographic gradients : is it the use of a non stationary covariance function (however is it new in the context of SWG?) ? And/or the use of climate zones ? By the way, I think the climate zones should be introduced before, and their use should be motivated. Here they appear for the first time l. 100 and we don’t really understand what they are, and why.

I got pretty much confused with the covariance function. Reading Sections 2.2 and 2.3, I understood that the covariance function was non stationary within the climate zone (given by eq 2), then its location-dependent parameters were interpolated in space to produce maps. But then it is written in the conclusion that the covariance function is first assumed to be stationary within the climate zones and then a non-stationary covariance function is subsequently obtained by convolution (??) of these stationary covariance. So I probably strongly misunderstood this part of modeling. Please consider clarifying.

I found Section 2.6 difficult to understand because all the steps of the procedure are actually described later in Section 3. More importantly, I haven’t understood what monthly total you use to condition at the virtual stations since there is no data there ? Is is with HCDP ? In which case, I find the comparison in Fig 6 unfair because of course we expect at the end to retrieve the right monthly totals since we used them to condition.

The temporal dependence seems not to be modeled at all. Are daily rainfall fields in this region almost independent ? Please justify this assumption.

Minor comments :
- eq 2 : can the parameters (gamma, nu, rho) take any value ?
- eq 4 : shouldn’t the parameters be \nu_i, \nu_j, \gamma_{1,i}, \gamma_{2,i}, \gamma_{1,i}, \gamma_{2,i}, \rho_{2,i}, \rho_{2,j} as in eq. 1 ? I guess your notations are that \nu=(\nu_i,\nu_j) but that’s not clear.
- in the interpolation process, how do you constrain the positive parameters (e.g. k, theta) to be >0 ?
- Section 2.6 : In step 4, «assess the uncertainty of rainfall interpolation (described in Section 3.4) » : I don’t see any uncertainty in Section 3.4. In step 5 « Identify areas where rainfall maps are highly uncertain» : actually in Section 3.5, the virtual stations cover all the island, not only the most uncertain parts.
- Fig 2 : the number on the color scales are to small
- l 265 : please specify what are the main differences with the benchmark BSNLG2022 ? Is it only the covariance non stationarity ?
- Section 3.4 : I guess that BSNLG2022 could not be used as benchmark because it’s multisite and not spatial ?
- Fig 5 : it’s confusing that the same color scale is used for positive and negative criteria (MB ; best color=green) and positive criteria (MAE, best color=blue).
- Fig. 6 : please consider adding the raingauge locations in the middle row
Citation: https://doi.org/10.5194/egusphere-2025-2181-RC2
- AC3: 'Reply on RC2', Lionel Benoit, 17 Oct 2025
  
  Dear Editor and Reviewers,
  Thank you for your detailed comments and suggestions about our manuscript entitled “A non-stationary trans-Gaussian model for daily rainfall over complex topography”.
  To capitalize on your propositions of improvement, we suggest to:
  (1) Thoroughly revise the introduction in order to better contextualize this study within the field of stochastic rainfall models, better define trans-Gaussian models, and better highlight the novelty of our approach.
  (2) Add a new section entitled “Example dataset: orographic precipitation on the Island of Hawai‘i” after the introduction in order to illustrate the impact of complex topography on daily rainfall fields before introducing our statistical model.
  (3) Improve the description of the construction of the non-stationary covariance function in section “Making the trans-Gaussian model non-stationary” and better explain the estimation of the covariance parameters in section “Model calibration”.
  (4) Add scatter plots of all target statistics in Figures 3 and 4 in order to complement the existing maps and provide a more quantitative assessment of model performance for stochastic rainfall generation.
  More details about the proposed improvements and a detailed response to all questions raised by the reviewers can be found in our point-by-point responses in the supplementary pdf documents. In these responses, the reviewer comments are in normal font and our responses are in italic.
  We hope that our responses will address your concerns, and that our proposed improvements will meet your expectations.
  Best regards,
  Lionel Benoit, on behalf of the authors.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2181-AC3
RC3:
'Comment on egusphere-2025-2181', Anonymous Referee #3, 19 Sep 2025

# Referee Report

## Overall assessment

The paper proposes a Gaussian model for rainfall that is spatially non-homogeneous (non-stationary) and trans-Gaussian. The topic is relevant and of interest. However, several aspects of the presentation and model specification require clarification to make the contribution and novelty fully clear.

The abstract could be more concise. The literature review on rainfall models appears overly centered on the authors’ own prior work; the opening could be broadened to include a wider range of related approaches.

It is not yet clear, from the abstract and introduction, what specific objectives the generator targets and which statistics are used to evaluate performance.

## Major comments

1. Novelty and positioning

- What is the genuinely new feature developed here? Gaussian models (including some forms of non-stationarity) have been studied for some time (e.g., Alliot 2009). The comparison to prior literature should be expanded to better highlight the novelty.

- l50: “fully non-stationary” — in what sense? Space only, or also time?

2. Modeling assumptions (end of p. 3)

- The choices of Gamma and Matérn are not justified. Why Gamma rather than, say, double-exponential or heavier-tailed alternatives? Why Matérn rather than exponential, Gneiting Matérn, or other forms? Please discuss strengths/weaknesses of these choices.

3. Definition and use of “trans-Gaussian” (l91)

- The term sounds new here, but the description resembles what several cited works already do (sometimes under labels like “censored Gaussian”). Clarify what is new in your definition/usage.

- The function $\Psi$ is introduced in Section 2.1 but does not reappear until the start of Section 2.2. Consider re-using or referencing it explicitly at the end of Section 2.1 for continuity.

- l93: Define the separation vector and the norm used. If altitude/mountain effects are important, indicate where they enter the model.

- As I understand it, there are two kinds of non-stationarity: (i) parameter variation by location $s$ and (ii) possible non-stationarity in the covariance. Should the covariance be non-isotropic?

- Eq. (2) lacks sufficient explanation; please expand.

- Eq. (3): I am not sure $N_s$ was defined. What optimizer is used for $l$ in Eq. (3)?

- Estimation strategy: Is there a risk in estimating the Gamma parameters first and then $C$? Should these be estimated jointly? Why use pairwise likelihood, and what guarantees does it provide in this context?

4. Structure of presentation (l135 and Section 2.4–2.6)

- This is the second mention of “climate zones,” but they have not been presented. Without this context, it is difficult to follow. Either present the model fully and then describe the application (e.g., fitting within each zone), or introduce the application earlier so the assumptions are easier to track.

- The paragraph before Section 2.4 lacks mathematical detail, which makes it harder to follow; the last sentence is particularly unclear.

- Section 2.6 mentions MCMC, but its role is not clearly explained. A brief conceptual explanation would help (and the Appendix would benefit from a few guiding sentences). Clarify the list of steps as well.

5. Evaluation metrics and visualization

- In Section 3.3, Comparing to another model is a good idea, but including a comparison with a model not authored by the same team would strengthen the case. Also, briefly recap the key differences that make BSNLG2022 distinct.

- Figure 3 is promising; the Wasserstein distance is an interesting choice, has it been used elsewhere in the literature of weather generators?

- I would also like to see an observed vs. simulated scatter plot, or an overall RMSE for the chosen statistics. Visual comparison across multiple maps is difficult and the lack of quantitative comparison is a limitation.

## Minor comments

- Clarify the meaning of “local scale” vs. “rain gauge” (l114 and l132).

- l146: “more scalable” — please quantify (e.g., $\mathcal{O}(N)$ vs. $\mathcal{O}(N^2)$).

## Recommendation

The paper addresses a relevant problem and has potential, but it requires clearer positioning, more explicit modeling justifications, and a more transparent estimation and validation strategy. I recommend a revision that addresses the points above.

Citation: https://doi.org/10.5194/egusphere-2025-2181-RC3
- AC2: 'Reply on RC3', Lionel Benoit, 17 Oct 2025
  
  Dear Editor and Reviewers,
  Thank you for your detailed comments and suggestions about our manuscript entitled “A non-stationary trans-Gaussian model for daily rainfall over complex topography”.
  To capitalize on your propositions of improvement, we suggest to:
  (1) Thoroughly revise the introduction in order to better contextualize this study within the field of stochastic rainfall models, better define trans-Gaussian models, and better highlight the novelty of our approach.
  (2) Add a new section entitled “Example dataset: orographic precipitation on the Island of Hawai‘i” after the introduction in order to illustrate the impact of complex topography on daily rainfall fields before introducing our statistical model.
  (3) Improve the description of the construction of the non-stationary covariance function in section “Making the trans-Gaussian model non-stationary” and better explain the estimation of the covariance parameters in section “Model calibration”.
  (4) Add scatter plots of all target statistics in Figures 3 and 4 in order to complement the existing maps and provide a more quantitative assessment of model performance for stochastic rainfall generation.
  More details about the proposed improvements and a detailed response to all questions raised by the reviewers can be found in our point-by-point responses in the supplementary pdf documents. In these responses, the reviewer comments are in normal font and our responses are in italic.
  We hope that our responses will address your concerns, and that our proposed improvements will meet your expectations.
  Best regards,
  Lionel Benoit, on behalf of the authors.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2181-AC2

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (20 Oct 2025) by Marie-Claire ten Veldhuis

AR by Lionel Benoit on behalf of the Authors (05 Dec 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (09 Dec 2025) by Marie-Claire ten Veldhuis

RR by Anonymous Referee #3 (19 Dec 2025)

Suggestions for revision or reasons for rejection

I would like to thank the authors for their careful revision of the manuscript and their responses to my comments. The new version is clearer and the added explanations help the reader better understand the modeling choices and the novelties of the approach.

Having now better grasped the strengths of the paper, I believe that since the work combines existing ideas (non-stationary covariance and non-stationary marginal distributions), the validation methodology could be strengthened to more precisely demonstrate the benefits of this combination.

## Minor comments

1. Notation and definitions (l102, p5)
- The notation $\mathcal{D}$ is introduced to denote the set of rain gauges but is not consistently used later in the manuscript, where it would be valuable to distinguish between the discrete set of rain gauges and the continuous spatial domain.
- On a related note, it is unclear in Equations 6 and 8 how $\Psi$ is known for arbitrary locations $s$ when the model is only trained on discrete stations. Please clarify whether this interpolation is addressed later; if so, it would be helpful to announce this when first introducing these equations.
- A notation for the continuous spatial domain would also be useful to avoid confusion between fitting at stations versus generating over the fine spatial domain. For instance, in Equations 3, 6, and 8, it is not entirely clear whether the variables are defined on the station set or on the spatial domain.

2. Section structure and terminology (Section 4)
- The subsections in Section 4 should be renamed more descriptively. It appears that Section 4.2 addresses multisite validation against BSNLG2022, while Section 4.3 focuses on spatial validation against CAI. More explicit titles would help readers navigate the comparisons.
- The terms "conditional" and "unconditional" are somewhat confusing in this context. It seems these refer to multisite versus spatial generation, respectively. Please make this distinction more explicit in the text.
- Throughout the plots, text, and tables, consider using abbreviated model names rather than generic phrases like "non-stationary trans-Gaussian model" and "the benchmark model." This would make it easier to distinguish when BSNLG2022 is used versus when CAI is the comparison model.
- Why is CAI not included in the model comparison table? Including it would provide a more comprehensive overview.

3. Visualization choices
- Make sure your color map are colorblind friendly. The "turbo"-like color map can distort the perception of color differences.
- Similarly, in Figure 6 whet is the point of showing positive and negative errors in the color bar but having the same dark color at the end of both sides?

## Unclear points regarding validation

Several aspects of the validation procedure require clarification:

1. Simulation strategy
- In the multisite validation, are the metrics computed using a single simulation, or are they averaged over multiple simulations? This should be stated explicitly.

2. Cross-validation procedure
- The training/validation split using cross-validation is not detailed enough. Are you applying this procedure to all stations? Is it a leave-one-out approach (removing one station at a time)? Are the validation metrics averaged over all folds? Additionally, is the benchmark model trained using the same cross-validation scheme to ensure a fair comparison?

3. Benchmark model selection
- BSNLG2022 is described as the benchmark model, but then CAI is used for spatial comparisons because BSNLG2022 is only multisite. Please clarify the rationale for this choice and how it affects the interpretation of results.

4. Choice of metrics
- The use of MB and MAE for validation is not well motivated. These metrics increase the number of quantities to examine, yet their specific value is not discussed. Moreover, they compare observed values with simulated values directly, whereas weather generators are typically evaluated on their ability to capture the statistics of observed values. Please discuss these choices in more detail or consider alternative metrics that better capture the statistical properties of the data.

## Suggestions for strengthening the validation

- The contribution of full non-stationarity versus the imposed weather regimes is not entirely clear. It is possible that much of the spatial variability is already captured by the weather regime structure. A benchmark comparison against the same model with stationary covariance (and possibly stationary marginals, but the interest of the latter is more intuitive) would more clearly demonstrate the benefit of incorporating non-stationary covariance beyond what is learned through the weather regimes alone.
- Additionally, it is unclear whether CAI and BSNLG2022 include non-stationarity (are empirical copula non stationary?); this information is not evident in Table C1 and should be clarified.
- The CAI model is not validated using BS and MCRPS. Including as a benchmark the stationnary version of your model could allow to easily compare these metrics against your full model.

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RR by Anonymous Referee #1 (15 Jan 2026)

RR by Anonymous Referee #2 (23 Jan 2026)

Suggestions for revision or reasons for rejection

The authors received three detailed and demanding reviews. They have done an excellent job of taking their comments into account. The article is now much clearer. In particular, the abstract has been completely rewritten and now presents the novelties and specifics of the article much more clearly. The method section has also been completely rewritten and reorganized, with more detail on certain points. I enjoyed reading this new version and the efforts made by the authors to improve their manuscript. I still have a few comments that should be easy to deal with:

- I still had trouble understanding the covariance modeling. In fact, I only understood it when I read the conclusion (l529-532). “Similarly, the covariance function of the latent field is first assumed to be stationary within relatively small climate divisions during the estimation of the covariance parameters, then the covariance parameters are assigned to the barycenter of each climate division, and finally these parameters are interpolated through space and incorporated in a model of non-stationary covariance.” This is very clear, and these three steps (stationary adjustment, assignment to the barycenter, interpolation) should be presented in this way in section 3, which remains unclear on this subject. For example, barycenter assignment only appears in brackets on line 201 (“preliminarily assigned to the barycenter of each climate division”).

- I appreciate that the authors have provided more details on conditioning to monthly totals, as requested by one of the reviewers. However, the MCMC procedure is quite technical and I think all the text from lines 261 to 285 could be moved to the appendix.

- I also appreciate the efforts made to justify the Gamma/Matern/geometric anisotropy choices, as requested by one reviewer. However, it should be noted that other choices could have been made without calling into question the main contribution of the article, which is the use of a fully non-stationary trans-Gaussian model (as indicated in line 59).

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ED: Publish subject to revisions (further review by editor and referees) (26 Jan 2026) by Marie-Claire ten Veldhuis

AR by Lionel Benoit on behalf of the Authors (06 Mar 2026) Author's response Author's tracked changes Manuscript

ED: Publish subject to revisions (further review by editor and referees) (10 Mar 2026) by Marie-Claire ten Veldhuis

ED: Referee Nomination & Report Request started (24 Apr 2026) by Marie-Claire ten Veldhuis

RR by Anonymous Referee #1 (18 May 2026)

RR by Anonymous Referee #3 (29 May 2026)

ED: Publish as is (01 Jun 2026) by Marie-Claire ten Veldhuis

AR by Lionel Benoit on behalf of the Authors (04 Jun 2026) Manuscript

Short summary

In mountainous regions the interactions between topography and prevailing winds generate orographic effects, which modulate rainfall occurrence and intensity depending on slope exposure, finally creating strong rainfall gradients. This study introduces a geostatistical model dedicated to rainfall mapping in mountainous areas, which explicitly account for possible orographic effects.