the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 19 Jun 2025
A non-stationary trans-Gaussian model for daily rainfall over complex topography
Abstract. The orographic effects that influence rainfall fields in mountainous regions depend on elevation and the exposure of the topography to prevailing winds. Transitions between wet and dry areas can occur within a few kilometers, creating strong horizontal gradients of various rainfall statistics such as the frequency of occurrence, the distribution of intensity and the structure of spatial correlation.
Most statistical models of daily rainfall assume spatial stationarity (i.e., the spatial homogeneity of rainfall statistics) and are therefore not well suited for studying the highly non-homogeneous characteristics of orographic rainfall. To overcome this limitation, we design a non-stationary trans-Gaussian geostatistical model for the analysis of daily rainfall fields over complex topography.
The modeling framework presented in this paper infers rainfall statistics from sparse rain gauge observations, simulates realistic rainfall fields after calibration and stochastically interpolates rain gauge observations to create rainfall maps. The performance of the model is assessed with data from the Island of Hawai‘i where extreme spatial gradients in rainfall are observed. The results presented in this paper demonstrate that a non-stationary trans-Gaussian model can skillfully reproduce orographic rainfall statistics as well as their variations in space.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-2181', Anonymous Referee #1, 15 Sep 2025
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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.
-
AC1: 'Reply on RC1', Lionel Benoit, 17 Oct 2025
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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.
-
-
RC3: 'Comment on egusphere-2025-2181', Anonymous Referee #3, 19 Sep 2025
# Referee Report## Overall assessmentThe 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 comments1. 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. $\mathcal{O}(N^2)$).## RecommendationThe 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.
-
AC2: 'Reply on RC3', Lionel Benoit, 17 Oct 2025
Peer review completion
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-2181', Anonymous Referee #1, 15 Sep 2025
-
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.
-
AC1: 'Reply on RC1', Lionel Benoit, 17 Oct 2025
-
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.
-
-
RC3: 'Comment on egusphere-2025-2181', Anonymous Referee #3, 19 Sep 2025
# Referee Report## Overall assessmentThe 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 comments1. 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. $\mathcal{O}(N^2)$).## RecommendationThe 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.
-
AC2: 'Reply on RC3', Lionel Benoit, 17 Oct 2025
Peer review completion
Journal article(s) based on this preprint
Model code and software
StochasticRainfallModel_Orography Lionel Benoit https://github.com/LionelBenoit/StochasticRainfallModel_Orography
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Lionel Benoit
Matthew P. Lucas
Denis Allard
Keri M. Kodama
Thomas W. Giambelluca
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(3018 KB) - Metadata XML
-
Supplement
(8081 KB) - BibTeX
- EndNote
- Final revised paper
See attached file