Spatial pattern regression for meteorological fields interpolation

Houssou, Vihotogbé; Carreau, Julie

doi:10.5194/hess-30-5791-2026

Articles | Volume 30, issue 18

https://doi.org/10.5194/hess-30-5791-2026

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

https://doi.org/10.5194/hess-30-5791-2026

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

Articles | Volume 30, issue 18

Research article

|

15 Sep 2026

Research article |

| 15 Sep 2026

Spatial pattern regression for meteorological fields interpolation

Vihotogbé Houssou and Julie Carreau

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Final revised paper (published on 15 Sep 2026)
Preprint (discussion started on 17 Apr 2026)

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2026-1702', Anonymous Referee #1, 10 May 2026

The manuscript describes a spatial interpolation method applied over North America. The method is described in detail and is suitable for the described applications. The validation is carried out in a reasonable way and the results are correctly interpreted by the authors. However, there are a number of issues that need to be addressed before this manuscript can be considered for publication.
Comments:
1. The SPR method described in the manuscript is not original. It was introduced years ago and is called Reduced Space Optimal Interpolation (RSOI, Schiemann et al. 2010). Please adjust your manuscript to acknowledge the RSOI reference and eventually point out the elements of originality of your work with respect to RSOI.
Ref:

Schiemann, R., M. A. Liniger, and C. Frei (2010), Reduced space optimal interpolation of daily rain gauge precipitation in Switzerland, J. Geophys. Res., 115, D14109, doi:10.1029/2009JD013047.
2. The direct backtransformation proposed at the beginning of section 3.5 is prone to introducing systematic errors in the final reconstructed field. When the background is a deterministic model, applying inverse transformations directly to the analysis can lead to systematic underestimation of precipitation, as described by Fletcher and Zupanski (2006). This occurs because the analysis-error variance at grid points must be considered in the inverse transformation. A correction method can be implemented using a Taylor series decomposition of the inverse transformation, as outlined in Fortin et al. (2015) and van Hyfte et al. (2023) for Box-Cox transformations. Alternatively, a more computationally intensive approach proposed by Erdin et al. (2012) involves applying the inverse transformation to 399 quantiles, equidistant in probability.

Please make the readers aware of this consequence of applying a direct inverse transformation.
Ref:

Fletcher, S.J. & Zupanski, M. (2006) A data assimilation method for log-normally distributed observational errors. Quarterly Journal of the Royal Meteorological Society, 132, 2505–2519. Available from: https://doi.org/10.1256/qj.05.222
Fortin, V., Roy, G., Donaldson, N. & Mahidjiba, A. (2015) Assimilation of radar quantitative precipitation estimations in the canadian precipitation analysis (capa). Journal of Hydrology, 531, 296–307.
van Hyfte, S., Le Moigne, P., Bazile, E., Verrelle, A. & Boone, A. (2023) High-resolution reanalysis of daily precipitation using AROME model over France. Tellus A: Dynamic Meteorology and Oceanography, 75, 27–49. Available from: https://hal.science/hal-04271427
Erdin, R., Frei, C. & Künsch, H.R. (2012) Data transformation and uncertainty in geostatistical combination of radar and rain gauges. Journal of Hydrometeorology, 13, 1332–1346.
3. Lines 84-88. This part is not clear. Does your method require a fixed station network over time?
4. Lines 89-92. This part is not clear. It seems that you are not using station data in your interpolation. Please rephrase this part.
5. It is not clear what procedure was used to select the data for the computation of PCA for daily data. Do you extract the PCA considering all daily data over the whole period? Do you consider only the data belonging to that specific day of the year? Please elaborate more on this point.
6. Can you elaborate a bit more on which strategy you used to fit the linear regression model of Eqs. (2)-(3)?

Citation: https://doi.org/10.5194/egusphere-2026-1702-RC1
- AC1: 'Reply on RC1', Vihotogbé Houssou, 09 Jun 2026
  
  We sincerely thank Referee #1 for their time, thorough review, and constructive comments on our manuscript. We have carefully considered all suggestions to improve the paper.
  Please find our detailed, point-by-point responses to all comments, along with an indication of how the manuscript will be revised, in the attached PDF file.
  
  Citation: https://doi.org/10.5194/egusphere-2026-1702-AC1
RC2:
'Comment on egusphere-2026-1702', Anonymous Referee #2, 15 May 2026

See PDF.

Citation: https://doi.org/10.5194/egusphere-2026-1702-RC2
- AC2: 'Reply on RC2', Vihotogbé Houssou, 09 Jun 2026
  
  We sincerely thank Referee #2 for their time, thorough review, and constructive comments on our manuscript. We have carefully considered all suggestions to improve the paper.
  Please find our detailed, point-by-point responses to all comments, along with an indication of how the manuscript will be revised, in the attached PDF file.
  
  Citation: https://doi.org/10.5194/egusphere-2026-1702-AC2

Peer review completion

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

ED: Submit a revised manuscript (25 Jun 2026) by Elena Toth

Dear Authors,
I personally believe that this kind of approach, exploiting the availability of RCM physically consistent simulations of the spatial fields to improve the spatial interpolation of sparse meteorological observations, is extremely promising.
Both Referees have acknowledged the relevance of the work and their constructive comments have suggested very useful revisions that will further valorise the research objectives and the novelty of the methodology: Ref#2’s comment has prompted a better definition of the research methodology and its potential, and Ref#1 has suggested the comparison with a previously presented similar methodology, and it is important to explain in the paper the differences from the proposed approach.
The Referees have both highlighted the need for several major revisions along the paper, necessary to clarify important aspects of the methodology and its soundness (in particular those on the back-transformation, on the role of the fixed station network, on the consistency of the application of SPR with the other methods, on the validation strategy and the ground truth). They have also asked for additional details (on the PCA and the regression fitting, on the impact of the station sampling) and for more discussion acknowledging some important limitations, in particular on the bias due to the smoothing represented by the evaluation against the RCM 11 km grids, and on the possible temperature biases due to inconsistencies between climatological periods.
I hope that the Referees will again help us in the revision process, explaining if they are satisfied with how you have addressed all their comments in your replies and with the revised version that you will submit.
Best wishes,
Elena Toth
University of Bologna
HESS Topical Editor

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AR by Vihotogbé Houssou on behalf of the Authors (15 Jul 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (22 Jul 2026) by Elena Toth

RR by Anonymous Referee #1 (27 Jul 2026)

Suggestions for revision or reasons for rejection

I would like to thank the authors for addressing the comments raised in my previous review. I appreciate the revisions they have made and am satisfied with their responses to my comments.

There is, however, one issue that remains unclear and that I believe is not given sufficient attention in the revised manuscript. To reproduce the reported results, readers need clear guidance on how to choose the number of principal components (or retained spatial patterns) required to achieve satisfactory model performance. At present, I could only find the following statement in Section 3.5.1:

"For SPR, the number of retained spatial patterns is selected as a percentage (10%–90%) of the maximum available patterns, allowing a consistent search across regions of different sizes."

This design choice is fundamental to the proposed methodology and has important implications for many potential applications. As such, it deserves a more thorough discussion.

For example, how is the optimal number of retained spatial patterns ultimately determined? At present, this information is difficult to identify in the manuscript. Is the selected number fixed throughout the study period, or does it vary from one sample or time step to another? If it varies, what are the implications for users interested in climatological applications, such as estimating long-term trends or return values? More generally, does allowing the number of retained principal components to change across samples affect the statistical properties, consistency, or interpretability of the resulting predictions?

I encourage the authors to expand the discussion of this aspect of the methodology and to provide sufficient detail to enable readers to reproduce the results and understand the implications of this important modeling choice.

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RR by Anonymous Referee #2 (19 Aug 2026)

ED: Publish subject to minor revisions (review by editor) (26 Aug 2026) by Elena Toth

AR by Vihotogbé Houssou on behalf of the Authors (27 Aug 2026) Author's response Author's tracked changes Manuscript

ED: Publish as is (01 Sep 2026) by Elena Toth

AR by Vihotogbé Houssou on behalf of the Authors (02 Sep 2026) Manuscript

Short summary

Spatial Pattern Regression (SPR) is a new way to reconstruct daily weather fields in regions with few measurement stations. Our approach combines information from past high-resolution simulations with available observations to produce more accurate maps of precipitations and temperature. Tests on both synthetic and real data show clear improvements over common methods, especially when stations are sparse, helping support better hydrological and climate studies.