How far can the statistical error estimation problem be closed by collocated data?

Vogel, Annika; Ménard, Richard

doi:https://doi.org/10.5194/npg-30-375-2023

Articles | Volume 30, issue 3

https://doi.org/10.5194/npg-30-375-2023

© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/npg-30-375-2023

© Author(s) 2023. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 30, issue 3

Research article

|

19 Sep 2023

Research article |

| 19 Sep 2023

How far can the statistical error estimation problem be closed by collocated data?

Annika Vogel and Richard Ménard

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Final revised paper (published on 19 Sep 2023)
Preprint (discussion started on 10 Oct 2022)

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2022-996', Ricardo Todling, 09 Nov 2022

Please find my review in the attached PDF.

Citation: https://doi.org/10.5194/egusphere-2022-996-RC1
- AC1: 'Reply on RC1', Annika Vogel, 17 Jan 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-996/egusphere-2022-996-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2022-996-AC1
RC2:
'Comment on egusphere-2022-996', Anonymous Referee #2, 26 Nov 2022

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-996/egusphere-2022-996-RC2-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2022-996-RC2
- AC2: 'Reply on RC2', Annika Vogel, 17 Jan 2023
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-996/egusphere-2022-996-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2022-996-AC2
EC1:
'Comment on egusphere-2022-996', Olivier Talagrand, 19 Dec 2022

The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2022/egusphere-2022-996/egusphere-2022-996-EC1-supplement.pdf

Citation: https://doi.org/10.5194/egusphere-2022-996-EC1
- AC3: 'Reply on EC1', Annika Vogel, 17 Jan 2023
  
  We thank the editor for the suggestions. We provide a detailed response to each of the referees separately and prepared a revised version of the manuscript. We hope that we could reply and adopt the manuscript in a sufficient way. In case of remaining questions or suggestions, we are open for further discussions.
  
  Citation: https://doi.org/10.5194/egusphere-2022-996-AC3

Peer review completion

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

AR by Annika Vogel on behalf of the Authors (18 Jan 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (26 Jan 2023) by Olivier Talagrand

RR by Ricardo Todling (13 Feb 2023)

RR by Anonymous Referee #3 (25 Feb 2023)

Suggestions for revision or reasons for rejection

This study addresses the challenge of accurately determining error statistics for data assimilation and other uses. The work presented here builds on a growing body of geophysical applications of estimating error statistics using collocated observations. A detailed mathematical theory is presented to demonstrate how three or more data sets, with limited assumptions and no knowledge of truth, may be used to simultaneously and/or sequentially calculate error covariances and error cross-covariances for all data sets. The authors present illustrative examples of its application and a useful conceptual model for how it could be applied.

I find the work to be quite complex, in a way that I appreciate. That is, the level of detail is sufficient without being overbearing. And I believe this adds an intriguing method (or at least considerable detail to existing methods) to the literature on error estimation from multiple data sets. It’s compelling to think on how one might apply the method to existing data, especially the determination of the three data sets that comprise the basic triangle and the subsequent order of the calculation. However, I think the authors have addressed this with sufficient detail for the present work.

I recommend a number of minor and technical edits to the text prior to publication.

Minor comments:

The use of scalar D_i for number of estimated error statistics and the matrix D_i;j for error dependency matrices and residual dependency matrices is unfortunate. I think it’s clear enough, and perhaps this is common notation in the literature of which I’m not aware. But, the authors may consider changing the scalar D_i notation for clarity.

Section 4.2 might be better titled as “Approximation for more than three datasets” as Section 4.1, which uses three datasets, already has “multiple” datasets.

Figs. 2, 3, and 4: the bottom axes of some of the subplots are cut off in this version. Please check that this is not the case in the final figures.

The “innocov” and “innocross” terms in algorithms A1 and A2 should be defined. They refer to the residual covariances and residual cross-covariances, correct? It’s not clear with this naming, so please either change the terms or provide a (brief) definition. Are these perhaps terms from a previous version of the manuscript?

Many instances of “were” being used in place of “where”.

Line 15: remove comma between “both” and “background”

Line 24: “raises” instead of “arises”

Line 27: remove “being”

Line 30: remove “and” (“but has only”)

Line 36: “optimality”?

Line 62: “where each approach”?

Line 69: spelling of “unknowns”

Line 114: suggest “as is the case”

Fig. 1 caption: spelling of “and” (instead of “ans”)

Line 257: suggest “other datasets and a”

Line 282: spelling of “datasets”

Line 318: should this be “D_i” instead of “U_i”?

Line 405, 448: perhaps “partial independence”?

Line 430: spelling of “cross-covariances”

Line 489: suggest “20,000” (comma instead of period)

Line 520: suggest “basic triangle are illustrated”

Fig. 3 caption: should be “but with a neglected”

Line 563: spelling of “solvability”

Fig. 6 caption: would it be better to state the distance as representing “the accuracy of assumed independence “?

Line 588: should be “affect” rather than “effect”

Line 639: “major challenge”

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RR by Anonymous Referee #2 (05 Mar 2023)

ED: Reconsider after major revisions (further review by editor and referees) (09 Mar 2023) by Olivier Talagrand

AR by Annika Vogel on behalf of the Authors (01 May 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (04 May 2023) by Olivier Talagrand

RR by Anonymous Referee #2 (22 May 2023)

RR by Anonymous Referee #3 (23 May 2023)

ED: Publish subject to minor revisions (review by editor) (24 May 2023) by Olivier Talagrand

AR by Annika Vogel on behalf of the Authors (05 Jun 2023) Author's response Author's tracked changes Manuscript

ED: Reconsider after major revisions (further review by editor and referees) (12 Jun 2023) by Olivier Talagrand

AR by Annika Vogel on behalf of the Authors (07 Jul 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (11 Jul 2023) by Olivier Talagrand

RR by Anonymous Referee #2 (13 Jul 2023)

ED: Publish as is (13 Jul 2023) by Olivier Talagrand

AR by Annika Vogel on behalf of the Authors (17 Jul 2023) Manuscript

Post-review adjustments

AA: Author's adjustment | EA: Editor approval

AA by Annika Vogel on behalf of the Authors (30 Aug 2023) Author's adjustment Manuscript

EA: Adjustments approved (30 Aug 2023) by Olivier Talagrand

Short summary

Accurate estimation of the error statistics required for data assimilation remains an ongoing challenge, as statistical assumptions are required to solve the estimation problem. This work provides a conceptual view of the statistical error estimation problem in light of the increasing number of available datasets. We found that the total number of required assumptions increases with the number of overlapping datasets, but the relative number of error statistics that can be estimated increases.