A methodology to obtain model-error covariances due to the discretization scheme from the parametric Kalman filter perspective

Pannekoucke, Olivier; Ménard, Richard; El Aabaribaoune, Mohammad; Plu, Matthieu

doi:https://doi.org/10.5194/npg-28-1-2021

Articles | Volume 28, issue 1

https://doi.org/10.5194/npg-28-1-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/npg-28-1-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 28, issue 1

Research article

|

14 Jan 2021

Research article |

| 14 Jan 2021

A methodology to obtain model-error covariances due to the discretization scheme from the parametric Kalman filter perspective

Olivier Pannekoucke, Richard Ménard, Mohammad El Aabaribaoune, and Matthieu Plu

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Final revised paper (published on 14 Jan 2021)
Preprint (discussion started on 08 May 2020)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC1: 'Comments', Anonymous Referee #1, 28 May 2020
- AC1: 'Answer to the referee 1', Olivier Pannekoucke, 14 Sep 2020
RC2: 'review of npg-2020-14', Anonymous Referee #2, 22 Jun 2020
- AC2: 'Answer to the referee 2', Olivier Pannekoucke, 14 Sep 2020
RC3: 'Referee comment', Anonymous Referee #3, 01 Jul 2020
- AC3: 'Answer to the referee 3', Olivier Pannekoucke, 14 Sep 2020

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Olivier Pannekoucke on behalf of the Authors (14 Sep 2020) Author's response Manuscript

ED: Referee Nomination & Report Request started (26 Sep 2020) by Wansuo Duan

RR by Anonymous Referee #2 (27 Sep 2020)

RR by Anonymous Referee #1 (09 Oct 2020)

Suggestions for revision or reasons for rejection

The authors have achieved a great job with this revised version. The presentation, the notations, the model assumption (typically approximation (27): neglecting Q in P^m computation), the mathematical and intuitive arguments which support this model are much clearer. There are no more ambiguities. The authors have answered all the questions of my review report. In particular, I thank the author for the reference to the work of Nicolis, which seems particularly relevant.

Yet, since I now well understood the main assumptions of the draft model, it raises other questions (or rather, it makes me reformulate some of my previous questions).
1) Mainly, I wonder what is the validity of the approximation (27). Actually, it may be a more severe assumption than assuming the decorrelation between model error and analysis error. I understand that all your methodology is built on top of this assumption (27); that relaxing this assumption has to be left for future works. And in my opinion, even if approximation (27) is debatable, this should not prevent your work to be publishable.
But, at least, you could check, in your numerical test case, the order of magnitude of the 3 terms of equation (21) : P^m (the “true” one, ie not the one approximated from (27)), Π^m and Q. You may focus on the matrix trace (mean of variances).
For instance, problems could appear if P^m~Q≫Π^m (when e.g. the already-discussed case ϵ^a≪1) or if P^m≪Π^m~Q. Indeed, Π^mcould be negative (even though it is probably positive in your diffusive example) in some cases and may balance Q in (27).
At page 15 (around line 45), orders of magnitude are compared, but it seems to me that no value is discussed for the “true” model error variance.
2) I do not understand the terminology “flow-dependent part of P^m” for Π^m and “climatological part /bias of P^m” for Q. It seems to me that both terms are flow-dependent, isn’t it? Is the bias terminology come from the type of ϵ_(q+1)^m (χ^a ) expression which depends on χ^t ? (ϵ_(q+1)^m (χ^a )≈∫_(t_q)^(t_(q+1)) [(v - U)· ∇ - κ∆] χ^a dt
But then, the same description/ terminology could be used for ϵ_(q+1)^m (χ^t ) and thus P^m because the ϵ_(q+1)^m (χ^t ) expression is similar. And P^m is the initial quantity of interest.

Typo and small corrections:

> Fig 1 : the « red » looks more like a salmon
> ^m is a the wrong place in equation 6
> Page 5, line 23, (2.1) => (21)
> Equation 47a, V => V^p

Referee Report: PDF

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RR by Anonymous Referee #3 (31 Oct 2020)

ED: Publish subject to minor revisions (review by editor) (11 Nov 2020) by Wansuo Duan

AR by Olivier Pannekoucke on behalf of the Authors (11 Nov 2020) Author's response Manuscript

ED: Publish as is (19 Nov 2020) by Wansuo Duan

AR by Olivier Pannekoucke on behalf of the Authors (24 Nov 2020)

Post-review adjustments

AA: Author's adjustment | EA: Editor approval

AA by Olivier Pannekoucke on behalf of the Authors (12 Jan 2021) Author's adjustment Manuscript

EA: Adjustments approved (13 Jan 2021) by Wansuo Duan

Short summary

Numerical weather prediction involves numerically solving the mathematical equations, which describe the geophysical flow, by transforming them so that they can be computed. Through this transformation, it appears that the equations actually solved by the machine are then a modified version of the original equations, introducing an error that contributes to the model error. This work helps to characterize the covariance of the model error that is due to this modification of the equations.