Weather persistence on sub-seasonal to seasonal timescales: a methodological review

Tuel, Alexandre; Martius, Olivia

doi:https://doi.org/10.5194/egusphere-2023-111

Preprints

https://doi.org/10.5194/egusphere-2023-111

Preprints

20 Feb 2023

| 20 Feb 2023

Weather persistence on sub-seasonal to seasonal timescales: a methodological review

Alexandre Tuel and Olivia Martius

Abstract. Persistence is an important concept in meteorology. It refers to surface weather or the atmospheric circulation either remaining in approximately the same state (stationarity) or repeatedly occupying the same state (recurrence) over some prolonged period of time. Persistence can be found at many different timescales; however, the sub-seasonal to seasonal (S2S) timescale is especially relevant in terms of impacts and atmospheric predictability. For these reasons, S2S persistence has been attracting increasing attention by the scientific community. The dynamics responsible for persistence and their potential evolution under climate change are a notable focus of active research. However, one important challenge facing the community is how to define persistence, from both a qualitative and quantitative perspective. Despite a general agreement on the concept, many different definitions and perspectives have been proposed over the years, among which it is not always easy to find one’s way. The purpose of this review is to present and discuss existing concepts of weather persistence, associated methodologies and physical interpretations. In particular, we call attention to the fact that persistence can be defined as a global or as a local property of a system, with important implications in terms of methods but also impacts. We also highlight the importance of timescale and similarity metric selection, and illustrate some of the concepts using the example of summertime atmospheric circulation over Western Europe.

Received: 30 Jan 2023 – Discussion started: 20 Feb 2023

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Journal article(s) based on this preprint

13 Sep 2023

Weather persistence on sub-seasonal to seasonal timescales: a methodological review

Alexandre Tuel and Olivia Martius

Earth Syst. Dynam., 14, 955–987, https://doi.org/10.5194/esd-14-955-2023,https://doi.org/10.5194/esd-14-955-2023, 2023

Short summary

Alexandre Tuel and Olivia Martius

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-111', Anonymous Referee #1, 24 Feb 2023

Reviewer Comment on: Weather persistence on subseasonal to seasonal timescales: a methodological review

Authors: Alexandre Tuel and Olivia Martius

Submitted to: egusphere 2023 111

Summary
The authors present a manuscript for a review article on persistence as the occurrence of approximately constant or recurrent atmospheric states in the subseasonal to seasonal (S2S) time range. Persistence determines damages, challenges understanding and promises predictab ility. The review considers various aspects of persistence: definitions, methodologies, and examples encompassed by this rather general notion

In their review, the authors undertake a challenge when they combine meteorological observations and a mathematical d efinition in terms of advanced statistical and dynamical approaches. While a general definition of persistence is already hard, the related notion recurrence is even more difficult to grasp and to distinguish. Not surprisingly, the review is weak when the authors attempt the almost impossible task to provide a comprehensive theoretical framework, but it gets stronger when it resorts to methodologies and meteorological examples. In particular t he variety of examples and their properties show the width of the seem ingly simple no tion persistence.
The authors are well aware of the difficulties involved and do not hide that. The review is wor th to read for two reasons: the collection of method ologies th at have been sugge sted to analyze persistence, and the huge amount of examples. In summary the authors provide a broad and useful overview with a lot of insight. Below I mention some aspects that I noticed.

Specific comments

Line 55: Since persistence needs a timescale for the definition this question is somehow circular: ‘At which timescale(s) does the persistence occur? There could be reference to Section 3.4.

Line 124, Eq (x(t))_t ∈ R^m what is the meaning of the subscript t together with the argument t? This should be explained.

Line 126, I recommend to use ‘state space’ instead of phase space (the often used notion phase space is reserved for Hamiltonian systems).

Line 136: To characterize the difference between precipitation and temperature an autocorrelation time would be appropriate here. This is better than ‘inertia’ since it is difficult to associate precipitation with an inertia, while temperature, on the other hand, has something like an inertia due to the heat capacity.

Lines 143, 155 in Section 3.1 Global, state and episodic persistence. It seems that global and state persistence are different concepts. Global persistence appears nothing else than stationarity which includes recurrence. Why is global persistence ‘strongly related to intrinsic system predictability’? Maybe these definitions could be useful: Does persistence of a state x(t) mean dx(t)/dt=0. And could global persistence be defined in terms of integrals or averages like d<x(t)>/dt=0? And are conservation laws useful?

Line 184: What is a ‘symmetrically, persistent state’?

Line 188: How can this sentence be understood: ‘However, state persistence only characterizes the average behavior of system states.’

Line 201: Section 3.2
Here is a good opportunity to define Lagrangian stationarity of a quantity ѱ(x,t) along a flow u, by ∂ѱ/∂t + u∙∇ѱ = 0, in comparison to the Eulerian stationarity with ∂ѱ/∂t = 0.

Line 216: The authors write ‘self-similarity of system values x(𝑡) with a metric’ but this notion can be confused with geometric self-similarity used in the definition of fractal objects for example. Or is that what the authors intend?

Line 300: write what the symbol E(…) denotes.

Line 348: On Long-range memory: mention the absence of a timescale.

Line 360: Write that the Hurst exponent H and d are related by H=2d+1, see the Table 1 in Franzke et al. (2020).

Line 382: Written is: ‘If temporal dependence is present’. This is unclear. Eq. (10) is the result a power-law in of ρ (Eq.(6)), line 350.

Line 490: Section 4.2.4. The role of the extremal index θ is difficult here. In extreme value statistics the extremal index is the inverse time scale with co-occurring extreme events, used to eliminate short term variability and to combine events. Why is this index a measure of stationarity? Is it correct that short term excursions are allowed with θ in Eq (13)?

Line 569: Please write that R_X₀ is the residence time for state x₀.

Citation: https://doi.org/10.5194/egusphere-2023-111-RC1
- AC1: 'Reply on RC1', Alexandre Tuel, 27 Mar 2023
  
  Dear reviewer, thank you for your comments. Please find our reply in the attached PDF.
  
  Citation: https://doi.org/10.5194/egusphere-2023-111-AC1
RC2:
'Comment on egusphere-2023-111', Abdel Hannachi, 03 Apr 2023

Review of "Veather persistence on sub-seasonal to seasonal

timescales: a methodological review" by Tuel and Martius,

submitted to Geoscientific Model Development.
The authors attempt to provide a 'comprehensive' review of weather

persistence on S2S timescales. The authors start by stating the

origin of persistence, i.e. stationarity/quasi-stationarity or

recurrence, along with the different methodologies. Stationarity

is then described in terms of globality (eg autocorrelation),

states (eg patterns) and episodic events. Lastly, they describe

recurrence as another facet of persistence via various

aspects/measures.

Recommendation

--------------
The paper exposes the subject of persistence in a detailed fashion.

It is quite useful to researchers working on S2S predictability,

especially early career scientists (eg PhD students).

I support its publication in Geoscientific Model Development

subject to some minor changes as detailed below.

Major Comments

--------------
The authors have put great effort in an attempt to bring together

all the different blocs of persistence. Persistence is a loose

concept as there is no definite definition for it, and the authors

propose to bring together its different facets. I had to go more

than once to get a clear picture of the different bits and pieces.

For example, in the 6 sections there are 13 subsections (excluding

subsub-subsections).
1. Perhaps to aid readers, especially early career researchers, an

extra figure showing a tree-like diagram linking the different

concepts of persistence would be welcome.
2. Some related papers are missing from this review. Two particular

references related to persistence and therefore predctability:

predictive oscillation patterns (Kooperberg and O'Sullivan 1996),

and a related paper: optimally interpolated patterns (Hannachi

2008), making use of the power spectra (ie autocorrelation.)
In global stationarity, thirs-order statistics based, eg on

bispectrum (Pires and Hannachi 2021) would complement the

persistence description.
In relation to extremes and persistence archetypal analysis

Hannachi and Trendafilov 2017, Chapman et a. 2022) also identifies

'quasi-stationary' states or regimes.

Minor Comments

--------------
1. Pg 1, abstract: delete acronym S2S. It is abbreviated in

section 1.
2. Pg 3, l75: you mean Fig. 1c.
3. Pg7, l175: delete 'of'
4. Pg 13, eq (15): may be use 0<|\alpha|<1 to include the case of

anti-persistence.

l358: add the following reference on short timescale of

precipitation (Hannachi 2014).

l367,369: may be 'persistence' is more convenient here

than 'stationary'.

5. Pg 14, last paragraph 4.1.2: Fig.3 does not seem to have been

mentioned.
6. Pg 15, l407: consider adding an earlier reference of Baur (1951)

on Grosswetterlagen.
7. Pg18, l461: add "by projecting simplified dynamics (eg quasi-

geostrophy) onto the leading modes of variability of the

GCM simulation" after "January conditions".
8. Pg 20, eq (13): x is not specified, and I think there is some

confusion in the original reference (Faranda et al.) I

think x represents the log of the distance between state

at time t {\bf x}(t) and {\bf x}_0.
9. Pg 23, eq(21): my understanding is that \alpha is a probability

(P(x_0) = P(x_{t+1}=x_0|x_t=x_0)), therefore eq(21) < 0,

and also log(\alpha)<0, please clarify.
10. Pg 31, l646: Ripley's K function was considered in Stephenson

et al. (2004), and Hannachi (2010).
11. Pg 31, l686: " ... as the variance of successive event counts

over an interval of ..."
12. Pg 37, l791: full stop before 'Note'

l793: 'too'

l801: use 'measures' instead of 'metrics'
Fig. 15: are a, b,c 1-day apart?
Section 5: I suggest renumbering/relabelling the subsections as:

5.1 Diagnostic methods

5.1.1 Window counts

5.1.2 Dispersion metrics

5.1.3 Ripley's K function

5.1.4 Distribution of inter-event times

5.2 Stochastic modeling of recurrence

5.2.1 Events series

5.2.2 Recurrence plots
References

----------
Baur F, 1951: Extended-range weather forecasting. In: T. F.

Malone, ed. Compendium of Meteorology, Amer. Met. Soc.,

Boston MA, pp. 814-33.

Chapman C C, D P Monselesan, J S Risbey, M Feng, and B M Sloyan,

2022: A large-scale view of marine heatwaves revealed by

archetype analysis. Nature Communication, 13, 7843.

Hannachi A, 2008: A new set of orthogonal patterns in weather

and climate: optimally interpolated patterns. J. Climate, 21,

6724-6738.

Hannachi A, 2010: On the origin of planetary-scale extratropical

winter circulation regimes. J. Atmos. Sci., 67, 1382-1401.

Hannachi A, 2014: Intermittency, autoregression and censoring: a

first-order AR model for daily precipitation. Meteorol. Appl.,

21, 384-397.

Hannachi A, and N Trendafilov, 2017: Archetypal analysis: mining

weather and climate extremes. J. Climate, 30, 6927-6944.

Kooperberg C, and F O'Sullivan, 1996: Prediction oscillation

patterns: A synthetic of methods for spatial-temporal

decomposition of random fields. J. Amer. Stat. Assoc., 91,

1485-1496.

Pires C, and A Hannachi, 2021: Bispectral analysis of nonlinear

interaction, predictability and stochastic modelling with

application to ENSO. Tellus A, 73, 1-30.

Stephenson D B, A Hannachi, and A O'Neill, 2004: On the existence

of multiple climate regimes. Q. J. R. Meteorol. Soc., 130,

583-605.

Citation: https://doi.org/10.5194/egusphere-2023-111-RC2
- AC2: 'Reply on RC2', Alexandre Tuel, 02 May 2023
  
  Dear Dr. Hannachi,
  Thank you for your comments. You may find our reply in the attached PDF.
  Best,
  Alexandre Tuel
  
  Citation: https://doi.org/10.5194/egusphere-2023-111-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-111', Anonymous Referee #1, 24 Feb 2023

Reviewer Comment on: Weather persistence on subseasonal to seasonal timescales: a methodological review

Authors: Alexandre Tuel and Olivia Martius

Submitted to: egusphere 2023 111

Summary
The authors present a manuscript for a review article on persistence as the occurrence of approximately constant or recurrent atmospheric states in the subseasonal to seasonal (S2S) time range. Persistence determines damages, challenges understanding and promises predictab ility. The review considers various aspects of persistence: definitions, methodologies, and examples encompassed by this rather general notion

In their review, the authors undertake a challenge when they combine meteorological observations and a mathematical d efinition in terms of advanced statistical and dynamical approaches. While a general definition of persistence is already hard, the related notion recurrence is even more difficult to grasp and to distinguish. Not surprisingly, the review is weak when the authors attempt the almost impossible task to provide a comprehensive theoretical framework, but it gets stronger when it resorts to methodologies and meteorological examples. In particular t he variety of examples and their properties show the width of the seem ingly simple no tion persistence.
The authors are well aware of the difficulties involved and do not hide that. The review is wor th to read for two reasons: the collection of method ologies th at have been sugge sted to analyze persistence, and the huge amount of examples. In summary the authors provide a broad and useful overview with a lot of insight. Below I mention some aspects that I noticed.

Specific comments

Line 55: Since persistence needs a timescale for the definition this question is somehow circular: ‘At which timescale(s) does the persistence occur? There could be reference to Section 3.4.

Line 124, Eq (x(t))_t ∈ R^m what is the meaning of the subscript t together with the argument t? This should be explained.

Line 126, I recommend to use ‘state space’ instead of phase space (the often used notion phase space is reserved for Hamiltonian systems).

Line 136: To characterize the difference between precipitation and temperature an autocorrelation time would be appropriate here. This is better than ‘inertia’ since it is difficult to associate precipitation with an inertia, while temperature, on the other hand, has something like an inertia due to the heat capacity.

Lines 143, 155 in Section 3.1 Global, state and episodic persistence. It seems that global and state persistence are different concepts. Global persistence appears nothing else than stationarity which includes recurrence. Why is global persistence ‘strongly related to intrinsic system predictability’? Maybe these definitions could be useful: Does persistence of a state x(t) mean dx(t)/dt=0. And could global persistence be defined in terms of integrals or averages like d<x(t)>/dt=0? And are conservation laws useful?

Line 184: What is a ‘symmetrically, persistent state’?

Line 188: How can this sentence be understood: ‘However, state persistence only characterizes the average behavior of system states.’

Line 201: Section 3.2
Here is a good opportunity to define Lagrangian stationarity of a quantity ѱ(x,t) along a flow u, by ∂ѱ/∂t + u∙∇ѱ = 0, in comparison to the Eulerian stationarity with ∂ѱ/∂t = 0.

Line 216: The authors write ‘self-similarity of system values x(𝑡) with a metric’ but this notion can be confused with geometric self-similarity used in the definition of fractal objects for example. Or is that what the authors intend?

Line 300: write what the symbol E(…) denotes.

Line 348: On Long-range memory: mention the absence of a timescale.

Line 360: Write that the Hurst exponent H and d are related by H=2d+1, see the Table 1 in Franzke et al. (2020).

Line 382: Written is: ‘If temporal dependence is present’. This is unclear. Eq. (10) is the result a power-law in of ρ (Eq.(6)), line 350.

Line 490: Section 4.2.4. The role of the extremal index θ is difficult here. In extreme value statistics the extremal index is the inverse time scale with co-occurring extreme events, used to eliminate short term variability and to combine events. Why is this index a measure of stationarity? Is it correct that short term excursions are allowed with θ in Eq (13)?

Line 569: Please write that R_X₀ is the residence time for state x₀.

Citation: https://doi.org/10.5194/egusphere-2023-111-RC1
- AC1: 'Reply on RC1', Alexandre Tuel, 27 Mar 2023
  
  Dear reviewer, thank you for your comments. Please find our reply in the attached PDF.
  
  Citation: https://doi.org/10.5194/egusphere-2023-111-AC1
RC2:
'Comment on egusphere-2023-111', Abdel Hannachi, 03 Apr 2023

Review of "Veather persistence on sub-seasonal to seasonal

timescales: a methodological review" by Tuel and Martius,

submitted to Geoscientific Model Development.
The authors attempt to provide a 'comprehensive' review of weather

persistence on S2S timescales. The authors start by stating the

origin of persistence, i.e. stationarity/quasi-stationarity or

recurrence, along with the different methodologies. Stationarity

is then described in terms of globality (eg autocorrelation),

states (eg patterns) and episodic events. Lastly, they describe

recurrence as another facet of persistence via various

aspects/measures.

Recommendation

--------------
The paper exposes the subject of persistence in a detailed fashion.

It is quite useful to researchers working on S2S predictability,

especially early career scientists (eg PhD students).

I support its publication in Geoscientific Model Development

subject to some minor changes as detailed below.

Major Comments

--------------
The authors have put great effort in an attempt to bring together

all the different blocs of persistence. Persistence is a loose

concept as there is no definite definition for it, and the authors

propose to bring together its different facets. I had to go more

than once to get a clear picture of the different bits and pieces.

For example, in the 6 sections there are 13 subsections (excluding

subsub-subsections).
1. Perhaps to aid readers, especially early career researchers, an

extra figure showing a tree-like diagram linking the different

concepts of persistence would be welcome.
2. Some related papers are missing from this review. Two particular

references related to persistence and therefore predctability:

predictive oscillation patterns (Kooperberg and O'Sullivan 1996),

and a related paper: optimally interpolated patterns (Hannachi

2008), making use of the power spectra (ie autocorrelation.)
In global stationarity, thirs-order statistics based, eg on

bispectrum (Pires and Hannachi 2021) would complement the

persistence description.
In relation to extremes and persistence archetypal analysis

Hannachi and Trendafilov 2017, Chapman et a. 2022) also identifies

'quasi-stationary' states or regimes.

Minor Comments

--------------
1. Pg 1, abstract: delete acronym S2S. It is abbreviated in

section 1.
2. Pg 3, l75: you mean Fig. 1c.
3. Pg7, l175: delete 'of'
4. Pg 13, eq (15): may be use 0<|\alpha|<1 to include the case of

anti-persistence.

l358: add the following reference on short timescale of

precipitation (Hannachi 2014).

l367,369: may be 'persistence' is more convenient here

than 'stationary'.

5. Pg 14, last paragraph 4.1.2: Fig.3 does not seem to have been

mentioned.
6. Pg 15, l407: consider adding an earlier reference of Baur (1951)

on Grosswetterlagen.
7. Pg18, l461: add "by projecting simplified dynamics (eg quasi-

geostrophy) onto the leading modes of variability of the

GCM simulation" after "January conditions".
8. Pg 20, eq (13): x is not specified, and I think there is some

confusion in the original reference (Faranda et al.) I

think x represents the log of the distance between state

at time t {\bf x}(t) and {\bf x}_0.
9. Pg 23, eq(21): my understanding is that \alpha is a probability

(P(x_0) = P(x_{t+1}=x_0|x_t=x_0)), therefore eq(21) < 0,

and also log(\alpha)<0, please clarify.
10. Pg 31, l646: Ripley's K function was considered in Stephenson

et al. (2004), and Hannachi (2010).
11. Pg 31, l686: " ... as the variance of successive event counts

over an interval of ..."
12. Pg 37, l791: full stop before 'Note'

l793: 'too'

l801: use 'measures' instead of 'metrics'
Fig. 15: are a, b,c 1-day apart?
Section 5: I suggest renumbering/relabelling the subsections as:

5.1 Diagnostic methods

5.1.1 Window counts

5.1.2 Dispersion metrics

5.1.3 Ripley's K function

5.1.4 Distribution of inter-event times

5.2 Stochastic modeling of recurrence

5.2.1 Events series

5.2.2 Recurrence plots
References

----------
Baur F, 1951: Extended-range weather forecasting. In: T. F.

Malone, ed. Compendium of Meteorology, Amer. Met. Soc.,

Boston MA, pp. 814-33.

Chapman C C, D P Monselesan, J S Risbey, M Feng, and B M Sloyan,

2022: A large-scale view of marine heatwaves revealed by

archetype analysis. Nature Communication, 13, 7843.

Hannachi A, 2008: A new set of orthogonal patterns in weather

and climate: optimally interpolated patterns. J. Climate, 21,

6724-6738.

Hannachi A, 2010: On the origin of planetary-scale extratropical

winter circulation regimes. J. Atmos. Sci., 67, 1382-1401.

Hannachi A, 2014: Intermittency, autoregression and censoring: a

first-order AR model for daily precipitation. Meteorol. Appl.,

21, 384-397.

Hannachi A, and N Trendafilov, 2017: Archetypal analysis: mining

weather and climate extremes. J. Climate, 30, 6927-6944.

Kooperberg C, and F O'Sullivan, 1996: Prediction oscillation

patterns: A synthetic of methods for spatial-temporal

decomposition of random fields. J. Amer. Stat. Assoc., 91,

1485-1496.

Pires C, and A Hannachi, 2021: Bispectral analysis of nonlinear

interaction, predictability and stochastic modelling with

application to ENSO. Tellus A, 73, 1-30.

Stephenson D B, A Hannachi, and A O'Neill, 2004: On the existence

of multiple climate regimes. Q. J. R. Meteorol. Soc., 130,

583-605.

Citation: https://doi.org/10.5194/egusphere-2023-111-RC2
- AC2: 'Reply on RC2', Alexandre Tuel, 02 May 2023
  
  Dear Dr. Hannachi,
  Thank you for your comments. You may find our reply in the attached PDF.
  Best,
  Alexandre Tuel
  
  Citation: https://doi.org/10.5194/egusphere-2023-111-AC2

Peer review completion

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

ED: Reconsider after major revisions (04 May 2023) by Christian Franzke

AR by Alexandre Tuel on behalf of the Authors (07 Jun 2023) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (08 Jun 2023) by Christian Franzke

RR by Abdel Hannachi (09 Jun 2023)

RR by Anonymous Referee #3 (30 Jun 2023)

Suggestions for revision or reasons for rejection

Review for: Weather persistence on sub-seasonal to seasonal timescales: a methodological review
Manuscript: #egusphere-2023-111

This study aims to synthesize the current mathematical methods for studying weather persistence, and also try to make it clear in terms of stationarity and recurrence.

It shows clearer improvements compared to the last revision.

Here are just a few comments for the authors to consider:

1. The added first figure did make the overall concept clearer. However, the author did not reference the figure in the main text. A related issue is whether the author could create another figure or modify Figure 1 to include all the sections, rather than just Section 4 and Section 5.
2. Regarding the indication of equations, there seems to be inconsistency in the author's approach. Sometimes the authors simply mention the equation number in the sentence, while other times they do not. To reduce the chance of misunderstanding, I suggest that the author maintain consistency when referring to specific equations. For example, they could use the format "Equation 1," "Equation 2," or another form such as "Eq1," "Eq2."
3. In relation to the research questions (L54-57), it may be beneficial to discuss or provide clarification regarding stationarity or recurrence in this section.

Minor:
1. L24: merge the citations.
2. L73: in spring 2019? – it seems you indicated 1979.
3. L100-120: I suggest the author rephase or emphasize main points at the beginning. It takes time to grasp the idea (could be my problem).
4. L233: is the ()t consistent with the previous one?
5. Figure 12: the caption: 1975-2004.
6. Figure 16: what are the differences between the different shadings?

Hide

ED: Publish subject to minor revisions (review by editor) (01 Jul 2023) by Christian Franzke

AR by Alexandre Tuel on behalf of the Authors (13 Jul 2023) Author's response Author's tracked changes Manuscript

ED: Publish subject to technical corrections (16 Jul 2023) by Christian Franzke

AR by Alexandre Tuel on behalf of the Authors (10 Aug 2023) Manuscript

Journal article(s) based on this preprint

13 Sep 2023

Weather persistence on sub-seasonal to seasonal timescales: a methodological review

Alexandre Tuel and Olivia Martius

Earth Syst. Dynam., 14, 955–987, https://doi.org/10.5194/esd-14-955-2023,https://doi.org/10.5194/esd-14-955-2023, 2023

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Alexandre Tuel and Olivia Martius

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Short summary

Weather persistence on sub-seasonal to seasonal timescales has been a topic of research since the early days of meteorology. Stationary or recurrent behaviour are common features of weather dynamics, and are strongly related to fundamental physical processes, weather predictability and surface weather impacts. In this review, we propose a typology for the broad concepts related to persistence and discuss various methods that have been used to characterise persistence in weather data.


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