Permutation Entropy and Complexity Analysis of Large-scale Solar Wind Structures and Streams

Kilpua, Emilia Katja Johanna; Good, Simon; Ala-Lahti, Matti; Osmane, Adnane; Koikkalainen, Venla

doi:https://doi.org/10.5194/egusphere-2023-2352

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https://doi.org/10.5194/egusphere-2023-2352

Preprints

24 Oct 2023

| 24 Oct 2023

Permutation Entropy and Complexity Analysis of Large-scale Solar Wind Structures and Streams

Emilia Katja Johanna Kilpua, Simon Good, Matti Ala-Lahti, Adnane Osmane, and Venla Koikkalainen

Abstract. In this work, we perform a statistical study of magnetic field fluctuations in the solar wind at 1 au using permutation entropy and complexity analysis. Slow and fast wind, magnetic clouds, interplanetary coronal mass ejection (ICME)-driven sheath regions and slow-fast stream interaction regions (SIRs) have been investigated separately. Our key finding is that there are significant differences in permutation entropy and complexity values between the solar wind types at larger timescales and little difference at small timescales. Differences become more distinct with increasing timescale, suggesting that smaller-scale turbulent features are more universal. At larger timescales, the analysis method can be used to identify localized spatial structures. We found that fluctuation properties in compressive structures (sheaths and SIRs) exhibit a clear locality. Our results shows that, in all cases apart from magnetic clouds at largest scales, solar wind fluctuations are stochastic with the fast wind having the highest entropies and low complexities. Magnetic clouds in turn exhibit the lowest entropy and highest complexity, consistent with them being coherent structures in which the magnetic field components vary in an ordered manner. SIRs, slow wind and ICME sheaths are intermediate to magnetic clouds and fast wind, reflecting the increasingly ordered structure. Our results also indicate that permutation entropy – complexity analysis is a useful tool for characterizing the solar wind and investigating the nature of its fluctuations.

Received: 12 Oct 2023 – Discussion started: 24 Oct 2023

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Emilia Katja Johanna Kilpua, Simon Good, Matti Ala-Lahti, Adnane Osmane, and Venla Koikkalainen

Status: closed

RC1:
'Comment on egusphere-2023-2352', Anonymous Referee #1, 20 Dec 2023

General comment:
The paper “Permutation Entropy and Complexity Analysis of Large-scale Solar Wind Structures and Streams” presents an extended characterization of the complexity of different solar wind structures such as ICMEs and SIRs along with fast and slow streams. Their stochastic character is investigated through permutation entropy and their complexity by means of the Jensen-Shannon complexity. The main results of this research is that plasma coherent structures such as ICMEs are the most complex (lowest entropy and highest complexity), whereas fast wind is the most stochastic (lowest complexity and highest entropy). Finally authors provide a local study of the Hurst exponent as a function of time and scales for the different stream categories. The paper is well organized, clear and very interesting. However at this stage I cannot recommend the publication as there are few issues that the authors should clarify.

Specific comments:
At paragraph 35, and throughout the paper, Authors present an extensive list of citations of previous permutation entropy applications in space physics studies. However, the paper by Raath et al. 2022 (https://doi.org/10.1029/2021JA030200open_in_new; R22) does not appear in the list. I recommend to include R22 in the bibliography, but, most importantly, to comment on the results, since R22 has several things in common with this study. For instance, in Fig. 12 of R22, Authors show a comparison between low-H data interval, fBM and ICME in the CH-plane. ICMEs are estimated through the method by Wang and Richardson 2004, thus compatible with the magnetic cloud set of this study. The analysis of R22 is based on Voyager 2 data and results are fairly in agreement with those of this study.
In the discussion Authors comment about the variability of the local Hurst exponent in terms of spectral slopes and intermittency. In my opinion there is something controversial in this part, since the relation between the Hurst exponent and the spectral slope β=2H+1 only holds under the condition of global self-similarity, as, for instance, for the fBM. Since the hallmark of turbulence is its multifractal character, conclusions about intermittency based on the Hurst exponent only cannot be fully consistent. The anomalous scaling, indeed, appears in high order structure functions S_q (say q > 4) and, although it is well established that intermittency varies with heliospheric distance and also varies among streams of different nature, a discussion about turbulence/intermittency without inspecting measures based on high-order statistics (i.e., kurtosis) is incomplete. For example, Authors observe intervals in the time-scale diagram where the Hurst exponent matches the scaling predictions by Kolmogorov or Kraichnan. However, it is not possible to characterize the turbulence by only looking at the first-order scaling exponent also in these cases. Such intervals, indeed, are also strongly intermittent and therefore high-order statistical measures are needed. So, I recommend revising the discussions about Hurst exponent and turbulence and the association between H and β, since the first scaling exponent ζ₁ coincides with H if and only if the scaling law is linear, i.e., ζ_q~ Hq (e.g., Flandrin, 1989).

Minor corrections:
Abstract: Since part of the discussion of the paper is made on the local Hurst exponent, I would recommend to mention this in the abstract
Line 104-105: Regarding the Brownian motion, please correct “square-root of time” with “time”.
Hurst exponent and permutation entropy are both indicated with H throughout the paper. I would recommend using different symbols for them to avoid any confusion.
Line 115: Authors write that “magnetic cloud time series are more consistent with larger Hurst exponents”, but also slow wind time series appear more similar to magnetic cloud than sheats or fast wind intervals.
Line 128: There is an extra factor H(P) in the definition of the Jensen-Shannon complexity measure, please remove it.
Line 147 and 201: 900 –> 600
Line 219: Authors write “the Hurst exponent is related to the first-order structure function as follows”. I would recommend to state explicitly that this relation holds if the scaling-law is linear, since «H is determined practically, by plotting S_q(τ) vs τ on a log-log plot, and taking the slope, which is equal to qH», as stated in Gilmoure et al. 2002.
Line 246: delete the extra-“the” at the end of the row
Line 324: Authors write that in the case of magnetic cloud "interpreting their nature in terms of the Hurst exponent could therefore be questionable". Please explain why.
Line 382: Authors conclude that "complexity-entropy analysis could reveal the occurrence of mesoscale structures in space plasmas at different scales". How this statistical analysis can be used to identify mesoscale structures? What the authors mean by mesoscale structures in this context?

Citation: https://doi.org/10.5194/egusphere-2023-2352-RC1
- AC1: 'Reply on RC1', Emilia Kilpua, 05 Feb 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-2352/egusphere-2023-2352-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-2352-AC1
RC2:
'Comment on egusphere-2023-2352', Anonymous Referee #2, 20 Dec 2023

I am recommending publication of this article because the graphs and values will probably be useful to future researchers.

I am, however, very disappointed in the level of analysis contained in this manuscript. It is simple a report of the values of complexity and entropy for the various data sets, with very little physics analysis and very little insight about the solar wind. This is a paper that a student could have written as a class project.

Citation: https://doi.org/10.5194/egusphere-2023-2352-RC2
- AC3: 'Reply on RC2', Emilia Kilpua, 16 Feb 2024
  
  We are glad that the reviewer recommends publication. On the second comment, we discuss in Section 4 the interpretations of our new findings in terms of current theories and findings of solar wind turbulence and previous studies. We note that the application of entropy and complexity analysis to the solar wind is a relatively young field. We intend to publish further studies that build on the foundations laid in this paper.
  
  Citation: https://doi.org/10.5194/egusphere-2023-2352-AC3
RC3:
'Comment on egusphere-2023-2352', Anonymous Referee #3, 03 Jan 2024
This work aims at characterizing time series of fast and slow solar wind, magnetic clouds, CME-driven sheaths and SIRs using permutation entropy, Jensen-Shannon complexity and Hurst exponent analyses. The study is original and innovative and worthy of prompt publication in ANGEO, following a few minor revisions that mainly concern adding clarifications and pertinent references in the manuscript, as well as reorganizing its structure.
Introduction (in agreement with Referee #1 remark on Abstract): Since part of the discussion of the paper is made on the local Hurst exponent, I would recommend devoting a paragraph to discussing this in the Introduction. For instance, I feel it would be fair to mention one of the first studies that used the Hurst exponent to study the geospace and specifically the geomagnetic activity, which was published in the same journal as the present manuscript under review (Balasis et al., 2006). In Balasis et al. (2006), the transition from anti-persistent to persistent behavior was associated with the occurrence of intense magnetic storms. Moreover, entropy analysis has also been used in several publications to study the near-Earth electromagnetic environment (for a recent review see Balasis et al., 2023).

Subsection 2.1: what is the time interval covered by the data considered in this study? For instance, do you analyze time series covering a full solar cycle? Please make this point clear here.

Subsection 2.2: at this instance the Hurst exponent along with the fBm model suddenly jumps into the manuscript to characterize the various types of solar wind time series. I think it would be making more sense to introduce the Hurst exponent together with the theory of the other analysis techniques of Permutation entropy and Jensen-Shannon complexity (as given in 2.3) and then move to Figure 1 together with the Results section. It is rather awkward to first apply the Hurst exponent and then introduce the related theory in Subsection 3.4. So, in my opinion, 2.3 and 3.4 should be combined in a common methodological section and presented before Section 3 of the Results.

Lines 284–285 read: “This trend was identified here in particular for the fast wind that also had throughout the investigated τ range the highest entropy and lowest complexity values.” I am a bit confused, if I understood well, with the suggested link between the highest entropy and lowest complexity, since in my (traditional?) perspective higher entropy values mean a lower organization or a less ordered state of the system under study, which in turn points to higher complexity values also. Therefore, higher entropy means higher complexity! Could you please comment upon this point?

Lines 285–286 read: “This could stem from the fact that the fast wind is permeated by Alfvénic fluctuations which are inherently stochastic in nature.” Why is that happening? please elaborate / explain a bit this point.

Last but not least, lines 339-340 read: “The exponents extending to the persistent regime (H > 0.5) were identified mostly in magnetic clouds and for the largest time-scales.” In previous Hurst exponent studies of geomagnetic activity indices, as well as corresponding solar wind variations (e.g., Balasis et al., 2006), persistency was associated with the occurrence of intense magnetic storms, i.e., with an extreme event. What could be a possible extreme event in your case?

References
Balasis, G., Daglis, I. A., Kapiris, P., Mandea, M., Vassiliadis, D., and Eftaxias, K.: From pre-storm activity to magnetic storms: a transition described in terms of fractal dynamics, Ann. Geophys., 24, 3557–3567, https://doi.org/10.5194/angeo-24-3557-2006, 2006.
Balasis, G., Balikhin, M.A., Chapman, S.C. et al. Complex Systems Methods Characterizing Nonlinear Processes in the Near-Earth Electromagnetic Environment: Recent Advances and Open Challenges. Space Sci Rev 219, 38 (2023). https://doi.org/10.1007/s11214-023-00979-7
Citation: https://doi.org/10.5194/egusphere-2023-2352-RC3
- AC2: 'Reply on RC3', Emilia Kilpua, 05 Feb 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-2352/egusphere-2023-2352-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-2352-AC2

Status: closed

RC1:
'Comment on egusphere-2023-2352', Anonymous Referee #1, 20 Dec 2023

General comment:
The paper “Permutation Entropy and Complexity Analysis of Large-scale Solar Wind Structures and Streams” presents an extended characterization of the complexity of different solar wind structures such as ICMEs and SIRs along with fast and slow streams. Their stochastic character is investigated through permutation entropy and their complexity by means of the Jensen-Shannon complexity. The main results of this research is that plasma coherent structures such as ICMEs are the most complex (lowest entropy and highest complexity), whereas fast wind is the most stochastic (lowest complexity and highest entropy). Finally authors provide a local study of the Hurst exponent as a function of time and scales for the different stream categories. The paper is well organized, clear and very interesting. However at this stage I cannot recommend the publication as there are few issues that the authors should clarify.

Specific comments:
At paragraph 35, and throughout the paper, Authors present an extensive list of citations of previous permutation entropy applications in space physics studies. However, the paper by Raath et al. 2022 (https://doi.org/10.1029/2021JA030200open_in_new; R22) does not appear in the list. I recommend to include R22 in the bibliography, but, most importantly, to comment on the results, since R22 has several things in common with this study. For instance, in Fig. 12 of R22, Authors show a comparison between low-H data interval, fBM and ICME in the CH-plane. ICMEs are estimated through the method by Wang and Richardson 2004, thus compatible with the magnetic cloud set of this study. The analysis of R22 is based on Voyager 2 data and results are fairly in agreement with those of this study.
In the discussion Authors comment about the variability of the local Hurst exponent in terms of spectral slopes and intermittency. In my opinion there is something controversial in this part, since the relation between the Hurst exponent and the spectral slope β=2H+1 only holds under the condition of global self-similarity, as, for instance, for the fBM. Since the hallmark of turbulence is its multifractal character, conclusions about intermittency based on the Hurst exponent only cannot be fully consistent. The anomalous scaling, indeed, appears in high order structure functions S_q (say q > 4) and, although it is well established that intermittency varies with heliospheric distance and also varies among streams of different nature, a discussion about turbulence/intermittency without inspecting measures based on high-order statistics (i.e., kurtosis) is incomplete. For example, Authors observe intervals in the time-scale diagram where the Hurst exponent matches the scaling predictions by Kolmogorov or Kraichnan. However, it is not possible to characterize the turbulence by only looking at the first-order scaling exponent also in these cases. Such intervals, indeed, are also strongly intermittent and therefore high-order statistical measures are needed. So, I recommend revising the discussions about Hurst exponent and turbulence and the association between H and β, since the first scaling exponent ζ₁ coincides with H if and only if the scaling law is linear, i.e., ζ_q~ Hq (e.g., Flandrin, 1989).

Minor corrections:
Abstract: Since part of the discussion of the paper is made on the local Hurst exponent, I would recommend to mention this in the abstract
Line 104-105: Regarding the Brownian motion, please correct “square-root of time” with “time”.
Hurst exponent and permutation entropy are both indicated with H throughout the paper. I would recommend using different symbols for them to avoid any confusion.
Line 115: Authors write that “magnetic cloud time series are more consistent with larger Hurst exponents”, but also slow wind time series appear more similar to magnetic cloud than sheats or fast wind intervals.
Line 128: There is an extra factor H(P) in the definition of the Jensen-Shannon complexity measure, please remove it.
Line 147 and 201: 900 –> 600
Line 219: Authors write “the Hurst exponent is related to the first-order structure function as follows”. I would recommend to state explicitly that this relation holds if the scaling-law is linear, since «H is determined practically, by plotting S_q(τ) vs τ on a log-log plot, and taking the slope, which is equal to qH», as stated in Gilmoure et al. 2002.
Line 246: delete the extra-“the” at the end of the row
Line 324: Authors write that in the case of magnetic cloud "interpreting their nature in terms of the Hurst exponent could therefore be questionable". Please explain why.
Line 382: Authors conclude that "complexity-entropy analysis could reveal the occurrence of mesoscale structures in space plasmas at different scales". How this statistical analysis can be used to identify mesoscale structures? What the authors mean by mesoscale structures in this context?

Citation: https://doi.org/10.5194/egusphere-2023-2352-RC1
- AC1: 'Reply on RC1', Emilia Kilpua, 05 Feb 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-2352/egusphere-2023-2352-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-2352-AC1
RC2:
'Comment on egusphere-2023-2352', Anonymous Referee #2, 20 Dec 2023

I am recommending publication of this article because the graphs and values will probably be useful to future researchers.

I am, however, very disappointed in the level of analysis contained in this manuscript. It is simple a report of the values of complexity and entropy for the various data sets, with very little physics analysis and very little insight about the solar wind. This is a paper that a student could have written as a class project.

Citation: https://doi.org/10.5194/egusphere-2023-2352-RC2
- AC3: 'Reply on RC2', Emilia Kilpua, 16 Feb 2024
  
  We are glad that the reviewer recommends publication. On the second comment, we discuss in Section 4 the interpretations of our new findings in terms of current theories and findings of solar wind turbulence and previous studies. We note that the application of entropy and complexity analysis to the solar wind is a relatively young field. We intend to publish further studies that build on the foundations laid in this paper.
  
  Citation: https://doi.org/10.5194/egusphere-2023-2352-AC3
RC3:
'Comment on egusphere-2023-2352', Anonymous Referee #3, 03 Jan 2024
This work aims at characterizing time series of fast and slow solar wind, magnetic clouds, CME-driven sheaths and SIRs using permutation entropy, Jensen-Shannon complexity and Hurst exponent analyses. The study is original and innovative and worthy of prompt publication in ANGEO, following a few minor revisions that mainly concern adding clarifications and pertinent references in the manuscript, as well as reorganizing its structure.
Introduction (in agreement with Referee #1 remark on Abstract): Since part of the discussion of the paper is made on the local Hurst exponent, I would recommend devoting a paragraph to discussing this in the Introduction. For instance, I feel it would be fair to mention one of the first studies that used the Hurst exponent to study the geospace and specifically the geomagnetic activity, which was published in the same journal as the present manuscript under review (Balasis et al., 2006). In Balasis et al. (2006), the transition from anti-persistent to persistent behavior was associated with the occurrence of intense magnetic storms. Moreover, entropy analysis has also been used in several publications to study the near-Earth electromagnetic environment (for a recent review see Balasis et al., 2023).

Subsection 2.1: what is the time interval covered by the data considered in this study? For instance, do you analyze time series covering a full solar cycle? Please make this point clear here.

Subsection 2.2: at this instance the Hurst exponent along with the fBm model suddenly jumps into the manuscript to characterize the various types of solar wind time series. I think it would be making more sense to introduce the Hurst exponent together with the theory of the other analysis techniques of Permutation entropy and Jensen-Shannon complexity (as given in 2.3) and then move to Figure 1 together with the Results section. It is rather awkward to first apply the Hurst exponent and then introduce the related theory in Subsection 3.4. So, in my opinion, 2.3 and 3.4 should be combined in a common methodological section and presented before Section 3 of the Results.

Lines 284–285 read: “This trend was identified here in particular for the fast wind that also had throughout the investigated τ range the highest entropy and lowest complexity values.” I am a bit confused, if I understood well, with the suggested link between the highest entropy and lowest complexity, since in my (traditional?) perspective higher entropy values mean a lower organization or a less ordered state of the system under study, which in turn points to higher complexity values also. Therefore, higher entropy means higher complexity! Could you please comment upon this point?

Lines 285–286 read: “This could stem from the fact that the fast wind is permeated by Alfvénic fluctuations which are inherently stochastic in nature.” Why is that happening? please elaborate / explain a bit this point.

Last but not least, lines 339-340 read: “The exponents extending to the persistent regime (H > 0.5) were identified mostly in magnetic clouds and for the largest time-scales.” In previous Hurst exponent studies of geomagnetic activity indices, as well as corresponding solar wind variations (e.g., Balasis et al., 2006), persistency was associated with the occurrence of intense magnetic storms, i.e., with an extreme event. What could be a possible extreme event in your case?

References
Balasis, G., Daglis, I. A., Kapiris, P., Mandea, M., Vassiliadis, D., and Eftaxias, K.: From pre-storm activity to magnetic storms: a transition described in terms of fractal dynamics, Ann. Geophys., 24, 3557–3567, https://doi.org/10.5194/angeo-24-3557-2006, 2006.
Balasis, G., Balikhin, M.A., Chapman, S.C. et al. Complex Systems Methods Characterizing Nonlinear Processes in the Near-Earth Electromagnetic Environment: Recent Advances and Open Challenges. Space Sci Rev 219, 38 (2023). https://doi.org/10.1007/s11214-023-00979-7
Citation: https://doi.org/10.5194/egusphere-2023-2352-RC3
- AC2: 'Reply on RC3', Emilia Kilpua, 05 Feb 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere-2023-2352/egusphere-2023-2352-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2023-2352-AC2

Emilia Katja Johanna Kilpua, Simon Good, Matti Ala-Lahti, Adnane Osmane, and Venla Koikkalainen

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Short summary

The solar wind is organised to slow and fast streams, interaction regions and transient structures originating from solar eruptions. Their internal characteristics are not well understood. The more comprehensive understanding of such features can give insight on physical processes governing their formation and evolution. Using tools from information theory we find that the solar wind shows universal turbulent properties on smaller scales while on larger-scale clear differences arise.


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