Spatiotemporal Recurrence Pattern of Earth&rsquo;s Polar Cap Variation During Geomagnetic Storms

Oludehinwa, Irewola Aaron; Olusola, Olasunkanmi Isaac; Velichko, Andrei; Anoke-Uzosike, Rose

doi:10.5194/egusphere-2025-2810

Preprints

https://doi.org/10.5194/egusphere-2025-2810

Preprints

07 Jul 2025

| 07 Jul 2025

Spatiotemporal Recurrence Pattern of Earth’s Polar Cap Variation During Geomagnetic Storms

Irewola Aaron Oludehinwa, Olasunkanmi Isaac Olusola, Andrei Velichko, and Rose Anoke-Uzosike

Abstract. This study examined the recurrence patterns of Earth's polar cap activities in response to various geomagnetic storm intensities. Time series data of polar cap indices (PCN and PCS) for great storm, severe storm, strong storm, moderate storm, and weak storm events were analysed. Nonlinear dynamics tools, including Recurrence Plot (RP), Recurrence Rate (RR), and Length of Diagonal Line (L), were applied to the polar cap variations to identify the recurring patterns associated with the various categories of geomagnetic storms. The RP, RR, and L effectively captured the distinct recurrence features in the PCN and PCS variations across different storm categories. During great storms, severe storms, and strong geomagnetic storms, the RPs unveils a strong deterministic structure for both PCN and PCS variations, whereas moderate and weak storms showed a rare deterministic structure of RP. Similarly, RR and L values were high during great storms, severe storms, and strong storms, however these indicators significantly decline during moderate and weak storms. These findings indicates that the recurrence density and deterministic behaviour in the polar cap activities, intensify with increased solar wind energy input into the magnetosphere.

Received: 13 Jun 2025 – Discussion started: 07 Jul 2025

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Irewola Aaron Oludehinwa, Olasunkanmi Isaac Olusola, Andrei Velichko, and Rose Anoke-Uzosike

Status: closed

RC1:
'Comment on egusphere-2025-2810', Anonymous Referee #1, 02 Aug 2025
This study (entitled “Spatiotemporal Recurrence Pattern of Earth’s Polar Cap Variation During Geomagnetic Storms” by Oludehinwa et al.) examine the recurrence patterns of the polar cap indices for the Northern and Southern Hemispheres (PCN and PCS, respectively) during geomagnetic storm events and their changes in response to the intensity of the geomagnetic storms. For this study, geomagnetic storms are categorized into five intensity groups (great storms, severe storms, strong storms, moderate storms, and weak storms) according to the minimum value in the SYM-H index during the interval of the corresponding geomagnetic storm events. For each of these five intensity groups, two geomagnetic storm events are selected to use Recurrence Plot (RP) and Recurrence Quantification Analysis (RQA) techniques. Based on the results in the RP for a total of 10 geomagnetic storm events in five intensity groups, as well as the results in the Recurrence Rate (RR) and average line length of the diagonal (L) obtained from the RQA for these events, the authors suggest that the recurrence density and deterministic behavior in the polar cap activities (which could be in terms of the ionospheric electric fields in the near-pole regions generated by the solar wind-magnetosphere interaction and therefore the transpolar ionospheric Hall currents related to these fields) intensify with increasing solar wind energy input into the magnetosphere.
Although the research topic in this study and the mathematical methods used here for data analysis are somewhat interesting, it is needed to address incomplete information and difficult-to-understand data selection methods for the analysis, as well as the typos and grammatical errors. Therefore, this reviewer suggests that this manuscript requires substantial changes before it can be reconsidered for publication. The relevant comments are provided below.

MAJOR COMMENTS
Identification method of geomagnetic storm events: The authors select two geomagnetic storm events for each of five categories divided based on the minimum value in the SYM-H index. Then,

It is unclear if there are any reasons that the authors examine ONLY TWO geomagnetic storm events for each of the five categories.

This reviewer recommends providing the start time and end time of the geomagnetic storm events (rather than the storm occurrence date) and the time when the SYM-H index is minimum in Table 1.

Recurrence plot analysis method

Equation 1 (x_i = (x₁, x₂, …, x_n), i = 1, 2, …, 1440) #1: Because this study focuses on the recurrence pattern of the variations in the (1-min resolution) polar cap indices “during geomagnetic storms”, the numbers of the time series data (i) in both the PCN and PCS indices for geomagnetic storm events can vary depending on the number of minutes in the time interval of the events. Therefore, all analyses used in this study could be performed using the polar cap indices during the geomagnetic storm intervals, instead of the storm occurrence dates. This reviewer cannot find any reasons that the numbers of data points in both polar cap indices are fixed to 1440 (equivalent to 1-day data). It is also worth noting that the X-axis range in Figure 1 (01:00-23:00 UT in this version, rather than 00:00-24:00 UT) as well as the X-axis range and Y-axis range in Figures 2-6 can vary according to the time interval of the events.

Equation 1 #2: It is difficult to understand what “x_i = (x₁, x₂, …, x_n)” means. Equation 1 needs to be rewritten more clearly.

Line 111: It is unclear which value is used as the threshold distance (ε).

Lines 125-126: How do the authors determine 15 as the embedding dimension (m = 15) and 6 as the time delay (τ = 6)?

Interpretation of recurrence plot

Section 3 #1: The results shown in Figures 2-6 can be provided in terms of visible elements such as diagonal lines, vertical/horizontal lines, clustering of points, and so on.

Section 3 #2: It is needed to elaborate on the locations in Figures 2-6 that the authors pointed out in the text (for example, “… it was seen that the polar cap signals exhibit some coherent structure in its underlying dynamics.” on Lines 147-148, “The RPs observation for these severe geomagnetic storms captured a wide distribution of recurrence points, with a high density of recurrence points observed in all the severe storms investigated.” on Lines 162-164, and so on).

Section 4 (Discussion and Conclusion): Only the discussion of the results obtained from the recurrence plots and from the recurrence quantification analysis (in terms of RR and L) has been provided in Section 4. It is also needed to provide the conclusion of this study in this section.

MINOR COMMENTS
Title: The abstract and main body of this manuscript lead to a question about why the authors put “Spatiotemporal” in the title.

Line 28: In “… the polar cap undergoes significant variations …”, please elaborate on the kind of variations the polar cap undergoes significantly.

Lines 31-32: In “One of the parameters that measures the activities of the Earth’s polar cap is the Northern and Southern Hemisphere Polar Cap indices (PCN and PCS).”, please elaborate on which activities of the Earth’s polar cap.

Lines 38-39: Detailed description about “some coherent structures, which are complex multiscale and chaotic” is needed. Moreover, it is unclear what “They” indicates.

Line 65: “… and 2024-2018 …” → “… and 2004-2018 …”

Line 68: “… middle latitude (MLT) …” → “… magnetic local time (MLT) …”

Line 76: Rewrite “Donner and co” in different form.

Lines 90-91: “In section 3, we unveiled our results and observation in section 4, the discussion and conclusion in section 5.” should be rewritten based on the main body of the manuscript.

Line 92: 2.0 → 2

Lines 108, 131, & 135: According to the title in Section 2 (Data Acquisition and the Concept of Nonlinear Dynamics), this section could be organized into two subsections by “Data Acquisition” and “Concept of Nonlinear Dynamics”.

Equation 3 & Line 124: What does “i” mean in “ε_i”, which has not appeared on Line 111?

Line 124, Line 134, & Line 140: Are the total number of considered states (N) on Line 124, the number of points (N) on Line 134, and the maximum possible line length in the RP (N) on Line 140 identical? Please also provide the values of N.

Lines 129-130: It is NOT necessary to mention “Laminarity (LAM)”, “Entropy (ENTR)”, “Determinism (DET)”, and “Trapping Time (TT)” unless these measures are used in this study. Please rephrase this sentence.

Equation 5: i (in the parentheses) → l

Line 139: How do the authors obtain the length of a diagonal line (l)?

Line 141 #1: 3.0 → 3

Line 141 #2: Please re-arrange the subsections in Section 3 based on the observations of the polar cap indices and the results obtained from the recurrence plot and recurrence quantification analysis techniques.

Line 148: Detailed descriptions about “some coherent structure in its underlying dynamics” and “these inherit coherent structures” are needed.

Line 188 & Line 203: magenta → red (or orange)?

Line 189 & Line 204: grey → purple?

Line 213: 4.0 → 4

Lines 228-229: What does “similarity dynamics” mean?

Lines 278-280: Delete repetition.

Lines 308-312: Delete repetition.

Throughout the manuscript,
As mentioned in the first and second minor comments, please specify what kind of “polar cap activities”, “polar cap variation”, and so on the authors describe.

Clear explanations about “deterministic structure” and “deterministic behavior” mentioned in this manuscript are also needed.

Quanaaq → Qaanaaq

The words “(EKL)”, “(PCN and PCS)”, “(PCN)”, “(PCS)”, “(RPs)”, “(RQA)”, “(L)”, and so on appear many times. Please find and correct the corresponding parts.

The average line length of the diagonal (L) and length of the diagonal (l) could be distinguishable.

The quality of this manuscript would be improved when the corrections of typos and grammar mistakes that have not been mentioned yet are made.
Citation: https://doi.org/10.5194/egusphere-2025-2810-RC1
- AC1: 'Reply on RC1', Oludehinwa Irewola, 23 Sep 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2810/egusphere-2025-2810-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-2810-AC1
RC2:
'Comment on egusphere-2025-2810', Anonymous Referee #2, 14 Nov 2025

The manuscript investigates spatiotemporal recurrence patterns in Earth's polar cap indices (PCN and PCS) during geomagnetic storms using recurrence plots (RPs) and recurrence quantification analysis (RQA). The work is relevant to space physics, particularly for understanding nonlinear magnetosphere–ionosphere coupling and polar cap dynamics under storm-time conditions. The topic is timely, and the authors reference a suitable body of literature, especially works on PC indices and recurrence-based techniques. However, the manuscript requires significant revision to be suitable for publication. The scientific motivation is promising, but several methodological, conceptual, and presentation issues need to be addressed. Key deficiencies relate to clarity of methodology, interpretation, and language/structure.

Some comments are:
The authors claim that recurrence analysis has not been applied to polar cap indices (PCN/PCS) in previous literature. In my opinion, this should be demonstrated more explicitly, for example: a short discussion stating what has been done with PC indices, what has been done with RQA in geomagnetism, and where the gap lies.
The present form of the manuscript reads more as a data-driven analysis with limited physical interpretation. The authors do not clearly state why recurrence analysis is the appropriate tool, nor what physical processes they expect to diagnose. In other words, what specific physical mechanisms might produce recurrence in PC indices?
The authors state that the recurrence analysis was computed using the embedding dimension (m=15) and time delay (\tau=6). Without justification, the results may not be meaningful. It should be clarified why/how such values have been chosen, especially the utilized such a high embedding dimension. Did the authors employ a standard method (e.g., False Nearest Neighbors for m or Average Mutual Information for \tau) to justify these specific values? Justifying these core parameters is essential for the reproducibility and validity of the nonlinear analysis.
A significant clarification is required regarding other parameters used in the recurrence analysis. For example, threshold (\epsilon) choice strongly influences RP density, RR, L, and all subsequent claims. Additionally, how robust are the obtained results to changes in these parameters (m, \tau, \epsilon, N)?
PCN/PCS amplitude increases during stronger storms, which automatically increases recurrence density unless the time series is normalized. Thus, the manuscript risks conflating amplitude effects with dynamical structure. I would recommend to normalize the PCN/PCS time series to avoid amplitude-driven RR inflation.
There are many grammatical errors, inconsistent formatting, repeated references, and ambiguous phrasing.

L110: (i,l) -> (i,j)
L131: “It is mathematically express as” ---> “… expressed as”
L214: "The RP results... reveals distinct patterns" ---> "... reveal ..."
L214: "Great storms, severe storms, and strong storms reveals..." ---> "… reveal ..."
L234: “...categories of suggests that..." The noun is missing after "categories of"
L138 (Eq.5): Shouldn't be P(l) not P(i) ?
Several references are duplicated (e.g., Oludehinwa 2018, Donner 2019).

Citation: https://doi.org/10.5194/egusphere-2025-2810-RC2
- AC2: 'Reply on RC2', Oludehinwa Irewola, 24 Nov 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2810/egusphere-2025-2810-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-2810-AC2

Status: closed

RC1:
'Comment on egusphere-2025-2810', Anonymous Referee #1, 02 Aug 2025
This study (entitled “Spatiotemporal Recurrence Pattern of Earth’s Polar Cap Variation During Geomagnetic Storms” by Oludehinwa et al.) examine the recurrence patterns of the polar cap indices for the Northern and Southern Hemispheres (PCN and PCS, respectively) during geomagnetic storm events and their changes in response to the intensity of the geomagnetic storms. For this study, geomagnetic storms are categorized into five intensity groups (great storms, severe storms, strong storms, moderate storms, and weak storms) according to the minimum value in the SYM-H index during the interval of the corresponding geomagnetic storm events. For each of these five intensity groups, two geomagnetic storm events are selected to use Recurrence Plot (RP) and Recurrence Quantification Analysis (RQA) techniques. Based on the results in the RP for a total of 10 geomagnetic storm events in five intensity groups, as well as the results in the Recurrence Rate (RR) and average line length of the diagonal (L) obtained from the RQA for these events, the authors suggest that the recurrence density and deterministic behavior in the polar cap activities (which could be in terms of the ionospheric electric fields in the near-pole regions generated by the solar wind-magnetosphere interaction and therefore the transpolar ionospheric Hall currents related to these fields) intensify with increasing solar wind energy input into the magnetosphere.
Although the research topic in this study and the mathematical methods used here for data analysis are somewhat interesting, it is needed to address incomplete information and difficult-to-understand data selection methods for the analysis, as well as the typos and grammatical errors. Therefore, this reviewer suggests that this manuscript requires substantial changes before it can be reconsidered for publication. The relevant comments are provided below.

MAJOR COMMENTS
Identification method of geomagnetic storm events: The authors select two geomagnetic storm events for each of five categories divided based on the minimum value in the SYM-H index. Then,

It is unclear if there are any reasons that the authors examine ONLY TWO geomagnetic storm events for each of the five categories.

This reviewer recommends providing the start time and end time of the geomagnetic storm events (rather than the storm occurrence date) and the time when the SYM-H index is minimum in Table 1.

Recurrence plot analysis method

Equation 1 (x_i = (x₁, x₂, …, x_n), i = 1, 2, …, 1440) #1: Because this study focuses on the recurrence pattern of the variations in the (1-min resolution) polar cap indices “during geomagnetic storms”, the numbers of the time series data (i) in both the PCN and PCS indices for geomagnetic storm events can vary depending on the number of minutes in the time interval of the events. Therefore, all analyses used in this study could be performed using the polar cap indices during the geomagnetic storm intervals, instead of the storm occurrence dates. This reviewer cannot find any reasons that the numbers of data points in both polar cap indices are fixed to 1440 (equivalent to 1-day data). It is also worth noting that the X-axis range in Figure 1 (01:00-23:00 UT in this version, rather than 00:00-24:00 UT) as well as the X-axis range and Y-axis range in Figures 2-6 can vary according to the time interval of the events.

Equation 1 #2: It is difficult to understand what “x_i = (x₁, x₂, …, x_n)” means. Equation 1 needs to be rewritten more clearly.

Line 111: It is unclear which value is used as the threshold distance (ε).

Lines 125-126: How do the authors determine 15 as the embedding dimension (m = 15) and 6 as the time delay (τ = 6)?

Interpretation of recurrence plot

Section 3 #1: The results shown in Figures 2-6 can be provided in terms of visible elements such as diagonal lines, vertical/horizontal lines, clustering of points, and so on.

Section 3 #2: It is needed to elaborate on the locations in Figures 2-6 that the authors pointed out in the text (for example, “… it was seen that the polar cap signals exhibit some coherent structure in its underlying dynamics.” on Lines 147-148, “The RPs observation for these severe geomagnetic storms captured a wide distribution of recurrence points, with a high density of recurrence points observed in all the severe storms investigated.” on Lines 162-164, and so on).

Section 4 (Discussion and Conclusion): Only the discussion of the results obtained from the recurrence plots and from the recurrence quantification analysis (in terms of RR and L) has been provided in Section 4. It is also needed to provide the conclusion of this study in this section.

MINOR COMMENTS
Title: The abstract and main body of this manuscript lead to a question about why the authors put “Spatiotemporal” in the title.

Line 28: In “… the polar cap undergoes significant variations …”, please elaborate on the kind of variations the polar cap undergoes significantly.

Lines 31-32: In “One of the parameters that measures the activities of the Earth’s polar cap is the Northern and Southern Hemisphere Polar Cap indices (PCN and PCS).”, please elaborate on which activities of the Earth’s polar cap.

Lines 38-39: Detailed description about “some coherent structures, which are complex multiscale and chaotic” is needed. Moreover, it is unclear what “They” indicates.

Line 65: “… and 2024-2018 …” → “… and 2004-2018 …”

Line 68: “… middle latitude (MLT) …” → “… magnetic local time (MLT) …”

Line 76: Rewrite “Donner and co” in different form.

Lines 90-91: “In section 3, we unveiled our results and observation in section 4, the discussion and conclusion in section 5.” should be rewritten based on the main body of the manuscript.

Line 92: 2.0 → 2

Lines 108, 131, & 135: According to the title in Section 2 (Data Acquisition and the Concept of Nonlinear Dynamics), this section could be organized into two subsections by “Data Acquisition” and “Concept of Nonlinear Dynamics”.

Equation 3 & Line 124: What does “i” mean in “ε_i”, which has not appeared on Line 111?

Line 124, Line 134, & Line 140: Are the total number of considered states (N) on Line 124, the number of points (N) on Line 134, and the maximum possible line length in the RP (N) on Line 140 identical? Please also provide the values of N.

Lines 129-130: It is NOT necessary to mention “Laminarity (LAM)”, “Entropy (ENTR)”, “Determinism (DET)”, and “Trapping Time (TT)” unless these measures are used in this study. Please rephrase this sentence.

Equation 5: i (in the parentheses) → l

Line 139: How do the authors obtain the length of a diagonal line (l)?

Line 141 #1: 3.0 → 3

Line 141 #2: Please re-arrange the subsections in Section 3 based on the observations of the polar cap indices and the results obtained from the recurrence plot and recurrence quantification analysis techniques.

Line 148: Detailed descriptions about “some coherent structure in its underlying dynamics” and “these inherit coherent structures” are needed.

Line 188 & Line 203: magenta → red (or orange)?

Line 189 & Line 204: grey → purple?

Line 213: 4.0 → 4

Lines 228-229: What does “similarity dynamics” mean?

Lines 278-280: Delete repetition.

Lines 308-312: Delete repetition.

Throughout the manuscript,
As mentioned in the first and second minor comments, please specify what kind of “polar cap activities”, “polar cap variation”, and so on the authors describe.

Clear explanations about “deterministic structure” and “deterministic behavior” mentioned in this manuscript are also needed.

Quanaaq → Qaanaaq

The words “(EKL)”, “(PCN and PCS)”, “(PCN)”, “(PCS)”, “(RPs)”, “(RQA)”, “(L)”, and so on appear many times. Please find and correct the corresponding parts.

The average line length of the diagonal (L) and length of the diagonal (l) could be distinguishable.

The quality of this manuscript would be improved when the corrections of typos and grammar mistakes that have not been mentioned yet are made.
Citation: https://doi.org/10.5194/egusphere-2025-2810-RC1
- AC1: 'Reply on RC1', Oludehinwa Irewola, 23 Sep 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2810/egusphere-2025-2810-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-2810-AC1
RC2:
'Comment on egusphere-2025-2810', Anonymous Referee #2, 14 Nov 2025

The manuscript investigates spatiotemporal recurrence patterns in Earth's polar cap indices (PCN and PCS) during geomagnetic storms using recurrence plots (RPs) and recurrence quantification analysis (RQA). The work is relevant to space physics, particularly for understanding nonlinear magnetosphere–ionosphere coupling and polar cap dynamics under storm-time conditions. The topic is timely, and the authors reference a suitable body of literature, especially works on PC indices and recurrence-based techniques. However, the manuscript requires significant revision to be suitable for publication. The scientific motivation is promising, but several methodological, conceptual, and presentation issues need to be addressed. Key deficiencies relate to clarity of methodology, interpretation, and language/structure.

Some comments are:
The authors claim that recurrence analysis has not been applied to polar cap indices (PCN/PCS) in previous literature. In my opinion, this should be demonstrated more explicitly, for example: a short discussion stating what has been done with PC indices, what has been done with RQA in geomagnetism, and where the gap lies.
The present form of the manuscript reads more as a data-driven analysis with limited physical interpretation. The authors do not clearly state why recurrence analysis is the appropriate tool, nor what physical processes they expect to diagnose. In other words, what specific physical mechanisms might produce recurrence in PC indices?
The authors state that the recurrence analysis was computed using the embedding dimension (m=15) and time delay (\tau=6). Without justification, the results may not be meaningful. It should be clarified why/how such values have been chosen, especially the utilized such a high embedding dimension. Did the authors employ a standard method (e.g., False Nearest Neighbors for m or Average Mutual Information for \tau) to justify these specific values? Justifying these core parameters is essential for the reproducibility and validity of the nonlinear analysis.
A significant clarification is required regarding other parameters used in the recurrence analysis. For example, threshold (\epsilon) choice strongly influences RP density, RR, L, and all subsequent claims. Additionally, how robust are the obtained results to changes in these parameters (m, \tau, \epsilon, N)?
PCN/PCS amplitude increases during stronger storms, which automatically increases recurrence density unless the time series is normalized. Thus, the manuscript risks conflating amplitude effects with dynamical structure. I would recommend to normalize the PCN/PCS time series to avoid amplitude-driven RR inflation.
There are many grammatical errors, inconsistent formatting, repeated references, and ambiguous phrasing.

L110: (i,l) -> (i,j)
L131: “It is mathematically express as” ---> “… expressed as”
L214: "The RP results... reveals distinct patterns" ---> "... reveal ..."
L214: "Great storms, severe storms, and strong storms reveals..." ---> "… reveal ..."
L234: “...categories of suggests that..." The noun is missing after "categories of"
L138 (Eq.5): Shouldn't be P(l) not P(i) ?
Several references are duplicated (e.g., Oludehinwa 2018, Donner 2019).

Citation: https://doi.org/10.5194/egusphere-2025-2810-RC2
- AC2: 'Reply on RC2', Oludehinwa Irewola, 24 Nov 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-2810/egusphere-2025-2810-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-2810-AC2

Irewola Aaron Oludehinwa, Olasunkanmi Isaac Olusola, Andrei Velichko, and Rose Anoke-Uzosike

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Short summary

This study examine the recurrence patterns in the Earth's polar cap variations in response to great, severe, strong, moderate, and weak storms. Complex system methods comprising Recurrence Plot (RP), Recurrence Rate (RR) and Length of diagonal line (L) was used to unveil the recurring patterns in the Earth polar cap. The recurrence density and deterministic behavior in Earth polar cap variations intensifies with increased solar wind energy input into the magnetosphere.


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