04 Aug 2021
04 Aug 2021
Quantifying the nonlinear dependence of energetic electron fluxes in the Earth's radiation belts with radial diffusion drivers
 ^{1}Departmenf of Physics, University of Helsinki, Finland
 ^{2}Space Science Institute, Colorado, USA
 ^{1}Departmenf of Physics, University of Helsinki, Finland
 ^{2}Space Science Institute, Colorado, USA
Abstract. In this study, we use mutual information to characterise statistical dependencies of seed and relativistic electron fluxes in the Earth's radiation belts on ultra low frequency (ULF) wave power measured on the ground and at geostationary orbit . The benefit of mutual information, in comparison to measures such as the Pearson correlation, lies in the capacity to distinguish nonlinear dependencies from linear ones. After reviewing the property of mutual information and its relationship with the Pearson correlation for Gaussian bivariates of arbitrary correlation, we present a methodology to quantify and distinguish linear and nonlinear statistical dependencies that can be generalised to a wide range of solar wind drivers and magnetospheric responses. We present an application of the methodology by revisiting the case events studied by Rostoker et al. (1998). Our results corroborate the conclusions of Rostoker et al. (1998) that ULF wave power and relativistic electron fluxes are statistically dependent upon one another. However, we find that observed enhancements in relativistic electron fluxes correlate modestly, both linearly and nonlinearly, with the ULF power spectrum when compared with values found in previous studies (Simms et al., 2014), and with values found between seed electrons and ULF wave power for the same case events. Our results are indicative of the importance in incorporating data analysis tools that can quantify and distinguish between linear and nonlinear interdependencies of various solar wind drivers.
Adnane Osmane et al.
Status: final response (author comments only)

RC1: 'Comment on angeo202147', Anonymous Referee #1, 28 Aug 2021
Summary:
The manuscript entitled “Quantifying the nonlinear dependence of energetic electron fluxes in the Earth’s radiation belts with radial diffusion drivers” by Osmane et al revisits a highly cited case study (Rostoker et al., 1998) (1) to quantify the strength of the statistical dependence between ULF wave power and seed and relativistic electron fluxes measured at GEO and (2) to test for the presence of nonlinear dependence (l.7073).
To do so, the mutual information and the Pearson correlation coefficient are computed for each combination of:
 1) (*) ground or (*) geosynchronous ULF wave power and
 2) (*) seed (130 keV) or (*) relativistic (1.2 MeV) electron fluxes at GEO
for the 2 events analyzed by Rostoker (26 days in 1993 and 91 days in 1994, l.203), with and without a 24 hr window moving average, for time offsets varying between 100hr and +100hr (see Fig. 4 Fig.11, with the max values for each instance summarized in Table 1). ULF data is Augsburg ULF indices. Electron fluxes are from the synchronous orbit particle analyzer (SOPA) instruments onboard multiple GEO spacecraft.
The findings are that “the Pearson and the mutual information are both statistically significant” (l. 327), corroborating Rostoker et al. (1998)’s finding that ULF wave power and relativistic electron fluxes are statistically dependent (l. 89). It is also found that the statistical dependence between ULF wave power and 130 keV flux is larger than for relativistic fluxes (l. 347). Because the “adjusted correlation” is found to be larger than the Pearson correlation, the results “indicate that the response of relativistic electron fluxes can be a combination of linear and nonlinear dependence” (l. 369). More generally, the paper calls for “incorporating data analysis tools that can quantify and distinguish between linear and nonlinear interdependencies of various solar wind drivers” (l.1213, l. 391392).
General Comments:
The work revisits cases discussed in a highly cited paper using a new approach (higher time resolution with a “comparable data set”  l.212, correlation and mutual information quantification). Proper credit is given to related works. The contribution is clear and it is compared and contrasted with other published results (e.g., l.326346). In that context, the work is fairly important: It represents an incremental advance to the current state of knowledge of the field.
That said, the written presentation of the work lacks conciseness. An effort in synthesis would greatly benefit the readability (thus accessibility and potential impact) of the manuscript. This should be done, if only to do justice to the research work.
Specific Comments:
1) Section 2 provides~ 100 lines of generalities (“a brief but selfcontained tutorial on the Shannon entropy and mutual information for a reader who is not familiar with information” l.8485). Yet, its value for the remainder of the manuscript is unclear. For instance, Fig. 1 provides the results of a numerical test that appears to be unrelated to the paper. In addition, the formulas used to determine the quantities plotted Fig.4Fig.11 are not explicitly provided. One way to address this comment could be to shorten the Section 2, retaining only the descriptions of the formulas that are used in the ULF/Eflux data analysis, and referring the interested reader to works already published on this topic (e.g. Wing et al., 2016 and references for information theory lectures). Another way could be to create an appendix for supplementary material.
2) The main results are presented as a list of 8 different figures (each composed of 4 different panels). Yet, there is little difference between the figures: They all appear to provide similar information (namely, that there exists a significant dependence between waves and particle fluxes). Fig.1 a)b) is enough to illustrate the approach and the point of the paper is conveyed in Table 1. In that context, everything else could be provided as supporting material. If all 32 panels are really necessary to convey the message of the work, this needs to be explained at the beginning of Section 4.

AC1: 'Reply on RC1', Adnane Osmane, 05 Oct 2021
We thank the reviewer for carefully reading the manuscript and providing us with a constructive feedback. Our answer to the specific comments (in bold and italic) can be found below.
Specific Comments:
1) Section 2 provides~ 100 lines of generalities (“a brief but selfcontained tutorial on the Shannon entropy and mutual information for a reader who is not familiar with information” l.8485). Yet, its value for the remainder of the manuscript is unclear. For instance, Fig. 1 provides the results of a numerical test that appears to be unrelated to the paper.
Figure 1 is provided to show the correspondence between the Pearson correlation and mutual information. We believe that everyone in our scientific community is familiar with the notion that a Pearson correlation greater than 0.8 is large. But the use of mutual information is fairly recent, and Figure 1 determines what values of mutual information are considered large, i.e., MI> 0.5. This clarification is added to the revised manuscript and discussion of Figures 411 will refer to Figure 1.
In addition, the formulas used to determine the quantities plotted Fig.4Fig.11 are not explicitly provided. One way to address this comment could be to shorten the Section 2, retaining only the descriptions of the formulas that are used in the ULF/Eflux data analysis, and referring the interested reader to works already published on this topic (e.g. Wing et al., 2016 and references for information theory lectures). Another way could be to create an appendix for supplementary material.
We understand. We will shorten section 2, and be more explicit about the equations used to plot Figures 411.
2) The main results are presented as a list of 8 different figures (each composed of 4 different panels). Yet, there is little difference between the figures: They all appear to provide similar information (namely, that there exists a significant dependence between waves and particle fluxes). Fig.1 a)b) is enough to illustrate the approach and the point of the paper is conveyed in Table 1. In that context, everything else could be provided as supporting material. If all 32 panels are really necessary to convey the message of the work, this needs to be explained at the beginning of Section 4.
We agree with the reviewer that Table 1 contains all the essential results for the paper, and that a reader could simply go through Table 1 and skip the figures. However, Table 1 is extracted from the information found in Figures 411, and while the shape of the statistical dependencies are similar, differences between the two Events exist and are noteworthy, such as the larger error for mutual information in the 1993 Event than for the 1994 Event due to a difference in data points. Keeping the figures in the manuscript makes the paper selfcontained and allows the reader to see the range of lags for which the mutual information is significantly different from zero. The revised manuscript contains this explanation at the beginning of Section 4 and inform the reader that the main results can immediately be found in Table 1.

RC2: 'Comment on angeo202147', Anonymous Referee #1, 28 Aug 2021
Please note the typo of the Referee comment https://doi.org/10.5194/angeo202147RC1
"Fig.1 a)b) is enough to illustrate the approach" should be "Fig.4 a)b) is enough to illustrate the approach".
Sorry about that

AC2: 'Reply on RC2', Adnane Osmane, 05 Oct 2021
We thank the reviewer for carefully reading the manuscript and providing us with constructive feedback. Our answer to the specific comments (in bold and italic) can be found below.
 Line 19, “Their study are...” > “Their studies are...”
It is now corrected.
 Line 61, the Balikhin et al. citation is missing the publication year
It is now corrected.
 Lines 99100, it is easy to think of scenarios where one can gain information about the likelihood of event X, given Y. However, it is not so easy to think of scenarios where one can lose information. If X and Y are unrelated then no information is gained about X given Y. Can the authors elaborate on this?
Indeed, if X and Y are not dependent on one another, we have not lost information. But if a variable X (e.g., ULF wave power) and Y (MeV electron fluxes) are dependent on one another under some conditions (e.g., large solar wind speed ), the removal of the conditions upon which the dependence is strong can result in a loss of information (reduction of mutual information), and thus a loss of knowledge. We have added this clarification to the text.
 Lines 168169, the sentence is a bit awkward. The authors probably want to say ...mutual information and Pearson correlation is an indication that the correlation should not be interpreted linearly (or something like that).
It is now corrected.
 Line 210, there should be a coma between “radial diffusion” and “is a leading”.
It is now corrected.
 Line 238, there should be a description that SOPA is an instrument on board of Los Alamos National Laboratory (LANL) spacecraft.
We have added this description.
 Line 256, “... and positive viceversa”. Should this be “...negative viceversa”?
Indeed, good catch.
 Line 261, should the value above the shaded area represents a mutual information that has least three (not six) sigma significance?
Yes, it is three sigma. We corrected it.
 Lines 260270 and Figures 4 and 5. One of the main differences between mutual information and correlation in Figures 4 and 5 is that mutual informations consistently have very pronounced secondary peaks at time offset around 100 h whereas the secondary peaks in the Pearson correlations appear to be less pronounced or less significant. Can the authors discuss this?
This is a good observation. Overall, we show that the Pearson correlation is missing out about 2030% of the statistical dependence due to its inability to capture nonlinearities. We would need to look at each of these peaks in isolation to quantify this, but we can postulate that the difference we see in secondary peaks has a similar explanation.
10 In mutual information plots, Figures 4a, 4c, 5a, and 5c, the secondary peaks probably correspond to negative correlations, as inferred from their Pearson correlation counterparts. The anticorrelations can also be seen in Figures 6 and 7. Can the authors explain this anticorrelation between F1.2 and Sgr and Sgeo at time offset 100 h? The anticorrelations between F130 and Sgr and Sge can also be seen in Figure 11 at about the same time offset.
In Figure 4, the secondary peak around 100h corresponds indeed to a small negative correlation. This negative correlation can be explained by a loss of relativistic electrons associated to an enhancement in ULF wave power. ULF wave power locally increase the magnitude of the mean magnetic field sampled by the particles. So one possible consequence is that locally cyclotron resonances can take place for particles with higher parallel velocity, and result in enhanced scattering and losses. This is beyond the scope of the paper and it is speculative, but it might be interesting for future study to search for ULF wave modulation of local waveparticle interactions in terms of mutual information.
 Lines 345346, the authors claim that their results show that quantitatively the dependence is modest. This claim is repeated on line 386 and elsewhere in the manuscript. Table 1 shows adjusted correlations of 0.6 to 0.8. In many studies of space science, correlations of 0.70.8 would be considered strong or very good. “modest” is probably a subjective term. Can the authors comment on what they would quantitatively consider modest or strong or weak correlations?
This statement certainly needs clarification. The statistical dependence between 130 keV and ULF wave power is NOT modest since a Pearson correlation of 0.78 and a mutual information of 0.67 is large. However, the Pearson correlation for the relativistic electron and ULF wave power, ranging between 0.4 and 0.59 is modest (significant, but not large) when compared to the statistical dependence between 130 keV and ULF wave power. When compared to previous results, e.g., Simms et al. find correlations ranging between 0.15 and 0.65 between the maximum electron fluxes 48120 hours after the beginning of a storm and average ULF wave power for a given phase storm. In our case, we correlate the maximum ULF wave power with the maximum electron fluxes measured over an hour interval. We have added this clarification to the revised manuscript and provided more details in our discussion of the results of Simms et al.

AC2: 'Reply on RC2', Adnane Osmane, 05 Oct 2021

RC3: 'Comment on angeo202147', Anonymous Referee #2, 04 Sep 2021
The manuscript presents an interesting study of the dependence of the radiation belt energetic electron fluxes on ULF waves. The study uses mutual information and Pearson correlation to compute the correlations between (1.2 MeV and 130 keV) electron fluxes and (Sgr, Sgeo) ULF power indices. The study derives an expression that relates mutual information to Pearson correlation coefficient. Their equation is very useful for comparing measures of linear and nonlinear correlations and for determining nonlinearities in the system.
Overall, the manuscript is well written and the study is well executed. It should be published. This reviewer only has minor comments and suggestions, as listed below.
 Line 19, “Their study are...” > “Their studies are...”
 Line 61, the Balikhin et al. citation is missing the publication year
 Lines 99100, it is easy to think of scenarios where one can gain information about the likelihood of event X, given Y. However, it is not so easy to think of scenarios where one can lose information. If X and Y are unrelated then no information is gained about X given Y. Can the authors elaborate on this?
 Lines 168169, the sentence is a bit awkward. The authors probably want to say ...mutual information and Pearson correlation is an indication that the correlation should not be interpreted linearly (or something like that).
 Line 210, there should be a coma between “radial diffusion” and “is a leading”.
 Line 238, there should be a description that SOPA is an instrument on board of Los Alamos National Laboratory (LANL) spacecraft.
 Line 256, “... and positive viceversa”. Should this be “...negative viceversa”?
 Line 261, should the value above the shaded area represents a mutual information that has least three (not six) sigma significance?
 Lines 260270 and Figures 4 and 5. One of the main differences between mutual information and correlation in Figures 4 and 5 is that mutual informations consistently have very pronounced secondary peaks at time offset around 100 h whereas the secondary peaks in the Pearson correlations appear to be less pronounced or less significant. Can the authors discuss this? The authors did not plot the absolute values of the correlation coefficients r, but one can sort of see this in the plots.
In mutual information plots, Figures 4a, 4c, 5a, and 5c, the secondary peaks probably correspond to negative correlations, as inferred from their Pearson correlation counterparts. The anticorrelations can also be seen in Figures 6 and 7. Can the authors explain this anticorrelation between F1.2 and Sgr and Sgeo at time offset 100 h? The anticorrelations between F130 and Sgr and Sge can also be seen in Figure 11 at about the same time offset.
 Lines 345346, the authors claim that their results show that quantitatively the dependence is modest. This claim is repeated on line 386 and elsewhere in the manuscript. Table 1 shows adjusted correlations of 0.6 to 0.8. In many studies of space science, correlations of 0.70.8 would be considered strong or very good. “modest” is probably a subjective term. Can the authors comment on what they would quantitatively consider modest or strong or weak correlations?

AC3: 'Reply on RC3', Adnane Osmane, 05 Oct 2021
We thank the reviewer for carefully reading the manuscript and providing us with constructive feedback. Our answer to the specific comments (in bold and italic) can be found below.
 Line 19, “Their study are...” > “Their studies are...”
It is now corrected.
 Line 61, the Balikhin et al. citation is missing the publication year
It is now corrected.
 Lines 99100, it is easy to think of scenarios where one can gain information about the likelihood of event X, given Y. However, it is not so easy to think of scenarios where one can lose information. If X and Y are unrelated then no information is gained about X given Y. Can the authors elaborate on this?
Indeed, if X and Y are not dependent on one another, we have not lost information. But if a variable X (e.g., ULF wave power) and Y (MeV electron fluxes) are dependent on one another under some conditions (e.g., large solar wind speed ), the removal of the conditions upon which the dependence is strong can result in a loss of information (reduction of mutual information), and thus a loss of knowledge. We have added this clarification to the text.
 Lines 168169, the sentence is a bit awkward. The authors probably want to say ...mutual information and Pearson correlation is an indication that the correlation should not be interpreted linearly (or something like that).
It is now corrected.
 Line 210, there should be a coma between “radial diffusion” and “is a leading”.
It is now corrected.
 Line 238, there should be a description that SOPA is an instrument on board of Los Alamos National Laboratory (LANL) spacecraft.
We have added this description.
 Line 256, “... and positive viceversa”. Should this be “...negative viceversa”?
Indeed, good catch.
 Line 261, should the value above the shaded area represents a mutual information that has least three (not six) sigma significance?
Yes, it is three sigma. We corrected it.
 Lines 260270 and Figures 4 and 5. One of the main differences between mutual information and correlation in Figures 4 and 5 is that mutual informations consistently have very pronounced secondary peaks at time offset around 100 h whereas the secondary peaks in the Pearson correlations appear to be less pronounced or less significant. Can the authors discuss this?
This is a good observation. Overall, we show that the Pearson correlation is missing out about 2030% of the statistical dependence due to its inability to capture nonlinearities. We would need to look at each of these peaks in isolation to quantify this, but we can postulate that the difference we see in secondary peaks has a similar explanation.
1 In mutual information plots, Figures 4a, 4c, 5a, and 5c, the secondary peaks probably correspond to negative correlations, as inferred from their Pearson correlation counterparts. The anticorrelations can also be seen in Figures 6 and 7. Can the authors explain this anticorrelation between F1.2 and Sgr and Sgeo at time offset 100 h? The anticorrelations between F130 and Sgr and Sge can also be seen in Figure 11 at about the same time offset.
In Figure 4, the secondary peak around 100h corresponds indeed to a small negative correlation. This negative correlation can be explained by a loss of relativistic electrons associated to an enhancement in ULF wave power. ULF wave power locally increase the magnitude of the mean magnetic field sampled by the particles. So one possible consequence is that locally cyclotron resonances can take place for particles with higher parallel velocity, and result in enhanced scattering and losses. This is beyond the scope of the paper and it is speculative, but it might be interesting for future study to search for ULF wave modulation of local waveparticle interactions in terms of mutual information.
 Lines 345346, the authors claim that their results show that quantitatively the dependence is modest. This claim is repeated on line 386 and elsewhere in the manuscript. Table 1 shows adjusted correlations of 0.6 to 0.8. In many studies of space science, correlations of 0.70.8 would be considered strong or very good. “modest” is probably a subjective term. Can the authors comment on what they would quantitatively consider modest or strong or weak correlations?
This statement certainly needs clarification. The statistical dependence between 130 keV and ULF wave power is NOT modest since a Pearson correlation of 0.78 and a mutual information of 0.67 is large. However, the Pearson correlation for the relativistic electron and ULF wave power, ranging between 0.4 and 0.59 is modest (significant, but not large) when compared to the statistical dependence between 130 keV and ULF wave power. When compared to previous results, e.g., Simms et al. find correlations ranging between 0.15 and 0.65 between the maximum electron fluxes 48120 hours after the beginning of a storm and average ULF wave power for a given phase storm. In our case, we correlate the maximum ULF wave power with the maximum electron fluxes measured over an hour interval. We have added this clarification to the revised manuscript and provided more details in our discussion of the results of Simms et al.
Adnane Osmane et al.
Adnane Osmane et al.
Viewed
HTML  XML  Total  BibTeX  EndNote  

597  65  15  677  4  0 
 HTML: 597
 PDF: 65
 XML: 15
 Total: 677
 BibTeX: 4
 EndNote: 0
Viewed (geographical distribution)
Country  #  Views  % 

Total:  0 
HTML:  0 
PDF:  0 
XML:  0 
 1