Relationship between cosmic ray intensity and Earth's magnetic field: Cross-wavelet analysis during intense and moderate geomagnetic storms

Cosmic rays (CRs) are high-energy particles, primarily protons and atomic nuclei,^1,2 that originate from the Sun, from outside of the solar system in our own galaxy, and from distant galaxies.^3–5 Protons account for nearly 90% of all incoming cosmic ray particles, with helium nuclei accounting for the remaining 9% and nuclei of heavier elements accounting for the remaining 1%.^6,7 The flux of cosmic rays is a crucial factor in atmospheric chemistry and has implications for climate modeling.

Early studies on cosmic rays suggest that CR constitute a time and spatial variable flux of fully ionized energetic atoms and sub-atomic particles with specific abundances and energies that continuously permeate the solar system reaching the Earth and producing ionization and nuclear reactions in the terrestrial atmosphere.^8–11 The geomagnetic field deflects CRs depending on their rigidity, and thus, the flux of the low rigidity component gets modified.¹² This modulation by the geomagnetic field is significant as it influences the radiation environment at aviation altitudes and poses risks to electronics and human health during polar flights.

Intense geomagnetic storms created by the solar activity can affect geomagnetic shielding and thus affect the interplanetary CR distribution.^13–16 These storms, often triggered by solar flares and coronal mass ejections, can enhance cosmic ray fluxes, leading to increased radiation exposure. Higher-energy cosmic rays continue to enter the geomagnetic field, but the vast majority are occupied in radiation belts of our planet.¹⁷ Upon entering the Earth's atmosphere, primary CR particles with energies below several hundred MeV/nuc are simply stopped and absorbed in the atmosphere due to ionization losses.⁶ However, if the energy of a primary particle is sufficiently high, it can interact with atmospheric nuclei and produce several secondary leptonic and hadronic particles and radiation in the form of a shower.¹⁸ These secondary particles include muons, neutrons, and other radiation components that can reach the ground and are detectable by neutron monitors and ground-based observatories.

In the past, several interesting works tried to look at the connection between cosmic rays and Earth magnetic field. For example, Bowen et al.¹⁹ investigated the intricate relationship between terrestrial magnetism and cosmic-ray intensity. By using measurement in Austria, they found both positive and negative correlations between the Earth's magnetic field and cosmic-ray ionization. Positive magnetic effects were most apparent during magnetic storms, suggesting that increases in the Earth's horizontal magnetic force led to concurrent rises in cosmic-ray ionization. Conversely, negative effects were observed in daily, diurnal, and seasonal analyses, with one significant finding showing that cosmic-ray ionization decreased as the horizontal magnetic force increased, a relationship particularly pronounced in seasonal variations. Mandrikova et al.²⁰ introduced a wavelet-based method for analyzing Earth's magnetic field variations and cosmic ray data, particularly during high solar activity periods. Using a combination of multiresolution wavelet decompositions and neural networks, they found that the most significant geomagnetic disturbances coincide with notable fluctuations in cosmic ray levels, indicating a direct correlation between solar-induced geomagnetic changes and cosmic ray intensities. Shea and Smart²¹ investigated the potential connection between cosmic rays, changes in the geomagnetic field, and climate changes. They analyzed geomagnetic cutoff rigidities over 400 years, finding significant variations that mirror solar cycle effects on cosmic radiation flux.

Hedgecock²² also found a significant solar cycle variations in the magnetic field's power levels, spectral density at zero frequency, and correlation lengths, correlating these fluctuations with changes in cosmic ray intensities. It was also found that the magnetic field's vector average decreases with higher solar activity, leading to increased directional fluctuations. Freeman et al.²³ have shown that the horizontal component of a geomagnetic field also exhibits rapid fluctuations on short time scales, which may have an impact on CR flux. Rather remarkably, Okpala and Egbunu²⁴ compared the variability of cosmic ray count rates from two mid-latitude neutron monitors (Hermanus and Rome) and two higher-latitude neutron monitors (Inuvik and Oulu) to geomagnetic stations near the neutron monitoring stations (NMs). According to their findings, solar activity is often substantially associated (greater than 0.75) with the mean strength of the geomagnetic field and the galactic cosmic ray count rate. Kharayat et al.²⁵ also investigated the association among cosmic ray intensity, interplanetary magnetic field, and geomagnetic storms during solar cycle 23 and found that the trajectories of the CRs are affected during a geomagnetic storm, leading to a decrease in the observed cosmic ray intensity.

CR intensities and fluxes, as well as secondary particles that arrive on the ground, are of interest to scientists working in many different fields. Over the last few decades, a large number of studies have been conducted to study the correlation of CR intensity (CRI) with solar wind parameters.^26–29 Their findings showed that the intensity of cosmic rays and geomagnetic activity are both influenced by solar and interplanetary parameters.³⁰ However, the association of horizontal component of the geomagnetic field and CR intensity has not been extensively studied. Since the amount of cosmic ray particles entering the atmosphere and reaching the surface is affected by Earth's magnetic field, it is our interest to understand the relationship between the CR intensity and the horizontal component of the geomagnetic field using modern statistical tools, such as cross-wavelet transform (XWT) and wavelet transform coherence (WTC).

This paper is organized as follows: We described the methodology in Sec. II. Results are discussed in Sec. III. Finally, we concluded our findings in Sec. IV.

We used two-minute resolution count rate data obtained from three neutron monitor (NM) stations, IRKT, YKTK, and HRMS, which are publicly available in the Neutron Monitor Database (NMDB) site (https://www.nmdb.eu/nest/). The chosen NM stations have varying cutoff rigidity, latitude, longitude, and local time (LT). The detailed information of all stations is given in Table I. The magnetometer data utilized in this study were obtained from the International Real-time Magnetic Observatory Network (https://intermagnet.org/data-donnee/data-eng.php). Also, the interplanetary parameters, the southward component of interplanetary magnetic field IMF-Bz (nT), solar wind speed V (km/s) are obtained from OmniWeb data repository (https://omniweb.gsfc.nasa.gov/form/dx1.html).

The cross wavelet transform (XWT) is a mathematical technique used to analyze and explore the relationships between two-time series.³⁴ It is built upon the continuous wavelet transform (CWT), which is a powerful tool for decomposing a time series into its time–frequency representation.³⁵ The XWT combines the CWTs of two different time series and provides valuable insights into their possible association. By analyzing the XWT, we can identify locations where the two series exhibit high common power, indicating regions of strong correlation or association. This offers a way to understand the shared dynamics or common features between the two-time series. Wavelet coherence (WTC) is a measure defined within the framework of the XWT. It quantifies the local correlation between the CWTs of the two-time series at each time–frequency point. By examining the WTC, we can uncover regions of locally phase-locked behavior, indicating that certain frequency components in the two-time series are consistently synchronized or correlated.³⁶ We implemented both XWT and WTC to study the relationship between CRI and dH in our study.

This section investigates the relationship between CRI and dH during two intense and one moderate geomagnetic storms using wavelet-based approaches (XWT and WTC).

Figure 1 presents an overview of space weather indicators during the September 6–10, 2017 geomagnetic storm, providing a visual representation of the temporal relationship between the solar events and the resulting geomagnetic disturbances. The variations in the interplanetary magnetic field (IMF) component Bz, plasma speed (km/s), Kp index, and Dst index are presented in this analysis. On September 8, 2017, at 07:40 UT, a powerful M8.1 class solar flare was recorded, accompanied by geomagnetic indices indicating a strong geomagnetic storm: Bz = −23 nT and Vx = 800 km/s, resulting in a Dst index of −142 nT. Following this event, on September 9, 2017, at 10:50 UT, an M3.7 class solar flare was observed, and the geomagnetic indices showed signs of recovery with Dst reaching −37 nT. A coronal mass ejection (CME) associated with the intense X9.3 solar flare on September 6, 2017, arrived earlier than anticipated, reaching Earth at 23:04 UT on September 7, 2017. This CME triggered a strong geomagnetic storm, which commenced at 23:25 UT and rapidly intensified into a severe geomagnetic storm around 23:50 UT. The effects of this CME led to the persistence of severe geomagnetic storm intensities on September 8, 2017, resulting in a Dst index of −142 nT. The impact of the September 6, 2017, CME also suggests the likelihood of another strong geomagnetic storm with severe conditions on September 9, 2017. However, as the effects of the CME diminish, conditions are assumed to gradually improve, transitioning to a minor geomagnetic storm by September 10, 2017.

Figure 2 represents the characteristics of daytime high-latitude geomagnetic disturbances and geomagnetic pulsations during the strong magnetic storm that occurred from August 24–28, 2018. This magnetic storm took place during the declining phase of the 24th solar activity cycle, which exhibited a very low level of solar flare activity. On August 26, 2018, a G3 geomagnetic storm was initiated by a solar filament eruption that occurred on August 20, 2018, at around 07:00 UT. The formation of a storm resulted from the accumulation of various factors, including weak coronal mass ejections (CMEs), transients, co-rotating interaction regions (CIR), and high-speed streams (HSSs).³⁸ During this event, the southward flow of the interplanetary magnetic field (IMF Bz) was observed with the value of −20 nT during its main phase, coinciding with a time of decrease in the Dst index value (−180 nT).^39,40 This indicated the influence of the IMF Bz component on the strength of the geomagnetic disturbance. The solar wind speed rose to 500 km/s, and the Kp-index reached around 7.5. Interestingly, during the recovery phase of the storm on August 27, 2018, the value of solar wind speed increased to around 700 km/s.

On February 18, 2020, a geomagnetic storm, induced by a coronal hole high-speed stream (CH HSS), caused the reduction of the Earth's magnetic field to low values, causing the Disturbed storm time index (Dst) index to fall to −52 nT. Figure 3 shows variations in solar wind parameters from February 16–20, 2020.

Bz component of the interplanetary magnetic field attained a maximum divergence of −12 nT as it extended southward. Around 19:00 UTC on February 18, the Bz component gradually moved northward, trended less southerly, and became more variable. Up until around midnight UTC on February 19, the speed of the solar wind was mostly 350 to 400 km/s, but it then picked up and finally exceeded 500 km/s. This may have happened as a result of a link to the coronal hole high-speed stream (CH HSS), which has a negative south polar polarity. Due to the poor connection with the south pole, and negative polarity CH HSS, the solar wind field is predicted to be increased and disrupted for the duration of February 19, 2020. Through February 21, 2020, the solar wind condition is projected to be somewhat disturbed due to persistent but diminishing CH HSS impacts.

Figure 4(a) depicts the variations in CR counts and dH across the HRMS, IRKT, and YKTK stations during the intense geomagnetic storm period of September 6–10, 2017. During the storm days (September 8–9), a clear decrement can be observed in both CR counts (with similar trends) and dH at all stations, with a smaller decrement at the HRMS station compared to the IRKT and YKTK stations. Here, YKTK is a station with low cutoff rigidity, and HRMS and IRTK are of high cutoff rigidity. This shows that during the storm, the decrement in CRI is higher at the stations with low cutoff rigidity. At the beginning of September 8, the start of the storm, a sudden dip in dH value is observed at both the IRTK and YKTK stations, but no such trend is seen at the HRMS station. However, the nature of dH corresponding to the peak decrement of CR counts shows a similar pattern, although the YKTK station exhibits a larger decrement in CR counts compared to the other two stations. This indicates that CR counts and dH can vary according to latitude. Figure 4(b) represents the variation in CR counts and dH during another storm event on February 16–20, 2020. We observe a decrement in CR counts at the YKTK station, with some lag in relation to dH. However, CR counts are found to have slightly increased at the IRKT station. No clear signature is noticed at the HRMS station. Similarly, Fig. 4(c) represents the variation in CR counts and dH during the geomagnetic storm event of August 24–28, 2018. A clear decrement in dH is observed at all stations, with some lag. The CR counts decreased at the beginning of the storm event on August 26, with the rate varying at different stations. Additionally, an increase in CR counts is noticed during the time corresponding to the maximum dip in dH at all stations. We observed distinct variations in the relative counts of cosmic ray intensity and the components of changes in the Earth's magnetic field over each station. The degree of change in the Earth's component of the magnetic field varies with the severity of the storm.⁴¹ The variations in cosmic ray counts also depend on the severity of the storm and are caused by geomagnetic storms.^31,42 In addition, the ring currents have a range of effects on how intense cosmic rays are seen on Earth. Measuring from the geomagnetic axis, its magnetic field is parallel to the geomagnetic field outside the ring current. The increased strength of the geomagnetic field outside of the ring current was the first suggested factor for a decrease in cosmic ray intensity, which typically coincides with a magnetic storm.⁴³ We can see such nature of CR counts and dH on their time series plot in Fig. 4.

Interestingly, cosmic ray counts are inversely related to solar wind speed, primarily due to the heliospheric magnetic field (HMF) carried by the solar wind. As the wind speed increases, the HMF strengthens, scattering and deflecting incoming cosmic rays more effectively, which reduces their intensity near Earth. This modulation process is a key aspect of solar-terrestrial interactions. Periods of high wind speed provide greater shielding against cosmic rays, leading to lower cosmic ray counts. Conversely, during periods of low wind speed, more cosmic rays are able to penetrate the heliosphere, resulting in higher cosmic ray counts. Additionally, the north and south orientations of the Bz component significantly influence cosmic ray counts. When the Bz component is oriented south, it reconnects with Earth's geomagnetic field, creating geomagnetic storms and disturbances that enhance magnetic shielding against cosmic rays, thereby reducing cosmic ray flux. Contrarily, when the Bz component is oriented northward, it weakens its coupling with Earth's magnetic field, allowing more cosmic rays to penetrate the magnetosphere and reach the ground, which increases cosmic ray counts.

In this work, we implemented XWT and WTC methods to see the relationship between CR counts and dH. One of the key advantages of the XWT is its ability to provide detailed information about the phase connection between the CR counts and dH time series during geomagnetic storms. Here, phase represents the relative timing or synchronization between CR counts and dH signals. In the context of the XWT, phase information reveals the timing relationships between the corresponding frequency components in the CR counts and dH time series. Whereas, WTC helps to provide a local correlation between the CWTs of CR counts and dH. This method reveals locally phase-locked behavior.³⁴ Finding correlated regions between signals that are generally uncorrelated is the most useful application of WTC. A temporal frequency map that the WTC approach produces tells us how strong the correlation is.

Figures 5–7 depict the relationship of CR counts and dH using XWT and WTC techniques during the storm periods of September 6–10, 2017, August 24–28, 2018, and February 16–20, 2020, respectively. Colors in the cross-wavelet analysis (XWT) plots show the amplitude of the power, whereas colors in the wavelet coherence analysis (WTC) plots represent the power of the coherence, which ranges from 0 to 1. Using the WTC approach to examine two-time series data (CR counts and dH, arrows pointing NE and SW indicate that the CR counts is leading, while arrows pointing NW and SE indicate that the dH is leading). The thick black outlines represent the 95 $%$ confidence level, while the region below the thin line represents the coefficient of variation. Arrows represent the relative phase connection, with arrows pointing right for the in-phase relationship, left for the anti-phase relationship, and straight up for the lag between CR counts and dH by 90°.

Figure 5 depicts the relationship between CR counts and dH during September 6–10, 2017, across the HRMS, IRKT, and YKTK stations. During the storm period (September 8–9), there is a noticeable correlation between CR counts and dH across all stations. Specifically, a common power spectrum is observed in the time series of CR counts and dH, ranging from 16 to 64 min and 128 to 1024 min for HRMS and IRKT stations. At the YKTK station, this common power spectrum occurs between 16 and 32 min and 256–1025 min. This suggests a significant common fluctuation between CR counts and dH during these periods, indicating a correlation between the two variables. Additionally, Fig. 7 reveals a long-term common power between CR counts and dH over 512–1024 min at the HRMS station. Furthermore, the analysis using WTC confirms a positive association between CR counts and dH during the storm period, specifically between 128 and 256 min at HRMS, 128–512 min at IRKT, and 128–1024 min at YKTK stations. Interestingly, WTC exhibits a negative association between CR counts and dH during the period 512–1024 at the IRTK station.

Figure 6 illustrates the XWT and WTC analysis of CR counts and dH from August 24–28, 2018, across the HRMS, IRKT, and YKTK stations. Notably, there are significant short-term common fluctuations between CR counts and dH spanning 8–64 min observed at all stations. Furthermore, a long-term common power between these two-time series data are observed within 128–1024 min. The WTC analysis indicates a positive association between CR counts and dH during the storm period, with power spectrum peaks occurring between 128 and 512 min at HRMS and 256–512 min at YKTK stations, respectively. However, a different relationship between CR counts and dH is observed at the YKTK station, where a negative association is seen between 128 and 512 min during the storm period. Additionally, a long-term association between CR counts and dH is revealed between the period of 512–1024 min at this station.

Figure 7 illustrates the correlation between CR counts and dH from February 16–20, 2020, across studied stations. Similar to previous events, a common power spectrum is observed between CR counts and dH. Significant short-term common fluctuations between these two-time series are visible within the periods of 8–128 min and 256–1024 min at all stations. The XWT plots depict mixed trends of association between CR counts and dH, including positive, negative, and lag effects, observed at different periods and stations. A clearer trend emerges from the WTC plots. The WTC analysis reveals an opposing nature between CR counts and dH during the storm period, with significant fluctuations and correlations occurring between the periods of 128–256 min at HRMS and IRKT stations. However, it indicates a positive association (with lag) between CR counts and dH over the periods of 128–256 min and 512–1024 min.

Overall, the XWT and WTC demonstrate that the relationship between CR counts and dH is distinct over different stations and different storm events. Furthermore, they show the varied common periodicity associated with CR counts and dH fluctuations over different stations throughout the selected geomagnetic storms. We think the reason behind the varying relationship between CR counts and dH during the different periods in XWT and WTC is that when the geomagnetic field is disturbed or weakened, it allows more cosmic rays to penetrate the Earth's atmosphere, increasing CR counts. Conversely, when the geomagnetic field is stronger, it provides a shielding effect, reducing the influx of cosmic rays and leading to lower CR counts. There can be other possible reasons, including the strength and orientation of the interplanetary magnetic field, the location of the observer, and local magnetic fields.⁴⁴ Cosmic radiation increases with altitude. It varies with latitude, indicating that it is made up of charged particles that are influenced by the Earth's magnetic field.^45–47 The Earth's magnetic field, in general, is only used to regulate the primary cosmic radiation that arrives at the top of the atmosphere, but it may also influence low-energy charged particles at aviation altitudes.^48,49 Two mechanisms regulate (vary) the flux (flow rate) of cosmic rays striking the Earth's upper atmosphere: the Sun's solar wind and the Earth's magnetic field.⁵⁰ The solar wind is the expanding magnetized plasma of the Sun, which decelerates incoming particles while also partially rejecting certain particles with energy less than roughly 1 GeV.⁵¹ Because of fluctuations in solar activity during its normal eleven-year cycle, the amount of solar wind is not constant. As a result, the intensity of modulation fluctuates with solar activity.⁵² The Earth's magnetic field also deflects certain cosmic rays, as evidenced by the fact that the strength of cosmic radiation varies with latitude, longitude, and azimuth.⁵³ Because of the polarity of the Earth's geomagnetic field and the positive charge majority in primary cosmic rays, the cosmic flux fluctuates from east to west; this is known as the east-west effect.⁵⁴ The longitude dependency resulting from the geomagnetic dipole axis not being parallel to the Earth's rotation axis can lead to such variations in CRI.⁵⁷ Also, the intensity of cosmic rays is lower near the equator than at the poles because the geomagnetic cutoff value is the largest at the equator. This is explained by the fact that charged particles prefer to flow in the direction of field lines rather than across them.⁵⁵ 

In this study, we considered two intense geomagnetic storm events on September 8, 2017, and August 26, 2018, and one moderate event on February 18, 2020, to examine the association between cosmic ray intensity and the horizontal components of Earth's magnetic field using neutron monitors and inter-magnet instruments for the same stations. The following conclusions were made from our work:

• During the intense geomagnetic storm on September 8, 2017, a significant decrease in cosmic ray intensity and the horizontal component of Earth's magnetic field was seen over all stations as compared to the other two storm events (August 26, 2018, and February 18, 2020). This decrease was more prominent over the station that has low cutoff rigidity (YKTK station) than the stations with high cutoff rigidity (HRMS and IRKT stations). So, the magnetic cutoff rigidity of the station is an important factor that affect the cosmic ray distribution.^48,52,56

• The association between cosmic ray intensity and horizontal components of Earth's magnetic field does not have a causal association over each station. This might be attributed to interplay between many variables, such as the interplanetary magnetic field, observer location, local magnetic fields,⁴⁴ and the geomagnetic dipole axis.

• The previous conclusions are well supported by the cross-wavelet analysis (XWT) and wavelet coherence analysis (WTC) of time series data of CRI and dH. XWT showed a common power spectrum of the fluctuation of CRI and dH within various periods during the geomagnetic storms. Whereas, WTC presented the correlation of those common powers in the time–frequency domain. XWT and WTC have been demonstrated as powerful tools to study the association between two-time series data.

In conclusion, this study shows that the association between cosmic ray intensity and the horizontal components of Earth's magnetic field is not quite obvious, and it depends on several factors such as the severity of the geomagnetic storm and the magnetic cutoff rigidity, as well as the latitude and altitude of the station. So, a further statistical analysis is needed to fully understand their association. We believe this study will serve as a crucial reference for future studies.

We acknowledge Neutron Monitor Database (https://www.nmdb.eu/nest/), International Real-time Magnetic Observatory Network (https://intermagnet.org/), and OmniWeb data repository (https://omniweb.gsfc.nasa.gov/) to provide a data for this work.

The authors have no conflicts to disclose.

Chali Idosa Uga: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Sujan Prasad Gautam: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Binod Adhikari: Conceptualization (equal); Data curation (supporting); Investigation (supporting); Methodology (supporting); Software (supporting); Supervision (supporting). Ashok Silwal: Conceptualization (supporting); Data curation (supporting); Investigation (supporting); Methodology (supporting); Software (supporting); Supervision (supporting); Visualization (supporting). Ashutosh Giri: Investigation (supporting); Methodology (supporting); Software (supporting); Supervision (supporting); Visualization (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available within the article.