A preliminary investigation of the empirical relationship between small-amplitude Forbush Decreases and solar wind disturbances

ABSTRACT

High-magnitude (amplitude |$\le -3{{\ \rm per\ cent}}$|⁠) Forbush decreases (FDs) are generally employed by researchers investigating the solar-terrestrial connection mechanisms. Although it has been observed that small-amplitude FDs are relatively important as they may be the outcome of the response of cosmic ray time-intensity variations to solar ejections that generate interplanetary and solar wind plasma disturbances, empirical relations between weak FDs and solar-terrestrial parameters are rarely tested. In an attempt to analyse the suggested strong connections between weak FDs and solar-terrestrial phenomena, we employed some comparatively more efficient, accurate, and highly sensitive versions of the recently developed computer FD event selection software. Large catalogues of low-amplitude (FD(per cent) ≥−3) Forbush events were selected from Apatity, Moscow, Newark, and Oulu NMs. These catalogues allow us to test, for the first time, the empirical relations between small-amplitude FDs and solar wind data. We find significant negative correlations between solar wind speed (SWS) and the small FDs at OULU, NWRK, and MOSC stations. While the relation at OULU is strong and statistically significant at 95 per cent confidence level, the weak correlation at NWRK and MOSC is only significant at a 90 per cent level. The negative correlation between the small events at OULU and the interplanetary magnetic field (IMF) is also significant at the a 95 per cent level. The relation between SWS and IMF and high-amplitude FDs were also tested, and the correlation coefficients were negative, strong, and statistically significant at a 99.9 per cent level of significance.

1 INTRODUCTION

Cosmic rays (CRs) that have their origin inside our Galaxy are referred to as Galactic cosmic rays (GCRs). They are energetic charged particles with energy ∼100 MeV–10 GeV (e.g. Yu et al. 2015) and are composed of 2 per cent electrons and 98 per cent atomic nuclei, with the latter made up (in number density) of 87 per cent protons, 12 per cent helium, and about 1 per cent heavier nuclei (Simpson 1983). GCRs are prevalent in the heliosphere with a flux that is nearly isotropic. However, their intensity experiences variabilities due to the modulation effect in the heliosphere attributed to solar-magnetic activity. At various time-scales, GCR flux is recurrently modulated, such as CR diurnal anisotropy (Okike 2021b), 27 d, and 11 yr long-term changes. There also exist abrupt or non-periodic modulations in GCR intensity. Ground level enhancements (GLEs) and Forbush decreases (FDs) are the outstanding non-periodic transient time-intensity changes of CR flux observed by ground-based neutron monitors (NMs) of the worldwide network (Oh, Yi & Kim 2008; Badruddin & Kumar 2015). The magnitude of the depressions varies with the different geomagnetic cut-off rigidity of each NM detector. Geomagnetic cut-off rigidity expresses the resistance experienced by a CR particle to permeate the terrestrial magnetic field (Lingri et al. 2016).

FDs, discovered by Scott E. Forbush (Forbush 1937, 1938), are characterized by short-term abrupt depressions in the intensity of CR flux and reach their optimum decrease in about a day and accompanied by a gradual recovery within a few days (Gopalswamy et al. 2014). They are grouped as either sporadic/non-recurrent or recurrent in accordance with the kind of interplanetary medium disturbances that generate them. Sporadic/non-recurrent FDs are those caused by ephemeral interplanetary events associated with coronal mass ejections (CMEs) and their interplanetary counterparts (ICMEs) streaming from the Sun, while recurrent FDs result from high-velocity streams from coronal holes (CHs) rotating together with the Sun (e.g. Cane 2000b; Richardson 2004; Badruddin & Kumar 2015; Melkumyan et al. 2018). On average, the non-recurrent events have large magnitudes and asymmetric profile relative to symmetric configurations and small amplitudes that characterizes the recurrent FDs (Lockwood 1971; Melkumyan et al. 2019).

Although the magnitude of FDs is a subject of controversy (Belov et al. 2001c; Okike & Collier 2011; Okike 2020d), events whose magnitudes are |$\ge 3{{\ \rm per\ cent}}$| are often regarded as strong or high-magnitude FDs (Van Allen 1993; Cane, Richardson & von Rosenvinge 1993, 1996; Belov 2008; Oh et al. 2008; Harrison & Ambaum 2010; Laken et al. 2012). It is true that some researchers who analyse the solar sources of FDs (Oh et al. 2008; Belov et al. 2001c, 2014; Okike et al. 2021a) attempt to connect the strength of Forbush events with the condition of the solar wind plasma that generate them, the reported quantitative magnitude of FDs are usually calculated using the count rate of the reference detectors. It has been demonstrated, using a few isolated FDs, that energy/rigidity of NMs determines the recovery time of FD event (Jamsen et al. 2007). This effort has recently been complimented by Lagoida et al. (2023) using a larger event sample from spacecraft measurement. Nevertheless, the recent statistical methods (e.g Okike & Nwuzor 2020) suggest that a number of other factors may also contribute significantly to the CR intensity variations at a given station.

Investigators analysing the solar origins of FDs submit that stronger or larger FDs are triggered by higher solar wind speed (SWS) and high magnitude interplanetary magnetic field (IMF), while the weak or small amplitude FDs tend to be driven by relatively lower velocity and weaker IMF solar wind. In earlier work on the CR intensity decreases at Calgary NM station, Badruddin., Venkatesan & Ananth (1991) observed that large magnitude FDs are caused by shocks associated with the helium-enhancement (He), and those not associated with He shocks show relatively a small decrease in CR intensity quantified as CR(per cent) > −3. FDs with CR (per cent) ≤ (−3, −3.5, and −5) are severally adopted as thresholds for selecting large FDs (see e.g. Pudovkin & Veretenenko 1995; Oh et al. 2008; Kristjansson et al. 2008; Okike et al. 2021a).

A close scrutiny of the rich body of literature on CR intensity variations shows that catalogues of small amplitude |$\le 3{{\ \rm per\ cent}}$| are rarely published. Among the existing catalogues prepared from single NMs including those of Lockwood (1990), Tinsley & Deen (1991), Cane et al. (1993), Pudovkin & Veretenenko (1995), Cane, Richardson & von Rosenvinge (1996), Oh et al. (2008), and Lee, Oh & Yu (2013), it is only the catalogue of Pudovkin & Veretenenko (1995) that contains FDs whose magnitudes are |$\le 3{{\ \rm per\ cent}}$|⁠. Out of the 65 FDs they selected, 14 FDs are of low amplitude (⁠|$\le 3{{\ \rm per\ cent}}$|⁠) while others are |$\ge 3{{\ \rm per\ cent}}$|⁠. It has been empirically (see Okike & Umahi 2019b; Okike 2019b, 2020d; Alhassan, Okike & Chukwude 2022a, for example) demonstrated that the reasons for the bias towards selection of high-magnitude FDs may be due to the inefficiency of the implemented manual event detection techniques. CR diurnal anisotropy has a significant effect on the amplitude of FDs, especially the low-amplitude ones. As previously observed by Barouch & Burlaga (1975), it would be difficult to detect the elusive small FDs using data from a single CR station without some sort of data transformation/filtering. The possibility of detecting very low-magnitude FDs from raw CR data, without data filtering/decomposition, was attempted for the first time by Okike (2020b) and Okike et al. (2020). In agreement with the suggestions of Bartels (1935) and Barouch & Burlaga (1975) regarding removal of the influence of CR diurnal anisotropy through harmonic analysis or numerical filtering, Okike (2020b) and Okike et al. (2020) equally accounted for the contributions from anisotropy in the same CR data. This allows the authors to empirically compare the predicted differences between FDs identified from raw (i.e. unprocessed) CR data with FDs selected from harmonically transformed CR data. fig. 2 of Okike (2020b) and fig. 6 of Okike et al. (2020) elegantly present the similarities and differences between the two signals (raw and Fourier-transformed CR data) and the FDs associated with them.

The alternative approach – averaging CR data over a number of CR stations – suggested by Barouch & Burlaga (1975) and attempted thereafter (Barouch & Burlaga 1975) has been pursued by some researchers. Using data from seven high-latitude NMs, Dumbovic et al. (2011), selected very low amplitude (0.3 per cent) FDs. Another team that has made significant contributions in this area is the Russian researchers at the Pushkov Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences (IZMIRAN). They have created a comprehensive list of FDs including the large and very low magnitude (0.3 per cent) events using the global survey method (GSM). This catalogue is published at one of the IZMIRAN websites (Forbush-effects and interplanetary disturbances (FEID)). Their event catalogue is very popular and has been used extensively in many FD-based investigations (e.g. Belov et al. 2001c, a, 2007, 2014; Lingri et al. 2016, 2019).

None the less, the method of combining CR data from different CR stations has been heavily criticized. Sandstrom (1968) warned that great care must be taken when attempting to combine CR-time intensity variations from different locations of the Earth. CR anisotropy which could change the amplitude and phase of FDs at different stations may have dramatic effects when FD signals are assimilated from different NMs. For effective results, only FDs that are worldwide in extent such that the changes in phases and amplitudes are simultaneous and proportional at all NMs may be combined. Unfortunately, detecting such symmetry in one FD event profile at different places/detectors is almost a utopia. This is due to the contamination from the ever-present and location-dependent CR anisotropy. And when the amplitude and phase of events are different at different NMs, the resultant event amplitude might be affected in many different ways. The amplitude may be unduly enhanced, diminished, or even completely destroyed when they are totally out of phase at the different stations. In fact, Jordan et al. (2011) referred to FD signals from different stations as ‘desperate’ data, criticizing the idea of combining them.

Fig. 1 (Lingri et al. 2022) gives a graphic illustration of CR time-intensity variations during the time of small (left-hand panel) and large (right-hand panel) FDs at different locations on the Earth. Though the narrow panels severely reduce the amplitudes of the flux variations, it does not require an expert in the field to observe that the event profiles differ at different NMs. Signals from the same stations are presented for each of the events on the two panels. The first two stations are labelled MGDN (i.e. Magadan) and NVBK (Novosibirsk). The level of intensity reductions differs for each of the FDs at the two stations. NVBK sees a higher intensity reduction than MGDN. A significant phase shift between the event profiles at the two stations is evident. MGDN seems to see event reduction earlier than NVBK. This implies that why one station is recording increasing intensity (a reference to MGDN at the left-hand panel), NVBK was observing depression in flux intensity.

The event onset time, indicated by the vertical lines (referencing the onset of sudden storm commencement at Earth), is clearly not the same at some of the stations. The four stations, juxtaposed very closely at the left-hand panel (KERG, INVK, TXBY, and YKTK), have different onset as well as significantly different flux profiles. The time of maximum increase at INVK coincides with the time of the first decrease at KERG, for instance. TXBY is recording maximum increase almost at the time YKTK is recording maximum intensity reduction. Further inspection of the graph may reveal other differences. Without further comments, we note, in agreement with Sandstrom (1968) and Jordan et al. (2011), that averaging FD signals for these events over these stations may not be suitable as the amplitudes of the resultant wave profile would be greatly biased/distorted. For example, it may not be possible to correctly rank FD events using data averaged from different stations (see also Okike et al. 2021a; Okike 2021b).

Kontor, Lyubimov & Pereslegina (1977) examined the characteristics of small FDs and their association with the general dependence of the intensity in 1965–1972. They noted that small FDs are due to the continuous structure of the streams of heterogeneous magnetized solar wind plasma. The work further considered and reported various structural forms of small FDs such as (a) conventional asymmetric form with a steeper fall phase relative to the recovery phase, (b) symmetrical form where the steepness of the decay is almost the same with the recovery, (c) the degenerate form which is the pronounced decrease of the recovery steepness down to zero, (d) the form in which the coupling coefficient assumes unity, less than unity or even negative due to > 30 MeV solar CRs in a given FD and, (e) a spike – the rapid short time increase of the steepness of the recovery phase which may even exceed the steepness of the intensity increases; its occurrence is occasional. Melkumyan et al. (2019) compared 350 recurrent and 207 sporadic FDs in solar cycles 23 and 24. The data is taken from the IZMIRAN data base. They reported that sporadic FDs dominate the data in the maxima of the cycles while recurrent FDs prevailed in the minimum between the cycles. Their investigation further showed that FD parameters (amplitude, rate of decrease, anisotropy) tended to be larger for sporadic events as compared to recurrent FDs.

1.1 Motivation for the current work

Okike (2021a) roundly criticized the manual technique of FD event identification in the current era of Big Data and the preponderance of sophisticated and extremely high-speed computers. In view of the paradigm shift from manual/punch card NM data acquisition (Moraal, Belov & Clem 2000; Shea & Smart 2000), a corresponding improvement in the methodical approaches to Forbush event selection is obviously expected. But a close inspection of the literature shows that this is not the case. Besides the various versions of FD event detection computer codes developed by our group (see Okike 2019a, b, 2020d, a; Okike & Umahi 2019a; Okike & Alhassan 2022; Alhassan et al. 2022a; Alhassan, Okike & Chukwude 2022b, for example), we are not aware of other publications that have fully automated Forbush event detection, selection, timing, amplitude estimation, and event cataloguing.

Although the semi-automated methods of Ramirez et al. (2013), IZMIRAN group (Belov 2008; Belov et al. 2018b, a; Abunina et al. 2020) and Light et al. (2020) reflect significant improvements over the commonly employed manual method, Alhassan et al. (2022a) observe that researchers using such methods are yet to deal with a number of potential pitfalls in the implemented software. A statistical approach to Forbush event detection is lacking in the method, for example. The semi-automated approach also follows the conventional manual case study approach which can only analyse one FD event at a time. The detectability of small-amplitude FDs is another challenge yet to be surmounted by users of the semi-automated technique. Though the global survey method of the IZMIRAN team could detect low amplitude FDs of the order of 0.3–0.5 per cent (see Belov et al. 2018), it is evident from Tables 1 and 2 that the current algorithm can detect very low-amplitude events (magnitude |$\ge -0.03{{\ \rm per\ cent}}$|⁠). Conventionally, FD magnitudes may be represented with their real (negative) (e.g Kristjansson et al. 2008; Okike 2021a; Okike & Alhassan 2022) or absolute (positive) (e.g. Lockwood 1990; Belov 2008; Okike 2021a; Okike & Alhassan 2022) values. In the current work, real or absolute values of FDs are likewise used interchangeably. The present algorithm is extremely sensitive to small FDs and can even detect events whose amplitude is as small as |$\ge -0.01{{\ \rm per\ cent}}$| (see also Okike 2020d).

Unlike other versions of our code which are somewhat biased towards the selection of large FDs, the algorithm used here is only designed for the analysis of weak FDs. The large catalogues of events identified (see Tables 1 and 2) allow us to investigate, for the first time, the statistical link between small Forbush events and solar wind/terrestrial parameters.

2 DATA SOURCE

The daily averaged CR intensity count rates data between 1996 to 2008 (solar cycle 23) from Apatity (APTY, 43.28°N, 42.69°E, rigidity = 5.6GV), Moscow (MOSC, 55.5°S, 37.3°E, rigidity = 0.77 GV), Newark (NWRK, 39.7°N, −75.8°W, rigidity = 2.09 GV) and Oulu (OULU, 65.0°N, 25.5°E, rigidity = 0.77 GV) NM stations, corrected for atmospheric pressure, sourced from IZMIRAN common website: http://cr0.izmiran.ru/mxco/main.htm were used. The site is hosted by the IZMIRAN team. This period is of interest as it presents several energetic events, particularly during its declining phase in 2003 October to November, 2005 January, and 2006 December (see e.g. Gopalswamy et al. 2014; Lingri et al. 2016). These stations are among the oldest stations that have continuous records of relevant observations. table 1 of Gil et al. (2015) shows that four of these NMs are NM64 type at sea level with an altitude of less than 300 m. MOSC NM detector is known for its sensitivity to the anisotropy of CRs, and due to its 24 NM counters, it is believed to have one of the most reliable statistical accuracies of the NMs operating close to sea level (Belov et al. 2018b). Some of the relevant characteristics of the two NMs are indicated by Alhassan., Okike. & Chukwude. (2021b) and the references therein. Daily mean SWS, IMF, and geomagnetic storm indices (Dst and ap) data are downloaded from the OMNI data base http://omniweb.gsfc.snasa.gov/ow.html. One of the merits of using daily averaged CR data is that the effects of CR diurnal anisotropy are reduced (Dumbovic et al. 2011; Belov et al. 2018b).

3 AUTOMATED EVENT SELECTION

Unlike the routine in the manual FD selection model in which portions of the time series data are culled out and plotted, a computer algorithm that is able to concurrently calculate FD event time, as well as its magnitude, is developed and applied in this work. The FD location software employed is comparable to the FD location code developed by Okike & Umahi (2019b). The major dissimilarities are that this current algorithm accepts raw CR data as an input signal and is designed to detect only small amplitude events. The code is written in a non-commercial computer software R (Team, R Core 2014). Originally, R was written by Robert Gentleman and Ross Ihaka, Department of Statistics, University of Auckland (see R Foundation for Statistical Computing Platform). The present code is equipped to select FDs from both raw (i.e. CR data downloaded from website in the original form they were provided by the principal investigator) and pre-processed (i.e. Fourier transformed) CR data. It is designed to search for depressions/turning points as well as the time of reductions in the raw CR data. The depressions are signatures of FD events. The two subroutines integrated into the current program are used to search for the FDs and the associated event time of occurrences.

Using the CR count means for the period of interest as the normalization threshold, the first subroutine determines the event amplitude whereas the second subroutine tracks the time of occurrence of the reductions. Okike & Umahi (2019b) adopted a predefined threshold or baseline (B) such that when B ≥−1, the code detects large FDs, and B ≤−0.5, it selects small FDs. With some technical improvements in the algorithm, a baseline of B ≤−0.01 is used to detect and select very small FDs (see Okike & Umahi 2019b, for more details on the normalization baseline of the code). Totals of 87 and 115 small-magnitude FDs (per cent) ≥−3) were respectively selected from APTY and MOSC NM stations for the period of 1996–2008. These catalogues are presented in Tables 1 and 2. Using the program-selected FD dates as input data to a simple coincident algorithm, 32 simultaneous FDs were detected at the two stations. Selection of the corresponding solar-terrestrial variables (IMF, SWS, Dst, and ap) to the FDs at APTY and MOSC as well as the simultaneous FDs at the two stations was achieved with the coincident algorithm. The selected FDs from the two NM stations and the 32 simultaneous events obtained along with their associated solar wind data are given respectively in Tables 1, 2, and 3.

4 RESULTS AND DISCUSSION

We present two-variable regression model results of our analysis based on CR(per cent) ≥−3 data displayed in Tables 1, 2, and 3. The results of the regression analysis between the magnitude of FDs and related solar-geophysical variables are given in Tables 4, 5, and 6.

While the scatter plot of FD magnitude and IMF intensity for APTY and MOSC data are displayed in panel a of Figs 2 and 3, that of simultaneous FDs are in Fig. 4. The results of the regression and correlation analysis are R² ∼0.03, 0.11; r ∼ 0.17, −0.33 with p − values of 0.14 and 0.038, respectively for APTY and MOSC data. For the simultaneous FD data, we obtained R² ∼0.0017, 0.04; r ∼ 0.04, −0.19 while the p − values are 0.83 and 0.29 corresponding to APTY and MOSC FDs. The link between these variables does not appear to be statistically significant at a 95 per cent confidence level.

The graphs of FD-SWS relation for APTY and MOSC data are shown in panel b of Figs 2 and 3. In Fig. 4, we also plotted that of the simultaneous FD catalogues. The analysis yields R² ∼0.01, 0.03; r ∼ 0.10, −0.17 with p − values of 0.39 and 0.07, respectively for APTY and MOSC data. For the simultaneous FD data, the regression analysis gives R² ∼0.0008, 0.02; r ∼ 0.03, 0.15 while the p − values are 0.88 and 0.41 corresponding to APTY and MOSC FDs. The 2D model employed here does not suggest statistically significant links among these parameters.

The magnitude of FD plotted against Dst for APTY and MOSC variables are laid out in panel c of Figs 2 and 3. Displayed in different panels of Fig. 4 is the graph of the simultaneous FD data set. The regression analysis give respectively R² ∼0.03, 0.08; r ∼ 0.18, 0.29 while the p − values are of the order of 0.09 and 1.9 × 10⁻⁰³ for APTY and MOSC data. For the simultaneous catalogue, we obtained R² ∼0.06, 0.01; r ∼ 0.25, 0.09; p − values 0.17 and 0.61, respectively for APTY and MOSC FDs. The 2D model employed here suggests a statistically significant link between FD and Dst when MOSC data alone is considered but a non-statistically significant correlation in the other data sets.

The amplitude of FD versus ap scatter plots for APTY and MOSC variables are displayed in panel d of Figs 2 and 3. In Fig. 4, we showed the relations for the simultaneous FD data set and IMF, SW, D_st, and ap. The 2D regression analysis give respectively R² ∼0.01, 0.08; r ∼ 0.12, −0.29 with the p − values as 0.29 and 1.8 × 10⁻⁰³ for APTY and MOSC data. For the simultaneous FDs, we found R² ∼ 7.41 × 10⁻⁰⁶, 0.00; r ∼ 0.003, 0.02; p − values 0.99 and 0.90, respectively for APTY and MOSC FDs. The result of the analysis here indicates a statistically significant correlation in the FD-ap relation from the MOSC data alone but does not suggest a statistically significant connection between these variables in the other catalogues.

We conducted regression analysis between FD_APTY and FD_MOSC of Table 3. This test yield R² ∼0.38; r ∼ 0.67 with the p − value as 2.04 × 10⁻⁰⁴. The result is statistically significant at a 95 per cent confidence level. This strong and highly statistically significant relationship may be a pointer to the efficiency of the R-based computer code deployed here.

The regression results between FD and IMF at APTY and MOSC stations are respectively r ∼ 0.17 and −0.33. The coefficients of correlation for the simultaneous FDs at the two stations are 0.04 and −0.19, respectively. This implies that IMF has a stronger influence on CR intensity variations at MOSC NM. Using comparatively high-magnitude FDs, Alhassan., Okike. & Chukwude. (2021a) reported correlation coefficients r ∼ 0.35 and 0.36 for simultaneous FDs at APTY and MOSC stations, respectively. Based on a catalogue of FDs created using 7 NM stations, Dumbovic et al. (2011) found a correlation coefficient result of 0.62 for FD-IMF relation. The comparatively weak correlation presented in the current work is quite interesting as it underscores the claim that weak FDs might also be the result of interactions of CMEs/ICMEs with the Earth’s magnetosphere.

The FD-SWS r ∼ 0.10, −0.17 and 0.03, 0.15 results for the APTY and MOSC separate data and simultaneous events respectively show a near-zero correlation for the APTY station but a mild direct and inverse relation for the MOSC catalogue. These results fairly reflect the previous findings of Singh & Badruddin (2007) and Bhaskar et al. (2016) and Okike (2020d) but contrary to Alhassan et al. (2021a).

The result of the FD-Dst relation shows that 18 per cent, 29 per cent, 25 per cent, and 9 per cent of CR flux intensity variations at APTY and MOSC NM stations can be attributed to the Dst parameter. The chance probability observed in the analysis of MOSC separate data further indicates that the result is highly significant at the 95 per cent confidence level. Our r values are lower than that obtained by Belov et al. (2001b) (r ∼ < 0.42) and Alhassan et al. (2021a), but comparable to those of Okike (2020d). The FD-ap statistical analysis reveals that 12 per cent, 29 per cent, 0.3 per cent, and 2 per cent derived from the coefficient of determination imply that geomagnetic storm index ap, plays a role in FD evolution. This reflects the submission of Alhassan et al. (2021b) in which 8 per cent of CR intensity variation was due to ap.

4.1 Result validation

• Step One: Correlation Test for low-amplitude FDs

  In view of the difficulties of detecting, timing, and calculating the amplitudes of weak or small FDs highlighted in Section 1, validation of the results presented in Section 4 is of utmost importance. Several steps are taken to achieve this. In addition to investigating the large FDs at APTY and MOSC stations (see Tables 6, 7 and 8), the analyses performed are repeated with both small and large FDs calculated from NWRK and OULU stations. For easy comparison with the outcome of analyses of APTY and MOSC data, the results for the small events are presented in different panels of Fig. 5 and Tables 4, 5, and 9. It can be inferred from Figs 4 and 5 that events at NWRK and OULU are more related than those at APTY and MOSC. While almost all the events used in the two-variable regression model register weak correlation, except the relation between APTY and D_st, almost all the similar relations in Fig. 5 improved significantly with some suggesting very strong and statically significant relations. The correlation coefficient for NWRK versus D_st and OULU versus D_st are respectively 0.36 and 0.46. The number of simultaneous FDs (48) at NWRK and OULU stations is also larger than those observed by APTY and MOSC (32). These differences/similarities between these groups of stations are interesting and could stimulate further research. It calls, for example, for a more critical look into the longitudinal and latitudinal distribution of CRs (Tezari et al. 2016; Okike & Umahi 2019b). If the strong connection between these small amplitude FDs is real, it would imply that small FDs, just like the big events, may also be relevant in weather prediction as observed by (Belov et al. 2018b).

• Step Two: Correlation Test for High-amplitude FDs

  The connection between large FDs identified from APTY and MOSC, and from NWRK and OULU and solar-terrestrial parameters are presented in different panels of Figs 6 and 7. The similarities between the strong relations in the two diagrams are quite convincing. This is unlike the case we have in Figs 4 and 5. We may interpret the result with regard to the influence of the location-dependent diurnal CR anisotropy. It appears that it has less impact on the large FDs. The results of the regression analyses are presented in Table 4 where it is clearly indicated that each of the relations is highly significant.

• Step Three: Correlation Test for all the FDs

  The analyses performed with the small FDs at APTY and MOSC stations are repeated with the sum total of all the FDs at the four stations including APTY (442 FDs) and MOSC (471), NWRK (457), and OULU (454). The result is presented in Table 7. The results obtained for the strong and simultaneous FDs are in agreement with those presented here.

• Step Four: Visual Comparison between FDs at APTY and MOSC

  Although the various correlation results presented are pointers to the validity of the new FD catalogues, such quantitative results do not give the reader the opportunity to visually judge the reported relationships. Fig. 8 shows the relationship between FDs at APTY and MOSC stations. There is a beautiful resemblance between the upper and lower panels of the graph. Almost all the identical big events in the two panels can be visually identified, suggesting that the difference in the number of FDs is mainly due to the small events. Some of the big events including the events of 31/10/2003 and 19/01/2005 are marked. The diagram shows that the solar cycle effects exhibited by CR flux may be completely attributed to FDs. Evidently, both large and small FDs concentrate at the solar maximum, defining the dome shape around 2000 and 2004.

• Step Five: Visual Comparison of FDs at OULU and IZMIRAN Website

  The FDs selected by the current program were also compared with those identified by the IZMIRAN group within the same period. This is presented in Fig. 9. The large events of 31/10/2003 and 19/01/2005 are also labelled here. These two events are respectively timed on 29/10/2003 and 18/01/2005 on the IZMIRAN website. It should be noted that the implemented software calculates the events time of minimum reduction whereas the GSM looks at the event onset time.

  The differences between FDs selected by the GSM and those from OULU are significant. While the differences in the number of events are understandable since CR of different time resolutions is expected to yield different numbers of FDs (Lockwood 1971). The major difference between the two catalogues lies in the number and magnitude of large FDs. While the large events of 31/10/2003 and 19/01/2005 are very popular and have been extensively discussed in the literature, the GSM result suggests that there are many other FDs that are larger than the event of 19/01/2005. But a review of the literature confirms that large event catalogues of Svensmark., Enghoff & Svensmark (2012), Kristjansson et al. (2008), Svensmark, Bondo & Svensmark (2009), and Svensmark et al. (2016) present the event of 19/01/2005 as second to the largest within the declining phase of Solar Cycle 23.

  Several potential problems that may affect the phase and magnitude of FD signals assimilated from different stations as highlighted in Section 1 may play a key role in the magnitude of FDs calculated using the GSM. There could be an extensive discussion on the observed differences between the FD catalogues presented in Fig. 9. Okike (2021a), in an attempt to explain the implications of using different baselines to calculate FD events, performed a detailed and rigorous analysis of FDs selected from isolated/culled CR data and those identified using the holistic approach (where the largest available volume of CR data from a station is employed). The use of different base periods as is applicable in the traditional case study/manual approach to FDs has some serious bias effect on the event magnitudes. The authors also emphasized the need and methods to remove the solar cycle effect in order to guarantee the correctness of FD ranking. Nevertheless, the form of analysis performed in the current work has its advantages too.

  Fig. 9, for example, allows us to test/visualize the impact of solar cycle effects and the use of different base periods on the magnitude/ranking of FDs. It also provides an ample opportunity to interpret our results in connection with the physical agents that cause FDs. First, the representation of solar cycle effects (the dome shapes) in Figs 8 and the lower panel of Fig. 9 was not evident in our previous results (see fig. 3 of Okike et al. 2020 and fig. 9 of Okike 2021a). It is because only large FDs are presented in those diagrams whereas both large and small FDs are included here. It does appear that the solar cycle phenomenon is driven by small FDs which seem to have a greater occurrence rate within the time of solar maximum.

  Second, the increased rate of FD occurrence around the solar maximum, indicated by the FDs selected from APTY, MOSC, and OULU stations, reflects the CME occurrence rate presented in fig. 3 of Riley et al. (2006), fig. 1 of Lamy et al. (2019) and fig. 8 of Gopalswamy (2016). We regard this as empirical evidence that CMEs are the progenitors of FDs. A large body of the literature (Cane et al. 1993; Cane et al. 1994; Cane 2000a; Belov et al. 2014) in astrophysics is dedicated to analysis associating FDs with CMEs as the causative agents.

  Third, the magnitude of large FDs (⁠|$\ge 5{{\ \rm per\ cent}}$|⁠) for Solar Cycle 19-23 (54 yr) was ranked in fig. 11 of Okike (2021a). The events of 31/10/2003 and 19/01/2005 are respectively given the rank of 2 and 7 in the diagram. This contrast sharply with the ranking of the events in the upper panel of Fig. 9. It would appear that the magnitude of the event of 29/10/2003 in the upper panel of Fig. 9 is seriously biased. It has been shown that the magnitude of FDs varies according to the base period used in the normalization stage. Table 1 of Okike (2021a) lists the magnitudes and number of FDs calculated using the static averages of separate years (2003 and 2005) and the combined years (2003–2005) as the normalization baselines. A close inspection of the table reveals some bewildering differences in the number as well as event magnitudes in the three periods. Specifically, reference was made to the magnitude of the event of 20/12/2003, 23/12/2003, and 29/12/2003. When the event magnitude was calculated with reference to the mean for the three-year period, the magnitudes of these three events were 4.34, 5.14, and 4.12 per cent, respectively. However, when the magnitudes were recalculated with reference to the static average for only 2003, the values obtained for the respective events were 0.06, 1.16, and 0.34 per cent. The magnitudes of the event of 31/10/2003 in the referenced table are 35.18 and 29.56 per cent respectively for the period of 2003 and 2003–2005. These should serve as a pointer to the reasons for the confusion and contradictions in the existing FD catalogues where the manual/non-holistic approach is employed to calculate the magnitude of FDs.

5 SUMMARY AND SUGGESTIONS FOR FUTURE WORK

There is a hypothesized link between short-term depressions in CR count rates and solar-terrestrial indices. Both are suggested to originate from common solar and interplanetary disturbances in form of CMEs and ICMEs (see e.g. Richardson & Cane 2011b; Badruddin & Kumar 2015). Investigation of their interrelationships is viewed as an interesting study in astrophysics. However, event selection remains a daunting challenge in FD-based solar-terrestrial investigations. The traditional manual FD selection technique is subjective and characteristically tedious. For example, it may not be successfully used to select small-amplitude FDs. Although Belov et al. (2018b) opined that it is quite difficult to identify Forbush effects (FEs) with a small amplitude using the data of a single NM, several recent works of our team (see Okike 2020d, 2021a; Okike et al. 2021b; Okike & Alhassan 2021, 2022, for example) confirm that fully automated approach is the way to go. The automated algorithm developed and implemented here selects small-amplitude events from four isolated NM stations. Large FDs from the stations are also identified and used for validation purposes. Many of the results presented showcase the implemented code as an improvement over the commonly known manual and semi-automated techniques. While Oh et al. (2008) report that the stronger FD events happen simultaneously whereas the weaker FDs are non-simultaneous, the current results show that many small-amplitude FDs do also happen at the same universal time. The 2D regression analysis performed on the FD lists from the two stations indicates a statistically significant correlation between the pairs of FDs, pointing to the reliability of the new FD catalogue.

It also suggests that the same solar agents may trigger the small FDs at the four stations. A number of the solar-terrestrial data register strong and statistically significant relations for NWRK and OULU stations. Such strong correlations are absent for APTY and MOSC stations. This is insightful as it calls for further investigation into the cause of the observed differences. The relation between small FDs at NWRK and OULU equally shows a stronger connection than those for APTY and MOSC. The same stronger connection is also apparent for the large/simultaneous FDs for NWRK and OULU. In order to better understand this phenomenon, similar analyses involving solar wind plasma as well as heliospheric (see Thomas, Owens & Lockwood 2014; Owens et al. 2014, 2015, for example) parameters may be conducted using a worldwide network of NMs.

The higher number of both small and large FDs detected from MOSC stations marks it as comparatively more sensitive than APTY, NWRK, and OULU stations. This agrees with the ranking of Moraal et al. (2000) (see table 1 of the paper). Several important calculations (see also Belov et al. 2018) testing the sensitivities of NMs have been performed using their response to small anisotropic and isotropic variations (1 per cent). NMs are usually ranked accordingly. If the current analysis is extended to the whole network of NMs, the very small event amplitude (<<1) which is detectable by the current code may be very useful in ranking of NMs.

The correlation and regression results, presented here should be viewed as a preliminary effort toward the evaluation of the possible relation between weak FDs and other solar/geophysical transient events. The catalogues of FDs, including the small events which happen simultaneously at the two stations (see Table 3), could be used to conduct a number of further investigations. For instance, the relation between CMEs/ICMEs and large FDs has been demonstrated by large volumes of published works (e.g. Mishra & Agarwal 2008; Cliver & Ling 2001; Richardson & Cane 2011a; Oh & Yi 2012; Kumar et al. 2013; Belov et al. 2014; Kumar & Badruddin 2014; Bhaskar et al. 2016; Melkumyan et al. 2018), whereas the proposed impact of CMEs/ICMEs on small FDs (Cane 2000a; Belov et al. 2018) have yet to be investigated in detail. Additionally, the presented event catalogues could be used for validation purposes. As indicated before, the IZMIRAN group publishes a number of small FDs (amplitude |$\le -0.3{{\ \rm per\ cent}}$|⁠) using data assimilated from a number of CR stations. Nevertheless, there are several significant differences between catalogues of FDs based on individual and array of NMs (Okike 2021a). In the light of several factors which could register important influence on the magnitude of small FDs at different locations on the Earth (see table 4 and fig. A.11 of Okike 2020c), small-amplitude Forbush events selected from disparate or an array/assimilated NM data (see Jordan et al. 2011; Okike 2021a) should be subjected to FD event consistency test (Okike et al. 2021b). This relevant disparity between the amplitudes of small FDs at different stations is interestingly suggested by the event of 27/06/1999 (the first event in Table 3). The amplitudes of the FDs are respectively 0.03 and 2.10 per cent at APTY and MOSC NM stations.

In spite of the apparent high efficiency and sensitivity of the present code, it is not without some challenges. One of the shortcomings of the implemented code is the use of raw CR data (see e.g. Okike 2020c; Ugwoke et al. 2022). Before reaching firm conclusions on the results of the present analysis, the Fast Fourier Transformation technique lately developed by Okike & Umahi (2019b) and deployed in several applications (see e.g. Okike 2020c, 2021a; Okike & Alhassan 2021, 2022; Alhassan et al. 2022b, a) would be used in future work to disentangle the superimposed effects of CR diurnal anisotropies on FDs. After subtracting the contribution from ejecta as suggested by Wibberenz et al. (1998), FD magnitudes selected from the resultant high-frequency signal maybe significantly different from the FD magnitudes identified from the raw data. The analyses performed using the current catalogues maybe repeated for comparative purposes.

ACKNOWLEDGEMENTS

We are grateful to the teams that host the websites http://cr0.izmiran.rssi.ru/ and https://omniweb.gsfc.nasa.gov/html/ow data.html from where we sourced the data for this article. To the non-commercial R software developers and all our friends on the R-mailing list ([email protected]), we remain indebted. The anonymous referee did a great job. His/her contributions are gratefully acknowledged.The first author received funding from the PanAfrican Planetary and Space Science Network (PAPSSN). PAPSSN is founded by the Intra-Africa Academic Mobility Scheme of the European Union under grant agreement no. 624224.

DATA AVAILABILITY

The data used in this work are available at http://cr0.izmiran.rssi.ru/, https://omniweb.gsfc.nasa.gov/form/dx1.html, and http://spaceweather.izmiran.ru/eng/fds2003.html.

References