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All issues Volume 14 (2024) J. Space Weather Space Clim., 14 (2024) 21 Full HTML
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J. Space Weather Space Clim.
Volume 14, 2024
Topical Issue - Space Climate: Long-term effects of solar variability on the Earth’s environment
Article Number 21
Number of page(s) 16
DOI https://doi.org/10.1051/swsc/2024015
Published online 21 August 2024

© J.J. Love et al., Published by EDP Sciences 2024

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

The solar-geospace storm of September 1–2, 1859, commonly known as the “Carrington event”, was extraordinary. Its occurrence influenced the historical development of notions of space physics and space weather (e.g., Cliver, 2006; Hudson, 2021). Today, it is widely regarded as a benchmark for extreme space weather hazards (e.g., Cliver and Dietrich, 2013; Lakhina and Tsurutani, 2018; Usoskin et al., 2023). The event started with a solar flare that was witnessed telescopically by Richard Carrington (1859) and Richard Hodgson (1859). Just 17.6 h later, ground-based magnetic observatories recorded the commencement of a magnetic storm (e.g., Cliver and Svalgaard, 2005). The short duration between the flare and the magnetic storm indicates that the Sun-Earth distance was traversed by an interplanetary coronal-mass ejection (ICME) of unusually high velocity (e.g., Cliver and Svalgaard, 2005). The resulting magnetic storm-induced interference on telegraph systems around the world (e.g., Boteler, 2006b) and spectacular aurorae in many nighttime skies (e.g., Green et al., 2006; Silverman, 2006; Hayakawa et al., 2018), some seen overhead at magnetic latitudes as low as 25.1°(e.g., Hayakawa et al., 2020a). This indicates that the Carrington magnetic storm was extremely intense. But quantification of its intensity has proven challenging because the amplitude and rapidity of geomagnetic field variation during the Carrington storm exceeded the recording capabilities of automatic analogue magnetometers then in operation at some observatories (e.g., Beggan et al., 2024).

In this context, researchers have concentrated on geomagnetic variation data reported from the Colaba observatory in Bombay (Mumbai), India (e.g., Tsurutani et al., 2003; Siscoe et al., 2006), where, at the time of the Carrington magnetic storm, geomagnetic field measurements were being made “by-eye” (Moos, 1910; Gawali et al., 2015). Although visual geomagnetic measurement methods have their limitations, they are not the same limitations suffered by automatic analogue magnetometers. Since Colaba was a low-latitude observatory, its data might be used to estimate Dst, a standard measure of global storm-time geomagnetic disturbance that is usually derived from data from four low-latitude observatories (Sugiura and Kamei, 1991; Karinen and Mursula, 2005). As the ring current strengthens, the horizontal component of the low-latitude geomagnetic field weakens, and Dst declines from a pre-storm, near-zero level to characteristically negative values (e.g., Daglis, 2006). A common measure of a magnetic storm’s overall intensity is the deepest, most negative value attained by Dst (e.g., Gonzalez et al., 1994). But given that magnetic storms are complicated phenomena, and given that the Carrington storm is one of the most intense storms in the history of direct geomagnetic-field measurement, it is perhaps not surprising that it has been challenging to accurately estimate the Carrington storm’s (global) intensity from old-fashioned disturbance data recorded (locally) at a single observatory.

In this report, we use recent observatory records of magnetic storms to develop a statistical model of storm intensity versus geomagnetic disturbance recorded locally at low-latitude observatories during magnetic storms. Using this model and a presentation by Hayakawa et al. (2022a) of the geomagnetic field data that are reported in the Colaba yearbook for 1859 (Fergusson, 1860), we estimate the most likely intensity of the Carrington storm (maximum –Dst), and, additionally, we obtain statistical measures of uncertainty (credibility intervals) on this intensity estimate. With a related model, we examine individual Colaba geomagnetic disturbance values reported for the Carrington storm. Important comparisons are made with Tsurutani et al. (2003), who estimated disturbance at Colaba during the Carrington storm, but with a dataset that was less complete than that of Hayakawa et al. Results inform fundamental understanding of extreme space-weather events (e.g., Hapgood, 2012), their occurrence frequency and future probability (e.g., Tsubouchi and Omura, 2007; Love, 2021), the hazards that extreme space-weather events represent for technological systems (e.g., Daglis, 2005), especially for long-line electricity transmission networks (e.g., Piccinelli and Krausmann, 2014; Ishii et al., 2021), and related risks for modern society (e.g., Baker et al., 2008; Eastwood et al., 2017; Oughton et al., 2019).

2 Local geomagnetic disturbance and Dst

Following Mursula et al. (2008), we define the ring-current index Dst as a discrete numerical sequence of hourly values calculated by averaging horizontal-component geomagnetic disturbance values acquired at N low-latitude observatories,

(1)

where, for each observatory n, hourly average disturbance is

(2)

Sqn(t) is a function representing solar-quiet variation; Hbn is a constant baseline. is a “boxcar” average of calibrated (“absolute”) sub-hourly horizontal-component magnetometer data Hn(tj) acquired within each full Universal-Time (UT) hour,

(3)

where the timestamp tj for each average is the centre time of each boxcar, the bottom of each UT hour (tj = 00:30, 01:30, etc.), and where the summation is over M sub-hourly samples. From 1957 until the 1980s, Dst was calculated using observatory hourly averages obtained from sub-hourly readings of analogue magnetograms; from the 1980s to the present time, Dst has been derived from digital 1-min-resolution magnetometer data. So, for example, an hourly average can be calculated by averaging M = 60 1-min Hn(tm) values, where tm ranges from 00:00 to 00:59 UT, or from 01:00 to 01:59 UT, etc. Importantly, in equation (2), a non-stormy waveform, is subtracted from each hourly average observatory hourly value, where Sqn(t) is a function that represents solar-quiet variation and Hbn is a constant baseline. In comparison to the intense storms considered in this analysis, the peak-to-peak amplitude of solar-quiet variation is very small, ~100 nT.

Local ground-level disturbance amplitude is normalized in equation (2) under the assumption that low-latitude geomagnetic disturbance around the world can be described by a time-varying external dipolar field, with factor 1/cosϕn, where ϕn is observatory geomagnetic latitude. Such a dipolar field would be generated by a westward-directed magnetospheric ring current (e.g., Dessler and Parker, 1959; Sckopke, 1966; Daglis, 2006). This current strengthens during the main phase of a magnetic storm, causing a decrease in the low-latitude strength of the geomagnetic field and a corresponding decrease in Dst(tj) from its near-zero quiet-time value (e.g., Loewe and Prölss, 1997), until it eventually reaches a minimum value (a maximum-negative value) that is often taken as a storm’s absolute “intensity” (e.g., Gonzalez et al., 1994). Whereas a symmetric ring current would generate low-latitude geomagnetic disturbance that is symmetric across local time, other current systems, such as partial-ring, field-aligned, magnetopause, magnetotail, and ionospheric currents, introduce local-time asymmetry in disturbance (e.g., Friedrich et al., 1999; Turner et al., 2000; Ohtani et al., 2001; Asikainen et al., 2010; Saiz et al., 2021). In terms of equation (1), at any given instance in time tj, local-time asymmetry introduces dispersion among the N observatory disturbance values that contribute to the average disturbance that is Dst(tj). This dispersion, in turn, introduces inaccuracy in estimates of Dst(tj), especially if Dst(tj) is estimated from disturbance from (say) just one observatory.

Commonly, Dst is based on data from four standard and long-running observatories that are widely separated in longitude: Hermanus (HER), South African National Space Agency (e.g., Kotzé, 2018), Kakioka (KAK), Japan Meteorological Agency (e.g., Minamoto, 2013), and Honolulu (HON) and San Juan (SJG), U.S. Geological Survey (USGS) (e.g., Love and Finn, 2011). Some analyses of early 20th-century magnetic storms have, due to limited data availability, calculated Dst for alternative sets of low-latitude observatories (e.g., Love et al., 2019a; Hayakawa et al., 2020c), and some versions of Dst incorporate data from additional magnetic observatories (e.g., Gjerloev, 2009; Mursula et al., 2011). Since the advent of digital magnetometers, it has been possible to calculate Dst with 1-min resolution (e.g., Iyemori et al., 2010; Gannon and Love, 2011).

Our definition of Dst, as given by equation (1), is not the same as that developed for the International Geophysical Year (IGY, 1957–1958), and which is today produced by the Kyoto World Data Center (WDC). The original formulation by Sugiura (1964) was subsequently changed by Sugiura and Kamei (1991), but neither of their formulations is based on proper trigonometry (Mursula et al., 2008; Love and Gannon, 2009; Mursula et al., 2011). Unfortunately, this error affects the entire Dst time sequence produced by Kyoto. The importance of this issue depends on the application at hand, but Sugiura’s averaging is incompatible with our analysis. For reference, in this study, we use the corrected 1-h-resolution index Dxt (1957-present) calculated using equation (1) by Oulu University using data from the four standard observatories (Karinen and Mursula, 2005).

3 The Colaba horizontal-component time sequence

From 1841 to 1872, variation in time t of the three components of the geomagnetic field vector, (horizontal component, declination, vertical component), was routinely monitored at the Colaba (CLA) observatory through a set of visual measurements (e.g., Gawali et al., 2015). The 1859 Colaba’s yearbook records that these measurements were made every day, except Sundays and holidays (Fergusson, 1860). Routinely, and during relatively calm geomagnetic conditions, horizontal-component measurements were made at the top of each Göttingen Astronomical Time (GöT) hour, 00:00, 01:00, 02:00, etc. Additional, more frequent auxiliary measurements, what the Colaba yearbook calls “disturbance observations”, were made during storms. Because both the hourly and auxiliary measurements are essentially instantaneous, they can be described as “spot” measurements. A time sequence of horizontal-component measurements (both hourly and auxiliary), each made at instances in time tj, provides a time sequence of variation in the strength of the horizontal geomagnetic field.

In this report, we use a version of the Carrington-storm Colaba data that has recently been presented by Hayakawa et al. (2022a). They converted Colaba yearbook data from (old) English units to nanoTeslas; they adjusted for temperature-related change in instrument response; and they made the data available in computer-readable format. For this analysis, we have adjusted the data from Göttingen Astronomical Time to Universal Time; UT = GöT – 40 min + 12 h (e.g., Boteler, 2006a). For both regular-hourly and auxiliary measurements, we plot in Figure 1 the local Colaba “spot” geomagnetic disturbance time sequence, during the Carrington storm,

(4)

thumbnail Figure 1

Time sequences of Colaba (CLA) disturbance values , equation (4), from 17:00 September 1 to 12:00 September 3, 1859, UT: regular hourly measurements (blue), auxiliary measurements (orange) made during periods of disturbance, UT-boxcar averages (black), the time of an apparent gap in the auxiliary measurements (grey). The hourly average centred on 06:30 UT is shown with and without the extreme hourly disturbance value at 06:20 UT. This figure can be compared with Hayakawa et al. (2022a). SC denotes sudden commencement; Gap denotes possible location of a data gap.

Here, are absolute horizontal intensities, SqCLA(t) is the solar-quiet function, each as reported by Hayakawa et al. (2022a). The geomagnetic latitude at Colaba in 1859 was ϕcla = 10.2°N.

It is useful to summarize some important details of the Colaba data that enter into our statistical analysis and our related discussion. Hourly measurements were made routinely in the days before the Carrington storm’s commencement; these hourly measurements were also made during all subsequent phases of the Carrington storm. The most positive disturbance at Colaba has a timestamp of 04:20 UT on September 2 (17:00 GöT on September 1). This represents geomagnetic activity prior to the storm’s sudden commencement – Greenwich Observatory reported that the storm commenced 20 min later, at 04:40 UT (Jones, 1955, his p. 102). After storm commencement, and during the storm’s main phase, auxiliary measurements were made, initially, every 15 min. The yearbook indicates that timestamps for the auxiliary horizontal-component measurements are 2 min after “full time” (Fergusson, 1860, his p. 154), which Hayakawa et al. (2022a) interpret to mean 2 min after the reference time given in the yearbook tables (measurements of other geomagnetic components were made on different schedules relative to the reference time). With this, auxiliary horizontal-component measurements have timestamps of 05:22, 05:37, 05:52 UT (18:02, 18:17, 18:32 GöT). No auxiliary measurement is reported, when one might have been expected, for 06:07 UT (18:47 GöT); this might be regarded as a “gap” (Hayakawa et al., 2022a, their Sect. 3 and Fig. 2). The next measurement, one of those normally made on the once-per-hour schedule, has a timestamp of 06:20 UT (19:00 GöT). This value is important. It is the deepest value of the Colaba time sequence, with  nT. For the next two hours, auxiliary measurements were made every 5 min, with the first coming just 2 min after the regular hourly measurement, with timestamps of 06:22, 06:27, 06:32, etc. UT (19:02, 19:07, 19:12, etc. GöT). The deepest of the auxiliary disturbance value  nT corresponds to a measurement at 06:27 UT (19:07 GöT). Then, for an hour and a half, during what was the early part of the storm’s recovery phase, measurements were made every 10 min, and they were made every 15 min for the remainder of the recovery phase.

thumbnail Figure 2

Relative residual differences , equation (6), calculated using Alibag (black, ABG), and (grey, open circle) Hermanus (HER), Kakioka (KAK), Honolulu (HON), San Juan (SJG), as a function of hourly average storm intensity Di, equation (5), calculated using the four standard Dst observatories (grey) HER, KAK, HON, SJG, for the selected storms listed in Table 1 (1989–2015). Also shown: average (black solid line) of the residual differences, and standard deviation contour lines of a normal distribution (dotted lines, ±1s, etc.).

In their analysis of the Carrington magnetic storm, Tsurutani et al. (2003) used the horizontal-component disturbance time sequence at Colaba as a proxy for a Dst time sequence of the Carrington magnetic storm, assuming, essentially, that from Colaba. Here, it is worth emphasizing that the dataset we are using, that presented by Hayakawa et al. (2022a) from the Colaba yearbook, is not that used by Tsurutani et al. Whereas the Colaba yearbook includes auxiliary measurements made as often as every 5 min during the Carrington storm, Tsurutani et al. only report that auxiliary measurements were made every 15 min. Whereas Hayakawa et al. obtained a solar-quiet SqCLA(t) waveform and baseline HbCLA using methods akin to those used in the standard estimation of Dst (Sugiura, 1964; Sugiura and Kamei, 1991), by averaging several quiet days of Colaba data from the month of August 1859, Tsurutani et al. used data from the day before the commencement of the Carrington storm to estimate a constant baseline without accommodation of solar-quiet variation. But, as noted by Hayakawa et al., the August 29 storm was still affecting the local horizontal-component variation prior to the occurrence of the September 2 storm, and this appears to have affected the baseline estimate of Tsurutani et al. As a result, the deepest disturbance value we quote here,  nT, is deeper than the –1600 nT (–1626 nT with latitude factor) value of Tsurutani et al.

4 Colaba hourly values and their accuracy

Tsurutani et al. (2003) used their estimate of the deepest disturbance value of –1600 nT (–1626 nT with latitude factor) at 06:20 UT as a spot measure of the Carrington storm’s deepest Dst value. Subsequently, Siscoe et al. (2006) noted that, consistent with our discussion in Section 2, Dst is traditionally a 1-h resolution index, with boxcar averaging performed over sub-hourly data within each UT hour. They suggested that, in contrast to Tsurutani et al., if the Colaba disturbance data are to be used to estimate the intensity of the Carrington storm, then those data should be averaged in each UT-hour so that comparisons can be made with the traditional definition of the Dst index.

In Figure 1, we plot the UT-hour averages of the Colaba disturbance data. The deepest average is for the 06:00–07:00 UT window, with a centered timestamp of 06:30 on September 2. This window encompasses the extreme disturbance value at 06:20 UT and eight auxiliary measurements at 06:22, 06:27, 06:32, …, 06:57 UT. In averaging these data, we consider two scenarios, one with and the other without the extreme disturbance value at 06:20 UT. This leads to and –902 nT for the 06:30 UT Colaba hourly average disturbance; refer to Figure 1. The range of these values encompasses the deepest (running-average) values of Hayakawa et al. (2022a), and –918 nT, again, depending on whether or not the extreme disturbance value at 06:20 UT is used.

Next, we check the accuracy of the UT-boxcar averages of the sparse and non-uniformly distributed Colaba disturbance data (for background, Love, 2009; Love et al., 2010). How representative are such hourly averages compared to, say, averages of data acquired densely and uniformly over a given hour? To answer this question, we examine hourly averages of modern observatory horizontal-component data acquired during extremely intense magnetic storms. We define the intensity of a storm i as the maximum −Dst value realized over the storm’s course

(5)

Where and are the start and end times of each storm. We identify 18 intense storms, the six most intense storms in the Oulu Dxt time sequence for each of solar cycles 22, 23, and 24, years 1989–2015; intensities for these storms are listed in Table 1. Then, for each of these storms, we obtain digital 1-min-resolution geomagnetic horizontal-component time sequences from the Alibag (ABG) observatory in India and the four Dst observatories (HER, KAK, HON, SJG). Note, to avoid interference from the electrification of Bombay (Mumbai) trams, Colaba geomagnetic monitoring operations were transferred to nearby Alibag in 1904 (e.g., Gawali et al., 2015). For the UT-hour of the maximum intensity of each storm ti and for each observatory n, we calculate the UT-boxcar hourly average of 60 1-min, values. Additionally, we subsample the 1-min geomagnetic time sequences in such a way as to mimic the sampling contributing to the most extreme hourly disturbance at 06:30 UT: like the regular hourly sampling at Colaba, we use for averaging horizontal-component values at the 20th minute of each UT hour, and, also, like the auxiliary sampling at Coloaba, we use values at minutes 22, 27, 32, …, 57. For the UT-hour of the maximum intensity of each storm ti and for each observatory n, we calculate their average . From these hourly averages, we form the relative residual differences

(6)

Table 1

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); and relative residual differences , equation (6), for Alibag (ABG), HER, KAK, HON, SJG, each for selected storms (1989–2015). Month (Mn), Day (Dy), Hour (Hr, UT). Data are shown in Figure 2. No values are listed if no observatory data are available.

These values are also listed in Table 1.

In Figure 2, we plot the relative residuals from Table 1 as a function of storm intensity Di. Importantly, the average of the relative residuals is very small, 0.0141, or about one per cent. This means that the averages are close to being unbiased estimates of the complete averages . The standard deviation, s, of the relative residuals is also small, 0.0200, or just two per cent. This means that the averages are, for our purposes, accurate estimates of the complete averages . We also note that the relative residuals do not show a significant trend across Di. In light of these observations, we have confidence in using UT-hourly averages of the Colaba observations.

5 Estimating Carrington storm intensity with error bars

In this section, we investigate the accuracy of estimates of Carrington-storm intensity based on disturbance data from the Colaba observatory. To that end, we examine the relationship between maximum –Dst realized during extremely intense magnetic storms and hourly averages of geomagnetic disturbance recorded at several low-latitude observatories during those storms. We identify 18 intense magnetic storms, the three most intense storms in the Oulu Dxt time sequence for each of solar cycles 19 through 24, years 1957–2015. Then, for the same 18 storms, we estimate hourly average horizontal-component disturbance time sequences for five observatories, Alibag and the four standard Dst observatories. For each storm i and each observatory n, we identify the maximum-negative geomagnetic disturbance value,

(7)

which we call a “local hourly average intensity”. In Table 2, we list the storm intensities Di and local hourly average intensities values for the 18 chosen storms. A few observations are worth making. The most intense storm (1957–2015) is that of March 1989 (e.g., Allen et al., 1989; Boteler, 2019); D1989 = 593 nT was attained at 01:30 UT on March 14. For that storm, local hourly average intensity values are only available for three (Hermanus, Kakioka, Honolulu) of the five observatories we consider because of data gaps at Alibag and San Juan. For Hermanus, the local intensity value occurred at the same time as D1989 (01:30 UT), whereas the value for each of Kakioka and Honolulu occurred one hour earlier (00:30, March 14).

Table 2

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); and hourly average local intensities , equation (7), calculated using Alibag (ABG) and HER, KAK, HON, SJG, each for selected storms (1957–2015). Month (Mn), Day (Dy), Hour (Hr, UT). Data in local-noon sector (09:00–14:59) are listed in bold. Data are shown in Figures 3 and 4.

An important issue to consider is the inconsistency of the local hourly average intensities listed in Table 2. Can we tell if differences between from pairs of observatories are statistically significant? We use a two-sample Kolmogorov-Smirnov algorithm (e.g., Press et al., 1992, Chapter 14.3; Bohm and Zech, 2010, Chapter 10.3.5) to calculate the probability (p-value) that local hourly average intensities from pairs of observatories could be independent statistical realizations from the same distribution. However, we recognize that the local intensities in each column of Table 2 are correlated – they correspond to the same 18 storms. Because tests of statistical significance assume that data are independent, this correlation in Table 2 must be removed to obtain meaningful probability estimates (e.g., von Storch, 1995; Wilks, 2006, Chapter 5). We remove it by bootstrap resampling (e.g., Efron and Tibshirani, 1994; Olea and Pawlowsky-Glahn, 2009). For a given pair of observatories, we make many random samplings with replacement of the local hourly average intensities, in each case, calculating a Kolmogorov-Smirnov p-value for pairs of observatories; we keep the median p-value of the resamplings as a suitable estimate of significance. These probabilities are listed in Table 3. We note, for example, that the probability that and could be realized from two independent statistical samplings of the same distribution is 0.4255. This is a relatively high probability, too high to confidently reject the hypothesis that the Kakioka and Honolulu local hourly average intensities are samples of the same distribution. On the other hand, the p-value for the discrepancy between and is lower at 0.0994, but even this is too high to confidently reject the hypothesis that the Alibag and San Juan local hourly average intensities are samples of the same distribution. More generally, the Kolmogorov-Smirnov p-values listed in Table 3 do not allow confident rejection of the hypothesis that the local hourly average intensities from one observatory and another observatory are samples of the same distribution. We return to the subject of Kolmogorov-Smirnov testing below when we examine the statistical consistency of datasets and models fitted to those datasets.

Table 3

Kolmogorov-Smirnov p-values calculated for pairs of hourly average observatory intensities , equation (7), for Alibag (ABG), Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG), for the i storms listed in Table 2 (1957–2015).

In seeking a plausible statistical estimate of the intensity of the Carrington storm based on local intensity data from Colaba, we need a model motivated by general physical principles. In this regard, we recognize that, fundamentally, space weather is the integrated result of the history of nonlinear (multiplicative) interaction of magnetospheric-ionospheric processes operating under external forcing by the solar wind (e.g., Vassiliadis et al., 2000; Boaghe et al., 2001). We assume, therefore, that Di and arise from a stochastic process involving the multiplication of many underlying random variables, and per the discussion in the Appendix, this would plausibly result in data that are lognormally distributed. Recognizing that if a population of random variables is lognormally distributed, then the logarithms of those variables will be normally distributed, we hypothesize that a given logarithmic local intensity ln can be generated by storms with a statistical distribution of logarithmic intensities ln D,

(8)

The mean of ln Di is the parameter ln , the standard deviation of ln Di is the parameter σ, and α is a bias parameter. For the storm intensities and local intensities listed in Table 2 (1957–2015), we obtain the parameters {ln α, σ} in the probability density function (8) by the method of maximum likelihood (e.g., Roe, 2001; Bohm and Zech, 2010). The joint probability density of the intensity data, given the model parameters, is the likelihood function

(9)

where multiplication is over the I storms contributing to the likelihood and the N observatories, with appropriate accommodation for missing values. The model parameters giving the maximum (the mode) of is the most likely solution,

(10)

(11)

In Figure 3, we plot the (log)normal density functions g(ln Di| ln , ln α, σ) and data from Table 2 separately (a) for data from Alibag and (b) for data from the four standard Dst observatories, Hermanus, Kakioka, Honolulu, and San Juan. We see, immediately, that the model based on Alibag data is similar to that based on data from the Dst observatories. In both cases, a lognormal model characterizes the intensity data across wide ranges in both ln and ln Di – the relationship between these two variables appears to be scale-invariant (see Appendix). For comparison, in Figure 3 we also plot the intensity data given in Table 4 for five superstorms, those of October 1903 (Hayakawa et al., 2020c), September 1909 (Hayakawa et al., 2019; Love et al., 2019b), May 1921 (Hapgood, 2019; Love et al., 2019a), March 1940 (Hayakawa et al., 2022b), and March 1946 (Hayakawa et al., 2020b); these data, which are not used in fitting the model, provide validation of the model (and its underlying hypotheses) for the most extreme values of and Di. As above, we perform Kolmogorov-Smirnov tests, but this time we use a one-sample algorithm to calculate the probability that the local intensity data are statistical realizations from the fitted models. We recognize that it is not meaningful to test for significance between a dataset and a model that has been fitted to those data – they are not statistically independent. We can, however, examine statistical significance under bootstrap resampling (e.g., Steinskog et al., 2007; Clauset et al., 2009, their Section 3.4; Corral and González, 2019, their Section 3.2). Bearing that in mind, we treat each of the two (log)normal models (a,b) shown in Figure 3 as hypothetical: we make multiple random samplings of the intensity data with replacement, and with each resampling, we calculate a Kolmogorov-Smirnov p-value against the model; median p-values from multiple resamplings are suitably unbiased estimates of significance. These bootstrap p-values are listed in Figure 3; these are not small enough to motivate rejection of the hypothesis that the intensity data are realized from a (log)normal processes.

thumbnail Figure 3

Contour lines of the (log)normal density function g(ln Di|ln , ln α, σ), equation (8): median (black solid line) and (black dotted lines) ±2σ, etc. intervals (dotted lines); intensities Di and from Table 2 (1957–2015): (a) Alibag (black, ABG) and (b) the four standard Dst observatories (grey, open circles), Hermanus (HER), Kakioka (KAK), Honolulu (HON), San Juan (SJG). Also shown: data from Table 4 (brown, green) for five other superstorms (1903, 1909, 1921, 1940, 1946). In each of (a, b) estimates of Carrington-storm D1859 are shown for Colaba with (blue) and without (orange) the extreme disturbance value at 06:20 UT. 68% (±1σ) credibility intervals in brackets. Bootstrap Kolmogorov-Smirnov probability, KS p.

Table 4

Hourly average superstorm intensities Di, equation (5), calculated for observatory (Obs) hourly average intensities , equation (7), for selected storms. Month (Mn), Day (Dy), Apia (API), Cape Town (CTO), Colaba (CLA), Coimbra (COI), Cuajimalpa (CUA), Hermanus (HER), Honolulu (HON), Kakioka (KAK), Mauritius (MRI), San Fernando (SFS), San Juan (SJG), Vieques (VQS), Watheroo (WAT), and Zi-Ka-Wei (ZKW). Data in local-noon sector (09:00–14:59) are listed in bold. Data are shown in Figures 3 and 4.

Importantly, we can use the models in Figure 3 to estimate the Carrington storm’s intensity D1859 and the corresponding accuracy of D1859 given the local hourly average intensity from Colaba. We recall from Section 4 our pair of estimates of local hourly average intensity during the Carrington storm, and 902 nT, depending on whether or not the extreme disturbance value at 06:20 UT is included in the averaging. In Figure 3(a), where the model is fitted to the Alibag intensity data, the bias parameter is α = 0.9407, and the dispersion parameter is σ = 0.1701. With this, we obtain D1859 = α ·  = 944 nT (blue) as the median intensity of all hypothetical storms generating a local hourly average intensity of  = 1004 nT. A 68% credibility interval for such a storm intensity is [D1859 · e−σ, D1859 · e+σ] = [796,1119] nT – for a width of 323 nT. On the other hand, D1859 = 848 nT (orange) for the median intensity of all hypothetical storms generating a local hourly average intensity of = 902 nT. A 68% credibility interval for such a storm intensity is [715,1005] nT – for a width of 290 nT. The two intervals, one for inclusion of the extreme value and one for its exclusion, are overlapping. We might, therefore, infer that the difference between the estimated Carrington storm intensities is insignificant. In Figure 3(b), for data from the Dst observatories, the model is very similar to that shown in Figure 3(a), for the Alibag data – the confidence intervals for α and σ are overlapping. This indicates that the models are not significantly affected by correlations between the local hourly average intensities from the various Dst observatories. This observation reinforces the confidence we have in our estimates of the intensity of the Carrington storm inferred from data from Colaba.

Siscoe et al. (2006, their Section 2) called attention to the important fact that, in terms of local time, the Colaba observatory was just about ideally situated for providing disturbance data that approximate Dst, because local disturbance tends to be greater (less) than Dst at local dusk (local dawn) and approximately equal to Dst at local noon and midnight (Chapman and Bartels, 1962, their Chapter 9.3; Love and Gannon, 2009, their Section 6.2). This indicates that at least part of the dispersion in Di seen in Figure 3 is due to data sampling across dawn-dusk asymmetry. Therefore, for similarity with Colaba measurements made around local noon, we now restrict ourselves to the intensities in Table 2 realized in the local-time sector 09:00 to 14:59. We also note the consistency seen in Figure 3(a) and (b), and so we treat the local noon-sector data from the Dst observatories and Alibag together and obtain a (log)normal model by maximum likelihood. In Figure 4, we plot corresponding results. The bias factor, α = 0.9603, is a bit closer to 1.0000, than for the results in Figure 3, meaning a smaller average difference between storm intensity Di and local hourly average intensity . The standard deviation, σ = 0.1204, is smaller than for the results in Figure 3, meaning more accurate estimates of Di for a given . Including the extreme disturbance value leads to a local hourly average intensity of  = 1004. From that, we obtain a median storm intensity of D1859 =  = 964 nT and a 68% credibility interval of [D1859 ∙ e–σ, D1859 ∙ e+σ] = [855,1087] nT – for a width of 232 nT. We note that the maximum (hourly average) – Dst estimate of ~850 nT given by Siscoe et al. is just outside and lower than our 68% credibility interval; on the other hand, the maximum (hourly average) –Dst estimate of ~1050 nT (Gonzalez et al., 2011) is within our 68% credibility interval. Excluding the extreme value at 06:20 UT leads to a local hourly average intensity of  = 902 nT. From this, we obtain a median storm intensity D1859 = 866 nT and a 68% credibility interval of [768,977] nT – for a width of 209 nT. These results represent our best estimates of the Carrington storm’s intensity and its statistical uncertainty.

thumbnail Figure 4

Same as Figure 3, except that, to mimic sampling performed at Colaba during the Carrington storm, intensities Di and are restricted to the local noon sector 09:00 to 14:59, bold values in Table 2.

6 Rarity of local intensity values

The spot disturbance value  = –1821 nT at 06:20 UT on September 2, 1859, is, among all the hourly and auxiliary values from Colaba, the most extreme. To put this into perspective, we examine modern 1-min-resolution observatory measurements recording 18 intense storms, years 1989–2015 (cycles 22–24) – the same 18 storms used in Section 4. For each storm i, we obtain its intensity Di. For each storm i and each observatory n, we estimate local disturbance sequences by using an algorithm specifically designed for separating 1-min-resolution from an observatory horizontal intensity sequence (Rigler, 2017); we apply a latitude factor as per equation (4). Then, per Section 4, to mimic the sampling schedule at Colaba, for each UT hour, we use for analysis the disturbance value at the 20th UT-minute of each hour, and like the Colaba auxiliary values, we use disturbance values at UT-minutes 2, 7, 12, …, 57. For each storm i and each observatory n, we denote these Colaba-like disturbance values as . From these samples, we identify the maximum negative geomagnetic disturbance value,

(12)

compare with equation (7). Finally, we restrict ourselves to local intensities realized in the local-noon sector. Individual values are listed in Table 5.

Table 5

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); local 1-min-resolution intensities, sampled as at Colaba, , equation (12), for Alibag (ABG) and HER, KAK, HON, SJG for selected storms, 1989–2015. Month (Mn), Day (Dy), Hour (Hr, UT), Minute (Mi, UT). Bold data are shown in Figure 5.

Next, we examine a model in which storms with a given logarithmic intensity ln Di will give rise to local logarithmic intensities ln that are normally distributed,

(13)

Here, the mean of ln is the parameter ln D + ln α = ln(Di · α), the standard deviation of ln is the parameter σ, and α is, again, a bias parameter. As per the Appendix, equation (13) can be recognized as the Bayesian dual of equation (8) and vice versa. Combining data from Alibag with those from the standard Dst observatories, we obtain the (log)normal model parameters {ln α, σ} in the probability density function (13) by the method of maximum likelihood. We plot results in Figure 5. The bootstrap-resampled Kolmogorov-Smirnov probability, developed per Section 5, is too high to reject the (log)normal hypothesis. For comparison, in Figure 5, we plot the extreme hourly intensity,  = 1821 nT, together with our corresponding estimate of the Carrington storm’s intensity, D1859 = 964 nT, with the inclusion of the extreme disturbance value at 06:20 UT. We also plot the extreme auxiliary intensity,  = 1386 nT, corresponding to a measurement at 06:27 UT, together with our corresponding estimate of the Carrington storm’s intensity, D1859 = 866 nT, with exclusion of the extreme disturbance value at 06:20 UT. The extreme hourly intensity value is close to 5.4σ from the median, whereas the extreme auxiliary intensity value is more than 4.0σ from the median.

thumbnail Figure 5

Contour lines of the (log)normal density function g(ln |ln Di, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities Di and from Table 5 (1989–2015): Alibag (black, ABG) and the four standard Dst observatories (grey, open circles), Hermanus (HER), Kakioka (KAK), Honolulu (HON), San Juan (SJG). The model enables estimation of an exceedance probability G(ln |ln Di, ln α, σ), equation (14), for a local-noon sector (09:00 to 14:59) intensity ln , equation (12) from a single low-latitude observatory, where data have been sparsely sampled as done at Colaba during the Carrington storm, given an hourly average storm intensity Di, equation (5). G estimated with (blue) and without (orange) the extreme disturbance value at 06:20 UT.

From the complementary cumulative, we obtain the exceedance probability

(14)

From this, we can estimate the probability that a storm with intensity D1859 = 964 nT would give rise to an hourly local intensity greater than  = 1821 nT. The probability is very low, G < 10−6, or 1/G > 106. In other words, under the (log)normal hypothesis, only one out of about a million storms with intensities like the Carrington storm would result in a local noon-sector hourly intensity greater than that reported from Colaba. Note, also, that the extreme auxiliary intensity at 06:27 UT would be a very unusual realization, G = 0.000027, for a storm with an intensity like that of the Carrington storm. These probabilities are so small they strain credulity.

At least two explanations need to be considered. First, it is possible that the geomagnetic disturbance recorded at Colaba during the Carrington storm was, in addition to being affected by the ring current, unusually affected by other current systems in the magnetosphere and ionosphere (e.g., Cliver and Svalgaard, 2005). Phenomenological analyses of modern observatory data recording recent magnetic storms suggest that dayside field-aligned currents may have significantly affected disturbance at Colaba (e.g., Cid et al., 2015; Ohtani, 2022), and recent numerical simulations suggest that such field-aligned currents might have resulted from unusually high solar-wind dynamic pressure and a dayside partial-ring current (e.g., Blake et al., 2021). Second, in circumstances like we have for the Carrington storm, with very unusual data originating from a single source, we should consider the possibility that there might be something wrong with the data. Indeed, Hayakawa et al. (2022a) call attention to the fact that the extreme local intensity value at 06:20 UT on September 2 is reported right after a hiatus or “gap” in the auxiliary measurements that were made during periods of unusual disturbance. As we note in Section 3, this gap is not explained in the Colaba yearbook. We wonder if the observatory workers were having difficulty obtaining accurate measurements during the extreme activity of the Carrington storm.

It is important to recognize that at least part of the seeming rarity of the extreme Colaba disturbance value at 06:20 UT is due to how the geomagnetic variation was sampled at Colaba – sparse spot sampling and only in the day sector. We examine modern 1-min-resolution observatory disturbance sequences from Alibag for 12 intense storms, years 2000–2015 (cycles 23–24). For each storm i, we obtain its hourly average intensity Di. For local intensity, we examine every minute of each storm, regardless of local time

(15)

compare this with equations (7) and (12). In Table 6, we list Di and values, and in Figure 6, we plot these values and model fits. In the same figure, we plot the extreme hourly intensity,  = 1821 nT, and storm intensity, D = 964 nT, corresponding to the inclusion of the extreme disturbance value at 06:20 UT, and the extreme auxiliary intensity, 1386 nT, and storm intensity, D = 866 nT, corresponding to exclusion of the extreme disturbance value at 06:20 UT. In comparison to the results shown in Figure 5, the small bias factor a = 0.8674 corresponds to a significant offset of median compared to Di – without the restrictions of subsampling, with more 1-min disturbance values from which to choose, local intensity values are higher than overall storm intensity. In this case, the probability that a storm with intensity D1859 = 964 nT would give rise to an hourly local intensity like the Colaba  = 1821 nT value is very low, G = 0.000251. This is still rare, but, we emphasize, that the probabilities listed in Figure 5 are more meaningful because they are sampled like the Colaba data.

thumbnail Figure 6

Contour lines of the (log)normal density function g(ln | ln Di, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities Di and from Table 6 (2000–2015): Alibag (black, ABG). G estimated with (blue) and without (orange) the extreme disturbance value at 06:20 UT.

Table 6

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, HER, KAK, HON, SJG; local 1-min-resolution intensities , equation (15), for Alibag (ABG) for selected storms, 2000–2015. Month (Mn), Day (Dy), Hour (Hr, UT), Minute (Mi, UT). Data are shown in Figure 6.

7 Conclusions

The uncertainty of the maximum –Dst of the Carrington storm is a primary factor affecting estimates of the occurrence frequency of extremely intense storms. So, for example, recalling results from Section 5, if one includes the extreme disturbance value at 06:20 UT in an hourly average of local-Colaba disturbance, then the median storm intensity of a storm giving rise to the observed disturbance at Colaba is D = 964 nT, with a 68% credibility interval of [855,1087] nT. Considering, then, the results of Love (2021), for a generalized extreme-value extrapolation of storms realized during solar cycles 14–24 (1902–2019), the most likely occurrence frequency of a storm with similar (or greater) intensity is 0.0107/solar cycle. For solar cycles with a typical duration of 10.73 years, this corresponds to a median wait time of 1003 years between such storms (also refer to Moriña et al., 2019). The 68% credibility interval is [0.0265,0.0038]/solar cycle, for a wait-time credibility interval of [405, 2824] years – an uncertainty in the occurrence frequency of almost a factor of seven due simply to the fact that the maximum –Dst of the Carrington storm is, itself, uncertain because it is inferred from observations made at a single observatory. If one were to prefer to exclude the extreme disturbance value at 06:20 UT because it might have been affected by non-ring currents or because it might be inaccurate, then one arrives at a median storm intensity of D = 866, [768, 977] nT. With generalized extreme-value extrapolation, the most likely occurrence frequency of storms with similar (or greater) intensity is 0.0241, [0.0536,0.0096]/solar cycle, for a median wait time of 445, [200,1118] years. This is more than a factor of two shorter than the case where we include the most extreme disturbance value.

At latitudes higher than Colaba, the interference that the Carrington storm brought to telegraph systems worldwide was plausibly due to geomagnetic disturbance generated by auroral-zone electrojet currents. Together with the storm’s main-phase increase in –Dst, the auroral oval expanded in latitude, increasing the geographic area that experienced high levels of geomagnetic disturbance, and increasing the exposure of grounded telegraph systems to geoelectric fields induced in the Earth (e.g., Boteler, 2006b). Reports of overhead aurora down to about 25 magnetic latitude (e.g., Hayakawa et al., 2020a) suggest that telegraph systems across all of Europe, most of North America, Asia, and Australia, and large parts of South America and Africa, were exposed to interfering geoelectric fields. More recent, albeit less intense, storms caused power-grid system interference across North America coincident with an overhead auroral oval (e.g., Love et al., 2022). Assuming that what has occurred in the past will occur again in the future, the magnitudes of these events inform estimates of the exposure that would be realized for present-day power-grid systems due to a storm with an intensity comparable to the Carrington storm (e.g., Oughton et al., 2019; Ebihara et al., 2021).

Ironically, our modern, technologically-based society would be at greater risk if the intensity of the Carrington storm were actually relatively low. To understand this, consider the level of widespread disruption of long-line telegraph systems caused by the Carrington storm. A storm of comparable intensity might be expected to deliver a considerable level of disruption to grounded power-grid systems. But storms of relatively low intensity (small maximum –Dst) occur more frequently than storms with relatively high intensity (large maximum –Dst). Therefore, if the intensity of the Carrington storm were (say) on the low side of our estimated credibility intervals, then we would expect widespread disruption of technological systems to occur more frequently than if the intensity of the Carrington storm were, instead, on the high side of our estimated credibility intervals. And, insofar as risk is, in a statistical sense, proportional to the probability of the hazard in a given duration of time (e.g., Kron, 2002; Smolka, 2006), then the risk associated with a Carrington-class storm is higher if the intensity of the Carrington storm was on the lower end of our credibility interval. Overestimating the intensity of the Carrington storm, and underestimating the risk of a future Carrington-class storm, might result in under-mitigation for possible impacts. In such a scenario, the occurrence of a Carrington-class storm could cause widespread disruption of technological systems that have been left unprotected. That could be costly. On the other hand, underestimating the intensity of the Carrington storm, and overestimating the risk of a future Carrington-class storm, might motivate over-mitigation for possible impacts. In such a scenario, expensive protections installed on technological systems might not be necessary. Really, to properly prioritize mitigation projects, what is needed is an accurate estimate of the intensity of the Carrington storm. Unfortunately, that is exactly what we do not have.

Acknowledgments

We thank R. D. Gold, K. A. Lewis, B. R. Shiro, and J. M. Carter for proof-reading a draft manuscript. We thank A. Guerrero and M. A. Hapgood for journal reviews. This work was supported by the U.S. Geological Survey, Geomagnetism Program. The editor thanks Antonio Guerrero and Mike Hapgood for their assistance in evaluating this paper.

Data availability statement

The Colaba data for the Carrington storm presented by Hayakawa et al. (2022a) are available from https://www.kwasan.kyoto-u.ac.jp/~hayakawa/data/. Data for the September 1909 storm are available from Love et al. (2019b, their supporting information). Data for the May 1921 storm are available from Love et al. (2019a, their supporting information). Data for the October 1903, March 1940, and March 1946 storms are available from H. Hayakawa on request (hisashi@nagoya-u.jp). The Oulu Dxt index is available from the University of Oulu, Finland (dcx.oulu.fi). Historical observatory data are available from the Kyoto WDC at https://wdc.kugi.kyoto-u.ac.jp, from the Edinburgh WDC at https://wdc.bgs.ac.uk, and from INTERMAGNET at https://www.intermagnet.org.

Appendix

We summarize some relevant statistical properties of the normal and lognormal distributions.

The lognormal distribution

By the central limit theorem of statistics, a population of random variables y generated from the addition of a large number of random and independent sub-variables, drawn from well-behaved but not necessarily identical distributions will be normally distributed and have the probability density

(A1)

where the parameters μ and σ are the mean and standard deviation, respectively, of the y variables. By simple corollary, if a population of positive random variables x is generated from the multiplication of a large number of positive, random, and independent sub-variables, drawn from well-behaved but not necessarily identical distributions then the variables lnx will be normally distributed with density

(A2)

where, now, μ and σ are the mean and standard deviation, respectively, of the lnx variables (not of the x variables). By the chain rule of differentiation,

(A3)

where the probability density function for a lognormally distributed variable x is

(A4)

(e.g., Aitchison and Brown, 1957; Crow and Shimizu, 1988). Even if the variables x have physical units, their logarithms are conventionally considered to be dimensionless (e.g., Matta et al., 2011).

Bayesian symmetry

Important insight concerning the normal distribution can be obtained from a simple interpretation of Bayesian probability, under which model parameters are assumed to be random variables that are related to the distribution of data (e.g., O’Hagan and Forster, 2004). We write Bayes’s equation,

(A5)

Here, g(y|μ, σ) is interpreted as the conditional probability density of the variable y, given an occurrence in the probability of the mean μ (we are interpreting σ to be a nonprobabilistic parameter). Conversely, g(y|μ, σ) is interpreted as the conditional “posterior” probability density of μ, given the probability of an instance of y (and the parameter σ). p(μ) is the (unconditional) “prior” probability density of μ. Typically, in working with Bayes’s equation, one uses the prior to summarize one’s preexisting knowledge or subjective prejudice concerning the parameter of interest. It is not our intent, here, to delve into the difficult philosophy that sometimes accompanies the choice of a prior, especially “informative” priors that significantly affect inference (e.g., Koenig et al., 2022). Seeking simplicity, we choose the “uninformative” prior suggested by Jeffreys (1961). This ensures that any inference is independent of the way the probability functions are parameterized. The Jeffreys prior for the mean of a normal distribution is just the unnormalized uniform distribution; in our case,

(A6)

In contrast, p(y), the “posterior” probability density of y, plays a passive role, one that ensures that equation (A5) is properly normalized. In our case,

(A7)

With these choices, we obtain,

(A8)

This simple-looking equation represents more than just a mathematical symmetry, it carries important interpretation: the parameter μ is normally distributed if the data y are normally distributed, and the two have identical standard deviations σ (e.g., Box and Tiao, 1992, their Section 1.3.1). If the variables ln x are normally distributed, then

(A9)

Scale invariance

Let us now consider a population of positive variables z (not necessarily normally or lognormally distributed). We can scale these variables by a positive factor θ and take their logarithms,

(A10)

The effect of such a scaling on the mean of the logarithms of the variables is straightforward; it is shifted,

(A11)

But the variance is unaffected by the same scaling,

(A12)

Because the variance of a constant is zero. Correspondingly,

(A13)

One can say that the variance (and the standard deviation) of the logarithms of the variables is scale-invariant (Lewontin, 1966). With this, we understand that equation (A2) has the property of scale-invariance,

(A14)

References

Cite this article as: Love JJ, Joshua Rigler E, Hayakawa H & Mursula K. 2024. On the uncertain intensity estimate of the 1859 Carrington storm. J. Space Weather Space Clim. 14, 21. https://doi.org/10.1051/swsc/2024015.

All Tables

Table 1

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); and relative residual differences , equation (6), for Alibag (ABG), HER, KAK, HON, SJG, each for selected storms (1989–2015). Month (Mn), Day (Dy), Hour (Hr, UT). Data are shown in Figure 2. No values are listed if no observatory data are available.

In the text
Table 2

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); and hourly average local intensities , equation (7), calculated using Alibag (ABG) and HER, KAK, HON, SJG, each for selected storms (1957–2015). Month (Mn), Day (Dy), Hour (Hr, UT). Data in local-noon sector (09:00–14:59) are listed in bold. Data are shown in Figures 3 and 4.

In the text
Table 3

Kolmogorov-Smirnov p-values calculated for pairs of hourly average observatory intensities , equation (7), for Alibag (ABG), Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG), for the i storms listed in Table 2 (1957–2015).

In the text
Table 4

Hourly average superstorm intensities Di, equation (5), calculated for observatory (Obs) hourly average intensities , equation (7), for selected storms. Month (Mn), Day (Dy), Apia (API), Cape Town (CTO), Colaba (CLA), Coimbra (COI), Cuajimalpa (CUA), Hermanus (HER), Honolulu (HON), Kakioka (KAK), Mauritius (MRI), San Fernando (SFS), San Juan (SJG), Vieques (VQS), Watheroo (WAT), and Zi-Ka-Wei (ZKW). Data in local-noon sector (09:00–14:59) are listed in bold. Data are shown in Figures 3 and 4.

In the text
Table 5

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); local 1-min-resolution intensities, sampled as at Colaba, , equation (12), for Alibag (ABG) and HER, KAK, HON, SJG for selected storms, 1989–2015. Month (Mn), Day (Dy), Hour (Hr, UT), Minute (Mi, UT). Bold data are shown in Figure 5.

In the text
Table 6

Hourly average storm intensities Di, equation (5), calculated using the four standard observatories, HER, KAK, HON, SJG; local 1-min-resolution intensities , equation (15), for Alibag (ABG) for selected storms, 2000–2015. Month (Mn), Day (Dy), Hour (Hr, UT), Minute (Mi, UT). Data are shown in Figure 6.

In the text

All Figures

thumbnail Figure 1

Time sequences of Colaba (CLA) disturbance values , equation (4), from 17:00 September 1 to 12:00 September 3, 1859, UT: regular hourly measurements (blue), auxiliary measurements (orange) made during periods of disturbance, UT-boxcar averages (black), the time of an apparent gap in the auxiliary measurements (grey). The hourly average centred on 06:30 UT is shown with and without the extreme hourly disturbance value at 06:20 UT. This figure can be compared with Hayakawa et al. (2022a). SC denotes sudden commencement; Gap denotes possible location of a data gap.
In the text
thumbnail Figure 2

Relative residual differences , equation (6), calculated using Alibag (black, ABG), and (grey, open circle) Hermanus (HER), Kakioka (KAK), Honolulu (HON), San Juan (SJG), as a function of hourly average storm intensity Di, equation (5), calculated using the four standard Dst observatories (grey) HER, KAK, HON, SJG, for the selected storms listed in Table 1 (1989–2015). Also shown: average (black solid line) of the residual differences, and standard deviation contour lines of a normal distribution (dotted lines, ±1s, etc.).
In the text
thumbnail Figure 3

Contour lines of the (log)normal density function g(ln Di|ln , ln α, σ), equation (8): median (black solid line) and (black dotted lines) ±2σ, etc. intervals (dotted lines); intensities Di and from Table 2 (1957–2015): (a) Alibag (black, ABG) and (b) the four standard Dst observatories (grey, open circles), Hermanus (HER), Kakioka (KAK), Honolulu (HON), San Juan (SJG). Also shown: data from Table 4 (brown, green) for five other superstorms (1903, 1909, 1921, 1940, 1946). In each of (a, b) estimates of Carrington-storm D1859 are shown for Colaba with (blue) and without (orange) the extreme disturbance value at 06:20 UT. 68% (±1σ) credibility intervals in brackets. Bootstrap Kolmogorov-Smirnov probability, KS p.
In the text
thumbnail Figure 4

Same as Figure 3, except that, to mimic sampling performed at Colaba during the Carrington storm, intensities Di and are restricted to the local noon sector 09:00 to 14:59, bold values in Table 2.
In the text
thumbnail Figure 5

Contour lines of the (log)normal density function g(ln |ln Di, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities Di and from Table 5 (1989–2015): Alibag (black, ABG) and the four standard Dst observatories (grey, open circles), Hermanus (HER), Kakioka (KAK), Honolulu (HON), San Juan (SJG). The model enables estimation of an exceedance probability G(ln |ln Di, ln α, σ), equation (14), for a local-noon sector (09:00 to 14:59) intensity ln , equation (12) from a single low-latitude observatory, where data have been sparsely sampled as done at Colaba during the Carrington storm, given an hourly average storm intensity Di, equation (5). G estimated with (blue) and without (orange) the extreme disturbance value at 06:20 UT.
In the text
thumbnail Figure 6

Contour lines of the (log)normal density function g(ln | ln Di, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities Di and from Table 6 (2000–2015): Alibag (black, ABG). G estimated with (blue) and without (orange) the extreme disturbance value at 06:20 UT.
In the text

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