Issue 
J. Space Weather Space Clim.
Volume 14, 2024
Topical Issue  Space Climate: Longterm 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 
Research Article
On the uncertain intensity estimate of the 1859 Carrington storm
^{1}
U.S. Geological Survey, Geomagnetism Program, Geologic Hazards Science Center, Denver, CO 80225, USA
^{2}
Institute for SpaceEarth Environmental Research, Nagoya University, Nagoya 4648601, Japan
^{3}
Institute for Advanced Research, Nagoya University, Nagoya 4648601, Japan
^{4}
RAL Space, Rutherford Appleton Laboratory, Harwell Campus, Science and Technology Facilities Council, Didcot OX11 0QX, UK
^{5}
Space Climate Group, Space Physics and Astronomy Research Unit, University of Oulu, PO Box 3000, 90014 Oulu, Finland
^{*} Corresponding author: jlove@usgs.gov
Received:
28
May
2023
Accepted:
14
May
2024
A study is made of the intensity of the Carrington magnetic storm of September 1859 as inferred from visual measurements of horizontalcomponent geomagnetic disturbance made at the Colaba observatory in India. Using data from modern observatories, a lognormal statistical model of storm intensity is developed, to characterize the maximumnegative value of the stormtime disturbance index (maximum –Dst) versus geomagnetic disturbance recorded at lowlatitude observatories during magnetic storms. With this model and a recently published presentation of the Colaba data, the most likely maximum –Dst of the Carrington storm and its credibility interval are estimated. A related model is used to examine individual Colaba disturbance values reported for the Carrington storm. Results indicate that only about one in a million storms with maximum –Dst like the Carrington storm would result in local disturbance greater than that reported from Colaba. This indicates that either the Colaba data were affected by magnetosphericionospheric current systems in addition to the ring current, or there might be something wrong with the Colaba data. If the most extreme Colaba disturbance value is included in the analysis, then, of all hypothetical storms generating the hourly average disturbance recorded at Colaba during the Carrington storm, the median maximum –Dst = 964 nT, with a 68% credibility interval of [855,1087] nT. If the most extreme Colaba disturbance value is excluded from the analysis, then the median maximum –Dst = 866 nT, with a 68% credibility interval of [768,977] nT. The widths of these intervals indicate that estimates of the occurrence frequency of Carringtonclass storms are very uncertain, as are related estimates of risk for modern technological systems.
Key words: Magnetic storm / Space weather / Extreme event / Historical event / Statistical analysis
© J.J. Love et al., Published by EDP Sciences 2024
This 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 solargeospace 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, groundbased 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 SunEarth distance was traversed by an interplanetary coronalmass ejection (ICME) of unusually high velocity (e.g., Cliver and Svalgaard, 2005). The resulting magnetic storminduced 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 “byeye” (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 lowlatitude observatory, its data might be used to estimate Dst, a standard measure of global stormtime geomagnetic disturbance that is usually derived from data from four lowlatitude observatories (Sugiura and Kamei, 1991; Karinen and Mursula, 2005). As the ring current strengthens, the horizontal component of the lowlatitude geomagnetic field weakens, and Dst declines from a prestorm, nearzero 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 geomagneticfield measurement, it is perhaps not surprising that it has been challenging to accurately estimate the Carrington storm’s (global) intensity from oldfashioned 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 lowlatitude 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 spaceweather events (e.g., Hapgood, 2012), their occurrence frequency and future probability (e.g., Tsubouchi and Omura, 2007; Love, 2021), the hazards that extreme spaceweather events represent for technological systems (e.g., Daglis, 2005), especially for longline 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 ringcurrent index Dst as a discrete numerical sequence of hourly values calculated by averaging horizontalcomponent geomagnetic disturbance values acquired at N lowlatitude observatories,
where, for each observatory n, hourly average disturbance is
Sq_{n}(t) is a function representing solarquiet variation; Hb_{n} is a constant baseline. is a “boxcar” average of calibrated (“absolute”) subhourly horizontalcomponent magnetometer data H_{n}(t_{j}) acquired within each full UniversalTime (UT) hour,
where the timestamp t_{j} for each average is the centre time of each boxcar, the bottom of each UT hour (t_{j} = 00:30, 01:30, etc.), and where the summation is over M subhourly samples. From 1957 until the 1980s, Dst was calculated using observatory hourly averages obtained from subhourly readings of analogue magnetograms; from the 1980s to the present time, Dst has been derived from digital 1minresolution magnetometer data. So, for example, an hourly average can be calculated by averaging M = 60 1min H_{n}(t_{m}) values, where t_{m} ranges from 00:00 to 00:59 UT, or from 01:00 to 01:59 UT, etc. Importantly, in equation (2), a nonstormy waveform, is subtracted from each hourly average observatory hourly value, where Sq_{n}(t) is a function that represents solarquiet variation and Hb_{n} is a constant baseline. In comparison to the intense storms considered in this analysis, the peaktopeak amplitude of solarquiet variation is very small, ~100 nT.
Local groundlevel disturbance amplitude is normalized in equation (2) under the assumption that lowlatitude geomagnetic disturbance around the world can be described by a timevarying external dipolar field, with factor 1/cosϕ_{n}, where ϕ_{n} is observatory geomagnetic latitude. Such a dipolar field would be generated by a westwarddirected 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 lowlatitude strength of the geomagnetic field and a corresponding decrease in Dst(t_{j}) from its nearzero quiettime value (e.g., Loewe and Prölss, 1997), until it eventually reaches a minimum value (a maximumnegative value) that is often taken as a storm’s absolute “intensity” (e.g., Gonzalez et al., 1994). Whereas a symmetric ring current would generate lowlatitude geomagnetic disturbance that is symmetric across local time, other current systems, such as partialring, fieldaligned, magnetopause, magnetotail, and ionospheric currents, introduce localtime 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 t_{j}, localtime asymmetry introduces dispersion among the N observatory disturbance values that contribute to the average disturbance that is Dst(t_{j}). This dispersion, in turn, introduces inaccuracy in estimates of Dst(t_{j}), especially if Dst(t_{j}) is estimated from disturbance from (say) just one observatory.
Commonly, Dst is based on data from four standard and longrunning 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 20thcentury magnetic storms have, due to limited data availability, calculated Dst for alternative sets of lowlatitude 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 1min 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 1hresolution index Dxt (1957present) calculated using equation (1) by Oulu University using data from the four standard observatories (Karinen and Mursula, 2005).
3 The Colaba horizontalcomponent 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, horizontalcomponent 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 horizontalcomponent measurements (both hourly and auxiliary), each made at instances in time t_{j}, provides a time sequence of variation in the strength of the horizontal geomagnetic field.
In this report, we use a version of the Carringtonstorm 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 temperaturerelated change in instrument response; and they made the data available in computerreadable 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 regularhourly and auxiliary measurements, we plot in Figure 1 the local Colaba “spot” geomagnetic disturbance time sequence, during the Carrington storm,
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, UTboxcar 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, Sq_{CLA}(t) is the solarquiet function, each as reported by Hayakawa et al. (2022a). The geomagnetic latitude at Colaba in 1859 was ϕ_{cla} = 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 horizontalcomponent 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 horizontalcomponent 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 onceperhour 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.
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 D^{i}, 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 horizontalcomponent 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 solarquiet Sq_{CLA}(t) waveform and baseline Hb_{CLA} 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 solarquiet variation. But, as noted by Hayakawa et al., the August 29 storm was still affecting the local horizontalcomponent 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 1h resolution index, with boxcar averaging performed over subhourly 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 UThour so that comparisons can be made with the traditional definition of the Dst index.
In Figure 1, we plot the UThour 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 (runningaverage) 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 UTboxcar averages of the sparse and nonuniformly 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 horizontalcomponent 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
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 1minresolution geomagnetic horizontalcomponent 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 UThour of the maximum intensity of each storm t^{i} and for each observatory n, we calculate the UTboxcar hourly average of 60 1min, values. Additionally, we subsample the 1min 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 horizontalcomponent 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 UThour of the maximum intensity of each storm t^{i} and for each observatory n, we calculate their average . From these hourly averages, we form the relative residual differences
Hourly average storm intensities D^{i}, 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 D^{i}. 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 D^{i}. In light of these observations, we have confidence in using UThourly averages of the Colaba observations.
5 Estimating Carrington storm intensity with error bars
In this section, we investigate the accuracy of estimates of Carringtonstorm 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 lowlatitude 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 horizontalcomponent disturbance time sequences for five observatories, Alibag and the four standard Dst observatories. For each storm i and each observatory n, we identify the maximumnegative geomagnetic disturbance value,
which we call a “local hourly average intensity”. In Table 2, we list the storm intensities D^{i} 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); D^{1989} = 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 D^{1989} (01:30 UT), whereas the value for each of Kakioka and Honolulu occurred one hour earlier (00:30, March 14).
Hourly average storm intensities D^{i}, 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 localnoon 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 twosample KolmogorovSmirnov algorithm (e.g., Press et al., 1992, Chapter 14.3; Bohm and Zech, 2010, Chapter 10.3.5) to calculate the probability (pvalue) 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 PawlowskyGlahn, 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 KolmogorovSmirnov pvalue for pairs of observatories; we keep the median pvalue 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 pvalue 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 KolmogorovSmirnov pvalues 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 KolmogorovSmirnov testing below when we examine the statistical consistency of datasets and models fitted to those datasets.
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 magnetosphericionospheric processes operating under external forcing by the solar wind (e.g., Vassiliadis et al., 2000; Boaghe et al., 2001). We assume, therefore, that D^{i} 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,
The mean of ln D^{i} is the parameter ln , the standard deviation of ln D^{i} 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
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,
In Figure 3, we plot the (log)normal density functions g(ln D^{i} 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 D^{i} – the relationship between these two variables appears to be scaleinvariant (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 D^{i}. As above, we perform KolmogorovSmirnov tests, but this time we use a onesample 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 KolmogorovSmirnov pvalue against the model; median pvalues from multiple resamplings are suitably unbiased estimates of significance. These bootstrap pvalues 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.
Figure 3 Contour lines of the (log)normal density function g(ln D^{i}ln , ln α, σ), equation (8): median (black solid line) and (black dotted lines) ±2σ, etc. intervals (dotted lines); intensities D^{i} 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 Carringtonstorm D^{1859} are shown for Colaba with (blue) and without (orange) the extreme disturbance value at 06:20 UT. 68% (±1σ) credibility intervals in brackets. Bootstrap KolmogorovSmirnov probability, KS p. 
Hourly average superstorm intensities D^{i}, 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 ZiKaWei (ZKW). Data in localnoon 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 D^{1859} and the corresponding accuracy of D^{1859} 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 D^{1859} = 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 [D^{1859} · e^{−σ}, D^{1859} · e^{+σ}] = [796,1119] nT – for a width of 323 nT. On the other hand, D^{1859} = 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 D^{i} seen in Figure 3 is due to data sampling across dawndusk asymmetry. Therefore, for similarity with Colaba measurements made around local noon, we now restrict ourselves to the intensities in Table 2 realized in the localtime sector 09:00 to 14:59. We also note the consistency seen in Figure 3(a) and (b), and so we treat the local noonsector 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 D^{i} 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 D^{i} 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 D^{1859} = = 964 nT and a 68% credibility interval of [D^{1859} ∙ e^{–σ}, D^{1859} ∙ 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 D^{1859} = 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.
Figure 4 Same as Figure 3, except that, to mimic sampling performed at Colaba during the Carrington storm, intensities D^{i} 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 1minresolution 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 D^{i}. For each storm i and each observatory n, we estimate local disturbance sequences by using an algorithm specifically designed for separating 1minresolution 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 UTminute of each hour, and like the Colaba auxiliary values, we use disturbance values at UTminutes 2, 7, 12, …, 57. For each storm i and each observatory n, we denote these Colabalike disturbance values as . From these samples, we identify the maximum negative geomagnetic disturbance value,
compare with equation (7). Finally, we restrict ourselves to local intensities realized in the localnoon sector. Individual values are listed in Table 5.
Hourly average storm intensities D^{i}, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); local 1minresolution 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 D^{i} will give rise to local logarithmic intensities ln that are normally distributed,
Here, the mean of ln is the parameter ln D + ln α = ln(D^{i} · 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 bootstrapresampled KolmogorovSmirnov 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, D^{1859} = 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, D^{1859} = 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.
Figure 5 Contour lines of the (log)normal density function g(ln ln D^{i}, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities D^{i} 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 D^{i}, ln α, σ), equation (14), for a localnoon sector (09:00 to 14:59) intensity ln , equation (12) from a single lowlatitude observatory, where data have been sparsely sampled as done at Colaba during the Carrington storm, given an hourly average storm intensity D^{i}, 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
From this, we can estimate the probability that a storm with intensity D^{1859} = 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 > 10^{6}. 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 noonsector 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 fieldaligned currents may have significantly affected disturbance at Colaba (e.g., Cid et al., 2015; Ohtani, 2022), and recent numerical simulations suggest that such fieldaligned currents might have resulted from unusually high solarwind dynamic pressure and a dayside partialring 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 1minresolution 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 D^{i}. For local intensity, we examine every minute of each storm, regardless of local time
compare this with equations (7) and (12). In Table 6, we list D^{i} 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 D^{i} – without the restrictions of subsampling, with more 1min 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 D^{1859} = 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.
Figure 6 Contour lines of the (log)normal density function g(ln  ln D^{i}, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities D^{i} and from Table 6 (2000–2015): Alibag (black, ABG). G estimated with (blue) and without (orange) the extreme disturbance value at 06:20 UT. 
Hourly average storm intensities D^{i}, equation (5), calculated using the four standard observatories, HER, KAK, HON, SJG; local 1minresolution 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 localColaba 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 extremevalue 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 waittime 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 nonring currents or because it might be inaccurate, then one arrives at a median storm intensity of D = 866, [768, 977] nT. With generalized extremevalue 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 auroralzone electrojet currents. Together with the storm’s mainphase 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 powergrid 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 presentday powergrid 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, technologicallybased 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 longline telegraph systems caused by the Carrington storm. A storm of comparable intensity might be expected to deliver a considerable level of disruption to grounded powergrid 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 Carringtonclass 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 Carringtonclass storm, might result in undermitigation for possible impacts. In such a scenario, the occurrence of a Carringtonclass 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 Carringtonclass storm, might motivate overmitigation 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 proofreading 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.kyotou.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@nagoyau.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.kyotou.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 subvariables, drawn from wellbehaved but not necessarily identical distributions will be normally distributed and have the probability density
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 subvariables, drawn from wellbehaved but not necessarily identical distributions then the variables lnx will be normally distributed with density
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,
where the probability density function for a lognormally distributed variable x is
(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,
Here, g(yμ, 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(yμ, 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,
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,
With these choices, we obtain,
This simplelooking 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
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,
The effect of such a scaling on the mean of the logarithms of the variables is straightforward; it is shifted,
But the variance is unaffected by the same scaling,
Because the variance of a constant is zero. Correspondingly,
One can say that the variance (and the standard deviation) of the logarithms of the variables is scaleinvariant (Lewontin, 1966). With this, we understand that equation (A2) has the property of scaleinvariance,
References
 Aitchison J, Brown JAC. 1957. The lognormal distribution: With special reference to its uses in economics. Cambridge University Press, Cambridge, UK. ISBN 9780521040112. [Google Scholar]
 Allen J, Sauer H, Frank L, Reiff P. 1989. Effects of the March 1989 solar activity. Eos Trans Am Geophys Union 70(46), 1479, 1486–1488. https://dx.doi.org/10.1029/89EO00409. [CrossRef] [Google Scholar]
 Asikainen T, Maliniemi V, Mursula K. 2010. Modeling the contributions of ring, tail, and magnetopause currents to the corrected Dst index. J Geophys Res Space Phys 115(A12): A12203. https://dx.doi.org/10.1029/2010JA015774. [CrossRef] [Google Scholar]
 Baker DN, Balstad R, Bodeau JM, Cameron E, Fennell JE, et al. 2008. Severe space weather events – Understanding societal and economic impacts. The National Academy Press, Washington, DC. ISBN 9780309141536. https://dx.doi.org/10.17226/12507. [Google Scholar]
 Beggan CD, Clarke E, Lawrence E, Eaton E, Williamson J, Matsumoto K, Hayakawa H. 2024. Digitized continuous magnetic recordings for the August/September 1859 storms From London, UK. Space Weather 22(3): e2023SW003, 807. https://doi.org/10.1029/2023SW003807. [CrossRef] [Google Scholar]
 Blake SP, Pulkkinen A, Schuck PW, Glocer A, Oliveira DM, Welling DT, Weigel RS, Quaresima G. 2021. Recreating the horizontal magnetic field at Colaba during the Carrington event with geospace simulations. Space Weather 19(5): e2020SW002, 585. https://dx.doi.org/10.1029/2020SW002585. [CrossRef] [Google Scholar]
 Boaghe OM, Balikhin MA, Billings SA, Alleyne H. 2001. Identification of nonlinear processes in the magnetospheric dynamics and forecasting of Dst index. J Geophys Res Space Phys 106(A12): 30047–30066. https://dx.doi.org/10.1029/2000JA900162. [CrossRef] [Google Scholar]
 Bohm G, Zech G. 2010. Introduction to statistics and data analysis for physicists. Verlag Deutsches ElektronenSynchrotron, Hamburg, Germany. ISBN 9783935702416. [Google Scholar]
 Boteler DH. 2006a. Comment on time conventions in the recordings of 1859. Adv Space Res 38(2): 301–303. https://dx.doi.org/10.1016/j.asr.2006.07.006. [CrossRef] [Google Scholar]
 Boteler DH. 2006b. The super storms of August/September 1859 and their effects on the telegraph system. Adv Space Res 38(2): 159–172. https://dx.doi.org/10.1016/j.asr.2006.01.013. [CrossRef] [Google Scholar]
 Boteler DH. 2019. A 21st century view of the March 1989 magnetic storm. Space Weather 17(10): 1427–1441. https://dx.doi.org/10.1029/2019SW002278. [CrossRef] [Google Scholar]
 Box GEP, Tiao GC. 1992. Bayesian inference in statistical analysis. John Wiley & Sons, New York, NY. ISBN 9781118033197. https://dx.doi.org/10.1002/9781118033197. [Google Scholar]
 Carrington RC. 1859. Description of a singular appearance seen in the Sun on September 1 1859. Month Notices Royal Astron Soc 20(1): 13–15. https://dx.doi.org/10.1093/mnras/20.1.13. [CrossRef] [Google Scholar]
 Chapman S, Bartels J. 1962. Geomagnetism, Volume 1. Oxford University Press, London, UK, 2 edn. [Google Scholar]
 Cid C, Saiz E, Guerrero A, Palacios J, Cerrato Y. 2015. A Carringtonlike geomagnetic storm observed in the 21st century. J Space Weather Space Clim 5: A16. https://dx.doi.org/10.1051/swsc/2015017. [CrossRef] [EDP Sciences] [Google Scholar]
 Clauset A, Shalizi CR, Newman MEJ. 2009. Powerlaw distributions in empirical data. SIAM Rev 51(4): 661–703. https://dx.doi.org/10.1137/070710111. [CrossRef] [Google Scholar]
 Cliver EW. 2006. The 1859 space weather event: Then and now. Adv Space Res 38(2): 119–129. https://dx.doi.org/10.1016/j.asr.2005.07.077. [CrossRef] [Google Scholar]
 Cliver EW, Dietrich WF. 2013. The 1859 space weather event revisited: Limits of extreme activity. J Space Weather Space Clim 3: A31. https://dx.doi.org/10.1051/swsc/2013053. [Google Scholar]
 Cliver EW, Svalgaard L. 2005. The 1859 solarterrestrial disturbance and the current limits on extreme space weather activity. Solar Phys 224: 407–422. https://dx.doi.org/10.1007/s112070054980z. [Google Scholar]
 Corral Á, González A. 2019. Power law size distributions in geoscience revisited. Earth Space Sci 6(5): 673–697. https://dx.doi.org/10.1029/2018EA000479. [CrossRef] [Google Scholar]
 Crow EL, Shimizu K, (Eds.). 1988. Lognormal distributions: theory and applications. Marcel Dekker, New York, NY. ISBN 9780824778033. [Google Scholar]
 Daglis IA, (Ed.). 2005. Effects of space weather on technology infrastructure. Springer, Dordrecht, The Netherlands. ISBN 9781402027543. https://dx.doi.org/10.1007/1402027540. [CrossRef] [Google Scholar]
 Daglis IA. 2006. Ring current dynamics. Space Sci Rev 124(1–4): 183–202. https://dx.doi.org/10.1007/s112140069104z. [Google Scholar]
 Dessler AJ, Parker EN. 1959. Hydromagnetic theory of geomagnetic storms. J Geophys Res 64(12): 2239–2252. https://dx.doi.org/10.1029/JZ064i012p02239. [CrossRef] [Google Scholar]
 Eastwood JP, Biffis E, Hapgood MA, Green L, Bisi MM, Bentley RD, Wicks R, McKinnell LA, Gibbs M, Burnett C. 2017. The economic impact of space weather: Where do we stand?. Risk Anal 37(2): 206–218. https://dx.doi.org/10.1111/risa.12765. [CrossRef] [Google Scholar]
 Ebihara Y, Watari S, Kumar S. 2021. Prediction of geomagnetically induced currents (GICs) flowing in Japanese power grid for Carringtonclass magnetic storms. Earth Planets Space 73. https://dx.doi.org/10.1186/s40623021014932. [CrossRef] [Google Scholar]
 Efron B, Tibshirani RJ. 1994. An introduction to the bootstrap. Chapman and Hall/CRC, New York, NY. https://doi.org/10.1201/9780429246593. [CrossRef] [Google Scholar]
 Fergusson EFT. 1860. Magnetical and meteorological observations made at the government observatory, Bombay, 1859. Bombay Education Society’s Press, Byculla, India. [Google Scholar]
 Friedrich E, Rostoker G, Connors MG, McPherron RL. 1999. Influence of the substorm current wedge on the Dst index. J Geophys Res Space Phys 104(A3): 4567–4575. https://dx.doi.org/10.1029/1998JA900096. [CrossRef] [Google Scholar]
 Gannon JL, Love JJ. 2011. USGS 1min Dst index. J Atmos SolarTerrestrial Phys 73(2): 323–334. https://dx.doi.org/10.1016/j.jastp.2010.02.013. [CrossRef] [Google Scholar]
 Gawali PB, Doiphode MG, Nimje RN. 2015. ColabaAlibag magnetic observatory and Nanabhoy Moos: The influence of one over the other. History Geo Space Sci 6(2): 107–131. https://dx.doi.org/10.5194/hgss61072015. [CrossRef] [Google Scholar]
 Gjerloev JW. 2009. A global groundbased magnetometer initiative. Eos Trans Am Geophys Union 90(27): 230–231. https://dx.doi.org/10.1029/2009EO270002. [CrossRef] [Google Scholar]
 Gonzalez WD, Echer E, Tsurutani BT, Clúa de González AL, Dal Lago A. 2011. Interplanetary origin of intense, superintensive and extreme geomagnetic storms. Space Sci Rev 158(1): 69–89. https://dx.doi.org/10.1007/s1121401097152. [CrossRef] [Google Scholar]
 Gonzalez WD, Joselyn JA, Kamide Y, Kroehl HW, Rostoker G, Tsurutani BT, Vasyliunas VM. 1994. What is a geomagnetic storm?. J Geophys Res 99(A4): 5771–5792. https://dx.doi.org/10.1029/93JA02867. [CrossRef] [Google Scholar]
 Green JL, Boardsen S, Odenwald S, Humble J, Pazamickas KA. 2006. Eyewitness reports of the great auroral storm of 1859. Adv Space Res 39(2): 145–154. https://dx.doi.org/10.1016/j.asr.2005.12.021. [CrossRef] [Google Scholar]
 Hapgood M. 2019. The great storm of May 1921: An exemplar of a dangerous space weather event. Space Weather 17(7): 950–975. https://dx.doi.org/10.1029/2019SW002195. [NASA ADS] [CrossRef] [Google Scholar]
 Hapgood MA. 2012. Prepare for the coming space weather storm. Nature 484: 311–313. https://dx.doi.org/10.1038/484311a. [CrossRef] [Google Scholar]
 Hayakawa H, Ebihara JRRY, Correia AP, Sôma M. 2020a. South American auroral reports during the Carrington storm. Earth Planets Space 72(1): 122. https://dx.doi.org/10.1186/s40623020012494. [CrossRef] [Google Scholar]
 Hayakawa H, Ebihara Y, Cliver EW, Hattori K, Toriumi S, et al. 2019. The extreme space weather event in September 1909. Month Notices Royal Astron Soc 484(3): 4083–4099. https://dx.doi.org/10.1093/mnras/sty3196. [CrossRef] [Google Scholar]
 Hayakawa H, Ebihara Y, Hand DP, Hayakawa S, Kumar S, Mukherjee S, Veenadhari B. 2018. Lowlatitude aurorae during the extreme space weather events in 1859. Astrophys J 869(1): 57. https://dx.doi.org/10.3847/15384357/aae47c. [CrossRef] [Google Scholar]
 Hayakawa H, Ebihara Y, Pevtsov AA, Bhaskar A, Karachik N, Oliveira DM. 2020b. Intensity and time series of extreme solarterrestrial storm in 1946 March. Month Notices Royal Astron Soc 497(4): 5507–5517. https://dx.doi.org/10.1093/mnras/staa1508. [CrossRef] [Google Scholar]
 Hayakawa H, Nevanlinna H, Blake SP, Ebihara Y, Bhaskar AT, Miyoshi Y. 2022a. Temporal variations of the three geomagnetic field components at Colaba Observatory around the Carrington storm in 1859. Astrophys J 928(1): 32. https://dx.doi.org/10.3847/15384357/ac2601. [CrossRef] [Google Scholar]
 Hayakawa H, Oliveira DM, Shea MA, Smart DF, Blake SP, Hattori K, Bhaskar AT, Curto JJ, Franco DR, Ebihara Y. 2022b. The extreme solar and geomagnetic storms on 1940 March 20–25. Month Notices Royal Astron Soc 517(2): 1709–1723. https://dx.doi.org/10.1093/mnras/stab3615. [CrossRef] [Google Scholar]
 Hayakawa H, Ribeiro P, Vaquero JM, Knipp MCGDJ, Mekhaldi F, et al. 2020c. The extreme space weather event in 1903 October/November: An outburst from the quiet Sun. Astrophys J Lett 897(1): L10. https://dx.doi.org/10.3847/20418213/ab6a18. [CrossRef] [Google Scholar]
 Hodgson R. 1859. On a curious appearance seen in the Sun. Month Notices Royal Astron Soc 20(1): 15–16. https://dx.doi.org/10.1093/mnras/20.1.15a. [CrossRef] [Google Scholar]
 Hudson HS. 2021. Carrington events. Annu. Rev Astron Astrophys 59(1): 445–477. https://dx.doi.org/10.1146/annurevastro112420023324. [CrossRef] [Google Scholar]
 Ishii M, Shiota D, Tao C, Ebihara Y, Fujiwara H, et al. 2021. Space weather benchmarks on Japanese society. Earth Planets Space 73(1): 108. https://dx.doi.org/10.1186/s40623021014205. [CrossRef] [Google Scholar]
 Iyemori T, Takeda M, Nosé M, Odagi Y, Toh H. 2010. Midlatitude geomagnetic indices ASY and SYM for 2009 (Provisional). Internal report of data analysis center for geomagnetism and space magnetism. Kyoto University, Japan. https://wdc.kugi.kyotou.ac.jp/aeasy/asy.pdf. [Google Scholar]
 Jeffreys H. 1961. Theory of probability. Clarendon Press, Oxford, UK. ISBN 9780198503682, 0198503687. [Google Scholar]
 Jones HS. 1955. Sunspots and geomagneticstorm data derived from Greenwich observations, 1874–1954. Her Majesty’s Stationery Office, London, UK. [Google Scholar]
 Karinen A, Mursula K. 2005. A new reconstruction of the Dst index for 1932–2002. Ann Geophys 23(2): 475–485. https://dx.doi.org/10.5194/angeo234752005. [CrossRef] [Google Scholar]
 Koenig C, Liu H, Schoot RVD, Depaoli S, eds. 2022. Moving beyond noninformative prior distributions: achieving the full potential of Bayesian methods for psychological research. Frontiers Media SA. ISBN 9782889742141, 2889742148. [CrossRef] [Google Scholar]
 Kotzé P. 2018. Hermanus magnetic observatory: A historical perspective of geomagnetism in southern Africa. History GeoSpace Sci 9(2): 125–131. https://dx.doi.org/10.5194/hgss91252018. [CrossRef] [Google Scholar]
 Kron W. 2002. Keynote lecture: Flood risk = hazard × exposure × vulnerability. In: Flood Defence ‘2002, vol 1, Wu B, Wang ZY, Wang GQ, Huang GH, Fang HW, Huang JC, (Eds.) Science Press, New York, NY. pp. 82–97. ISBN 9781880132548. [Google Scholar]
 Lakhina GS, Tsurutani BT. 2018. Super geomagnetic storms: Past, present and future. In: Extreme space weather: Origins, predictability, and consequences, chap. 7, Buzulukova N, (Ed.) Elsevier, Amsterdam, The Netherlands. pp. 157–185. ISBN 9780128127001. https://doi.org/10.1016/C20160037695. [Google Scholar]
 Lewontin RC. 1966. On the measurement of relative variability. Syst Zool 15(2): 141–142. https://dx.doi.org/10.2307/sysbio/15.2.141. [CrossRef] [Google Scholar]
 Loewe CA, Prölss GW. 1997. Classification and mean behavior of magnetic storms. J Geophys Res 102(A7): 14209–14213. https://dx.doi.org/10.1029/96JA04020. [CrossRef] [Google Scholar]
 Love JJ. 2009. Missing data and the accuracy of magneticobservatory hour means. Ann Geophys 27(9): 3601–3610. https://www.anngeophys.net/27/3601/2009/. [CrossRef] [Google Scholar]
 Love JJ. 2021. Extremeevent magnetic storm probabilities derived from rank statistics of historical Dst intensities for solar cycles 14–24. Space Weather 19(4): e2020SW002, 579. https://dx.doi.org/10.1029/2020SW002579. [CrossRef] [Google Scholar]
 Love JJ, Finn CA. 2011. The USGS Geomagnetism Program and its role in space weather monitoring. Space Weather 9(7): S07001. https://dx.doi.org/10.1029/2011SW000684. [Google Scholar]
 Love JJ, Gannon JL. 2009. Revised Dst and the epicycles of magnetic disturbance: 1958–2007. Ann Geophys 27(8): 3101–3131. https://doi.org/10.5194/angeo2731012009. [CrossRef] [Google Scholar]
 Love JJ, Hayakawa H, Cliver EW. 2019a. Intensity and impact of the New York Railroad superstorm of May 1921. Space Weather 17(8): 1281–1292. https://dx.doi.org/10.1029/2019SW002250. [CrossRef] [Google Scholar]
 Love JJ, Hayakawa H, Cliver EW. 2019b. On the intensity of the magnetic superstorm of September 1909. Space Weather 17(1): 37–45. https://dx.doi.org/10.1029/2018SW002079. [CrossRef] [Google Scholar]
 Love JJ, Lucas GM, Rigler EJ, Murphy BS, Kelbert A, Bedrosian PA. 2022. Mapping a magnetic superstorm: March 1989 geoelectric hazards and impacts on the United States power systems. Space Weather 20(5): e2021SW003, 030. https://dx.doi.org/10.1029/2021SW003030. [Google Scholar]
 Love JJ, Tsai VC, Gannon JL. 2010. Averaging and sampling for magneticobservatory hourly data. Ann Geophys 28(11): 2079–2096. https://dx.doi.org/10.5194/angeo2820792010. [CrossRef] [Google Scholar]
 Matta CF, Massa L, Gubskaya AV, Knoll E. 2011. Can one take the logarithm or the sine of a dimensioned quantity or a unit? Dimensional analysis involving transcendental functions. J Chem Educ 88(1): 67–70. https://dx.doi.org/10.1021/ed1000476. [CrossRef] [Google Scholar]
 Minamoto Y. 2013. Availability and access to data from Kakioka Magnetic Observatory, Japan. Data Sci J 12: G30–G35. https://dx.doi.org/10.2481/dsj.G040. [CrossRef] [Google Scholar]
 Moos NAF. 1910. Colaba magnetic data, 1846 to 1905. Part I: Magnetic data and instruments. Government Central Press, Bombay, India. [Google Scholar]
 Moriña D, Serra I, Puig P, Corral Á. 2019. Probability estimation of a Carringtonlike geomagnetic storm. Sci Rep 9(1): 2393. https://dx.doi.org/10.1038/s41598019389188. [CrossRef] [Google Scholar]
 Mursula K, Holappa L, Karinen A. 2008. Correct normalization of the Dst index. Astrophys Space Sci Trans 4(2): 41–45. https://dx.doi.org/10.5194/astra4412008. [CrossRef] [Google Scholar]
 Mursula K, Holappa L, Karinen A. 2011. Uneven weighting of stations in the Dst index. J Atmos SolarTerrestrial Phys 73(2): 316–322. https://doi.org/10.1016/j.jastp.2010.04.007. [CrossRef] [Google Scholar]
 O’Hagan A, Forster J. 2004. Kendall’s advanced theory of statistics. In: Bayesian Inference, 2nd edn, vol 2B, Arnold, London, UK. ISBN 9780470685693. [Google Scholar]
 Ohtani S. 2022. New insights from the 2003 Halloween Storm into the Colaba 1600 nT magnetic depression during the 1859 Carrington storm. J Geophys Res Space Phys 127(9): e2022JA030, 596. https://dx.doi.org/10.1029/2022JA030596. [Google Scholar]
 Ohtani S, Nosé M, Rostoker G, Singer H, Lui ATY, Nakamura M. 2001. Stormsubstorm relationship: Contribution of the tail current to Dst. J Geophys Res Space Phys 106(A10): 21199–21209. https://dx.doi.org/10.1029/2000JA000400. [CrossRef] [Google Scholar]
 Olea RA, PawlowskyGlahn V. 2009. KolmogorovSmirnov test for spatially correlated data. Stoch Environ Res Risk Assess 23(6): 749–757. https://dx.doi.org/10.1007/s0047700802551. [CrossRef] [Google Scholar]
 Oughton EJ, Hapgood M, Richardson GS, Beggan CD, Thomson AWP, et al. 2019. A risk assessment framework for the socioeconomic impacts of electricity transmission infrastructure failure due to space weather. Risk Anal 39(5): 1022–1043. https://dx.doi.org/10.1111/risa.13229. [CrossRef] [Google Scholar]
 Piccinelli R, Krausmann E. 2014. Space weather and power grids – A vulnerability assessment. European Union, Luxembourg. ISBN 9789279439711. https://dx.doi.org/10.2788/20848. [Google Scholar]
 Press WH, Teukolsky SA, Vetterling WT, Flannery BP. 1992. Numerical Recipes in Fortran 77, 2nd edn. Cambridge University Press, Cambridge, UK. ISBN 9780521430647. [Google Scholar]
 Rigler EJ. 2017. Timecausal decomposition of geomagnetic time series into secular variation, solar quiet, and disturbance signals. U.S. Geological Survey OpenFile Report, 2017–1037. https://dx.doi.org/10.3133/ofr20171037. [Google Scholar]
 Roe BP. 2001. Probability and statistics in experimental physics. Undergraduate texts in contemporary physics. SpringerVerlag, New York, NY. ISBN 9781468492965. https://dx.doi.org/10.1007/9781468492965. [Google Scholar]
 Saiz E, Cid C, Guerrero A. 2021. The relevance of local magnetic records when using extreme space weather events as benchmarks. J Space Weather Space Clim 11: 35. https://doi.org/10.1051/swsc/2021018. [CrossRef] [EDP Sciences] [Google Scholar]
 Sckopke N. 1966. A general relation between the energy of trapped particles and the disturbance field over the Earth. J Geophys Res 71(13): 3125–3130. https://dx.doi.org/10.1029/JZ071i013p03125. [CrossRef] [Google Scholar]
 Silverman SM. 2006. Comparison of the aurora of September 1/2 1859 with other great auroras. Adv Space Res 38(2): 136–144. https://dx.doi.org/10.1016/j.asr.2005.03.157. [CrossRef] [Google Scholar]
 Siscoe GL, Crooker NU, Clauer CR. 2006. Dst of the Carrington storm of 1859. Adv Space Res 38(2): 173–179. https://dx.doi.org/10.1016/j.asr.2005.02.102. [CrossRef] [Google Scholar]
 Smolka A. 2006. Natural disasters and the challenge of extreme events: Risk management from an insurance perspective. Philos Trans Royal Soc London Ser A 364: 2147–2165. https://dx.doi.org/10.1098/rsta.2006.1818. [Google Scholar]
 Steinskog DJ, Thøstheim DB, Kvamstø NG. 2007. A cautionary note on the use of the KolmogorovSmirnov test for normality. Month Weather Rev 135(3): 1151–1157. https://dx.doi.org/10.1175/MWR3326.1. [CrossRef] [Google Scholar]
 Sugiura M. 1964. Hourly values of equatorial Dst for the IGY. Ann Int Geophys Year 35: 9–45. [Google Scholar]
 Sugiura M, Kamei T. 1991. Equatorial Dst index 1957–1986. IAGA Bulletin, 40, International Service of Geomagnetic Indices Publication Office, SaintMaurdesFossess, France. [Google Scholar]
 Tsubouchi K, Omura Y. 2007. Longterm occurrence probabilities of intense geomagnetic storm events. Space Weather 5(12). https://dx.doi.org/10.1029/2007SW000329. [CrossRef] [Google Scholar]
 Tsurutani BT, Gonzalez WD, Lakhina GS, Alex S. 2003. The extreme magnetic storm of 1–2 September 1859. J Geophys Res 108(A7). https://dx.doi.org/10.1029/2002JA009504. [CrossRef] [Google Scholar]
 Turner NE, Baker DN, Pulkkinen TI, McPherron RL. 2000. Evaluation of the tail current contribution to Dst. J Geophys Res 105(A3): 5431–5439. https://dx.doi.org/10.1029/1999JA000248. [CrossRef] [Google Scholar]
 Usoskin I, Miyake F, Baroni M, Brehm N, Dalla S, et al. 2023. Extreme solar events: Setting up a paradigm. Space Sci Rev 219(8): 73. https://dx.doi.org/10.1007/s11214023010181. [CrossRef] [Google Scholar]
 Vassiliadis D, Klimas AJ, Valdivia JA, Baker DN. 2000. The nonlinear dynamics of space weather. Adv Space Res 26(1): 197–207. https://doi.org/10.1016/S02731177(99)010509. [CrossRef] [Google Scholar]
 von Storch H. 1995. Misuses of statistical analysis in climate research. In: Analysis of climate variability: Applications and Statistical techniques. von Storch H, Navarra A, (Eds.) SpringerVerlag, New York, NY. pp. 11–25. ISBN 9783662031698. [CrossRef] [Google Scholar]
 Wilks DS. 2006. Statistical methods in the atmospheric sciences. Elsevier, Amsterdam, The Netherlands ISBN 9780127519661. [Google Scholar]
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
Hourly average storm intensities D^{i}, 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.
Hourly average storm intensities D^{i}, 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 localnoon sector (09:00–14:59) are listed in bold. Data are shown in Figures 3 and 4.
Hourly average superstorm intensities D^{i}, 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 ZiKaWei (ZKW). Data in localnoon sector (09:00–14:59) are listed in bold. Data are shown in Figures 3 and 4.
Hourly average storm intensities D^{i}, equation (5), calculated using the four standard observatories, Hermanus (HER), Kakioka (KAK), Honolulu (HON), and San Juan (SJG); local 1minresolution 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.
Hourly average storm intensities D^{i}, equation (5), calculated using the four standard observatories, HER, KAK, HON, SJG; local 1minresolution 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.
All Figures
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, UTboxcar 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 
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 D^{i}, 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 
Figure 3 Contour lines of the (log)normal density function g(ln D^{i}ln , ln α, σ), equation (8): median (black solid line) and (black dotted lines) ±2σ, etc. intervals (dotted lines); intensities D^{i} 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 Carringtonstorm D^{1859} are shown for Colaba with (blue) and without (orange) the extreme disturbance value at 06:20 UT. 68% (±1σ) credibility intervals in brackets. Bootstrap KolmogorovSmirnov probability, KS p. 

In the text 
Figure 4 Same as Figure 3, except that, to mimic sampling performed at Colaba during the Carrington storm, intensities D^{i} and are restricted to the local noon sector 09:00 to 14:59, bold values in Table 2. 

In the text 
Figure 5 Contour lines of the (log)normal density function g(ln ln D^{i}, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities D^{i} 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 D^{i}, ln α, σ), equation (14), for a localnoon sector (09:00 to 14:59) intensity ln , equation (12) from a single lowlatitude observatory, where data have been sparsely sampled as done at Colaba during the Carrington storm, given an hourly average storm intensity D^{i}, equation (5). G estimated with (blue) and without (orange) the extreme disturbance value at 06:20 UT. 

In the text 
Figure 6 Contour lines of the (log)normal density function g(ln  ln D^{i}, ln α, σ), equation (13): median (black solid line) and ±2σ intervals (dotted lines); intensities D^{i} 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 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.