Statistics of the largest geomagnetic storms per solar cycle (1844–1993)

A previous application of extreme-value statistics to the first, second and third largest geomagnetic storms per solar cycle for nine solar cycles is extended to fourteen solar cycles (1844–1993). The intensity of a geomagnetic storm is measured by the magnitude of the daily aa index, rather than the half-daily aa index used previously. Values of the conventional aa index (1868– 1993), supplemented by the Helsinki Ak index (1844–1880), provide an almost continuous, and largely homogeneous, daily measure of geomagnetic activity over an interval of 150 years. As in the earlier investigation, analytic expressions giving the probabilities of the three greatest storms (extreme values) per solar cycle, as continuous functions of storm magnitude (ad), are obtained by least-squares fitting of the observations to the appropriate theoretical extreme-value probability functions. These expressions are used to obtain the statistical characteristics of the extreme values; namely, the mode, median, mean, standard deviation and relative dispersion. Since the Ak index may not provide an entirely homogeneous extension of the aa index, the statistical analysis is performed separately for twelve solar cycles (1868–1993), as well as nine solar cycles (1868–1967). The results are utilized to determine the expected ranges of the extreme values as a function of the number of solar cycles. For fourteen solar cycles, the expected ranges of the daily aa index for the first, second and third largest geomagnetic storms per solar cycle decrease monotonically in magnitude, contrary to the situation for the half-daily aa index over nine solar cycles. The observed range of the first extreme daily aa index for fourteen solar cycles is 159–352 nT and for twelve solar cycles is 215–352 nT. In a group of 100 solar cycles the expected ranges are expanded to 137–539 and 177–511 nT, which represent increases of 108% and 144% in the respective ranges. Thus there is at least a 99% probability that the daily aa index will

1 Introduction Siscoe (1976) applied the statistics of extremes to the ®rst, second and third largest geomagnetic storms in nine solar cycles (viz. 11 to 19), as measured by the average half-daily aa index. His study is extended here to fourteen solar cycles (viz. 9 to 22) using the average daily aa index for the interval 1844±1993. The conventional daily aa index (Mayaud, 1980) is available electronically through the National Geophysical Data Center, Boulder, Colorado, for the interval 1868±1993, which extends by almost three solar cycles the time-interval considered by Siscoe. Moreover, the daily aa index has recently been extended backwards in time by two solar cycles (1844±1868), using hourly measurements of magnetic declination made at the Helsinki Magnetic Observatory during the interval 1844±1880 (Nevanlinna and Kataja, 1993). Daily values of the``essentially equivalent'' Helsinki magnetic activity index Ak are available electronically through the Finnish Meteorological Institute, Helsinki (Nevanlinna, 1995). Therefore, it is now possible to investigate the statistics of the ®rst, second and third largest geomagnetic storms per solar cycle over a 150-year interval (1844±1993), using essentially homogeneous daily values of the aa index of geomagnetic activity.
However, since the conventional daily aa index is consistently homogeneous for only twelve solar cycles (viz. 11 to 22, as discussed in Sect. 2), the statistics of the largest geomagnetic storms per solar cycle are also studied for the shorter 126-year interval (1868±1993), as well as the 100-year interval (1868±1967) considered by Siscoe (1976). It is also important to consider all three cases separately because there are 137 missing daily values of the Ak index in the interval 1844±1867, which might just possibly in¯uence the results for fourteen solar cycles. However, the majority of these missing values (83) lie in the interval 19 July 1856±9 October 1856, which is close to sunspot minimum (1856.0).

Derivation of the aa and Ak indices
The aa index is derived from hand-scaled magnetograms from two almost antipodal observatories (at invariant magnetic latitudes of approximately AE50 ); one in the United Kingdom and the other in Australia (Mayaud, 1980;Menvielle and Berthelier, 1991). The subauroral observatories that have contributed to the derivation of the aa index are Greenwich (1868±1925), Abinger (1926± 1956) and Hartland (1957 ± present) for the Northern Hemisphere; Melbourne (1868±1919), Toolangi (1920± 1978) and Canberra (1979 ± present) for the Southern Hemisphere.
For each 3-h interval (00:00-03:00, 03:00-06:00, etc.), K indices (Mayaud, 1980;Menvielle and Berthelier, 1991;Joselyn, 1995) are derived for the two antipodal observatories. The K index is a quasi-logarithmic number between 0 and 9 that is assigned to the end of these speci®ed 3-h intervals. It is derived by measuring the maximum deviation [in nanoteslas (nT)] of the observed ®eld from the expected quiet-time level, for each of the three magnetic-®eld components (Joselyn, 1995). The largest of the three maxima at each observatory is converted to a K index by using a look-up table appropriate to that observatory. The K indices measured at the two antipodal observatories are then converted back into amplitudes and an individual aa index is the average of the two amplitudes, weighted to allow for the small dierence in latitude of the northern and southern observatories, or for the slight changes in the locations of the two antipodal observatories.
Since an individual three-hourly value of the aa index is derived from just two K indices, it provides only an approximate indication of the actual level of planetary geomagnetic activity. However, half-daily, or daily, averages of the aa index give an acceptably accurate indication of geomagnetic activity on a global scale and over a signi®cantly longer time-interval than any other index of geomagnetic activity (Menvielle and Berthelier, 1991). Mayaud (1973) published a unique 100-year (1868±1967) series of three-hourly aa indices, which is based on K indices that were all measured by the author himself to ensure the homogeneity of the time-series. The aa indices have continued to be published subsequently, in order to provide a rapidly available worldwide index of geomagnetic activity that is physically meaningful on a half-daily or daily time-scale. Nevanlinna and Kataja (1993) have essentially extended the aa index backwards in time by more than two solar cycles (viz. 9 and 10); namely, from 1 July 1844 to 31 December 1867. Their daily index, designated Ak, has been derived from hourly readings of declination (D) made at the Helsinki Magnetic Observatory (60 10.3¢ N, 24 59.0¢ E) during the interval 1844±1880 . Speci®cally, three-hourly K indices and daily Ak amplitudes have been computed from declination values using an algorithm developed for the automatic production of K indices in the case of modern digital data (Sucksdor et al., 1991). To ensure that the Helsinki K indices are as close as possible to the real ones, the percentage occurrence rate of K values in each bin was adjusted to be the same as the corresponding distribution at the present-day NurmijaÈ rvi Observatory, which is only 40 km from Helsinki. This was achieved by varying the Helsinki u 9 lower limit u 9 ; reasonably good agreement was found by ®xing u 9 200 nT. Then the three-hourly K indices were converted into daily Ak amplitudes in the range 0±400 nT.

Attributes and limitations of the extended aa index
The aa index is used in the statistical study of the largest geomagnetic storms per solar cycle because it spans a much longer time-interval than any other index of geomagnetic activity (Mayaud, 1980;Menvielle and Berthelier, 1991). In addition, the aa index comprises a consistently uniform, homogeneous time-series, at least since 1868, as required by the statistical analysis. For the period of overlap (1868±1880) between the conventional aa index and the Helsinki Ak index, the linear relationship between the monthly means of the two indices is 0X90 AE0X36 1X07 AE0X02 Ak, with a linear correlation coecient of 0.96 (Nevanlinna and Kataja, 1993). Owing to the local-time character of magnetic disturbances at Helsinki, Nevanlinna and Kataja (1993) have expressed strong reservations about comparing the daily aa and Ak indices. However, if days of missing Ak values are simply ignored, the corresponding linear relationship between the daily values of these indices is 2X71 AE0X19 0X93 AE0X01 ek, with a linear correlation coecient of 0.83. This correlation is highly signi®cant for 4415 daily values (there are 150 missing daily values of Ak in the interval 1 January 1868±30 June 1880), according to Student's t-test (Weatherburn, 1952). However, since the linear relationship between the daily values of aa and Ak yields scaled (``corrected'') values of Ak that are smaller than the original values if ek b 38X7, the actual tabulated (``uncorrected'') values of Ak are used in this study of extremes in geomagnetic activity. Siscoe (1976) elected to use the half-daily aa index for two reasons: (1) it is referred to universal time rather than``storm time'', which avoids any possible ambiguity relating to the actual de®nition of storm time; and (2) a time-scale of 12 h is intermediate between the substorm time-scale $ 1 h and the storm time-scale $ 24 h. Hence the half-daily aa index ®lters out substorm variations but retains the storm variation. The daily aa index is used in this paper because half-daily values of the Ak index are not available; use of half-daily values of the aa index would therefore eliminate solar cycles 9 and 10 from the present investigation. However, the statistical study of the ®rst, second and third largest geomagnetic storms per solar cycle is undertaken separately for solar cycles 11 to 22 and for solar cycles 11 to 19, again using the average daily aa index. The latter set of results permits strict comparisons with the results derived by Siscoe (1976) using the half-daily aa index.

Theory of the statistics of extremes
As noted by Siscoe (1976), in every solar cycle there is a largest, second largest and third largest geomagnetic storm. The magnitudes of these three extreme storms in each solar cycle, as measured by the aa index of geomagnetic activity, can be considered to be stochastic variables with some distribution of probabilities. In the theory of the statistics of extremes, it is shown that the probability distribution functions of extreme values have a common form for a wide range of phenomena (Fisher and Tippett, 1928;Gumbel, 1942Gumbel, , 1954Gumbel, , 1958Court, 1952). This common form applies if the probability that a given observation is as large as a certain magnitude x, or larger than this magnitude, decreases asymptotically as exp(Ax) or faster. In practice, it is not necessary to know the probability distribution for the entire set to verify that this condition holds. It suces to con®rm that the observed extremes comply with the form of the extreme-value distribution function appropriate to this condition (Gumbel, 1942(Gumbel, , 1954. Following such con®rmation, the free parameters in each distribution function can be found by a least-squares ®t to the observations. This procedure yields analytic expressions for the probabilities of the ®rst three extreme values as continuous functions of the magnitude (x). These three analytic probability functions can be used to ®nd the probabilities of extremes beyond the observed ranges.
If U m x denotes the (asymptotic) probability that a given observation of an extreme aa in a solar cycle is less than x, where m 1Y 2 and 3 for the ®rst, second and third largest values, the theory of extremes gives for U m x the following functional forms (Gumbel, 1954(Gumbel, , 1958 U 1 x expÀ expÀy 1 Y U 2 x expÀ2 expÀy 2 1 2 expÀy 2 Y U 3 x expÀ3 expÀy 3 1 3 expÀy 3 9a2 expÀ2y 3 Y 1 where the reduced variate y m is de®ned by the linear relation If the probability function for all aa in a solar cycle were known, the constants m and m could be found directly from this function. Since it is not known, however, m and m must be found by ®tting to the observations. An important requirement in the theory of the statistics of extremes is that the extreme values should be independent. Therefore, no more than one extreme value is selected from any single storm period. In practice, this independence is achieved by ensuring that extreme values are separated in time by at least 30 days (i.e. by more than one synodic solar-rotation period). Following Siscoe (1976), it is assumed that any 80-year modulation of extreme-event amplitudes, produced by the approximately 80-year period in solar activity, is suciently small to be neglected.

Application of the statistics of extremes to geomagnetic storms
Values of the daily aa index for the ®rst, second and third largest maxima in geomagnetic activity for solar cycles 9 to 22 are given in Table 1, together with their dates of occurrence. As noted previously, there are 137 missing daily values of the aa ek index during solar cycles 9 and 10. The three maxima for solar cycle 22 are based on (®nal) daily values of the aa index up to the end of 1993. However, the validity of these maxima has been checked by inspecting the tabular values of the (provisional) daily aa index published in the Solar-Geophysical Data prompt reports (National Geophysical Data Center, Boulder, Colorado), to ensure that no larger values occurred in the interval 1 January 1994±30 April 1996. For solar cycles 11 to 19, all nine dates in Table 1 that specify the ®rst maximum in the daily aa index (per solar cycle) are identical to the corresponding dates presented by Siscoe (1976, cf. his Table 1) for the half-daily aa index. Similarly, seven of the dates of the second maximum are identical, whereas only two of the dates of the third maximum are identical. According to both the``turning-point'' and``phaselength'' tests of randomness (Kendall and Stuart, 1976), each of the three sets of extreme values of the daily aa index presented in Table 1 is not signi®cantly dierent from a random time-series if the hypothesis of randomness is tested at the level 0X01 (the ®rst and second maxima are not signi®cantly dierent from random time-series if tested at the level 0X05). Empirical relationships between U m x and x m 1Y 2Y 3 are derived by ®rst arranging the three random sets of extreme aa values in order of increasing magnitude, as shown in Table 2 for solar cycles 9 to 22. (Although not presented here, similar tables of ordered extreme aa values can be constructed for solar cycles 11 to 22 and 11 to 19.) Then the observed values of the probabilities U m x are given by U m x nx À 1ax, where nx is the ordinal number associated with each of the observed values of x and x is the total number of solar cycles. However, following the procedure recommended by Gumbel (1954) and Krumbein and Lieblein (1956), this de®nition of U m x can be replaced by U m x nxax 1. This procedure is useful for small samples because it enables all x values to be used and it does not alter the results signi®cantly. Thus Eq. 1 can be used to ®nd the values of y m corresponding to each observed value of x, which can then be tested to see if they satisfy the linearity condition implied by Eq. 2.
The results are shown in Fig. 1, where the respective symbols u, R and e are used to denote points de®ning the ®rst, second and third largest geomagnetic storms per solar cycle. Parts a, b and c of the ®gure present results for solar cycles 9 to 22, 11 to 22 and 11 to 19, respectively. Straight-line ®ts to the data points by the method of least squares show that the linear relation- ships are quite well satis®ed, at least for the ®rst and second largest geomagnetic storms (i.e. m 1 and 2). By comparing Fig. 1c with the corresponding ®gure in the paper by Siscoe (1976), it is found that the linear relationship is slightly less well satis®ed for the daily than the half-daily aa index. The discrepancy between the daily and half-daily results for solar cycles 11 to 19, measured in terms of the sums of squares of residuals, is most pronounced for the third largest geomagnetic storm m 3. According to the theory of extremes, the probabilities of the three largest daily aa indices per solar cycle are now determined by Eq. 1, together with the following speci®c linear relationships for solar cycles 9 to 22 The corresponding linear relationships for solar cycles 11 to 22 are Finally, the linear relationships for just solar cycles 11 to 19 are y 1 1X887x À 267X6a100Y y 2 1X055x À 236X7a100Y y 3 0X897x À 176X2a100X 5 Table 3 presents the observed means and AE the observed standard deviations for the ®rst, second and third largest geomagnetic storms in solar cycles 9 to 22 (cf. Table 2), 11 to 22 and 11 to 19. Figure 1 and Table 3 permit comparisons to be made between the dierent results obtained for solar cycles 9 to 22, 11 to 22 and 11 to 19. It should be emphasized again that results for solar cycles 11 to 22 are based entirely on the use of the consistently homogeneous aa index, and extend the results obtained by Siscoe (1976) for solar cycles 11 to 19 by including the three most recent solar cycles. Results for solar cycles 9 to 22 involve the inclusion of two additional solar cycles prior to 1868, which requires the use of the Helsinki Ak index. Although the daily Ak index is intended to provide an extension of the daily aa index backwards in time to 1844 (Nevanlinna and Kataja, 1993), it may not provide a completely homogeneous extension of this latter index.
Visual inspection of the statistical data presented in Fig. 1 and Table 3 suggests that the results for solar cycles 11 to 22 are essentially intermediate between those for solar cycles 9 to 22 and 11 to 19. However, it appears that the noticeably smaller values of the aa index for the largest (and possibly second largest) geomagnetic storms in solar cycles 9 and 10 (cf.

Statistical characteristics of the extreme values
The standard statistical parameters of the reduced variate y m m 1Y 2Y 3, namely the mode, median, mean and standard deviation can be determined from the distribution functions (Gumbel, 1958). Using these standard values of y m , the associated values of x can be derived from Eqs. 3, 4 and 5. Numerical values of these standard statistical parameters for the daily aa index are presented in Table 4 for solar cycles 9 to 22. Pairs of numbers in parentheses are the corresponding numerical values for solar cycles 11 to 22 (®rst number) and 11 to 19 (second number): this abbreviation is adopted throughout the remainder of the paper. Each second value in parentheses can be compared directly with the equivalent numerical value for the half-daily aa index presented in Table 3 of the paper by Siscoe (1976). The numbers in Table 4 indicate that the most probable and average values of the daily aa index for the three largest storms are well separated. This statement is true for the full (9 to 22) and restricted (both 11 to 22 and 11 to 19) number of solar cycles, and is in agreement with the conclusion reached by Siscoe (1976) for the halfdaily aa index. However, the relative dispersion standard deviation /mode) of the daily aa index is essentially independent of m for solar cycles 9 to 22, whereas it increases signi®cantly as m increases from 1 to 3 for solar cycles 11 to 19, as in the case of the half-daily aa index. For solar cycles 11 to 22, the relative dispersion increases signi®cantly as m increases from 1 to 2 but then only increases slightly as m increases from 2 to 3. These results suggest that, as the number of solar cycles increases from 9 to 14, the relative dispersion tends to an almost constant value, at least for m 1 to 3. The relative dispersion is important because it de®nes the number of solar cycles required to have an appreciable probability that the extreme value is very far from the modal value. If the relative dispersion is comparatively small, a larger number of solar cycles is required. Conversely, if the relative dispersion is comparatively large, a smaller number of solar cycles suces.
The mean and standard deviation in Table 4 are for an in®nite number of solar cycles; the expected mean and standard deviation are smaller for a ®nite number of cycles. For example, in the case m 1 for 14 (12,9) cycles, the mean is 270.4 (287.4, 293.6) and the standard deviation is 66.2 (53.7, 49.2). These expected values for 14 (12, 9) solar cycles are in remarkably good agreement with the observed values presented in Table 3, namely 270.5 (287.4, 293.6) and 65.2 (53.0, 49.6).

Probabilities of the three largest geomagnetic storms per solar cycle
Equations 1±5 can be used to determine the (asymptotic) probabilities of the three largest geomagnetic storms per solar cycle as functions of the aa index. Figure 2a±c shows plots of the probability U m x (m 1Y 2 and 3) that a given observation of an extreme geomagnetic storm in a solar cycle is less than aa. The continuous, dashed and dotted curves refer to the ®rst m 1, second m 2 and third m 3 largest geomagnetic storms. Following the convention adopted for Fig. 1, a, b and c of Fig. 2 refer, respectively, to solar cycles 9±22, 11±22 and 11±19.
For the ®rst largest geomagnetic storms m 1, the continuous curves in b and c of Fig. 2 show a very similar variation of Ux with increasing . Conversely, the continuous curve in Fig. 2a is signi®cantly dierent to the continuous curves in b and c, particularly for the lower aa values `300. This discrepancy almost certainly arises from the noticeably lower observed values of the ®rst largest geomagnetic storms for solar cycles 9 and 10 (cf. Table 1). For the second largest geomagnetic storms m 2, the dashed curves in Fig. 2a±c exhibit a rather more gradual change in the variation of Ux with aa as the number of solar cycles increases from 9 to 14. Finally, for the third largest geomagnetic storms m 3, the dotted curves in Fig. 2a±c reveal only minor dierences in the variation of Ux with aa as the number of solar cycles increases from 9 to 14. Notwithstanding the detailed dierences between the comparable curves presented in Fig. 2a±c, the general trend of the variation of U m x (m 1Y 2 and 3) with increasing aa does not depend critically on the number of solar cycles included in the statistical analysis. The probabilities presented in Fig. 2 can be used to calculate the expected ranges of the ®rst, second and third largest geomagnetic storms (m 1Y 2 and 3) for a given number of solar cycles. The growth in the expected ranges of extreme values of the daily aa index as a function of the number of cycles can be estimated from the equations 1 x 1 À Ux À1 , the expected number of cycles required to have one cycle with an extreme equal to or greater than x , and 2 x Ux À1 , the expected number of cycles required to have one cycle with an extreme less than x (Gumbel, 1958). This procedure is strictly valid, of course, only if the number of cycles is large enough to eliminate the possibility of a biased sample. The functions 1 x and 2 x are plotted in Fig. 3 for the three largest extremes m 1Y 2Y 3: Fig. 3a refers to solar cycles 9 to 22, Fig. 3b refers to solar cycles 11 to 22 and Fig. 3c refers to solar cycles 11 to 19. In each case the curves labelled 1 and 2 meet at the point where 1 2 2, which is located at the median value of the extreme daily aa index.
The continuous, dashed and dotted curves in Fig. 3a can be interpreted as implying that for 14 solar cycles (9 to 22) all but two of the observed extreme values of the daily aa index are expected to be con®ned to the ranges 173X4` 1`4 07X6Y 134X4` 2`3 52X1 and 107X7` 3`2 74X9, where the subscripts denote values of m. For each of the three ranges m 1Y 2Y 3, one observed extreme is expected to lie outside the range to the right and one outside to the left. This theoretical expectation is not realized precisely for the observed extremes presented in Table 2 because these extremes do not lie exactly on the least-squares ®ts to the data points, which are used to derive Fig. 3a, as is clear from Fig. 1a. As the number of cycles increases, however, the ranges indicated in Fig. 3a become increasingly better representations of the true expected ranges. For a group of 100 solar cycles, the corresponding statistical ranges for the continuous, dashed and dotted curves are 136X9` 1`5 38X6Y 93X3` 2`4 54X4 and 73X5 3`3 46X9. Similarly, the equivalent statistical ranges for observations over 12 solar cycles (11 to 22) are 176X5` 1`5 11X1Y 114X8` 2`4 55X0 and 77X0` 3`3 58X8 (cf. Fig. 3b); the equivalent statistical ranges for observations over 9 solar cycles are 186X7` 1`5 11X4Y 123X0` 2`4 83X1 and 61X3` 3`3 91X2 (cf. Fig. 3c).
If the expected dilatation for a given number of cycles is measured by the ratio of the largest to the smallest value in the range, then for 100 cycles the ratios are 3.9 for 1 , 4.9 for 2 and 4.7 for 3 . These ranges and dilatations are based on measured values of the daily index for solar cycles 9 to 22. The corresponding dilatations based on measured values of the daily index for solar cycles 11 to 22 are 2.9, 4.0 and 4.7. Similarly, the dilatations based on measured values of the daily index for just solar cycles 11 to 19 are 2.7, 3.9 and 6.4. These latter three values can be compared directly with the equivalent values derived implicitly by Siscoe (1976) for the half-daily index, namely 2.2, 3.4 and 7.3. Therefore, the expected dilatations of both the Fig. 3a±c. Curves de®ning the expected number of solar cycles required to ®nd one cycle with an extreme value of equal to or greater than the abscissa ( 1 branch) and to ®nd one cycle with an extreme value of less than the abscissa ( 2 branch) for the three largest m 1Y 2 and 3 geomagnetic storms per solar cycle: a solar cycles 9±22, b solar cycles 11±22 and c solar cycles 11±19. The continuous, dashed and dotted curves refer to the ®rst m 1, second m 2 and third m 3 largest geomagnetic storms, respectively daily and half-daily indices for solar cycles 11 to 19 vary signi®cantly as m increases from 1 to 3. Conversely, the variation of the expected dilatations of the daily index as m increases is appreciably less for all fourteen solar cycles (9 to 22) than for the subset of nine solar cycles (11 to 19). An intermediate situation prevails for twelve solar cycles (11 to 22), although the behaviour as m increases is probably closer to that for nine solar cycles (11 to 19). This last result again suggests that the noticeably smaller values of the index for the largest (and possibly second largest) geomagnetic storms in solar cycles 9 and 10 may be exerting a signi®cant in¯uence on the statistical results.
More generally, the curves in Fig. 3a, which are based on the extreme values of the daily index for solar cycles 9 to 22, indicate that the expected range decreases signi®cantly as m increases from 1 to 3. The curves in Fig. 3b, which are based on the extreme values of the daily index for solar cycles 11 to 22, indicate that the expected range again decreases signi®cantly as m increases from 1 to 3, although there is a very small increase as m increases from 1 to 2. Conversely, the curves in Fig. 3c, which are based on the extreme values of the daily index for solar cycles 11 to 19, indicate that the expected range varies less as m increases but is again greatest for m 2. Therefore, for the interval 1844±1993 (i.e. solar cycles 9 to 22), the expected ranges of the daily index for the ®rst, second and third largest geomagnetic storms per solar cycle decrease monotonically in magnitude. This result is the exact opposite of the one found by Siscoe (1976) for the expected ranges of the half-daily aa index during the shorter interval 1868±1967 (i.e. for solar cycles 11 to 19). The expected ranges of the daily aa index for this shorter interval exhibit a more neutral behaviour.
The statistical procedures discussed in this section provide only a rather limited opportunity of testing the predictive accuracy of extreme-value statistics, as applied to the three largest geomagnetic storms per solar cycle. Based on observations of the three largest values of the daily index per solar cycle for the 100-year interval 1868±1967 (i.e. solar cycles 11±19; cf. Table 1), Eqs. 1 and 5 can be used to estimate the expected ranges for 12 and 14 solar cycles (cf. Fig. 3c) The theoretical expected ranges for the ®rst, second and third largest geomagnetic storms per solar cycle are 219X4` 1`3 97X0Y 168X3` 2`3 72X6 and 106X9` 3`2 96X9, for 12 solar cycles; and 216X2` 1`4 05X5Y 164X0 2`3 81X1 and 102X7` 3`3 04X3, for 14 solar cycles. In the case of the largest geomagnetic storms [viz.
1 ] for solar cycles 11±22, one observed extreme (5 August 1972) is less than 219.4, as required by the theory, but no observed extreme is greater than 397.0. In the case of the largest geomagnetic storms for solar cycles 9±22, three observed extremes (20 December 1847, 9 April 1858and 5 August 1972 are less than 216.2 and no observed extreme is greater than 405.5 This last result suggests once again that the noticeably smaller values of the index for the ®rst largest geomagnetic storms in solar cycles 9 and 10 may be exerting a signi®cant in¯uence on the statistical results. As noted previously, however, the theoretical expectation is not realized in practice even for the observed extremes of solar cycles 11 to 19 (cf. Table 1) because these extremes do not lie exactly on the least-squares ®ts to the data points, as is clear from Fig. 1c. Moreover, considerable circumspection must be exercised in any attempt to use the statistics of extremes to predict the largest geomagnetic storms in the next solar cycle. The great merit of the theory of extremes is that it enables increasingly better estimates to be made of the true expected ranges as the time-interval for the prediction becomes increasingly long compared with the timeinterval for which observations exist. On the basis of the results presented in this paper, it can be claimed with con®dence that there is at least a 99% probability that the daily aa index satis®es the condition `550 for the largest geomagnetic storm in 100 solar cycles, as implied by Fig. 3a±c. The corresponding condition for the largest geomagnetic storm in 500 solar cycles (i.e. approximately ®ve-and-a-half millennia) is `650.

Conclusions and discussion
A previous study of the largest geomagnetic storms per solar cycle for the 100-year interval 1868±1967 (Siscoe, 1976) is extended to the 150-year interval 1844±1993. The intensity of a geomagnetic storm is classi®ed in this paper by the magnitude of the daily index rather than the half-daily index. The latter index was preferred by Siscoe (1976) because a time-scale of 12 h is intermediate between the substorm time-scale $ 1 h and the storm time-scale $ 24 h. However, inclusion of the essentially equivalent Helsinki magnetic activity index ek (Nevanlinna and Kataja, 1993), which is available only on a daily basis for the interval 1844±1868, inevitably restricts the present study to the daily aa index of magnetic activity. Since the ek index may not provide an entirely homogeneous extension of the index, the statistical analysis is performed separately for twelve solar cycles (1868±1993), as well as nine solar cycles (1868±1967).
Many other studies of geomagnetic storms and magnetic activity have been undertaken using dierent magnetic indices and classi®cation criteria (e.g. Gosling et al., 1991;Tsurutani et al., 1992;Taylor et al., 1994;Silbergleit, 1996Silbergleit, , 1997. However, these studies usually relate geomagnetic storms to satellite measurements of various physical properties of the solar wind and interplanetary magnetic ®eld. Such studies are necessarily restricted to just the last few solar cycles, whereas the present investigation uses the only magnetic index that spans 14 solar cycles. An important conclusion of the application of extreme-value statistics to the daily index for the longer interval (1844±1993) is that the relative dispersion (= standard deviation/mode) does not vary signi®cantly between the ®rst m 1, second m 2 and third m 3 largest geomagnetic storms per solar cycle (cf. Table 4). Over the shorter time-interval (1868±1967), however, the relative dispersion varies signi®cantly (viz. increases monotonically as m increases) for both the daily and half-daily indices. Conversely, for the longer interval (1844±1993), the expected ranges of the daily index for the ®rst, second and third largest geomagnetic storms per solar cycle decrease monotonically in magnitude as m increases (cf. Fig. 3a). This latter result is the exact opposite of the result derived by Siscoe (1976) for the shorter interval (1868±1967), during which the corresponding expected ranges of the half-daily index increase monotonically in magnitude as m increases. The expected ranges of the daily index for this shorter interval (1868±1967) exhibit a more neutral behaviour; indeed, the largest expected range occurs for m 2, as it does in the case of the intermediate interval (1868±1993).
The statistical results derived in this paper can be used to place an upper limit on the largest geomagnetic storm that is likely to occur in a much longer period of time, which may be imagined to extend either backwards into the past or forwards into the future. For example, there is at least a 99% probability that the daily index satis®es the condition `550 for the largest geomagnetic storm in 100 solar cycles. The corresponding condition for 500 solar cycles (about 5500 years) is `650. Such predictions of the largest geomagnetic storm that might occur in the future may be of some value to experimentalists designing sensitive instruments to be¯own on spacecraft exploring the near-Earth environment. Similar predictions can be made for the lower limit of the largest geomagnetic storm in 100 (or 500) solar cycles. Siscoe (1976) used a lower limit appropriate to the half-daily index to infer that it is unlikely that any solar cycle in recent history would have passed without the occurrence of at least one geomagnetic storm that produced auroral displays visible at mid-latitudes, provided the statistics (for the longer interval) were the same as those for nine solar cycles (viz. 11 to 19). An analogous argument is adduced at the end of this section in a discussion of the remarkable conjugate auroral observations during the night of 16 September 1770 (Willis et al., 1996).
By comparing the results obtained in this paper with those presented by Siscoe (1976), it is clear that results derived by applying the statistics of extremes to the ®rst, second and third largest geomagnetic storms per solar cycle depend on the actual time-interval considered. The results also depend, though to a lesser extent, on whether the daily or half-daily index is used as a measure of the intensity of a geomagnetic storm. Both these dependencies are to be expected for a statistical analysis based on a relatively small number of solar cycles. Nevertheless, the main scienti®c conclusions reached by Siscoe (1976) are con®rmed by the present investigation. In particular, it remains unlikely that any solar cycle in recent history would have passed without one geomagnetic storm that produced auroral displays visible at mid-latitudes, provided the statistics were the same as those for the last 14 solar cycles.
In this context, it is instructive to apply the preceding statistical analysis to observations of the aurora australis on the night of 16 September 1770, recorded by Banks (Beaglehole, 1962) and Parkinson (1773) on board HMS Endeavour during the ®rst voyage of Captain James Cook to Australia (Eather, 1980). Both recorded descriptions of the aurora australis refer to a red light or glow in the southern sky accompanied by rays, or stripes, of a brighter coloured light extending directly upwards. This auroral display was observed at around 23:00 LT, at which time HMS Endeavour was about 10 S of the equator between Timor and the island of Savu. Observations of the aurora borealis on the same night, and the two following nights, have been recorded in Chinese provincial histories, which refer to auroral displays seen in the northern sky from several sites in China about 40 N of the equator (Willis et al., 1996). These observations provide the earliest example yet known of conjugate auroral sightings and they probably occurred during an intense geomagnetic storm.
According to the statistical analysis given here, 164` 1`4 35 for a group of 21 solar cycles (cf. Fig. 3a), which corresponds to (backwards) extrapolation over the time-interval 1767±1993 (McKinnon, 1987). Following the reasoning adduced by Siscoe (1976), 164 is taken to represent the expected minimum largest geomagnetic storm for 21 cycles. This particular value of is almost equal to the values for the geomagnetic storms of 24 September 1847, 9 April 1858 and 26 May 1967 (cf. Table 1). As noted in the catalogue published by Angot (1896), the storm of 24 September 1847 produced an aurora that was visible from Greenwich 51X29 NY 0X00 E, while the storm of 9 April 1858 produced an aurora that was visible from Senftenberg 51X31 NY 14X01 E and Breslau (Wrocøaw; 51X05 NY 17X00 E, amongst other European cities. The more recent storm of 26 May 1967, which is understandably much better documented (e.g. Findlay et al., 1969), produced an aurora that was visible from many locations in the United States of America and Canada. One observation from as far south as Alabama % 35 N, and several nearly overhead observations from Washington DC 38X55 NY 77X00 W, were reported. Therefore, since there is a clear correspondence between the latitude of overhead auroral displays and the magnitude of the index (Legrand and Simon, 1989), it seems highly likely that the remarkable conjugate auroral observations on the night of 16 September 1770 (Willis et al., 1996) were associated with an intense geomagnetic storm. This conclusion is corroborated by the fact that auroral displays were also observed from several places in both Europe (Angot, 1896) and Japan (Matsushita, 1956;Willis et al., 1996) on the following night.
Topical Editor K.-H. Glaûmeier thanks M. Lester and R. Friedel for their help in evaluating this paper.