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NASA/TP—2006–214711

An Examination of Selected GeomagneticIndices in Relation to the Sunspot CycleRobert M. Wilson and David H. HathawayMarshall Space Flight Center, Marshall Space Flight Center, Alabama

December 2006

National Aeronautics andSpace AdministrationIS20George C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama35812

https://ntrs.nasa.gov/search.jsp?R=20070021477 2018-05-07T08:13:21+00:00Z

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�

NASA/TP—2006–214711

An Examination of Selected Geomagnetic Indices in Relation to the Sunspot CycleRobert M. Wilson and David H. HathawayMarshall Space Flight Center, Marshall Space Flight Center, Alabama

December 2006

Nat�onal Aeronaut�cs andSpace Adm�n�strat�on

Marshall Space Fl�ght Center • MSFC, Alabama 35812

��

Ava�lable from:

NASA Center for AeroSpace Informat�on7115 Standard Dr�ve

Hanover, MD 21076 –1320301– 621– 0390

Th�s report �s also ava�lable �n electron�c form at<https://www2.st�.nasa.gov>

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TABLE OF CONTENTS

1. INTRODUCTION ......................................................................................................................... 1

2. RESULTS AND DISCUSSION .................................................................................................... 3 2.1 Geomagnet�c Ind�ces: aa, Ap, and NDD ................................................................................ 3 2.2 aa (adjusted) Geomagnet�c Index ........................................................................................... 10 2.3 S�ngle-Var�ate and B�var�ate F�ts Based on Parametr�c M�n�mum Values ............................. 16 2.4 Cycl�c Averages and Sums ...................................................................................................... 19 2.5 Apmax, ND (Ap ≥ 100) and ND (Ap ≥ 200) .............................................................................. 24 2.6 Est�mat�ng Ap, NDD, and Apmax Pr�or to 1932 ..................................................................... 29 2.7 Epoch Analyses ....................................................................................................................... 32

3. CONCLUSION .............................................................................................................................. 36

REFERENCES .................................................................................................................................... 39

�v

LIST OF FIGURES

1. Var�at�on of annual averages of sunspot number, R (panel (a)); aa-geomagnet�c �ndex, aa (panel (b)); Ap-geomagnet�c �ndex, Ap (panel (c)); and number of d�sturbed days, NDD (panel (d)). See text for deta�ls .................................................... 4 2. Scatterplots of aa versus R (panel (a)); Ap versus R (panel (b)); and NDD versus R (panel (c)). All values of the geomagnet�c �nd�ces l�e on or above theidentifiedregressionlines ............................................................................................. 5

3. Var�at�on of the �nterplanetary components aaI (panel (a)); ApI (panel (b)); and NDDI (panel (c)). The th�ck l�nes are 2-yr mov�ng averages of the res�dual �nterplanetary components ................................................................................................. 6

4. H�stograms of the frequency of occurrences for E(aaImax) (panel (a)) and E((aaImax (last)) (panel (b)), relat�ve to the elapsed t�me �n years from E(Rmax) ..................................................................................................................... 7

5. Scatterplots of Rm�n (cycle n + 1) versus selected max�mum �nterplanetary components for cycle n (panels (a)–(c)) and Rmax (cycle n + 1) versus selected max�mum �nterplanetary components for cycle n (panels (d)–(f)). See text for deta�ls .............................................................................................................. 8

6. Scatterplots of Rm�n (cycle n + 1) versus selected last-occurr�ng max�mum �nterplanetary components for cycle n (panels (a)–(c)) and Rmax (cycle n + 1) versus selected last-occurr�ng max�mum �nterplanetary components for cycle n (panels (d)–(f)). See text for deta�ls ................................................................................... 9

7. Scatterplots of Rm�n2 (cycle n + 1) versus selected 2-yr mov�ng averages of the max�mum �nterplanetary components for cycle n (panels (a)–(c)) and Rmax2 (cycle n + 1) versus 2-yr mov�ng averages of the max�mum �nterplanetary components for cycle n (panels (d)–(f)). See text for deta�ls ............................................. 10

8. Var�at�on of selected parametr�c d�fferences, 1901–2000. See text for deta�ls .................. 11

9. Var�at�on of selected parametr�c d�fferences, 1932–2000. See text for deta�ls .................. 12

10. Scatterplot of aa (adjusted) versus R .................................................................................. 13

11. Var�at�on of aaI (adjusted). The th�ck l�ne �s the 2-yr mov�ng average of the res�dual �nterplanetary component ........................................................................... 14

12. Scatterplots of Rm�n2 (cycle n + 1) versus aaI (adjusted)max2 (cycle n) (panel (a)) and Rmax2 (cycle n + 1) versus aaI (adjusted)max2 (cycle n) (panel (b)). See text for deta�ls .............................................................................................................. 15

v

LIST OF FIGURES (Continued)

13. Cycl�c var�at�on of selected parameters. Not�ce the strong secular �ncreases �n Rm�n, Rmax, and aa ...................................................................................................... 17

14. Scatterplots of Rmax versus selected m�n�mum values of the parameters ........................ 18

15. Scatterplots of Rmax(observed)versusselectedbivariatefits.Seetextfordetails .......... 19

16. Var�at�on of cycl�c averages of selected parameters .......................................................... 20

17. Scatterplots of cycl�c averages of selected parameters aga�nst Rm�n (left panels) and Rmax(rightpanels).Thenumberedfilledcirclesrefertoindividualsunspotcycles........ 22

18. Scatterplots of add�t�onal cycl�c averages and sums aga�nst Rm�n (left panels) and Rmax (r�ght panels) ..................................................................................................... 23

19. Scatterplot of NSSC versus R ............................................................................................ 24

20. Var�at�on of the annual max�mum da�ly Ap value, Apmax ................................................ 25

21. Scatterplot of Apmax versus R ........................................................................................... 25

22. Scatterplot of Apmax versus Ap ......................................................................................... 26

23. Sem�-annual var�at�on of Apmax ....................................................................................... 27

24. H�stogram of the frequency of occurrences of E(Apmax) relat�ve to the elapsed t�me �n years relat�ve to E(Rmax) ...................................................................................... 27

25. Cycl�c var�at�on of Apmax (panel (a)) and ND (Ap ≥ 100) and ND (Ap ≥ 200) (panel (b)) ........................................................................................................................... 28

26. Scatterplots of Apmax versus Rm�n and Rmax (panels (a) and (b), respect�vely) and ND (Ap ≥ 100) versus Rm�n and Rmax (panels (c) and (d), respect�vely) ................... 29

27. Comparisonofcycle23parametricvalues(filledcircles)andcyclicaverages (cycles 11–22 or 16–22) based on epoch analyses us�ng E(Rmax) as the temporal marker. See text for deta�ls ................................................................................................ 33

28. Comparisonofcycle23parametricvalues(filledcircles)andcyclicaverages (cycles 11–22 or 16–22) based on epoch analyses us�ng E(Rmax)n + 1. See text for deta�ls ........................................................................................................................... 35

29. Parametr�c values for 2005 and 2006 ................................................................................ 38

v�

LIST OF TABLES

1. Selected s�ngle-var�ate regress�ons for 1932–2005 (n = 74) ................................................... 30

2. Selected b�var�ate regress�ons for 1932–2005 (n = 74) ........................................................... 30

3. Parametr�c est�mates for 1868–1931 ...................................................................................... 31

v��

LIST OF ACRONYMS AND NOMENCLATURE

aa ant�podal geomagnet�c �ndex

<aa> cycl�c average of aa

aa90 90-percent �nterval for aa

<aa>90 90-percent �nterval for <aa>

aa (adjusted) aa adjusted value, equal to aa (observed) + 3 for aa values pr�or to 1957

<aa> (adjusted) cycl�c average of adjusted aa

aaCHL/FRD the proxy aa �ndex based on two 3-hr �ntervals (0–6 UT) from Cheltenham and Freder�cksburg

aaI �nterplanetary component of aa (= aa – aaR)

aaI (adjusted) �nterplanetary component of adjusted aa

aaI (adjusted) max2 max�mum value of 2-yr mov�ng average of aaI (adjusted)

aaImax max�mum value of aaI

aaImax2 max�mum value of 2-yr mov�ng average of aaI

aaImax (last) max�mum value of aaI (last), the last peak �n the decl�ne of the cycle

aam�n m�n�mum yearly average aa value �n the v�c�n�ty of cycle m�n�mum

aam�n (adjusted) m�n�mum yearly average adjusted aa value

aaR sunspot cycle component of aa

Ap da�ly equ�valent planetary ampl�tude geomagnet�c �ndex

<Ap> cycl�c average of Ap

Ap (adjusted) adjusted Ap

ApI �nterplanetary component of Ap (= Ap – ApR)

v��

LIST OF ACRONYMS AND NOMENCLATURE (Continued)

Apmax max�mum value of Ap

ApImax max�mum value of ApI

ApImax2 max�mum value of 2-yr mov�ng average of ApI

ApImax (last) max�mum value of ApI (last), the last peak �n the decl�ne of the cycle

Apm�n m�n�mum yearly average Ap value

ApR sunspot cycle component of Ap

bv b�var�ate

cl confidencelevel

dmax max�mum da�ly value of R dur�ng year

E(aam�n) epoch of aam�n for cycle n

E(aam�n)n + 1 epoch of aam�n for cycle n + 1

E(aaI max) epoch of aaI max

E(aaI max (last)) epoch of aaI max (last)

E(Apmax) epoch of Apmax

E((Apmax)m�n)n + 1 epoch of (Apmax)m�n for cycle n + 1

E(Apm�n)n + 1 epoch of Apm�n for cycle n + 1

E(NDDm�n)n + 1 epoch of NDDm�n for cycle n + 1

E(NSSCm�n)n + 1 epoch of NSSCm�n for cycle n + 1

E(Rmax) epoch of sunspot max�mum ampl�tude

E(Rmax)n + 1 epoch of sunspot max�mum ampl�tude for cycle n + 1

E(Rm�n) epoch of sunspot m�n�mum ampl�tude

E(Rm�n)n + 1 epoch of sunspot m�n�mum ampl�tude for cycle n + 1

�x


IHV Inter-Hour Var�ab�l�ty (geomagnet�c �ndex)

K geomagnet�c �ndex scaled sem�logar�thm�cally

n sunspot cycle number; sample s�ze

ND (Ap ≥ 100) number of days when Ap ≥ 100

ND (Ap ≥200) numberofdayswhenAp ≥ 200

NDD number of d�sturbed days (days when Ap ≥ 25)

<NDD> cycl�c average of NDD

NDDI �nterplanetary component of NDD (= NDD – NDDR)

NDDI max max�mum value of NDDI

NDDI max2 max�mum value of 2-yr mov�ng average of NDDI

NDDI max (last) max�mum value of NDDI (last)

NDDm�n m�n�mum yearly NDD

NDDR sunspot cycle component of NDD

NH Northern Hem�sphere

NSD number of spotless days

NSSC number of sudden storm commencements

<NSSC> cycl�c average of NSSC

<NSSC>90 90-percent �nterval of <NSSC>

R sunspot number

<R> cycl�c average of R

R90 90-percent �nterval of R

<R>90 90-percent �nterval of <R>

r coefficientofcorrelation

r2 coefficientofdetermination

x


RM max�mum ampl�tude (smoothed monthly mean sunspot number)

RM90 90-percent �nterval for RM

Rm m�n�mum ampl�tude (smoothed monthly mean sunspot number)

Rm90 90-percent �nterval for Rm

Rmax max�mum yearly average sunspot number (max�mum ampl�tude)

Rmax2 max�mum ampl�tude of 2-yr mov�ng average of R

Rmax90 90-percent �nterval for Rmax

Rm�n m�n�mum yearly average sunspot number (m�n�mum ampl�tude)

Rm�n2 m�n�mum ampl�tude of 2-yr mov�ng average of R

Rm�n90 90-percent �nterval for Rm�n

RY.12 correlationcoefficientforbivariatefit

sd standard dev�at�on

se standard error of est�mate

SH Southern Hem�sphere

SY.12 standarderrorofestimateforbivariatefit

UT un�versal t�me

x �ndependent var�able

y dependent var�able

Δ1, 2, 3, 4, 5 d�fferences 1, 2, 3, 4, 5

ΣNDD sum of NDD over solar cycle

ΣNSSC sum of NSSC over solar cycle

ΣNSSC90 90-percent �nterval for <NSSC>

1

TECHNICAL PUBLICATION

AN EXAMINATION OF SELECTED GEOMAGNETIC INDICES IN RELATION TO THE SUNSPOT CYCLE

1. INTRODUCTION

Over the years, a number of geomagnet�c �nd�ces have been dev�sed to descr�be the var�at�on of thegeomagneticfieldinresponsetothechangingconditionsofthesolarwindatEarth.1,2 A few of these �nd�ces, �n fact, have been shown to rel�ably pred�ct the strength of the follow�ng sunspot cycle several years �n advance.3–14 It �s th�s aspect that w�ll be exam�ned more closely �n th�s Techn�cal Publ�cat�on.

Th�s study �s d�v�ded �nto seven results sect�ons. Sect�on 2.1 exam�nes the observed yearly average values of the aa and Apgeomagneticindices,aswellasthenumberofdisturbeddays(NDD),definedasthose days when the Ap �ndex equals or exceeds 25, all �n relat�on to annual sunspot number (R). The aa �ndex was �ntroduced by Mayaud �n the early 1970s15,16andisdefined as the average of 3-hr K �nd�ces convertedtotheamplitudeofthefieldattwonearlyantipodalobservatories(inEnglandandAustralia).The aa �ndex, as der�ved by Mayaud, �s ava�lable from 1868 to the present. Although Nevanl�nna and Kataja17 have prov�ded an extens�on for the aa �ndex back to 1844 us�ng Hels�nk� magnet�c records, these extended values have not been used �n the present study. Ap, or the da�ly equ�valent planetary ampl�tude, �s a da�ly �ndex of geomagnet�c act�v�ty that �s determ�ned us�ng a l�near scale rather than the quas�-logar�thm�c scale of the K �nd�ces. Values for �t and NDD are ava�lable from 1932 to the present. (Yearly values of R, aa, and Ap can be found onl�ne at ftp://ftp.ngdc.noaa.gov/STP/.18)

Sect�on 2.2 exam�nes the effects of suggested changes to the observed aa record. Svalgaard, Cl�ver, and Le Sager19 der�ved a new geomagnet�c �ndex called the Inter-Hour Var�ab�l�ty (IHV) �ndex for �nvest�gat�ons of the long-term var�ab�l�ty of the solar w�nd-magnetosphere system and used �t to reconstruct the observed aa record. For the �nterval 1957–2000, they found yearly averages of the�r proxy aa �ndex (based on observat�ons at Cheltenham/Freder�cksburg dur�ng the two 3-hr per�ods between 0 and 6 hr UT) and the IHV �ndex to be v�rtually the same, wh�le for the �nterval preced�ng 1957, there was cons�derable d�fference between the �nd�ces. Hence, they concluded that values of the observed aa �ndex pr�or to 1957 m�ght be �naccurate.

Sect�on 2.3 exam�nes annual m�n�mum values of aa, Ap, and NDD (aam�n, Apm�n, and NDDm�n, respect�vely) for each cycle �n relat�on to Rmax (the annual max�mum ampl�tude of the sunspot cycle), bothassingle-variateandbivariatefits(thelatterfitalsousingRm�n, the annual m�n�mum ampl�tude of thesunspotcycle).Single-variateandbivariatefitspreviouslyhavebeenshowntoreliablypredictthesizeof the ongo�ng sunspot cycle some 2 to 3 years �n advance.5

2

Sect�on 2.4 exam�nes cycl�c averages (m�n�mum-to-m�n�mum based on R) of aa, Ap, NDD, and NSSC (the number of sudden storm commencements), denoted <aa>, <Ap>, <NDD>, and <NSSC>, respect�vely, and the cycl�c sums (the total number recorded over the sunspot cycle) of NDD and NSSC, denoted ΣNDD and ΣNSSC, respect�vely, all �n relat�on to the cycl�c average of R, denoted <R>, and Rm�n and Rmax.

Sect�on 2.5 exam�nes Apmax and the number of days when Ap equals or exceeds 100 and 200, denoted ND (Ap ≥ 100) and ND (Ap ≥ 200), respect�vely.

Section2.6 identifies single-variate andbivariate regressionsofAp, NDD, and Apmax aga�nst aa and aa and R, respect�vely, for the �nterval of 1932–2005. These regress�ons are then used to prov�de est�mates of Ap, NDD, and Apmax pr�or to 1932.

Sect�on 2.7 looks at epoch analyses of sunspot number and the var�ous geomagnet�c parameters (aa, Ap, NDD, Apmax, and NSSC) relat�ve to E(Rmax) and E(Rm�n) to determ�ne when onset for cycle 24 should be expected.

3

2. RESULTS AND DISCUSSION

2.1 Geomagnetic Indices: aa, Ap, and NDD

F�gure 1 d�splays the annual var�at�on of sunspot number R �n (panel (a)), aa (panel (b)), Ap (panel (c)), and NDD (panel (d)). For R and aa, values are g�ven for 1868–2005, wh�le for Ap and NDD, values are g�ven for 1932–2005. The lower numbers refer to sunspot cycles 11–23. The th�n vert�cal l�nes refer to the occurrences of the sunspot m�n�mum year for each of the sunspot cycles.

Inspectionoffigure1revealstwoimportantaspectsofsolar/geomagneticactivity:First,therehasbeen a general r�se �n R and aa over the years and second, wh�le R usually �s s�ngle peaked, aa (and Ap and NDD)generallyhastwoormorepeaks,withthefirstbeingcloselyassociatedwiththerising/maximumphase of the sunspot cycle and the other(s) occurr�ng dur�ng the decl�n�ng port�on of the sunspot cycle, often occurr�ng just pr�or to the onset of the new cycle. Hathaway, W�lson, and Re�chmann20 prev�ously have shown that the secular trend �n R �s long-term, extend�ng from the Maunder m�n�mum. Also, Cl�lverd et al.,21 us�ng two long-runn�ng European stat�ons (Sodankylä and N�emegk), have reconstructed the long-term aa �ndex and concluded that the trend �n aa �s real, �nferr�ng a long-term �ncrease �n solar coronal magneticfieldstrength.Indeed,WilsonandHathaway22 have reported that, on the bas�s of 10-yr mov�ng averages, R and aa are h�ghly correlated (r = 0.933).

In order to expla�n the later-occurr�ng peaks �n the aa �ndex (wh�ch nearly always have been the largest of the cycle), Feynman23 suggested that two components compr�se the aa �ndexone that �s �n phase and correlated d�rectly w�th the sunspot cycle and the other (the res�dual) that �s out of phase and assoc�ated w�th �nterplanetary d�sturbances from the Sun (h�gh-speed solar w�nd streams and the l�ke24). F�gure 2 plots aa versus R (panel (a)), Ap versus R (panel (b)), and NDD versus R(panel(c)).Theidentifiedl�nes �n each panel are those that allow for a determ�nat�on of the �nferred sunspot cycle component for the geomagnet�c �nd�ces by pass�ng a l�ne through the two lowest po�nts. Hence, all geomagnet�c �ndex values l�e e�ther on or above these l�nes. By subtract�ng th�s component from the observed value of the geomagnet�c �ndex, one �nfers the value of the res�dual �nterplanetary component. (The techn�que employed here �s sl�ghtly d�fferent from that employed �n Hathaway, W�lson, and Re�chmann13 and Hathaway and W�lson.14)

4

1880 1900 1920 1940 1960 1980 2000

200

100

11 12 13 14 15 16 17 18 19 20 21 22 23

11 12 13 14 15 16 17 18 19 20 21 22 23

17 18 19 20 21 22 23

17 18 19 20 21 22 23

Ann

ual M

ean

Su

nspo

t Num

ber,

R

40

20

0

0

Ann

ual M

ean,

aa

Geo

mag

netic

Inde

x, a

a

30

10

20

Ann

ual M

ean,

Ap

Geo

mag

netic

Inde

x, A

p

150

50

0

0

100

Ann

ual N

umbe

r of

Dis

turb

ed D

ays,

ND

D

Calendar Year

1880(b)

(c)

(d)

(a)

1900 1920 1940 1960 1980 2000

1940 1960 1980 2000

1940 1960 1980 2000

F�gure 1. Var�at�on of annual averages of sunspot number, R (panel (a)); aa-geomagnet�c �ndex, aa (panel (b)); Ap-geomagnet�c �ndex, Ap (panel (c)); and number of d�sturbed days, NDD (panel (d)). See text for deta�ls.

5

aaR 5.77789 0.08229R

1868–2005

1932–2005

1932–2005

1000 200

1000 200

1000 200

20

0

40

Annual Mean Sunspot Number, R

Ann

ual M

ean

Geo

met

ric In

dex,

aa

ApR 7.07302 0.02605R

NDDR 7.92684 0.12337R

15

0

30

Ann

ual M

ean

Geo

met

ric In

dex,

Ap

100

0

200

Ann

ual N

umbe

r of D

istu

rbed

Day

s, N

DD

(a)

(b)

(c)

F�gure 2. Scatterplots of aa versus R (panel (a)); Ap versus R (panel (b)); and NDD versus R (panel (c)). All values of the geomagnet�c indiceslieonorabovetheidentifiedregressionlines.

6

F�gure 3 plots the res�dual �nterplanetary components of aaI (panel (a)), ApI (panel (b)), and NDDI (panel (c)), all plotted as the th�n jagged l�nes. The th�cker, smoother l�nes are 2-yr mov�ng averages25 of the res�duals (us�ng a we�ght�ng of 1:2:1), and, as before, the lower numbers refer to the �nd�v�dual sunspot cycles 11–23 and the th�n vert�cal l�nes mark the occurrences of the sunspot m�n�mum years.

10

0

20

30

1880(a)

(b)

(c)

11 12 13 14 15 16 17 18 19 20 21 22 23

17 18 19 20 21 22 23

17 18 19 20 21 22 23

1900 1920

Calendar Year

1940 1960 1980 2000

Inte

rpla

neta

ry C

ompo

nent

of t

heaa

Geo

mag

netic

Inde

x, a

a I

10

5

0

15

40

20

0

60

80

100

120

Inte

rpla

neta

ry C

ompo

nent

of t

he A

p G

eom

agne

tic In

dex,

Ap I

Inte

rpla

neta

ry C

ompo

nent

of

the

ND

D, N

DD

I

ApI = Ap – ApRApR = 7.07302 + 0.02605R

NDDI = NDD – NDDRNDDR = 7.92684 + 0.12337R

aaI = aa – aaRaaR = 5.77789 + 0.08229R

1940 1960 1980 2000

1940 1960 1980 2000

F�gure 3. Var�at�on of the �nterplanetary components aaI (panel (a)); ApI (panel (b)); and NDDI (panel (c)). The th�ck l�nes are 2-yr mov�ng averages of the res�dual �nterplanetary components.

7

F�gure 4 shows h�stograms mark�ng the frequency of occurrences for the max�mum ampl�tude of aaI (E(aaI max, panel (a)) and for the last occurrences of the local max�mums of aaI (E(aaI max (last), panel (b)), wh�ch often also turns out to be E(aaI max) for most of the cycles (11, 12, 14, 15, 17, 18, 21, and 22), �n years relat�ve to the occurrences of E(Rmax). For E(aaI max), two cycles (11 and 16) have E(aaI max) 2 yr after E(Rmax); four cycles (12, 13, 19, and 23) have E(aaI max) 3 yr after E(Rmax);fivecycles(14,15,18, 21, and 22) have E(aaI max) 5 yr after E(Rmax); and two cycles (17 and 20) have E(aaI max) 6 yr after E(Rmax). For E(aaI max (last)), the bulk of the cycles (13, 14, 15, 18, 21, 22, and 23) have E(aaI max (last)) 5 yr after E(Rmax).

Freq

uenc

y of

Occ

uren

ce fo

r E(a

a Imax

)

Cyc

les

11 a

nd 1

6

Cyc

les

12, 1

3, 1

9, a

nd 2

3

Cyc

les

14, 1

5, 1

8, 2

1, a

nd 2

2

Cyc

les

17 a

nd 2

0

2

0

4

6

8

4 621 3 5 74 621 3 5 7

Elapsed Time in Years Past E(Rmax)Elapsed Time in Years Past E(Rmax)

Freq

uenc

y of

Occ

uren

ce fo

r E(a

a Imax

(las

t))

Cyc

le 1

1

Cyc

le 1

2

Cyc

le 1

6

Cyc

les

13,

14,

15,

18,

21,

22,

and

23

Cyc

les

17, 1

9, a

nd 2

0

2

0

4

6

8

(a) (b)

F�gure 4. H�stograms of the frequency of occurrences for E(aaImax) (panel (a)) and E((aaImax (last)) (panel (b)), relat�ve to the elapsed t�me �n years from E(Rmax).

F�gure 5 d�splays scatterplots of Rm�n (cycle n + 1) versus aaI max (cycle n) (panel (a)); Rm�n (cycle n + 1) versus ApI max (cycle n) (panel (b)); Rm�n (cycle n + 1) versus NDDI max (cycle n) (panel (c)); Rmax (cycle n + 1) versus aaImax (cycle n) (panel (d)); Rmax (cycle n + 1) versus ApI max (cycle n) (panel (e)); and Rmax (cycle n + 1) versus NDDI max (cycle n) (panel (f)). In each panel, the small, downward-po�nt�ng arrow marks the parametr�c value for cycle 23. For panels (b), (c), (e), and (f), nostatisticallysignificant(confidencelevel(cl) ≥95 percent) l�near correlat�on can be �nferred between the parameters. For these, only the mean and standard dev�at�on (sd) are shown. However, for panels (a) and(d),statisticallysignificantpositivelinearcorrelationsareinferred(thoserelatedtoaaI max). Thus, g�ven aaI max = 26.1 for cycle 23, one �nfers Rm�n = 13 ± 4.9 and Rmax = 183.7 ± 46.3 for cycle 24, the next sunspot cycle. Both pred�ct�ons are 90-percent pred�ct�on �ntervals so, there �s only a 5-percent chance that Rm�n w�ll be smaller than 9.1 and Rmax w�ll be smaller than 137.4. L�kew�se, there �s only a 5-percent chance that Rm�n w�ll be larger than 17.9 and Rmax w�ll be larger than 230.

8

100 0 020 30

26.1

5

0

10

15

aaImax (cycle n)

(a) (b) (c)

(d) (e) (f)

5 10 15

13

ApImax (cycle n)

50 100

97

NDDImax (cycle n)

mean = 146.7 sd = 30

mean = 146.7 sd = 30

mean = 9.8 sd = 3.2

mean = 9.8 sd = 3.2

Rm

in (c

ycle

n+

1)

26.1

y

y

100

0

200 y = 9.969 + 6.655xr = 0.801, r 2 = 0.641se = 25.56, cl>99.8%

y = –2.288 + 0.585xr = 0.740, r 2 = 0.547se = 2.73, cl>99%

Rmax (24)90 = 183.7 ± 46.3

Rmin (24)90 = 13 ± 4.9

13 97

Rm

ax (c

ycle

n+

1)

5

0

10

15

100

0

200

5

0

10

15

100

0

200

100 0 020 30 5 10 15 50 100

F�gure 5. Scatterplots of Rm�n (cycle n + 1) versus selected max�mum �nterplanetary components for cycle n (panels (a)–(c)) and Rmax (cycle n + 1) versus selected max�mum �nterplanetary components for cycle n (panels (d)–(f)). See text for deta�l.

F�gure 6 dep�cts scatterplots of Rm�n (cycle n + 1) versus aaI max (last) (cycle n) (panel (a)); Rm�n (cycle n + 1) versus ApI max (last) (cycle n) (panel (b)); Rm�n (cycle n + 1) versus NDDI max (last) (cycle n) (panel (c)); Rmax (cycle n + 1) versus aaI max (last) (cycle n) (panel (d)); Rmax (cycle n + 1) versus ApI max (last) (cycle n) (panel (e)); and Rmax (cycle n + 1) versus NDDImax (last) (cycle n) (panel (f)). For panels (b)and(c),nostatisticallysignificantlinearcorrelationcanbeinferredbetweentheparameters.Forthese,only the mean and sdareshown.Fortheotherpanels,statisticallysignificantpositivelinearcorrelationsare �nferred. Thus, g�ven aaI max (last) = 15 for cycle 23 (an assumed value, s�nce the values for 2006 and 2007 are presently unknown, true also for ApImax (last) and NDDI max (last)), one �nfers Rm�n = 7.2 ± 5 and Rmax = 117.5 ± 41.2 for cycle 24. Also, g�ven ApI max (last) = 5.8 and NDDI max (last) = 31 for cycle 23, one �nfers Rmax = 110.1 ± 38.4 and Rmax = 96.2 ± 35.6, respect�vely, for cycle 24. These pred�ct�ons clearly d�ffer w�th the aforement�oned pred�ct�ons based on aaImax for cycle 23.

F�gure 7 shows scatterplots of Rm�n2 (cycle n + 1) versus aaI max2 (cycle n) (panel (a)), where thesubscript2referstothe2-yrmovingaverage(showninfig.3asthethickersmootherlines);Rm�n2 (cycle n + 1) versus ApI max2 (cycle n) (panel (b)); Rm�n2 (cycle n + 1) versus NDDI max2 (panel (c));

9

100 0 020 30

15

5

0

10

15

aaImax (last) (cycle n)

(a) (b) (c)

(d) (e) (f)

5 10 15

5.8

ApImax (last) (cycle n)

50 100

31

NDDImax (last) (cycle n)

mean 9.8 sd 3.2

mean 9.8 sd 3.2

Rm

in (c

ycle

n1)

15

y yy

y

100

0

200y 14.235 6.885xr 0.846, r 2 0.716se 22.72, cl 99.9%

y –1.347 0.567xr 0.732, r 2 0.536se 2.76, cl 99%

Rmin (24)90 117.5 41.2

y 59.585 8.713xr 0.844, r 2 0.712se 18.02, cl 95%

Rmax (24)90 110.1 38.4

y 60.867 1.140xr 0.864, r 2 0.747se 16.69, cl 95%

Rmax (24)90 96.2 35.6

Rmin (24)90 7.2 5

5.8 31

Rm

ax (c

ycle

n1)

100

200

100

200

5

0

10

15

5

0

10

15

100 0 020 30 5 10 15 50 100

F�gure 6. Scatterplots of Rm�n (cycle n + 1) versus selected last-occurr�ng max�mum �nterplanetary components for cycle n (panels (a)–(c)) and Rmax (cycle n + 1) versus selected last-occurr�ng max�mum �nterplanetary components for cycle n (panels (d)–(f)). See text for deta�ls.

Rmax2 (cycle n + 1) versus aaI max2 (cycle n) (panel (d)); Rmax2 (cycle n + 1) versus ApI max2 (cycle n) (panel (e)); and Rmax2 (cycle n + 1) versus NDDImax2 (cycle n) (panel (f)). Only panels (a) and (d) are foundtocontainastatisticallysignificantpositivelinearcorrelation,bothhavinghighercoefficientsofcorrelat�on (r) and smaller standard errors (se)thanforeitheroftheunsmoothedplots(figs.5and6).Thus,g�ven aaI max2 = 18.6 for cycle 23, one �nfers Rm�n2 = 16.9 ± 4.8 and Rmax2 = 146.7 ± 30.5 for cycle 24, both pred�ct�ons be�ng 90-percent pred�ct�on �ntervals. Hence, there �s only a 5-percent chance that cycle 24’s Rm�n2 w�ll be smaller than 12.1 or larger than 21.7, and there �s only a 5-percent chance that cycle 24’s Rmax2 w�ll be smaller than 116.2 or larger than 177.2. The 90-percent pred�ct�on �nterval for cycle 24’s Rm�n2 suggests that �ts Rm�n (the annual average value) w�ll be about 10.4 ± 3 and �ts Rm (12-mo mov�ng average, or smoothed monthly mean sunspot number m�n�mum ampl�tude) w�ll be about 9.4 ± 2.9. S�m�larly, the 90-percent pred�ct�on �nterval for cycle 24’s Rmax2 suggests that �ts Rmax (the annual average value) w�ll be about 157.5 ± 31.8 and �ts RM (12-mo mov�ng average, or smoothed monthly mean sunspot number max�mum ampl�tude) w�ll be about 162.9 ± 33.5. Thus, these values suggest that cycle 24’s act�v�ty w�ll be greater than average (and larger than was seen for cycle 23), s�m�lar �n s�ze to that exper�enced �n cycles 21 and 22.14,26,27

10

50 10 15 20

18.6

5

0

10

20

15

aaImax2 (cycle n)

(a) (b) (c)

(d) (e) (f)

ApImax2 (cycle n)

8.7

NDDImax2 (cycle n)

mean = 16.3 sd = 1.6

mean = 137.5 sd = 26.6

mean = 137.5 sd = 26.6

mean = 16.3 sd = 1.6

Rm

in2

(cyc

le n+

1)

18.6

y

y

100

150

200

50

y = –3.639 + 1.106xr = 0.880, r 2 = 0.775se = 2.65, cl>99.9%

Rmin2 (24)90 = 16.9 ± 4.8

y = –2.737 + 8.034xr = 0.906, r 2 = 0.820se = 16.83, cl>99.9%

Rmax2 (24)90 = 146.7 ± 30.5

Rm

ax2

(cyc

le n+

1)

100

150

200

50

100

150

200

50

5

0

10

20

15

5

0

10

20

15

50 10 15 250 50 75 100

50 10 15 20 50 10 15 250 50 75 100

57.8

8.7 57.8

F�gure 7. Scatterplots of Rm�n2 (cycle n + 1) versus selected 2-yr mov�ng averages of the max�mum �nterplanetary components for cycle n (panels (a)–(c)) and Rmax2 (cycle n + 1) versus 2-yr mov�ng averages of the max�mum �nterplanetary components for cycle n (panels (d)–(f)). See text for deta�ls.

2.2 aa (adjusted) Geomagnetic Index

Recallfromfigure1thelong-termriseintheobservedaa �ndex over t�me, wh�ch �s also apparent �n R, th�s �nd�cat�ng a strong correlat�on between the parameters. Indeed, as prev�ously ment�oned, W�lson and Hathaway22 have noted that on the bas�s of 10-yr mov�ng averages, aa and R are h�ghly correlated (r = 0.933). Also, as prev�ously ment�oned, Svalgaard, Cl�ver, and Le Sager19 �n the�r reconstruct�on of the observed aa �ndex us�ng the IHV �ndex found that, for the �nterval of 1957–2000, both parameters are v�rtually �dent�cal, wh�le for the �nterval preced�ng 1957, the reconstructed aa (based on the IHV �ndex) l�es above the observed aa. Hence, they concluded that the observed aa pr�or to 1957 m�ght be �naccurate. (In 1957, the Northern Hem�sphere (NH) magnetometer used �n der�v�ng the aa �ndex was moved from Ab�nger, England, to �ts present locat�on �n Hartland, England.)

11

1 2 3 4 5

0

5

–5

5

0

–5

aaC

HL/

FRD

– IH

V,

2aa

– IH

V,

3

1.

2.

3.

4.

5.

Values of IHV priorto 1915 are reduced by30%, as advised by Svalgaard et al.

SH magnetometermoved from Melbourneto Toolangui.

NH magnetometermoved from Greenwichto Abinger.

NH magnetometermoved from Abingerto Hartland.

SH magnetometer moved from Toolanguito Canberra.

Notes:

1900(a)

(b)

(c)

1910 1920 1930 1940 1960 1970 1980 19901950 2000–5

0

5

aa –

aa C

HL/

FRD

,1

1900 1910 1920 1930 1940 1960 1970 1980 19901950 2000

1900 1910 1920 1930 1940 1960 1970 1980 19901950 2000

Calendar Year

F�gure 8. Var�at�on of selected parametr�c d�fferences, 1901–2000. See text for deta�ls.

F�gure 8 shows the d�fferences aa – aaCHL/FRD,Δ1 (panel (a)); aaCHL/FRD – IHV,Δ2 (panel (b)); and aa – IHV, Δ3 (panel (c)), where aa �s the observed aa �ndex value; aaCHL/FRD �s the der�ved aa value based on observat�ons at Cheltenham/Freder�cksburg dur�ng the two 3-hr per�ods between 0 and 6 hr UT (wh�ch represented the observed or proxy aa �ndex �n the Svalgaard, Cl�ver, and Le Sager study19), and IHV �s the reconstructed aa value, these latter two values taken from Svalgaard, Cl�ver, and Le Sager. Acrossthetopofthechartaretimeticks1–5,correspondingtothenotesgiventotherightofthefigure.For example, note 1 says that values of IHV pr�or to 1915 are reduced 30 percent, as adv�sed �n Svalgaard, Cl�ver, and Le Sager; note 2 states that the Southern Hem�sphere (SH) magnetometer was moved from Melbourne, Austral�a, to Toolangu�, Austral�a, and so forth. Today, the NH magnetometer �s located at Hartland, England, and the SH magnetometer �s located at Canberra, Austral�a. These stat�ons are now determiningtheofficialaa �ndex.

Inspectionoffigure8revealsthateachofthedifferencesappearstoberelativelystablefrom1957onward,averagingabout2.3forΔ1,0.1forΔ2,and2.3forΔ3. For the earl�er port�on of the record, 1901–1956,Δ1isfoundtoaverageabout1.8,Δ2about–3.1,andΔ3 about −1.3. The d�fferences of the means, especiallyforΔ2andΔ3,arefoundtobestatisticallysignificant.Hence,anoffset-correctionappearstobeneededfortheearlierinterval(1901–1956),onebeingabout3forΔ2andabout3.6forΔ3.

12

F�gure 9 repeats the analys�s, but th�s t�me compar�ng aa, Ap, and IHV. Plotted are the d�fferences aa – Ap(Δ4) and Ap – IHV(Δ5).Thenotesatthetopofthechartrefertothenotesexplainedinfigure8.Fortheinterval1957–2005,Δ4averagesabout9.1andΔ5 averages about −6.7, wh�le for the earl�er �nterval of 1932–1956, they average, respect�vely, 6.8 and −7.5. So, �t appears that, relat�ve to Ap, the aa �ndex needs to be offset-corrected by 2.3 un�ts and, relat�ve to IHV, aa should be offset-corrected by about 3.1 un�ts (= 2.3 + 0.8).

19401930(a)

(b)

1950

4 5

1960 1970 1980 1990 20000

5

10

15

0

–5

–10

aa–

Ap,

4

Ap

–IH

V,

5

Calendar Year

19401930 1950 1960 1970 1980 1990 2000

F�gure 9. Var�at�on of selected parametr�c d�fferences, 1932–2000. See text for deta�ls.

F�gure 10 replots aa versus R, but th�s t�me add�ng �n an offset value of 3 to the observed values of aa for all values of aa pr�or to 1957. In truth, wh�le sl�ght offset-correct�ons m�ght be attr�butable after each movement of the NH or SH magnetometers, for s�mpl�c�ty sake, a un�form correct�on of 3 �s used to account for the general d�fference �nferred between the earl�er (pr�or to 1957) and later aa datasets.

13

1000 200

20

0

40

aa (a

djus

ted) aaR (adjusted) 8.83123 0.06254R

R

F�gure 10. Scatterplot of aa (adjusted) versus R.

F�gure 11 replots the aaI component of the adjusted aa dataset,followingtheformatusedinfigure3. Wh�le the values of aaI (adjusted) are sh�fted upward, as compared to that shown for aaIinfigure3,thereremainsnoticeableinbothfiguresanincreaseinthevalueofaa. For example, for the �nterval 1868–1956, the observed (unadjusted) aa averaged 17.17, wh�le for the �nterval 1957–2005, �t averaged 23.98, an �ncrease of nearly 40 percent. Us�ng the adjusted aa,forthefirstinterval,itaveraged20.17,ascomparedto 23.98 for the second �nterval, an �ncrease of about 19 percent. For the same two �ntervals, aaR averaged 9.69 and 12.06, nearly a 25-percent �ncrease; aaR (adjusted) averaged 11.79 and 13.61, an �ncrease of about 15 percent; aaI averaged 7.48 and 11.92, an �ncrease of about 59 percent; and aaI (adjusted) averaged 8.38 and 10.37, an �ncrease of about 24 percent. Thus, the �ncrease �n geomagnet�c act�v�ty appears real, be�ng seen �n both the sunspot cycle component and the �nterplanetary component, regardless of whether the data have been adjusted or not.

14

5

0

10

15

20

25

18801860

11 12 13 14 15 16 17 18 19 20 21 22 23

1900 1920

Calendar Year

1940 1960 1980 2000

aaI (

adju

sted

)

aaI (adjusted) aa (adjusted) – aaR (adjusted)aaR (adjusted) 8.83123 0.06254R

aa (adjusted) aa 3 for 1868–1956aa for 1957–2005

F�gure 11. Var�at�on of aaI (adjusted). The th�ck l�ne �s the 2-yr mov�ng average of the res�dual �nterplanetary component.

F�gure 12 plots Rm�n2 (cycle n + 1) versus aaI (adjusted) max2 (cycle n) (panel (a)), where the latter parameter refers to the max�mum 2-yr mov�ng average of the aaI (adjusted)asplottedinfigure11,foreachcycle; and Rmax2 (cycle n + 1) versus aaI (adjusted) max2 (cycle n) (panel (b)). The downward-po�nt�ng arrow g�ves the value of aaI (adjusted) max2 for cycle 23, equal to 16.9. Thus, for cycle 24, the 90-percent pred�ct�on �nterval for Rm�n2 = 16 ± 4.7 and Rmax2 = 143.2 ± 23.2. These values are sl�ghtly lower than what was gleaned us�ng the observed aa (unadjusted) values. In terms of Rm�n and Rm, the 90-percent pred�ct�on �ntervals for cycle 24 are 9.8 ± 2.9 and 8.8 ± 2.8. S�m�larly, �n terms of Rmax and RM, the 90-percent pred�ct�on �ntervals for cycle 24 are 153.8 ± 24.7 and 159 ± 25.5.

15

10 15 2050(a)

(b)

5

0

10

15

20

aaI (adjusted)max2 (cycle n)

100

50

150

200

y

y

Rm

ax2

(cyc

le n

1)R

min

2 (c

ycle

n1)

y –8.294 1.435xr 0.890, r 2 0.793se 2.58, cl 99.9%

Rmin2 (24)90 16 4.7

y –42.347 10.981xr 0.947, r 2 0.897se 12.77, cl 99.9%

Rmax2 (24)90 143.2 23.2

16.9

16.9

10 15 2050

F�gure 12. Scatterplots of Rm�n2 (cycle n + 1) versus aaI (adjusted)max2 (cycle n) (panel (a)) and Rmax2 (cycle n + 1) versus aaI (adjusted)max2 (cycle n) (panel (b)). See text for deta�ls.

16

2.3 Single-Variate and Bivariate Fits Based on Parametric Minimum Values

F�gure 13 d�splays cycl�c values of Rm�n (panel (a)); Rmax (panel (b)); aam�n (panel (c)); Apm�n (panel (d)); and NDDm�n (panel (e)) for cycles 12–23. For aam�n, Apm�n, and NDDm�n, these are the m�n�mum values found �n the v�c�n�ty of sunspot m�n�mum (w�th�n 1 yr follow�ng sunspot m�n�mum). Th�s caveat �s necessary because cycle 21 actually had lower values �n 1980 dur�ng the max�mum ampl�tude phase of the sunspot cycle, the only cycle �n the ent�re record to have such an anomaly. Also, for aam�n, two l�nes are plotted: the values based on the observed record and the values based on the adjusted record. Additionally,statisticallysignificantregressionsaregivenforRm�n, Rmax, and aam�n (both observed and adjusted). For Apm�n and NDDm�n, only the mean and sd are g�ven.

F�gure 13 clearly shows that over the course of cycles 12–23, there has been an apparent secular r�se �n Rm�n, Rmax, and aam�n. On the bas�s of these �nferred trends, one est�mates the 90-percent pred�ct�on �nterval for each of the parameters to be Rm�n = 12.3 ± 4.8 (or Rm = 11.2 ± 4.6), Rmax = 167.9 ± 55 (or RM = 173.3 ± 56.5), aam�n (observed) = 20.6 ± 4.6, and aam�n (adjusted) = 20.4 ± 5.2.

F�gure 14 d�splays s�ngle-var�ate scatterplots of Rmax versus Rm�n (panel (a)); aam�n (panel (b)); Apm�n (panel (c)); and NDDm�n (panel (d)). Thus, observat�on of the m�n�mum values of these parameters allows (at least for aam�n, Apm�n, and NDDm�n) for the pred�ct�on of Rmax for the ongo�ng sunspot cycle some 2–3 yr �n advance. (Although ne�ther plot of Rmax versus Rm�n isstatisticallysignificant,thatbasedon cycles 12–23 or that based on cycles 17–23, the regress�ons are shown because they w�ll be used �n the nextfigureaspartofabivariatefit.)

Fromfigure14,itshouldbeobviousthat,ifcycle24hasanaam�n of about 20, as suggested by the l�near secular trend, then, clearly, one should expect an Rmax of about 160 or so for cycle 24. Such a value also suggests that Apm�n should measure about 11 and NDDm�n about 25 or so. Through July 2006, Ap has averaged only about 7, compared to 13.6 for the year 2005, and the total number of d�sturbed days amounts to only 9, compared to 43 for the year 2005. Hence, e�ther these values w�ll substant�ally �ncrease as the year progresses, espec�ally �n the fall of the year, or cycle 24’s values w�ll fall well below the means. It �s �nterest�ng to note that based on a presumed repeat�ng pattern of “below-above-above-below” observed �n Apm�n and NDDm�n, one expects both Apm�n and NDDm�n to be “above” the�r respect�ve means, suggest�ng that the latter half of 2006 m�ght be geophys�cally qu�te act�ve.

17

5

0

10

15

Apm

in

5

0

10

15

20

25

30

35

ND

Dm

in

50

100

200

150R

max

5

0

10

15

20

25

aam

in

24222018161412

Sunspot Cycle Number

Adjusted

5

0(a)

(b)

(c)

(d)

(e)

10

15

Rm

in

y

y

yy'

y = –7.316 + 0.819xr = 0.762, r 2 = 0.580se = 2.62, cl>99.5%

y = –24.709 + 8.025xr = 0.711, r 2 = 0.505se = 30.02, cl>98%

Rmin (24)90 = 12.3 ± 4.8

Rmax (24)90 = 167.9 ± 55

y ' = 1.166 + 0.802xr = 0.736, r 2 = 0.542se = 2.81, cl>99%

aamin (24)90 = 20.4 ± 5.2

y = –6.709 + 1.138xr = 0.870, r 2 = 0.757se = 2.44, cl>99.9%

aamin (24)90 = 20.6 ± 4.6

mean = 9.7 sd = 1.8

mean = 20.4 sd = 9.3

F�gure 13. Cycl�c var�at�on of selected parameters. Not�ce the strong secular �ncreases �n Rm�n, Rmax, and aa.

18

50

100

150

200R

max

Rmin5 10 15 20 25

y ' (adjusted)y

aamin

50 10 1550

100

150

200 y

Rm

ax

5 10 15 20 25 30 35

y

NDDminApmin

50(a) (b)

(c) (d)

10 15 050

100

150

200

50

100

150

200

y12–23 = 77.033 + 5.521xr = 0.525, r 2 = 0.276se = 36.320, cl>90%

y = 14.206 + 7.686xr = 0.890, r 2 = 0.793se = 19.427, cl>99.9%

y = 0.791 + 14.589xr = 0.874, r 2 = 0.764se = 15.988, cl>98%

y = 89.110 + 2.594xr = 0.803, r 2 = 0.644se = 19.595, cl>95%

y ' = –32.343 + 9.736xr = 0.940, r 2 = 0.883se = 14.588, cl>99.9%

y17–23 = 148.034 – 0.644xr = –0.071, r 2 = 0.005se = 32.795, cl>90%

y12–23

y17–23

F�gure 14. Scatterplots of Rmax versus selected m�n�mum values of the parameters.

F�gure 15 shows scatterplots of Rmax (observed) versus Rmax (predicted)onthebasisofspecificbivariate fits, where bivariate fit 1 (bv1) �s based on Rm�n and aam�n (panel (a)); bv2 on Rm�n and aam�n (adjusted) (panel (b)); bv3 on Rm�n and Apm�n (panel (c)); and bv4 on Rm�n and NDDm�n (panel (d)).Forthevariousfits,theonehavingthesmalleststandarderrorofestimateisbv4. Thus, once Rm�n and NDDm�n are observed for cycle 24 (perhaps �n 2007 or 2008), an alternate means for pred�ct�ng Rmax for cycle 24 w�ll be ava�lable (hav�ng a 90-percent pred�ct�on �nterval measur�ng about ±15.5 un�ts of sunspot number).

19

50(a) (b)

(c) (d)

100 150 20050

100

150

200

yR

max

(obs

erve

d)

Rmax (predicted, bv1)

50 100 150 200

y


50 100 150 20050

100

150

200

y

Rm

ax (o

bser

ved)


50 100 150 200

y


50

100

150

200

50

100

150

200

y 1.088 – 6.820Rmin 12.298aaminr 0.963, r 2 0.928se 12.091

y 11.590 – 4.147Rmin 17.419Apminr 0.972, r 2 0.944se 8.696

y 121.378 – 5.877Rmin 3.665NDDminr 0.978, r 2 0.956se 7.698

y –40.418 – 2.428Rmin 11.386aamin (adjusted)r 0.955, r 2 0.912se 13.372

F�gure 15. Scatterplots of Rmax(observed)versusselectedbivariatefits. See text for deta�ls.

2.4 Cyclic Averages and Sums

F�gure 16 d�splays cycl�c averages and sums, from sunspot m�n�mum to sunspot m�n�mum, of <R> (panel (a)); <aa>, both the observed and adjusted (panel (b)), where the dashed l�ne represents the adjusted values; <Ap> (panel (c)); <NDD> (panel (d)), where the dashed l�ne refers to ΣNDD; and <NSSC> (panel (e)), where the dashed l�ne refers to ΣNSSC. (<R> for cycle 11 excludes the year 1867 value of R and all parameters for cycle 23 exclude the year 2006 values.)

For <R>, <aa>, <aa> (adjusted), <NSSC>, and ΣNSSC, all display statistically significantupward l�near secular trends, g�ven by the regress�on equat�ons. Thus, on the bas�s of the �nferred trends, the 90-percent pred�ct�on �nterval of the parameters for cycle 24 �s <R> = 82.4 ± 29.5, <aa> = 26.5 ± 4, <aa> (adjusted) = 26.2 ± 4.2, <NSSC> = 35.2 ± 6.1, and ΣNSSC = 358 ± 62. Also, because of the presumed “below-above-above-below” pattern, one expects <Ap> to be above 14.8, <NDD> to be above 54.4, and ΣNDD to be above 563.

20

10

15

40

50

60

70

<Ap>

< ND

D>

15

20

25

30

35

<NSS

C>

400

500

600

700

150

200

250

300

350

ΣN

SSCΣ

ND

D

24222018


161412

10

20

15

25

<aa>

,<aa>

(adj

uste

d)

(adjusted)

y´y

yy’

y = 5.457 + 1.240xr = 0.832, r 2 = 0.693se = 3.37, cl>99.9%

y´ = 10.846 + 0.638xr = 0.751, r 2 = 0.565se = 2.33, cl>99.5%

<NSSC> (24)90 = 35.2 ± 6.1

<aa> (24)90 (adjusted) = 26.2 ± 4.2

y´ = 90.000 + 11.154xr = 0.798, r 2 = 0.637se = 34.26, cl>99.8%

ΣNSSC (24)90 = 358 ± 62

50

0(a)

(b)

(c)

(d)

(e)

100

<R>

y

y = 2.468 + 3.331xr = 0.637, r 2 = 0.406se = 16.41, cl>98%

<R> (24)90 = 82.4 ± 29.5

mean = 54.4 sd = 10.6

mean = 563 sd = 86

mean = 14.8 sd = 1.8

y = 3.495 + 0.958xr = 0.870, r 2 = 0.757se = 2.23, cl>99.9%

<aa> (24)90 = 26.5 ± 4.0

F�gure 16. Var�at�on of cycl�c averages of selected parameters.

21

F�gures 17 and 18 show compar�sons of each of the parameters aga�nst Rm�n (left panels) and aga�nst Rmax(rightpanels).Allstatisticallysignificantregressions(cl ≥ 95percent)areidentified.Hence,once Rm�n has been observed for cycle 24, one can use these regress�ons to est�mate the cycl�c parametr�c averages (and sums). L�kew�se, once Rmax has occurred, one can use these regress�ons to est�mate w�th greater accuracy the cycl�c parametr�c averages (and sums). (One can also use the secular est�mates to est�mate Rm�n and Rmax for cycle 24. The secular est�mate for <R> suggests that Rm�n w�ll be about 14 and Rmax about 160; the secular est�mate for <aa>, whether adjusted or not, suggests Rm�n w�ll be about 15 and Rmax about 185; the secular est�mate for <NSSC> suggests Rm�n w�ll be about 15 and Rmax about 190; and the secular est�mate for ΣNSSC suggests Rm�n w�ll be about 15 and Rmax about 200.)

Apeculiaritynotedinfigure18isthecyclicsplitseenin<NSSC>andΣNSSC. Namely, cycles 11–16 and 17–23 appear to be grouped d�st�nctly from each other. Such group�ngs re�nforce the not�on that cycles of late have been more robust as compared to earl�er cycles. If the trend cont�nues, then cycle 24, l�kew�se, should be expected to be another robust cycle.26,28–30 (Another pecul�ar�ty �s the apparent “below-above-above-below” pattern not�ceable �n <Ap>, <NDD>, and ΣNDD versus Rm�n or Rmaxinfigs.17and18,which,ifreal,suggestsaboveaveragevaluesforcycle24.)

F�gure 19 shows the scatterplot of yearly counts of NSSC versus R. Pla�nly, years of h�gher R are assoc�ated w�th years of h�gher NSSC, and v�ce versa, although even dur�ng per�ods of low R (near sunspot m�n�mum), one st�ll expects to see about 10 or more sudden storm commencements.

22

y

y

y

100

50

0(a)

(b)

(c)

(d)

(e)

20

15

10

25<R

><a

a>

25

10

20

15

1519

17

19

18

22

21

17

23

23

20

20

18 21 22

<aa>

(adj

uste

d)<A

p>

40

45

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65

5 1510Rmin Rmax

50 150 200100

y

y

y

y

y

19

18

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21

17

23

20 1723

20

222118

19

<ND

D>

40

45

50

55

60

65

0 50

10

15

25

20

15

20

15

10

25

100

50

mean = 14.8 sd = 1.8

mean = 54.4 sd = 10.6

y = 13.790 + 0.853xr = 0.738, r 2 = 0.544se = 3.01, cl>99.5%

y = 35.583 + 3.345xr = 0.609, r 2 = 0.371se = 16.86, cl>95%

y = –0.314 + 0.506xr = 0.981, r 2 = 0.962se = 3.68, cl>99.9%

y = 8.832 + 0.093xr = 0.859, r 2 = 0.737se = 2.42, cl>99.9%

y = 12.994 + 0.074xr = 0.884, r 2 = 0.782se = 1.69, cl>99.9%

y = 7.180 + 0.054xr = 0.907, r 2 = 0.823se = 0.74, cl>99.9%

y = 13.099 + 0.291xr = 0.827, r 2 = 0.684se = 6.46, cl>95%

y = 17.600 + 0.583xr = 0.654, r 2 = 0.427se = 2.62, cl>98%

F�gure 17. Scatterplots of cycl�c averages of selected parameters aga�nst Rm�n (left panels) and Rmax(rightpanels).Thenumberedfilledcirclesrefertoindividualsunspotcycles.

23

18

1922

21

Cycles 11–16

yy

y y

2017

23

400(a)

(b)

(c)

500

ΣN

DD

ΣN

SSC

600

700

15

35

30

25

20

200

300

400

< NSS

C>

18

1922

21

20 17

23

50 10 15

Rmin

500 100 150 200

Rmax

Cycles 11–16Cycles 11–16

Cycles 11–16

Cycles17–23

Cycles 17–23

Cycles 17–23

Cycles17–23

400

500

600

700

15

35

30

25

20

200

300

400

y = 19.144 + 1.052xr = 0.672, r 2 = 0.452se = 4.48, cl>98%

y = 12.060 + 0.123xr = 0.839, r 2 = 0.703se = 3.42, cl>99.9%

y = 212.331 + 9.570xr = 0.652, r 2 = 0.425se = 43.11, cl>98%

y = 168.594 + 0.945xr = 0.685, r 2 = 0.470se = 41.27, cl>99%

mean = 562.9 sd = 85.9

mean = 562.9 sd = 85.9

F�gure 18. Scatterplots of add�t�onal cycl�c averages and sums aga�nst Rm�n (left panels) and Rmax (r�ght panels).

24

1000 200

20

0

40

60

80

NSS

C

R

1868–2005

ΣNSSC = 3645y

y = 11.907 + 0.251xr = 0.865, r 2 = 0.748se = 6.621, cl>99.9%

F�gure 19. Scatterplot of NSSC versus R.

2.5 Apmax, ND (Ap ≥ 100) and ND (Ap ≥ 200)

F�gure 20 d�splays the yearly Apmax for the �nterval 1932–2005. As before, the numbers at the bottom of the chart refer to sunspot cycles 17–23 and the th�n vert�cal l�nes refer to the epochs of sunspot m�n�mum for each cycle. Not�ceable �s that all cycles have two or more peaks of act�v�ty, w�th the largest Apmax values usually occurr�ng after Rmax (two except�ons: cycles 18 and 22). Also, all cycles have peak values �n excess of Ap = 100 and all but one (cycle 20) had peak values exceed�ng Ap = 200, w�th the h�ghest Apmax (= 280) occurr�ng �n cycle 19 �n November 1960 (after �ts Rmax). For cycle 23, �ts h�ghest Apmax occurred �n October 2003 (= 204), the s�xth h�ghest value on record.

F�gure 21 dep�cts the scatterplot of Apmax versus R.Althoughthereisastatisticallysignificantpos�t�ve correlat�on between the parameters, assoc�at�ng h�gher (lower) Apmax w�th h�gher (lower) R, one clearly sees that even when the sunspot cycle �s near solar m�n�mum there �s st�ll opportun�ty for large Ap. For example, for the 15 yr when R < 20, Apmax spanned 38 (cycle 23) to 202 (cycle 22), w�th one-th�rd of the years hav�ng Apmax ≥ 100.

25

1940 1950 1970 1990

17 18 19 20 21 22 23

1960 1980 2000

100

50

0

200

150

300

250

Ann

ual M

axim

um D

aily

Ap

Valu

e, A

pmax

Calendar Year

F�gure 20. Var�at�on of the annual max�mum da�ly Ap value, Apmax.

100 125 150 1750 25 50 75 200

y

50

0

100

150

250

200

Apm

ax

R

300

y = 85.532 + 0.569xr = 0.542, r 2 = 0.294se = 45.267, cl >99.9%

F�gure 21. Scatterplot of Apmax versus R.

26

F�gure 22 d�splays the scatterplot of Apmax versus Ap. Pla�nly, h�gher (lower) Apmax assoc�ates w�th h�gher (lower) Ap and the assoc�at�on �s stronger (r = 0.67) than �s found for R (r = 0.54). For Ap < 10, as yet, there has never been an Apmax ≥ 100. (82 has been the largest value of Apmax for Ap < 10.)

50 10 15 20 25

y

50

0

100

150

250

300

200

Apm

ax

Ap

y = –5.854 + 9.018xr = 0.669, r 2 = 0.447se = 40.020, cl>99.9%

F�gure 22. Scatterplot of Apmax versus Ap.

F�gure 23 shows the frequency of occurrence for E(Apmax),definedhereastheepochofApmax occurrence,duringtheyear.Apparentisasemi-annualvariationofthegeomagneticfield,withtwoprom�nent peaks near the equ�noxes and m�n�ma near the solst�ces.31

F�gure 24 g�ves the frequency of occurrence of E(Apmax) relat�ve to E(Rmax),definedhereastheepoch of sunspot max�mum occurrence, for cycles 17–23. Most of the cycles have the�r Apmax value 3 to 4 years after E(Rmax), as has prev�ously been noted.

F�gure 25 d�splays the cycl�c var�at�on of Apmax (panel (a)), and ND (Ap ≥ 100) and ND (Ap ≥ 200) (panel (b)) for cycles 17–23. Clearly, cycle 19 has been the most geomagnet�cally act�ve cycle on record, hav�ng the h�ghest Apmax (= 280) and the largest ND (Ap ≥ 100) and ND (Ap ≥ 200) (39 days and 5 days, respect�vely). All cycles have had at least 13 days (cycle 20) w�th Ap ≥ 100, w�th cycle 23 exper�enc�ng 22 such days so far, �nclud�ng one day of Ap ≥ 200.

27

Freq

uenc

y of

Occ

uren

ce fo

r E(A

pmax

)

J F M A M J J A S O N D

4

0

8

12

16

Months of Year

F�gure 23. Sem�-annual var�at�on of Apmax.

Freq

uenc

y of

Occ

uren

ces

of E

(Apm

ax)

–2 –1 0 1 2 3 4 5

2

1

0

4

3

Elapsed Time in Years Relative to E(Rmax)

F�gure 24. H�stogram of the frequency of occurrences of E(Apmax) relat�ve to the elapsed t�me �n years relat�ve to E(Rmax).

28

ND (Ap≥ 200)

10

0

20

30

40

Num

ber o

f Day

s pe

r Cyc

le

ND (Ap≥ 100)

180

200

220

240

260

280

Apm

ax

2422201816


2422201816(a)

(b)

F�gure 25. Cycl�c var�at�on of Apmax (panel (a)) and ND (Ap ≥ 100) and ND (Ap ≥ 200) (panel (b)).

29

F�gure 26 dep�cts scatterplots of Apmax versus Rm�n (panel (a)); Apmax versus Rmax (panel (b)); ND (Ap ≥ 100) versus Rm�n (panel (c)); and ND (Ap ≥ 100) versus Rmax (panel (d)). Of the plots, only thelatterisfoundtoshowastatisticallysignificantpositivelinearregression.Thus,giventhesizeofanongo�ng sunspot cycle, one can crudely est�mate the number of days dur�ng that cycle when Ap ≥ 100.

(a) (b)

(c) (d)

10 1550

300

250

200

150

Apm

ax

150 20010050

Rmin Rmax

300

250

200

150

mean = 222.3 sd = 33.1

mean = 222.3 sd = 33.1

30

40

20

10

0

ND

(Ap

≥ 10

0)

y

30

40

20

10

010 1550 150 20010050

mean = 22.4 sd = 7.0

y = –7.600 + 0.214xr = 0.801, r 2 = 0.642se = 5.41, cl >95%

F�gure 26. Scatterplots of Apmax versus Rm�n and Rmax (panels (a) and (b), respect�vely) and ND (Ap ≥ 100) versus Rm�n and Rmax (panels (c) and (d), respect�vely).

2.6 Estimating Ap, NDD, and Apmax Prior to 1932

Tables 1 and 2 �dent�fy s�ngle-var�ate (aga�nst aa or aa (adjusted)) and b�var�ate (R and aa or aa (adjusted)) regress�ons, respect�vely, for Ap, NDD, and Apmax for the �nterval 1932–2005. All regressionsarehighlystatisticallysignificantandmightproveuseful fordeterminingestimatesofAp, NDD, and Apmaxfortheearlier interval1868–1931.Itshouldbenotedthat thebivariatefitsoffernosignificantimprovementoverthesingle-variatefitsbasedonaa or aa (adjusted) alone.

30

Table 1. Selected s�ngle-var�ate regress�ons for 1932–2005 (n = 74).

Correlation r r 2 se cl (%)

Ap = –2.906 + 0.761aaAp = –4.806 + 0.808aa (adjusted)NDD = –63.375 + 5.078aaNDD = –77.194 + 5.438aa (adjusted)Apmax = –37.966 + 7.121aaApmax = –47.403 + 7.213aa (adjusted)

0.9470.9700.9260.9570.6570.642

0.8960.9420.8570.9160.4310.412

1.320.99

10.327.91

40.5941.26

>99.9>99.9>99.9>99.9>99.9>99.9

Table 2. Selected b�var�ate regress�ons for 1932–2005 (n = 74).

Correlation RY.12 R 2Y.12 SY.12

Ap = –2.684 + 0.733aa + 0.006RAp = –4.510 + 0.775aa (adjusted) + 0.007R NDD = –64.855 + 5.279aa – 0.044R NDD = –78.434 + 5.600aa (adjusted) – 0.037RApmax = –27.488 + 5.636aa + 0.332RApmax = –35.376 + 5.667aa (adjusted) + 0.352R

0.9480.9730.9290.9590.7160.711

0.9000.9960.8630.9200.5120.506

1.270.94

10.157.73

37.8538.11

Table 3 prov�des the est�mates of Ap, NDD, and Apmax for 1868–1931 us�ng the aforement�oned regressionfits.Forsimplicity,thetabularentriesforNDDandApmax have been rounded to the nearest wholenumber.Eachparametriccolumncontainstwonumbers:Thefirstisthevaluebasedonthesingle-variatefitandthesecondisbasedonthebivariatefit.Asanexample,for1868,theentriesforAp are 10.9 (single-variatefit) and10.9 (bivariatefit); forAp (adjusted), they are12.3 (single-variatefit) and12.2(bivariatefit),andsoon.Toobtainthe90-percentpredictionintervalfortheestimate,itisapproximately1.668 t�mes the appropr�ate se for the single-variatefits and1.668 times the appropriateSY.12 for the bivariatefits (given in tables1and2).Thus, forAp, the 1868 est�mate �s 10.9 ± 2.2(single-variatefit)and 10.9 ± 1.5(bivariatefit);forAp(adjusted) �t �s 12.3 ± 1.7(single-variatefit)and12.2 ± 1.6, and so on. (The negat�ve numbers �n table 3 for NDD should be �nterpreted as 0 + 1.668 se or 0 + 1.668 SY.12, where se equals 10.3 or 7.9 for the single-variate fits, dependent upon using the observed or adjusted data,respect�vely; SY.12equals10.2or7.7forthebivariatefits,dependentuponusingtheobservedoradjusteddata.

31

Table 3. Parametr�c est�mates for 1868–1931.

Year Ap Ap (adjusted) NDD NDD (adjusted) Apmax Apmax (adjusted)

1868 10.9 10.9 12.3 12.2 29 30 38 39 92 88 106 81

1869 12.9 13.0 14.4 14.5 42 42 52 52 110 114 124 126

1870 14.0 14.4 15.6 16.0 49 46 60 57 120 144 134 156

1871 13.3 13.6 14.8 15.1 45 43 55 53 114 130 128 142

1872 15.1 15.2 16.7 16.8 57 55 68 67 130 139 145 151

1873 12.5 12.5 13.9 13.9 39 39 49 49 106 108 120 119

1874 8.2 8.3 9.4 9.4 11 10 19 18 67 70 80 80

1875 5.6 5.6 6.7 6.6 –7 –7 0 0 42 41 55 51

1876 4.3 4.3 5.3 5.3 –15 –15 –9 –9 30 30 43 39

1877 3.9 3.9 4.8 4.8 –18 –18 –13 –13 25 27 38 36

1878 2.6 2.6 3.4 3.4 –27 –27 –22 –22 13 14 26 24

1879 2.4 2.5 3.3 3.3 –28 –28 –23 –23 12 14 25 23

1880 5.8 5.9 6.9 7.0 –5 –6 2 1 44 48 57 58

1881 7.4 7.6 8.6 8.7 6 5 13 12 59 67 80 78

1882 14.5 14.5 16.1 16.0 53 53 64 64 125 121 139 132

1883 10.4 10.5 11.8 11.8 26 25 34 34 87 92 101 103

1884 7.8 8.0 9.0 9.2 8 7 16 15 62 73 76 84

1885 8.8 8.9 10.1 10.1 15 14 23 22 72 77 85 87

1886 12.8 12.6 14.3 14.0 41 43 51 53 109 97 123 107

1887 9.6 9.4 10.9 10.6 20 21 28 29 79 69 93 79

1888 8.8 8.6 10.1 9.8 15 16 23 24 72 62 85 71

1889 6.6 6.5 7.7 7.5 0 1 7 8 51 45 64 55

1890 5.2 5.2 6.3 6.2 –9 –9 –3 –2 38 35 51 45

1891 10.0 10.0 11.4 11.2 23 23 32 32 83 80 97 91

1892 15.5 15.5 17.2 17.1 60 60 71 71 134 133 149 145

1893 10.0 10.3 11.4 11.6 23 21 32 30 83 97 97 108

1894 12.8 13.0 14.3 14.4 42 41 52 51 109 115 124 126

1895 10.9 11.0 12.2 12.3 29 28 38 37 91 96 105 107

1896 10.7 10.7 12.1 12.0 28 28 37 37 90 88 103 98

1897 7.4 7.4 8.5 8.5 5 5 13 13 58 57 72 67

1898 8.6 8.5 9.8 9.7 13 14 21 22 70 67 83 77

1899 7.1 7.0 8.2 8.1 3 3 10 11 55 50 69 60

1900 2.8 2.9 3.7 3.7 –25 –26 –20 –20 15 18 28 28

1901 1.7 1.7 2.5 2.5 –33 –33 –28 –28 5 7 18 17

1902 2.0 2.1 2.9 2.9 –30 –31 –26 –26 8 11 22 20

1903 6.1 6.2 7.2 7.2 –3 –3 4 4 47 48 60 58

1904 5.9 6.1 7.0 7.1 –5 –6 2 2 45 52 58 62

1905 8.4 8.6 9.7 9.8 12 11 20 19 68 78 82 88

1906 6.5 6.7 7.6 7.8 0 –2 7 6 50 60 64 71

1907 9.3 9.4 10.5 10.6 18 17 26 25 76 83 90 94

1908 10.0 10.1 11.4 11.3 23 23 32 32 83 84 97 95

1909 10.1 10.1 11.4 11.4 24 24 32 32 84 84 98 94

32

Table 3. Parametr�c est�mates for 1868–1931 (Cont�nued).

Year Ap Ap (adjusted) NDD NDD (adjusted) Apmax Apmax (adjusted)

1910 10.4 10.3 11.8 11.5 26 27 34 35 87 77 101 87

1911 9.1 8.9 10.4 10.1 17 18 25 26 75 64 88 73

1912 3.8 3.8 4.7 4.7 –19 –19 –13 –13 25 23 38 33

1913 3.6 3.6 4.6 4.5 –20 –20 –14 –14 23 21 36 31

1914 5.4 5.4 6.4 6.3 –8 –8 –2 –1 40 37 53 47

1915 9.0 9.0 10.2 10.2 16 15 24 24 73 76 87 87

1916 12.2 12.2 13.6 13.6 37 37 47 47 103 103 117 114

1917 10.9 11.3 12.3 12.6 29 27 38 36 92 110 106 121

1918 13.5 13.6 15.0 15.0 46 45 56 56 115 120 129 132

1919 14.1 14.1 15.7 15.6 50 51 61 61 122 120 136 131

1920 10.4 10.4 11.8 11.6 26 26 34 35 87 84 101 94

1921 9.7 9.6 11.0 10.8 20 21 29 30 80 74 93 84

1922 11.3 11.1 12.7 12.4 32 33 41 42 95 83 109 93

1923 4.9 4.8 5.9 5.8 –12 –11 –5 –5 35 32 48 42

1924 4.8 4.8 5.8 5.8 –12 –12 –6 –6 34 35 47 45

1925 7.0 7.1 8.1 8.2 3 2 10 9 55 61 68 71

1926 12.2 12.2 13.6 13.6 37 37 47 47 103 105 117 116

1927 9.7 9.9 11.0 11.2 21 20 29 29 80 89 94 100

1928 10.5 10.7 11.8 12.0 26 25 35 34 87 98 101 82

1929 11.8 11.9 13.2 13.2 35 34 44 44 100 103 113 114

1930 18.8 18.4 20.6 20.2 81 84 94 96 165 145 180 156

1931 9.8 9.7 11.1 10.9 21 22 30 31 81 74 95 84

2.7 Epoch Analyses

F�gure 27 shows the compar�sons between yearly averages of var�ous parametr�c values for cycle 23 (thefilledcircles)andthemeanoftheparametricvalues(theline)forcycles11–22(forR, aa, and NSSC) or 16–22 (Ap, NDD, and Apmax) for elapsed t�me �n years from E(Rmax) on the bas�s of epoch analyses. For R (panel (a)), cycle 23 appears to closely m�m�c the behav�or of the mean of cycles 11–22, suggest�ng, perhaps, that m�n�mum ampl�tude for cycle 24 (�ts onset) w�ll occur �n year 7 follow�ng cycle 23’s E(Rmax), th�s year correspond�ng to 2007. The proport�on of cycles hav�ng E(Rmax)n + 1 sooner than year 7 �s 3/12, wh�le the proport�on hav�ng E(Rmax)n + 1 �n year 7 �s 5/12 and the proport�on hav�ng E(Rmax)n + 1 �n year 8 �s 4/12. All cycles hav�ng E(Rmax)n + 1 �n year 8 are cycles of longer per�od (m�n�mum-to-m�n�mum duration≥135moonthebasisof12-momovingaveragesorsmoothedmonthlymeansunspotnumber),wh�le all cycles hav�ng E(Rmax)n + 1 �n year 7 are cycles of shorter per�od (m�n�mum-to-m�n�mum durat�on ≤126mo).ThosehavingE(Rmax)n + 1 �n years 5 or 6 are a m�xed bag, w�th cycles 15 and 16 be�ng cycles of shorter per�od and cycle 12 be�ng of longer per�od. Now, m�dway through 2006 (year 6), R has averaged only about 16. Th�s value should become smaller as the year progresses, although �t already �s smaller than was seen �n 5 of the past 6 cycles for compar�son year 6 follow�ng E(Rmax). Also, �t �s nearly w�th�n the 90-percent pred�ct�on �nterval for Rm�n based on cycles 11–22 (Rm�n90 = 7 ± 7). So, presently, one cannot d�scount that year 2006 m�ght ult�mately be the m�n�mum ampl�tude year for cycle 24 mark�ng �ts

33

0

(a) (b) (c)

(d) (e) (f)2 4 6 8

E(Apmin)n + 1

MeanCycles 16–22

MeanCycles 16–22

MeanCycles 11–22 Mean

Cycles 11–22

MeanCycles 16–22

Cyc

le 1

6

Cyc

le 1

8

Cyc

les

17, 1

9, 2

1, a

nd 2

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Cyc

le 2

0

5

10

15

25

20

Ap

Elapsed Time in Years From E(Rmax) Elapsed Time in Years From E(Rmax) Elapsed Time in Years From E(Rmax)

E(Rmin)n + 1

Cyc

le 1

6

Cyc

les

12 a

nd 1

5

Cyc

les

17, 1

8, 1

9, 2

1, a

nd 2

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les

11, 1

3, 1

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nd 2

0

0 2 4 6 8

MeanCycles 11–22

50

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200

150

R

0 2 4 6 8

E(NDDmin)n + 1

Cyc

le 1

6

Cyc

les

17, 1

8, 1

9, 2

1, a

nd 2

2

Cyc

le 2

0

20

0

40

60

80

100

120

ND

D

E(aamin)n + 1

Cyc

le 1

6

Cyc

les

12, 1

5, a

nd 1

8

Cyc

les

13, 1

4, 1

7,19

, 21,

and

22

Cyc

les

11 a

nd 2

0

0 2 4 6 8

10

0

20

40

30

aa E(NSSCmin)n + 1

Cyc

les

12, 1

3, 1

5, 1

7, 1

9, a

nd 2

1

Cyc

le 1

6

Cyc

les

14, 1

8, 2

0, a

nd 2

2

Cyc

le 1

1

0 2 4 6 8

20

0

40

80

60

NSS

C

0 2 4 6 8

E((Apmax)min)n + 1

Cyc

les

16, 1

8, 1

9, a

nd 2

2

Cyc

les

17 a

nd 2

1

Cyc

le 2

0

100

0

200

300

Apm

ax

Figure27.Comparisonofcycle23parametricvalues(filledcircles)and cycl�c averages (cycles 11–22 or 16–22) based on epoch analyses us�ng E(Rmax) as the temporal marker. See text for deta�ls.

onset. (It should be noted that, as yet, no new cycle spots have been seen, th�s be�ng a cruc�al parameter for herald�ng the onset of a new sunspot cycle.32–34)

For aa (panel (b)), �t too has m�m�cked the mean for cycles 11–22 rather closely, except for the value at year 3 from E(Rmax) wh�ch �s the h�ghest on record, be�ng nearly 2.5 standard dev�at�ons above the mean. The bulk of the cycles (6/12) have had E(aam�n)n + 1 �n year 8, correspond�ng to 2008, suggest�ng that m�n�mum ampl�tude occurrence for cycle 24 should be expected �n 2007, s�nce E(aam�n) usually follows E(Rm�n) by 1 yr (true for 9 of 12 cycles).

34

For NSSC (panel (c)), aga�n, �t too m�m�cs the mean for cycles 11–22, except now the value for year 3 from E(Rmax) �s lower than any of the past 6 cycles, but w�th�n the observed range of values (9–49) for all known cycles. Half of the sunspot cycles had E(NSSCm�n)n + 1 �n year 6 from E(Rmax), correspond�ng to 2006. S�nce NSSC �s strongly correlated (r = 0.87) aga�nst R (fig.19),itmaybethat2006m�ght turn out to be the m�n�mum ampl�tude year for cycle 24. (It should be noted that cycle 21 had three consecut�ve years of NSSC = 20, �n years 5, 6, and 7 from E(Rmax). Here, year 6 has been used as the m�n�mum value year.)

For Ap (panel (d)), cycle 23’s values generally have been below that of the mean for cycles 16–22, except for the peak �n year 3 from E(Rmax). However, whereas the peak at year 3 for aa �s of record value, the value at year 3 for Ap �s only the th�rd largest, below that wh�ch was seen for cycles 19 (23.7) and 21 (22.6).

For NDD (panel (e)), l�ke Ap, the values generally have been below the mean for cycle 16–22, except for year 3 from E(Rmax). The value of 113 d�sturbed days, however, matches the h�ghest ever seen, hav�ng occurred tw�ce before �n cycle 18 dur�ng years 4 and 5 from E(Rmax).

For Apmax (panel (f)), cycle 23’s values have been above the mean for every year except year 2 from E(Rmax). Its max�mum value (Apmax = 204) occurred �n year 3.

Figure28issimilartofigure27,exceptnowthecomparisonepochisE(Rm�n)n + 1, mean�ng the epoch of the next cycle’s m�n�mum ampl�tude Rmin.Thefilledcirclesrefertocycle23’sparametricvalues,drawn presum�ng that onset for cycle 24 w�ll be 2007, and the l�nes represent the parametr�c means. For R (panel (a)), the 90-percent pred�ct�on �nterval for R �n year –1 relat�ve to E(Rm�n)n + 1 �s R90 = 13.9 ± 11.1 and, as already been ment�oned, for year 0 relat�ve to E(Rm�n)n + 1 �t �s 7 ± 7. (If eventually the year 2006 �s recogn�zed as the m�n�mum ampl�tude year for cycle 24, then cycle 23’s parametr�c values must be moved 1 yr to the r�ght �n all panels.)

For aa (panel (b)), yearly values for cycle 23 are expected to cont�nue to decl�ne through E(Rm�n)n + 1 (actually to the year follow�ng the onset year for cycle 24). Certa�nly, �f 2007 represents the m�n�mum ampl�tude year for cycle 24, then E(aam�n)n + 1 for cycle 24 should be expected �n 2008. (The proport�on of cycles hav�ng E(aam�n) �n the year follow�ng E(Rm�n) �s 9/12.)

For NSSC (panel (c)), all yearly values for cycle 23 are above the mean, except for year – 4 (2003). Its value (NSSC = 11) �s more than two standard dev�at�ons below the mean and �s outs�de the range of prev�ously observed cycles, 14 – 49. On the other hand, one should note the sharp decrease �n the mean at year –3. Should the low value of NSSC for the year 2003 be better assoc�ated w�th year −3 rather than – 4, th�s m�ght be an �nd�cat�on that the m�n�mum ampl�tude year for cycle 24 �s 2006 rather than 2007. Assum�ng that the 2003 value of NSSC �s year –3, �ts value �s now only one standard dev�at�on below the mean and �s w�th�n the range of prev�ously observed cycles, 9–37. The proport�on of cycles hav�ng E(NSSCm�n)n + 1 at year –1 �s 4/12 and 5/12 at year 0. (If one assumes the 2003 value represents the year –3 value from E(Rm�n)n + 1, then the last filled circle, the 2005 value, occurs at year –1 from E(Rm�n)n + 1.)

35

For Ap (panel (d)), all yearly values for cycle 23 are below the mean, except for year – 4 (2003). Its value (Ap = 21.7) �s w�th�n the range of prev�ously observed cycles, whether one presumes �t to be year – 4 (12.7 – 23.7) or year –3 (14.4 – 22.3). The proport�on of cycles hav�ng E(Apm�n)n + 1 at year 1 relat�ve to E(Rm�n)n + 1 �s 5/7.

For NDD (panel (e)), all yearly values for cycle 23 are below the mean, except for year – 4 (2003), wh�ch �s outs�de the range of prev�ously observed cycles (32–107). Its value (NDD = 113), however, �s w�th�n the range of prev�ously observed cycles for years –3 (52–113) and –2 (40–113). The proport�on of cycles hav�ng E(NDDm�n)n + 1 at year 1 relat�ve to E(Rm�n)n + 1 �s 7/7.

For Apmax (panel (f)), all yearly values for cycle 23 have been above the mean, except for year –5 (2002), although �t �s w�th�n the range of prev�ously observed cycles (73–236). Sh�ft�ng the 2002 value (78) to year – 4, however, places �t outs�de the range of prev�ously observed cycles (136–280). The proport�on of cycles hav�ng E((Apmax)m�n)n + 1 �s 3/7 for year 0 and 4/7 for year 1 relat�ve to E(Rm�n)n + 1.

E(Apmin)n + 1

MeanCycles16–22

MeanCycles11–22

MeanCycles11–22

MeanCycles11–22

MeanCycles16–22

MeanCycles16–22

Cyc

les

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nd 1

8

Cyc

les

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9, 2

0, 2

1, a

nd 2

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5

10

15

25

20

Ap

–6(a) (b) (c)

(d) (e) (f)

–4 –2 0

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100

200

150

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Elapsed Time in Years From E(Rmin)n + 1 Elapsed Time in Years From E(Rmin)n + 1 Elapsed Time in Years From E(Rmin)n + 1

–6 –4 –2 0

10

0

20

40

30

aa

E(NDDmin)n + 1

Cyc

les

16, 1

7, 1

8, 1

9, 2

0,

21

, and

22

20

0

40

60

80

100

120

ND

D

E((Apmax)min)n + 1

Cyc

les

18, 1

9, a

nd 2

2

Cyc

les

16, 1

7, 2

0, a

nd 2

1

100

0

200

300

Apm

axE(aamin)n + 1

Cyc

les

11, 1

2, 1

5, 1

6, 1

7,

18

, 20,

21,

and

22

Cyc

les

13, 1

4, a

nd 1

9

20

40

80

60

NSS

CE(NSSCmin)n + 1

Cyc

les

17, 1

9, 2

0, a

nd 2

1

Cyc

les

12, 1

4, 1

5, 1

6, 1

8, a

nd 2

2

Cyc

les

11 a

nd 1

3

0–6 –4 –2 0

–6 –4 –2 0 –6 –4 –2 0 –6 –4 –2 0

Figure28.Comparisonofcycle23parametricvalues(filledcircles) and cycl�c averages (cycles 11–22 or 16–22) based on epoch analyses us�ng E(Rmax)n + 1. See text for deta�ls.

36

3. CONCLUSION

Th�s study has exam�ned a var�ety of geomagnet�c �nd�ces �n relat�on to the s�zes and occurrences ofminimumandmaximumamplitudesofsunspotcycles.Thestudyconfirmsthatbothsunspotnumberand the aa geomagnet�c �ndex have �ncreased over t�me, such that cycles of late are among the largest on record. Wh�le both sunspot number and the aa �ndex �s strongly correlated, there are d�fferences between them. Namely, each sunspot cycle usually has a s�ngle peak, wh�le the aa �ndex usually has two or more peaks, often w�th the major peak occurr�ng dur�ng the decl�n�ng port�on of the sunspot cycle.

The aa �ndex was decomposed �nto two componentsone assoc�ated d�rectly w�th sunspot number R and the other be�ng the res�dual �nterplanetary component due to coronal holes and the l�ke. Plots of the res�dual �nterplanetary component, however, cont�nue to d�splay mult�ple peaks dur�ng the decl�n�ng port�on of the sunspot cycle, w�th the largest usually be�ng the last peak just pr�or to sunspot m�n�mum for the next sunspot cycle. The peaks, whether us�ng the largest or the last-occurr�ng peak, strongly correlate w�th both the m�n�mum and max�mum ampl�tudes of the follow�ng cycle, typ�cally several years �n advance. Unfortunately, the mult�ple peaks dur�ng the decl�ne of cycle 23 present a d�lemma for est�mat�ng the max�mum ampl�tude of the next cycle. Based on the max�mum value of the res�dual for cycle 23, cycle 24 w�ll have Rm�n90 = 13 ± 4.9 and Rmax90 = 183.7 ± 46.3, wh�le based on the last observed res�dual peak, cycle 24 w�ll have Rm�n90 = 7.2 ± 5 and Rmax90 = 117.5 ± 41.2, y�eld�ng an overlap of 8.1–12.2 for Rm�n and 137.4–158.7 for Rmax. The d�lemma can be m�t�gated by apply�ng a 2-yr mov�ng average to the res�duals, thereby y�eld�ng pred�ct�ons of Rm�n90 = 10.4 ± 3 and Rmax90 = 157.5 ± 31.8 for cycle 24, these annual averages equ�valent to 9.4 ± 2.9 and 162.9 ± 33.5, respect�vely, expressed as smoothed monthly mean sunspot number.

In recent years, wh�le stud�es ma�nta�n that the aa �ndex has �ncreased substant�ally over t�me, compar�sons of �t w�th the IHV �ndex suggest that values of the aa �ndex pr�or to 1957 m�ght be somewhat inaccurate. In 1957, the NHmagnetometer used for deriving the official aa �ndex was moved from Ab�nger, England, to �ts present locat�on �n Hartland, England. The offset for the earl�er aa values appears to measure about 3. Hence, by s�mply add�ng 3 to the observed values of aa, one can adjust aa, br�ng�ng the two d�sparate datasets �nto closer agreement. Do�ng so and repeat�ng the analyses results �n pred�ct�ons of Rm�n90 = 9.8 ± 2.9 and Rmax90 = 153.8 ± 24.7 for cycle 24, these values equ�valent to 8.8 ± 2.8 and 159 ± 25.5, respect�vely, expressed as smoothed monthly mean sunspot number. Us�ng the adjusted values for aa does not significantly alter the predictions ofminimumandmaximumamplitude for cycle 24,although �t does reduce the uncerta�nty by 3 percent for Rm�n and 22 percent for Rmax.

Once parametr�c cycl�c m�n�mum values are observed (usually, the m�n�mum values of the geomagnet�c �nd�ces occur �n the year follow�ng sunspot m�n�mum), one can use these m�n�mum values of the geomagnet�c �nd�ces to deduce w�th h�gher prec�s�on (±15.5 un�ts of sunspot number) the s�ze fortheongoingcycle,eitheronthebasisofsingle-variateorbivariatefits(wherethebivariatefitsalso�ncorporate Rm�n). If the epoch of sunspot m�n�mum �s the year 2006, one should expect the m�n�mum �n the geomagnet�c �nd�ces �n 2007. On the other hand, �f the epoch of sunspot m�n�mum �s delayed unt�l

37

2007, then the m�n�mum �n the geomagnet�c �nd�ces should not be expected unt�l 2008, w�th sunspot max�mum ampl�tude expected about 2–3 yr later. The best pred�ctor of sunspot max�mum ampl�tude after the onset of the new cycle appears to be the one �ncorporat�ng both NDDm�n and Rm�n (bv4), be�ng Rmax90 = 121.4 – 5.877 Rm�n + 3.665 NDDm�n ± 15.5 and hav�ng r = 0.978 and r2 = 0.956, mean�ng that 95.6percentofthevariancecanbeexplainedbythebivariatefit.

Each of the parameters Rm�n, Rmax, and aam�n d�splays a secular r�se over cycles 12–23, such that values for cycle 24 are Rm�n90 = 12.3 ± 4.8 (or Rm90 = 11.2 ± 4.6), Rmax90 = 167.9 ± 55 (or RM90 = 173.3 ± 56.5), and aa90 = 20.6 ± 4.6. Secular r�ses �n Ap, NDD, or Apmax cannot be detected, ow�ng to the brev�ty of these records (only about half as long as that for aa).

F�gure 29 shows the values for seven cycle-related parameters for the �nterval of January 2005–July 2006, �nclud�ng R, aa, Ap, NDD, Apmax, NSD (the number of spotless days28,30,35 dur�ng the month), and dmax (the max�mum da�ly value of sunspot number dur�ng the month) (panels (a) through (g), respect�vely). Plotted are the monthly means of each parameter, w�th the hor�zontal l�ne drawn dur�ng the year 2005 represent�ng the yearly parametr�c average. To the r�ght are yearly parametr�c averages (90-percent probab�l�ty �ntervals) dur�ng the sunspot m�n�mum year. Clearly, values now be�ng exper�enced �n the year 2006 suggest that cycle m�n�mum �s �mm�nent; the yearly averages for 2006 are now fall�ng w�th�n the 90-percent probab�l�ty �ntervals �nd�cat�ve of the sunspot m�n�mum year.

In add�t�on to the secular r�ses found for Rm�n, Rmax, and aam�n, cycl�c averages (m�n�mum-to-minimum)forsomeoftheparametersarefoundtoshowstatisticallysignificantsecularincreasesaswell.For cycle 24, <R>90 = 82.4 ± 29.5, <aa>90 = 26.5 ± 4, <NSSC>90 = 35.2 ± 6.1, and ΣNSSC90 = 358 ± 62. Also, based on a presumed �nherent behav�or (“below-above-above-below”), <Ap> w�ll be >14.8, <NDD> > 54.4, and ΣNDD >563 for cycle 24. Furthermore, cycl�c averages for cycle 24 can be deduced from cycle 24’s Rm�n and Rmax,oncetheyareobserved.Anunexpectedfindingisthat<NSSC>andΣNSSC scatterplots cluster �nto two d�st�nct cycle group�ngs: 11–16 and 17–23.

Concern�ng Apmax, all cycles are found to have had at least 1 day of Apmax ≥ 100 and all but one (cycle 20) had at least 1 day of Apmax ≥ 200. Opportun�t�es for large Apmax rema�n throughout the solar cycle, �nclud�ng near sunspot m�n�mum. For the 15 yr when R was <20, Apmax spanned 38 to 202, w�th one-th�rd of the years hav�ng at least 1 day of Apmax ≥ 100. Apmax ≥ 100 has never been seen when Ap < 10. Once Rmax for a sunspot cycle has been observed, the number of days that Apmax ≥ 100 can be eas�ly est�mated.

OnthebasisofthestatisticallysignificantregressionsbetweenAp, NDD, and Apmax aga�nst aa dur�ng the �nterval 1932–2005, est�mates can be made for the earl�er �nterval 1868–1931.

Epoch analyses for each of the parameters suggest that sunspot m�n�mum year for cycle 24 probably �s the year 2007, although one cannot, as yet, rule out the year 2006 as the sunspot m�n�mum year. For someoftheparameters,abetterfitisfoundusingtheyear2006asthesunspotminimumyear,whileforothers,thebetterfitisfoundusingtheyear2007asthesunspotminimumyear.

38

7 6.6*

14.5 9.9*

JJ

29.8

J

20

0(a)

(b)

(c)

(d)

(e)

(f)

(g)

40

Calendar Month-Year *90% Intervals for E(Rmin)

2005 2006

Raa

39.7 22.2*

55.4

50

0

100

Apm

ax

22.9 11.9*

56.8

50

0

100

dmax

23.2

20

0

40

11 2.7*

Ap

13.6

20

10

0

30

17 10*NSD

1.0

20

10

0

30

2.5 1.2*ND

D 3.620

10

0

F�gure 29. Parametr�c values for 2005 and 2006.

39

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42

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. 239-18298-102

Unclassified Unclassified Unclassified Unl�m�ted

An Exam�nat�on of Selected Geomagnet�c Ind�ces �n Relat�onto the Sunspot Cycle

Robert M. W�lson and Dav�d H. Hathaway

George C. Marshall Space Fl�ght CenterMarshall Space Fl�ght Center, AL 35812

Nat�onal Aeronaut�cs and Space Adm�n�strat�onWash�ngton, DC 20546–0001

PreparedbytheScienceandExplorationResearchOffice,ScienceandMissionSystemsOffice

Unclassified-UnlimitedSubject Category 92Ava�lab�l�ty: NASA CASI 301–621–0390

Prev�ous stud�es have shown geomagnet�c �nd�ces to be useful for prov�d�ng early est�mates for the s�ze of the fol-low�ng sunspot cycle several years �n advance. Exam�ned �n th�s study are var�ous precursor methods for pred�ct�ng the m�n�mum and max�mum ampl�tude of the follow�ng sunspot cycle, these precursors based on the aa and Ap geo-magnet�c �nd�ces and the number of d�sturbed days (NDD), days when the da�ly Ap �ndex equaled or exceeded 25. Also exam�ned �s the yearly peak of the da�ly Ap �ndex (Apmax), the number of days when Ap≥100,cyclicaveragesof sunspot number R, aa, Ap, NDD, and the number of sudden storm commencements (NSSC), as well the cycl�c sums of NDD and NSSC. The analys�s y�elds 90-percent pred�ct�on �ntervals for both the m�n�mum and max�mum ampl�tudes for cycle 24, the next sunspot cycle. In terms of yearly averages, the best regress�ons g�ve Rm�n = 9.8 ± 2.9 and Rmax = 153.8 ± 24.7, equ�valent to Rm = 8.8 ± 2.8 and RM = 159 ± 25.5, based on the 12-mo mov�ng average (or smoothed monthly mean sunspot number). Hence, cycle 24 �s expected to be above average �n s�ze, s�m�lar to cycles 21 and 22, produc�ng more than 300 sudden storm commencements and more than 560 d�sturbed days, of wh�ch about 25 w�ll be Ap≥100.Onthebasisofannualaverages,thesunspotminimumyearforcycle24willbeeither2006 or 2007.

52

M–1178

Techn�cal Publ�cat�onDecember 2006

NASA/TP—2006–214711

Sun, sunspot cycle, solar cycle pred�ct�on, geomagnet�c �nd�ces

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NASA/TP—2006–214711

An Examination of Selected GeomagneticIndices in Relation to the Sunspot CycleRobert M. Wilson and David H. HathawayMarshall Space Flight Center, Marshall Space Flight Center, Alabama

December 2006

National Aeronautics andSpace AdministrationIS20George C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama35812

an examination of selected geomagnetic indices in relation ... · pdf filean examination of...

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