using the inflection points and rates of growth and decay ...using the inflection points and rates...
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NASA/TP—2008–215473
Using the Inflection Points and Rates of Growth and Decay to Predict Levels of Solar ActivityRobert M. Wilson and David H. HathawayMarshall Space Flight Center, Marshall Space Flight Center, Alabama
September 2008
National Aeronautics andSpace AdministrationIS20George C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama35812
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NASA/TP—2008–215473
Using the Inflection Points and Rates of Growth and Decay to Predict Levels of Solar ActivityRobert M. Wilson and David H. HathawayMarshall Space Flight Center, Marshall Space Flight Center, Alabama
September 2008
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 An Overv�ew of Cycl�c Behav�or (Cycles 12–23) ............................................................. 3 2.2 Correlat�ve Results .......................................................................................................... 11
3. SUMMARY ......................................................................................................................... 34
REFERENCES ......................................................................................................................... 36
�v
v
LIST OF FIGURES
1. Var�at�on of R for elapsed t�me �n months t = 0–132 mo from E(Rm) ........................ 4
2. Var�at�on of ΔR for elapsed t�me �n months t = 0–132 mo from E(Rm) ...................... 6
3. Var�at�on of 12-mma of ΔR for elapsed t�me �n months t = 0–126 mo from E(Rm) ... 7
4. Var�at�on of parametr�c values over cycles 12–23 ....................................................... 8
5. Var�at�on of spec�fic t�m�ng s�gnatures based on ΔR .................................................. 9
6. Var�at�on of spec�fic t�m�ng s�gnatures based on 12-mma of ΔR ................................ 10
7. Scatter plot of RM versus ΔRmaxpos ........................................................................ 13
8. Scatter plot of RM versus slope at E(ΔRmaxpos) ...................................................... 14
9. Var�at�on of r2(ΔR(t)) ................................................................................................ 16
10. Scatter plots of RM versus ΔR(t = 10) and RM versus ΔR(t = 25) .............................. 17
11. Scatter plot of RM versus (12-mma of ΔR)maxpos ................................................... 18
12. Var�at�on of r2(12-mma of ΔR(t)) .............................................................................. 19
13. Scatter plots of RM versus 12-mma of ΔR(t = 8) and RM versus 12-mma of ΔR(t = 22) .............................................................................................................. 20
14. Scatter plot of PER versus ΔRmaxneg ...................................................................... 21
15. Scatter plot of Rm(cycle n + 1) versus ΔRmaxneg ...................................................... 22
16. Scatter plot of RM(cycle n + 1) versus ΔRmaxneg ...................................................... 23
17. Scatter plot of PER versus (12-mma of ΔR)maxneg .................................................. 24
18. Scatter plot of Rm(cycle n + 1) versus (12-mma of ΔR)maxneg .................................. 25
19. Scatter plot of RM(cycle n + 1) versus (12-mma of ΔR)maxneg ................................. 26
20. Scatter plots of PER versus ΔR(T = 30) and 12-mma of ΔR(T = 31) ........................... 27
v�
21. Scatter plots of Rm(cycle n + 1) versus ΔR(T = 31) and 12-mma of ΔR(T = 27) ........... 28
22. Scatter plots of RM(cycle n + 1) versus ΔR(T = 14) and 12-mma of ΔR(T = 44) .......... 29
23. Scatter plots of SlopeASC versus ΔRmaxpos and (12-mma of ΔR)maxpos and of SlopeDES versus ΔRmaxpos and (12-mma of ΔR)maxpos .............................. 31
24. Scatter plots of SlopeDES versus ΔRmaxpos, (12-mma of ΔR)maxpos, and SlopeASC ............................................................................................................. 32
25. Var�at�on of cycle 23’s R and 12-mma of ΔR for t = 0–135 (May 1996–August 2007) ........................................................................................... 33
v��
LIST OF TABLES
1. Average parametr�c values for selected group�ngs of cycles ........................................ 5
2. Selected parametr�c values for sunspot cycles 12–23 ................................................... 11
3. Correlat�ons aga�nst sunspot cycle number ................................................................. 12
v���
LIST OF ACRONYMS
12-mma 12-mo mov�ng average
NOAA Nat�onal Ocean�c and Atmospher�c Adm�n�strat�on
�x
NOMENCLATURE
a y-�ntercept �n the regress�on equat�on
ASC ascent durat�on (elapsed t�me from E(Rm) to E(RM))
<ASC> mean ascent durat�on
b slope �n the regress�on equat�on
cl confidence level
DES descent durat�on (elapsed t�me from E(RM) to E(Rm) of the follow�ng cycle)
E(Rm) epoch of Rm
E(RM) epoch of RM
E(ΔRmaxneg) epoch of ΔRmaxneg
E(ΔRmaxpos) epoch of ΔRmaxpos
P probab�l�ty
PER per�od (cycle durat�on)
<PER> mean cycle per�od
r coeffic�ent of l�near correlat�on
r2 coeffic�ent of determ�nat�on
R smoothed monthly mean sunspot number
Rm sunspot m�n�mum ampl�tude
<Rm> mean Rm
RM sunspot max�mum ampl�tude
<RM> mean RM
x
NOMENCLATURE (Continued)
ΔR month-to-month change �n R
ΔRmaxneg max�mum negat�ve rate of change �n R
ΔRmaxpos max�mum pos�t�ve rate of change �n R
sd standard dev�at�on
se standard error of est�mate
t elapsed t�me �n months from E(Rm)
t(1) elapsed t�me �n months from E(Rm) to E(ΔRmaxpos) or E((12-mma of R)maxpos)
t(2) elapsed t�me �n months from E(ΔRmaxpos) or E((12-mma of R)maxpos) to E(RM)
t(3) elapsed t�me �n months from E(RM) to E(ΔRmaxneg) or E((12-mma of R)maxneg)
t(4) elapsed t�me �n months from E(ΔRmaxneg) or E((12-mma of R)maxneg) to E(Rm) of the follow�ng cycle
T elapsed t�me �n months from E(RM)
x �ndependent var�able �n the regress�on equat�on
y dependent var�able �n the regress�on equat�on
1
TECHNICAL PUBLICATION
USING ThE INFLECTION POINTS AND RATES OF GROwTh AND DECAY TO PREDICT LEvELS OF SOLAR ACTIvITY
1. INTRODUCTION
Sunspot cycles are convent�onally descr�bed us�ng 12-mo mov�ng averages (12-mmas) of monthly mean sunspot number, where (s�nce 1981) the offic�al number �s now determ�ned by the Sunspot Index Data Center <http://s�dc.oma.be/�ndex.php3>,1 located at the Royal Observatory of Belg�um (�n Brussels). Formerly, �t was ma�nta�ned at the Sw�ss Federal Observatory �n Zur�ch, Sw�tzerland. The m�n�mum value of the 12-mma of monthly mean sunspot number �s called the sunspot m�n�mum ampl�tude (Rm) and �ts occurrence denotes the epoch of Rm (E(Rm)). L�kew�se, the max�mum value of the 12-mma of monthly mean sunspot number �s called sunspot max�mum ampl�tude (RM) and �ts occurrence denotes the epoch of RM (E(RM)). In real�ty, sunspot m�n�-mum and max�mum are better p�ctured as be�ng broad �ntervals of t�me of several years �n length when sunspot numbers are predom�nantly lower and h�gher, respect�vely, rather than be�ng spec�fic �nstances �n t�me. Wh�le th�s �s true, �t �s conven�ent to use the m�n�mum and max�mum values and the�r epochs of occurrence to descr�be the general character�st�cs of a sunspot cycle, such as �ts rela-t�ve s�ze, ascent durat�on (ASC), descent durat�on (DES), and cycle length (called per�od (PER)).
Follow�ng Rm, the sunspot number gradually �ncreases �n value, typ�cally reach�ng a max�-mum rate of growth about 1–2 yr after E(Rm) and atta�n�ng RM about 3–5 yr after E(Rm). The po�nt of max�mum pos�t�ve rate of growth �n sunspot number represents the ascend�ng �nflect�on po�nt, and �ts numer�cal value has proven to be useful for est�mat�ng the later occurr�ng RM.2–10 L�kew�se, dur�ng the decl�n�ng port�on of the sunspot cycle, there �s a second �nflect�on po�nt (the descend�ng �nflect�on po�nt), occurr�ng typ�cally about 6–7 yr after Rm, wh�ch seems to be related to the per�od of the ongo�ng sunspot cycle.2
The purpose of th�s Techn�cal Publ�cat�on �s to reexam�ne the role of the �nflect�on po�nts and rates of growth and decay �n sunspot cycle pred�ct�on on the bas�s of the behav�ors of cycles 12–23, the most rel�ably determ�ned sunspot cycles.11–13
2
3
2. RESULTS AND DISCUSSION
2.1 An Overview of Cyclic Behavior (Cycles 12–23)
F�gure 1 d�splays the mean cycle curve, based on epoch analys�s and cycles 12–23 (the smoothed th�ck l�ne) for elapsed t�me �n months (t) from E(Rm), t = 0–132 mo. The th�n l�nes plotted above and below the mean cycle curve represent the upper and lower 90-percent pred�ct�on l�m�ts about each monthly value, respect�vely. The occurrence dates of E(RM) relat�ve to E(Rm), wh�ch determ�nes the ASC for each of the cycles, are noted across the top, and the actual RM values for the cycles are noted to the r�ght. The respect�ve PER for each of the cycles �s noted near the bottom r�ght, where cycle 23’s PER �s not shown s�nce �t rema�ns ongo�ng. It should be noted, however, that the Nat�onal Ocean�c and Atmospher�c Adm�n�strat�on (NOAA) Solar Cycle 24 Pred�ct�on Panel14 has pred�cted cycle 24’s offic�al onset to occur about March 2008 ± 6 mo, �nferr�ng that cycle 23’s PER w�ll measure about 141 ± 6 mo; hence, �t w�ll be a cycle of longer cycle length, l�ke cycles 12–14 and 20 (the first new cycle spot was reported1 �n January 2008).
Inspect�on of figure 1 suggests that cycles 12–23 m�ght be loosely grouped e�ther �nto three arb�trary groups based on relat�ve s�ze (large-ampl�tude cycles (18, 19, 21, and 22), average-ampl�-tude cycles (15, 17, 20, and 23), and small-ampl�tude cycles (12, 13, 14, and 16)) or poss�bly �nto two groups based on cycle length (short-per�od cycles (15–19, 21, and 22) and long-per�od cycles (12–14, 20, and 23)).15 Table 1 g�ves averages and standard dev�at�ons (�n parentheses) for Rm, RM, ASC, and PER for each of these group�ngs. It �s found that large-ampl�tude cycles tend to r�se more qu�ckly to RM (shorter ASC), be of shorter PER, and have h�gher Rms than average-s�ze cycles, wh�ch �n turn have h�gher/shorter values as compared to small-ampl�tude cycles. L�kew�se, short-per�od cycles tend to r�se more qu�ckly to RM and to have h�gher Rms and RMs than long-per�od cycles.
F�gure 2 shows the mean cycle curve of the month-to-month rate of change �n the smoothed monthly mean sunspot number (R), based on epoch analys�s and cycles 12–23 (the smoothed th�ck l�ne) for elapsed t�me �n months from E(Rm), t = 0–132 mo. The occurrence dates for each cycle’s max�mum pos�t�ve rate of change dur�ng the ascend�ng port�on of the sunspot cycle are across the top, and the actual values of the max�mum pos�t�ve rate of change dur�ng the ascend�ng port�on of the sunspot cycle are to the upper r�ght. The occurrence dates for each cycle’s max�mum negat�ve rate of change dur�ng the descend�ng port�on of the sunspot cycle are at the bottom, and the actual values of the max�mum negat�ve rate of change dur�ng the descend�ng port�on of the sunspot cycle are to the lower r�ght.
Based on the aforement�oned group�ngs (ampl�tude and per�od), �t �s found that large-ampl�-tude cycles tend to have h�gher rates of change both dur�ng the ascend�ng (9.5(1.4)) and descend�ng (–7.5(1.1)) port�ons of the sunspot cycle as compared to cycles of average s�ze (6.7(1.3) and –5.2(0.3)) and cycles of smaller ampl�tude (4.7(1.1) and –4.7(0.9)). Also, cycles of shorter per�od tend to have h�gher rates of change dur�ng both the ascend�ng (8.5(2.0)) and descend�ng (–6.5(1.5)) port�ons of the sunspot cycle as compared to cycles of longer per�od (4.9(1.0) and –4.7(0.8)). The t�m�ng of the
4
20 40 60 80 100 120 140
20
40
60
80
100
120
140
160
180
200
mean 12–23
PER
E(RM)
22 15 18 17 19 12 14 20 13
16 21
22 18 21 17 13 1216
14, 2
015
19, 2
3
RM
19
21
2317
20
15
14
1216
13
22
18
t, Elapsed time in months from E(Rm)
R (s
moo
thed
mon
thly
mea
n su
nspo
t num
ber)
5-377441(F1)F�gure 1. Var�at�on of R for elapsed t�me �n months t = 0–132 mo from E(Rm).
5
Table 1. Average parametr�c values for selected group�ngs of cycles.
Group Cycles <Rm> <RM> <ASC> <PER>*Amplitude: Large 18, 19, 21, 22 8.9 (4.2) 169.0 (22.1) 40.5 (5.4) 121.8 (4.2) Average 15, 17, 20, 23 5.6 (3.8) 114.0 (7.3) 46.8 (2.6) 128.3 (10.4) Small 12-14, 16 3.9 (1.7) 76.2 (9.8) 52.8 (6.4) 134.3 (9.3)Period: Short 15–19, 21, 22 6.6 (4.3) 139.8 (41.4) 44.1 (7.1) 121.9 (3.3) Long 12–14, 20, 23 5.5 (3.3) 91.6 (23.8) 50.2 (5.6) 139.0 (2.9)
* Excludes Cycle 23
max�mum pos�t�ve rate of change �n R (ΔRmaxpos) and the max�mum negat�ve rate of change of R (ΔRmaxneg) are found to span 14–42 and 51–83 mo, respect�vely, and to average 27.8 (10.1) and 71.4 (9.5) mo, respect�vely.
F�gure 3 dep�cts the 12-mma of the month-to-month rate of change shown �n figure 2. The th�ck l�ne �s the mean curve and the th�n l�nes above and below the mean curve represent the 90- percent pred�ct�on l�m�ts. Smooth�ng removes a cons�derable amount of the chopp�ness that appears �n figure 2. The occurrences of the 12-mma of ΔRmaxpos are across the top, the �nd�v�dual values of the 12-mma of ΔRmaxpos are to the upper r�ght, the occurrences of the 12-mma of ΔRmaxneg are across the bottom, and the �nd�v�dual values of the 12-mma of ΔRmaxneg are to the lower r�ght.
Based on the aforement�oned group�ngs (ampl�tude and per�od), �t �s found that large-ampl�-tude cycles tend to have h�gher rates of change both dur�ng the ascend�ng (6.8(1.3)) and descend�ng (–4.6(0.6)) port�ons of the sunspot cycle as compared to cycles of average s�ze (3.9(0.3) and –3.0(0.4)) and small-ampl�tude cycles (2.7(0.6) and –2.3(0.5)). Also, cycles of shorter per�od tend to have h�gher rates of change dur�ng the ascend�ng (5.5(1.9)) and descend�ng (–3.8(1.2)) port�ons of the sunspot cycle as compared to cycles of longer per�od (3.1(0.8) and –2.6(0.7)). The t�m�ng of the 12-mma of ΔRmaxpos and ΔRmaxneg �s found to span 14–41 and 66–96 mo, respect�vely, and �s found to aver-age 23.7 (7.8) and 76.8 (8.3) mo, respect�vely.
F�gure 4 dep�cts the cycl�c var�at�on of the prev�ously d�scussed parameters. The hor�zontal l�ne �n each subpanel �s the mean. The standard dev�at�on (sd) �s also g�ven. It �s not�ceable that cycles of late (cycles 18–23) have h�gher averages of Rm, RM, ΔRmaxpos, ΔRmaxneg, 12-mma of ΔRmaxpos and 12-mma of ΔRmaxneg as compared to earl�er cycles (cycles 12–17), and the d�ffer-ences �n the�r means are stat�st�cally �mportant. Thus, cycles of late have been more robust than earl�er cycles and the h�gher parametr�c values may result from a long-term secular �ncrease over t�me.16,17
F�gure 5 shows the cycl�c var�at�on of spec�fic t�m�ng parameters, where t(1) �s the elapsed t�me �n months from E(Rm) to the epoch of ΔRmaxpos (E(ΔRmaxpos)), t(2) �s the elapsed t�me �n months from E(ΔRmaxpos) to E(RM), t(3) �s the elapsed t�me �n months from E(RM) to the epoch of ΔRmaxneg (E(ΔRmaxneg)), and t(4) �s the elapsed t�me �n months from E(ΔRmaxneg) to E(Rm) of the next cycle. F�gure 6 shows the same t�m�ng parameters but us�ng the 12-mma values of ΔRmaxpos and ΔRmaxneg. In both figures, the hor�zontal l�nes are the means (g�ven to the r�ght �n
6
20 40 60 80 100 120 140
8
6
4
2
0
–2
–4
–6
–8
10
mean 12–23
E( Rmaxneg)
E( Rmaxpos during ASC)R
(rate
of c
hang
e in
R)
22
15, 1
9 12
14
17, 1
8, 2013 23 1621
1816, 1
9, 20
22 2112 13 14, 1
723 15
Rmaxneg
13
12
17
2122
19
15, 20, 231614
18
Rmaxpos
211518
1922
141213
23
2017 16
t, Elapsed time in months from E(Rm)5-377441(F2rev)
∆Δ
Δ
Δ
Δ
F�gure 2. Var�at�on of ΔR for elapsed t�me �n months t = 0–132 mo from E(Rm).
7
20 40 60 80 100 120
–4
–6
–2
0
2
4
6
8
10
mean 12–23
E(12-mma of Rmaxpos)
13 12, 2
019
, 22
16 18 17 15
14, 2
1, 23
19
22
1821
14
12
1315
2023
16
17
t, Elapsed time in months from E(Rm)
12-m
ma o
fR
5-377441(F3)
16, 2
1, 1322 13 18 16 19
15 20
1412
13
22
2023
15, 16
141712
2118
19
(12-mma of Rmaxpos)(12-mma of
Rmaxneg)
E(12-mma of Rmaxneg)
Δ
ΔΔ
Δ
Δ
F�gure 3. Var�at�on of 12-mma of ΔR for elapsed t�me �n months t = 0–126 mo from E(Rm).
8
1412
1618
2022
24
100
150
200 50 51015
Suns
pot C
ycle
Num
ber
RMASCPER Rm
5-37
7441
(F4)
40 305060130
mean
= 12
8.1sd
= 9.2
mean
= 46
.7sd
= 7.0
mean
= 11
9.7sd
= 41.9
mean
= 4.4
9sd
= 1.97
mean
= 7.0
sd= 2
.3
mean
= –3
.28sd
= 1.14
mean
= –5
.8sd
= 1.5
mean
= 6.1
sd= 3
.8
110
150
1412
1618
2022
24
10 8 628 6 4 24
Suns
pot C
ycle
Num
ber
12–mma of Rmaxpos Rmaxpos
1412
1618
2022
24
00 –2 –4–4 –6–2 –8 –10–6
Suns
pot C
ycle
Num
ber
12–mma of Rmaxneg Rmaxneg
Δ
ΔΔ
(a)
(b)
(c)
(d)
(e)
(h)
(g)
(f)
Δ
F�gure 4. Var�at�on of parametr�c values over cycles 12–23.
9
1412 16 18 20 22 24
20
40
20
40
20
40
40
60
80
mean = 27.8sd = 10.1
mean = 18.8sd = 13.5
mean = 25.8sd = 9.3
mean = 57.2sd = 16.0
Based on R
Sunspot Cycle Number
t(1)
t(2)
t(3)
t(4)
5-377441(F5)
Δ
(a)
(b)
(c)
(d)
F�gure 5. Var�at�on of spec�fic t�m�ng s�gnatures based on ΔR.
10
1412 16 18 20 22 24
20
40
20
40
20
40
40
60
80
mean = 23.7sd = 7.8
mean = 23.0sd = 11.4
mean = 30.2sd = 6.8
mean = 51.4sd = 9.6
Based on 12-mma of R
Sunspot Cycle Number
t(1)
t(2)
t(3)
t(4)
5-377441(F6)
Δ
(a)
(b)
(c)
(d)
F�gure 6. Var�at�on of spec�fic t�m�ng s�gnatures based on 12-mma of ΔR.
11
each subpanel along w�th the sd). No s�gn�ficant d�fferences �n the t�m�ng parameters are apparent, compar�ng earl�er cycles w�th those of late. Thus, on average, the greatest pos�t�ve rate of growth �n R occurs about 2 yr after Rm and about 1.5–2 yr before RM, and the greatest negat�ve rate of growth �n R occurs about 2–2.5 yr after RM and about 4–5 yr before succeed�ng cycle Rm. Table 2 prov�des a conven�ent summary of the cycl�c parametr�c values and the�r occurrences relat�ve to E(Rm).
Table 2. Selected parametr�c values for sunspot cycles 12–23.
Cycle Rm(E(Rm)) RM(E(RM)) ASC PER ΔRmaxpos(t) ΔRmaxneg(t)12-mma of
ΔRmaxpos(t)12-mma of
ΔRmaxneg(t )12 2.2 (12–1878) 74.6 (12–1883) 60 135 4.1 (14) –4.1 (83) 2.47 (19) –2.55 (88)13 5.0 (03–1890) 87.9 (01–1894) 46 142 4.6 (16) –3.6 (57) 3.36 (14) –1.70 (69)14 2.6 (01–1902) 64.2 (02–1906) 49 139 3.9 (39) –5.4 (51) 2.00 (20) –2.08 (96)15 1.5 (08–1913) 105.4 (08–1917) 48 120 8.5 (42) –5.3 (75) 3.82 (41) –2.69 (72)16 5.6 (08–1923) 78.1 (04–1928) 56 121 6.3 (22) –5.5 (82) 3.07 (23) –2.70 (78)17 3.4 (09–1933) 119.2 (04–1937) 43 125 6.4 (39) –4.7 (73) 4.25 (35) –2.47 (73)18 7.7 (02–1944) 151.8 (05–1947) 39 122 8.6 (33) –6.9 (73) 5.95 (31) –4.30 (71)19 3.4 (04–1957) 201.3 (03–1958) 47 126 10.8 (22) –6.2 (75) 8.69 (21) –4.79 (79)20 9.6 (10–1964) 110.6 (11–1968) 49 140 6.4 (22) –5.3 (73) 3.97 (19) –3.21 (74)21 12.2 (06–1976) 164.5 (12–1979) 42 123 8.1 (24) –8.4 (74) 5.83 (20) –4.02 (78)22 12.3 (09–1986) 158.5 (07–1989) 34 116 10.6 (21) –8.3 (64) 6.92 (21) –5.45 (66)23 8.0 (05–1996) 120.8 (04–2000) 47 – 5.5 (40) –5.3 (77) 3.57 (20) –3.43 (78)
Recall from figure 4 that there �s the h�nt that cycles of late have been more robust than earl�er cycles and that th�s behav�or may be the result of a long-term secular �ncrease over t�me. Table 3 prov�des a conven�ent summary of the stat�st�cs, compar�ng parametr�c values aga�nst sun-spot cycle number. Indeed, all parameters except PER appear to correlate well aga�nst sunspot cycle number. Table 3 g�ves the coeffic�ent of l�near correlat�on (r), the coeffic�ent of determ�nat�on (r2) (a measure of the amount of var�ance expla�ned by the regress�on), the y-�ntercept �n the regress�on equat�on (a), the slope of the �nferred regress�on (b), the standard error of est�mate (se), and the con-fidence level (cl), where cl >95 percent �s cons�dered stat�st�cally �mportant (cl >90 percent �s of mar-g�nal stat�st�cal �mportance). Project�ons (90-percent pred�ct�on �ntervals) for cycle 24 based on the �nferred regress�ons are also g�ven. Hence, before cycle 24 has offic�ally started, presum�ng the val�d�ty of the �nferred regress�ons, cycle 24 �s expected to have Rm = 11.5 ± 4.5 (th�s w�ll be exceeded low, for R measured 5.9 �n September 2007), RM = 166.9 ± 64.1, ASC = 39.3 ± 10.7 mo, ΔRmaxpos = 9.4 ± 3.7, ΔRmaxneg = –7.6 ± 2.0, 12-mmaΔRmaxpos = 6.55 ± 3.03, and 12-mmaΔRmaxneg = –4.83 ± 1.43. Pre-sum�ng cycle 23 has a PER = 141 mo (�mply�ng E(Rm) for cycle 24 �n March 2008), PER for cycle 24 �s est�mated to be about 125 ± 18 mo.
2.2 Correlative Results
2.2.1 ΔRmaxpos
The role of ΔRmaxpos as a pred�ctor for RM w�ll be exam�ned �n th�s sect�on. F�gure 7 shows the scatter plot of RM versus ΔRmaxpos, where ΔRmaxpos �s recalled as the monthly ΔRmaxpos
12
Table 3. Correlat�ons aga�nst sunspot cycle number.
Parameter r r2 a b se cl (%)Cycle 24
(Predicted)**Rm 0.780 0.608 –8.248 0.821 2.504 >99.5 11.5 ± 4.5RM 0.594 0.353 20.868 6.085 35.375 >95 166.9 ± 64.1ASC –0.591 0.349 66.614 –1.140 5.902 >95 39.3 ± 10.7PER* –0.510 0.260 152.045 –1.409 8.306 <90 118.2 ± 15.2ΔRmaxpos 0.558 0.312 0.620 0.364 2.029 >90 9.4 ± 3.7ΔRmaxneg –0.685 0.470 –0.806 –0.283 1.108 >98 –7.6 ± 2.012-mma of ΔRmaxpos 0.578 0.334 –1.031 0.316 1.674 >95 6.55 ± 3.0312-mma of ΔRmaxneg –0.746 0.556 0.861 –0.237 0.789 >99.5 –4.83 ± 1.43
*Excludes cycle 23 (presuming PER = 141 for cycle 23, then the numbers are, respectively, –0.223, 0.050, 139.446, –0.587, 9.663, <90, and 125.3 ± 17.5) **90-percent prediction interval
(growth �n R) dur�ng the ascend�ng port�on of the sunspot cycle (the ascend�ng �nflect�on po�nt). The number bes�de each of the filled c�rcles �dent�fies the spec�fic sunspot cycle. As an example, cycle 19 (the uppermost filled c�rcle) had �ts greatest pos�t�ve change �n R (= 10.8) between t = 22 (R = 98.5) and t = 23 (R = 109.3) and �t �s th�s number that has been correlated w�th �ts later occurr�ng RM (= 201.3). The th�ck stra�ght l�ne (denoted y) �s the �nferred regress�on l�ne and the th�n vert�cal and hor�zontal l�nes are the med�ans for the two parameters. The �nferred regress�on �s computed as y = 11.365 + 15.519x and has r = 0.867, y�eld�ng r2 = 0.755 (mean�ng that about three-fourths of the var�ance �n RM can be expla�ned by the �nferred regress�on based on the behav�or of ΔRmaxpos), and se = 21.8. The regress�on �s found to be stat�st�cally s�gn�ficant at better than the 0.1-percent level of s�gn�ficance (or cl >99.9 percent). The result of F�sher’s exact test for 2×2 cont�ngency tables �s also g�ven,18 wh�ch �nd�cates that the probab�l�ty (P) of obta�n�ng the observed result, or one more suggest�ve of a departure from �ndependence (chance), �s P =12.1 percent. It �s apparent then that by mon�tor�ng the month-to-month rate of change �n R, the later occurr�ng RM can be cont�nually est�mated. For ΔRmaxpos ≤6.4, �t �s ant�c�pated that RM ≤121; for ΔRmaxpos >6.4, �t �s ant�c�pated that RM >105.
Instead of us�ng ΔRmaxpos as the est�mator for the later occurr�ng RM, RM can be compared aga�nst the slope at ΔRmaxpos, where the slope �s computed as R at E(ΔRmaxpos) m�nus Rm d�v�ded by the elapsed t�me �n months between E(ΔRmaxpos) and E(Rm). F�gure 8 d�splays the scatter plot of RM versus the slope at E(ΔRmaxpos). As before, the number bes�de the filled c�rcles �dent�fies the sunspot cycle number, the th�ck l�ne �s the �nferred regress�on, the th�n l�nes are the med�ans, and the results of l�near regress�on analys�s and F�sher’s exact test for the 2×2 cont�ngency table are g�ven. As an example, cycle 19 (the uppermost filled c�rcle) had ΔRmaxpos = 10.8 occurr�ng at t = 22 mo. The slope for cycle 19 �s then computed to be (98.5 – 3.4)/22 = 4.32, and �t �s th�s number that �s correlated w�th the later-occurr�ng RM (= 201.3).
Clearly, figure 8 prov�des a much-�mproved pred�ct�on of RM as compared to us�ng fig-ure 7 because of the large reduct�on �n se (reduced by half, from 21.8 to 10.6), but not know�ng exactly when ΔRmaxpos occurs presents a problem. For example, cycles 14, 15, 17, and 23 had the�r actual ΔRmaxpos values very late �n the�r ascents (t = 39, 42, 39, and 40 mo, respect�vely),
13
5-377441(F7)
2 4 6 8 10
14
12
13
16
20
23 17
18
21
22
19
15
Rmaxpos during ASC
y = 11.365 + 15.519xr = 0.869, r2 = 0.755se = 21.773, cl > 99.9%
200
150
RM
100
50
1 5
4 2⇒ P = 12.1%
y
Δ
F�gure 7. Scatter plot of RM versus ΔRmaxpos.
14
5-377441(F8)
1 2 3 4 5
14
16
13
12
15
20
21
19
23
22
17
18
Slope at E( Rmaxpos)
y = 16.847 + 41.227xr = 0.970, r2 = 0.941se = 10.648, cl > 99.9%
200
150
100RM
50
0 6
6 0⇒ P = 0.1%
y
Δ
F�gure 8. Scatter plot of RM versus slope at E(ΔRmaxpos).
15
although smaller local peaks occurred earl�er �n the�r ascents (3.2 at t = 16 and 3.5 at t = 30 for cycle 14, 5.4 at t = 17 for cycle 15, 4.8 at t = 24 and 5.4 at t = 33 for cycle 17, and 5.2 at t = 20 for cycle 23). Us�ng these earl�er values, cycle 14’s RM would have been pred�cted to be about 61 (at t = 16) and 66 (at t = 30), both values very close to �ts actual RM = 64; cycle 15’s RM would have been pred�cted to be about 95 (at t = 17), close to but below �ts actual RM = 105; cycle 17’s RM would have been pred�cted to be about 86 (at t = 24) and 95 (at t = 33), both close to but below �ts actual RM = 119; and cycle 23’s RM would have been pred�cted to be about 92 (at t = 20), close to but below �ts actual RM = 121. Because the 90-percent pred�ct�on �ntervals assoc�ated w�th each pred�cted RM �s ± 39.5, all of the actual RM values would have fallen w�th�n the pred�ct�on �nterval.
Instead, us�ng the est�mated slope at the t�me of the earl�er peaks, �t would have been pre-d�cted that cycle 14’s RM to be about 60 (at t = 16) and 72 (at t = 30), cycle 15’s RM to be about 84 (at t = 17), cycle 17’s RM to be about 91 (at t = 24), and cycle 23’s RM to be about 91 (at t = 20). Because the 90-percent pred�ct�on �ntervals assoc�ated w�th each pred�cted RM �s ± 19.3, only the actual RM values for cycles 14 and 15 would have fallen w�th�n the pred�ct�on �nterval; cycles 17 and 23 would have fallen just outs�de the upper l�m�t of the pred�ct�on �nterval (by about 10 un�ts of sunspot number).
Because many of the cycles e�ther had the�r ΔRmaxpos or alternat�ve early cycle local peaks before t = 30, �t may be that RM can be pred�cted on the bas�s of the collect�ve behav�or of ΔRmaxpos �n compar�son to RM before t = 30 mo. F�gure 9 shows the var�at�on of the r2s determ�ned month-by-month from E(Rm) based on a compar�son of RM and the month-to-month change �n R(ΔR)(t) for t = 0–26 mo. It �s found that the peak r2 occurs at t = 25 mo; although, as early as t = 14 mo, ΔR(t) prov�des a stat�st�cally mean�ngful est�mate for RM (the earl�est est�mate appears to be at t = 10 mo, but �t has a rather large se, equal to ± 34 mo). In figure 9, the cross-hatched pattern s�gn�fies those ts when a stat�st�cally mean�ngful correlat�on �s �nferred between RM and ΔR(t).
F�gure 10 d�splays the scatter plots of RM versus ΔR(t = 10) (panel (a)) and ΔR(t = 25) (panel (b)). Clearly, by mon�tor�ng ΔR(t), an �ncreas�ngly accurate est�mate for RM can be effected, espe-c�ally at t = 25 mo. Such an approach would have suggested �ts RM would l�kely be about 105.6 ± 33.6 (the 90-percent pred�ct�on �nterval) for cycle 23.
2.2.2 12-mma of ΔR
The role of the 12-mma of ΔR as a pred�ctor for RM w�ll be exam�ned �n th�s sect�on. F�gure 11 shows the scatter plot of RM versus the max�mum pos�t�ve value of the 12-mma of ΔR. The structure of the chart follows that used �n figures 7, 8, and 10. F�gure 11 �s str�k�ngly s�m�lar to figure 8, both hav�ng r = 0.97, r2 = 0.94, and se = 10.6. Thus, by mon�tor�ng the 12-mma of ΔR, �ncreas�ngly accurate pred�ct�ons of the later occurr�ng RM can be effected. The d�sadvantage of the 12-mma of ΔR(t) as compared to ΔR(t) �s that �t lags that of ΔR(t) by several months, so �ts use as a pred�ctor of RM appears more appropr�ately to be that of a confirmatory role.
F�gures 12 and 13 are equ�valents to figures 9 and 10, but based on the compar�son of RM versus the 12-mma of ΔR(t). From figure 12, �t �s found that as early as t = 8 mo, the value of the 12-mma of ΔR(t) prov�des a stat�st�cally mean�ngful est�mate for RM, w�th the best pred�ctor
16
5-377441(F9)
5 10 15 2520t, Elapsed time in months from E(Rm)
RM = 57.641+16.524 R (t = 25)r = 0.907, r 2 =0.822, se =18.560, cl >99.9%
RM = 64.252+27.068 R (t = 10)r = 0.635, r 2 = 0.403, se = 33.976, cl >95%
cl ≥ 95%1.00
r2 (R(
t))
0.50
11
2
2
Δ
Δ
Δ
F�gure 9. Var�at�on of r2(ΔR(t)).
occurr�ng at t = 22 mo. For cycle 23, such an approach would have suggested �ts RM l�kely would be 111.3 ± 25.8 (the 90-percent pred�ct�on �nterval).
2.2.3 ΔRmaxneg
The role of ΔRmaxneg as a pred�ctor of PER for the current cycle and of Rm and RM for the follow�ng cycle w�ll be exam�ned �n th�s sect�on. F�gure 14 d�splays the scatter plot of PER versus ΔRmaxneg. The structure of the chart follows that of prev�ous charts. Based on F�sher’s exact test for 2×2 cont�ngency tables, �t �s found that the P of obta�n�ng the observed result, or one more suggest�ve of a departure from �ndependence (chance), �s P = 17.5 percent. Based on l�near regress�on analys�s, a stat�st�cally �mportant correlat�on �s �nferred to ex�st between PER and ΔRmaxneg, hav�ng r = 0.66, r2 = 0.44, se = 7.2, and cl >95 percent. From the known value of ΔRmaxneg for cycle 23 (= –5.3 at t = 77 mo or 30 mo past E(RM)), �t �s �nferred that PER for cycle 23 should be about 130 ± 13 mo (the 90-percent pred�ct�on �nterval). Thus, on the bas�s of the val�d�ty of the �nferred regress�on, �t �s noted that there �s only a 5-percent chance that cycle 23 w�ll pers�st longer than 143 mo, �nd�cat�ng E(Rm) for cycle 24 should be expected before May 2008.
17
5-37
7441
(F10
)
12
38
102
46
2017
15
12
14
16
18
19
21
22
1323
13
1520
1612
14
2317
1822
21
19
y
y
R (t
=10)
R (t=
25)
⇒ P
= 4.
0%
⇒ P
= 0.
1%
y=64
.252+
27.06
8xr=
0.635
,r2 =
0.403
se=3
3.976
,cl>
95%
200
RM100
06
60
y=57
.641+
16.52
4xr=
0.907
,r2 =
0.822
se=1
8.560
,cl>
99.9%
15 1
5
ΔΔ
(a)
(b)
F�gure 10. Scatter plots of RM versus ΔR(t = 10) and RM versus ΔR(t = 25).
18
5-377441(F11)
108642
20
15
13
16
12
14
1723
19
21
2218
(12-mma of R) maxpos
y =26.935+20.662xr =0.970, r 2 =0.942se=10.630, cl >99.9%
200
150
100
RM
50
1 5
5 1⇒ P = 4.0%
y
Δ
F�gure 11. Scatter plot of RM versus (12-mma of ΔR)maxpos.
19
5-377441(F12)
5 10 15 2520t, Elapsed time in months from E(Rm)
RM =53.701+39.863 (12-mma of R) (t =8)r =0.621, r 2 =0.386, se =34.462, cl >95%
RM =39.648+20.078 (12-mma of R) (t =22)r =0.946, r 2 =0.895, se =14.245, cl >99.9%
cl ≥95%1.00
0.50
r2 (12
-mm
a of
R(t)
)
1
2
1
2
Δ
Δ
∆
F�gure 12. Var�at�on of r2(12-mma of ΔR(t)).
F�gures 15 and 16 show scatter plots of Rm(n + 1) and RM(n + 1) versus ΔRmaxneg, respec-t�vely. Ne�ther plot �s �nferred to be stat�st�cally �mportant. Thus, wh�le ΔRmaxneg seems to prov�de an �nd�cat�on for the expected length of an ongo�ng cycle, �t does not prov�de any �nd�cat�on as to the expected s�ze of the Rms and RMs for the follow�ng cycle.
2.2.4 (12-mma of ΔR)maxneg
The role of (12-mma of ΔR)maxneg as a pred�ctor of PER for the current cycle and of Rm and RM for the follow�ng cycle w�ll be exam�ned �n th�s sect�on. F�gure 17 d�splays the scatter plot of PER versus (12-mma of ΔR)maxneg. As found for PER versus ΔRmaxneg (fig. 14), a stat�st�-cally �mportant correlat�on �s �nferred between PER and (12-mma of ΔR)maxneg, hav�ng r = 0.63, r2 = 0.39, se = 7.5, and cl >95 percent. From the known value of (12-mma of ΔR)maxneg for cycle 23 (= –3.43 at t = 78 mo or 31 mo past E(RM)), �t �s �nferred that PER for cycle 23 w�ll be about 127 ± 14 mo (the 90-percent pred�ct�on �nterval). Thus, on the bas�s of the �nferred regress�on, there �s only a 5-percent chance that cycle 23 w�ll pers�st longer than 141 mo, �nd�cat�ng E(Rm) for cycle 24 should be expected before Apr�l 2008.
20
12
38
102
46
20
2317
15 1214
16
19
21 1822
1313
1520
1612
14
2317
1822
21
19y
y
12-m
ma o
f R
(t=2
2)
⇒ P
= 28
.4%
⇒ P
= 4.
0%
200
RM100
15 1
5
y=53
.701+
39.86
3xr=
0.621
,r2 =
0.386
se=3
4.462
,cl>
95%
y=39
.648+
20.07
8xr=
0.946
,r2 =
0.895
se=1
4.245
,cl >
99.9%
24 2
4
5-37
7441
(F13
)
12-m
ma o
f R
(t=8
)Δ
Δ
F�gure 13. Scatter plots of RM versus 12-mma of ΔR(t = 8) and RM versus 12-mma of ΔR(t = 22).
21
–8 –10–2
150
125
100
–4 –6
13
15
20
16
12
19
1817
14
22
21
y
R maxneg
PER
⇒ P = 17.5%
4 2
1 4
y =150.729+3.909xr =0.661, r2 =0.437se =7.235, cl >95%
PER(23)90 =130.0± 13.3
Cycle 23 (–5.3)
5-377441(F14)
Δ
F�gure 14. Scatter plot of PER versus ΔRmaxneg.
F�gures 18 and 19 show scatter plots of Rm(n + 1) and RM(n + 1) versus (12-mma of ΔR) maxneg, respect�vely. Ne�ther plot �s �nferred to be stat�st�cally �mportant. Thus, wh�le (12-mma of ΔR)maxneg seems to prov�de an �nd�cat�on for the expected length of an ongo�ng cycle, �t does not prov�de any �nd�cat�on as to the s�ze of the Rms and RMs of the follow�ng cycle.
2.2.5 ΔR(T ) and 12-mma of ΔR(T )
The roles of ΔR(T) and 12-mma of ΔR(T) for elapsed t�mes �n months from E(RM), T, as pred�ctors of PER, Rm(n + 1), and RM(n + 1) w�ll be exam�ned �n th�s sect�on. F�gure 20 shows the scatter plots of PER versus ΔR(T = 30) (panel (a)) and PER versus 12-mma of ΔR(T = 31) (panel (b)). The two plots represent the best fits of PER versus ΔR(T) and PER versus 12-mma of ΔR(T) for T = 0–40 mo. Apparently, the reason for the success of these �nferred regress�ons �s the fact that ΔR(T) and the 12-mma of ΔR(T) are near the�r max�mum negat�ve values at these
22
–2 –4 –6 –8 –10
19, 20
22, 23
20, 21 21, 22
17, 18
16, 17
15, 16
14, 15
18, 19
13, 14
12, 13
y
Rmaxneg
⇒ P = 60.8%
10
15Rm
(cyc
len+
1)
5
y = –0.014 – 1.122xr = –0.462, r 2 = 0.213se = 3.520, cl <90%
3 3
3 2
5-377441(F15)
Cycle 23 (–5.3)
Δ
F�gure 15. Scatter plot of Rm(cycle n + 1) versus ΔRmaxneg.
elapsed t�mes relat�ve to E(RM). Both fits are stat�st�cally �mportant. Based on the known value of ΔR(T = 30) dur�ng the descend�ng port�on of cycle 23 (= –5.3), �t �s �nferred that �ts PER w�ll be about 122 ± 14 mo (the 90-percent pred�ct�on �nterval). Based on the known value of the 12-mma of ΔR(T = 31) dur�ng the descend�ng port�on of cycle 23 (= –3.43), �t �s �nferred that �ts PER w�ll
23
5-377441(F16)
–10–8–6–4–2
15, 16
14, 15
16, 1722, 23
21, 22
19, 20
12, 13
13, 14
18, 19
20, 21
17, 18
Rmaxneg
y = 43.178 – 13.930xr = –0.522, r2 = 0.273se = 37.197, cl <90%
200
150
100
50
RM(c
ycle
n+1)
2 43 2
⇒ P = 39.2%
y
Cycle 23 (–5.3)
Δ
F�gure 16. Scatter plot of RM(cycle n + 1) versus ΔRmaxneg.
24
–6
150
125
100
–3
13
15
20
16
12
19
18
17
14
22
21
y
(12–mma of R)maxneg
PER
⇒ P = 17.5%
4 2
1 4
y =143.750+ 4.790xr =0.627, r 2 =0.393se =7.532, cl >95%
PER(23)90=127.3±13.8
Cycle 23 (–3.43)
5-377441(F17)
Δ
F�gure 17. Scatter plot of PER versus (12-mma of ΔR)maxneg.
be about 125 ± 12 mo (the 90-percent pred�ct�on �nterval). S�nce cycle 23 has already pers�sted for 135 mo, E(Rm) should be expected very soon. However, because new cycle spots typ�cally occur e�ther s�multaneously w�th up to several months pr�or to E(Rm) and because the first confirmed new cycle spot for cycle 24 occurred �n January 2008, �t appears that E(Rm) for cycle 24 w�ll occur out-s�de these 90-percent pred�ct�on �ntervals. (The �nterval from May 1996–August 2007 corresponds to 135 mo. Because cycle 23 undoubtedly w�ll have a PER more l�ke cycles 12–14 and 20 rather than cycles 15–19, 21, and 22, �nclus�on of cycle 23’s PER w�ll greatly weaken the �nferred regress�ons. Hence, the �nferred correlat�ons are probably a fluke, w�th cycles more l�kely d�str�buted preferen-t�ally as short- and long-per�od cycles, rather than follow�ng the �nferred l�near regress�on l�nes.)
F�gure 21 shows the scatter plots of Rm(n + 1) versus ΔR(T = 31) (panel (a)) and Rm(n + 1) versus 12-mma of ΔR(T = 27) (panel (b)), the most stat�st�cally �mportant correlat�ons. Wh�le the first plot (panel (a)) �s only of marg�nal stat�st�cal �mportance (cl >90 percent), the latter one
25
–6
10
15
5
–3
y
(12–mma of R)maxneg
Rm(c
ycle
n+1)
⇒ P = 19.7%5
2
1
3
y =1.194 –1.618xr =–0.515, r2 =0.265se =3.402, cl <90%
Rm(24)90 =6.7± 6.2
Cycle 23 (–3.43)
5-377441(F18)
Δ
F�gure 18. Scatter plot of Rm(cycle n + 1) versus (12-mma of ΔR)maxneg.
26
–6
100
200
150
50
–3
y
(12–mma of R)maxneg
RM(c
ycle
n+1)
⇒ P = 17.5%42
14
y = 72.767–15.625xr = –0.453, r 2 = 0.205se = 38.885, cl <90%
RM(24)90 =126.4 ± 71.3
Cycle 23 (–3.43)
5-377441(F19)
Δ
F�gure 19. Scatter plot of RM(cycle n + 1) versus (12-mma of ΔR)maxneg.
27
–8–6
–4–2
20
13
15
19 17
2216
12
14
150
125
100
18
20
21
y
R(t =
30)
PER
⇒ P
= 17
.5%4
21
4
y=13
3.31+
2.191
xr=
0.650
,r2 =
0.423
se=7
.338,
cl>9
5%
PER(
23) 90
=121
.7±
13.5
Cycle
23 (–
5.3)
–8–3
13
15
19
18
22
16
1214
150
125
100
17
20
21
y
12–m
ma o
f R
(t=3
1)PER
⇒ P
= 17
.5%4
21
4
y=13
9.603
+4.29
6xr=
0.720
,r2 =
0.519
se=6
.718,
cl>9
8%
PER(
23) 90
=124
.9±
12.3
Cycle
23 (–
3.43)
5-37
7441
(F20
)
ΔΔ
(a)
(b)
F�gure 20. Scatter plots of PER versus ΔR(T = 30) and 12-mma of ΔR(T = 31).
28
5-37
7441
(F21
)
20
–2–4
–6–8
12, 1
3
16, 1
713
, 14
14, 1
5
18, 1
9
22, 2
3
21, 2
2
19, 2
0
17, 1
8
15, 1
620/21
R(T
=31)
y=3.9
43–0
.901x
r=–0
.573,
r2 =0.3
29se
=3.25
3,cl
>90%
14 12 10 8 6 4 2Rm(cyclen+1)
Rm(cyclen+1)
24
32
⇒P
= 39
.2%
y
Rm(2
4)90
=6.8
± 6.0
Cycle
23 (–
3.2)
01
–1–2
–3–4
–5
12, 1
3
16, 1
7
17, 1
8
13, 1
4
14, 1
5
18, 1
9
22, 2
3
21, 2
2
19, 2
0
15, 1
6
20, 2
1
12–m
ma o
f R
(T =
27)
y=0.1
91–2
.590x
r=–0
.791,
r2 =0.6
26se
=2.42
7,cl
>99.5
%
14 12 10 8 6 4 2
15
41
⇒P
= 6.7
%
y
Rm(2
4)90
=7.6
± 4.4
Cycle
23 (–
2.88)
ΔΔ
(a)
(b)
F�gure 21. Scatter plots of Rm(cycle n + 1) versus ΔR(T = 31) and 12-mma of ΔR(T = 27).
29
(panel (b)) �s found to be h�ghly stat�st�cally �mportant (cl >99.5 percent). Based on the known value (= –3.2) of cycle 23’s ΔR(T = 31), the Rm for cycle 24 �s est�mated to be about 7 ± 6 (the 90-percent pred�ct�on �nterval); wh�le, based on the known value (= –2.88) of cycle 23’s 12-mma of ΔR(T = 27), the Rm for cycle 24 �s est�mated to be about 8 ± 4 (the 90-percent pred�ct�on �nterval). Based on the 2×2 cont�ngency table, Rm(24) can be expected to be ≥5.6. The lowest value of the 12-mma of R observed to date dur�ng the decl�n�ng port�on of cycle 23 measures 5.9 (September 2007), so obv�-ously E(Rm) for cycle 24 �s most �mm�nent.
Attempts to find a stat�st�cally �mportant correlat�on between RM(n + 1) and ΔR(T) and between RM(n + 1) and the 12-mma of ΔR(T) proved fru�tless. However, for completeness sake, fig-ure 22 �s �ncluded, wh�ch shows the scatter plots of RM(n + 1) versus ΔR(T = 14) (panel (a)) and RM(n + 1) versus the 12-mma of ΔR(T = 44) (panel (b)). Based on the known value (= 1.9) of cycle 23’s ΔR(T = 14), the RM for cycle 24 �s est�mated to be about 139 ± 67 (the 90-percent pred�ct�on �nterval); wh�le, based on the known value (= –1.88) of cycle 23’s 12-mma of ΔR(T = 44), the RM of cycle 24 �s est�mated to be about 137 ± 69).
5 3 1 –1 –3 –5
18, 19
22, 23
20, 21
21, 22
17, 18
16, 1719, 20
15, 16
14, 15
13, 14
12, 13
y
R(t =14)
⇒ P = 17.5%
RM(c
ycle
n+1)
RM(c
ycle
n+1)
50
100
150
200
y =119.834+10.030xr =0.539, r2 =0.291se =36.735, cl >90%
4 21 4
5-377441(F22)
Cycle 23 (1.9)
PER(24)90 =138.9 ± 67.3
0 –1 –2 –3
18, 19
22, 23
20, 2121, 22
17, 18
16, 1719, 20
15, 16
14, 15
13, 14
12, 13
y
12–mma of R(t = 44)
⇒ P = 39.2%
50
100
150
200y =88.482 – 25.643xr = –0.515, r 2 =0.266se =37.377, cl <90%
2 43 2
Cycle 23 (–1.88)
PER(24)90 =136.7± 68.5
ΔΔ
(a) (b)
F�gure 22. Scatter plots of RM(cycle n + 1) versus ΔR(T = 14) and 12-mma of ΔR(T = 44).
30
2.2.6 SlopeASC and SlopeDES
F�gure 23 d�splays the scatter plots of SlopeASC (panel (c)) and SlopeDES (panel (a)) versus ΔRmaxpos and SlopeASC (panel (d)) and SlopeDES (panel (b)) versus (12-mma of ΔR)maxpos, where SlopeASC �s defined as (RM – Rm)/ASC and SlopeDES �s defined as ((Rm for cycle n + 1) – (RM for cycle n))/DES for cycle n. All correlat�ons are �nferred to be h�ghly stat�st�cally �mportant, espec�ally SlopeDES versus (12-mma of ΔR)maxpos. Est�mates (90-percent pred�ct�on �ntervals) of SlopeDES for cycle 23 are made based on the observed values of ΔRmaxpos and (12-mma of ΔR)maxpos.
F�gure 24 shows scatter plots of SlopeDES versus ΔRmaxneg (panel (a)), (12-mma of ΔR) maxneg (panel (b)), and SlopeASC (panel (c)). Aga�n, all correlat�ons are �nferred to be h�ghly stat�st�-cally �mportant, espec�ally SlopeDES versus SlopeASC. Est�mates (90-percent pred�ct�on �ntervals) of SlopeDES for cycle 23 are made based on the observed values of ΔRmaxneg, (12-mma of ΔR)maxneg, and SlopeASC.
Of the var�ous est�mates for SlopeDES, the best est�mate (smallest 90-percent pred�ct�on �nter-val) appears to be the one based on (12-mma of ΔR)maxpos. Hence, cycle 23’s SlopeDES l�kely w�ll measure about –1.15 ± 0.29.
2.2.7 Cycle 23 Behavior: May 1996–August 2007
F�gure 25 d�splays the 12-mma of R (panel (b)) for elapsed t�mes t = 0 (May 1996)–135 (August 2007). The 12-mma of ΔR for elapsed t�mes t = 0 (May 1996)–128 (January 2007) �s also shown. For conven�ence, the values of Rm, RM, ASC, DES, PER, (12-mma of ΔR)maxpos, and (12-mma of ΔR)maxneg and the dates of occurrence of E(Rm), E(RM), E((12-mma of ΔR)maxpos), and E((12-mma of ΔR)maxneg) are g�ven. Thus, RM for cycle 23 measured 120.8 (occurr�ng at t = 47, Apr�l 2000) and the value of R at t = 135 mo (August 2007) measured 6.1. The current SlopeDES �s (6.1 – 120.8)/88 = –1.30, wh�ch �s w�th�n the w�ndow of the est�mated value for cycle 23’s SlopeDES (= –1.15 ± 0.29), �nferr�ng that E(Rm) for cycle 24 �s �mm�nent. Recall from figure 6 that the t�me (t(4)) from E((12-mma of ΔR)maxneg) to E(Rm) of the follow�ng cycle averages about 51 mo (w�th sd = 10), �nferr�ng that E(Rm ) for cycle 24 should follow November 2002 (t = 78) by about 51 ± 18 mo (the 90-percent pred�ct�on �nterval), or that there �s a 95-percent P that cycle 24 w�ll have �ts offic�al onset before September 2008. (S�nce R �s now known through September 2007, the actual SlopeDES �s –1.29.)
31
3 6 9
20
17
15
14 16
18
19
13
23
5
4
3
2
112
22
21
y
Rmaxpos
Slop
e ASC
y = –0.439 + 0.429xr = 0.883, r2 = 0.780se = 0.566, cl >99.9%
SlopeDES (23)90 = –1.08 ± 0.40
⇒ P = 12.1%1
4
5
2
20
17 15
16
1413
12
2122 22
19
18
–3
–2
–1
0
y Slop
e DES
y =0.036 – 0.202xr = –0.926, r2 =0.858se = 0.217, cl >99.9%
⇒ P = 4.5%4
0
2
5
3 6 9
20
17
1416
18
19
13
2315
12
22
21
y
(12–mma of R)maxpos
y = 0.083+ 0.551xr =0.951, r 2 = 0.904se = 0.372, cl >99.9%
⇒ P = 4.0%1
5
5
1
Cycle 23 (5.5)
5-377441(F23)
SlopeDES (23)90 = –1.15 ± 0.2920
1715
16
1413
12
21
19
18
y
y = –0.262 – 0.249xr = –0.967, r2 = 0.936se = 0.156, cl >99.9%
⇒ P = 17.5%4
1
2
4Cycle 23 (3.57)
ΔΔ
(a) (b)
(c) (d)
F�gure 23. Scatter plots of SlopeASC versus ΔRmaxpos and (12-mma of ΔR)maxpos and of SlopeDES versus ΔRmaxpos and (12-mma of ΔR)maxpos.
32
–3–6
–9
20
1715
1214
16
18
19
13
22 21y
Rmax
neg
⇒ P
= 17
.5%
0
SlopeDES –3–2–1
y=–0
.038+
0.236
xr=
0.694
,r2 =
0.482
se=0
.397,
cl>9
8%Sl
ope DE
S(23
) 90=
–1.29
±0.73
4 12 4
Cycle
23 (–
5.3)
–1–2
–3–4
–5–6
20
171512
14
16
18
19
13
2221
y
(12–
mm
a of
R)m
axne
g
⇒ P
= 17
.5%
y=–0
.164+
0.379
xr=
0.864
,r2 =
0.746
se=0
.282,
cl>9
9.9%
Slop
e DES(
23) 9
0=–1
.46±
0.52
4 12 4
Cycle
23 (–
3.43)
12
34
5
2017
15
1214
16
18
19
13
2221
y
Slop
e ASC
⇒ P
= 1.
3%
y=–0
.355–
0.408
xr=
–0.92
5,r2 =
0.856
se=0
.206,
cl>9
9.9%
Slop
e DES(
23) 9
0=–1
.33±0
.38
5 01 5
Cycle
23 (2
.40)
5-37
7441
(F24
)
ΔΔ
(a)
(b)
(c)
F�gure 24. Scatter plots of SlopeDES versus ΔRmaxpos, (12-mma of ΔR)maxpos, and SlopeASC.
33
20 40 60 80 100 120 140
100
–5
–4
–3
–2
–1
0
1
2
3
4
5
50
(12–mma of R)maxpos =3.57E((12–mma of R)maxpos): January 1998 (t =20) (12–mma of R)maxneg= –3.43
E((12–mma of R)maxneg): November 2002 (t =78)
Rm = 8.0E(Rm): May 1996
RM = 120.8E(RM): April 2000
Cycle 23
ASC = 47 DES ≥ 88 PER ≥ 135
t, Elapsed time in months from E(Rm)
12–m
ma o
f R12
–mm
a of
R
5-377441(F25)
Δ
(a)
(b)
F�gure 25. Var�at�on of cycle 23’s R and 12-mma of ΔR for t = 0–135 (May 1996–August 2007).
34
3. SUMMARY
Var�ous methods,4,19–22 those us�ng geomagnet�c precursor �nformat�on �n part�cular, have been used to successfully pred�ct RM several years �n advance of �ts occurrence. For cycle 24, the next sunspot cycle, precursor techn�ques suggest that �t w�ll have an RM equal to about 130 ± 14 (see W�lson and Hathaway22 and references conta�ned there�n), peak�ng about 44 ± 5 mo after E(Rm), or about November 2011 (± 5 mo) �f the offic�al start of cycle 24 �s, �ndeed, March 2008, as pred�cted by the NOAA Solar Cycle 24 Pred�ct�on Panel.14 Th�s pred�ct�on for the s�ze of cycle 24’s RM �s more support�ve of the h�gher consensus pred�ct�on of the NOAA panel (140 ± 20) than �ts lower consensus pred�ct�on (90 ± 10), where the two consensus pred�ct�ons ar�se from consequences of two d�fferent dynamo-related techn�ques. So, �t �s w�th great ant�c�pat�on23 that solar researchers awa�t the start and r�se of cycle 24.
Once a cycle offic�ally gets underway, pred�ct�ons of the level of future act�v�ty based on autoregress�ve techn�ques24,25 and other curve-fitt�ng methods6,9,19 can be employed. As found �n th�s Techn�cal Publ�cat�on and �n prev�ous stud�es,2,6,7,9,19 the s�ze (RM) of the cycle can be better pred�cted about 2 yr �nto the r�se of a sunspot cycle, espec�ally once �ts ascend�ng �nflect�on po�nt has been clearly d�scerned.
In th�s Techn�cal Publ�cat�on, �t has been noted that, on the bas�s of cycles 12–23 (the most rel�ably determ�ned sunspot cycles), cycles can be loosely grouped �nto three groups based on RM or two groups based on cycle durat�on. Large-ampl�tude cycles (cycles 18, 19, 21, and 22) tend to have h�gher values of Rm, r�se to RM more qu�ckly (shorter ASC), and have shorter cycle durat�on (shorter PER) than e�ther average-ampl�tude cycles (cycles 15, 17, 20, and 23) or small-ampl�tude cycles (cycles 12, 13, 14, and 16). L�kew�se, short-per�od cycles (cycles 15–19, 21, and 22) tend to r�se more qu�ckly to RM and to have h�gher values of Rm and RM than long-per�od cycles (cycles 12–14, 20, and 23). It �s also noted that, over the course of cycles 12–23, there have been stat�st�cally �mpor-tant secular r�ses �n Rm and RM and �n the values of R at the ascend�ng (more pos�t�ve) and descend-�ng (more negat�ve) �nflect�on po�nts. On the bas�s of the secular �ncreases, est�mates have been made for cycle 24; namely, �ts Rm should measure about 11.5 ± 4.5, �ts RM about 167 ± 64, �ts ASC about 39 ± 11 mo, the value of �ts ΔRmaxpos about 9.4 ± 3.7, the value of �ts ΔRmaxneg about –7.6 ± 2.0, the value of �ts 12-mmaΔRmaxpos about 6.55 ± 3.03, and the value of �ts 12-mmaΔRmaxneg about –4.83 ± 1.43. Its PER w�ll measure about 125 ± 18 mo.
Th�s study has shown that the later occurr�ng RM can be cont�nually est�mated by mon�-tor�ng the month-to-month rate of change �n R. For values of ΔRmaxpos ≤6.4, �t �s expected that RM ≤121, wh�le for values of ΔRmaxpos >6.4 �t �s expected that RM >105. Based on the slope deter-m�ned at ΔRmaxpos, RM can be pred�cted w�th even h�gher prec�s�on (se = 10.6). It was found that, as early as 10 mo after E(Rm), the value of ΔR(t) can be used to est�mate the later occurr�ng RM, although the best result �s determ�ned about 2 yr �nto the cycle. Use of the 12-mma of ΔR to pred�ct RM prov�des s�m�lar accuracy as compared to us�ng the slope at ΔRmaxpos. Values of the (12-mma of ΔR)maxpos ≤4.25 strongly suggest RM ≤121, wh�le h�gher values suggest a larger RM.
35
Both ΔRmaxneg and �ts 12-mma value appear to correlate w�th PER, such that cycle 23’s PER �s expected to pers�st no longer than 143 mo, �nferr�ng that cycle 24 should have �ts offic�al start no later than May 2008. An exam�nat�on of ΔR and �ts 12-mma value relat�ve to E(RM) suggests that Rm for cycle 24 w�ll measure about 7.6 ± 4.4. R measured 5.9 �n September 2007, so Rm for cycle 24 �s expected very soon. The (12-mma of ΔR)maxpos suggests that the SlopeDES for cycle 23 w�ll measure about –1.15 ± 0.29. The computed current SlopeDES for cycle 23 �s –1.29 through September 2007 and �s shr�nk�ng w�th the passage of t�me. On average, the length of t�me �n months from (12-mma of ΔR)maxneg to Rm �s 51 ± 18 mo (the 90-percent pred�ct�on �nterval). Thus, there �s only a 5-percent chance that E(Rm) for cycle 24 w�ll occur after August 2008.
36
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37
13. Waldme�er, M.: The Sunspot-Activity in the Years 1610–1960, Schulthess & Co., Zur�ch, Sw�tzerland, 1961.
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38
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Using the Inflection Points and Rates of Growth and Decay to Predict Levels of Solar Activity
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
Prepared for the Science and Exploration Vehicle Office, Science and Mission Systems Office
Unclass�fied-Unl�m�tedSubject Category 92Ava�lab�l�ty: NASA CASI 301–621–0390
The ascending and descending inflection points and rates of growth and decay at specific times during the sunspot cycle are examined as predictors for future activity. On average, the ascending inflection point occurs about 1–2 yr after sunspot minimum amplitude (Rm) and the descending inflection point occurs about 6–7 yr after Rm. The ascending inflection point and the inferred slope (including the 12-mo moving average (12-mma) of ΔR (the month-to-month change in the smoothed monthly mean sunspot number (R)) at the ascending inflection point provide strong indications as to the expected size of the ongoing cycle’s sunspot maximum amplitude (RM), while the descending inflection point appears to provide an indication as to the expected length of the ongoing cycle. The value of the 12-mma of ΔR at elapsed time T = 27 mo past the epoch of RM (E(RM)) seems to provide a strong indication as to the expected size of Rm for the following cycle. The expected Rm for cycle 24 is 7.6 ± 4.4 (the 90-percent prediction interval), occurring before September 2008. Evidence is also presented for secular rises in selected cycle-related parameters and for preferential grouping of sunspot cycles by amplitude and/or period.
48
M–1240
Techn�cal Publ�cat�onSeptember 2008
NASA/TP—2008–215473
Sun, sunspot cycle, solar activity, sunspot cycle prediction, inflection points, sunspot growth rates
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NASA/TP—2008–215473
Using the Inflection Points and Rates of Growth and Decay to Predict Levels of Solar ActivityRobert M. Wilson and David H. HathawayMarshall Space Flight Center, Marshall Space Flight Center, Alabama
September 2008
National Aeronautics andSpace AdministrationIS20George C. Marshall Space Flight CenterMarshall Space Flight Center, Alabama35812