NASA Technical Memorandum 4759

/ -,,

Statistical Technique for Intermediate and

Long-Range Estimation of 13-Month

Smoothed Solar Flux and Geomagnetic Index

K.O. Niehuss, H.C. Euler, Jr., and W.W. Vaughan

September 1996

NASA Technical Memorandum 4759

1

Statistical Technique for Intermediate and

Long-Range Estimation of 13-Month

Smoothed Solar Flux and Geomagnetic Index

K.O. Niehuss and H.C. Euler, Jr.

Marshall Space Flight Center • MSFC, Alabama

W.W. Vaughan

University of Alabama in Huntsville ,, Huntsville, Alabama

National Aeronautics and Space Administration

Marshall Space Flight Center • MSFC, Alabama 35812

September 1996
ACKNOWLEDGMENTS

The authors thank Leonard W. Howell, Jr., Electromagnetics and Aerospace Environments

Branch, Robert L. Holland, Computer Sciences Corporation, for technical contributions, Steve

Smith, Computer Sciences Corporation, for computer programming support, and Margaret B.

Alexander, Electromagnetics and Aerospace Environments Branch, for reviewing this

memorandum.

ii
FOREWORD

This document provides the methodology used in the development of the MSFC 13-month

smoothed solar flux and geomagnetic index intermediate (months) and long-range (years) statistical

estimation technique. These estimates are provided as input to orbital lifetime models.

Section 1 is an introduction and reason the estimates are determined. Section 2 is the history of the

development of the computer program that calculates the estimates. Section 3 contains the

methodology of the calculation technique. Section 4 describes the flow of the computer program to

generate the estimates. Section 5 discusses the engineering use of the computer program's products.

Appendix A gives the linear least squares model development. Appendix B is an overview of the

cubic connection between the predicted cycle and the mean. Appendix C provides the rationale for

the statistics used to calculate the percentile values. Appendix D provides the McNish-Lincoln and

quantile computer programs. Appendix E contains an example of the computer program product

provided in the monthly Marshal Space Flight Center solar activity memorandum. Appendix F gives

an assessment of the estimation technique accomplished for a number of past solar cycles.

Questions or comments, contact NASA Marshall Space Flight Center Systems Analysis and

Integration Laboratory, Electromagnetics and Aerospace Environments Branch, Chief, Steven D.

Pearson (205) 544-2350.

iii
iv
TABLE OF CONTENTS

1.0 Introduction ............................................................................................... 1

2.0 Background ............................................................................................... 1

3.0 MSFC Lagrangian Linear Regression Methodology ................................................. 2

3.1 Development of Fi0.7 and Ap DataBases .................................................. : ........ 2

3.1.1 13-Month Smoothed Solar Flux (Fl0.7) DataBase ........................................... 3

3.1.2 13-Month Smoothed Geomagnetic Index (A,) Data Base ................................. 6

3,2 Modified McNish-Lincoln Linear Regression Method ............................................

4.0 Application of MSFC Lagrangian Linear Regression Technique ................................... 8

4.1 F_0.7and A p Preprocessor Computer Program ................................................... 10

4.2 Modified McNish-Lincoln Linear Regression Computer Program ............................. 12

4.3 Statistical Estimation Output Report Computer Program ........................................ 14

5.0 Engineering Use of MSFC Solar Activity Statistical Estimation Technique Products ............ 15

6.0 Concluding Remarks .................................................................................... 16

7.0 References ................................................................................................ 17

8.0 Bibliography ............................................................................................. 18

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Appendix F

Linear Least Squares Model Development ............................................... A-1

Cubic Connection From Present Predicted Cycle to Mean of Next Cycle ............ B- 1

Estimates of Percentiles for the 13-Month Smoothed Solar Flux Data ................ C-1

Modified McNish-Lincoln and Quantile Subroutine Examples ........................ D-1

MSFC Monthly Solar Activity Memorandum Example .................................. E-1

MSFC Lagrangian Linear Regression Technique Historical Plots ...................... F-1

V
2-1

3-1

3-2

3-3

3-4

4-1

4-2

4-3

4-4

5-1

C-1

C-2

C-3

C-4

C-5

C-6

C-7

C-8

C-9

C-10

C-11

C-12

C-13

C-14

C-15

E-1

E-2

F-1

F-2

F-3

F-4

F-5

F-6

LIST OF FIGURES

Modified McNish-Lincoln Linear Regression Method ........................................ 2

R versus F10.7with Curve Fit from Conversion Equation ................................... 4

13-Month Smoothed (Converted and Observed) Solar Flux (F!0.7) ......................... 4

Ci versus Ap with Curve Fit from Conversion Equation .................................... 6

13-Month Smoothed (Converted and Recorded) Geomagnetic Index (Ap) ................ 7

MSFC_ La_grangian Linear Regression Technique Block Diagram ........................... 9

F10.7 and Ap Preprocessor Computer Program Block Diagram. ............................ 11

Modified McNish-Lincoln Linear Regression Computer Program Block Diagram ...... 13

Report Program Block Diagram ................................................................ 14

MSFC Spacecraft Orbital Lifetime Prediction Model Block Diagram ...................... 15

Arbitrary Probability Density Function f(x) and Its Cumulative Distribution F(x) ..... C-2

13-Month Smoothed Lagrangian Flux ....................................................... C-3

Mean and Standard Deviation Trend Lines .................................................. C-3

Relative Frequency Histogram ................................................................ C-5

Frequency Polygon ............................................................................ C-5

Comparison of Frequency Polygon Methods for Q(0.95) ................................ C-7

Comparison of Frequency Polygon Methods for Q(0.976) ............................... C°7

Quantile Plot .................................................................................... C-9

Interpolated Quantiles ......................................................................... C- 10

Quantile Method ............................................................................... C- 11

Relationship Between the Empirical Distribution Function, F_, and the Quantile

Function, Q .................................................................................... C- 12

Relative Frequency Histogram and Probability Density Function of p_ ................ C-17

Relative Frequency Histogram and Probability Density Function for PH .............. C-17

Relative Frequency Histogram and Probability Density Function for P2_ .............. C- 18

Quantile Method: Q(0.95) .................................................................... C-19

Plot of Recent Monthly Mean and 13-Month Smoothed Solar Flux ...................... E-9

Estimate of 13-Month Smoothed Solar Flux for Balance of Cycle 22,

23, and Beginning of Cycle 24 ................................................................ E-9

Minimum to Minimum Cycle 20 MLLRT Model Results Evaluation ..................... F-2

Maximum to Maximum Cycle 20 MLLRT Model Results Evaluation .................... F-3

Minimum to Minimum Cycle 21 MLLRT Model Results Evaluation ..................... F-4

Maximum to Maximum Cycle 21 MLLRT Model Results Evaluation .................... F-5

Minimum to Minimum Cycle 22 MLLRT Model Results Evaluation ..................... F-6

Maximum to Maximum Cycle 22 MLLRT Model Results Evaluation .................... F-7

vi
LIST OF TABLES

Table

3-1

C-1

C-2

E-1

E-2

E-3

Solar Cycle Elements Information in Data Base ............................................... 5

Simulated Data and Sample Means and Variances ......................................... C- 16

Means and Variances of the Beta Distribution ............................................. C-16

Recent Monthly Mean Solar Activity Values ................................................. E-3

13-Month Smoothed 10.7-cm Solar Flux .................................................... E-4

Estimates of 13-Month Smoothed Solar Activity for Balance of

Cycle 22, Cycle 23, and Beginning of Cycle 24 ............................................ E-5

vii
ap

Ap

X,

C

Ct

Ct

CREAT_BLOCK

COMB_YLGINT

COV

CUBFIT

EDF

F 10.7

Flo.7

FNDPRCNT

FORTRAN

FORECAST

FRMEMO

G JR

INFLBS

INTERP

LINCMC

m

MEMO

MERGE

MLLRT

MSFC

NASA

P

Pj

m

P

pdf

PCHMAX

PCHMIN

LIST OF SYMBOLS AND ACRONYMS

3-Hourly Value of Geomagnetic Index

Daily or Monthly Mean Value (depending on use) of Geomagnetic Index

13-Month Smoothed Geomagnetic Index

Regression Coefficient

Magnetic Character Figure

13-Month Smoothed Magnetic Character Figure

Create File Subroutine

Lagrangian Interpolation Subroutine

Covariance

Cubic Curve Fit Subroutine

EmpiricalDistribution Function

Daily or Monthly Mean Value (depending on use) of 10.7-cm Solar Radio

Noise Flux

13-Month Smoothed 10.7-cm Solar Radio Noise Flux

Find the Percentile Subroutine

Formula Translation (computer language)

Modified McNish-Lincoha Linear Regression Program

Memorandum Printout Subroutine

Inverse Matrix Subroutine

Find-Cubic-Inflection-Point Subroutine

Linear Interpolation Subroutine

Modified McNish-Lincoln Subroutine

Number of Points per Solar Cycle

Memorandum

Merge Subroutine

MSFC Lagrangian Linear Regression Technique

Marshall Space Fight Center

National Aeronautics and Space Administration

Period Related to Maximum to Maximum or Minimum to Minimum of Solar

Cycles

Period Related to Maximum to Maximum or Minimum to Minimum ofj _

Solar Cycle

Mean Period

Probability Density Function

Smoothing Subroutine for Maximum to Maximum

Smoothing Subroutine for Minimum to Minimum

viii
PRCNTILE

PREPROC

Q

QUANTILE

R

R

Ro

Rmearlo

m

ARo

RANKIT

REPORT

t

to

VAR

,g

(I

Percentiles Subroutine

F10.7 and Ap Preprocessor Program

Quantile

Quantile Subroutine

Wolf's Relative Sunspot Number

13-Month Smoothed Sunspot Number

Predicted 13-Month Smoothed Sunspot Number

Mean of observed 13-Month Smoothed Sunspot Number for n Cycles at

Predicted Time Point for R p

Difference between Rmear_ and Rp as Calculated by Modified McNishLincoln Linear Regression Method

Observed 13-Month Smoothed Sunspot Number at Prediction Time Point

Mean of observed 13-Month Smoothed Sunspot Number for n Cycles at

Prediction Time Point for R o

Difference between Rmean. and R° at Prediction Time Point

Arrange Data in Ascending Order Subroutine

Report Program

Time

Time from Method Initialization Time Point (Prediction Time Point)

Variance

Mean Time Increment

Standard Deviation

ix

Technical Memorandum

Statistical Technique for Intermediate and Long-Range

Estimation of 13-Month Smoothed Solar Flux and

Geomagnetic Index

1.0 Introduction

Because no generally accepted solar physical model is available to accurately predict

future solar activity, Marshall Space Flight Center (MSFC) developed a 13-month smoothed

solar flux and geomagnetic index intermediate (months) and long-range (years) statistical

estimation technique'. The reason for issuing intermediate and long-range solar activity

estimates is the need for updated inputs to the upper atmosphere density models used for

satellite orbital lifetime predictions and performance requirement analyses 2. Mission analysis

and planning for future spacecraft launches and on-orbit operations require estimates of

orbital lifetimes, altitudes, inclinations, and eccentricities. This report documents the MSFC

13-month smoothed solar flux and geomagnetic index intermediate and long-range statistical

estimation technique, referred to as the MSFC Lagrangian Linear Regression Technique

(MLLRT).

2.0 Background

Excellent surveys of various future solar activity estimation techniques were

presented by Vitinskii v, Scissum 4, Slutz, et al. 5, and Donnelly 6. Neural Network

applications have been more recently made by Macpherson et al. 7, and Calvo et al. 8 and

others. The linear regression method, applied by McNish and Lincoln 9, was modified by

Boykin and Richards 1°, and Avaritt 1_. A later improvement applying the modified McNish

and Lincoln linear regression method to develop the MSFC Lagrangian Linear Regression

Technique was accomplished by Holland and Vaughan 1.

Yu. I. Vitinskii 3 conducted an extensive survey and analysis of solar activity

prediction methods. While recognizing the magnitude of the problem and encouraging

studies of active processes taking place on the Sun to solve it, he reiterated the current status:

"...we have shown that the reliability of the results obtained using these methods still leaves

much to be desired." His analysis, however, showed the linear regression method usually

gives accurate results to a year in advance. For several-years-in-advance, the linear

regression method becomes increasingly less accurate.

McNish and Lincoln 9 suggested that the estimation of a sunspot cycle's future

behavior, based on the mean approximation of all past cycles, could be improved by adding

to the mean a correction proportional to the departure of the current value of the cycle from

the mean cycle. They also recommended the method not be used for making future

projections longer than one year. Using a data base with two additional solar cycles, Boykin

and Richards t° modified the McNish-Lincoln linear regression method so the 13-month

smoothed relative sunspot number (R) could be estimated for 10 years in advance, at

quarterly rather than yearly intervals. Figure 2-1 illustrates this method that is also applicable

to 13-month smoothed solar flux (F_07) and geomagnetic index (Ap).
150

100

Z

[-

0

Z

U.1

50

[-

0

0

0

0

/ w -x, \x

Ro / r / / _ t_"-'-PREDICTED"\\ CYCLE

/ / I

' MEAN OF ALL_..._

I PAST CYCLES

_to_ II,meano I

1 2 3 4 5 6 7 8

TIME FROM MINIMUM (YEARS)

Rp = _neanp + ARp

m

ARp = CAR o

COy (Rn,Rp) C=

VAR (Ro)

Figure 2-1. Modified McNish-Lincoln Linear Regression Method.

Holland, and Vaughan _determined that better statistical estimations are possible, in a chi

square sense, by selecting the start and end of each set (solar cycle) at the maximum (or

minimum) and normalizing the data sets using a Lagrangian linear regression statistical

technique. This determination and initialization of the modified McNish-Lincoln linear

regression method at the cycle's maximum or minimum establish the current MSFC statistical

technique for intermediate and long-range estimation of F_07 and A p.

3.0 MSFC Lagrangian Linear Regression Methodology

m

Sections 3.1 and 3.2 present the methodology used to develop the F_07 and Ap, data

bases and the modified McNish-Lincoln linear regression method used in the MSFC

Lagrangian Linear Regression Technique (MLLRT).

3.1 Development of El07 and Ap Data Bases

In order to use the modified McNish-Lincoln linear regression method to estimate

future solar flux and geomagnetic index, Ft07 and A p, data bases were developed. Two data

bases for F_07 and Ap were developed depending on what estimates are relative to time point

in the present cycle. Solar cycle minimum estimates use a maximum to maximum initialized

data base. Note when using the maximum to maximum data base, the cycles are identified by

cycle number at maximum initialization date. For example, maximum to maximum cycle 18

starts in May 1947 with' the first half of cycle 18 and the last half of cycle 19. Solar cycle

maximum estimates use a minimum to minimum initialized data base. To develop a data base

select the time of all maximum points of the _0.7 for each cycle, and use these same time
m

points for the Ap, or depending on the desire estimate, select the time of all minimum points

of the F107 and Ap for each cycle. Determine all periods, Pj, related to maximum to

maximum (or minimum to minimum) for the cycles. Calculate the mean period, P, for

cycles initialized at maximum (or minimum). Divide all cycle periods by P. This gives the

mean time increment, xj = P/P, to be used for the j,h cycle. Note that zj may be less than,

equal to, or greater than one month. Finally, consider each cycle a group of data with m

approximately equal to the number of points in P. To obtain an F_07 or Ap value

corresponding to each of the m points per cycle requires interpolation since points are not

selected at precisely one month intervals for all cycles. A more detailed description of the

data bases development is presented in sections 3.1.1 and 3.1.2.

3.1.1 13-Month Smoothed Solar Flux (FiaT) Data Base

Although some researchers believe they have sufficient reason to separate the data for

sunspot cycles 1 through 8 from the total data base, the MSFC Lagrangian Linear Regression

Technique for estimation of future F_07 uses the observed data for all observed cycles.

Including cycles 1 through 8 provides information applicable to the apparent behavior of the

cycle period and to the overall magnitude during this time frame as well as a larger data base

for statistical estimates. The measured F10.7 data base was extended back to 1749 by using

Wolf's relative sunspot values, R, and a R to F_07 conversion equation. R is defined by the

equation:

R=k(lOg + f) (1)

where R is the Wolf number, k is a correction factor to equalize counts from different

observers, g is the number of groups visible on a given day, and f is the number of single

spots observed on a given day 12. The R values were smoothed using the Zurich 13-month

smoothing equation:

Ri_ 6 + Ri+ 6 -

(2)

where i indicates the month of interest. This smoothing technique was developed by the

Swiss Federal Observatory, Zurich, Switzerland _2. Once R values are smoothed to R

values, the following equation (developed by MSFC in collaboration with Jack Slowey of the

Smithsonian Astrophysical Observatory) converts recorded R data to Fj0v data:

F107 = 49.4 + 0.97R + 17.6e (-°°35_) (3)

__ d

Equation (3), derived from recorded I=107and R data from 1947 to 1978, has a linear

correlation coefficient of 0.98. Figure 3-1, a plot of equation (3) with R and 1=107data to

1995, shows that equation (3) is still adequate for today's applications.

3
2.50

t¢3

8t

o 2t_

E

&

t_

U. 1N)

_ lot}

.5O

. ::::_i _:!':"_!-!i.,_" J

l.......................................................... i......................... i.............................

I i i i i i i i i i I i , i i I , , , , I , , , ,

0 50 1 (X) 1.50 200 250

13-Mouth Smoothed R from ]947 to ]995

-- m

Figure 3-1. R versus F1o.7with Curve Fit from Conversion Equation.

Since 1947, observed values of daily solar flux are used to compute mean monthly F1o.7

values. Use equation (2), replacing R with Flo.7 , tO calculate the Flo.7. Figure 3-2 is a plot of

I

converted and observed FIo.7 data for all cycles since 1749.

250

200

O

"_ 150

_ 100

,..-i

50

1749

.......... t...,o ........

I I i I i I I I I I I I i I I I I I I I I

1799 1849 1899 1949 1999

Year

Figure 3-2. 13-Month Smoothed (Converted and Observed) Solar Flux (_o.7)-

All solar flux cycles in the regression technique data base prior to 1947 use, as the

starting point, the established R maximum or minimum. After 1946, the maximum or

minimum is based on the observed F10.7maximum or minimum values computed from the

4
measured Flo7 values _. Table 3-1 has the minimum and maximum starting point years and

period of each cycle with associated converted or observed Fi07 values. Data format is year

with months in decimal form, i.e., January is 0.000 and December is 0.917. The equation to

calculate the months is

month decimal value = (month number- 1)/12 (4)

Table 3-1. Solar Cycle Elements Information in Data Base

Year/Month

Cycle of

Minimum

I 1755.083

2 1766.417

3 1775.417

4 1784.667

5 1798.250

6 1810.500

7 1823.333

8 1833.833

9 1843.500

10 1855.917

11 1867.167

12 1878.917

13 1890.167

14 1902.000

15 1913.500

16 1923.583

17 1933.667

18 1944.083

19 1954.250

20 1964.750

21 1976.417

22 1986.667

Year/Month

of

Maximum

1750.250

1761.417

1769.667

1778.333

1788.083

1805.083

1816.333

1829.833

1837.167

1848.083

1860.083

1870.583

1883.917

1894.000

1906.083

1917.583

1928.250

1937.250

1947.333

1958.167

1970.500

1981.333

1989.417

Summary:

Median (50%)

Mean

Standard Deviation (a)

Minimum

Solar Flux

Value*

70.66

72.13

70.06

71.26

68.22

67.00

67.03

70.12

71.78

68.23

69.11

67.85

69.01

68.02

67.54

69.30

68.36

70.33

69.69

72.59

73.27

72.90

69.50

69.75

1.93

Maximum

Solar Flux

Value*

139.87

134.13

162.02

203.26

186.47

100.26

99.84

120.35

191.99

177.61

144.97

185.82

123.04

135.50

113.54

152.07

126.32

165.30

214.39

245.45

156.34

204.55

213.11

156.34

160.70

39.92

Rise Time

in

Year/Month

6.334

3.250

2.916

3.416

6.833

5.833

6.500

3.334

4.583

4.166

3.416

5.000

3.833

4.083

4.083

4.667

3.583

3.250

3.917

5.750

4.916

2.750

4.08

4.38

1.21

Fall Time

in

Year/Month

4.833

5.000

5.750

6.334

10.167

5.417

7.000

4.000

6.333

7.834

7.084

8.334

6.250

8.000

7.417

6.000

5.417

6.833

@917

6.583

5.917

5.334

6.33

6.49

1.36

Min. to Min.

Period

(Year/Month)

11.334

9.000

9.250

13.583

12.250

12.833

10.500

9.667

12.417

il.250

11.750

11.250

11.833

11.500

10.083

10.084

10.416

10.167

10.500

11.667

10.250

11.25

11.03

1.19

Max. to Max.

Period

(Year/Month)

11.167

8.250

8.666

9.750

17.000

11.250

13.500

7.334

10.916

12.000

10.500

13.334

10.083

12.083

11.500

10.667

9.000

10.083

10.834

12.333

10.833

8.084

10.83

10.87

2.12

* Converted prior to 1947 and observed after 1946

The converted and observed FI07 data are Lagrangian interpolated to normalize the

data for the 132 months from the maximum or minimum cycle starting dates. The data is

stored by month and cycle number to construct a data base for use in the modified McNishLincoln linear regression method.

5
3.1.2 13-Month Smoothed Geomagnetic Index (Ap)Data Base

Because the measured geomagnetic index (Ao) data base is relatively short (1932 to

1996), it was extended back to 1884 using mean mofithly magnetic character figure (Ci) data.

The mean monthly C_ data is converted to 13-month smoothed data using equation (2) and

replacing R with C i. Once the mean monthly C_ is converted to 13-month smoothed

magnetic character figure (C_), use equation (5) to convert the extended record of C, data to

Ap values.

mp = 2.8068e 2393_' (5)

This equation, derived from recorded _3C, and Ap data from 1932 to 1963,_ has a_linear

correlation coefficient of 0.80. Figure 3-3 is a plot of equation (5) with C_ and Ap from

1932 to 1963.

30

_e_ i _. =2.8068e(2393E:i _ ': ! _ A'_¢:_?

o .......,.. ............... ......................................................

5 _ _ _ _ I _ _ _ _ I , _ _ _ I _ _ _ _ I , , _ _

0.40 0.50 0.60 0.70 0.80 0.90 1.0

13-Month Smoothed C from 1932 to 1963

i

Figure 3-3. C, versus A 0 with Curve Fit from Conversion Equation.

After 1931, the measured values of daily Ap are used to compute the mean monthly value.

Use equation (2), replacing R with A_, to calculate Ap. Figure 3-4 is a plot of converted and

recorded A_ data for all cycles since 1884.

6
30

1,¢

25

_ 2o

lO

d,

5

1884 1904 1924 1944 1964 1984 2004

Ycai"

Figure 3-4. 13-Month Smoothed (Converted and Recorded) Geomagnetic Index ( A p).

The converted and recorded Ap data are Lagrangian interpolated to normalize the data

for the 132 months from the maximum or minimum cycle starting dates. The data is stored

by month and cycle number to construct a data base for use in the modified McNish-Lincoln

linear regression method.

3.2 Modified McNish-Lincoln Linear Regression Method

The MLLRT uses the Boykin and Richards 1°modified McNish-Lincoln linear

regression method (for the linear least squares method development see appendix A) and an

appropriately constructed data base that starts at the maximum or minimum to estimate the

balance of the present cycle where the cycle is def'med from minimum to minimum or

maximum to maximum. This method is summarized in the following steps:

1. Mean F10.7or Ap is calculated from the completed cycles in the F10.7or Ap data

base constructed using the Lagrangian interpolated data points for use in the McNish-Lincoln

linear regression method. This mean also estimates F10.7or Ap for the next cycle with P.

2. McNish-Lincoln linear regression method produces a statistical estimate for the

rest of the present cycle using one linear coefficient. The period for the present cycle, for

which estimates of solar activity are being calculated, is the P.

3. Since, for the present cycle, only 21 or 22 corresponding points are available for

a linear regression fit of the estimated point to the last observed point, to justify calculating a

standard deviation based on a normal distribution function is difficult. This non-normal

distribution function produces upper and lower bounds that can and do go below the

parameter physical limits. Despite being a non-normal distribution, the data are standardized

to make calculations easier. The actual distribution of deviations from the smoothed linear

regression line and mean line is divided by the standard deviation and used to determine the

upper and lower bounds at predetermined percentile levels. Upper and lower bounds are
calculated by the Quantile method. The equation used is:

Q(x i) = i/(n+ 1) (6)

where Q is quantile, i equals 1 through the total number of completed cycles, and n is equal

to the total number of completed cycles. Once the quantile is calculated the percentile is:

Percentile(y) = 100.0 Q(x) (7)

Rationale for using the Quantile method is discussed in appendix C.

4. Between the upper and lower bounds discussed in step 3 is the "error space" on a

two-dimensional plot of F_07 or Ap versus time, t.

4.0 Application of MSFC Lagrangian Linear Regression Technique

-- D

The computer programs to calculate statistical estimates for future Flo7 and A p cycles

and to perform subsequent statistical analyses were developed in FORTRAN. The following

sections describe how the programs implement the MLLRT. MLLRT consists of three main

programs: (1) the _07 and Ap preprocessor computer program (PREPROC), (2) modified

McNish-Lincoln linear regression computer program (FORECAST), and (3) statistical

estimate output report computer program (REPORT). These programs are used to produce

the MSFC monthly solar activity memorandum data (see appendix E). Figure 4-1 is a

summary block diagram showing the flow of files through these programs.
Readin Input

(FLXINPUT.MAX, FLXINPUT. MIN,

APINPUT.MAX, APINPUT.MIN )

Run F,o,andA PreprocessorProgram

(PI_.PROC_to make13-month

smoothed data

Output files from PREPROC and Input

files to FORECAST

(FLUXMAX. DAT, FLUXMIN. DAT,

APMAX. DAT, APMIN. DAT )

L

Run Modified McNish-Lincoln Linear

Regression Program (FORECAST) to

estimate the rest of present cycle

(min. to min. or max. to max. )

Output files from FORECAST and Input

Files to REPORT

(FLXOUT. DAT and APOUT. DAT)

Run Report Program (REPORT) to

combine F ,0.,and A data for the solar

memorandum.

Output file from REPORT [

(MEMO)

Figure 4-1. MSFC Lagrangian Linear Regression Technique Block Diagram.

9
-- m

4.1 F!o.7 and Ap Preprocessor Computer Program

PREPROC uses mean monthly R, F107, Ci, and Ap data to supply the linear regression

program with 13-month smoothed data initialized at the cycle's maximum or minimum solar activity

dates. Table 3-1 summarizes the initialization F_07 dates for present and past solar cycles. Data are

initialized at the F_07 maximum (minimum) dates to enable the program to better estimate duration of

the present cycle and when its minimum (maximum) will occur. Once the minimum (maximum) is

determined, the preprocessor supplies the linear regression computer program with 13-month

smoothed data initialized at the minimum (maximum) solar activity date for the computer program to

provide a better estimate for the "new" present cycle and occurrence of its maximum (minimum).

Steps in section 3. l are the methodology for the preprocessor computer program. How individual

files are generated for FORECAST is given in the next paragraph.

PREPROC starts with the monthly mean values of R and Fi07 input data (flxinput.max or

flxinput.min) or the C i and A v input data (apinput.max or apinput.min). The flxinput.max or

flxinput.min data have two sources: the monthly mean R data from M. Waldmeier's book 12,and the

National Research Council of Canada monthly mean F10_7 data. Both sets of data are smoothed by

equation (2) in section 3.1.1 and R data are converted to F_o7 by equation (3). The apinput.max or

apinput.min data have two sources: the monthly mean geomagnetic C_ data from the Handbook of

Geophysics and Space Environments _3and the Institute for Geophysics in Gottingen, Germany

monthly mean ALdata. Both sets of data are smoothed by equation (2) in section 3.1.1 and C i data

are converted to Ap data by equation (5). Once the F_07 or Ap sets are complete, the data are

grouped into cycles then normalized to 132 points by a 3-point Lagrange interpolation subroutine

(COMB_YLGINT). Ft07 or Ap output from the preprocessor program is used in a create file

subroutine (CREAT_BLOCK) which formats the normalized data into a new output file

(fluxmax.dat or fluxmin.dat) or (apmax.dat or apmin.dat) for FORECAST. Figure 4-2 is a

summary block diagram showing the flow of files through PREPROC and its subroutines to create

the input file for FORECAST.

10
Read monthly R, F,o.7,

C_, and A_ Input Data

(FLXINPUT.MAX,

FLXINPUT. MIN,

APINPUT. MAX,

APINPUT. MIN)

l

Calculate the 13-

Month Smoothed

Values

Convert either R to

F,o., or C_to A P

1

Lagrangian

Interpolation

Subroutine

(COMB_YLG_T)

l

Create New File

Subroutine

(CREAT_BLDCK)

Figure 4-2.

Output files

(FLUXMAX. DAT,

FLUXMIN. DAT,

APMAX. DAT,

APMIN. DAT )

F_0.7and Ap Preprocessor Computer Program Block Diagram.

11
4.2 Modified McNish-Lincoln Linear Regression Computer Program

FORECAST estimates F_07 and Ap for the rest of the minimum to minimum or

maximum to maximum present cycle using the methodology discussed in section 3.2. A

description of generating individual files via FORECAST follows.

D

FORECAST reads the F10.7 or Ap data (fluxmax.dat, fluxmin.dat or apmax.dat,

apmin.dat) and standardizes completed cycles data. F_07 or Ap data enter the modified McNishLincoln subroutine (LINCMC) which first calculates the mean of the completed cycles.

LINCMC calculates next the difference between the actual and mean used in calculating the

variance matrix (A) and covariance matrix (B). LINCMC uses an inverse matrix subroutine

(G JR) to calculate the A _ matrix and subsequently the regression coefficient matrix (C). Using

the C matrix LINCMC estimates the rest of the present cycle (step 2 section 3.2). Using the

historical data base, the next cycle is estimated (step 1 section 3.2). Percentiles subroutine

(PRCNTILE) calculates the 95.0 and 5.0 percentile values for the present cycle. PRCNTILE

divides the difference data by the standard deviation then calculates the desired quantile value by

the quantile subroutine (QUANTILE). QUANTILE arranges the data in ascending order in

subroutine (RANKIT) then calculates the quantile values (equation 6 in section 3.2 step 3).

PRCNTILE enters the find-the-percentile-subroutine (FNDPRCNT) and calculates the needed

percentile value. FNDPRCNT determines the location of the desired quantile value then uses a

linear interpolation subroutine (INTERP) if the value is not the exact quantile calculated by

QUANTILE. Once the quantile values are determined, each is multiplied by 100 for the

percentile value. FORECAST next enters a smoothing subroutine (PCHMAX or PCHMIN).

The find-cubic-infection-point subroutine (INFLBS) locates the cubic inflection point between

the McNish-Lincoln estimate and the best statistical estimate. The percentiles have a cubic

inflection point also. The cubic curve fit subroutine (CUBFIT) smoothes the curve between the

two estimates (see appendix B for cubic connection theory). Once smoothed, the final

combined data are formatted for plots and tables. Figure 4-3 is the modified McNish-Lincoln

linear regression computer program block diagram. Appendix D is a listing of LINCMC and

QUANTILE subroutines.

12
Read in Input

(FLUXMAX. DAT,

FLUXMIN. DAT,

APMAX. DAT,

APMIN. DAT )

?

Find the Standard

Deviation of the

completed cycles

1

McNish-Lincoln

Subroutine

(LINCMC)

1

Smooth out

discontinuity

Subroutine

(PCHMAX or

PCHMIN)

l

Output files

(FLXPLOT. DAT or

APPLOT. DAT;

FLXPRED. DAT or

APPRED. DAT;

FLXOUT. DAT or

APOUT. DAT;

FLXCHECK. DAT or

APCHECK. DAT)

Find-InflectionPointSubroutine

(INFLBS)

,k

Cubic Curve Fit

I Subroutine

[ (CUBFIT)

I Calculate the

Mean of the

completed

solar cycles

4

I Calculate the

Diffference

between the

Actual and the

Mean

Calculate

Variance

Matrix (A) and

Covariance

Matrix (B)

Calculate the

Inverse of A

Subroutine

(G JR)

Calculate

Regression

Coefficient (C)

Calculate the

Predicted

values for the

remainder of

the present

cycle

Calculate the

Percentiles

Subroutine

(PRCNTILE)

__ Divide the

difference by

the standard

deviation

Calculate the

Quantile

Subroutine

(QUA/[TILE)

Find-theArrange the

data in

ascending

order

Subroutine

(RANKIT)

Calculate the

Quantile

Values

PercentileSubroutine ] Determine the I

(FNDPRCNT) | location of the I

Use this twice _ data for the

once for upper | percentile

bound and | selected I

second time ] ,[

for lower I , _.*

bound [ ]_ Linear. ]

/ ' ] Interp°lati°n I

I Subroutine ]

Figure 4-3. Modified McNish-Lincoln Linear Regression Computer Program Block Diagram.

13
4.3 Statistical Estimation Output Report Computer Program

REPORT combines the F_o7 and Ap data, reads the FORECAST output files

(fluxout.dat and apoutput.dat), and merges the data via the merge subroutine (MERGE). The

memorandum printout subroutine (FRMEMO) takes the data from MERGE and prints the table

for the MSFC monthly solar activity memorandum. Figure 4-4 is a summary block diagram of

the report program; appendix E is an example of the memorandum.

Read Input Data

(FLXOUTPUT, and

APOUTPUT)

Subroutine

(MERGE)

Memorandum Printout I

Subroutine I

(FRMEMO)

Output file

(MEMO)

Figure 4-4. Report Program Block Diagram.

Appendix E provides a sample of the MLLRT computer program output during a

maximum to maximum cycle. The present 132-month period cycle initialization month is June

1989 (cycle 22 maximum) and the F_07 value is 213.1. Using the statistical Lagrangian data

base for the converted and observed smoothed solar flux, the estimate of the 95, -50, and 5

percentile values for F_0.7is arranged in a monthly sequence through the balance of cycle 22,

through cycle 23, and into the first half of cycle 24. Obtainable from the 95, -50, and 5

percentile Fj0v values for the present cycle is an estimate of the range for the upcoming solar

flux cycle maximum values. The range estimate given for the 132 months from the maximum

of cycle 23 to the maximum of cycle 24 is determined from the mean (-50%) cycle and the

upper (95%) and lower (5%) bounds based on the converted and observed F]07 data base

(currently 21 cycles). The tie-in for the present to next cycle uses a cubic connection from the

nearest inflection points on the rising leg of cycle 23 to the maximum of the 95, -50, 5

percentile values of the flux for all cycles. This defines the maximum of cycle 23 and the

extension of the estimate into the next cycle. Tie-ins accomplished with a cubic curve fit explain

the relatively smoothed appearance of the curve in this tie-in area. The memorandum in

appendix E, presents the 95 and 5 percentile values for the past 21 cycles as a matter of

reference relative to the mean (-50%) period used in the statistical estimation computer program.

m

Appendix E discussion above also applies to the Ap estimates.

14
5.0 Engineering Use of MSFC Solar Activity Statistical Estimation Technique

Products

n

MLLRT's future F_0.7and Ap estimates are intended mainly for use as inputs to upper

atmosphere models, i.e., Marshall Engineering Thermosphere Model, (NASA CR-179359 and

CR- 179389 ), NASA/MSFC Global Reference Atmosphere Model, (NASA TM-4715), and

others. Figure 5-1, for example, depicts inputs related to the MSFC spacecraft orbital lifetime

model.

ESTIMATED ORBITAL STATE

FUTURE VECTOR

SOLAR FLUX

BEHAVIOR

_FLrI'URE

GEOMAGNETIC

\ INDEX

MODEL DENSITY

ORBITAL

LIFETIME

PREDICTED

',RAFT

ORBITAL

LIFETIME

ATHTUDE

TIMELINE

BALLISTIC

C

TIMELINE

Figure 5-1. MSFC Spacecraft Orbital Lifetime Prediction Model Block Diagram TM.

Since there is no method for intermediate and long-term predictions of daily F_0.7 and ap,

orbital lifetime models use the F_o.7 and Ap estimates 2. Orbital lifetime estimates, control

analysis programs, et al., require a specific date to associate with the future estimate of F10.7and

Ap to compute corresponding atmospheric density. Future estimated F10.7 and Ap data points

should be identified with the First day of the given month.

For spacecraft projects requiring a minimum risk design lifetime orbital altitude(s) and

a specified control capability, the 95 percentile estimates of F_o.7and Ap are recommended.

Taking into account the short-term (days) dynamics, these estimates permit the design of a

statistically conservative spacecraft lifetime and control capability at a given orbital altitude(s).

The lifetime determination should be based on the most current intermediate and long-range

statistical estimate of the future solar flux and geomagnetic index consistent with the critical

project development decision time points prior to planned launch of the spacecraft.

Changes in orbital density associated with short-term variations in the daily F10.7and ap

required as inputs to the upper aunospheric models, are not represented by the F_o.7 and Ap

statistical estimates given in the MSFC monthly solar activity memorandum. Future changes in

total atmospheric density cannot be estimated with any acceptable degree of statistical confidence

15
using existing techniques. Representative data sets, based on past daily Fl0 7 and a v values, may

be utilized to compute this dynamic component of the orbital altitude densi_.

The design requirements for solar activity and associated on-orbit density values are

specified in the appropriate spacecraft and space vehicle project design requirements

documentation. Consult these documents for this information.

6.0 Concluding Remarks

Became no physical solar activity prediction models are accepted generally by the

scientific community, MSFC developed the MLLRT. MLLRT provides Fl0._ and Ap values for

input parameters to the MSFC orbital altitude neutral aunosphere models noted in section 5.0.

Although the atmosphere models are built on inputs of F10.7 and ap, the _0.7 and Ap values are

the lowest level of smoothing with any acceptable level of confidence that lends itself to

reasonably accurate statistical prediction.

The technique utilized by MSFC is based on a small sample size and a nongamsian

distribution of F_0.7and Ap. To estimate future activity MSFC uses a mean cycle and deviations

derived from previous cycles, combined with a one-term linear regression, to estimate future

activity of the present cycle. Since intermediate term (months) estimates are more accurate than

long range (years) estimates, the data base is initialized at both established maximum and

minimum values of solar cycle activity. Based on data from previous cycles, the probability is

90 percent that the actual future F_o.7or Av values will be within the current MLLRT computer

program output for estimated 95 and 5 percentile values. The computer program may be

adjusted to accommodate any desired percentiles between 95 and 5.

To provide an assessment of the MLLRT computer program's products, appendix F

presents plots (Figure F- 1 through F-6) of the _0.7 future estimates from the MLLRT for a 5-

year period from date of estimation for several different solar cycles. Three of the solar flux

estimation periods are for 1 I-year periods and began at the minimum between cycles 19-20, 20-

21, and 21-22 and three at the maximum of cycles 20, 21 and 22.

16
.

.

.

5.

.

.

.

.

10.

11.

12.

13.

14.

References

Holland, R. L., and William W. Vaughan: Lagrangian Least-Squares Prediction of

Solar Flux (Fi0.7). Journal of Geophysical Research, Vol. 89, No. A1, pp. 11-16,

January 1, 1984.

Dreher, P. E., and A. T. Lyons: Long-Term Orbital Lifetime Predictions. NASA

Technical Paper 3058, George C. Marshall Space Flight Center, Huntsville, Alabama,

October 1990.

Vitinskii, Yu, I.: Solar-Activity Forecasting. Translated from Russian, Israel Program

for Scientific Translations, Jerusalem. NASA TT F-289, Clearing House for Federal

Scientific and Technical Information, Springfield, Virginia, 1965.

Scissum, J.: Survey of Solar Cycle Prediction Models. NASA TM-53593, 1967.

Slutz, R. J., Gray, T. B., West, M. L., Stewart, F. G., and Leftin, Margo: Solar

Activity Prediction. NASA CR- 1939, November 1971.

Donnelly, R. F. (Ed.): Solar-Terrestrial Predictions Proceedings, vols. 1, 2, 3, and 4,

U. S. Government Printing Office, Washington D. C., 1979.

Macpherson, K.P., A. J. Corwayan J.C. Brown: Prediction of Solar and Geomagnetic

Activity Data Using Neural Networks. Journal of Geophysical Research, Vol. 100,

No. A 1 I, pp. 21,735-21,744, November 1,1995.

Calvo, R. A., H. A. Ceecalto, and R. D. Piacentini: Neural Network Prediction of

Solar Activity. AstrophysicJournal, Vol. 444, pp. 916-920, May 10, 1995.

McNish, A. G., and Lincoln, J. V." Prediction of Sunspot Numbers. Transactions,

American Geophysical Union, 30, 1949, p. 673.

Boykin, E. P., and Richards, T. J.: Application of the Lincoln McNish Technique to

the Prediction of the Remainder of the Twentieth Sunspot Cycle. Lockheed Missiles

and Space Company, Technical Memorandum 54/30-89, Huntsville, Alabama, March

1966.

Avaritt, D." Sunspot Prediction Program (Sunspot Deck). Program Document

3F0360, Computer Sciences Corporation, Huntsville, Alabama, 1972.

Waldmeier, M.: The Sunspot Activity in the Years 1610-1960. Zurich Schulthess and

Company, Switzerland, 1961.

Valley, Shea L. (Ed.): Handbook of Geophysics and Space Environments, McGrawHill Book Company, New York, April 1965.

Dreher, P. E., Little, R. P., and Wittenstein, G.: Skylab Orbital Lifetime Prediction

and Decay Analysis NASA, Technical Memorandum 78308, Huntsville, Alabama,

November 1980.

17
8.0 Bibliography

Bumba, V. and J. Uleczek (Ed.): Basic Mechanism of Solar Activity. D. Reidel Publishing

Company, Boston, MA, ISBN#90-277-0680-8, 1976.

Eckstein, Herbert P.: Solar Activity: A Bibliography. RSIC-706, Redstone Scientific

Information Center, U.S. Army Missile Command, September 1967.

Jordan, Stuart (Ed.): The Sun as a Star. NASA SP-450, 1981.

Mayland, P.N.: Derivation, Meaning, and Use of Geomagnetic Indices. American Geophysical

Union, Washington, D.C., ISBN#0.87590-022-4, 1980.

Rubashev, B.M. : Problems of Solar Activity. NASA TT F-244, December 1964.

Schove, D. Justin (Ed.): Sunspot Cycles. Hutchinson Ross Publishing Company,

Stroudsburg, Penn, ISBN# 0-87933-424-X, 1983.

Vorchishin, K. S. and M. I. Stodilko: The Statistical Stability of Solar Cycling. Astron. J.,

Vol. 59, No. 6, pp. 1171-1183, 1983.

Wilson, Robert M.: Sunspot Variation and Selected Associated Phenomena: A Look at Cycle 21

and Beyond. NASA TM-82474, February 1982.

Wilson, Robert M.: A Comparative Look at Sunspot Cycles. NASA TP-2325, May 1984.

Xanthakis, J. (Ed): Solar Activity and Related Interplanetary and Terrestrial Phenomena.

Springer-Verlag Publishers, New York, New York, ISBN#3-540-06314-5, 1973.

18
APPENDIXA

LinearLeastSquaresModelDevelopment

RobertL. Holland

ComputerSciencesCorporation

Huntsville,Alabama

From the Lagrangianinterpolatedcycles,therearecurrently21 completecyclesfrom

maximumto maximumpeaks.Thereare132interpolatedvaluesin eachof thesecycles.In the

following developmentof themodifiedMcNish-Lincolnlinearregressionmethodeachpoint in

thedatawill bereferredto asRij wherei refersto thei 'hpoint within a cycle(1 to 132)andj

refersto thecyclenumber(1 to 22). Othervariableswill bedefinedastheyappear.

Theexpectedvalueof R' at p-pointsbeyondm, wherem is the numberof observed

valuesin thecurrent cyclecanbe written as a linear combinationof any numberk < m

coefficients times the deviations from the means of the k-known 13-month smoothed monthly

values within the current cycle.

Holland, Rhodes, and Euler A-1included the time derivatives in the linear combination

model. In the following development, the derivatives will be omitted since their results

showed that including more than one coefficient and the derivative did not improve the

confidence level nor the goodness of fit. For k=2, the model requires 2 coefficients and 2

independent variables and likewise for k < m. The k = 2 model will be developed for

simplicity then specialized for k = 1, which is the MLLRT model in use at MSFC. The model

may be presented classically as

z' = ax + by (A-l)

where x and y are the two independent variables, z' is the dependent variable, and a and b are

the linearly-related (regression) coefficients.

z' is the model predicted value of the actual z data with N known (xj, yj, zj)

corresponding points. It is required to find the coefficients a and b which best fit the data in a

least squares to minimize the sum of the squares of the deviations of the chosen model (A-l)

predictions from the data. Substituting the data points xj, yj in the right side of equation (A- 1)

gives a model predicted z'j, i.e.,

z'j = axj + byj, (A-2)

A-1
1 < j < N (the number of known data points).

Now it is required to minimize the sum of the squared differences (deviations) of these

predicted z'. from the data zj. These deviations dj are J

dj = zj - z'j = zj -(axj + byj) (A-3)

There are N of these deviations for each corresponding point within the 132-point

Lagrangian interpolated cycle. Denoting dj, xj, yj, zj and z'1 as n-dimensional vectors defined

as

d

d2

,dN

Z

r

Z 1

Z 2

,ZN

Z !

Z'

k,ZN

(- -,

X I

X 2

X = y =

,,,XN _

_yj

Y2

,YN

then equation (A-3) can be written in vector-matrix notation as

d = z - (ax + by) (A-4)

The sum of the squares of the deviations of the data (z) from the model prediction (ax +

by) are obtained by taking the dot product of d with itself• This gives

d2 = d • d = [z -(ax + by)]. [z -(ax + by)] (A-5)

carrying our this dot product gives

d2 - z ,, z- 2z * (ax • by) + (ax * by). (ax ,, by)

---- Z 2 - 2az * x - 2bz • y + (ax • by) 2

A-2
or

d 2 = z 2 - 2az • x - 2bz • y + a2x 2 + 2abx • y + b2y 2

To minimize d z, the partial derivatives of d 2 with respect to a and b must be zero.

these derivatives gives

(A-6)

Performing

_d 2

= -2z • x + 2ax 2 + 2bx • y = 0 (A-7)

3a

3d 2

= -2z • y + 2by 2 + 2ax • y = 0 (A-8)

3b

Rearranging terms and dropping the 2's gives

ax 2 + bx • y = z• x (A-9)

ax • y +by 2 = zo y

or in the adopted vector-matrix notation this becomes

(A-IO)

(::/.:(X2x.,

$ $ $ (A-11)

v (M) c

Thus from the vector-matrix equation

v = (M) c (A-12)

the c coefficient vector is determined by multiplying equation (A-12) through by the inverse of

(M), i.e.,

(M)-lv = (M)-I(M)e _ e = (M)-lv (A-13)

notice that (M) 1 and v on the right are all known data.

For a single independent variable x, then b=0 and there is no y variable. In this special

one coefficient case, equation (A-11) becomes

A-3
Z • X

z • x = ax 2 or a = "--'2--- (A-14)

x

It now remains to associate the foregoing with the actual solar flux Fro7

geomagnetic index Ap data and the statistical method (MLLRT) currently in use at MSFC.

x _ ARm, AR m = R m - Rm,

y ----> ARm_l, ARm_1 = Rm_ I - Rm_l,

1 N

and Ri = N_R0_-I 1 < i < 132

(Rml

R m2

also R m =

\RmN)

CARrel

ARmx

AR m -

\ARmN

and

ARm+ p = Cm+p AR m (A-15)

where

t

AR m * AR m

C =

AR m • AR m

m is the number of known points within the current cycle, p is any point beyond m within the

132-point cycle. Since the motivation in the MLLRT model is for projecting beyond the N

known data points and beyond m in the current cycle, then p>m for the N+I current cycle.

A-4

m is the most current 13-month smoothed month number from the beginning of the current

cycle• a ---->c and b --->o are the coefficients corresponding with c in the MLLRT model which

is a one coefficient model, z is any 21-point data vector from the 132 in the cycle, excluding

AR m and AR m_l. z is the model prediction for any of the known points within any of the

known cycles or for the points in the current cycle.

With the above association, the MLLRT model can now be presented as
From equation (A-15) the MLLRT one coefficient model is

Rm+p, N+I = Rm+p q- Cm+pARm, N+I

where

and

- 1 N

Rm+ p = --ZRm+p,j

N j=_

m

ARm = Rm.N+I Rm

(A-16)

Reference

A-I. Holland, R. L., C. A. Rhodes, and H. C. Euler: Lagrangian Least Squares Prediction of Solar

Activity. NASA TM-82462, George C. Marshall Space Flight Center, Huntsville, Alabama,

April 1982.

A-5

APPENDIX 13

CubicConnectionFromPresentPredictedCycleto Meanof NextCycle

Robert L. Holland

Computer Sciences Corporation

Huntsville, Alabama

The following steps describe the procedure for developing the cubic connection from

present predicted cycle to mean of next cycle:

1. Depending on when the present cycle was initialized, select the last inflection point

of the estimated (present) cycle (tl, F 0 near the 24-month point before the next cycle maximum

or minimum. The 24-month limit was used to avoid a sudden rise or fall in the transition gap,

since more than one inflection point exist in the prediction data. This should be done with at

least a five-point numerical scheme for determining where the second derivative goes to zero B-1

One such scheme is Sterling's in reference B2

d2F_ _ 1 [ FN+2 4FN+ , 5F N

--_)N (At) z _ I- _ 2

4v , v _2] _- - . (B-l)

3 12

2. Select the maximum (or minimum) on the mean cycle (tm, Fro). This is normally the

input, but it can also be determined numerically.

3. The second derivative should be zero at (t I, F_), and the first derivative is zero at (tm,

Fm). A cubic curve is the lowest degree polynomial which can be determined that will go

through the two points and have these properties at the two points.

4. The coefficients of the polynomial

F(t)= at 3 + bt 2 +ct + d (B-2)

are to be determined from the following four linear equations and will have the required four

properties:

6at_ + 2b = 0 {second derivative is zero at inflection point (B-3)

3a t 2 + 2bt m +c = 0 {first derivative is zero at maximum or minimum (B-4)

a t_ + b t_ + ct_ + d = F_ {curve goes through the inflection point (tt,, F 0, (B-5)

2

at 3 + b t m + ct m+ d = F m {curve goes through the maximum (or minimum)

point (t m, Fro) (B-6)

There are four linear equations in four coefficients which may be written in vector/matrix

notation and solve for a, b, c, and d. Define vector F, matrix (M) and vector of coefficients to

be determined e as follows:

B-1
F -= FI c-= and

Fm

r 6tl 2 0 !)

3t 2 2t m 1

(M)= 3 2

tl tI tl

3 2

tm t m tm

The corresponding vector/matrix equation for equations (B-3) through (B-6) above is

(M)c = F

To solve for c multiply the inverse (M) _ i.e.

(M)"(M)c = (M)"F

then c = (M)IF.

Note that all the elements of (M) _ and F are known data.

The solution for c becomes

(B-7)

(B-8)

1

--b

3t I

Fm-FI

Lg2t m 2 2

-2t2 +3 t_ 3 tl + 2tmtI

+tm-t:/b

FI-_.3(2,2i-2tmtl+t2 m)b

(B-9)

B-1.

B-2.

References

Waldmeier, M.: The Sunspot Activity in the Years 1610-1960. Zurich Schulthess and

Company, Switzerland, 1961.

Scarbrough, J. B .: Numerical Mathematical Analysis. Johns Hopkins Press, Baltimore,

Maryland, 1962, pp. 133-134.

B-2
APPENDIX C

Estimates of Percentiles for the 13-Month Smoothed Solar Flux Data

Leonard W. Howell, Jr.

Electromagnetics and Aerospace Environments Branch

Systems Analysis and Integration Laboratory

George C. Marshall Space Flight Center

National Aeronautics and Space Administration

The intent of the MLLRT Model and the periodic memos was to provide NASA programs with

an estimation of the most probable solar activity trends, based on statistical analysis of the

historical record, with confidence bounds placed on that estimate. However, it has turned out

that many of the most important applications required estimates for the high extreme values, and

so the 97.7 percentile envelope was used in this application. For this reason a re-examination of

the methodology for evaluating the percentile estimates associated with these bounding values

was undertaken and some changes have been made. This appendix summarizes the results of

this evaluation and the rationale for the revised approach.

Percentiles are the probability, expressed as a percentage, that a variable X will be less than a

specified value Xp. To eliminate repetition on the factors of 100 in the mathematical expressions

which follow, we denote this probability as a fraction, or "quantile", rather than a percentile. The

only difference between percentile and quantile is that percentile refers to a percent of the set of

data and quantile refers to a fraction of the set of data. For example, for a standard normal

distribution, 1.96 is the 0.975 quantile and is the 97.5 th percentile thus p= Pr{X < Xp} and it

ranges from zero to one. The inverse function Q(p)=xp is also useful to yield the value in the

range (F10.7 flux, in this case) associated with a given probability (p value). The concept of a

quantile is illustrated in Figure C-1 for an arbitrary probability density function f(x) and its

cumulative distribution F(x).

C-I
f(x)

Arbitrary Probability Density Function

and Quantile xo

pF(x)

X Xp

0

X Xp

Cumulative Distribution

and Quantile xp

Figure C-1. Arbitrary Probability Density Function fix) and Its Cumulative Distribution F(x).

Historically the MSFC solar activity memos have reported p values of 0.5, the median or "best

estimate," and bounding values of 0.023 and 0.977. These are easy to interpret for the engineering

community because of their relationship to the "2-sigma" probabilities of the Normal

distribution, even though there are strong indications that the Normal distribution is not

applicable to this data set.

Calculation of percentile probabilities is a well defined process in cases were there is a large data

set or where, for other reasons, the sample is known to follow a specified frequency distribution.

The probability is the integral under the frequency distribution from the lower limit of the

distribution, often -0% to xp. For the problem at hand, however, where the data set consists of

only 21 data points and the appropriate distribution is unknown, the probability can only be

estimated. One must select one of the various possible methods. Direct calculation of a precise

value which "truly" represents the situation is not possible.

For illustration we will use the 13-month smoothed Lagrangian flux data set illustrated in Figure

C-2. The 13-Month Smoothed Lagrangian Solar Flux data is calculated from the monthly solar

flux data that is first smoothed by the Zurich smoothed equation. The 13-month smoothed time

series data is then distributed into groups of cycle data. The data then is interpolated by a

Lagrangian method into 21 "peak-to-peak" solar cycles, with each cycle consisting of 132 points

(months) as depicted in Figure C-2. It should be noted that the data is not really data in the

usual statistical sense of "observations", but it is actually the output from multiple layers of

processing of the original observations. This fact is not relevant to the issues addressed in this

appendix, but it is important to the physical interpretation of the solar model results. The

processing tends to mask some of the actual variations and properties of the original data.

C-2
250

200

z

'_ 150

v.-

100

40 60 80 100

Month

Figure C-2. 13-Month Smoothed Lagrangian Flux.

120 140

I

When observing this data, it is important to note that two cycles appear to run side-by-side and

on the high side from month 80 to just under month 100, and then again briefly around months

115-118. Secondly, the mean and standard deviation trendlines can be constructed from the data

and are shown in Figure C-3.

L'_-Mean giandard Deviation

180.0 -1

160.0

140.0

120.0

100.0

E 80.0

6oo ,oo 20.0 .................. ......

0.0 i t

0 20 40 60 80 100 120 140

Month

Figure C-3. Mean and Standard Deviation Trend Lines.

C-3
General

Two methods for estimating percentiles are developed and compared. Both methods presented

make no attempt to analytically model an underlying distribution and thus represent strictly

empirical approaches. The first method discussed makes use of frequency histograms to

construct cumulative frequency polygons and is similar to the previous method. The second

method presented is referred to as the Quantile method in the statistical literature and will be

compared to the cumulative frequency polygon method.

Method 1. Cumulative Frequency Polygons

Graphical techniques can be easily implemented and they are very valuable in exploratory data

analysis and in combination with formal statistical tests. In the former the objective is to

discover particular features of the underlying distribution such as outliers, skewness or kurtosis

(i.e., thickness of the tails of the distribution). The saying that a picture is worth a thousand

words is especially appropriate for this type of analysis.

The frequency polygon is a many sided figure that is often used as an approximation of the

underlying probability density function of a random variable and is constructed from a histogram

of the data. Construction and use of the frequency polygon is well documented in the literature,

and the interested reader is referred to Kohler [C-l], Downie and Heath[C-2], and Kendall and

Stewart [C-3]. To draw the frequency polygon, the same set of coordinates is used as for the

histogram, but this time the class mark, or midpoint of each class width, is identified (as the

average of the two class limits) and a dot is positioned above it at a height equal to the relative

frequency density. The dots are then connected by straight lines. To complete the polygon,

straight lines are also drawn from the dots above the first and last class marks, respectively, to

points on the horizontal axis a distance W below the first class midpoint and a distance W above

the last class midpoint, where W is the class width.

This method is illustrated using the 21 data points defined by the first month of the 21 solar

cycles. The data in rank order is given below:

S={98.3, 99.9, 107.1,120.5, 123.2, 125.4, 132.9, 133.2, 145, 152.1,155, 162.2, 162.3, 177.4,

185.8, 186.4, 192, 202.2,203.6, 203.7, 245.1}.

First, the sample mean is computed as 157.8 and the standard deviation as 39.7; these points are

visible as the initial points (extreme left hand end) of the curves in Figure C-3. Next, the

standardized data set is constructed as zi=(xi- 157.8)/39.7 which gives

Sstandardized : { - 1.498, - 1.458, - 1.276, -0.939, -0.871, -0.816, -0.627, -0.619, -0.322,

-0.143, -0.07, 0.111, 0.114, 0.494, 0.706, 0.721,0.862, 1.119, 1.154, 1.157, 2.2 }

The frequency histogram and polygon are then constructed as illustrated in Figure C-4 for this

standardized data set.

C-4
U.td_ .j

0.2

C.1S

0.1

C.05

-4 -3 -2 -1 0

Standardized Data

I

3

Figure C-4. Relative Frequency Histogram.

The cumulative frequency polygon or "less-than" ogive is then constructed directly from the

polygon, as shown in Figure C-5. Quantiles of interest are then approximated using linear

interpolation. More sophisticated methods may be applied by first smoothing the frequency

polygon and then using higher order interpolation methods if desired.

[

i ---O_Frequency Polygon ---l---Cumulative Frequency Polygon l

.........

0.8

0.7

_ 0.6

0.!!

a.

0.2

0.1

-3 -2 -1 O 1 2 3

Standardized Data

Figure C-5. Frequency Polygon.

Quantiles may be obtained from Figure C-5 using linear interpolation which are then transformed

back to the original units of the data by multiplying by the standard deviation and then adding the

C-5
mean.For example,Q(0.95)is 1.996x39.7+ 157.8= 237,or, in termsof percentiles,237is the

95thpercentile.

Discussion

When constructing the relative frequency polygon, one must first construct the relative frequency

histogram. Constructing histograms usually requires a certain amount of personal judgment. For

example, the number of classes to be used and the corresponding class width is a matter of

personal choice. General guidelines recommend between five and twenty classes; a smaller

number sacrifices too much detail, a larger one retains too much of it. Many statisticians use

Sturgess's rule to determine the number of classes k, where Sturgess's rule gives k=1+3.3

logl0(n), where n is the sample size. Another rule found in Panofsky and Brier [C-4] suggests

k=5 logl0(n). For the data set under consideration, n=21 so the two rules give 5.5 and 6.6, and so

k=6 is a reasonable number of classes to begin with.

Next, the class width W must be determined. Assuming that the observations have been put in

rank order so that xl < x2 < ... < Xn, then the range of the data is defined to be R=-xn-Xl, and hence

the class width W must be greater than or equal to R/k in order to include all the data in the

frequency count. However, if one simply sets W=R/k then the endpoints of the data (xl and Xn)

will correspond to the class limits of the first and last class, respectively, which is usually not

appropriate. In fact, the general guidelines are that the class midpoints should roughly

correspond to the mean of the data points that fall within each class. However, this in practice

can only be achieved in some approximate sense so that it is very unlikely that two statisticians

would ever arrive at the same histogram parameters for the same data set.

This is a very important consideration when implementing a general purpose algorithm for

estimating quantiles and so we next explore the sensitivity of quantile estimation with respect to

the class width, keeping the number of classes fLxed at k=6 for this data set. Two methods of

class width selection are compared.

The first method selects W=R/(k -1) which forces the endpoints of the data to be the midpoints

of the first and last class. This method is applied across the 132 months, at each month

constructing the associated cumulative frequency polygon and then estimating Q(0.95) and

Q(0.976). A second method is chosen where the mean of the data (which is zero for the

standardized data set) corresponds to one of the class midpoints and W=(R+I)/k as

recommended in Downie and Heath [C-2] which insures that xl and x21 do not correspond to the

first and last class limits. Thus, with W defined, the number of classes set at k=6, and one class

midpoint defined, all class limits are automatically determined and the histogram and

corresponding cumulative polygon can be constructed and the quantiles of interest estimated.

Q(0.95) and Q(0.976) are compared for these two methods in Figures C-6 and C-7; the monthly

mean curve is included in each figure as a reference curve. Notationally, the graph labeled

Polygon2 refers to the W=R/(k - 1) method. Now, in both of these figures we note that while the

two graphs are fairly close to each other, they nevertheless differ in a number of places. Most

notably, however, is that each graph has some unusual jagged sections in one region but does not

necessarily have a jagged section in the graph produced by the other method in the same region.

C-6
I _I_ - - -2- - Po ygonl (0_95) - Polygon2(0 9fi) i

• l

250.0

)¢

230.0

210.0

190.0

170.0

150.0

130.0

110.0

90.0

70.0

50.0

w--

I I I I I

0 20 40 60 Month 80 100 120 140

Figure C-6. Comparison of Frequency Polygon Methods for Q(0.95).

300.0

_Mean ...... Polygon1 (0.976) .Polygon2(0.976)_

X

I

M.

250.0

200.0

150.0

100.0

50.0

0.0 I [ [ I I I

0 20 40 60 Month 80 100 120 140

Figure C-7. Comparison of Frequency Polygon Methods for Q(0.976)

This clearly points out how sensitive quantile estimation is to the choice of the class width W.

Thus, given that the selection of W (and even k, the number of classes) is quite subjective and

variations in W produce different and even jagged results in Q, it is strongly recommended that

quantiles NOT be estimated using frequency polygons. However, this does not detract from

their use for other purposes, such as their graphical value.

C-7
Method 2: Quantile Plots

Chambers, et. al. [C-5], give a succinct discussion of using quantile plots and the quantile

function in their book Graphical Methods for Data Analysis. As with Method 1, these methods

will be illustrated using the 21 data points defined by the first month of the 21 solar cycles. The

actual data presented in rank order is given below:

S--{98.3, 99.9, 107.1,120.5, 123.2, 125.4, 132.9, 133.2, 145, 152.1,155, 162.2, 162.3, 177.4,

185.8, 186.4, 192, 202.2, 203.6, 203.7, 245.1},

and recall that the sample mean is 157.8 and the standard deviation is 39.7.

The authors in [C-5] next construct an operational definition of quantile and give supporting

rationale for its functional form. Starting with a set of rank ordered data given as xi for i--1 to n,

with xl < x2 < ... < Xn , let p represent any fraction between 0 and 1. Then the quantile Q(p)

corresponding to the fraction p is defined to be xi whenever p is one of the fractions pi=(i-0.5)/n,

for i= 1 to n.

Thus, for the 21 observations in S, the following table constructed, where pi=(i-0.5)/21 for i=l to

21.

I , [ 00241 0071i 0 119] 0 167/ 0214] 02621 03101 0357] 0405 I 0452[ 05001

P.i...... _" __-"...... '" " • _ - -. -'--- '" - • " _ " _ "

[Q(pi) _ 98.3_ 99.9I 107.1 120.5 123.2_ ! 25.4_[, 132_91 133.21 145.0[ 152.11 155.0[

_;, _ _o_ .... ............................ _,_2_ _ _._#: _ _ ..... _ _ -_.............. _ _ _ _;__ ......

[p__._0.548_ 0.5951 0.643 0.691I 0.73_ 0.786_ 0.8331 0.8811 0.929i 0976l

FQ(p_ i5_162.21 162.3[ 177.4! 185.8/ 186.41 192.0! 202.2/ 203.6/ 203.7/ 245.11

This may then be presented as a quantile plot as depicted in Figure C-8.

C-8
......

I

25O

230

m

_ 210

190

170

Q.

0

150

130

o_

110

0

9O

7O

m

m

m m

m m

m

m

m

m

m

mint

m m

m

m m m

0 0.2 0.4 0.6 0.8

Fraction of Data

Figure C-8. Quantile Plot.

So far, the quantile function is defined only for certain discrete values of p, namely Pi. If

necessary, the definition of Q(p) is extended within the range of the data by linear interpolation.

This means connecting consecutive points with straight line segments, as show in the Figure C-9

below. Symbolically, if p is a fraction fofthe way from Pi tO Pi+l, then Q(p) is defined as

Q(p)=(1-f)Q(pi) + f Q(pi+l)

This cannot be used to define Q(p) outside the range of the data, where p is smaller than 0.5/n or

larger than (1-.5/n). As the authors in [C-5] state:

"Extrapolation is a tricky business; if we must extrapolate we will play safe and define

Q(p)=x] for p<pl and Q(p)=xn for P>Pn, which produces the short horizontal segments at

the beginning and end of the Figure C-9."

C-9
Fraction of Data, p

Figure C-9. Interpolated Quantiles.

In practical terms, quantiles smaller than 0.238 (percentiles smaller than 2.38%) or larger than

0.976 (percentiles larger than 97.6%) are beyond the resolution of the .sample size under

consideration (n=21) and cannot be derived from the data without further assumptions. Thus,

for this data set, x1=98.3 is the 0.0238 quantile and x21=245.1 is the 0.976 quantile. This result

can be generalized to say that for any data set consisting of 21 observations, the minimum value

xl is the 0.0238 quantile and the maximum value x/i is the 0.976 quantile.

In order to go beyond these limits, one approach we considered was to learn if the theoretical

endpoints of the distribution of data, corresponding to Q(0) and Q(1), were known or could be

estimated using physical laws. Linear interpolation could then be used to approximate the very

small or very large quantiles of interest. Unfortunately this approach did not produce usable

bounds.

The quantile method is applied to the 21 solar cycles at each of the 132 months to give Q(0.95)

and Q(0.976), i.e., the 95 th and 97.6 th percentiles. These results are depicted in Figure C-10.

C-10
[ Mean --Q(0,95) ' - - _ ' ' Q(0.976)

250.0

I'" _ "., Q(0.976) ,

i J

• "_ • 5 )

2°°°1 '%. ,7"

I N. ,'/

X

150.0 _, ,I

, 100.0

i

i 50.0 ! _ L t _

0 20 40 60 Month 80 100 120 140

Figure C- 10. Quantile Method

Note that Q(0.95) and Q(0.976) are roughly identical in the months 80-95 and months 115-118

where we previously noted that two cycles appear to run side-by-side and on the high side. This

is because Q(0.95) is computed by interpolating between x20 and x21 which are roughly equal to

each other in these regions. Furthermore, Q(0.95) yields a somewhat smoother graph than

Q(0.976) as one would expect because Q(0.95) is determined by linear interpolation between x20

and Xzj (which is a form of averaging and hence smoothing), while Q(0.976) is always the

maximum of the data (x21) and is therefore more sensitive.

The Quantile Method and Program Risk

As stated previously, Chambers, et al.[C-5], define the quantile Q(p) corresponding to the

fraction p to be xi whenever p is one of the fractions pi=(i-0.5)/n, for i=l to n. This definition of

Q is widely used and the authors support their choice of the functional form of Q as follows:

"Why do we take Pi to be pi=(i-0.5)/n and not, say i/n ... or several other reasonable

choices? We will mention only that when we separate the ordered observations into

two groups by splitting exactly on an observation, the use of the function (i-0.5)/n

means that the observation is counted as being half in the lower group and half in the

upper group."

This is a very interesting, if not compelling, argument in favor of this functional form, and would

usually be used without further consideration in most applications, especially in regions away

from the endpoints of the data. However, it is important to look at the application of these

results to NASA programs and the inherent program risks associated with their use. For

example, as discussed in the preceding paragraphs, this functional form for Q suggests that if we

select the largest observation from a sample of size 21, then we define it to be the 0.976 quantile,

or in programmatic terms, we are stating that the risk is only 2.4% and this result is based on a

C-11
sampleof size 21, without any probability distribution assumptions! Becausesignificant

programdesignand planningdecisionsmay potentially be basedon thesenumbers,a deeper

investigationof their propertiesis appropriate.

First,notethatthequantilefunction describedaboveis associatedwith a specialcaseof the socalledempiricaldistributionfunction(EDF),which is astepfunctiongenerallydefinedby

i--c

Fn(xi) - for 0 < c < 1 (C-l)

n-2c+l

for the ordered observations xl < x2 ... -< Xn. Figure C-11 illustrates the inverse relationship

between the quantile function, Q, and the empirical distribution function, F,, where Pi is defined

to be pi=(i-c)/(n-2c+ 1).

Relationship Between the Empirical Distribution Function, F.,

and the Quantile Function, Q,: Q is the Inverse of F.

Pn

Fn(xi)=pi

P3

P_

Pl

Q(_)=x_

z.a -D_

• ¢

I I I I

X1 X2 X3 Xi Xn

Ordered Observations

Figure C-11. Relationship Between the Empirical Distribution Function, F,, and the Quantile

Function, Q.

The objective at this point is to select a value of c that makes the most sense for our application.

However, it must be understood that whatever methods employed to select c must be quite

general in that they are distribution free, i.e., there will be no probability distribution

assumptions made. This is necessary because the underlying probability distribution of the solar

flux under consideration is obviously not known; otherwise the quantiles would also be known

exactly and there would be no reason for analysis. With this in mind, a branch of statistics

referred to as "order statistics" will be used.

C-12
For a sample of n values xl, x2, ..., x_ with a probability density function f(x) and cumulative

distribution F(x), and where the observations have been arranged in rank order so that xl < x2 ...

< xn, then the probability density function of the ith order statistic xi is given by the following

equation, as derived in Kendall and Stewart [C-3],

n_

g(x, ) = {F(x, ) ]2-' {1 - F(x, ) }"-' f(x, ) (C-2)

(i 1) _(n i)_

Thus, for any distribution with density function fix) and cumulative distribution F(x), if a

random sample of size n is selected from this distribution and the observations are arranged in

increasing order, the above equation defines the probability density function of the ith order

statistic.

For example, the probability density function of the minimum or smallest observation is obtained

by setting i=l in Equation C-2, giving g(x_ ) = n{1 - F(x_ )},-1 f(x_ ). Similarly, the probability

distribution of the maximum or largest observation is obtained by setting i=n in Equation C-2,

giving g(x,,)= n{F(x,)} n-' fix,,). Now, in the solar flux study, nature (and processing of the

data) provides a random sample of size n (n=21 at present) for each of 132 months, and then

they are arranged in rank order to construct the quantile plot. Since a linear combination of these

ranked observations is to be used to provide a quantile associated with a particular p-value, the

distribution of the p-values that are determined by a linear combination of the order statistics

shall be investigated.

That is, if a linear combination of the ranked observations is used to estimate a particular quantile

xp of interest, the question "what is the distribution of p-values provided by this linear

combination of the ordered observations?" must be answered. For example the maximum

observation x21 is begin considered to estimate a p-value in the neighborhood of 0.97. Therefore,

the following question is to be answered: what is the distribution of p-values obtained when x21

is used to estimate the percentile of interest?

Imagine that as nature generates or picks the 21 observations from the unknown distribution, the

maximum value x21 is selected and the question is "what is the p-value associated with this

particular x21?" The answer is F(x20, i.e., the area under the density function fix) curve and to

the left of x21 as illustrated in Figure 1. Repeating this sampling process of selecting another set

of 21 observations and picking the maximum value x21 of each of these sets of 21 observations,

one would like to know the distribution of the p-value associated with this set of x21's. That is,

what is the probability distribution of p21=F(x21) and more generally, the distribution of pi=F(xi)

for each of the order statistics?

First, note that the p's and the x's are in the same order since p is a non-decreasing function of x,

and furthermore 0 < Pl <'P2 .-. < Pn < 1 since F is bounded by 0 and 1. As discussed above, the

objective at this point is to discover the probability distribution of the order statistics Pi • To

help answer this question, a famous result in probability theory, often called the Probability

C-13
IntegralTransformationp=F(x), is used. This theorem proves that the distribution of p is the

standard uniform distribution, independent of the distribution F(x). That is, f(p)=l for 0 _< p <1

and F(p)=p. In fact, this is the underlying theory behind random number generation on digital

computers, which is often use in Monte Carlo simulations. The computer generates a standard

uniform random number u between 0 and 1 and then solves the equation x=Fq(u) to generate

random numbers from any specified distribution F(x).

Inserting this result in the general formula for the distribution of the i_ order statistic given in

Equation C-2, the probability density function of Pi is obtained and is seen to be Beta

distribution:

n! i-t ).-i

f(p,)=(i_l)!(n_i)!p_ (I-p, , 0<p. <1, (C-3)

i

with mean and variance given by /.t =

n+l

and o2 = i(n-i +1)

(n + 1)2(n +2)"

This is a very important result because ifc is set to 0, i.e., c=0 in Equation (C-l), then

i

Fn (xi) - n + 1 for i = 1, 2,...,n (C-4)

which is identical to the mean of the distribution ofpi and explains why c=0 is called the "mean"

plotting position. For example, suppose for the solar sample of size 21 the largest observed

value x21 is used as the quantile. The question is, what is P2_? By the first definition of Q it

would be p2t=Q(x20=(21-.5)/21--0.976. However, if Q is defined as Q(xi)=i/(n+l), then the

definition of a quantile is forced to coincide with the mean of the distribution of Pi, so that

p21=21/22=0.954. To see which definition makes the most sense for the present application, the

distribution of P21 may be analyzed by setting i=n=21 in the Beta distribution above, getting

f(p21) =21 P2_, 20 0 < p < 1 (C-5)

As noted above, the mean is given by 21/22=0.954, and the median is given by 24"_'T=0.968.

Furthermore, a 90% tolerance interval for P21 is determined in a similar manner and is given by

{0.867, 0.998} with a midpoint of 0.932. In comparing the above two definitions for Q, the

choice (i-0.5)/n gives a result closer to the median of the distribution, but the choice of i/(n+l)

yields the more reasonable result, in that it exactly matches the mean of the distribution and is

also more central to the midpoint of the interval. A similar argument by symmetry holds for the

minimum order statistic xl and its associated p_.

For quantiles not corresponding to xl and x21, linear interpolation should be used as suggested by

Chambers, et. al.,[C-5] between the appropriate values, and the quantile function will be defined

C-14
as Q(xi)=i/(n+l). Furthermore,the Beta distribution can then be used to give the exact

distribution of the interpolatedor "smoothed"p -values. As we've seenabove,the meanswill

alwaysmatch.

Theinterestedreadermayfind acomputersimulationhelpful in visualizingtheseresults. A

generalprocedurefor conductingacomputersimulationis outlinedbelowandanexampleis

provided:

Step 1: Generate21randomnumbersfrom any specified probability distribution. For example,

generate 21 numbers from a standard normal distribution, where the random number generator

used in this computer program generates a single standard normal distribution by summing 12

uniformly distributed random numbers and subtracting 6. This insures that the p-values obtained

in Step 2 are not the same uniform random numbers used to generate the random sample in the

first place.

Step 2: Order the 21 random numbers in ascending order. Then for each random number,

compute its p-value, i.e., the area under the normal distribution and to the left of it. Store these

results as Run 1 as in Table C-1.

Step 3: Repeat steps 1 and 2. This is done 5000 times in this example.

Step 4: Analyze the results. Relative frequency histograms of the p_, p_, and P2_ data in Table

C-1 are constructed and compared to the Beta distribution in Equation C-3. These results are

illustrated in Figures C-12, C-13, and C-14, where the Beta distribution is the smoothed curve

shown with the histogram, showing excellent agreement between simulation results and the

theoretical model. Use Table C-2 to compare between the means and variances of the simulated

results for p_, Pil, and P2_ and corresponding means and variances of the Beta distribution given in

Equation C-3.

C-15
'..O

O'. ',O

O

I_o o.

i r.- I

i"O

_D C)

j to')

c:; I c_

('-¢

(-4 C,

C) C,

- _ k,- i ,,n i

I _) _ C

),,0

u_ C

o_l 0_ C

O C

E) O C

{M 0') C

O E:

CI) (.0 C

O C

cO _ C

a0 C

O C

q9 I_

I% r'-- C

r.,. C

c) c

(.o oJ C

r_ c

c) c

_o_ocE)

O C

0') C

('9 O) C

_c:; c

q,d '_" C

__ _c

O C

O C

_ _ C

"_ O C

_ oc

_ '_" C

O C

_ C

5 6c

(1) _ C

,._ O C

C

_o _

"1 CN C

_ oC

2) .

t'N _

_ _ C

O C

6 •

L) O9 C

E: 0") _ C

(_

_= O C

(_ . . _

> .)--

O_ C

'_E OJ (::_ C

_ •

O C

e-"

_ ,_ c

"- O C

O

e- _ -r"

C-16
25

15

10

5

0

0.1 0.2

I I d I

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Pl

Figure C-12. Relative Frequency Histogram and Probability Density Function of P l.

>-

3.5

3

2.5

2

1.5

1

0.5

0

0

t

0.1 0.2 0.3 0.4 0.5

P_I

i

!,q

0.6 0,7 0.8

I

0.9

Figure C-13. Relative Frequency' Histogram and Probability Density Function for P_1.

C-17
25

20

15

10

1

[

[

5

l

0

0 0.2 0,4 0.6 0.8 1

p21

Figure C-14 Relative Frequency Histogram and Probability Density Function for P21.

Continuing with a discussion of program risks, it is standard practice to talk about the Mean Time

Between Failures (or the occurrence of some disastrous event), and hence using the mean i/(n+l)

equates correctly to this terminology. A similar argument in favor of i/(n+l) versus (i-0.5)/n by E.

J. Gumbel [C-6] states the following about the method of using (i-0.5)/n in a discussion of return

periods:

the method " ... claims that an event which has already happened once in n years will

occur, in the mean, in 2n years. If the extreme observation has economic consequences, as

in the case of floods, the danger factor is heavily under-estimated. The compromise is

misleading where the plotting problem is of most interest."

He continues on with his discussion of return periods of extreme events and concludes that i/(n+ 1)

is the most appropriate to use for these applications.

In conclusion, it is recommended that the following methods be used when using the quantile

method for the present application:

(1) Use the approach outlined by Chambers [C-5], but set Q(pi)=xi when pi=i/(n+ I )

instead of (i-0.5)/n

(2) use the linear interpolation method when necessary as previously discussed

(3) recognize that percentiles smaller than 4.5% or larger than 95.4% are beyond the

resolution of the sample size under consideration (n=21 ) and cannot be derived from the

data without further assumptions.

(4) use Q(0.95) instead of Q(0.954) because it is more common and will be smoother as a

result of interpolation, as previously discussed.

Using these methods for the data set under consideration, Q(0.95)=240.96. Figure C-15 shows

Q(0.95) with these recommendations implemented. The mean is provided as a reference curve.

C-18
I Mean• : "-Q(o.95)I

25O

200

_' 150

0')

0

0 100

50

I I I I I I

0 20 40 60 80 100 120

P

Figure C- 15. Quantile Method: Q(0.95)

140

Recommendations

Two methods of estimating percentiles are investigated in this report. The quantile method and the

relative frequency polygon method make no attempt to analytically model an underlying

distribution and thus represent strictly empirical approaches. Of these two methods, the relative

frequency polygon method is not recommended for use in estimating percentiles because of the

sensitivity of Q(p) to the class width and resulting jagged edges in Q.

Therefore, it is recommended that the Quantile method be used to estimate percentiles as long as

they are within the resolution of the sample size. As stated earlier, this means that percentiles

smaller than 4.5% and larger than 95.4% are beyond the resolution of the sample size under

consideration (n=21)and cannot be derived from the data. Furthermore, we learned that Q(0.95)

yields a somewhat smoother graph because of the linear interpolation between x20 and x21 while

Q(0.954) is always the maximum of the data (x20 and is therefore more sensitive. Therefore, it is

recommended that the 95 tnpercentile be estimated in the present application when using the quantile

method.

C-19
References

C-1. Kohler, Heinz, Statistics for Business and Economics, Scott, Foresman and Company,

Glenview, Illinois, 1988, pp. 49-89.

C-2. Downie, N. M., and Heath, R. W., Basic Statistical Methods, Harper and Brothers, New

York, 1959, pp. 16-28.

C-3. Kendall, Maurice G, and Stuart, Allan, The Advanced Theory of Statistics, Hafner

Publishing Company, New York, 1963,

C-4. Panofsky, Hans A., and Brier, Glenn W., Some Applications of Statistics to Meteorology,

The Pennsylvania State University, University Park, PA, 1958, pp.2-13.

C-5. Chambers, John M., et. al., Graphical Methods for Data Analysis, Wadsworth International

Group and Duxbury Press, 1983, pp. 1-100.

C-6. Gumbel, E. J., Statistics of Extremes, Columbia University Press, New York, 1960, pp. 4-

34.

C-20
APPENDIXD

ModifiedMcNish-LincolnandQuantileSubroutineExamples

SUBROUTINE LINCMC (R,N,NU2)

INTEGER*2 p,Q, IDG,K,I,J,M,MUPRIM,MU2PI,MU2,MUI,MU,NUI,NU2,I2,NU N

REAL*8 UPPER_PERCENTILE,LOWER_PERCENTILE,R(132,25),RMEAN(264),

IRPRED(264),ERRUP(264),ERRDN(264),DR(132,25),A(20,20),AINV(20,20 ,

2APRIME(100),B(132,25),C(132,25),JDUM(20),DETA(2),SIGM(132),SIGMA

COMMON /BLKI/ I2

COMMON /IDG/ IDG

COMMON /ANS/ RMEAN,RPRED, ERRUP,ERRDN, SIGM

COMMON /COUNT/ NU,NUI,MU,MUI,MU2,MU2PI,M

COMMON /PRCNT/ UPPER_PERCENTILE,LOWER_PERCENTILE

EQUIVALENCE (A(I,I),AINV(I,I),APRIME(1))

MUI=MU2-MU+I

NU=NU2-NUI+I

MUPRIM=M-MU2

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE

WRITE

WRITE

WRITE

WRITE

44,400

44,410

44,410

44,420

44,410

'YOU ARE IN LINCMC SUBROUTINE'

'M: ',M,'N: ',N,'NUI: ',NUI

'NU2= ',NU2, 'MU= ',MU, 'MU2= ',MU2

'UPPER_PERCENTILE: ',UPPER_PERCENTILE,

'LOWER_PERCENTILE= ',LOWER_PERCENTILE

'MUI: ',MUI 'NU= ',NU, 'MUPRIM: ',MUPRIM

ii0

DO ii0 J=NUI,N

WRITE (44,440) 'R(',J, ' '

WRITE (44,490) (R(I,J),I=I,M)

CONTINUE

WRITE (44 400) 'GOING INTO DO LOOP 130'

END IF

120

130

DO 130 I=I,M

RMEAN(I)=0.

DO 120 J=NUI,NU2

RMEAN(I)=RMEAN(I)+R(I,J)

CONTINUE

RMEAN(I)=RMEAN(I)/NU

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,400) 'FORMAT OF WRITE IS I,RMEAN'

WRITE (44,460) (I,RMEAN(I),I=I,M)

WRITE (44,400) 'GOING INTO DO LOOP 150'

END IF

D-I
140

150

160

170

180

190

200

210

220

230

240

DO 150 I=I,M

DO 140 J=NUI,N

DR (I, J)=R (I, J) -RMEAN (I)

CONTINUE

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

DO 160 J=NUI,N

WRITE (44,440) 'DR(',J, ') '

WRITE (44,490) (DR(I,J),I=I,M)

CONTINUE

WRITE (44,400) 'GOING INTO DO LOOP 190'

END IF

DO 190 K=I,MU

DO 180 Q=I,MU

A(K,Q)=0.

DO 170 J=NUI,NU2

A (K, Q) :A (K, Q) +DR(MUI+K-I, J) *DR(MUI+Q-I, J)

CONTINUE

CONTINUE

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

DO 200 Q=I,MU

WRITE (44,440) 'A(',Q, ')'

WRITE (44,510) (A(K,Q),K=I,MU)

CONTINUE

WRITE (44,400) 'GOING INTO DO LOOP 230'

END IF

DO 230 P:I,MUPRIM

DO 220 Q=I,MU

B(P,Q) =0.

DO 210 J=NUI,NU2

B (P, Q) :B (P, Q) +DR (MU2+P, J) *DR (MUI+Q-I, J)

CONTINUE

CONTINUE

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

DO 240 Q:I,MU

WRITE (44,440) 'B(',Q,')'

WRITE (44,510) (B(P,Q),P:I,MUPRIM)

CONTINUE

WRITE (44,400) 'GOING INTO DO LOOP 260'

END IF

K=0

DO 260 J=I,MU

DO 250 I=I,MU

K=K+I

APRIME(K)=A(I,J)

D-2
250 CONTINUE

260 CONTINUE

270

28O

290

300

310

320

330

340

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,400) 'FORMAT OF WRITE IS K,APRIME'

WRITE (44,460) (I,APRIME(I),I=I,K)

END IF

DETA (i) :3.

DETA (2) :0.

CALL GJR (A,MU,MU,MU,MU,*390,JDUM,DETA)

IF (DETA(1).EQ.0.) GO TO 390

K=MU*MU+I

DO 280 J=MU,I,-I

DO 270 I=MU,I,-I

K:K-I

AINV(I, J) =APRIME (K)

CONTINUE

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

DO 290 J:I,MU

WRITE (44,450) 'AINV(',J, ') '

WRITE (44,500) (AINV(P,J),P=I,MU)

CONTINUE

WRITE (44,400) 'GOING INTO DO LOOP 320'

END IF

DO 320 P=I,MUPRIM

DO 310 K=I,MU

C(P,K)=0.

DO 300 Q=I,MU

C(P,K)=C(P,K)+B(P,Q) *AINV (Q, K)

CONTINUE

CONTINUE

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

DO 330 Q:I,MU

WRITE (44,440) 'C(',Q, ')'

WRITE (44,500) (C(P,Q),P=I,MUPRIM)

CONTINUE

WRITE (44,400) 'GOING INTO DO LOOP 360'

END IF

DO 360 P=I,M

IF (P.GT.MU2) GO TO 340

RPRED(P)=R(P,NU2+I)

GO TO 360

RPRED(P)=0.

D-3
35O

360

DO 350 K=I,MU

RPRED(P) :RPRED (P) +C (P-MU2, K) *DR(MUI+K-I, NU2+I)

CONTINUE

RPRED (P ) =RPRED (P ) +RMEAN (P )

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,400) 'FORMAT OF WRITE IS I,RPRED'

WRITE (44,460) (I,RPRED(I),I:I,M)

END IF

MU2PI=MU2+I

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,430) 'MU2PI= ',MU2PI

END IF

C WHEN I2=3 THE SUBROUTINE BYPASSES THE SIGMA TO CALCULATE

C 95% AND 5% USING EMPERICAL STATISTICS WITH QUANTILE FUNCTION.

C

IF (I2.EQ.3) THEN

CALL PRCNTILE (DR,C,NU2)

IF (IDG.EQ.I.OR. IDG.EQ.2) THEN

WRITE

WRITE

WRITE

44,410)

44,410)

44,420)

WRITE

WRITE

WRITE 44

WRITE 44

WRITE 44

WRITE 44

WRITE 44

WRITE 44

WRITE (44

WRITE (44

END IF

44 410)

44 400)

M: ',M, 'N= ',N, 'NUI: ',NUI

NU2= ',NU2,'MU= ',MU,'MU2: ',MU2

UPPER_PERCENTILE= ',UPPER_PERCENTILE,

LOWER_PERCENTILE= ',LOWER_PERCENTILE

MUI= ',MUI, 'NU= ',NU, 'MUPRIM= ',MUPRIM

FORMAT OF WRITE IS I,RMEAN'

460) (I,RMEAN(I),I=I,2*M)

400) 'FORMAT OF WRITE IS I,RPRED'

460) (I,RPRED(I),I=I,2*M)

400) 'FORMAT OF WRITE IS I,ERRUP'

460) (I,ERRUP(I),I=I,2*M)

400) 'FORMAT OF WRITE IS I,ERRDN'

460) (I,ERRDN(I),I=I,2*M)

400) 'YOU ARE EXITING LINCMC SUBROUTINE'

RETURN

ELSE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,400) 'GOING INTO DO LOOP 380'

D-4
370

380

WRITE (44,400) 'FORMAT OF WRITE IS P,SIGMA'

END IF

DO 380 P:MU2PI,M

SIGMA:0.

DO 370 J:NUI,NU2

SIGMA:SIGMA+(R(P,J)-RPRED(P))**2

CONTINUE

SIGMA=SQRT(SIGMA/(NU-MU+I))

ERRUP(P)=RPRED(P)+2.*SIGMA

ERRDN(P)=RPRED(P)-2.*SIGMA

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,480) P,SIGMA

END IF

IF (ERRDN(P).LT.0.) ERRDN(P)=0.

CONTINUE

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,470) (P,ERRUP(P),ERRDN(P),P=MU2PI,M)

WRITE (44,400) 'IF I2 NOT EQUAL TO 3 RETURN TO

END IF

END IF

RETURN

390 WRITE (6,*) 'MATRIX WAS SINGLR'

STOP

400 FORMAT

410 FORMAT

420 FORMAT

430 FORMAT

440 FORMAT

450 FORMAT

460 FORMAT

470 FORMAT

480 FORMAT

490 FORMAT

500 FORMAT

510 FORMAT

)

2X,A7,I4,1X A7,I4,1X,A7

2X,AIS,F5.2 IX,AI8,F5.2

2X,A7,I5)

2X,A3,I3,A2

2X,A6,I3,A2

6(IX, I3,1X,F8.2))

3(IX, I3,1X,F8.2,1X,F8.2

2X, I4,1X,F8.2)

12F6.1)

(5 IX,F9.6))

(8(IX,F8.2))

I4)

END

MAIN '

SUBROUTINE QUANTILE (DDR,NU2,P)

C .......................................................................

C THIS SUBROUTINE RANKS THE DATA IN ASCENDING ORDER THEN ALLOCATES

C THE QUANTILE DISTRIBUTION FOR THE DATA GIVEN.

C ........................................................................

C

C .......................................................................

C THE DATA IS DIMENSIONED STARTING AT ZERO WHEN THE DIMENSION

D-5
INTEGER*2 NU2,P,K, IDG, I,M,MU2PI,MU2,MUI,MU,NUI,NU, I2

REAL*8 DDR(50),RNKANS(50),C,QUANT(50)

COMMON /BLKI/ I2

COMMON /IDG/ IDG

COMMON /ANS2/ RNKANS,QUANT

COMMON /COUNT/ NU,NUI,MU,MUI,MU2,MU2PI,M

WRITE

WRITE

WRITE

WRITE

END IF

IF(IDG.EQ.I.OR.IDG.EQ.2)THEN

(44,120) 'NU2= ,NU2

(44,110) 'WRITE FORMAT IS I,DDR'

(44,140) (I,DDR I),I=I,NU2)

(44,110) 'CALL RANKIT'

C ........................................................................

C CALLS RANKIT SUBROUTINE TO SET THE DATA IN ASCENDING ORDER

C .......................................................................

CALL RANKIT (DDR,NU2,RNKANS)

IF(IDG.EQ.I.OR.IDG.EQ.2)THEN

WRITE (44,110) 'WRITE FORMAT IS I,RNKANS'

WRITE (44,140) (I,RNKANS(I),I=I,NU2)

END IF

C .......................................................................

C CALCULATE THE QUANTILE FUNCTION SEE APPENDIC C OF TM TBD FOR

C EQUATION.

C .......................................................................

DO I=I,NU2

QUANT(I)=(FLOAT(I)-.5)/(FLOAT(NU2))

QUANT(I)=FLOAT(I)/(FLOAT(NU2)+I.0)

END DO

C ...... WRITE STAEMENTS FOR TROUBLE SHOOTING PROGRAM .....................

IF (IDG.EQ.I.OR.IDG.EQ.2) THEN

WRITE (44,120) 'P: ',P

WRITE (44,110) 'FORMAT OF WRITE IS K,RNKANS,QUANT'

WRITE (44,130) (K,RNKANS(K),QUANT(K),K:I,NU2)

WRITE(44,6000) 'CALL FNDLAMBDA'

END IF

RETURN

C ........ FORMAT STATEMENTS ..............................................

ii0 FORMAT (A50)

120 FORMAT (2X,A7,I5)

130 FORMAT (IX, I3,1X,FS.2,FS.2)

D-6
140 FORMAT (6(IX, I3,1X, FS.2))

END

D-7

APPENDIX E

MSFC Monthly Solar Activity Memorandum Example

EL23 (Example)

TO:

FROM:

SUBJECT:

Distribution

EL23/Chief, Electromagnetics and Aerospace Environments Branch

Solar Activity Inputs for Upper Atmospheric Models Used in Programs

to Estimate Spacecraft Orbital Lifetime

The solar activity information in this memorandum is provided as input data for upper

atmospheric models to insure compatibility between calculations made for spacecraft orbital

lifetime predictions. The Marshall Engineering Thermosphere Model (NASA CR-179359

and CR-179389), as well as the NASA/MSFC Global Reference Atmospheric Model-1995

Version (NASA TM-4715), require inputs of the 10.7-cm solar flux (Fl07) and the

geomagnetic index (A 0) to compute atmospheric density. Statistical estimates are provided

for the future 13-month smoothed values of these parameters.

To provide a better statistical estimate of the forthcoming solar minimum and the temporal

profile for cycle 23, the MSFC linear regression program has been revised. The MSFC

linear regression program is based on the Lagrangian least-squares statistical technique.

This technique is described in the papers "Lagrangian Least-Squares Prediction of Solar

Flux (Fm7)", Journal of Geophysical Research, Vol. 89, Number A1, Pages 11 through

16, January 1, 1984. A more detailed explanation of the MSFC linear regression computer

program and modifications that took place in 1995 and 1996 are in the paper "Statistical

Technique for Intermediate and Long-Range Estimation of 13-Month Smoothed Solar Flux

and Geomagnetic Index" (NASA TM-TBD) which is being processed for publication.

MSFC's linear regression program uses the 13-month smoothed 10.7-cm solar flux (F _07)

cycles 1 through 21 converted and observed data base and the 13-month smoothed

geomagnetic index (Ao) cycles 13 through 21 converted and observed data base. The

program estimates the intermediate-term (months) and long-term (years) behavior_of the 13-

month smoothed solar flux (F roT) and 13-month smoothed geomagnetic index (A v)' up to

132 months into the future, initialized from the cycle 22 June 1989 maximum. Once cycle

23 has been established from observed solar activity data, the solar flux program and

geomagnetic index program will be re-initialized at the minimum for cycle 23. The mean

cycle period of 11 years (132 months) will be assumed for cycle 23. Cycle 24 will then be

estimated using the solar flux statistics for cycles 1 through 22 and geomagnetic index

statistics for cycles 13 through 22.

The changes of orbital density associated with short-term (days) variations in the daily

10.7-cm solar flux and the 3-hourly geomagnetic index, required as inputs to the upper

atmospheric models, are not represented by the 13-month smoothed statistical estimates

E-1
given in these tables. This dynamic component of the total atmospheric density cannot be

estimated into the future with any acceptable degree of statistical confidence using existing

techniques. Representative data sets, based on past daily 10.7-cm solar flux and 3-hourly

geomagnetic index values, may be utilized to compute this dynamic component of the

orbital altitude density.

The design requirements for solar activity and associated on-orbit density values are

specified in the appropriate spacecraft and space vehicle project design requirements

documentation. These documents should be consulted for this information.

Enclosed are the following tables and figures:

Table E-1 (Enclosure 1) contains values for the recent monthly 10.7-cm solar flux and

geomagnetic index.

Table E-2 (Enclosure 2) contains the mid-point calculated values of the 13-month smoothed

observed 10.7-cm solar flux.

Table E-3 (Enclosure 3) contains the statistical estimate of 13-month smoothed 10.7-cm

solar flux and geomagnetic index for balance of cycle 22, cycle 23, and the beginning of

cycle 24.

Figure E- 1 (Enclosure 4) is a plot of solar cycle 22 monthly mean and 13-month smoothed

observed 10.7-cm solar flux.

Figure E-2 (Enclosure 4) is a plot of the 13-month smoothed 10.7-cm solar flux statistical

estimates for solar cycles 22, 23, and beginning of cycle 24.

The 50 percentile values in table E-3 and figure E-2, at and beyond maximum of cycle 23,

are approximated by the arithmetic mean and given with 95 percentile and 5 percentile

values.

The information in this memorandum is based upon data received from the National

Research Council of Canada for the Series C 10.7-cm solar flux data and the Institute for

Geophysics in Gottingen, Germany for the geomagnetic index data.

Questions on the contents of this memorandum may be addressed to Harold Euler at

(205) 544-2282.

APPROVAL:

4 Enclosures

E-2
TABLE E- 1

RECENT MONTHLY MEAN SOLAR ACTIVITY VALUES

Solar Flux

Fi0.7 (Series C)

Geomagnetic

Index Ap

1994 January 115.0 15

February 99.6 30

March 90.4 24

April 79.1 29

May 79.9 26

June 77.3 14

July 80.5 11

August 76.1 8

September 79.1 8

October 87.7 22

November 80.9 14

December 84.1 13

1995 January 82.7 11

February 85.6 15

March 85.1 15

April 77.7 16

May 75.6 18

June 75.7 11

July 73.8 8

August 73.8 9

September 72.0 13

October 77.9 16

November 74.2 9

December 72.6 9

1996 January 74.5 9

February 71.5 10

March 70.7 11

April 69.1 * 12*

May 69.7* 7*

June 69.5* 6*

Solar flux in units of 10 4 JANSKY (where one JANSKY equals 10.26W m 2 Hz _ Bandwidth)

* Provisional

NASA/MSFC/EL23 E-3 ENCLOSURE 1
TABLEE-2

13-MONTHSMOOTHED10.7-CMSOLARFLUX

MONTH 1989 1990 1991 1992 1993 1994 1995

JANUARY 190.2 200.4 205.5 181.8 125.6 92.7 80.6

FEBRUARY 194.1 200.5 206.3 174.8 123.0 91.2 80.2

MARCH 199.7 198.7 205.9 168.5 120.6 90.2 79.9

APRIL 204.4 195.6 206.8 162.9 118.1 89.3 79.2

MAY 209.3 192.4 207.1 158.8 114.8 88.2 78.5

JUNE 213.1 189.9 207.4 154.2 111.3 86.7 77.7

JULY 212.6 190.4 207.7 146.6 109.6 84.5 76.9

AUGUST 209.7 193.9 206.8 138.9 107.6 82.5 76.0

SEPTEMBER 207.2 198.3 203.9 133.7 103.9 81.7 74.8

OCTOBER 206.3 200.6 199.7 130.5 100.4 81.4 73.8

NOVEMBER 206.1 201.2 195.4 128.2 97.5 81.2 73.2"

DECEMBER 203.3 202.7 188.9 127.3 94.8 81.0 72.7*

*Provisional

NOTE: TABLE E-2 contains the 13-month smoothed 10.7-cm solar flux at the mid-point computed from

the National Research Council of Canada, Ottawa and Penticton Series C observed monthly values.

NASA/MSFC/EL23 E-4 ENCLOSURE 2
TABLE E-3 ESTIMATES OF 13-MONTH SMOOTHED SOLAR ACTIVITY FOR

BALANCE OF CYCLE 22, CYCLE 23, AND BEGINNING OF CYCLE 24

TIME 10.7-CM SOLAR FLUX (Fio7) GEOMAGNETIC INDEX

PERCENTILE PERCENTILE

95.0% 50% 5.0% 95.0% 50% 5.0%

1996 0003

1996 0837

1996 1670

1996 2503

1996 3337

1996 4170

1996 5003

1996 5837

1996 6670

1996 7503

1996 8337

1996 9170

1997 0003

1997 0837

1997.1670

1997.2503

1997.3337

1997.4170

1997 5003

1997 5837

1997 6670

1997 7503

1997 8337

1997 9170

1998 0003

1998 0837

1998 1670

1998 2503

1998 3337

1998 4170

1998.5003

1998.5837

1998.6670

1998.7503

1998.8337

1998.9170

1999.0003

1999.0837

1999.1670

1999.2503

1999.3337

1999.4170

1999.5003

1999.5837

1999.6670

1999.7503

1999.8337

1999.9170

2000.0003

2000.0837

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

74.1

75 1

76 5

78 3

80 1

82 4

85 1

87.8

91.2

96.0

100.9

105 4

109 3

112 3

115 4

119 4

123 7

128.4

134 1

140 1

144 8

148 1

150 1

152 1

154.9

158.2

161.5

165.1

170.3

175.6

178.8

180.4

182 0

183 6

185 1

186 6

188 1

189 6

191.0

192.3

193.6

194.9

196.0

197.1

198.1

199.0

199.8

200.5

201.1

201.5

72 6

72 5

72 6

72 8

73 0

73 3

73.7

74.2

74.8

75.7

76.9

78.2

79.5

80.7

82 1

83 5

85 1

86 7

88 6

9O 8

93 0

95 1

97.2

99.2

101.4

103.7

105.9

108.2

110.6

113 0

115 5

118 0

120 4

122 9

125 2

127 6

129.9

132.2

134.3

136.4

138.4

140.3

142.1

143.8

145.3

146.7

147.9

149.0

149.9

150.6

71.3

70.2

69.5

68.5

67.6

66.8

66.2

65.2

64.3

63.4

62.5

61.7

60 9

6O 5

60 4

60 2

60 3

60 6

6O 8

61.3

62.3

63.3

64 9

66 8

68 3

69 6

71 4

73 1

73 7

74 3

76 9

78 5

8O 0

81 6

83 1

84.6

86.0

87.5

88.9

90.2

91.5

92.7

93.8

94.9

95.8

96.7

97.5

98.2

98.8

99.2

10.3

10.8

11.3

ii 8

12 1

12 5

12 8

13 0

13 1

13 0

13 5

14 5

15 0

15 3

15 7

15 9

15 9

16.2

16.4

16.4

16.5

16.8

16.9

16.4

15.8

15.6

15.7

15 7

16 0

16 3

16 5

16 4

16 2

16 3

16.4

16.5

16.6

16.7

16.8

16.9

17.0

17.1

17.2

17.3

17.3

17 .4

17.5

17.5

17.5

17.6

i0.i

i0.i

10.2

i0.2

10.2

10.2

i0.i

i0 2

i0 2

i0 2

i0 3

i0 4

i0 5

10.6

10.8

ii.0

ii.i

ii.i

11.2

11.4

11.7

12 0

12 4

12 7

12 8

12 9

13 1

13 3

13.4

13 .5

13 5

13 5

13 3

13 3

13 3

13 2

13 2

13 2

13 2

13 2

13 1

13 1

13 .i

13 1

13 1

13 1

13 0

13 0

13 0

13 0

9.7

9.3

8.8

8.4

8.0

7.8

7.7

7.7

7.9

8.1

8.4

8.8

8.8

8 7

8 7

8 5

8 5

8 5

8 7

8 9

9 1

9 5

i0 0

i0 6

i0 9

i0 4

i0 4

i0 4

I0 4

i0 4

i0 7

ii 0

Ii 2

ii 2

ii 3

ii 4

11.5

11.5

11.6

11.7

11.8

11.8

11.9

11.9

12.0

12.0

12.1

12.1

12.1

12.2

NASA/MSFC/EL23 E-5 ENCLOSURE 3

(I of 4)
TABLEE-3 ESTIMATESOF 13-MONTHSMOOTHEDSOLARACTIVITYFOR

BALANCEOFCYCLE22, CYCLE23, ANDBEGINNINGOFCYCLE24

TIME 10.7-CM SOLARFLUX (Fio.v) GEOMAGNETICINDEX

PERCENTILE PERCENTILE

95.0% 50% 5.0% 95.0% 50%

(Xp)

5.0%

2000.1670

2000.2503

2000.3337

2000.4170

2000.5003

2000.5837

2000.6670

2000.7503

2000.8337

2000.9170

2001.0003

2001.0837

2001.1670

2001.2503

2001.3337

2001.4170

2001.5003

2001.5837

2001.6670

2001.7503

2001.8337

2001 9170

2002 0003

2002 0837

2002 1670

2002 2503

2002 3337

2002 4170

2002 5003

2002 5837

2002.6670

2002.7503

2002 8337

2002 9170

2003 0003

2003 0837

2003 1670

2003 2503

2003 3337

2003 4170

2003 5003

2003 5837

2003 6670

2003 7503

2003 8337

2003 9170

2004 0003

2004 0837

2004 1670

2004.2503

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

201

202

202

239

234

230 8

230.3

230.0

227.8

225.6

224 8

223 7

222 6

219 3

214 8

211 2

205 9

200 9

196 3

191.4

187.0

182.3

177.9

176.3

176.1

176.7

177.8

177.3

175.1

171.4

166.1

161.6

160 0

159 0

157 4

154 7

150 4

144 8

139 0

133 3

128 5

124 6

121 5

119 3

117 9

116 4

117 6

llq 2

116 3

116 8

9 151.2

1 151.5

2 151.6

0 158.1

3 156.1

154.2

152.7

151.1

149.7

148.1

146.6

144.9

143.0

140.8

138.7

137 0

135 8

134 7

133 7

132 8

131 4

129 7

127.8

126.0

124.4

123.0

121.9

120.5

118.6

116.7

114.8

113.2

111.8

110.3

108.9

107.4

105.6

103.9

102.5

101.3

i00.i

98.8

97.6

96.6

95.5

94.6

93.5

92.4

91.6

90.9

99.6

99.8

99.8

98.5

97.8

96.7

95.3

93.7

92.5

91.6

90.8

90.0

89.0

88.3

87.4

86.0

85.3

84 7

83 7

82 4

81 0

80 1

79 5

78 8

78 0

77.1

76.1

75.2

74.3

73 3

72 3

71 8

71 5

71 1

70 9

71 0

70 9

70.7

70.5

70.4

70.3

70.6

71.0

71.1

71.3

71.1

70.6

70.3

69.9

69.5

17 6

17 6

17 6

20 0

19 7

19 3

19 0

18 8

18 6

18.6

18.5

18.6

18.8

19 3

19 9

2O 8

21 3

21 1

2O 8

21.3

22.4

22.6

22 2

22 0

22 1

22 7

23 3

23 3

23 3

22 8

21 2

21 3

21.2

21.0

20.5

20.0

19 4

18 7

17 9

17 6

17 9

18 3

18 5

19 1

19 6

20 0

20.4

20.6

20.9

21.6

13.0

13.0

13.0

15.2

15.2

15.3

15 5

15 6

15 6

15 6

15 6

15 6

15 6

15 7

15 8

16 0

16.2

16.3

16.2

16 3

16 5

16 7

16 9

16 8

16 8

17.0

17.3

17.3

17.2

17.2

17 0

16 7

16 7

16 6

16 4

16 2

15 8

15 6

15 3

15 1

14 9

14 7

14 6

14 6

14 6

14.8

15.0

15.1

15.4

15.6

12.2

12.2

12.2

11.9

12.2

12.2

12 3

12 5

12 1

ii 5

ii 4

ii 4

ii 4

11.4

11.5

11.5

11.4

11.3

ii.i

ii.i

ii 2

ii 7

12 4

12 5

12 6

12 7

12 8

13 2

13 9

14.1

13 7

13 2

13 1

13 0

12 8

12 6

12 3

12 3

12.2

12.2

12 .i

11.9

12.0

11.5

ii.i

11.4

11.4

11.4

11.6

11.7

NASA/MSFC/EL23 E-6 ENCLOSURE 3

(2 of 4)
TABLEE-3

TIME

ESTIMATESOF 13-MONTHSMOOTHEDSOLARACTIVITYFOR

BALANCEOFCYCLE22, CYCLE23, ANDBEGINNINGOFCYCLE24

10.7-CM SOLARFLUX (Fio7) GEOMAGNETICINDEX

PERCENTILE PERCENTILE

95.0% 50% 5.0% 95.0% 50%

(Xp)

5.0%

2004.3337

2004.4170

2004.5003

2004.5837

2004.6670

2004.7503

2004.8337

2004.9170

2005.0003

2005.0837

2005.1670

2005.2503

2005.3337

2005 4170

2005 5003

2005 5837

2005 6670

2005 7503

2005 8337

2005.9170

2006.0003

2006.0837

2006.1670

2006.2503

20_6.3337

2006.4170

2006.5003

2006.5837

2006.6670

2006.7503

2006.8337

2006.9170

2007.0003

2007.0837

2007.1670

2007.2503

2007.3337

2007.4170

2007.5003

2007.5837

2007.6670

2007.7503

2007 8337

2007 9170

2008 0003

2008 0837

2008 1670

2008 2503

2008 3337

2008 4170

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

NASA/MSFC/EL23

117.3

117.6

117.4

116.7

115.5

113.9

110.7

105.3

99 5

98 3

98 8

99 9

I00 4

98 8

95 8

93.2

91.7

91.6

91.1

90.7

90.0

89.1

88.4

87.5

86.1

84.6

82.5

80.0

77.7

77.4

77.0

77.5

78.4

80.0

82.1

84.0

86.4

89.3

92.3

95 8

i01 2

106 7

112 0

116 6

120 1

123 6

127 8

132.5

137.4

143.4

89.9 69.2

89.0 68.9

88.0 68.6

86.9 68.2

85.9 67.9

85.1 67.7

84.1 67.6

83.0 67.5

82.0 67.5

81.0 67.3

80.2 67.2

79.7 67.2

79.2 67.2

78.6 67.2

78.1 67.1

77.6 67.1

77 .i 67.0

76.5 67.0

76.0 67.0

75.6 67.0

75 .I 67.0

74.7 67 .I

74.4 67.3

74.1 67.3

73.7 67.4

73.3 67.5

72.9 67.6

72.5 67.7

72.1 68.0

71.9 68.0

71.7 68.0

71.6 68.0

71.5 67.9

71.6 67.9

71.8 67.7

72.0 67.6

72.3 67.7

72.7 67.6

73.0 67.6

73.5 67.6

74.2 67.9

75.2 68.1

76.3 68.5

77.3 68.5

78.5 68.8

79.7 69 .i

81.0 69.3

82.5 69.5

84.1 70.2

85.9 71.0

E-7

21

21

22

22

22

23

23

24

24

8

8

0

1

4

0

6

1

3

24 1

23.4

22.5

21.7

21.4

20.9

20.5

19.9

19.0

17.9

16.8

16.2

16.1

16.4

16.6

16.3

16.1

15.9

15.4

15.2

15 3

15 8

16 0

16 2

16 4

16 5

16.8

17.0

17 .i

17.1

16.8

16.5

15.7

15.0

15.5

15.5

15.7

15.9

16.0

16.2

16.4

15.8 11.7

16.0 11.6

16.3 11.6

16.6 11.9

16.9 11.9

17.0 ii 9

17.0 12 4

16.9 12 8

16.8 12 8

16.5 12 7

16.0 12 8

15.6 12 8

15.3 12 8

14.9 12 4

14.5 11.8

14.1 11.2

13.6 10.5

13.1 9.9

12.7 9.4

12.3 8.9

12.0 8 5

11.8 8 3

11.8 8 0

11.6 7 6

11.4 7 3

11.3 7 2

11.3 7.1

11.2 7.2

11.2 7.4

11.2 7.6

11.2 7.8

11.3 7.9

11.3 7.9

11.3 8.1

11.3 8.1

11.2 8.1

11.2 8.1

Ii.i 8.2

ii.i 8.2

ii.0 8 1

ii.0 8 0

10.9 8 0

10.9 8 0

10.9 8 2

10.9 8 2

ii.0 8 3

ii.i 8 2

11.2 8.3

11.2 8.4

11.3 8.6

ENCLOSURE 3

(3 of 4)
TABLEE-3

TIME

2008 5003

2008 5837

2008 6670

2008 7503

2008 8337

2008 9170

2009.0003

2009.0837

2009 1670

2009 2503

2009 3337

2009 4170

2009 5003

2009 5837

2009 6670

2009 7503

2009 8337

2009 9170

2010 0003

2010 0837

2010 1670

2010.2503

2010.3337

2010.4170

2010 5003

2010 5837

2010 6670

2010 7503

2010 8337

2010 9170

2011 0003

2011 0837

2011 1670

2011 2503

2011 3337

ESTIMATESOF 13-MONTHSMOOTHEDSOLARACTIVITYFOR

BALANCEOFCYCLE22, CYCLE23, ANDBEGINNINGOFCYCLE24

10.7-CM SOLARFLUX (Fio.7) GEOMAGNETICINDEX

PERCENTILE PERCENTILE

95.0% 50% 5.0% 95.0% 50%

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

JUN

JUL

AUG

SEP

OCT

NOV

DEC

JAN

FEB

MAR

APR

MAY

149.7

154.6

158.2

160.3

162.1

164.4

167.7

171.4

174.9

179.5

185.0

188 6

190 3

190 7

191 0

193 5

196 6

199 1

202 7

208 4

212 9

215 1

218 6

223 7

226 5

228.2

230.2

232.2

235 2

238 5

240 7

240 1

239 8

241 6

241 6

87.9

90.1

92.2

94.2

96.2

98.6

100.9

103.0

105.3

107.5

109.9

112.2

114.7

117.3

119.6

121.6

123.2

124.9

126 8

128 8

131 0

133 1

135 5

138 3

141 0

143 2

145 6

148 2

150 7

152.9

154.8

156.6

158.8

161.0

161.0

71 8

72 5

73 7

74 2

74 4

74 7

74 9

75.0

74.9

74.8

74.7

75 0

75 3

75 6

76 6

77 4

77 7

78.7

80.0

81.5

83.5

84.8

85.8

87.7

89.5

9O 8

93 2

95 7

97 5

98 7

98 7

98 1

97 7

99 1

99 1

16.4

16 5

16 8

16 8

16 3

15 8

15 6

15 7

15 8

16 2

16 5

16.9

17.4

17.7

17.7

17 9

18 2

17 9

17 3

17 6

18 4

18 6

19 1

19 9

19 6

19.7

19.9

20.3

20 5

2O 5

2O 8

21 1

21 6

22 1

22 1

11.5

ii .7

12.0

12.3

12.5

12.7

12.9

13.1

13.3

13.4

13.5

13.6

13.7

13 7

13 6

13 6

13 7

13 7

13 7

13 8

13 8

13 9

14 2

14 4

14 3

14.2

14.3

14.3

14.4

14.6

14.8

15.0

15.2

15.5

15.5

5.0%

8 9

9 1

9 6

i0 1

i0 4

I0 4

i0 3

10.4

10.5

10.6

10.7

ii .2

12.I

12 0

Ii 4

i0 9

i0 6

i0 5

i0 9

II 0

i0 5

10 5

i0.7

i0.9

ii.0

I0 7

i0 8

i0 5

i0 4

II 0

ii 4

ii 4

ii 5

ii 7

ii 7

NASA/MSFC/EL23 E-8 ENCLOSURE3

(4 of 4)
250

230

210

•_- 190

170

150

L

ell

t_ 130

110

|

,"* 90

70

I

! I !

[-- 4_--i 3-Month smootlaed ]

-----13_ Monthly Mean

| , , •

1992

50

1987 1988 1989 1990 1991 1993 1994 1995 1996 1997

Date

Figure E- 1. Plot of Recent Monthly Mean and 13-Month Smoothed Solar Flux

_=

t...

r,_

"6

O

o

250

200

150

100

50

Figure E-2.

Actual Data 1 Coefficient Linear Model Using Cycle 1

..... 95 Percentile Value Through 21 Data Base and Most Recent

50 Percentile Value Observed 13-Month Smoothed Solar Flux (See

0 ..... 5 Percentile Value Table E-2)

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Date

Estimate of 13-Month Smoothed Solar Flux for Balance of Cycle 22, 23, and

Beginning of Cycle 24

NASA/MSFC/EL23 E-9 ENCLOSURE 4

APPENDIX F

MSFCLagrangianLinearRegressionTechniqueHistoricalPlots

MSFCLagrangianLinearRegressionTechnique(MLLRT) HistoricalPlotsweredevelopedto

provideanassessment of theMLLRT overthelastthreecycles.Theplotsaremadestartingonthe

monthandyearspecifiedoneachplot. Theseplotsaremadefor onemonthfor eachyearduring

cycles20,21,and22. Theplotsshowtheestimatesasin theMSFCMonthlySolarActivity

memorandumwith theobserved13-MonthSmoothedF10.7 added. Each estimate plot is for five years.

This appendix contains also the completed cycle plots for the minimum to minimum cycles and the

maximum to maximum cycles. Concerning the maximum to maximum cycle plots, the cycles are

identified by cycle number at maximum initialization date. For example, maximum to maximum cycle

20 starts on July 1970 and finishes at the maximum estimate of cycle 21. Concerning the minimum to

minimum cycle plots, the cycles are identified by cycle number at minimum initialization date, e. g.,

minimum to minimum cycle 20 starts on October 1964 and finishes at the minimum estimate of cycle

21.

F-1
I

...... L..._.Li.....! ....i'..-.._--.-.i---_-

iiiii "ii i;

- o- ; :

...... i'.i......i.....i..- ........... !_....

i' i>i !..

'ib__-ii: :iiT! .....

_.t.....i......i.....i..._iiil.i .....

_i i i i l Ni

_iiiiiil

_atd_

i . /"!

l",i! i,

...._'i........ 'i_Y! ...........

_..... '_"X_;..........

_ i\_

...... J............ i....:_.......

.... J .... t .... i ....

_Ql:l

_- .,.._ii /i:!I_-

.......... ....... ,,,ii....

............ il ...../-i--.fi.......

if"l"i,

ill ,,_i

......._,_ '>'i/'i iT_'''''T .......T.......

...... _. ...,_!.-:,_% ....... _....... _.......

i__; _./

"-.'........ ,--..._-_..--_,_------e" .......

i if i .'i_i'

4--/--i ....... _..... i:/-. .i........

....... i<:_'-..;...... i........

/ i ,,'i!i i i

.:,vi....,:.;.._:..._ ....... :_.. _.......

•_- ./V .i-

_i i i .X ' ::/

__......._.._.._...._! .....

_i i : i !

...... i/_i, .,:i: ......... _ i:_l ...._°

...... i_......i.....i!. , , .__...... ...:....._ '_'......_'

.....:I_,..._....Y...i.-.-ii.......!......

- ,,..'I ,,'

0

"_ "_ _= o

>>_,

._.____

= 6"_

0 ,

.-1 ,

.......... _............. _.................. _......... i I.....

:>-, i i i i

i _ :

..........i..................i................. i-

"_ .......... !............".............. -....... _....._'......L

:: _i° _" !/ __ -

I........ ,-'_--_ .............. _-i' .......... _---I"_

I _ " "_ _I

I_"_2

" ............. _.....:...............

................... 'i_-"__ _""_'"__:_-' ......... !__i-

.... _ .... _ .... I , _ ,,

L'OI

0

°.,,_

uJ

(D

0

E--,

_J

0

tj

.,.._

o

,,.._

:E

LL

L_

I;-2
,.iI

...... i...... i...... !...... i ]!....

i i,/:_.?'Ii.if

....... !----.-_'-----i-/-i ..... _;i......

_-.__ i...... _.

h,,,i....i....i....i....i....i....

i .i;;!Ii_;

--"! "i........ ;:." "j_!i_ ........

............ ;/----.;I-.!-....-.:t .........

I ! i'?_ " '\i

_.__,"i, :_ .!

_"i :; i .ii........

• i [ '," i

i _ ..1.-r"i

' _ ,'i l i i

<s".."Y__:_;: ...... _...

LDI___

f / !_-

_7 ,sl ,,: !/'/

,_.!_j_ --._

...... i..... .........

l

_,"-_. i _ ', i i"_

.....i.....i-'t-i ......i......_>-,;----i+-

......!....... i....i......!"_....... i'?i _-_

--_--i ....... i....i......i......i....... ':-ii-._

_i i i i i :

"'_'"_ ....... i........ _....... i...... I'i"

._........ i........ _....... _-..i

....... i........ i........ i....... !..... _ii ......

i i i ii,!ii,

....... i........ i...... i7"i,;"7.....

i i _" i,'i. I

_ ii i i_._':'I i.,.. "_-i .......

F-3

l.l.l

l.T.l

O

c_ c_,.._ 0

_ _-_

I i

i I

I

I'

I

LOI

0

r,#3

0

[-

0

>-,

q2

°_

o

°_

¢.0
.........

\ i ,,= :

I

.... I .... I .... I .... i,,,, _

( _.,iIi

'_--i........ '_-:,!:,_:_;i .........

_" i "\ ', \::

.... i .... i .... i .... i .... _

.!....... i....... i........ i--i.----i .........

",i _"., i__ i

'_, ', ,. _,_ _

......... i'%, ..:i_ii ......... -_

i i "\ ::',, i_ L__... i......... __.,.....t __

i _,1,,,_

ii.........

,_...i.... i.... i.... i....!

..... _.....i.....;._.._.,.',. ,/ ......

..... 4.... b..--i:-....;,..-.i-....i....--i .....

:_,'!1!ii _

......... !_.,x: ....._.... :....._... ..".....

ililili

_ i.-j

A

....... "Y '" /---i ........... ,

1"/! i

__ __ -

0

_o

_'_

Z ,,

J

,..'i l'........ i...

_-..-# ........ i ............. _........

,-" ! I : =""

•

i h _-

LOI

.4

d

o

° ...._

;>

¢_

"cI

o

t'_

;>.,

°...,

. ,..._

o

_L

..-,

iT.,

F--4
_i i i i i _.:i

...... _'_"ibttbi:"t ...... "tl]

i l...... i... i....... i:_i:.:!I1_

ill i i [ ii" iltl

,: I ! .[/11

[ :: [/i ,'f

......!------_-:-i---,-_-i---)::---i--.--_

,,_i- ___.._ ]

.....,,!.. .,_ -i i i : "

_Oq=l

_i!i!i.i,'_

i i--1--ii,i._i '._- _-

......:....._......:f._;...,_. .......

i

i ]i,Z ii,' i '/!_-;i ,' _ _ _

i i 'ki i i,,i'.

_---,,4,._ _ i

..... i i-i-i--i-;--:' i _

:::::: :1_

'w!!!t!! ...... :1

_iiiiii

i iiiii! f:_ FTFFT_ ....

_!iiiii

_iiii!!i

_ _ _ _ _ _ _ _ _-

[ i i i i'\iii

...... _ii ..... ....... i...... i... i....i----'i--ii._ 1 ...... 1.... [.............. i;_'

...... i....... i. I...... i...... I,:_H

!i....... i.. i... i--_;;_--1_

0

0

°_o_

_'_

Z ,,

it

F-5

LOt

0",

0

q3

;>

0

c'q

0

o_

_t_
,,'_._ /

...... irY-!:-j

j_-,- .:....\_/ .......... riii...i

"-x...... i--i...... __---.....A ...... i.......

"_i ",i/ i

_",,!_ i

........ i 'i .... _ ....

_ _ __ _-

......... _....... _.,:.....i.,:__.._.i ........ I_

.... ,..i.... f,........4_ ......... _........ I:,_

_i....... _,;'_-F_i

.... i .... L.... ] .... I .... _

_ _ __ __ -

_°n_I

...... _-i......... "ii...... ii........

I......... ....... ...... l,:

..._......... i......... _......... ". (_...

m

O

O

c_

•=,.___

_ ,,

r_ ,

•_,,t i_-

.... _.............. i......... T i_,,, ....

a, .... _................ ................. _............... i..._ o.., .......

', i if,,_

•_ ..!............. !............... ! ........ _.......... _ --- _ _ _ ;'_._

_, i.... i " i • if >-, .... _........... !.#:........ _.....t./_......_f....._ ...

: i ! _ " =_ .............. _........ ,_........... __...

#' : : i.. " ..... _.....!.......... __.....! ....... o...

....... _-_............ _----i ....... ._--.

............... !.......... _,,.i._ _ _,......... ,_y......

............... _.......... __.._._..._ ....... -I

.............. i:............. __i................ _-............... .... l'........... """_i 1

....i....i....i....i....

L-OIzI

F-6

0

°,,._

ILl

¢)

0

o,l

©

0

_4

U_
!

El i ! i 11!/ : _

i! ,_':!_i?;7"?" ! i i 1

_-.i i .,_.i.

'i iVi:_?ii ....._-

...iJ!,"/.i i i i i..... __

1_

Lo_I

_i i i i i#:. il

-L--,i--i-..-.i-..ii....._

si i i i !/ i"! i': i iJ, ",i _ i

! ! ! __. 8_ I I-

....... _.,, i........ i---_-_-,,-----i _',., ...... ;7,---i: i",.i-__

ii:j ................ i....... i,:_,_i_ _/'_ ..... _

i ...... L....i.....l.....L..\i.._!.... !_

_. i ! ,'...i'Ji !

F-7

m

a=

o

o

r_0

._ ._ _ ,--

_'_

Z , ,

M

.................... _........ +.......... :........ ,,£........ :....... /.........

......... ,i..1.I........... ,,': ............ i-.!: ...........

d i ,i I

........"_" "c'"i....................i"_ ............. l'7......_ .......

'.4.,.. i "..'-7 -

.............. +_i....... _.... i............... _!_ ............

.............. __ .................... i........ _-:_ ...... __-_, • . . ,.. .............. _-i .................... i................... _'"l ....

"_i.................... i................. _"J_:

_._,, ',_.....,J_

_._,...._.. ......._

...... _i-..._. _

.... _ .... i .... I ....

L'OI .4

_D

©

O

[--

_D

q2

It.

b.O o..._


REPORT DOCUMENTATION PAGE FormApprove_

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

gathering end maintaining the data needed, end completing end reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Oireclorete for Information Operation and Reports, 1215 Jefferson

Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget. Paperwork Reduction Project (0704-0188). Washington, DC 20503

1. AGENCY USE ONLY (Leave Blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

September 1996 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Statistical Technique for Intermediate and Long-Range Estimation

of 13-Month Smoothed Solar Flux and Geomagnetic Index

6. AUTHORS

K.O. Niehuss, H.C. Euler, Jr., and W.W. Vaughan*

7.PERFORMINGORGANIZATIONNAMES(S) ANDADDRESS(ES)

George C. Marshall Space Flight Center

Marshall Space Flight Center, Alabama 35812

9.SPONSORING/MONITORING AGENCYNAME(S)ANDADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546

8. PERFORMING ORGANIZATION

REPORT NUMBER

M-815

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-4759

11. SUPPLEMENTARY NOTES

Prepared by Systems Analysis and Integration Laboratory, Science and Engineering Directorate

*University of Alabama in Huntsville, Huntsville, Alabama

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category 18

12b, DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This report documents the Marshall Space Flight Center (MSFC) 13-month smoothed

solar flux (F10.7) and geomagnetic index (Ap) intermediate (months) and long-range (years)

statistical estimation technique, referred to as the MSFC Lagrangian Linear Regression

Technique (MLLRT). Estimates of future solar activity are needed as updated input to upper

atmosphere density models used for satellite and spacecraft orbital lifetime predictions. An

assessment of the MLLRT computer program's products is provided for 5-year periods from

the date estimates were made. This was accomplished for a number of past solar cycles.

14. SUBJECT TERMS

natural space environment, solar activity, solar flux, geomagnetic index,

statistics, estimates

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified Unclassified

7540-01-280-5500

_'U.S, GOVERNMENT PRINTING OFFICE--1996--733-109&40048

15. NUMBER OF PAGES

81

16. PRICE CODE

A05

20. LIMITATION OF ABSTRACT

Unlimited

Standard Form 298 (Rev. 2-89)

National Aeronautics and

Space Administration

Code JTT

Washington, DC

20546-0001

Official Business

Penalty for Private U_, $300

Postmaster: If Undeliverable (Section 158 Postal Manual), Do Not Return
ERRATA

NASA TECHNICAL MEMORANDUM 4759

STATISTICAL TECHNIQUE FOR INTERMEDIATE AND LONG-RANGE

ESTIMATION OF 13-MONTH SMOOTHED SOLAR FLUX AND GEOMAGNETIC

INDEX

by K.O. Niehuss, H.C. Euler, Jr., W.W. Vaughan

September 1996

°

.

°

,

.

°

Page 12, second paragraph, ninth line:

Change ...calculates the 95.0 and 5.0 percentile...

To read ...calculates the desired percentile...

Page 12, second paragraph, twenty second line:

Change...plots and tables. Figure 4-3 is...

To read ...plots and tables. However, before the Ap data are formatted for plots and tables, the A.._

approximate percentile by ratio subroutine (RATIO) approximates the desired percentile values ff Ap is

above or below the possible calculated percentile value by the quantile method. Figure 4-3 is...

Page 13,Figure4-3,firstcolmnn:

[ I X_Appro m te

PerPcex/tileby Ratio

Subroutine

fRAT_O)

1

[ Outpu.tf'des ]

Page 14, last paragraph:

Change Appendix E ... Ap estimates.

To read Appendix E discussion above applies to _ Ap estimates except for the percentile values.

Since the Ap number of cycles is smaller than the F,_, number of cycles (10 cycles instead of 22), the

upper and lower bounds will be calculated at 90 and 10 percentile instead of 95 and 5 percentile. To

give the lifetime predictiofi models consistent input data, and to compensate for the small A_ data

base, an approximated 95 and 5 percentile was calculated using a ratio technique. The comp_uter

program calculates the 5 percent value from the 80 percent value range using a ratio of the data. Then

adding and subtracting the 5 percent value to and from the 90 and 10 percenliles gives the

approximated 95 and 5 percentiles. The approximated percentiles for the A, make a very small

difference in the lifetime calculations from upper atmosphere model orbital density values and provide

Ap percentiles inputsconsistentwiththe Fm._percentiles.

Page D-.6, DO Loop of Quantile Subroutine, twenty-second through twenty-fifth lines:

Change DO I-I,NU2

END DO

To read QUANTNUM -- NU2-NUI+I

DO I-NUI, NU2

QUANT (I) --FLOAT (I+I-NUI) / (FLOAT (QUANTNUM) +I. 0 )

END DO

Page E-2, twenty-third line:

After ...values.

Add the sentence... Due to the small sample size, the 95 percentile and 5 percentile values for the

geomagnetic index are approximated.