NASA CR-61355. ATLANTIC T.ROP ICAL CYCLONE STATISTICS. By Harold L.
...... P - N F d O . m . . * . . -. N d d O N O - 4. I mmmmumrl. N. I1 3 .NP-mOIO .N.
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NASA CONTRACTOR REPORT NASA CR-61355
ATLANTIC T.ROP ICAL CYCLONE STATISTICS By Harold L. Crutcher U. S. Department of Commerce National Oceanic and Atmospheric Administration Environmental Data Service, National Climatic Center Asheville, North Carolina 28801 July 1971
P r e p a r e d for
a
NASA-GEORGE C. MARSHALL SPACE Marshall Space Flight Center, Alabama 35812
FLIGHT C E N T E R
TECHNICAL REPORT STANDARD T I T L E P A G E 12. GOVERNMENT ACCESSlON NO. 13. RECIPIENT'S CATALOG NO.
.
REPORT NO.
.
T I T L E AND S U B T I T L E
NASA CR-61355 5.
REPORT D A T E
J u l y 1971
ATLANTIC TROPICAL CYCLONE STATISTICS
6. P E R F O R M l N G ORGANIZATION CODE
8. P E R F O R M I N G ORGANIZATlON R E P O R T
AUTHOR(S)
'.
1
Harold L. Crutcher 10. WORK U N l T NO.
P E R F O R M l N G ORGANlZATlON N A M E AND ADDRESS
1.
U. S. Department of Commerce N a t i o n a l Oceanic and Atmospheric Administration Environmental Data S e r v i c e , N a t i o n a l C l i m a t i c Center A s h e v i l l e , North C a r o l i n a 28801 2.
1 1 . CONTRACT OR GRANT NO.
H- 76789 13. T Y P E O F REPORT & P E R I O D COVEREl
SPONSORING AGENCY N A M E AND ADDRESS
C o n t r a c t o r Report
N a t i o n a l Aeronautics and Space Administration Washington, D. C. 20546
14. SPONSORlNG AGENCY CODE
I 5 . S U P P L E M E N T A R Y NOTES
Technical Coordinator: 6.
S. C. Brown, Aerospace Environment D i v i s i o n , Aero-Astrodynamic Laboratory, Marshall Space F l i g h t Center, Alabama
ABSTRACT
S t a t i s t i c a l c l i m a t o l o g i e s of North A t l a n t i c , Caribbean and Gulf of Mexico t r o p i c a l cyclones are p r e s e n t e d . These are s t r a t i f i e d according t o season, geographical l o c a t i o n , and selected t i m e i n t e r v a l s . The s t a t i s t i c s a r e derived by approximating t h e d i s t r i b u t i o n of t r o p i c a l cyclone movements by t h e b i v a r i a t e normal d i s t r i b u t i o n . The a p p l i c a b i l i t y of t h e b i v a r i a t e normal and b i v a r i a t e "t" d i s t r i b u t i o n s i n d e s b r i b i n g t h e t r o p i c a l cyclone movements f o r t h e above areas i s examined by performing Chi-square goodness of f i t c a l c u l a t i o n s f o r f o u r t e e n areas. I n g e n e r a l , t h e b i v a r i a t e "t" model provides a b e t t e r f i t t o t h e data. For example, a t t h e .05 level of s i g n i f i c a n c e , the b i v a r i a t e "t" model i s r e j e c t e d i n t h r e e of t h e f o u r t e e n areas, w h i l e t h e b i v a r i a t e normal model i s rejected i n e i g h t areas. S i n c e t h e b i v a r i a t e ' I t " d i s t r i b u t i o n a s y m p t o t i c a l l y approaches t h e b i v a r i a t e normal d i s t r i b u t i o n f o r l a r g e d a t a s a m p l e s t h e d i f f e r e n c e may b e a t t r i b u t e d t o l i m i t e d d a t a s a m p l e s . It i s concluded t h a t t h e b i v a r i a t e normal d i s t r i b u t i o n , t h e g e n e r a l , provides a u s e f u l model f o r d e p i c t i n g t h e movements of t r o p i c a l cyclones. An accompanying p u b l i c a t i o n provides t a b l e s which may b e used t o o b t a i n p'robabilit i e s t h a t an e x i s t i n g t r o p i c a l cyclone w i l l be w i t h i n a s e l e c t e d t a r g e t area a t t h e end of p r e s c r i b e d t i m e i n t e r v a l s . These p r o b a b i l i t i e s l i k e w i s e may b e computed by use of t h e F o r t r a n I V program included i n t h e p r e s e n t p a p e r as a n appendix.
17. KEY WORDS
t r o p i c a l cyclones s t a t i s t i c a l climatologies b i v a r i a t e normal d i s t r i b u t i o n cyclone movements
MSFC
- Form 3292 (May 1969)
-
Unc 1as s i f i e d Un 1i m i t ed D, GEISSLER
FOREWORD This work w a s sponsored under Cross S e r v i c e O r d e r No. H76789 by t h e Aerospace Environment D i v i s i o n , Aero-Astrodynamics Laboratory, Marshall Space F l i g h t Center, because t h e N a t i o n a l Aeronautics and Space Administrat i o n m a i n t a i n s i n s t a l l a t i o n s and conducts a c t i v i t i e s along t h e A t l a n t i c and t h e Gulf of Mexico c o a s t a l r e g i o n s - - r e g i o n s a f f e c t e d by t r o p i c a l cyclones. The s i z e and complexity of many s p a c e v e h i c l e s make r a p i d movement i m p o s s i b l e and demand l e n g t h y on-pad checkout procedures. Thus , t h e v e h i c l e and much ground s u p p o r t equipment must b e maintained i n a storm-vulnerable c o n f i g u r a t i o n f o r perhaps 30 days b e f o r e launch. S i n c e t h i s s t u d y should a l s o f i n d w i d e a p p l i c a t i o n i n a number o f n e t e o r o l o g i c d o r g a n i z a t i o n s , i t i s b e i n g d i s t r i b u t e d t o several o f f i c e s i n t h e N a t i o n a l Weather Service, t h e A i r Weather S e r v i c e of USAF, and t h e Navy Weather S e r v i c e .
iii
TAELE OF CONTENTS
References
- -- - - - - - --
--
-
- - -- - - - - -
-
- -- -
Appendix I
-
The B i v a r i a t e Normal D i s t r i b u t i o n
Appendix I1
-
Determination of Model F i t
Appendix I11
-
B i v a r i a t e S t a t i s t i c s of North A t l a n t i c T r o p i c a l
15
Cyclone Movements (1899-1969) ( 1 , J ) Coordinates Appendix I V
-
E l e c t r o n i c Computer Program t o I n t e g r a t e t h e B i v a r i a t e Normal D i s t r i b u t i o n over an Offset Circle (Fortran I V
Accompanying Study
-
-
IBM
360/65)
B i v a r i a t e Bormal Offset C i r c l e P r o b a b i l i t y Tables
w i t h Offset E l l i p s e Trar,sformations and A p p l i c a t i o n s t o Geophysical Data, CAL Report XM-2464-G-1, Cornell Aeronautical Laboratory, I n c .
York. Authors:
3 volumes,
, Buffalo , Wew
C . Groenewoud, D. C. Hoaglin, John A.
V i t a l i s and E. L. Crutcher.
iv
1967.
Introduction The b i v a r i a t e normal d i s t r i b u t i o n has been used p r e v i o u s l y i n t h e study of t r o p i c a l storms &d/or h u r r i c a n e s by Haggard and o t h e r s (1965)~ Haggard and o t h e r s (1967), and Hope and Neumann (1968, 1969, 1 9 7 0 ) . This d i s t r i b u t i o n i s d i s c u s s e d i n numerous t e x t s and t e c h n i c a l papers relating t o statistics.
Appendix I provides r e f e r e n c e s t o some of t h e
developmental work and reviews t h e t h e o r e t i c a l basis f o r t h e d i s t r i b u tion.
R e s u l t s of t e s t s d e s c r i b e d i n Appendix I1 i n d i c a t e t h a t d i s t r i -
b u t i o n s of t r o p i c a l cyclone movement v e c t o r s when s e l e c t i v e l y s t r a t i f i e d can be d e s c r i b e d by t h e b i v a r i a t e normal model. The purpose of t h i s r e p o r t i s t o summarize some of t h e information contained i n o b s e r v a t i o n s o f t r o p i c a l cyclones t o provide guides for f o r e c a s t e r s and t h e many p r i v a t e and government o r g a n i z a t i o n s which a r e a f f e c t e d by t h e s e storms.
The r e s u l t s p r e s e n t e d i n Appendix I11 a r e
s t a t i s t i c a l c l i m a t o l o g i e s o f t r o p i c a l cyclone movements s t r a t i f i e d accordi n g t o season (June-July; August; September; October; November-May) g e o g r a p h i c a l l o c a t i o n ( five-degree l a t i t u d e by five-degree l o n g i t u d e (1
s q u a r e s " ) , and s e l e c t e d t i m e i n t e r v a l s (12-, 24-,
96-hours )
36-, 48-,
72- and
.
Copies of b i v a r i a t e p r o b a b i l i t y t a b l e s and a p p l i c a t i o n s by Groenewoud and o t h e r s
(1967) a r e being d i s t r i b u t e d w i t h
this report.
These, along
w i t h t h e s t a t i s t i c a l c l i m a t o l o g i e s i n d i c a t e d above, a l l o w t h e u s e r t o
make p r o b a b i l i t y s t a t e m e n t s concerning f u t u r e storm movements for planning
or d e c i s i o n making purposes.
Appendix I V provides t h e F o r t r a n I V
-
IBM 360/65 program which e a s i l y w a s adapted f o r u s e on t h e CDC 6600 and RCA S p e c t r a 70/45 computers t o provide t r o p i c a l cyclone s t r i k e p r o b a b i l i t i e s which w i l l appear i n a second paper.
The procedures
and m a t e r i a l p r e s e n t e d h e r e should n o t r e p l a c e p r e s e n t f o r e c a s t i n g techniques b u t should be used as a source of a d d i t i o n a l information.
Data Source The s t a t i s t i c s p r e s e n t e d h e r e a r e based on data t a k e n from t h e N O M , EDS, N a t i o n a l Climatic C e n t e r ' s Card Deck 993 ( T r o p i c a l Cyclone Deck). The p r e p a r a t i o n of t h i s deck w a s funded by t h e Commander, Naval Weather S e r v i c e Command, Washington, D. C .
The data a r e , f o r t h e most p a r t ,
t a k e n from t h e c h a r t s of North A t l a n t i c T r o p i c a l Cjrclones p r e s e n t e d by Cry and o t h e r s (1959) and Cry (1965).
A complete d e s c r i p t i o n of t h i s
deck i s a v a i l a b l e i n a r e f e r e n c e manual a v a i l a b l e a t t h e N a t i o n a l Climatic Center.
The p e r i o d of r e c o r d used h e r e i s 1899-1969.
This
deck c o n t a i n s t h e l a t i t u d e and l o n g i t u d e p o s i t i o n s ( i n degrees t o t e n t h s ) of storm c e n t e r s a t OOZ and 122.
A 1 1 movement v e c t o r s were c a l c u l a t e d
using t h e positions at t h e s e times.
Only storms c l a s s i f i e d as a " t r o p i -
> 34 k n o t s ) and o r i g i n a t i n g i n t h e c a l storm" or "hurricane" (winds -
North A t l a n t i c Ocean were t r e a t e d . cyclones."
These w i l l be r e f e r r e d t o as " t r o p i c a l
Movements f o r t h e p e r i o d s when t h e s e storms were c l a s s i f i e d
as " t r o p i c a l depressions" or were e x t r a t r o p i c a l a r e n o t included. Computation of S t a t i s t i c s
(a) Stratifications The data were s t r a t i f i e d according t o time and l o c a t i o n of occurrence.
2
The y e a r w a s d i v i d e d i n t o f i v e seasons: October; and November-May.
June-July; August; September;
This c l a s s i f i c a t i o n s e p a r a t e s p e r i o d s which
t e n d t o e x h i b i t d i f f e r e n t c h a r a c t e r i s t i c s i n storm movement or i n t h e geographical l o c a t i o n of storm development.
'Geographical s t r a t i f i c a t i o n
was achieved by d i v i d i n g t h e North A t l a n t i c and a d j a c e n t a r e a s i n t o s e p a r a t e five-degree l a t i t u d e by five-degree l o n g i t u d e a r e a s o r "squares.
Figure 1 shows t h e s e squares and i l l u s t r a t e s t h e scheme used t o i d e n t i f y them.
The t h r e e o r f o u r d i g i t number p l o t t e d i n each square g i v e s t h e
c o o r d i n a t e s of t h e southwest corner of t h e square.
The l a s t two d i g i t s ,
when m u l t i p l i e d by f i v e , g i v e t h e l o n g i t u d e i n d e g r e e s . d i g i t s g i v e t h e l a t i t u d e i n degrees.
For example, t h e four d i g i t number
1010 i n d i c a t e s t h e a r e a between 1 0 and
and 50 degrees west l o n g i t u & .
The preceding
15 degrees n o r t h l a t i t u d e and 45
That i s , 10°N and 50°W l o c a t e s t h e south-
w e s t c o r n e r of t h e s q u a r e . (b)
Coordinate Transformations
The l a t i t u d e - l o n g i t u d e p o s i t i o n s of t h e storm c e n t e r s were transformed i n t o p o s i t i o n s i n t h e orthogonal T , J g r i d system c u r r e n t l y used a t t h e N O M , NWS, N a t i o n a l Meteorological Center.
This g r i d c o n s i s t s of a
square g r i d superimposed on a p o l a r s t e r e o g r a p h i c p r o j e c t i o n o f t h e Northern Hemisphere.
The t r a n s f o r m a t i o n e q u a t i o n s a r e : I = B[sin ( A ' ) ]
+ 24
J = Brcos ( A ' ) ]
+ 26
where
3
"
8
, I ,
, , , ,
9
, I , ,
( , , ,1
8
9-1
UI
R cu
c-
w (u
$I !n cu
m rl
Ln (u
4
X = longitude (degrees) I$ = l a t i t u d e ( d e g r e e s ) B = 31.2043 [ c o s ($')/(l + s i n
($'))I
This g r i d e l i m i n a t e s t h e c u r v a t u r e e f f e c t s p r e s e n t i n a l a t i t u d e l o n g i t u d e system.
-
Figure 2 shows t h e I,J g r i d o v e r l a i d on a p o l a r s t e r e o g r a p h i c map of t h e North A t l a n t i c Ocean and surrounding area.
The f o l l o w i n g t a b l e
g i v e s t h e approximate d i s t a n c e e q u i v a l e n t t o one g r i d l e n g t h f o r various l a t i t u d e s . Latitude
One Grid Length (Kilometers )
10°N
241
20°N
273
30°N
308
40°N
334
Conversion from g r i d i n t e r v a l s t o k i l o m e t e r s .
(c)
Computations
Movement v e c t o r s i n terms of (1,J) c o o r d i n a t e s were compiled f o r h
e l a p s e d times of 1 2 , 24, 36,
48, 72 and 96 hours.
A l l t h e movement
v e c t o r s o r i g i n a t i n g i n a given Square were t r a n s l a t e d such t h a t t h e i n i t i a l p o s i t i o n s were moved t o t h e c e n t e r of t h e square.
The b i v a r i a t e
s t a t i s t i c s were computed f o r t h e s t r a t i f i c a t i o n s i n d i c a t e d p r e v i o u s l y by u t i l i z i n g t h e machine program, Winds A l o f t Summary (1963).
5
t
6
The p e r t i n e n t s t a t i s t i c s , i n c l u d i n g b o t h p o l a r and component forms of t h e means, are l i s t e d i n Appendix 111.
These are:
(1) R e s u l t a n t d i r e c t i o n of storm movement ( d e g r e e s )
(2)
Magnitude of t h e r e s u l t a n t storm movement
(3)
and
- (e)
- (D,)
( 4 ) Mews of t h e components of storm movement
- (GI and E )
( 5 ) and ( 6 ) Standard d e v i a t i o n s along t h e major and minor axes of
-
the distribution
(s
and s ) . b
a
These are c a l l e d SIGX and SIGY
i n t h e t a b l e s and example a p p l i c a t i o n s by Groenewoud, Hoaglin, V i t a l i s and Crutcher (op. c i t . ) .
( 7 ) The angle of r o t a t i o n measured counterclockwise from t h e I axis
-
($)
( 8 ) The number of o b s e r v a t i o n s - ( n ) These parameters were computed from t h e following e x p r e s s i o n s :
e
n = Arctan
n AIi/
C
i=1
D
r
=
AI =
J n c [(
i=1
n ( C
AJi
i=1
2
AI.) 1
AI,)/n
7
+ (
n C i=l
nJi)
2
l/n
2
where
(a) A I = I
-
o
I
f
and A J = J
f
-
J
o
where t h e s u b s c r i p t s o and f
i n d i c a t e t h e i n i t i a l and f i n a l p o s i t i o n s , r e s p e c t i v e l y . (Note t h e r e v e r s a l of I
0
and I
f
i n t h e formulation of A I .
This m o d i f i c a t i o n makes t h e s i g n s of t h e components a g r e e w i t h t h e s t a n d a r d meteorological c o o r d i n a t e system.)
A12)/n-1)
-
n (( C
AI)2/n(n-1))
i=1
i=1
(sAI i s t h e s t a n d a r d d e v i a t i o n along t h e I a x i s )
( s A Ji s t h e s t a n d a r d d e v i a t i o n along t h e J a x i s )
r
(rAIAJ
(e) K
1
n
n
n
1
i s t h e c o r r e l a t i o n c o e f f i c i e n t o f t h e I and J components)
and K
2,
t h e e i g e n v a l u e s , are t h e r o o t s of t h e determinant
w i t h K1 being t h e l a r g e r .
8
The determinant expanded i s (‘AI
2
- ~ ) ( 2 s- ~K)~ -
s
AI
2s
AJ
2r 2 = 0 AIAJ
or
r K =
7
(sA12+ s
2
AJ
) It: JisA12
+
s
AJ
2)2
- 4sAI2 sAJ2 (l-rAIAJ
Examples A p p q d i x I11 c o n t a i n s a l i s t i n g of t h e b i v a r i a t e s t a t i s t i c s needed t o d e f i n e t h e d i s t r i b u t i o n of storm movements. t h e movement s t a t i s t i c s for two s q u a r e s .
Here, each page c o n t a i n s
A l l seasons and t i m e i n t e r v a l s
a r e included except for c a s e s w i t h l e s s t h a n f i v e o b s e r v a t i o n s . Figure 3 i l l u s t r a t e s how t h e s t a t i s t i c s are used t o c o n s t r u c t probability ellipses. considered.
Here, Square 2518 ( t h e n o r t h c e n t r a l Gulf area) i s
The d a t a show t h e end p o i n t of t h e 24-hour movements when
a l l o r i g i n a t e a t t h e c e n t e r of t h e s q u a r e .
The s t a t i s t i c s computed from t h e s e d a t a are:
n
=
-AI =
s
a
s
b $
73 -.20
=
1.08
=
.65
=
3.5O
9
The season i s September.
-
( s e e page 111-27)
ua cv
0
a
M d
rl
v)
d
d
10
u
d
The p r o b a b i l i t y e l l i p s e s a r e c o n s t r u c t e d through t h e following s t e p s . Since t h e s t a t i s t i c s were computed i n t h e 1,J c o o r d i n a t e system, t h i s system must be used i n t h e s t e p s i n d i c a t e d . Locate t h e mean of t h e movements. Construct an I,J c o o r d i n a t e system such t h a t t h e o r i g i n c o i n c i d e s w i t h t h e mean. Rotate t h i s c o o r d i n a t e system counterclockwise from t h e I - a x i s through t h e a n g l e $. Select the probability value desired. S e l e c t t h e a p p r o p r i a t e m u l t i p l i e r from Figure Multiply s
a
4.
and s ( t h e s t a n d a r d d e v i a t i o n along t h e b
major and minor a x i s ) by t h i s m u l t i p l i e r . Let t h e s e d i s t a n c e s d e f i n e t h e l e n g t h of t h e semi-major and semi-minor axis. Construct t h e e l l i p s e d e s c r i b e d by t h e s e d i s t a n c e s . I n F i g u r e 3 t h e .25, .50 and
.75 p r o b a b i l i t y e l l i p s e s are drawn.
The
mean movement v e c t o r , along w i t h t h e s i z e , shape and o r i e n t a t i o n of t h e p r o b a b i l i t y e l l i p s e s , g i v e s a c l e a r p i c t u r e of' how t h e storm movements are d i s t r i b u t e d .
Here t h e d a t a i n d i c a t e a l a r g e v a r i a t i o n i n t h e
d i r e c t i o n of movement.
The p r o b a b i l i t y e l l i p s e s r e l a t e the same i n f o r -
mation by t h e east-west o r i e n t a t i o n of t h e major a x i s .
A d d i t i o n a l P u b l i c a t i o n s and Future Work This i s t h e f i r s t of a series of p u b l i c a t i o n s d e a l i n g w i t h t r o p i c a l storm movement s t a t i s t i c s and s t r i k e p r o b a b i l i t i e s .
The proposed
1,. 00 * 90
R
e
80
e
70
e
60
c1
*rl
d
2cd .5Q P
0 k
a
.40
* 30
e
20
e
10 0
I
I
I
I
I
I
I
I
I
I
1 .'QO 2 :00 Mu1t i p l i e r (M) Figure
4 Radii for various probability ellipses.
3
The major and minor axes are given by aa(M) and ob(M) e For example, the multiplier (M) for a .5O probability e l l i p s e i s 1.18. (Adapted from National Weather Records Center, Winds Aloft Summary, 1963) e
12
t i t l e s and sponsoring agencies f o r t h e s e f u t u r e p u b l i c a t i o n s a r e as
A t l a n t i c T r o p i c a l Cyclone S t r i k e P r o b a b i l i t i e s (For S e l e c t e d S t a t i o n s and t h e Month of September)
-
Division , Aero-Astrodynamics Laboratory
Aerospace Environment Marshall Space F l i g h t
Center, NASA, H u n t s v i l l e , Alabama. A t l a n t i c T r o p i c a l Cyclone S t r i k e P r o b a b i l i t i e s , (Volume I , 24-Hour Movements ; Volume 11, 48-Hour Movements ; Volume I11 72-Hour Movements )
-
Commander, Naval Weather Service Command,
Washington, D. C. A t l a n t i c T r o p i c a l Cyclone Mean Vector Charts
-
Commander,
Naval Weather S e r v i c e Command, Washington, D. C.
Future work may be extended t o :
(a)
Use of H o t e l l i n g ' s "T2" t e s t t o d e l i n e a t e a r e a s of s i m i l a r
or d i s s i m i l a r storm movement i n time and space. (b)
Development of a t h e o r e t i c a l model t o permit use of p r i o r conditions.
(c)
Development of c l a s s i f i c a t i o n and d i s c r i m i n a t i o n or c l u s t e r i n g techniques t o i s o l a t e homogeneous time-space groups.
This
w i l l be an e x t e n s i o n t o ( a ) above. Summary The b i v a r i a t e normal d i s t r i b u t i o n i s used as a model t o d e s c r i b e t h e movements of t r o p i c a l cyclones f o r s t a t e d p e r i o d s from s p e c i f i e d posit i o n s ( s e e Appendix 11).
Due t o t h e s m a l l number of c a s e s , t h e d i s t r i -
butions a r e d e s c r i b e d b e t t e r by t h e b i v a r i a t e t - d i s t r i b u t i o n .
13
As t h e
b i v a r i a t e normal d i s t r i b u t i o n i s approximated by t h e b i v a r i a t e "t" with an i n c r e a s i n g number of o b s e r v a t i o n s , it i s assumed t h a t t h e b i v a r i a t e normal d i s t r i b u t i o n model can be used t o compute v a l i d movement s t a t i s t i c s and s t r i k e p r o b a b i l i t i e s .
The bivariate s t a t i s t i c s of t r o p i c a l cyclone movements are computed and p r e s e n t e d .
Sample s i z e s range from 5 t o almost 100.
Obviously,
more confidence should be p l a c e d i n t h o s e s t ? . t i s t i c s which are based on t h e l a r g e r sample s i z e s .
S t r i k e p r o b a b i l i t i e s m a y be computed by
t h e u s e r from t a b l e s which accompany t h i s paper as a s e p a r a t e publicat i o n or by means of an e l e c t r o n i c computer program included as an appendix.
Acknowledgments Acknowledgment i s made t o Dr. S. Kaufman and Mr. C. Groenewoud of Cornell Aeronautical Laboratory for permission t o u s e t h e i r F o r t r m I V program t o compute t h e s t r i k e p r o b a b i l i t i e s .
The cooperation of t h e
N a t i o n a l Weather S e r v i c e ' s Computer Division f o r h e l p i n t h e a c t u a l computing of t h e p r o b a b i l i t i e s i s acknowledged a l s o . The a u t h o r expresses h i s a p p r e c i a t i o n f o r t h e s i g n i f i c a n t c o n t r i b u t i o n s made by t h e following personnel of t h e N a t i o n a l Climatic Center: Messrs. Frank Quinlam and Glenn O'Kelley developed t h e necessary computer programs; Messrs. Danny F u l b r i g h t and Grant Goodge performed much of t h e work connected w i t h t h e t e s t i n g of models; Mr. Ray Hoxit provided e d i t o r i a l a s s i s t a n c e ; Mr. Robert Courtney performed t h e necessary d r a f t i n g ; and Mrs. Margaret Larabee typed t h e manuscript.
14
REFERENCES Cry, G . W . ,
W. H. Haggard and H. S. White,
1959:
"North A t l a n t i c
T r o p i c a l Cyclones," Technical Paper No. 36, U. S. Weather Bureau, Washington, D. C. Cry, G . W . ,
1965:
, 214
pp.
"Tropical Cyclones of t h e North A t l a n t i c Ocean,"
Technical Paper No. 55, U. S. Weather Bureau, Washington, D. C . ,
148 pp. Groenewoud, C . ,
D. C. Hoaglin, John A. V i t a l i s and H . L. Crutcher,
1967: B i v a r i a t e Normal Offset Circle, P r o b a b i l i t y Tables w i t h O f f s e t E l l i p s e Transformations and Applications t o Geophysical
Data, CAL Report XM-24644-1, Laboratory, I n c . Haggard, W i l l i a m H . ,
, Buffalo,
3 volumes.
Cornell Aeronautical
New York.
Harold L. Crutcher and Georgia C. Whiting, 1965:
"Storm S t r i k e P r o b a b i l i t i e s , ' I paper p r e s e n t e d a t t h e Fourth Technical Conference on Hurricanes and T r o p i c a l Meteorology,
M i a m i , F l o r i d a , November 22-24, Haggard, W i l l i a m H . ,
Harold L. Crutcher, R. F. Lee and F. T . Quinlan,
1967 : "Hurricane Recurvature and S a t e l l i t e Photography , I 1 paper p r e s e n t e d at t h e F i f t h Technical Conference on Hurricanes and T r o p i c a l Meteorology, Caracas, Venezuela, November 20-28.
15
Hope , John R. and Charles J. Meumann ,
1968:
" P r o b a b i l i t y of T r o p i c a l
Cyclone Induced Winds a t Cape Kennedy," Technical Memorandum WBTM SOS-1, Weather Bureau, ESSA, U. S. Department of Commerce ,
S i l v e r Spring, Maryland,
67 pp.
Hope, John R. and Charles 5 . Neumann, 1969:
"Climatology of A t l a n t i c
T r o p i c a l Cyclones by Two and One-Half Degree Latitude-Longitude Boxes ," Technical Memorandum WBTM-SR
44, Weather
Bureau , Southern
Region, ESSA, U. S. Department of Commerce, F o r t Worth, Texas,
48 PP. Hope, John R . and Charles 5. geumann, 1970:
"An o p e r a t i o n a l technique
f o r r e l a t i n g t h e movement of e x i s t i n g t r o p i c a l cyclones t o p a s t
tracks," Monthly Weather Review, Volume 98 (12), pp. 925-933.
U. S. Department of Commerce, National Weather Records Center, 1963: Winds Aloft Summary.
THE BIVARIATE NORMAL DISTRIBUTION
APPENDIX I.
Bravais
(1846) provides t h e f i r s t e x t e n s i o n from t h e u n i v a r i a t e t o Maxwell (1859) , Bertrand (1888), Pearson
bivariate d i s t r i b u t i o n .
( l g O O ) , and S t r u t t (1919) provide f u r t h e r e x t e n s i o n s , and B a r t l e t t (1934) d i s c u s s e s v e c t o r r e p r e s e n t a t i o n s i n samples.
The following
d i s c u s s i o n i s t a k e n i n p a r t from Crutcher (1959). A v e c t o r d i s t r i b u t i o n i s s a i d t o be normal i f t h e p r o b a b i l i t y d e n s i t y
has a m a x i m u m at some p o i n t and f a l l s o f f i n a l l d i r e c t i o n s as f ( x , y ) = exp
x2
where Q i s d i s t r i b u t e d as
v-dimensional d i s t r i b u t i o n .
(-%Q)
with v degrees of freedom f o r t h e For t h e 2-dimensional d i s t r i b u t i o n
L
-1
and i s d i s t r i b u t e d as r e p l a c e Q. specified I
i
x2
with 2 degrees of freedom.
x2
may be used t o
The p r o b a b i l i t y t h a t a p o i n t l i e s i n s i d e t h e e l l i p s e f o r a
x2
i s t h e n F ( x 2 q 2 ) = P. P
be determined.
For a given p r o b a b i l i t y P ,
- or xp becomes t h e v e c t o r Then dx2 P
x2
P
can
r a d i u s t o determine
t h e p r o b a b i l i t y e l l i p s e contour corresponding t o p r o b a b i l i t y P.
Eq. 1 t h e n r e p r e s e n t s a b i v a r i a t e normal d i s t r i b u t i o n where v i s 2 , X
and Y a r e orthogonal components,
and p UX
t h e components,
and
CI
X
CI
Y
Y
are t h e r e s p e c t i v e means of
are t h e s t a n d a r d d e v i a t i o n s of t h e r e s p e c t i v e
components, and pxy i s t h e c o r r e l a t i o n between t h e components.
I- 1
If pxy
equals 1, t h e d i s t r i b u t i o n i s a degenerate b i v a r i a t e d i s t r i b u t i o n
which i s not encountered i n p r a c t i c e . t h e variances
and
0 X
are e q u a l and p
0
Y
v
X
)2
equals zero.
The expression
Xy
(1) reduces t o exp (-R2/o R2 = (X-p
The o p p o s i t e extreme occurs when
2,
where
+ (Y-p
Y
and 2oX2 = 20
)2
Y
The d i s t r i b u t i o n i s t h e n c i r c u l a r .
= CT
v
These two form t h e l i m i t s of t h e
d i s t r i b u t i o n , t h a t i s , t h e s t r a i g h t l i n e and t h e c i r c u l a r .
Since t h e
c o r r e l a t i o n between components i s o f t e n z e r o , t h e c i r c u l a r form f r e q u e n t l y
w i l l be encountered.
Now, if w
and w Y
equals X
e q u a l s (Y-p ) 2 / o 2 , e x p r e s s i o n (1) Y Y
reduces t o
and i f p
XY
i s zero reduces t o
f ( x , y ) = exp {-1/2[wx2 + w 2 1 ) Y L e t t i n g w2 = w
+ w
X
Y
2 , Eq.
(4)
( 4 ) becomes (5)
f(x,~= ) exp (-w2/2)
which i s t h e familiar c e n t r a l Rayleigh ( S t r u t t ) d i s t r i b u t i o n (1919)i f only d i s t r i b u t i o n of t h e magnitudes i s considered and t h e v e c t o r mean i s zero. If t h e d i s t r i b u t i o n i s e l l i p t i c a l , t h e n p
may b e s i g n i f i c a n t l y XY
d i f f e r e n t from z e r o .
I n t h i s c a s e t h e axes may be r o t a t e d through t h e
angle $ t o a new axes along which t h e components a r e n o t c o r r e l a t e d .
1-2
The v a l u e s for t h e components i n t h e new c o o r d i n a t e system may be o b t a i n e d from Equations
(6a) and ( 6 b ) . X' = X s i n
J,
+
Y cos J,
Y' = Y s i n
$
-
X cos J,
Y' = Y s i n J,
-
while t h e means may be expressed as
Here
)I
-
-
+
X cos
(measured counterclockwise from t h e p o s i t i v e X axis) i s given
as
+
= ( 1 / 2 ) Arctan [ 2 p
u
0
XYXY
/(ux2-oy2)]
S t a n d a r d i z a t i o n of t h e new v a r i a t e s X' and Y' provides Equation ( 5 ) as Equation ( 7 )
(7)
f ( x , y ) = exp ( - ( w 1 l 2 / 2 ) and i s a measure of t h e s t a n d a r d i z e d r e s u l t a n t of t h e X' and Y' components.
Thus, t h e mean of a normal v e c t o r d i s t r i b u t i o n c o i n c i d e s
w i t h t h e p o i n t of maximum p r o b a b i l i t y .
I n s t a n d a r d i z e d form, t h e
p r o b a b i l i t y i s p r o p o r t i o n a l t o exp [-(w2/2)
1.
Expression (1)i s completely d e f i n e d by f i v e parameters: (%
and
Y
) , t h e two v a r i a n c e s ( o
coefficient (p
X
XY
)
.
and
CT
Y
2,
, and
two means
the correlation
Moreover , t h e s e parameters d e f i n e t h e p r o b a b i l i t y
d e n s i t y as a f u n c t i o n only of t h e v e c t o r v a r i a b l e .
I- 3
REFERENCES
B a r t l e t t , M. S . , 1934:
"The v e c t o r r e p r e s e n t a t i o n of a sample,"
Proc. Camb. P h i l . Soc., Bertrand, J.
, 1888:
Note s u r l a p r o b a b i l i t e
Troisieme n o t e s u r l a p r o b a b i l i t e du t i r a l a
c i b l e , " Comp. Rend. ,,&I
1846:
327-340.
"Calcul des p r o b a b i l i t e s :
du t i r a l a c i b l e :
Bravais , A.,
30, pp.
pp. 387-391 and pp. 521-522.
"Analyse mathematique SUT l e s p r o b a b i l i t e s d e s
e r r e u r s de s i t u a t i o n d'un p o i n t , " Mem. p r e s e n t e s par d i v e r s s a v a n t s , Acad. S e i . , P a r i s , Mem. Sov. Etrang. Crutcher, H. L.
, 1959:
2,
pp. 255-332.
"Upper Wind S t a t i s t i c s C h a r t s of t h e Northern
Hemisphere," NAVAER SO-1C-535,
Volumes I and 11.
U. S. Navy,
O f f i c e of t h e Chief of Naval Operations. Maxwell, J. C . ,
1859:
" I l l u s t r a t i o n s of t h e dynamical t h e o r y of g a s e s .
P a r t 1. On t h e motions and c o l l i s i o n s of p e r f e c t l y e l a s t i c spheres."
Pearson, K.
P h i l . Mag., 30, pp. 19-32.
, 1900: "On
t h e c r i t e r i o n t h a t a given system of d e v i a t i o n s
from t h e probable i n t h e case of a c o r r e l a t e d system of v a r i a b l e s
i s such t h a t it can reasonably b e supposed t o have a r i s e n from a r a n d a n sampling."
P h i l . Mag.
, 50,
S t r u t t , J. (Lord Rayleigh) , 1919:
If
pp.
157-175.
On t h e problem of random v i b r a t i o n s
and of random f l i g h t s i n one, two or t h r e e dimensions."
Mag.,
3'7,
pp. 321-3470
1-4
Phil.
APPENDIX 11. A.
DETERMINATION OF MODEL FIT
Determination of F i t t o t h e B i v a r i a t e Normal D i s t r i b u t i o n .
This s e c t i o n d e s c r i b e s t h e t e s t i n g of t h e v a l i d i t y of t h e assumption t h a t t r o p i c a l cyclone movement d i s t r i b u t i o n s a r e b i v a r i a t e normal. Crutcher (1957, 1958) made t h i s assumption i n work on e x t r a - t r o p i c a l cyclones.
Here t h e assumption was supported by t h e demonstration t h a t
t h e component d i s t r i b u t i o n s i n themselves were d i s t r i b u t e d normally. Though t h i s i s a necessary c o n d i t i o n , i . e . , t h a t t h e marginal d i s t r i b u t i o n s be d i s t r i b u t e d normally, it i s not a s u f f i c i e n t c o n d i t i o n .
It
may be i n f e r r e d from Hald's (1952) s u g g e s t i o n (page 602) t h a t a twodimensional
x2
This w a s t h e b a s i s f o r t h e assumption
t e s t may be made.
of b i v a r i a t e normality f o r wind d i s t r i b u t i o n s as used by Crutcher (1959). The reasonableness of t h i s assumption i s e v i d e n t when t h e expected f r e q u e n c i e s a r e compared w t h observed f r e q u e n c i e s . Though it may be a d v i s a b l e at times t o go t o t h e u n c o r r e l a t e d forms f o r purposes of t h i s t e s t , t h e g e n e r a l c a s e i n which t h e c o r r e l a t i o n i s not
zero may be used. Appendix I.
This i s Q or
x2
o b t a i n e d from expression (1)i n
It i s r e p e a t e d h e r e .
~2 = [1/(1-rxy2)]
( X - X ) ~ / S ~ ~ ](x-X)(y-Y)/s -[~~ s
L
XY
X Y
I+[ (Y-Y)2/sy J
where t h e sample e s t i m a t e s of t h e parameters r e p l a c e t h e p o p u l a t i o n parameters. Now, t h e use of t h e normal d i s t r i b u t i o n implies t h a t a r e l a t i v e l y l a r g e number of o b s e r v a t i o n s w a s a v a i l a b l e .
11- i
This i s n o t always t h e case i n
t r o p i c a l cyclone d a t a s t r a t i f i e d by season and by five-degree l a t i t u d e by five-degree l o n g i t u d e squares.
Therefore, t h e b i v a r i a t e t - d i s t r i b u -
t i o n model w a s i n v e s t i g a t e d a l s o . B.
Determination of F i t t o t h e B i v a r i a t e Student -&Distribution.
The r a t i o n a l e h e r e i s t h a t i f t h e t r o p i c a l cyclone movements a r e b i v a r i a t e t and as t h e b i v a r i a t e t a s y m p t o t i c a l l y approaches t h e b i v a r i a t e normal, t h e non-rejection of t h e t - d i s t r i b u t i o n would permit t h e assumption of b i v a r i a t e normality i n t h e computation of storm s t r i k e
or t a r g e t s t r i k e p r o b a b i l i t i e s .
The m u l t i v a r i a t e t - d i s t r i b u t i o n a l s o
approaches t h e m u l t i v a r i a t e normal d i s t r i b u t i o n a s y m p t o t i c a l l y j u s t as i n t h e u n i v a r i a t e and t h e b i v a r i a t e c a s e s .
The m u l t i v a r i a t e form i s
i n d i c a t e d f o r t h e t - d i s t r i b u t i o n by Krishnaiah and o t h e r s (1969) ,
) ~ others. S t e f f e n s (1968), John ( ~ 9 6 1 and
Let x 1 7 x
2,
x
,.. . ,xV
be
d i s t r i b u t e d j o i n t l y as a v - v a r i a t e normal w i t h zero means, common unknown v a r i a n c e rs2 , and known c o r r e l a t i o n m a t r i x D = ( p
. .>.
1J
Let vs2/cs2 be a
chi-square v a r i a t e w i t h v degrees of freedom d i s t r i b u t e d independently
,.. . ,t
. ,xy.
of x1 y x2 y x 3 y . .
Then t h e j o i n t d i s t r i b u t i o n of t l , t 2 t 3
V
where ti = x . / s i s known t o be a c e n t r a l v - v a r i a t e t - d i s t r i - b u t i o n , Dunn 1
and Massey (1965). Let random v a r i a b l e s x,y have a b i v a r i a t e normal d i s t r i b u t i o n with means pl,
u2 and v a r i a n c e s
0
2 7
1
CT
2
2,
respectively, then v s
b o t h a r e independent of x , y and have a freedom where s
X
and s Y
x2
and v s 2 / a 2 2 Y
d i s t r i b u t i o n w i t h v degrees of
a r e e s t i m a t e s of a
follows t h a t ti = ( x i - p i ) / s i ,
2/012
X
1
and a2 2 , r e s p e c t i v e l y .
where % r e p l a c e s 1-1 and i = 1,2 and each i i
11-2
It
has a Student t - d i s t r i b u t i o n .
The j o i n t d e n s i t y f u n c t i o n following
S t e f f e n s (1968) i s
I-
P r o b a b i l i t i e s a s s o c i a t e d with t h i s f u n c t i o n may be e v a l u a t e d f o r v degrees of freedom and v a r i o u s v a l u e s of t u s i n g t h e t a b l e s developed C r i t i c a l v a l u e s of t a l s o have been t a b u l a t e d
by S t e f f e n s (op. c i t . ) . by degrees of freedom.
Values of t f o r a given p r o b a b i l i t y l e v e l a r e
determined by i n t e r p o l a t i o n using t h e v a l u e s of S t e f f e n s ' I n t e g r a l I1
and h i s t a b u l a r d a t a .
The expression
I1 = (1 - P)/4
(3)
where P = p r o b a b i l i t y l e v e l , gives t h e proper value t o use i n determining
t when v a l u e s of I, have been p l o t t e d a g a i n s t t. For example, u s i n g a p r o b a b i l i t y of
.bo and 75 degrees of freedom I, = (1 -
. 4 0 ) / 4 = .15
and i n t e r p o l a t i o n i n S t e f f e n s ' t a b l e s y i e l d s a v a l u e f o r t C.
(4) 1
= t 2 = .gob.
T e s t i n g of Models for T r o p i c a l Cyclone Movement
Figure 11-1 shows t e n geographic five-degree l a t i t u d e by five-degree longitude squares i n t h e southern North A t l a n t i c and Gulf of Mexico a r e a s . These a r e a s were s e l e c t e d t o t e s t t h e b i v a r i a t e normal and t - d i s t r i b u t i o n f u n c t i o n models f o r t h e 12-hour t r o p i c a l cyclone movements during September f o r t h e p e r i o d 1899-1969.
The s e l e c t e d geographic a r e a s a r e shown i n
black.
11- 3
35
30
2.5
!(r
15 i5'
85'
Figure 11-1
80'
75'
70'
65"
60'
Tests of null hypotheses for the bivariate normal and bivariate tdistribution f o r 12-hour movemats of tropical cyclones during the month of Septeniber. The geographic areas are s h m i n black. The number of tropical cyclone movements i s shown i n the upper l e f t corner of each square. A single asterisk or double asterisk indicates rejection of the null hypothesis for the bivariate normal and the bivariate t-distribution respectively. The rejection l e v e l of c1 = 0.05 involved 4 degrees of freedom as ten equiprobability intervals were selected as class intervals. Period of Record 1899-1969.
11-4
A s t a n d a r d model was used f o r t e s t i n g a l l s q u a r e s , i . e . , random v a r i a b l e s A1,AJ were s t a n d a r d i z e d r e s u l t i n g i n means 0 and v a r i a n c e s 1. A r o t a t i o n of a x i s w a s performed t o remove c o r r e l a t i o n .
By d e f i n i t i o n of t h e t
v a r i a t e , t h e s e s t a n d a r d i z e d random v a r i a b l e s have a j o i n t d i s t r i b u t i o n which i s t h e b i v a r i a t e t - d i s t r i b u t i o n .
A
x2
t e s t of goodness of f i t w a s made f o r each of t h e b i v a r i a t e normal
For more d e t a i l s %he r e a d e r i s r e f e r r e d
and b i v a r i a t e t - d i s t r i b u t i o n s . t o Crutcher and F a l l s
(1971).
The g e n e r a l procedure i s t h e following.
The d i s t r i b u t i o n i s s e t up w i t h t e n s h e l l s . , each s h e l l h o l d i n g , t h e o r e t i cally., t e n p e r c e n t of t h e volume. e l l i p t i c a l or c i r c u l a r .
The s h e l l s may be r e c t a n g u l a r , s q u a r e ,
A v a i l a b i l i t y of p o l a r t a b l e s f o r t h e normal
d i s t r i b u t i o n and t h e a v a i l a b i l i t y of r e c t a n g u l a r t a b l e s f o r t h e t - d i s t r i b u t i o n p e r m i t s t h e use of e l l i p t i c a l c y l i n d r i c a l s h e l l s f o r t h e f i r s t md square c y l i n d r i c a l s h e l l s f o r t h e second.
The expected f r e q u e n c i e s
f o r each s h e l l t h e n a r e n/10 and may be expressed as Ei.
An a c t u a l count
of t h e end p o i n t s of t h e observed v e c t o r s f a l l i n g i n s i d e each s h e l l t h e n i s made.
This may be expressed as Oi.
squared and t h e square i s d i v i d e d by E t h e t e n q u o t i e n t s a r e added.
The d i f f e r e n c e , ( O i i
-
Ei)
, is
This i s done f o r . each s h e l l and
This i s expressed as
(5) i=1
The q u a n t i t y X2 i s d i s t r i b u t e d as
x2,
Pearson (1900).
The b i v a r i a t e
frequency s u r f a c e i s f i t t e d w i t h two means, two v a r i a n c e s , one c o r r e l a t i o n , and a f i x e d volume, causing a loss of s i x degrees of freedom.
11-5
As there
a r e t e n s h e l l s and s i x degrees of freedom are l o s t , X2 i s d i s t r i b u t e d as
x2
w i t h f o u r degrees of freedom.
Figure 11-2 shows e q u i p r o b a b i l i t y e l l i p s e s and r e c t a n g l e s of 0.40 and
0.50 f o r t h e normal and t - d i s t r i b u t i o n , r e s p e c t i v e l y .
The September
12-hour cyclone movements a r e i n d i c a t e d by t h e d o t s from t h e i n t e r s e c t i o n of t h e I , J c o o r d i n a t e s a t t h e c e n t e r of t h e Square 2512.
With 30 t r o p i c a l
c y c l o n e movements and t e n s h e l l s , t h r e e d o t s a r e expected i n each s h e l l . There are two i n t h e e l l i p t i c a l s h e l l and two i n t h e r e c t a n g u l a r s h e l l where boundaries a r e 0.40 and 0.50 p r o b a b i l i t y r e c t a n g l e s .
The contribu-
t i o n of each s h e l l t o X2 f o r each d i s t r i b u t i o n i s (3-2)2/3 or 0.333. This i s done f o r a l l t e n e l l i p t i c a l s h e l l s or r e c t a n g u l a r s h e l l s , t h e n t h e t o t a l X2 i s found f o r each c a s e . appropriate decision c r i t e r i a f o r
c1
The n u l l hypothesis H :
the
x2
= 0.05 w i t h f o u r degrees of freedom.
w a s t e s t e d against t h e a l t e r n a t e
0
hypothesis H : a
This t h e n i s compared a g a i n s t t h e
where
c1
= 0.05.
s t a t i s t i c obtained i s l e s s t h a n
Here
x2 ( a , 4 ) ’
x2 b Y 4 )
i s 9.488.
When
t h e n u l l hypothesis t h a t
t h e b i v a r i a t e normal d i s t r i b u t i o n shows a reasonable f i t t o t h e a c t u a l d a t a d i s t r i b u t i o n i s not r e j e c t e d .
Table 11-1 g i v e s t h e r e s u l t s of t h e t e s t i n g of t h e n u l l hypothesis for t h e two d i s t r i b u t i o n s .
An a s t e r i s k denotes r e j e c t i o n of t h e n u l l hypo-
t h e s i s f o r t h e b i v a r i a t e normal d i s t r i b u t i o n , while a double a s t e r i s k i n d i c a t e s r e j e c t i o n of t h e n u l l hypothesis f o r t h e bivariate t - d i s t r i b u tion.
The b i v a r i a t e normal d i s t r i b u t i o n model i s r e j e c t e d f i v e t i m e s ,
while t h e b i v a r i a t e t - d i s t r i b u t i o n i s r e j e c t e d t w i c e o u t of t e n . a s t e r i s k s are shown a l s o on Figure 11-1.
11-6
The
-
~~
SQUARE 2512
e
Figure 11-2
Distribution of t r o p i c a l cyclone 12-hr movements for September, years 1899-1969 i n the I, J grid system. The .40 and .5O probab i l i t y e l l i p s e s and rectangles for the b i v a r i a t e normal and the b i v a r i a t e Student t - d i s t r i b u t i o n respectively are shown. The probability of a t r o p i c a l cyclone occurring a t h i n the bands defined by t h e e l l i p s e s or the rectangles i s .lo. The number of movements i s 30.
11-7
Both models are r e j e c t e d i n t h e Square 2018 j u s t n o r t h of Yucatan and between Yucatan and Cuba.
Examination of t h e d a t a i n d i c a t e s some b i -
modality which i s e v i d e n t when t h e five-degree square i s broken down i n t o two and one-half degree s q u a r e s .
T r o p i c a l cyclones, i f i n t h e
n o r t h e r n p a r t , t e n d t o move n o r t h , w h i l e t h o s e i n t h e southern p a r t t e n d t o move west. I n Square 2514 t h e s l o w moving storm of September 18-21, 1964, cont r i b u t e d g r e a t l y t o X2 due t o s e v e r a l movements i n t h e 0.30 t o 0.40 p r o b a b i l i t y band.
Though t h e following i s c o n j e c t u r e , t h i s could be
due t o e s t i m a t e s of movement being e q u a l i z e d by t h e a n a l y s t over
several periods.
Table 11-1. Chi-square
(x2)
t e s t f o r f i t of t r o p i c a l cyclone
12-hour movements d u r i n g September. degrees of freedom =
P e r i o d 1899-1969
4 , c r i t i c a l v a l u e of
Square
1512
B i v a r i a t e normal (rejected%)
69 54
14.91 15.60
30
8.00
6.37
2515 2518
76 75 76
3012
31
3014 3512 3514
74 37 36
2018
2512 2514
II- 8
* *
11.00
*
3.47 7.38 11.68
*
13.43
W
5.67
= 9.488.
X2
X2
No. o f Obs.
x2 (a,4)
a = 0.05
B i v a r i a t e "tI' (rejected"")
5.93 14.52
8"
9.33 16.89
**
7.27 8.21 8.68 4.92 8.14 6.22
Table 11-2 g i v e s t h e r e s u l t s of t e s t i n g t h e same n u l l hypothesis f o r s e l e c t e d samples f o r time i n t e r v a l s g r e a t e r t h a n twelve hours. d a t a from a l l seasons except November-May a r e .used.
Here
The b i v a r i a t e
normal model i s r e j e c t e d i n t h r e e of t h e f o u r c a s e s , while t h e b i v a r i a t e I?
t11 model i s r e j e c t e d only once.
I n g e n e r a l , t h e s e r e s u l t s agree with
t h o s e i n d i c a t e d by Table 11-1. Table 21-2.
Chi-square
( x 2 ) test
f o r f i t of s e l e c t e d samples
of t r o p i c a l cyclone movements f o r time i n t e r v a l s g r e a t e r t h a n 12 hours.
4,
P e r i o d 1899-1969, a = 0.05, degrees of freedom =
c r i t i c a l value of
x2 (01,4)
=
9.488.
'
X2
2518
24
June -J u l y
37
5.43
4.35
3015
36
September
54
15.99
8.96
1516
48
October
44
15.09
*
21.91
2015
72
August
39
11.51
*
5.36
.
B i v a r i a t e Normal ( r ejetted")
B i v a r i a t e "t" (rejected**)
Square
Season
No. of Obs
x2
Time (Hrs)
**
Table 11-3 provides t h e approximate p r o b a b i l i t i e s t h a t t h e computed v a l u e s given i n Tables 11-1 and 11-2 would be exceeded by chance.
x2
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APPENDIX I V Program t o i n t e g r a t e t h e b i v a r i a t e normal d i s t r i b u t i o n over an o f f s e t circle.
b
The following program r e p r e s e n t s a v a r i a b l e increment numerical i n t e g r a t i o n method as a p p l i e d t o t h e i n t e g r a l of an e l l i p t i c b i v a r i a t e normal d e n s i t y over an o f f s e t circle.
It w a s developed and programmed by Dr. S. Kaufman
and C. Groenewoud of Cornell Aeronautical Laboratory, I n c . It i s reproduced here with t h e permission of t h e authors. Persons wishing t o use t h e program should t r y t o reproduce two t e s t cases before applying it t o o t h e r s i t u a t i o n s . These t e s t cases a r e :
Test Case I1
Test Case I SIGX
=
2
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5.0
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IV-1
0.84203
PROGRAM B I N D C D I M E N S I O N VO ( 15 1 3 G ( 5 r 5 r 1 5 1 r S ( 41 1001 FDRMAT ( 5 F 1 0 . 3 ) 1002 F O R M A T ( 2 X r b H S I G X r 2 X ~ 6 HS I G Y 8 2 X 1 6 H H r2Xr6H K r 2 X ~ 6 H R r 2 X r 6 H PROB 1 1004 F O R M A T ( I l O 0 E 1 0 . 2 ) 1005 F O R M A T ( 5 F 8 . 3 r F 8 . 5 1 1006 F O R M A T ( 3 E 1 0 . 2 ) 1007 F O R M A T ( 2 X 1 5 H S ( 1 ) = F 6 * 4 r 2 X , 5 H S ( 2 ) = F 6 , 4 0 2 X , 5 H S ( 3 ) = F 6 o 4 r 2 X r 5 H ~ ( 4 ~ a F 6 o * 4 r 2x3 2 HPsF6 4 1 1008 F O K M A T ( 2 X 1 5 E 1 5 . 8 ) 1009 F O R M A T ( 2 X r 1 5 H S D M E B O D Y G O O F E D ) 2001 F O R M A T ( 2 X r 2 H M = I 2 r 4 X ~ Z H I ~ I 2 ~ 2 E 1 5 ~ 8 ) ZOO2 FORMAT ( 6 E 1 5 . 5 ) 2003 F O R M A T ( 2 X r 2 H V V l O F 6 . 2 ) 2004 F O R M A T ~ 2 X r 3 H I A ~ I 2 r 3 X r 3 H I B ~ I 2 r 3 X 1 3 H F 4 r E 1 5 . 8 r 3 X r 5 ~ ~ E L X ~ E l 5 ~ 8 ~ 2005 F O R M A T ( 2 X r 3 H I A = I 2 $ 2 X 9 3 H I B = I 2 r 2x1 5 H I N D X = I 2 r Z X 0 b H I N O X 2 - 1 2 r Z X r 4 H I N O g I *2/ 1 2006 F O R M A T ( 2 x 1 5 H I PO§. I ~ ~ ~ X , ~ H S I G X = F ~ . I I I ZG X Y a~F~4 .H lSr 2x1 3 H C H o F 4 * 1 > 2 X r *3HCK=F4.1//) 2007 F O R M A T ( Z X r / / ) 2008 F O R M A T ( l H 1 ) DO 4 K K K Z l r l l DO 3 111=183 3 READ (5r1006) ( G ( I I I J J J J ~ K K K ) P J J J = ~ ~ ~ ~ 4 CONTINUE CON§T=10.**8 RT2=SQRT(2,0) CRlPI=O*3989422804 READ (5r1004) I P R I N T r E R R R E A D (5r1004) N C A S E 00 700 I C A S E - 1 r N C A S E READ (5r1001) S I G X I S I G Y ~ C H ~ C K ~ R W R I T E ( 6,2008 1 DO 10 I s 1 0 4
*
S(I)=O 10 C O N T I N U E
14 15
16 20
RZrR*R RR2=R/RT2 IPl?S=l GO T O (1%163 1%1 6 ) r I P O S BOTTOM = ( C K + R R 2 ) / S I G Y TOP = ( C K + R ) / S I G Y GO T O 20 BCITTOM: (CK-R)/SIGY TOP = ( C K - R R 2 ) / S I G Y SRmSIGX/RRZ
IV-2
IV-3
IV-4
C =Cl*SR + G ( I A r I B r 6 ) D l = G ( I A # I B # 7 ] *SR + G ( I A I I B , ~ ) D2=Dl*SR + G ( I A r I B r S ) D =D2*SR + G ( I A r I B r 1 0 ) E= G ( I A # I B # l l ) F 1 = A * E L L +B F2= F l * E L L + C F3- F2*ELL+D F4= F3*ELL+E DELX=((360*0*ERR*SR)/F4)*~~~~5 WRITE ( 6 r 2 0 0 4 1 I A I I B ~ F ~ ~ D E L X 90 X 2 = X + U E L X V5=X2 GO TO 1 8 2 9 1 I F ( I N D X o E Q e I N D X 2 ) GO TO 92 IN0 = 0 X2 = 'VO( INDX) 92 C A L L E L I P S E ( I P O S r S IGXI SIGYAICH~CKI R a X 2 9 W2 1 Xl=(X+X2)/2.0 CALL ELIPSE ( I P O S I S I G X I S I G Y I C H ~ C K J R ~ X ~ ~ W ~ ) C A L L NOR ( W a P H I ) Y=(X*X)/2*0 QA= P H I * E X P ( - Y ) C A L L NOR ( W l r P H I ) Y=(X1*X1)/2*0 QB=4oO*PHI*EXP(-Y) C A L L NOR ( W 2 r P H I ) Y=(X2*X2)/2.0 QC = P H I * E X P ( - Y ) Q=((X2~X)/b0O)*(QA+QB+QC)~CRTPI S(IPOS)= S(IPOS)+Q
x=x2 w=w2 I F O ( o G E o V 6 ) GO TO 200 I F ( I N O m E Q . 0 ) G O TO 9 3 GO TO 90 93 I N D X = I N D X + l IN0 = 1 XDX=X+aOOI C A L L E L I P S E ( I P O S I S I G X r S I G Y r CHI C K r RIXDXI C A L L AB(XDX8WDWr I A ) I B ) GO T O 8 8 180 J = l 1800 v o v = v o ( J ) IF(VB.LT.VOV) GO T O 1 8 1 IF(J.EQ.15) GO T O 84 J=J+1 GO T O 1800
IV-5
WDW 1
3
P
e
181 INDX=J GO TO 8 6 182 J = l 1820 VOV=VO( J 1 I F ( V B o L T o V O V 1 GO TO 1 8 3 IF(JeEQ.15) GO T O 8 4 J=J+1 GO T O 1 8 2 0 183 I N D X 2 = J GO T O 9 1 200 GO T O ~ 3 0 0 r 4 0 0 r 5 0 0 r 6 0 0 ) rI P O S 306 I P O S = I P O S + l G O TO 1 4 400 I P O S = I P O S + l AAA = S I G X S IG X = S I GY SIGYZAAA B B B =CH CH = CK CK = B B B GO T O 1 4 500 I P U S = I P O S + 1 GO T O 1 4 600 Z l = ( C H + R R 2 ) / S I G X Z2=(CH=RR2)/SIGX Z3=(CK+RR2)/SIGY Z4=(CK-RR2)/SIGY C A L L NOR ( Z l r A A 1 ) C A L L NOR ( Z 2 r A A 2 ) C A L L NOR ( Z 3 r A A 3 ) C A L L NOR ( 2 4 r A A 4 ) P=(AAl=AA2)*(AA3--AA41 PROB= S ( 1 ) - S ( Z ) + S ( 3 ) - S ( 4 ) - P WRITE(6r2007) W R I T E (60 1007) S ( 1 ) r S ( 2 1 r 5 ( 3 I r S ( 4 1 r P W R I T E ( 6 r 2007) WRITE(br1002) W R I T E ( 6 3 1005) S I G Y r S I G X * C K r C H r R r P R O B C I N T E R C H A N G E S I N A B O V E S T A T E M E N T A R E I N T E N T I O N A L 0 S E E 400-500r 700 C O N T I N U E STOP END S U B R O U T I N E E L I P S E ( N I S I G X r S I G Y r C H a C K r RI X r W 1 R A D ~ = R ~ ~ ~ - ( X ~ S I G X U C H ) ~ ~ ~ , RAD=SQRT(RAD2) GO TO ( 1 0 1 2 O r 1 0 3 2 O ) r N . > 10 W = ( C K + R A D ) / S I G % ) GO TO 30 20 W = ( C K - R A D ) / S I G Y 30 R E T U R N END
