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1 Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble Covariance. Part II: Employment of Filtered States in Hybrid Ensembles. Justin G. McLay 1 NRC/Naval Research Laboratory, Monterey, California Jonathan E. Martin University of Wisconsin, Madison, Wisconsin (Submitted to Monthly Weather Review, November 29, 2004) 1 Corresponding Author: Justin G. McLay, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Monterey, CA, 93943-5502. E-mail: [email protected].

Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

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Page 1: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

1

Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble

Covariance. Part II: Employment of Filtered States in Hybrid Ensembles.

Justin G. McLay1

NRC/Naval Research Laboratory, Monterey, California

Jonathan E. Martin

University of Wisconsin, Madison, Wisconsin

(Submitted to Monthly Weather Review, November 29, 2004)

1 Corresponding Author: Justin G. McLay, Naval Research Laboratory, 7 Grace Hopper Ave., Stop 2, Monterey, CA, 93943-5502. E-mail: [email protected].

Page 2: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

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28Acknowledgments This research was supported by a grant from the University of Wisconsin-Madison.

Page 29: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

29References

McLay, J. G., and J. E. Martin, 2005: Using filtering to mitigate stochastic model errors’ effect

on ensemble covariance. Part I: Evaluation of a prototype filtering scheme. Mon. Wea.

Rev., submitted.

Wilks, D. S., 1995: Statistical Methods in the Atmospheric Sciences. Academic Press, 467 pp.

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1

30Table Captions Table 1. Sequence of operations in the probabilistic forecast verification procedure.

Table 2. Best-case Brier score results arranged by gridpoint pair and event. See text and

Table 3 for interpretation.

Table 3. Interpretation of the column of three entries in a given box of Table 2.

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Tabl

e 1.

Seq

uenc

e of

ope

ratio

ns in

the

prob

abili

stic

fore

cast

ver

ifica

tion

proc

edur

e.

1.Fo

r a g

iven

ope

ratio

nal e

nsem

ble,

cre

ate

vario

us v

ersi

ons

of h

ybrid

en

sem

ble

(A.1

.5, A

.1.1

0, A

.1.1

5,…

., A

.2.5

, A.2

.10,

….,

A.4

.5, A

.4.1

0,…

. et

c.).

2.D

eriv

e pr

obab

ilist

ic fo

reca

sts

for a

ll fo

ur d

icho

tom

ous

even

ts fr

om th

e op

erat

iona

l ens

embl

e an

d ea

ch o

f the

hyb

rid e

nsem

bles

.

3.P

erfo

rm s

teps

1 a

nd 2

for 2

21 o

pera

tiona

l ens

embl

es.

4.U

se th

e B

rier s

core

and

rela

tive

oper

atin

g ch

arac

teris

tic to

ver

ify th

e pr

obab

ilist

ic fo

reca

sts,

then

look

for t

he v

ersi

on o

f hyb

rid e

nsem

ble

that

pr

ovid

es th

e fo

reca

sts

that

are

mos

t im

prov

ed re

lativ

e to

the

oper

atio

nal

fore

cast

s.

5.D

o al

l the

abo

ve fo

r joi

nt h

eigh

t dis

tribu

tions

ass

ocia

ted

with

five

diff

eren

t gr

idpo

int p

airs

.

31

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Tabl

e 2.

Bes

t-cas

e B

rier S

core

resu

lts a

rran

ged

by g

ridpo

intp

air a

nd e

vent

. Se

e te

xt a

nd T

able

3 fo

r int

erpr

etat

ion.

Even

t

1.20 2

Grid

poin

tPa

ir

66

(-,-)

(+,-)

(-,

+)

(+

,+)

1 2 3 4 5

2.19 4

2.30 4

1.5’ 1

4.5’ 3

4.8’ 3

3.6 8 1

3.15 7 2

3.5 1

1.19 3

1.12 5 2

4.30 2

2.8’ 3

3.10

3

4.15

’8 38 2.5 4

2.17 4

4.32 5 25

2.21 8 48

2.21 6 46

32

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Tabl

e 3.

Int

erpr

etat

ion

of th

e co

lum

n of

thre

e en

tries

in a

giv

en b

ox o

f Tab

le 2

.

Ent

ry 1

Ent

ry 2

Ent

ry 3

the

num

ber o

f eve

nts

for t

he g

iven

grid

poin

tpai

r for

whi

ch th

e hy

brid

ver

sion

spe

cifie

d in

Ent

ry 1

is a

ble

tosi

mul

tane

ousl

y yi

eld

som

e fo

rm o

f pos

itive

rela

tive

impr

ovem

ent

–va

lues

are

roun

ded

to th

e ne

ares

t int

eger

–po

sitiv

e va

lues

less

than

5 a

re re

pres

ente

d w

ith a

chec

k m

ark

the

perc

enta

ge re

lativ

e im

prov

emen

t atta

ined

desi

gnat

ion

of th

e hy

brid

ver

sion

–po

sitiv

e va

lues

gre

ater

than

5 a

re h

ighl

ight

ed in

light

gre

y.

33

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34Figure Captions

Figure 1. Relative operating characteristic 2x2 contingency table. The four possible

contingencies are a “hit”, “miss”, “false alarm”, and “correct rejection”. As an example

of the table’s interpretation, if an event is forecast and is ultimately observed, then a “hit”

is recorded. An event is considered to be forecast if the ensemble-derived probability for

the event exceeds a pre-specified threshold value. The allowable threshold values are a

function of the number of members in a given ensemble. For instance, if an ensemble has

three members, then the allowable threshold values are 0.0, .33, .67, and 1.0.

Figure 2. Illustration of possible hybrid ensemble configurations. The area within each

panel describes a two-dimensional sample space. Operational ensemble members and

pair-wise samples are represented as dots and X’s, respectively. a) Configuration A. b)

Configuration B. See the text for further interpretation of these configurations.

Figure 3. Orientation of the gridpoint pair referenced in the idealized experiments. The

horizontal position of a particular gridpoint is indicated by a black dot.

Figure 4. Example of a table that provides α as a function of error variance and nf.

Values of error variance (m2) are given along the top of the table, and values of nf are

given along the left-hand side.

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35Figure 5. Two examples of α as a function of nf from the experiment where the errors are

drawn from a normal distribution with µ = 0 m, σ2 = 400 m2, and ρ = .4. a) Results

obtained using the 18 February 2003 ensemble data. Values of nf are given along the

bottom of the plot. A bar describes the value of α for a given value of nf according to the

scale along the left-hand side of the plot. A dotted line denotes the height of a bar

associated with α = .5. b) Results obtained using the 21 July 2003 ensemble data.

Otherwise same as for a).

Figure 6. Ensemble joint distribution of 500 hPa geopotential height at two gridpoints.

Possible values of height Ф1 (m) at a gridpoint located at 170.0W longitude, 52.5N

latitude define the abscissa, and possible values of height Ф2 (m) at a gridpoint located at

22.5W longitude, 60.0N latitude define the ordinate. Each point on the plot depicts a

particular ensemble member’s specific values of height at the two gridpoints. The

ensemble members are associated with the 14 May 2003 NCEP GFS 192h ensemble.

The grey point corresponds to the control ensemble member, and the black points

correspond to the perturbed ensemble members.

Figure 7. As for Figure 6, except that the dotted lines identify the climatological values

of 500 hPa geopotential height c1 and c2 at gridpoints 1 and 2, respectively, for 14 May

2003. Also, the labels S1, S2, S3, and S4 indicate the sample subspaces described in the

text.

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36Figure 8. As for Figure 7, except that the pair of signs in each parentheses are the signs

of the verifying 500 hPa geopotential height anomalies at gridpoints 1 and 2 in the case

that the point corresponding to verification falls into a given sample subspace. The first

entry in parentheses corresponds to the sign of the height anomaly at gridpoint 1, and the

second entry corresponds to the sign of the height anomaly at gridpoint 2.

Figure 9. a) Composite 500 hPa geopotential height (m) anomaly pattern associated with

the negative phase of the Pacific North American (PNA) pattern. Composite based upon

NCEP/NCAR Reanalysis data for the month of January and the years 1968, 1969, 1971,

1972, 1982, and 1989. Contours every 15m. Negative height anomalies ≤ -15m shaded.

Figure created using NOAA-CIRES/Climate Diagnostics Center interactive website. b)

Horizontal positions of two particular gridpoints located at the 500 hPa vertical level.

The position of each gridpoint is indicated by a black dot. The designation of each

gridpoint is indicated by the number below the dot

Figure 10. Orientations of the five different gridpoint pairs referenced in the analysis.

The horizontal position of a particular gridpoint is indicated by a black dot, and the two

gridpoints in a given pair are connected by a black dotted line. The designation of a

given gridpoint pair is indicated by the number alongside the black dotted line associated

with the given pair.

Figure 11. a) Composite 500 hPa geopotential height (m) anomaly field associated with

the positive phase of the Pacific North American (PNA) pattern. Composite based upon

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37NCEP/NCAR Reanalysis data for the month of January and the years 1977, 1981, 1983,

1985, 1988, and 1992. Contours every 15m. Negative height anomalies ≤ -15m shaded.

b) Composite 500 hPa geopotential height (m) anomaly field associated with the positive

phase of the North Atlantic Oscillation (NAO) pattern. Composite based upon

NCEP/NCAR Reanalysis data for the month of January and the years 1984, 1986, 1987,

1988, 1989, 1990, 1991, 1993, and 1994. Contours and shading same as for a). Figures

created using NOAA-CIRES/Climate Diagnostics Center interactive website.

Figure 12. Percentage relative improvement in Brier score derived from hybrid version

A.2.17 for gridpoint pair 5 for each of the four dichotomous events (-,-), (+,-), (-,+), and

(+,+). The four boxes represent the partitioned sample space (refer to Fig. 8), and the

relative improvement is superposed on the sample subspace corresponding to each event.

Positive relative improvement less than 5.0% is represented with a check mark.

Figure 13. ROC curves of the operational ensemble and a given hybrid ensemble for a

particular gridpoint pair and event. The solid (dashed) line is the ROC curve of the

operational (given hybrid) ensemble. Each open circle (dot) defines the operational

(given hybrid) ensemble’s false alarm rate and hit rate for a specific probability threshold.

For reference, the dotted line is the ROC curve of an ensemble with zero skill.

a) ROC curves for the operational ensemble and hybrid

ensemble version A.2.21, gridpoint pair 1, event (-,+).

b) ROC curves for the operational ensemble and hybrid

ensemble version A.1.20, gridpoint pair 3, event (-,+).

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38c) ROC curves for the operational ensemble and hybrid

ensemble version A.1.19, gridpoint pair 4, event (+,+).

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Figu

re 1

. R

elat

ive

oper

atin

g ch

arac

teris

tic 2

x2 c

ontin

genc

y ta

ble.

The

four

pos

sibl

e co

ntin

genc

ies a

re a

“hi

t”, “

mis

s”, “

fals

eal

arm

”, a

nd “

corr

ect r

ejec

tion”

. A

s an

exam

ple

of th

e ta

ble’

s int

erpr

etat

ion,

if a

n ev

ent i

s for

ecas

t and

is u

ltim

atel

y ob

serv

ed, t

hen

a “h

it” is

reco

rded

. A

n ev

ent i

s con

side

red

to b

e fo

reca

st if

the

ense

mbl

e-de

rived

pro

babi

lity

for t

he e

vent

exc

eeds

a p

re-s

peci

fied

thre

shol

d va

lue.

The

allo

wab

le th

resh

old

valu

es a

re a

func

tion

of th

e nu

mbe

r of m

embe

rs in

a g

iven

ens

embl

e. F

or in

stan

ce, i

f an

ense

mbl

e ha

s thr

ee m

embe

rs, t

hen

the

allo

wab

le th

resh

old

valu

es a

re 0

.0, .

33, .

67, a

nd 1

.0.

Even

t Fo

reca

st ?

Yes

N

o

Even

t O

bser

ved

?

Yes

No

Hit

(H)

Fals

eA

larm

(F)

Cor

rect

Rej

ectio

n (R

)

Mis

s (M

)

39

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Figu

re 2

. Ill

ustra

tion

of p

ossi

ble

hybr

id e

nsem

ble

conf

igur

atio

ns.

The

area

with

in e

ach

pane

l des

crib

es a

two-

dim

ensi

onal

sa

mpl

e sp

ace.

Ope

ratio

nal e

nsem

ble

mem

bers

and

pai

r-w

ise

sam

ples

are

repr

esen

ted

as d

ots a

nd X

’s, r

espe

ctiv

ely.

a)

Con

figur

atio

n A

. b)

Con

figur

atio

n B

. Se

e th

e te

xt fo

r fur

ther

inte

rpre

tatio

n of

thes

e co

nfig

urat

ions

.

X

X

XX

X

XX

X

X a)b)X

X

XX

X

XX

X

XX

XX

40

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Figu

re 3

. O

rient

atio

n of

the

grid

poin

tpai

r ref

eren

ced

in th

e id

ealiz

ed e

xper

imen

ts.

The

horiz

onta

l pos

ition

of a

par

ticul

ar

grid

poin

tis i

ndic

ated

by

a bl

ack

dot.

41

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Figu

re 4

. Ex

ampl

e of

a ta

ble

that

pro

vide

s αas

a fu

nctio

n of

err

or v

aria

nce

and

n f.

Val

ues o

f err

or v

aria

nce

(m2 )

are

gi

ven

alon

g th

e to

p of

the

tabl

e, a

nd v

alue

s of n

far

e gi

ven

alon

g th

e le

ft-ha

nd si

de.

100

22

5

400

90

0

1

600

250

0

5

0

.26

0.2

4

0

.27

0.2

4

0

.25

0.

20

10

0.51

0.

600.

65

0.69

0.

70

0.7

6

15

0.39

0.

51

0.58

0.

68

0.70

0

.73

20

0.59

0.

65

0.69

0.

71

0.77

0

.77

25

0.26

0.

25

0.25

0.

24

0.22

0

.21

30

0.20

0.

22

0.22

0.

21

0.21

0

.21

σ2

n f

42

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Figu

re 5

. Tw

o ex

ampl

es o

f αas

a fu

nctio

n of

nff

rom

the

expe

rimen

t whe

re th

e er

rors

are

dra

wn

from

ano

rmal

dis

tribu

tion

with

µ=

0 m

, σ2

= 40

0 m

2 , a

nd ρ

= .4

. a)

Res

ults

obt

aine

d us

ing

the

18 F

ebru

ary

2003

ense

mbl

e da

ta.

Val

ues o

f nfar

e gi

ven

alon

g th

e bo

ttom

of t

he p

lot.

A b

ar d

escr

ibes

the

valu

e of

αfo

r agi

ven

valu

e of

nfac

cord

ing

to th

e sc

ale

alon

g th

e le

ft-ha

nd si

de o

f the

plo

t. A

dot

ted

line

deno

tes t

he h

eigh

tof

a b

ar a

ssoc

iate

d w

ith α

= .5

. b)

Res

ults

obt

aine

d us

ing

the

21 Ju

ly 2

003

ense

mbl

e da

ta.

Oth

erw

ise

sam

e as

for a

).

5

10

1

5

20

2

5

30

α

.2.4.6.8

5

10

1

5

20

2

5

30

α

.2.4.6.8

n fn f

a)b)

43

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Φ(1

70.0

W, 5

2.5N

)1

Φ(22.5W, 60.0N) 2

Figu

re 6

. En

sem

ble

join

t dis

tribu

tion

of 5

00 h

Page

opot

entia

lhei

ght a

t tw

o gr

idpo

ints

. Po

ssib

le v

alue

s of h

eigh

t Ф1

(m) a

t a

grid

poin

tloc

ated

at 1

70.0

W lo

ngitu

de, 5

2.5N

latit

ude

defin

e th

e ab

scis

sa, a

nd p

ossi

ble

valu

es o

f hei

ght Ф

2(m

)at a

grid

poin

tloc

ated

at

22.

5W lo

ngitu

de, 6

0.0N

latit

ude

defin

e th

e or

dina

te.

Each

poi

nt o

n th

e pl

ot d

epic

ts a

par

ticul

ar e

nsem

ble

mem

ber’

s spe

cific

va

lues

of h

eigh

t at t

he tw

o gr

idpo

ints

. Th

e en

sem

ble

mem

bers

are

ass

ocia

ted

with

the

14 M

ay 2

003

NC

EPG

FS 1

92h

ense

mbl

e.

The

grey

poi

nt c

orre

spon

ds to

the

cont

rol e

nsem

ble

mem

ber,

and

the

blac

k po

ints

cor

resp

ond

to th

e pe

rturb

ed e

nsem

ble

mem

bers

.

44

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Φ(1

70.0

W, 5

2.5N

)1

Φ(22.5W, 60.0N) 2

S 3S 4

S 1S 2

Figu

re 7

. A

s for

Fig

ure

6, e

xcep

t tha

t the

dot

ted

lines

iden

tify

the

clim

atol

ogic

alva

lues

of 5

00 h

Page

opot

entia

lhei

ght c1

and c 2

at g

ridpo

ints

1 an

d 2,

resp

ectiv

ely,

for 1

4 M

ay 2

003.

Als

o, th

e la

bels

S1, S 2

, S3,

and S 4

indi

cate

the

sam

ple

subs

pace

s des

crib

ed in

th

e te

xt.

c 2

c 1

45

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Φ(1

70.0

W, 5

2.5N

)1

Φ(22.5W, 60.0N) 2(-

,+)

(-,-)

(+,+

)

(+,-)

c 1

c 2S 3

S 4 S 2S 1

Figu

re 8

. A

s for

Fig

ure

7, e

xcep

t tha

t the

pai

r of s

igns

in e

ach

pare

nthe

ses a

re th

e si

gns o

f the

ver

ifyin

g 50

0 hP

age

opot

entia

lhe

ight

ano

mal

ies a

t grid

poin

ts1

and

2 in

the

case

that

the

poin

t cor

resp

ondi

ng to

ver

ifica

tion

falls

into

a g

iven

sam

ple

subs

pace

. Th

e fir

st e

ntry

in p

aren

thes

es c

orre

spon

ds to

the

sign

of t

he h

eigh

t ano

mal

y at

grid

poin

t1, a

nd th

e se

cond

ent

ry c

orre

spon

ds to

the

sign

of t

he h

eigh

t ano

mal

y at

grid

poin

t2.

46

Page 47: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

Figu

re 9

. a) C

ompo

site

500

hPa

geop

oten

tialh

eigh

t (m

) ano

mal

y pa

ttern

ass

ocia

ted

with

the

nega

tive

phas

e of

the

Paci

fic N

orth

A

mer

ican

(PN

A) p

atte

rn.

Com

posi

te b

ased

upo

n N

CEP

/NC

AR

Rea

naly

sis d

ata

for t

he m

onth

of J

anua

ry a

nd th

e ye

ars 1

968,

19

69, 1

971,

197

2, 1

982,

and

198

9. C

onto

urs e

very

15m

. N

egat

ive

heig

ht a

nom

alie

s ≤-1

5m sh

aded

. Fi

gure

cre

ated

usi

ng

NO

AA

-CIR

ES/C

limat

e D

iagn

ostic

s Cen

ter i

nter

activ

e w

ebsi

te.

b) H

oriz

onta

l pos

ition

s of t

wo

parti

cula

rgrid

poin

tslo

cate

d at

the

500

hPa

verti

cal l

evel

. Th

e po

sitio

n of

eac

hgr

idpo

inti

s ind

icat

ed b

y a

blac

k do

t. T

he d

esig

natio

n of

eac

hgr

idpo

inti

s ind

icat

ed

by th

e nu

mbe

r bel

ow th

e do

t.

b)1

2a)

b)

47

Page 48: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

Figu

re 1

0. O

rient

atio

ns o

f the

five

diff

eren

t grid

poin

tpai

rs re

fere

nced

in th

e an

alys

is.

The

horiz

onta

l pos

ition

of a

parti

cula

r gr

idpo

inti

s ind

icat

ed b

y a

blac

k do

t, an

d th

e tw

o gr

idpo

ints

in a

giv

en p

air a

re c

onne

cted

by

a bl

ack

dotte

d lin

e. T

he d

esig

natio

n of

a g

iven

grid

poin

tpai

r is i

ndic

ated

by

the

num

ber a

long

side

the

blac

k do

tted

line

asso

ciat

ed w

ith th

e gi

ven

pair.

1 2

43

5

48

Page 49: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

Figu

re 1

1. a

) Com

posi

te 5

00 h

Page

opot

entia

lhei

ght (

m) a

nom

aly

field

ass

ocia

ted

with

the

posi

tive

phas

e of

the

Paci

fic N

orth

A

mer

ican

(PN

A) p

atte

rn.

Com

posi

te b

ased

upo

n N

CEP

/NC

AR

Rea

naly

sis d

ata

for t

he m

onth

of J

anua

ry a

nd th

e ye

ars 1

977,

198

1,

1983

, 198

5, 1

988,

and

199

2. C

onto

urs e

very

15m

. N

egat

ive

heig

ht a

nom

alie

s ≤-1

5m sh

aded

. b)

Com

posi

te 5

00 h

Page

opot

entia

lhe

ight

(m) a

nom

aly

field

ass

ocia

ted

with

the

posi

tive

phas

e of

the

Nor

th A

tlant

ic O

scill

atio

n (N

AO

) pat

tern

. C

ompo

site

bas

ed u

pon

NC

EP/N

CA

R R

eana

lysi

s dat

a fo

r the

mon

th o

f Jan

uary

and

the

year

s198

4, 1

986,

198

7, 1

988,

198

9, 1

990,

199

1, 1

993,

and

199

4.

Con

tour

s and

shad

ing

sam

e as

for a

). F

igur

es c

reat

ed u

sing

NO

AA

-CIR

ES/C

limat

e D

iagn

ostic

s Cen

ter i

nter

activ

e w

ebsi

te.

b)

a)b)

49

Page 50: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

Figu

re 1

2. P

erce

ntag

e re

lativ

e im

prov

emen

t in

Brie

r sco

re d

eriv

ed fr

om h

ybrid

ver

sion

A.2

.17

for g

ridpo

intp

air 5

for e

ach

of th

e fo

ur d

icho

tom

ous e

vent

s (-,-

), (+

,-), (

-,+),

and

(+,+

). T

he fo

ur b

oxes

repr

esen

t the

par

titio

ned

sam

ple

spac

e (r

efer

to F

ig. 8

), an

d th

e re

lativ

e im

prov

emen

t is s

uper

pose

d on

the

sam

ple

subs

pace

cor

resp

ondi

ng to

eac

h ev

ent.

Posi

tive

rela

tive

impr

ovem

ent l

ess t

han

5.0%

is re

pres

ente

d w

ith a

che

ck m

ark.

Φ(1

20.0

W, 4

5.0N

)1

Φ(80.0W, 40.0N) 2(-

,+)

(-,-)

(+,+

)

(+,-) 6.3

50

Page 51: Jonathan E. Martinmarrella.meteor.wisc.edu/McLay_and_Martin2005II.pdf · -,. 0/1) 2 34/ )* ,-2 2 5768,- 9 )* : (9 ; = )? *2 @a=b (,- >5 () 6: b ( 5 ; (,c (9 ,d,-e1,

b)

fals

e al

arm

rate

hit rate

c)

fals

e al

arm

rate

hit ratea)

fals

e al

arm

rate

hit rate

51Fi

gure

13.

RO

C c

urve

s of t

he o

pera

tiona

l ens

embl

ean

d a

give

n hy

brid

ens

embl

e fo

r a p

artic

ular

grid

poin

tpai

ran

d ev

ent.

The

solid

(das

hed)

line

is th

e R

OC

cur

ve o

fth

e op

erat

iona

l (gi

ven

hybr

id) e

nsem

ble.

Eac

h op

enci

rcle

(dot

) def

ines

the

ope

ratio

nal (

give

n hy

brid

)en

sem

ble’

s fal

se a

larm

rate

and

hit

rate

for a

spec

ific

prob

abili

ty th

resh

old.

For

refe

renc

e, th

e do

tted

line

is th

eR

OC

cur

ve o

f an

ense

mbl

e w

ith z

ero

skill

.a)

RO

C c

urve

s for

the

oper

atio

nal e

nsem

ble

and

hybr

iden

sem

ble

vers

ion

A.2

.21,

grid

poin

tpai

r 1, e

vent

(-,+

).b)

RO

C c

urve

s for

the

oper

atio

nal e

nsem

ble

and

hybr

iden

sem

ble

vers

ion

A.1

.20,

grid

poin

tpai

r 3, e

vent

(-,+

).c)

RO

C c

urve

s for

the

oper

atio

nal e

nsem

ble

and

hybr

iden

sem

ble

vers

ion

A.1

.19,

grid

poin

tpai

r 4, e

vent

(+,+

).