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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. 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].
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28Acknowledgments This research was supported by a grant from the University of Wisconsin-Madison.
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.
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.
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
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
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
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.
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.
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
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 (-,+).
38c) ROC curves for the operational ensemble and hybrid
ensemble version A.1.19, gridpoint pair 4, event (+,+).
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
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
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
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
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
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
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
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
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
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
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
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
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
(+,+
).