jonathan e. martinmarrella.aos.wisc.edu/mclay_and_martin2005i.pdf · 1 using filtering to mitigate...
TRANSCRIPT
1
Using Filtering to Mitigate Stochastic Model Errors’ Effect on Ensemble
Covariance. Part I: Evaluation of a Prototype Filtering Scheme.
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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29Acknowledgments
This research was supported by a grant from the University of Wisconsin-Madison.
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33Figure Captions Figure 1. Conceptual depiction of forecast trajectories evolved with the dynamics of the
real atmosphere and of a numerical model. The light grey striped shading represents an
initial PDF, and the grey dot represents an initial state sampled from the PDF. Black
solid arrow (dashed arrow) is the forecast trajectory evolved from the initial state using
the dynamics of the real atmosphere (of a numerical model). Time increases from left to
right, as indicated by the axis at the bottom of the figure.
Figure 2. Conceptual depiction of errors introduced by a numerical model. Same as for
Fig. 1, except that the open circle (black dot) represents the state at time t on the forecast
trajectory evolved with the dynamics of the real atmosphere (of a numerical model). The
difference between the two states, represented as the thick solid grey arrow, defines the
errors introduced into the forecast as of time t by the numerical model.
Figure 3. Illustration of the nature of the error component associated with systematic
processes. The box encompasses a two-dimensional forecast sample space for some time
t. Each open circle mi (black dot ni) represents the forecast state obtained by evolving
some initial state si forward in time using the dynamics of the real atmosphere (of a
numerical model). A grey arrow represents the error associated with each ni.
Figure 4. Illustration of the nature of the error component associated with stochastic
processes. Symbology the same as for Figure 3.
34Figure 5. Two simple illustrations of the effect of averaging two dissimilar vectors.
Each panel represents a two-dimensional space, with coordinate axes given as thin solid
lines. In each panel, two dissimilar vectors are represented as solid black arrows. The
average of the two dissimilar vectors is represented as a dotted black arrow. Note that the
average vector has smaller magnitude than either of the two vectors it derives from.
Figure 6. Several conceptualizations of the scenario wherein the possible true states are
all grouped near a small proportion of the filtered states. Each panel encompasses the
sample space for some ensemble forecast. In each panel the forecast state associated with
a given ensemble member is represented as a large dot, and the pair-wise samples
associated with each pair of ensemble members are represented as X’s. The continuum
of possible true states is defined with grey shading. In each of these examples the event
E1 is very likely to occur just because the possible true states are all grouped near the
pair-wise samples with asterisks.
Figure 7. Composite range of 192h 500 hPa geopotential height rmse (m) for a) the basic
operational ensemble and b) the distribution of pair-wise filtered states. In each case the
composite range of rmse is delimited by a grey shaded box and the composite median
rmse is depicted with a dashed line. The composite lower bound of the overall
operational ensemble’s range of rmse is depicted with the dotted line.
Figure 8. Number of pair-wise filtered states whose 192h 500 hPa geopotential height
rmse is less than that of the best overall operational member, for each of the 361
35ensembles available between 21 December 2002 and 21 December 2003. Each bar
represents the number of filtered states for a specific ensemble. The bars are arranged in
chronological order, beginning with the bar for the 21 December 2002 ensemble in the
upper left corner and proceeding left-to-right and top-to-bottom. The labels ‘J’, ‘F’, ‘M’,
etc. identify the bars associated with the ensembles on the first day of January, February,
March, and so forth. The number of filtered states associated with a given bar is
determined by the scale on the ordinate of each rectangular plot. The bold horizontal line
indicates the average over all 361 ensembles of the number of pair-wise filtered states
whose rmse is less than that of the best overall operational member.
Figure 9. The best pair-wise filtered state’s percentage improvement in 192h 500 hPa
geopotential height rmse relative to the rmse of the best overall operational member, for
each of the 361 ensembles available between 21 December 2002 and 21 December 2003.
Each bar represents the percentage improvement for a specific ensemble. The percentage
improvement associated with a given bar is indicated by the scale on the ordinate of each
rectangular plot. The bold horizontal line indicates the average over all 361 ensembles of
the best filtered state’s percentage improvement in rmse. Layout otherwise the same as
for Figure 8.
Figure 10. Number of best filtered states whose percentage improvement, α, in 192h 500
hPa geopotential height rmse relative to the rmse of the best overall operational member
lies within a specified range. The abscissa indicates the number of best filtered states.
The ranges of percentage relative improvement are given along the ordinate. Bars with
36light (dark) grey shading are associated with best filtered states that yield positive
(negative) percentage relative improvement.
Figure 11. Mean and median anomaly projection value, and mean anomaly projection
value plus or minus one standard deviation, as a function of forecast rank. Forecast rank
is given on the abscissa and anomaly projection value is given on the ordinate. The “+”
(“•”) symbols indicate the mean (median) anomaly projection value associated with each
forecast rank. The upper (lower) sequence of diamond symbols denotes for each forecast
rank the mean anomaly projection value plus (minus) one standard deviation. The grey
horizontal dashed line indicates the mean anomaly projection value for all forecast ranks.
Figu
re 1
. C
once
ptua
l dep
ictio
n of
fore
cast
traj
ecto
ries e
volv
edw
ith th
e dy
nam
ics o
f the
real
atm
osph
ere
and
of a
num
eric
al
mod
el.
The
light
gre
y st
riped
shad
ing
repr
esen
ts a
n in
itial
PD
F, a
nd th
e gr
ey d
ot re
pres
ents
an
initi
al st
ate
sam
pled
from
the
PDF.
Bla
ck so
lid a
rrow
(das
hed
arro
w) i
s the
fore
cast
traj
ecto
ry e
volv
ed fr
om th
e in
itial
stat
e us
ing
the
dyna
mic
s of t
he re
al
atm
osph
ere
(of a
num
eric
al m
odel
). T
ime
incr
ease
s fro
m le
ft to
righ
t, as
indi
cate
d by
the
axis
at t
he b
otto
m o
f the
figu
re.
t 0tim
e
37
Figu
re 2
. C
once
ptua
l dep
ictio
n of
err
ors i
ntro
duce
d by
a n
umer
ical
mod
el.
Sam
e as
for F
ig. 1
, exc
ept t
hat t
he o
pen
circ
le (b
lack
dot
)re
pres
ents
the
stat
e at
tim
e t o
n th
e fo
reca
st tr
ajec
tory
evo
lved
with
the
dyna
mic
s of t
he re
alat
mos
pher
e (o
f a n
umer
ical
mod
el).
The
diff
eren
ce b
etw
een
the
two
stat
es, r
epre
sent
ed a
s the
thic
k so
lid g
rey
arro
w, d
efin
es th
e er
rors
intro
duce
d in
to th
e fo
reca
st a
sof t
ime
tby
the
num
eric
al m
odel
. t 0t
time
38
Figu
re 3
. Ill
ustra
tion
of th
e na
ture
of t
he e
rror
com
pone
nt a
ssoc
iate
d w
ith sy
stem
atic
pro
cess
es.
The
box
enco
mpa
sses
a
two-
dim
ensi
onal
fore
cast
sam
ple
spac
e fo
r som
e tim
e t.
Eac
h op
en c
ircle
mi
(bla
ck d
ot n
i) re
pres
ents
the
fore
cast
stat
e ob
tain
ed b
y ev
olvi
ng so
me
initi
al st
ate
s ifo
rwar
d in
tim
e us
ing
the
dyna
mic
s of t
he re
al a
tmos
pher
e (o
f anu
mer
ical
mod
el).
A
gre
y ar
row
repr
esen
ts th
e er
ror a
ssoc
iate
d w
ith e
ach
n i.
m4
n 4
m3
n 3
m2
n 2
m1
n 1
39
Figu
re 4
. Ill
ustra
tion
of th
e na
ture
of t
he e
rror
com
pone
nt a
ssoc
iate
d w
ith st
ocha
stic
pro
cess
es.
Sym
bolo
gyth
e sa
me
as fo
r Fi
gure
3.
m1
n 1
n 5m
5
m2
n 2m
4 n 4
m3
n 3
40
Figu
re 5
. Tw
o si
mpl
e ill
ustra
tions
of t
he e
ffec
t of a
vera
ging
two
diss
imila
r vec
tors
. Ea
ch p
anel
repr
esen
ts a
two-
dim
ensi
onal
spac
e, w
ith c
oord
inat
e ax
es g
iven
as t
hin
solid
line
s. In
eac
h pa
nel,
two
diss
imila
r vec
tors
are
repr
esen
ted
as
solid
bla
ck a
rrow
s. T
he a
vera
ge o
f the
two
diss
imila
r vec
tors
is re
pres
ente
d as
a d
otte
d bl
ack
arro
w.
Not
e th
at th
e av
erag
e ve
ctor
has
smal
ler m
agni
tude
than
eith
er o
f the
two
vect
ors i
t der
ives
from
.
b)a)
41
Figu
re 6
. Se
vera
l con
cept
ualiz
atio
ns o
f the
scen
ario
w
here
in th
e po
ssib
le tr
ue st
ates
are
all
grou
ped
near
a
smal
l pro
porti
on o
f the
filte
red
stat
es.
Each
pan
el
enco
mpa
sses
the
sam
ple
spac
e fo
r som
e en
sem
ble
fore
cast
. In
eac
h pa
nel t
he fo
reca
st st
ate
asso
ciat
ed w
ith
a gi
ven
ense
mbl
e m
embe
r is r
epre
sent
ed a
s a la
rge
dot,
and
the
pair-
wis
e sa
mpl
es a
ssoc
iate
d w
ith e
ach
pair
of
ense
mbl
e m
embe
rs a
re re
pres
ente
d as
X’s
. Th
e co
ntin
uum
of p
ossi
ble
true
stat
es is
def
ined
with
gre
y sh
adin
g. I
n ea
ch o
f the
se e
xam
ples
the
even
t E1
is v
ery
likel
y to
occ
ur ju
st b
ecau
se th
e po
ssib
le tr
ue st
ates
are
all
grou
ped
near
the
pair-
wis
e sa
mpl
es w
ith a
ster
isks
.
X
X*
XX*
X
X
X*X*X*
XXX
XX
X
X*X
X
42
Figu
re 7
. C
ompo
site
rang
e of
192
h 50
0 hP
age
opot
entia
lhei
ght r
mse
(m) f
or a
) the
bas
ic o
pera
tiona
l ens
embl
e an
d b)
the
dist
ribut
ion
of p
air-
wis
e fil
tere
d st
ates
. In
eac
h ca
se th
e co
mpo
site
rang
e of
rmse
is d
elim
ited
by a
gre
y sh
aded
box
and
the
com
posi
te m
edia
n rm
seis
dep
icte
d w
ith a
das
hed
line.
The
com
posi
te lo
wer
bou
nd o
f the
ove
rall
oper
atio
nal e
nsem
ble’
s ran
ge o
f rm
seis
dep
icte
d w
ith th
e do
tted
line.
120
110
100 90 80 70 60
a)b)
rmse
43
Figu
re 8
. N
umbe
r of p
air-
wis
e fil
tere
d st
ates
who
se 1
92h
500
hPa
geop
oten
tialh
eigh
t rm
seis
less
than
that
of t
he b
est o
vera
ll op
erat
iona
l mem
ber,
for e
ach
of
the
361
ense
mbl
es a
vaila
ble
betw
een
21 D
ecem
ber 2
002
and
21 D
ecem
ber 2
003.
Eac
h ba
r rep
rese
nts t
he n
umbe
r of f
ilter
ed st
ates
for a
spec
ific
ense
mbl
e.
The
bars
are
arr
ange
d in
chr
onol
ogic
al o
rder
, beg
inni
ng w
ith th
eba
r for
the
21 D
ecem
ber 2
002
ense
mbl
e in
the
uppe
r lef
t cor
ner a
nd p
roce
edin
g le
ft-to
-rig
ht
and
top-
to-b
otto
m.
The
labe
ls ‘J
’, ‘F
’, ‘M
’, et
c. id
entif
y th
e ba
rs a
ssoc
iate
d w
ith th
e en
sem
bles
on
the
first
day
of J
anua
ry, F
ebru
ary,
Mar
ch, a
nd so
forth
. Th
e nu
mbe
r of f
ilter
ed st
ates
ass
ocia
ted
with
a g
iven
bar
is d
eter
min
ed b
y th
e sc
ale
on th
e or
dina
te o
f eac
h re
ctan
gula
r plo
t. T
he b
old
horiz
onta
l lin
e in
dica
tes t
he
aver
age
over
all
361
ense
mbl
es o
f the
num
ber o
f pai
r-w
ise
filte
red
stat
es w
hose
rmse
is le
ss th
an th
at o
f the
bes
t ove
rall
oper
atio
nal m
embe
r.
44
Figu
re 9
. Th
e be
st p
air-
wis
e fil
tere
d st
ate’
s per
cent
age
impr
ovem
ent i
n 19
2h 5
00 h
Page
opot
entia
lhei
ght r
mse
rela
tive
to th
e rm
seof
th
e be
st o
vera
ll op
erat
iona
l mem
ber,
for e
ach
of th
e 36
1 en
sem
bles
ava
ilabl
e be
twee
n 21
Dec
embe
r 200
2 an
d 21
Dec
embe
r 200
3.
Each
bar
repr
esen
ts th
e pe
rcen
tage
impr
ovem
ent f
or a
spec
ific
ense
mbl
e. T
he p
erce
ntag
e im
prov
emen
t ass
ocia
ted
with
a g
iven
bar
is
indi
cate
d by
the
scal
e on
the
ordi
nate
of e
ach
rect
angu
lar p
lot.
The
bold
hor
izon
tal l
ine
indi
cate
s the
ave
rage
ove
r all
361
ense
mbl
es
of th
e be
st fi
ltere
d st
ate’
s per
cent
age
impr
ovem
ent i
n rm
se.
Layo
ut o
ther
wis
e th
e sa
me
as fo
r Fig
ure
8.
45
Figu
re 1
0. N
umbe
r of b
est f
ilter
ed st
ates
who
se p
erce
ntag
e im
prov
emen
t, α,
in 1
92h
500
hPa
geop
oten
tialh
eigh
t rm
sere
lativ
e to
th
e rm
seof
the
best
ove
rall
oper
atio
nal m
embe
r lie
s with
in a
spec
ified
rang
e. T
he a
bsci
ssa
indi
cate
s the
num
ber o
f bes
t filt
ered
st
ates
. Th
e ra
nges
of p
erce
ntag
e re
lativ
e im
prov
emen
t are
giv
enal
ong
the
ordi
nate
. B
ars w
ith li
ght (
dark
) gre
y sh
adin
g ar
e as
soci
ated
with
bes
t filt
ered
stat
es th
at y
ield
pos
itive
(neg
ativ
e) p
erce
ntag
e re
lativ
e im
prov
emen
t.
020
4060
8010
012
014
016
0
range of percentage improvement
α≤
-10%
-10%
< α
≤-5
%
-5%
< α
< 0
%
0% <
α<
5%
5% ≤
α<
10%
10%
≤α
num
ber o
f bes
t filt
ered
sta
tes
46
fore
cast
rank
anom
aly
proj
ectio
nva
lue
Figu
re 1
1. M
ean
and
med
ian
anom
aly
proj
ectio
n va
lue,
and
mea
n an
omal
y pr
ojec
tion
valu
e pl
us o
r min
us o
ne st
anda
rd d
evia
tion,
as
a fu
nctio
n of
fore
cast
rank
. Fo
reca
st ra
nk is
giv
en o
n th
e ab
scis
sa a
nd a
nom
aly
proj
ectio
n va
lue
is g
iven
on
the
ordi
nate
. Th
e “+
”(“
•”) s
ymbo
ls in
dica
te th
e m
ean
(med
ian)
ano
mal
y pr
ojec
tion
valu
e as
soci
ated
with
eac
h fo
reca
st ra
nk.
The
uppe
r (lo
wer
) se
quen
ce o
f dia
mon
d sy
mbo
ls d
enot
es fo
r eac
h fo
reca
st ra
nk th
e m
ean
anom
aly
proj
ectio
n va
lue
plus
(min
us) o
ne st
anda
rd
devi
atio
n. T
he g
rey
horiz
onta
l das
hed
line
indi
cate
s the
mea
n an
omal
y pr
ojec
tion
valu
e fo
r all
fore
cast
rank
s.
47