REDU

D-R K IGM -POI T KALM
FILTER FOR
G OPHY I
D T
IMIL TION

Manoj K Kizhakkeniyil
. (Ph ' ic , Math malic

M. c. ( hy i

DI

NAT RALR

h mi try), Mang I rc Uni er it , 2002
a niv r ity. 2

BMITT D I PAR I L F L ILLM
T F
M
T
R TH D
R
F
PHY
T R OF PHIL
IN
R
AND ENV IR
MENTAL TUDIE

UNIV R ITY OF NORTH RN BRITISH COLUMB IA

June 20 15

© Manoj K Ki zhakkeniyiL 20 15

Abstract

Th main goal )f my rPs •c-ucb was to d ,·plop a pt act 1cal sdwnw for t hC' sigma-1 oint
K alm an :filtC'r (c PI\ ) for its npplicrttJon in a n'r listi c limat C' morlC'l. LargC' ('Ol1l p11t i-\tional xp ns has 1 en an obstacl to appl~·ing thP ,

K

to a high-dnnC'nsional sys-

t m. I addre. ' d this i. su 1 .\· dew•loping ·u1 ach·anc d . I KF da t a-assimila hon syst rm.
~1~· work also adclressed several oth r factor. - relatPcl to thC' practical implementation

of PKF . Th main 1 je ·tives f thi res arch were to: (i ) investigate' two mPthocls to
con truct are lucecl-rank igma-point m1sc nted K a lman filter (RR PCKF ): (ii ) propos a localizations heme forth

PKF : and (iii ) implement RR ' PCKF in a realistic

climate model.
I pr ent two methods to approximat

the

rror covanance by a n' luced-rank ap-

proximation. In the first m ethod , truncated singnlar-value d ecomposit ion (TSVD) is
applied on th error-covaria n ce matrix calculated in the data : p a e (RR Pl:KF (D))
whil in the second m ethod TSVD is a pplied

n th

error- covaric nee mat ri..x ·alcu-

lc t.<>d in tlw C'nsnnhk. p are' (RR SP KF (E)). T bC' n w i-\lgorithms an' first tC'stC'd on
the Lorenz-96 model, a one-climcnsi n al a tmos1 h ric "toy·· m o d el. The pt'rformancc
of both r nk-re lu ction m thocls a re clos t o th at of th e full-rank

PKF .

I propose a lo alization m thod for RR P ' I '~ ( ~ ). The rcsnlts from mmwrical rxperimcuts ou the Lmcuz-< G model showed that wlH'll llH' localizntwn and inflation
v:nc implemented. the optlm,d cstmwt c wa:-. <ldll(,'Yed \Yl t 11 n fimte mtml)('l of ~1gma

1 < in t s.

The realistic mod llnscd 111 th1s stndv was tlw ZPhiak- 'mw (Z ')modeL an intPrmrdiatr complex1t~· conpl' l

l ·iiio, outlwrn Osnllation ( ~

·s )) pr<'didion m )dC'l .

he RR PCKFs arr implPmrnt •d for t lw Z ' modrl \\'It h tlw assunila! ion of src1 snrfac t rmpera t nre anomaht>s. Thr rrsul t s showc·d 1hat hot h R I\ SPl.' KF ( D nnd ~ ) wPrc
able to correctly analyze th pha.<>c' and intPnsJt\' of all major EI\SO c·vpnts during the
stud.v period ,~·ith rdatiYc>ly similar estimation accuracv.
We

c mpared against

urtlwrmorP. th' RRSPL'KF

ns mbl . quare-mot filter (En ~ RF ). showing that tbr ov('rall

analysi skill of RR PCKF and En 'RF arr c·omparahlc> to Pnf'h ot hC'r lmt t hr fornH'r
1s mor robust than th latt r.

11

Conte nts

tra t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Li t

f

a 1

. . . . . . .

.

. . .

.

. . . .

.

. . .

.

. . . .

. . . .

Li t of Fig ur

.

. .

VI

X

A cknow 1 d g 1n nt

2

.

V IJ

lo a r

1

. . .

1

X ll

Introduction .

1

1.1

ompa t Overview

1

1.2

Background

2

1.3

Data assimilation

7

1.4

l\ Iotiva tion .

1~

1.5

Obj ective.

16

1.6

Outline.

1

A ssimilation m ethods a nd Models used in this study

21

2.1

Introdu ·tion . . . .

~1

2.2

The ret i l r view

:.. 2

2.2.1

Th framework

[ ~ nKF

2.2.2

ormula tion of

n RF .

lll

2.2.3

2.

1gma p int uu~ce nt 'd Kalmau fill 'l' ( ' PCKF)

2
31

Iod ls
2 .. 1

2.. 2

model

.2

Z0l iak- mw mocl '1 .

;:3.

h

Lor nz-

n

3

p li ati n

n an

n h

r n z-

m

1

3

.1

Introclurtion

~35

:3 .2

R duccd-rank ' igmc -pomt I\ alman nlt ('1 (R R . Pt TKF)

3G

.2.1

RR PCK (D): T

.2.2

RR PCI\ ( "): T,'\'D m th

11li ·ation
..J

nth Lornz-

.. .

.7

n~ •mblP ~pcH ' (' .

,g
41

modPl

XJ rim nt 1 set up . . . .
.4 .1

3.4.2
3.4.3
3.

in th0 data SJHH '('

\ TD

42

4

en ration of truth and observation.
Assimilation met hods ancl 1 r J

.n

clure

rror .

44

R ulL and di cus ·i n

44

3 ..5.1

naly ·i

tat

-stirnation

exp nm nts

with

RRSPCKF(D)

and

RRSPCKF (E) . . .
3.5.2
3.6

4

S nsiti\it)' exp riment with the RRSPCKF (E) nwtlwcl.

Summary

.......... .

49

53

Localization in Sig m a Point Kalman Filte r

57

4.1

Introduction . . . . . . . .

57

4 .2

Inbreed ing. filt er divergen c a nd spuriou

orr lati on

5

4.3

ovariance lo alization

59

4.4

ovari n e inflation ..

60

4.5

L calization in R R PUKF(E)

Gl

4.6

Numerical experiments . . . .

G4
lV

4.6.1

"'xp r rim ent al set up ..

G4

4 .. 2

R sult s and dis usswn

G

4.7

ummary
li

5

f RR

n

7

. . . . . . . . . . .

d 1

Ill e ll

7

5. 1

In t ro clur t 1011 . .

7G

.2

:.Iod r l and met hods .

77

r:

5.

5.2. 1

Z hw k- 'anr ).lodPl

5.2.2

:\Id hoc b and PXI t'l llllPntal "'I up

R sult s and disc ussiOn

D

.J \ 'prsJon

77

.

7

. . . . . . . . ..

5 .. 1

r nsiti,·it \' XI rim nts w1t h RR . ' P ' KF (D )

5.. 2

nnp arison of RR Pl' K (D ) and

PCI\:F

5

Forecast skill
5. A

01

omp nrison b t w rn the RR. PCI\:F (D ) and RR. 'I ' KF ( ~) ap-

92

pr aches . . . . . . . . . . . . . . . . . . . .

5.3.5
5 ...1

6

Comparison h tw en the RR . ' PCKF ( ) and

Summary'

n. 'RF

9G

............ .

. 1()()

Conclusions and Future direct ions

. 103

G.l

Introduct ion . . .

. 10.

6.2

Gen ral overview

. 10..1

6.3

Concluding summary

. l OG

G.4

Future dir ctions ..

. 111

6.4 .1

Implem nt a1ion of RR P KF (E ) f r a G C I

. 11 2

6.4 .2

A hybrid as. imihtion by coupling 3D-Var / 4D-Var + SPKF .

. 11'

6.4 .3

on- a.ussian st tistics in SP I F

. . . . . . . . . . . . . . .

. ll.J

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

v

List of Tabl

.1

ompu tat ion tim in ~P ond~ fnt \'c-lrlOll. dc1 t a as~imila t ion nw! hods 011
c Lmux P
with a 2.0 H7 Int PI I en! nnn Dnal 'or' Jnoc · c·s~or. . . . .
7

.J ~3 mP PXJ)('llllH'lll lllllllber for tlw exprnm nts tf grnc>rate Tahl -1.2 and T c1hlr .: lPSpPctivC'ly. GG

.J.l

Lo alizntlOn

-1 .2

h ciqwnd nc of tnn nwan R).I. 'E on tllP lo('ahzatwn nHlins ancl
numb r of sigma I omt.. Thr mfintion codficirnt i~ ().():3, . . . . . . . . 72

..J..

h d p ndence of l irn nwan R.:Vl 'E on t h inflation coefficient and
numl r of sigma I oints. The localization radius i~ G.
. 74

5.1

E T 0 stat e ·timation: Exi nm nt summary

xpPnm

nt dPtcub .

.J _ cmd

Vl

2

List of Figur

1.1

The clrp ncl ncr on tlw nutwl condition fm tlH' LorPn7-- >· modf'l . Hf'n'
four f; ri >: of 'r -C'OOUlma t ) of I h
()1('117- ;: nwdPl dl\'(' l g f' markedly
over im flom . mall dlff 1 nc · lll th llll 1nl cowlltwn . . . . . . . .
l

G

.1

Th tim m·c' raged r 'latl\'C' R~L . . as c1 fnnrt 1011 of nmnhC'l' of sigma
I oints (en. mhl m mb rs ) n:Pd for tlw Lorenz-: G modc'l . . . .
. 4G

.2

mparison h tw ' 11 the tru, val\! and ::uHllYsis fm \'Hiiahlc' Xl witl1
tim step for (a) the fllll-rank FCKF . (h ) R . I l'KF (D ) with~ l sigma
I inL (en , mbl m mh rs ) and (c) RR ' PCKF (E ) vnth 31 sigma pomt s. 47

3.3

The variation of th root mean-. qm1n' d rror v ' I tlw 4() variables with
tim :t p for (a) th fnll-rank FCKF . ( b ) RR P ' KF (D ) with 31 sigma
1 oints (en emhl meml rs ). and (c) RR P TKF (E ) v.·ith . 1 sigma point s. 4

3.4

Th time m an of th e R~l E a a function of tnmcated modPs (ensPmbl size) for RR PCKF (E ) m thod .
. . . . . . . . . . . . . . . . . . . 50

3.5

ompari on between the true valu and analysis of RRSFCKF (E ) for
th variabl Xl. The analysis from th e tnmca ted m des of 3 ..5. anrl. 7
a re hown in Figures 3.5a-3 .. ·.indicating a poor estimation skill in all
three cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 1

3.6

Comparison between the true valn and analysis of RR PUKF (E) for
the vaial le Xl. Th analysis fr m th full-rank PCKF and t run atrcl
modes of 41, 1, 21 and 11 a re .' hown in Figur s 3.6a 3.Ge. . . . . . .

3.7

The variation of the RJ\ISE ov r the 40 variablPs with tim stei for the
diff rent trun a ted modes (i.e., ensemble siz ) for the RRSP'CKF (E )
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5:3

3.

Th varian explained by the truncated mod (ensembles) with re~pect
Loth tot al varian for the RR. P KF ( ) method.
. . . . . . . . . . 54

Vll

.1

h tim mran R .0. I rrror of t lw R l . 'PC I\ ( ) sdwmr R.'i a f1m t 1011
hr r c>snlt s are shown
f numlwr of sigma pomt~ ( ns mhl mPmbPrs)
. G7
fm drffeH'llt srzc (S.r). ystrm . . . . . . . . . .

. J. .2

hr tnn mean R::\ 1 c'rror oftlH' RR.'P1.TK ·( ) sdwnw as a fnnctwn
of nnmlwr of . igma points ( C'U~c·mhlc' llH'lll hPr~ ) w lwn localiza t ron 1s
llf-iPcl. Tlw r . nlt ~ ar "'hm\·n for drffPI rnt Sl7C' (.YT) ~)·st c>m . . . . . . . . G

. J. .3

Th time nwan R::\1 PilCH of tlw RR . T> ' !\: ( ) '->dwmr d:-> a functmn
of nmnlwr of ~igma pmnt" ( C'll'-iC'mhlP lllC'mlH'l:-, ) wlwn hot h localrzatwn
and inAation c11 e us d . Tlw 1 r~ul s HIP show11 fm diffe1 c' nt SIZP (S,)
. 70
~yst m . . . . . . . .

. J. .. J.

Th tinw mran R::\ 1 rrror of tlw RR . ' I 'KF( ) :-,dwmP a:-> a fmwtion
of localization radm .. Th mflation p,u cUIH'tC'l 1~ 0.03 . . . . . . . . . . . 71

.J: . '-

The tim mran R::\ 1 Nror of t lH' R R . ' I · 1\ ( ) f-idwnw a:-; a fnnct iou
of inflation co fficiPnt . Tlw loca liza iou r cHllll" 1s
. . . . . . . . . . .

5.1

, raphical d 'Pi ·tion of tlw 0:'1iio rc'o wns (from
Environm ntal Prediction ( T 'EP)) . . . .

·cltwnal

T

'pntPrs for

2

5.2

T h root m an-squared rror (R::\1 ' ) of ' ' T (5°. '-5°
120-170°\\')
I et \\'een anal)· is and truth as a fnnct i()n of number of Pnsc•mblPs nsPd .

5. 3

T he R.0. 1 E of
T . bet wren the analysis and t rn t h a v<'ra gC'd ovC'r
the' pC'riod 1071- 2000 for tlw RR , P\.'KF (D) whC'n diffNPnt nnmlH'r~ of
truncat d m de ( l ) are used. Th number of C'nsemblc>s is eqnal to
(2 l 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

G

T h correlation coefficient of S TA between the analysis and truth m·eraged over t h e period 1971- 2000 for t h e RRSPl' K F (D) wh n cliffetcnt
numl ers of tru ncated modes ( l ) are us d. The number of nsembles
is equ a l to (2l+ 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

5. -!

T.

5.5

The R 1SE of t h e Nino-3.4 index of t h analysis against t he obser ved
counterpa rt. The p a n el (a) is fo r RR P C KF (D ) when -!1 ensembles
a re used and p a n el (b ) is fo r th e full-ran k SP KF . . . . . . . . . . .

5.6

The RM SE a nd corr lation coefficient of SS TA b etween t h e a n a lysis
and truth for full-r ank SP KF ai d RR PUKF (D) wit h -11 ensembles for th p riod 1971 2000 (a- d ). Figur 5.G( ) an l (f) ar r the
<..l if-fcrcn cc in SS T A RM SE a nd condat ion coeHicicnt of PC Kl~ aml
RRSP KF (D ) wh ere the s t atist ically signifi cant a reas (at 05 Vc conlid n ee level) a r sh ad cl . . . . . . . . . . . . . . . . . . . . . . . . . .
V lll

)9

5.7

n -tim st p for ca~t forth T1ii
r gJOn (. 0 , -f 0 •• 120- 170°\ \' ). -1
incl.iratrs the 'tiio- .<-1 inclrx of tlw ohservatwn and thr solll hnr mrlicat r. tlw ·tno- A mrlr · of a one month lrnd forrcast of RR . P L' K ( )
\\'it h ..J 1 rnsemhlrs. . . . .
. . . . .
. . . . . . . . . . . . . . . . . . 90
Th ·1iio-' -1 fore cas t R~L ~ and C<HtPlc1tion wtth 12-month !Pad limP
for RR . Pl.KF (D) fm tlw c a~r~ w1l h drffr 1rnt 111!1111H'I of <'llsrmhlc' mrm. 91
hrrs .

r::

'

.

Th e root l11C'Hll-"f111HlPd P llOI ( R~L '") of.'.'
(5 S-.Sn r. 120- 170°\\')
lwtwe n analYsis and tn1th ,\~ <l fmH twn of thP mtmlH'l of c•nsc•mhlPs
usrd for the RR PC!\ (D ) and RR , P 'I\F( .. ) mPthocls . . . . . . . .

5.10 Th R~L E and condatwn codfic wnt of .. T h PhH'<'ll t lw analysis
and truth for RR . PCKF ( .. ) and RR .' P ' 1\ (D ) w1th 11 <'ns<'mhlPs (ad ). (r) and (f) cH the chffPn'llCP m R~L E and m c()rrdat ion codficiPnt
of
T betwrC'n RR ' · K ( ) an d RR .' Pl'KF (D ). lw sta iJsi ica llv
signifi ccml a rC'as ( t ( .srlr conficl r ncP 1 \'el) ,ur s ha d Pcl . . . . . . . . .

1

5.11 Th Pxplainrd nuiann' (o/r) for tllC' RR.'Pl' KF (D and E) lllf'thods as
a fun t ion of muuber of ensc'mhles nsecl. . . . . . . . . . . . . . . . . . 95
5.12 Th R~l cliHen.'llCP l>t'hH't'll t h :'\ u-w-3 .-! mdPx of t IH• d llctlys is a~rllm,j
the ohserv d ount rp art for (a) RR 'PCKF (D ) an d (b) En ' RF wh('ll
41 ens mhles arc n 'eri. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.13

TA R~1 E c nd conrlation corfficirnt for En RF and RR PlTKF (D )
m ethod for the p riocl. 1971 2000 \vh n <:1:1 ensembles a rc usrd (a- d ) .
. (e) c nd (f ) (U'P the diffPrence in R).L E and conPlatio n c·opffi(iPnt of
S TA of EnSRF and RR Pl KF (D ) wh er e the stati. ti cally . ignific a nt
areas (at 9 o/c confidenc e leYel) ar shad d. . . . . . . . . . . . . . . .

9

5 .14 Niiio-3.4 index forecast RI\ l SE and correlation with 12 m onth 1 c-l d time
for RRSPUKF (E ). RRSP"CKF (D ) and En RF . The vertical error b a rs
ar the sampling errors at 95 o/c. confidence inten·al obtainrd u ing bootstrap experiments at ea h 1 ad time. . . . . . . . . . . . . .. _ . . . . 99

lX

Glossary

3D-V r

Dm1 n. wnal \'anal H nal

4D-Var

4 Dinwnsicmal \ "anat wnal

M

At mo. ph ric ; rwral

1rculatwn :\Iodf'l

BL E

Best Lin ar Tnl ia.c>cl

:-.timclt

E KF

En.' mbl

cljustmcnt K alman

ENSO

El Tiiio

uth rn Oscillation

EnKF

Ensembl K alman Filt er

EnSRF

Ensemll

EKF

Extended K alman Filt er

ESSE

Error

ETKF

Ensembl Tran form Kalman Filt er

GCM

Global Circulation

LDE05

Lamont- Doherty

LETKF

Local En embl Transform K alman Filtrr

iltPr

quare-Root Filt r

ubspace Stati ti a l Estimation

1od l
arth Observatory, mod l version 5

X

M

cramc -.cnrral
hservat ion

1r ulat ion ~ I odrl

\'S tem Smnd a t1on

Tat 10nal ' put rrs for

XJWnm nt

11\'lronm Pnt al Prc,rlict ion

WP
KF

Rrdncrd -R a nk . 1gm a- p omt l Tnsr<'lll d K a lma n Flltrr

( ) Rrdncrd-R a nk . 1gm a- p om l ' nscrnl cl K ellm a n

J}(Pr - D e~ ta

RR P

K

RR P

KF ( ) R edu · d-1 a nk ' 1gm a- pmnt ' n sc<nl< d K a lm a n FlltN - " llSPmhh'

IR

'c>qu ent wl Imp ort a nt R ·~a m plin ~

P DKF

igm a-p oint

' nt re~l D ifiPr nc<' K a lnw n F ilt c·r

PKF

i g me~- point

K a lm a n

PPF

igm a-p int P a rticl ' Filt r

SPUKF

igma-p oint C n sc nt c~ d K alma n Filt r r

SSTA

a

ilt C'r

urface Temp ra ture a n o m aly

TLM

T a ngent Linear l\ Io d el

TSVD

Trun a t ed Singu la r-Value Decomposition

SVD

Singu la r-Value D com 1 o ilion

TPCA

Trun at d P rincipal-Comr on nt Ana lysis

zc

Zebiak-Cane

Xl

Acknowledgements

irst and for r nwst. I would llkP to <'XIH Pss my d<'<'JH'S( gra t it ud<' t o my sup c>rvrsor
Dr. Yc umin a ng and co-: np 1 vism 1 I Pt ' 1 .Jac·kson fo r t h ' 11' gmd a ncP . m ·nt oring,
patienc . c mst aut 'nco urag<'lll 'llt . msrght and know]pdg<' mrp m t c•cl to lll <' dmmg my
stud)' p eri od a t ' ·B . lw ir k C'C'll and ,·rgorous anH lf'mH' ohs<•r \'(1\ ron <'nlight C'IlS m e
not only in this th esis hut al:o gr,·r moi i\'at ton for mv fnt nrP stmh ·
I would also lik to tak this opp ort um tv to acknowl 'dgP and express m~· gra trtml<'
to my gradu ate . lll n ·isorv commrt t <' m m b rs D1. 't <' p hPn Drry. Dr. :\.lich a ·l
:1illingham an l r. Li ang lwn for t h 'ir input and suggPs t ions along t lH· Wd,V. as w<'ll
c: , f r t a kin g tim t
re, ·iew t lus v.:m k.
T hi. tlw. i. gr<'a tly 11C'nd1t s h·om Youmin 's conr. c'. at 1: ':\13 '. fro m whi ch J ]('an H'd
m an)' h elpf 1l . t ati ·ti ·al m t hocl. an l t o ls . Thanks t o Dr . P Pt <' r .J ackson . Dr. KPn
Ott er and Dr. P hil Bur t n forth y O J n d my mind with int 1 stin g sciPilt ifi c topics
of ph.vsi ·aL biological and : cia l scienc s dnring m~· conrsr st mly. I h ave ab o b enefi ted fr om t he co ur e of Dr . P t r J ck on a nd Dr . t eplw u Dr ry on dynami cal
m t eorology.
p ecial thanks t o Dr. Ziwang Den g for his h lpfnl ·omments, sugges ti ons. and disu ion in my m anusc ri1 ts. Iany thanks t o Dr. Yanj ie 'heng ancl Dr. D akP 'hen
f r providing m e the la t t Zehi ak- Cane m ocl l.
I am grat efu l for the numeri cal ·amput ation SUPI ort of Dr. J ean \\'ang at thC' HP C
Lab of
BC . A sp ecial word f gra tit ud t o all the m emb ers of m)' research group .
D r . Ziwang Deng, Dr. Yanji Cheng, J aison mb aclan , \\'aqa.r Youn a:·, iraj 1 Islam ,
Xiaoqin Yan and Dejian Yang f r their friendships and cliscu. sions on r search .
I\1y deep est th anks go to my wond rful family nd m ar\'elous friend (of wh m th re
are simply too m any to m ention by n am ) for their monum nt aL unwavering supp ort
and encouragem ent .

e

Finally it w llcl 1 r emL' [ m e n t to say thank ) 'OU to
TB and lllEmb crs of it s
st aff for their assistan c in num rous administrative and fin an ial m att rs. , pecial
thanks to NB Physi s d pa.rtment for hiring m as a Te·1 'hin g ssist ant. I rrall.v
enjoyed my teaching lut ies.

Xll

his work is sup1 ortrcl 1 ~·. the

'an ad a (
British

T

atural , c1rn rs and n gmccrin g RrsC'mch onncil of
R ') D1~con'ry p1 oguu11 awl Paciiic 'cut \U , . C HH lua1 <' \ holarship by

olmnh1a's }.hms1 ry of

T

cl\'ancPd ~ dn catinn.

Xlll

Chapter 1

Introduction

1.1

Compact Ove rvie w

Th ge phy. ical

~·stem is not a

out roll d laborat or,v xperiment. Therdore modC'ls

are an imp ort ant tool to understand th m chanisms in gE oph~·sica l systems . Th('~·
describ how th comr onent. of the art h syst m changPs nsing mathematical eqnations representing the laws f 1 hysi sand data collect d from obs rvatious. Advanced
m dels, such as coupl d general irculation models help to und stand many proc sses
and mechanism f the eart h's ·limat system . They ar nsed to simulat

th past

and pr s nt climat as well as for future proj ect ions. I\ Iodern w ather forecasting is
ba d on numerical weather 1 r diction ( V:P ) mod ls. Th mod ls require current
state of the aLmosphere/oc an called as the initial
futur . The initial conditions are ol t ained from
is haoti an l higly s nsiliv t.

nclitions to int erp late in to

bs rvations. The \\' ather s.\·stnu

initial condit ions. Data assimilation helps to pro-

1

lu

high quaht.v imtwl oncl1t10n. )rtunallv comhmmg th

mod '1 output wtlh lh

bscrvat ion. .

her' c: r \'a rion~ sdwmrs for data assimilfllwn.

filter ( PK

1:-, a uciivati\' '-lcs:-,. dclcllllllllStH. Kalman filt<'l optumd for nouliw'al'

stat -. pa

st nnatwn.

R., imibtion met hods

he ,

'igma-r oint 1\fllmnn

K ~ has manv orh·ant agPs m·er othr1 popular dal a-

uch a" cnt..Pmh]P Kalm nn filtPt (Eu KF ) and Em.;PmblP ~qlmrr­

root filter ( n, RF) .

Large omputal ional <'XpPnsr has IH'<'ll an ohst adP \\·hilr· apph·mg SP KF I o a highdimensional system ~nch a.<> tlH' <><THllH' g ·n 'Ial r·tnnldtwn llH><lPl (0 ; '~ I ) or almosph ric general cucnla t ion nwdc>l (.A ; '~I)

lw ()\'('1 clll goal of this chsspr( at ion is

to rl.e,·elop adnmced. PKF data-as:-,imilation m·tlwd for its application in a n·alistH·
limat
d

mod l. Th pr sPntation of this clwptPr is orgamz<·d as follows. ,'ect iou 1.2

rib , the broad prol lem domain addre:srcl l v the bod.v of \vork prc>sc>nt rd m this

dis. rtati n.
Thi

ti n 1.

reviews th general concC'pts and goals of data assimilation.

ect ion aL o contain a compa t lit 1 ture ov ITi w of d1ffer nt data-assimilation

m thod .

ft r thi ·. the moti,·ation for this st ndy is given in . Pelion 1...1 . Ohj('ctives

of my st udy i. given in

ction 1.5. Finally. an outline of the dissertation is prm·iclrd

in Se tion 1.6.

1.2

Background

The realistic model I used in this s t ucly is the LDE0 5 vers10n of th
(Zel iak and
prediction . El

an e, 19 7· Chen et aL, 2004) for El

ino

ZC moclel

outhern Oscillation ( ~

T

0)

ino South rn OscillaL ion is the , trongest signal in the ' 'ariahility of

the glob l climate sys t em on th

int erannual time-scales. It occ ur in the tropi cal

Pacific Occ>an irreg;ularly at intervals of 2- 7 y < rs and influences thr global chmatr
2

).

through tel connect ions (\Yang C't al., 1
the strongest in
1 ains <
'l.lld

0 vrars of data r cordin)2;

h

rv nt m 1

7 1

was

'ri1o-rrlatf'd unpacts indmlrd lwavv

fiooclmg al ong t lH' n><t:t of 'c•nt ral . \nH'l H n knding t u <'Xt<'ll~lV<' damngC'

top opl 's llv s. ass t~. crops and mfnvlructmP Dwnghts in 11wm·
wrr also as:-;o ·iatPd with

sum co1mlriPs

I -n]o . It had an impa ton thP WPathcr con h twns monw l

the glohr throngh trlrcnnnPclwn~ and oc·can ClllTPUt s. TlHTr \\TIC' JWrsrsiC'nl forest
fi r es. above normal trmp tatun• and h 'lnw nmn1c1l JH'PcipiL\twn IrportPd in Canada
influ need byE:\ 0 (llsr h C't ell.. 1

). ThP cHTllral<' pn•dictwn of Pxtn·nH' wPnt lwr

C\'Cn t s and :-;casonal c llmat H' ph<'nonH'ml 1..., \ ' PI:: impot t ant fm ~oc H'ty.

he I imdv

is<' prrclictwn of \\"<'<lt lwr x r nws '"ill ~ig nificant ly rmrmruzP thr loss

and morr pr

f li\·es and <'conomic im1 act~ .

Pr dieting the future stat
tat

of th

of a ph\'Sical-~vstrm rrcpurrs a modC'l to pr >pagalc' thC'

s.\ ·st m through tim

y. t m (Hunt

t aL 2007).

ns v:ell as knowlPdgc of thP cunC'nt state of thP

11 model are. in general. a simplification of the rc·al-

world system, . ).loci ls c re u, e l to refrr to a larg
hyp ot h

rmmlwr of things . from a srt of

e to a seal d replica of the physical-s)·stem. The main fnnct ion~ of modc·ls

in lucie understanding and xplaining the m chanism and proces.'es of the> concC'nwd
system. predicting t h e futur

tate of th S)'Stem even b fore the observation is made

a nd cl rivation of new principles. Collected observation. h lp to improve the mo lel
structure aHCl fine tun the model (Sh nk and Franklin. 2001). Genernlly. modeL- can
be ·lassificd into either thcorcticaJ or i:3tatisti cal models c.lcp 'lldiug on how the:· c-u-e
m ade and their inten l cl us . Theoretical mod els are usua lly built either b efore or
after th data a re collected although the)' a re m a,iuly u eel as ad h oc m o d ls. The
m ain 1 urpose of a theoretical mod el i explan a t or.y, to un l rs t and th

mechanism

a nd I ro cesses of the concern d system and a lso for pre li ting th futur e state of the'
syst em . Stati. ti al models are pos t ho c models; th y ar us d for th e interpret at ion

3

of the data. arc ronstructrd during the anc lYsl~ of !h data rollr ted and the~· ar
typically without a m clwnism ( lwnk c nd Ft ankhn, 2001). R Pgrpssion models arc
Throrrtira! modP!s nm HI plv to a widr nmg

example of statistical modrls

of

syst rms and oft n gl\'<' in-d 1 th knowl dg ahont tlw sysiPm compcHPd to statistical
models. I\ lost of thr th 'Ol'Ptiral modPls <HC' <'XJH' C'S:-.Pd in thr form of nwtlH'matica!
cqna t ions.
artlH.;~·st rm

modds ar

g<>nrralh· t lH'OITt iced modd~ that <'On( am ma I hem a I ical

equations rrprC'sc>nting tlw ph~·<>1r~ of thr atmo...,phnr and occ'an. .:\ l orP oft c'n tlwy
d

not havr anal~·tir solut10ns and arc' ~oh· d nunwrically (.J ockPL 2012) . T lw main

advantage of1L'ing a th oreticalmodc>l ovPt a statisllC'almodPl is that. tlH' mH lPrlying
m

h·:mi -m that th former model1s hasrd on doC's not chang' from place to placf' or,

m

th r v:ord. it i · "intellectual!~· transportahl ,. (.'lwnk and

ranklin. 2001) . Sta-

tistical model , how v r. are based on "'mpirical relat ionsh1ps deri vc'd from the past
timement i

ries data and th statistical rrlation ma~· not hold wPll heumse the environo variable. For example. th El Tiiio vent of 19 2

3 could not lw prC'clictC'd

well becau · at that time stati 'ti a l models w re usecl to predict the seasonal climate.
Z hiak and Cane (19 7) clevel peel a th oretical model for E T'0. appl:ving the ba,sic
equati ons
and t h

f physics such a

rev.,;ton's laws of motion. the laws of thermodynamics

ontinuity equation to demonstrate the air-sea intera tions in the tr pic:al Pa-

cific. T h m odel d moust rat ed t he prospects of m aking a long-term sJat-ioual fon: cast
(Webster and P almer , 199 7).

Modern weather fore asts use num rical weat h r predi ction ( T\i\TP ) mo leis. which are
built up on t h e g neral prin cipl

of fl uid d yn amics an l t hermodyn amics. T he earth-

syst m m odels conta in p artial cliff rential equat ions t h at gov r n t he geop hysical fluid
syst ems. These p ar tial diiT r nLial equation do not h m·e an a h ·ti solutions ancl thcv

4

res lv d nnmrri ally (Jo k l. 2012) to ohtam futnr
condition clPscrihing th

statr. starting from an mitwl

rnrrrnt stat' of th sY"'trm (\\'ang rt al .. 2000). B fmr we'

obtain the solutiOn. this 1111tial conditiOn nmst lw prondPrl. which . along with tbe
equations in the models. rontrob thC' C'\·olntwn of tlw :-;olution t Ul.JE'Clory m space and
t im e.

The spnsitiv d pendC'nc on lw initial conditwn (Lm<'nl . Hl ·:3) ran lJp SC'C'n in FJgur

1.1 in the model solution of th

(Lor nz. 1
Z). th

Lmt'll7- . mode·!

\\'lwn tlw LorC'llz-G:3 model

) is mitiated from arlnt1auh· cloo.;(' altc·rnatiVC' mitial coll(llt10ns (X. Y .

model solution: sprPad cmt from ·ach other aftpr a cPrtaiu llttmhn of stpps.

If we ronsidrr tlw mod 1 solut1o11. tMtC'd ftom thr initial condition(:> 0 . !J.O ..J.O) as
th

s

true stat . then the closer the othrr nntial cowlitwns an· to (!J. O. 5.0 .. 0) . the·

longer it tak s for thrir soluti ns to :pr ad awav from the lnw stalP.

his indicatE's

th importan e of I r paring thP initial conditions with high qnalit .Y and accurac.v to
pr diet th futur

tate of a haot ic s~·stcm such a: w at hc•r. Data assim1lat ion hPlps

to prepare high q mlity initial condition . m rging the information from tlw modd as
well as the ob

rvation. (\\'ang et aL 2000: Kalnay. 2003).

Earth-system model

simulat

climate in detail. hut at thP same time they have

limit ations too. The uncertainti . in th model I recliction can arise from the inherent nonlinearity of the atmospheric sy tem that makes t h
1963, 2006). in ompl te 1 hy i

system cha ti · (Lorenz.

( .g., cloud phy ics) , errors in sp cifying th bound-

a ry condition or due to th errors in the initial condition (Stott and K et tl borough,
2002).

Both the mod ls and observations ontain errors. Som of the model limit at ions an'
errors in mod lling th dynami s of the earth-system , lineariza tion . errors in numerical a r proximations and errors in the paramet erization of the su h-grid scale ph~·s ical
5

--------

I

20

-

15

I

X=5 00 , Y:.5 00 . Z=5 00
X=5 01, Y=-5 01 Z=-5 01
X=505.Y= 505 :Z= 5 05
X:5 20 .Y= 5 20 ;Z= 5 20

10

\

5

I
I

)(

0

I

I
I

-5

I

- 10

I
I

-15

I
I

-20

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time steps

--Fignrc 1.1: The dt•p<' n< knc<' on t lw i m t 1a 1 t'OIH h t ion for t lw Lon·n;- ·:J 111< H k I. H <'ll '
four scri o of X -coorcli nat<' of t lw Lorl'llZ- ·:J Ill<" 1e 1 d 1wrJ\<' nHil k<'< lh· owr t llll <' from

~mall diffPn'nC '~ in lw initial concbtwn .
procPsses. For applications such ao wm t h<' l fon•c,"' t i11!\ alH I dim at<' 1" c<hct 1011. 1t IS
not po:sihl

to haw ohs<•rYa tions of all the y,nial1k at all the nu "II' I !(ll< 1 poin h.

The oboen·ation: are limitPd in spal'<' aJl(\ time. :\lon•owL tlw stat<' has to IH' <•stimat ed from t h available ob~erva t ion. .

ourcc'~ of ini t inl-concli t ion f'rror~ arP 1wnt ns<>

of f wer o hserYa lions than the numher of nw<l<' 1 grid point o in hot h spatia 1 <1PnsitY
and tPmporal frequrnc~· for all the varialles. nrors in the clat<1 (instrnnwntal enor~
and r preO<·n tat i yen ess c rro rs) . crro rs in q nal it,· con t ro I. nro rs in o h J<'<·ti w anal, ·sis
anl missing varialles (T<·ixc·irn c•t nl.. 20()/ ). Tlw limit ations in tlw accurate' ckpiclion of t lw at mospheric stat<' can lw attr ihn ted to im p<•rfrct mo< k Is . im] ll'rf<'d and
inaccurat r obsPrvations ancl the inherPnt chaotic nat nn of tlH s~·strm.
In short . becm.1sc> of the' C'rrors in the clyuamicalmocld and obsc'n·ntion~ tlH'r<' \:-; n nc><'d

t o 1 c la n

th inf rmation ft om t h m p ro p rlv.

estima ting the s t at

ptnna l rs1Ima t 10n IS th r pro Pss of

f a ~)·s t m with mmimal r n or. hy com b inin g o bse rva ti ons a nd

a m a th em a ti calmod rl \Ylt h t h I I rrsprc(J\'P stat ist ical nnc<'tl aint .v.

lw a ppllca twu s

of opt imal ('S timaJ ion r a npps fi 0111 \\"('at lwr fm ('( r\Stmg ( () artifi ial int d h ,!!,('lH'('. Th
m os t comm mly nsrd m t hods fm opt 1m ill ('S( 1mat ion mP t h' nwt hod of !C'ast sqn a rPs.
tl1P m aximum hkc>hhoo cl and tlw r cnrsl\'P Bm·r~ian f'..,I Jmation . In Pa r t h sciPllCP. thf'
purpose of opt im al Psti m a t ion is to providP hPst <'stimaiC's of l lH' stc1tP of tlw OC('Hl1
or a tm osph re .

h s prov1d<' 1JC'I(P r <'siimatp · than can 1H' ohtain<'d hy nsing so h,ly

bsern1 tiona l d ata or mtmc'ricalmodf'l~

In Pc-lrth ~(H'IH ' P. optnual Pslinw ti ou of th

st a t e of t h oc em >r at mosph 'r<' 1. known clS data as~mul(1 t H m

W' of I h' main goa h-.

of d a t a as. imila t ion in ' \ \'P 1. to pwnde t hC' initial ('OlHll t 10n~ fm wc·al h PJ prPdict ion
m od els.

1.3

Data assimilation

D a t a assimilati n is used t o op t imall)· stim ate t he st at e of a physical-system using
imp rfect num ri a l m od ls a ncl n oisy obs r vat ion:. In gen eral. a d a t a-a,s similation
syst em con sist s of three p rt. : a m a th em at i ·a l m od el. a . t of obs(' rvation s and a
d at a-a. simila tion m etho l (R obinson a nd Lermusiau x. 200 2). The assimilati o n

f the

obs r vation int o th num rical m o del h elp t o adj n t th d a t a dyn amically, improving
th quality
mical syst
. of the estima t ed st a te . The m ath m a tical st a te of the dyna
.
. em
a re describ ed by as t of va ri a b les kn own as th st at e vc:tria hles . The dynami cs lin k t h
ol ervation s with oth r st at

vari c 1 les and influence them . The estimat d . tate or

n alysis an b e u s cl as the initial conditions for the for cast . The m ain a pplicat ion s
of lat a assimila tion include: (i) estimating th

7

urrent s t at e of the a tmosp h('ric nnd

o

amc vanabl s. \Vhtch an l

ns o as the mttwl conottwn fm th

fnt11rc forecast;

ano (ii) obtaining a goon tra.Jc tm\ of the past dtm(\tP hv tt. r of all thr ol srrvations
available.

Tumcncal WC'athcr 1 rPrlJ (IOlllS c111 initial-\·alnC' prohlC'm . Data HSSllllllation plodllC'C'S
an am1h·sis. ,,·htch 1s a maximnm-hkPllhood Pst irunt c of atmosplwr1c and oc'<'<mic stat <'S
and srrvps as opt nnalm1tial condition. to in it ralll<' t lw nnmPncalmodPls. Da I a assimilation an also h used fm pnramrt Pr <'" t llll<l t ion ( 0 1 ·Pill. :200 1) In t hP PHd v days of

1\\\'P. the computational cost to lH' paid for mwlysis \\'cl.s lltPlf'\'clllt in co11ttast to thP

c st of a 2-1-h ur forC'ca.' t. Hm,·c'V<'l. ,,·it h cldvancPd comptltaticmnl faciliiH'S . highly
lev loped data-assimilation nwt hods. cllld a va~ t ll<'l work of o l>sPrva ious. I lw f'OS t
of th

com1 utati m · rPquirC'd h~· t lw assuntlat JOn 1.' t vpH·allv t hf' cost of a 2 -hom

[ re ast (Talagrand. 1

- : \\'ang et al.. 2000). In ocPanographv. data assimil(ltion

ha be ome an important to 1 for st nd~·ing the ocean ·ircnlatwn .

'omparc>cl to th('

atmosph re we hav mu ·h [ wer observ tion.' m the ocean. and also t hr dirPct nwaurem nts can be d ifficult and exp n. ive in th o f'an. Data assimilation combinc>s
o ean model with ol erYation . with th rPsults. callc->d tllP analysis. providing morC'
complete and accurate information about the o ·ean compared to \vhat we get from
model simu lation or ol servations alone (Talagrand. 1997: Fukumori. 2012).

Data assimilat ion is a mathematically rigorous process in whi ·h the or timal stat

of

t h syst m ise timated as a weighted com! ination of model for casts and obs rYations.
T he weight. a r

d t er m ined according to th

estimated uncert ainti s of both the

m odel forecast a nd obser vation data. In g neral. th
assim ilation can 1 e grou1 eel into two typ es: (i) t h
T alagr and , 19 6;

meth dologies used for data

variational meth cl (DinlPt and

ourtier eL a l. . 1994, 199 ); a nd (ii) th sequ ,ntial method (K alman.

1960; K alman and Bucy, 1961; K alnay, 2003; Auroux an d Blum , 200 ).

h variational m thad i, bet don optlmal-controlthror)·.

h most popular m tho l

in this approach 1, thr adjoint mrthod. It s mam ohjrct1vr 1st
tion that mcasUH'S tll<' m1::-.fit IH't\\.l'l'll thl'
ob, n·ations (Dim t and
assimilation \'l·indov,·.

alagu-md. 1

mimm1/jr a rost fnnr-

<'llllHH(d s<'<!ll<'lH <' of 111odd :-.tatcs aw l

. Tnlagraml. 2012) avmlal>l

J\'C'r

a giVC'll

he popnlar \'C.Uia t wna1 mPt hods nsPd iu T\ \ 'P ar<' t hrPC'- and

four-dinwnsional \'ariatwnal assmulat1on. ltsnalh· dPnntPd as ~3 1 -Var and ...J:D- ar. r<'sprciively. In the nuiationc l assmul(l(Ion m thod. thP information C'ontanwd in thr
data arr I ropagat eel fonvmd 111 t llllC' t hrongh t hr nonlin('Hr rnodf'l (fon•cast modd)
and 1 ropagatrd backward m tim' though tlw tangPnl lmcar modd ( L:\1) ( alagrand.
2012). whi ·his compntationally \"Pn· PXJWnsln'

h

3D- \'a r m t hod has 1H'en nsPd in many op<'ra t wnal W('cl t lH•r-prC'Clict wn C'Pll t rC's.

In 3D-Var. th for ·a-"t and obsrrvatwn 'rror: arC' l1 ·at('(l as tim(•-mvariant and Gaussian. des ri bed h~· stat ionar.\· ~rror-covariancrs (Talagrancl. 1097). Th<· mam limit at 1011
of the 3D-Var i,' that it do ,' not consiclrr the dynaruicall.v clC'pPndPnt errors of the·
nonlin ar cl~·nami . and it as~ umes that all oh ·ervations are made at t lw anal.vsis t inw.
The 4D-Var. c n. id reel as an advanced vrrsion of the>. D-Var. seeks th init1nl comlition that leads to a for ca t fitting b st "·ith the obsen·ations in a givC'n assimilation
window. In 4D-Var. the backgr und stat is compar d to the observation at the exact
tim

of th

observation in the assimilation window and the ha ·kground-covarianc

matrix is implicitly evolv d within the given a, similation window.

Th

nunmuza-

tion of th' 4D-Var cost function. which m ca::mrcs the misfit between the temporal
scx]UC'ncr of modd . t atrs and o hsC'rvations, rrquirC's finding t hr p;rRcliPnt using t lw
adjoint mod 1 (Yang et al., 2007). Th main disadvanta ge of 4D-Var in m teorological and oceanographkal a1 plicatim1s is the om1lexity in the c mputations of the
adjoint of Lhe assimilating mocld an l various observation OJ rat ors, their valid at ions
an l maintenance (Talagran<i, 2012).

9

h . qn nt1al m t hod 1s l as d on slnna t 1011 them v. for xample. l h 1 alman-hased
filte t

(K alman. 1

200 ).

0: l 'alman and Bn Y. 1

<'qnrntial-assimilation m thods prm·Hi

1: Kalnav. 200.:

nronx and Blmn,

th r.stimatP of th stat

of tllC' sys-

t m sequ ntially n.ing a prohahihstH' framewmk. <md tlwy also ptm·Idc' tlw estimalr
of th unrertaint\' a~soc1e1t d \\'Ith th stat

npdatP ,'pqJH'ntial-assmulatwnmPthocls

are C'a.s y to implement because there Js no lH'C'd 1o dC'n\'(' t lH' ad.Jmnt modd . TlH'sP
method, estimat th stater cur"!\. ]y m t\Yo s Pps. fhP fi1~t s tc'p IS tlw fml'cast stPp
to get th prior-c'rror statistics . and tlw second siPp IS tlw npdaiC' step tha mcorporates the observation mformatwn and updatP the pli<H distnlmtJou and pn>vH[(' tlw
post rior-c'tTor statL ·tics (B rtino PIal. ~OCr ) .
I\'

nr of the mam fpalnn' of th recnr-

, timator. such as the Kalmcln filt<'r is t lu1 tlwv '"'flti'·>fv the :viarkov propC'rt:v. it

implies that futur

state of the S)'S(Pm is ~olPly based m1tlw pH'sC'nl stale and arP

ind p ncl nt of the past stat ,' ( mhadan. 201. ) .

Optimal interpolati n ( I) is a fairl)· simpl sPquential data-assimilation mC'thocl ('Ommonly used for operational r\YP (Dale)·. 1991: Kalna:v. 2003). Optimal intPrpolation
is a minimum varian e estimator. In OI. similar to . D- Var. the backgronnd covarianc u d is constant in tim and i. obtained fr m a long-term stat is tical mean . One
of th main disadvantage of OI is that the tationary 1 ackground-covariance should
represent the variability in th whole assimilation domain at all tim (Bertino et al..
2003).

T he K alma n filter is on e of t h m ain sequential data-assim ilation methods. In Kalman
filt rs, t h goal is to fi nd t h e best linear unbiased e. timate (BLLJE). which i a sum d
as t h e linear combina tions of prior estim ate and obs rva t ions. \\'hen the model is
lin ar nd rr rs are assum d to b e Gaussian , I alman fi lters give the optimal solution
(Bert ino ct. al. , 2003). For a weakl)·-nonlinear sy Lem . K alma n filt rs can perform

10

qmt

'"' ll l ut for a highly n nlinrcr mod 'l thr solutwn of thr K alman

not

1 timal ( li~·oslu ....... 00.5)

lltrr is

The adnmc<'d derrYntrY<'~ of 1\.almcm filtc'l~ han' hc<'n

d " lo1 co for tlwir apphcat wn m nonlm m st ima t ion ~uch a~ t h Pnscmhlr I -ahmm
fil t C'r (EnKF) (

\Tll.

n. 1

-! ). "·h1rh usrs ), l ontc' 'arlo samplmg to estimfiiP tlw prior-

Prror. tcrti . tirs . ~ f ore infmm<1 ion ahn11t th
ar giYc'n in

n altrrnnt

e twn 1 1 and

K <tlm , n filtc'J and its ndwmrc'd vari<wts

lwptc>r ~·

mrthod i~ nsmg partid filtNs (Yan LPc'll\\'Pll awl ~ ,·c·nsPtL 1 DG :

son ami . ndrrson. 1

. ),hllc>r Pt al . 1

lH lrr-

.. Krnnan. :wrn: \'all Lc'P II\\'P n . 200. ) . TlH'

main ndnllltagc of palt1 lc hltc·r~ 1~ that th<'Y ai<' : ·.tutal)l< · <'\'<'11 wll<'n the· dywunics
nn' nonlinrar r nd thr nror statr-, w: nu' non- ; ,ll h~I;m

In partH }C' filtc'rs thf'rc' is

no n rC'd to linrnrizC' rlw nonlinr<u mndd

In pnrticlc' filtc'I"' . the mode'! pmhahilit:v-

d n ity functi n is reprrsentecl h.\' a cllscret

set of C'nsPmhlc'~ callC'd .. partidps". TbC'

probal ility-o n._ it~· function is rstimatPcl hy propagating thC' partH'lc·s forward in timr'
through th n nlin arm d 1. \\"h n the ohsen·ation!-. are a\·ailahlP the· partidPs an· rrampled incorporating th ol s n·ation information (Ya n LPrnwen. 2000). In a particlf'
filter. a finite numl er of parti le will lead to filter clegenrracy ( t h
size r clue

rffectiv pal ticlr

to a very small numb r of partid . ) and the filt r fails. Howcvrr . a large

number of par ticl sis computational!~, v r.v expensive c ncl thi. limits the a1 plication
of p art icle filt er in geophysical- ystems (van Leeuwen. 2012).

Sequentia l data-a imilation m t ho ds l ased on ensemble methods are becoming
mor

p op ular over t h

Yariat ion a l data- ass imilation m thocl

bee a n c of t h eir flow-

d ependen t (d y n a mically ch anging inst ead of a stationary) stru cture. n

adjoint nor

TL:Ms a r e r equired , th ir high r p otential for parallelizat ion and th y ar eas)· to code
(H a mill , 2006 ). In th s quenha l algori t hms the stat e can be up latrcl whenever the
observation s a re avail a bl and n ot r equired t o r est a r t t h e assimilat ion )' lc.

11

1.4

M tiv ti n

The l alman filt rr

ncl iL d ri ,·a l j,·

han hr n \nddv , pplird in rllmosphcric and

occamc sc1 n s m rrcrnt \'C'cHS ( \'C'ns 11. :._()(1:1.
Drng ancl

·Prg 'r. 2001: RohPrl PI al.. 200 :

ang. 200 : ~ \'Pll~<'n. ~DO ). ThP sL-mdMd Kalman Fill<'l (1\almcm. 1 GO)

is a lin m an8lys1s :y::.t m ha~ don Bm·p< thC'CHPm c-md gl\'Ps tlw BLl' ~ making n sP

of 1ast obser\'ation. and clnuumc. of the' modPl. TlH' application of thP stauda rd
Kalman filt r 1. not pract1c,1l fm \\\'P modp]c.., hPCclll~f' of it. lnrgP lllllllh 1 of dPgH'Ps
of fr

d m and non lin ant\' (. l1 \·o. lu . :._()().'). , m hadclll and

aug. 20()0).

Th ex t n led I alnwn fill r (EKF ) is an c'xpall'-llOll of t lw Kalmc-tu fill <'L in whjch
lin arization i. rmrlo~·~d to approximat
tional Kalman filt 'r equo.tiou.
The main drawback

f

nonlinem systPm~. In thP

ell<' applit'd afte1

KF i.- th

KF . the tradi-

lin'auzi11g all the wmliiH'Cll uwd<'ls.

rcquirC'ment of a .Ja cobian or t lH' TL.0.1 for t hC'

linrarizntion of t h<' nonlinrm modrls. "·hich i. rxtl lllC'lv diffi('nlt to impknlC'nt in a
large-dim nsional sy tem ( .J uli r ann .hlmann. 1 07: Ambadan and Tang. 2009). l11
addition the EKF r tain only th fir t-01 dcr statistical moments in evaluat ing Prror
statistics, often leading to underestimation in rror covariance and subsequently to
ub-optimal fil ering (Evensen. 1992: Gauthier et al.. 1993: " lillcr et al.. 1994).

In t ad, the EnKF, prop sed by Evensen (1994:), estimates th error stati.'tic.- using
nsembles generated 1 y int egrating the nonlinear m d 1 (Ev nsen, 1094:: Houtekamcr
and Mitchell , 199 ).

nlike EKF. EnKF n ith r requires lin arizati n of the cl~·nam­

i al model nor negl cis th c ntril ution from higher-ord r t rms to error statistics
(Ev ns n , 1997). EnKF and its clerivativ s hav 1 ccom p w rful and widel~· used
meth ds within Lhe cla.La-assimilation community lue t.o th ir algorithmic simplicity.

12

the relat.i e ea~ of implemcntatwn cls well as ~Ifmdahl<> compntational cost ( nclcrson. 2001: \Yhitaker and Ham1lL 2002:

\'C'll"'

n. 200. ,

n Pxt ensrvP re\'lPW of \ h h t rat u r ' on

i u. 200 ) .

h n, nd Sn)·dpr. 2007. Lr and
n l · F can h

found in ~ vc'nsPn

(200 . 2009)'

o avoid thP nnrlC'r- stnnation of forPcasl-<'llOI co\'<lllrtllcc• elm• to tlw dc>nc·asc• lll ensemble> sprc>act th c hs n·atrons <ll'C' oftPn JH'ltllllH·d 111 tlw standard ·nl\F (B wgc>rs
et al.. 1

: Houtdcalll 'l and ~IJtclwll. 1 c ) Ho\\'P\'C'l. \\'hitakPr and Hannll (20 02 )

found t hal t h

1wrt mlwcl ohsrn·at lOlls can net as cUI r~ddrt mnal -,onn·c· of samplmg

error. making the c>stimatP of mwh·..,1. -c·nm covar iancc• ]pss cH cnrai P. ThP\' dPvPlopf'd

tlH' ('ll.C'lllhlC' :-;qnarC'-[(JOt fi!tC'l (En TIF ) to cl\'Oid tlH· pPrtmhation of nl>sc•n'rltions.
which u.'e' the traditional Kalman gai n to updatP thP rnsc•mhlP mean and tlw reformulated Kalman gain to upclat e t hP dc'\'Ja t wn from t h nwan. For t b ' samP PnsPmhlP
ize. En RF can yi ld lwtt r results than tlw
ti nal exp ns

nl\F withm1t any additional compnta-

(\\'hitaker and Hamill. 2002: Tippett

t al.. 200. ). ThC're arf' diffr'lPnt

formulation of the quare-root filter that hav been p1 oposc>d including an ensPmblf'
adjustment Kalman filter (EAKF ) ( nd r:::;on. 2001). an ensembl transform Kalman
filt r (ETKF) (Bi hop

t aL 2001) apart from the En RF (\\.hitaker and Hamill.

2002). It i not clear if there i any partie 1lar formul a tion of t h

. quare'-root filtPr

that is better than the others (Tir pett et al.. 2003).

A main challeng in EnKF and EnSRF is choosing an ap1 ropriat
small ns mble siz ofien results in ina ·curat

representation of th

ens mble size.
rr r-cm·anan ·

matrix whereas a large ens mble size is not computationally feasil l
dimensional sy terns. Therefore, s m
how can we g nerat

specifi

for high-

questions are likely to arise such as

with a greater degree of ac ·uracy. th

finite samples for the

optimal estimate of pr diction-error covarianc ? Furthermor , what is t h - percent agp

13

of th

. timat l rror van a n

th

tn1

c unt 1part for a givPn rnsPmhlr

m t hod. em rent h· usC'cl fm genPI a t m g pns mhlr m mh ers m t h s tandard

h
~ ni\

r le t rl. t

are un a ble to a n s wrr thrs qu rs t w n s ().I a n oj Pt ctl.. 201 : T a ng ct a l.,

not her con
fun ctions.

wh nth

111 in

th

tra dition a l

s ar gn ed 111 T a ng a nd

nl"

1 ~ 1b

01-±a ).

t n 'a tnw nt of nonlnwa r mPasurc'm ent

mhadan (~ 00 ) (l S wdl a.s

mra.s urr m cn! fun ct iO n 1~ non lnwm. tlw C' ll11 P ll1

n l\

ang PI a l. (2 014a ).
a lgonthm cont a ins

an implicit assumption . th ' fo recas t of tlw m n~ n rc' nH ' lll fnn c! ion is unlna srd or the'
m ean of t h r forrca.s t r qu a l. t o t h r fou 'ca.., t of t hP llH'an. Tlw Hnplwit ass tmlptwn
ould

au s

errors in ps tmwting th

K ellman gain ( ang a nd

mh ad a n . 200!) :

ang

et al.. 201.±·1).

R e ntly.

mba d a n a nd T a n g (2 00 ) as wdl a:-. Lno a nd ::\ Joroz (2 009 ) int rodn C'c'd a

nrw rn. rmhl e-hast>rl filt r r t o th P fi Pld of Pnrtb . ci n rrc, nsin g tlw si m p!P Lm f' llZ s~·s tf'DJ ,

whi hi

c ll ed th e .igma-p oint 1\:a lm a n filt 1 (. PKF ). Th

. PKFs an ' a d Pri vative-]pss

K alma n filt er optima l for n onlincc r . t a t -sp ace est im ati on (Jnli Pr et al.. 1995: .Jnlier
a nd

hlm ann. 2002 . 200-±: Va n ler ).lerwe Pt al. . 2004 ). In . 'PKF the error st a tisti cs

and mod 1 . t at e a re an al~·zed using th

tr a nsf rma tion of a d et erminis ticall.v ch o-

en minimal set of weight ed sample p oint s called sigm a p oint s. which ca pt urc's thr
tru e m ean and cova ria nce of the prior random Ya riable com1 let el.v and t h e p os t erior
m ean and covariance to the e ond order a nd up t o third order for G au.-sian input s
(Julier et al. , 1995 : Juli r , 2002; Va n cler T\1erwe . 2004 ). The

PKF famil~· inclu l s

sigma-poi11t unscented K alman f-ilter (SPUKF ) lJasccl on ::-;calccl unscented tra nsformation (Juller et al. , 1995 ; Julier , 2002) , sigm a-point rC'ntrn,l diffrr ncr K rtlman filtrr
(SPCDKF) based on Stirlings interpolation formulas (N0rgaanl rt al.. 199 . 2000;
Ito and Xiong 2000) and th ir sqnar -root variant s (Van ler T\Ierwe a nd \\'an . 2001:
Vander T\1erw , 2004). This tudy is only confined to the SPUI\:F .

14

hallcng m implem nting PK
omputational xp ns
an

for a rrahsttr high-dun ns1 mal s)·strmts th larg

( an and You . ... ()() )

or a high -dnnrns1 m a l S)"S tPm snch as

an or atmosphe1r global circula tiOn modrl ( ~ ·;..r). it IS n o t compntationallv

f as ibl t

satiSf)· th rrqmrrm nt th a t th nmuh Pt of s1g m H pmnt s should h r grC'atPr

than twir thr numbrr of svs t r m s ta t c>s ( h a ndta...,<'ka r c• t a l.. 200 . H a n a nd Li . 200 :
mhadan and

a n g. 200 : Lu o a nd ;.. rorn 7. 200 . ). Fm nn pl Pmc'nling SP KF . thc> s tat<'

vrrtor i, r clefinrd through sta tr a u gnH' llt a t ion b)· <<>JH<\I C' na tm g tlH' modd s t at s,
pro cess n oisr and obsc>r n 1t1 m nm~P.

lu~ ,,.111 f111 t lw1 11H' l< 'c1S(' tlw ll1lllilH't of sigma

point. an l m ak

P l\F nn ~ uit a l>l e fm t ra li:t JC' < lnna t P m o dc· l ~. \\"hc•n tlw p1o cps~

is a ldi t i ,.

the comput a ti o n a l co mpl Pxi t)· f'an lw r<'dll ('<•d hv nsm g t}H· non-

no is

augment d st at
ar

\ "E'

·tor (\ "a nd ' r ).Icrwr cmd \Yc1n .•JlOl: Fa n a nd ) "e m . 200 ).

lwrP

om rrdu c d-ord r sigm a-p oint nwthocb usin g snnpl x sigm a p oint s (.J nli Pr. 200 2.

200 ) . whi h can d eer a

sigm a p oint s t o L

1 for an L dinwn~wn a l svs t Pm . This

numb r i. still comput a tion all:v inf asibl .

R ecent!)·. Amba d a n and T a n g (2009) su ggcs tcct a snbs pan- proj ct wn a pproach for tlw
dim n ·iona l compr

ion l )" th a pplica tion of tnmca t ecl princip a l-comp on C'nt an alysis

(T P CA ) on the multidimension a l sigm a-p oint sp ace. In this m ethod th P sigm a p oint s
in the princip al-compon nt space. v,·hich will r t a in the m a in feat urc's of th origin al
sigm a-point p a e, a re used to a p r roxima t e the error prop agati on . Luo and ). Ioroz
(2009 ) suggest ed tha t a sub p a e a pproach b ased on singula r-value d eromposit ion
(SVD ) can b e appli d on the

rror- ovariance matrix to red u ce th

sigm a point s.

Anoth er meth od fo r rank-red uction is t h C h lesky b as d d composit ion of th r rrorcovaria n ce m atrix a nd subsequ ent rank-reduct ion of t he squar -root m atrix (St ev:art ,
199 · Ch andr as ka r et al. , 200 ) .

T h e ensembl formulation of t he l\ a lman filtr r and it s , ·ari a nts h as b en an actiYe a r a

15

of r s arch in data-a~:-.1mila t ion probl'm~ lll gc'o~c H'lH <' b 't a w.;c' of it:-. flow- l 'pcudc'lll
rr r structur

and ra."

of im1l m ntation .

hP mam di:-.c-Hh·antagc of adwmcPd

I nlmau hltcL hkc 'nKF cUld En RF 1s that SPkctiou of th<' uullcd c·usc·mblc slZ<' lS
not detrrmimstic and th' rPqmn'mc•nt of a grc·atPI sJ/P of Pnsc·mhks forth' cHTnralC'

a1 proximal10n of th m 'cUl and \'Clllc nc 'of th

}HHH' and postPllOl

chstrihttllOll

h'

._PI<F is a cl t rmmist1c dPII\'<lltv -lPss E ,dman flltr'l optimal for nonlinC'ar statC'space cstnnatwn pro hi m .

. PKF has lH'C' ll a p Pat :-;\tC'CPss in t lw fi(']d

c. timation in man~· m ns of c·nglll<'<'llllg . PK

of optimal

h,ts a gi<'al pntPnt1c1.l in the fiPld of

o ean graphy anclmctcorolog_v But tlw largP compttlcllional PX{H'll'-i lws IH'Pll a main
chall nge in HPI l~·ing t lw

PK

to a high-dmwu wnal , \·st PilL

hP rc·qwrPm nt that

the mnnl <'r of sioma points . hould h' grPat 'l thau (\\'H 'C' thC' umuhPr of svstc·m state·s
i ' a disaclvantag for it: apphcatwnm fC'ahsti · climatC' wodds . Tlns lH'C'PssialPs the·
r quirement f d vcloping a 1 racti ·ally implt-m ntahl' a1 proxnnatwn f n , 'PKF.

1.5

Objectives

G iv n th

computat i nal constraint. of applying a full-rank

PCKF for clat a-

assimilation problems in oceanography and m teorology, the data-assimilation comm unity is quite interested in the develor m nt of a reduced-rank

PCK F meth l for

data assimilation. I address t hi import ant issue by developing an ad vancecl

PK F

data-assim ilat ion m ethod . Specifically, I:

1. invesLigat e two m t hods to construct a redu ced- rank. sigma-point unscentrd
Kalman filt er (RRSP K F);

2. develop a localization s ·h em e a nd s tudy it s impacts on t h e perfonnan C' of

16

RR Pl.K ; and

implcmrnt t h R R.._ r r 1\F lll a l Pahc..,t ](' dima ((' mod d . which IS. ( () lllV knowledge. thr first trial in the data-a'-'">llllllation c·omm11111tv

To ill\'C:-.t Jgatc the hr:-.t ohjt•ctJ\'C 1 proposC' two uwtlwcb of Ic\Uk rC'dt l dlO!l~
method~

(

of rank rPdnct ion Pmplov tlw t IlllH 'r\ t Pel singular-vahw dC'composit ion

\ TD) to factonze th cm·miamr lll<\ttrx cmd n·duC'<' it::; 1ank th1ough tnlllf'H(JOll.

In t h
th

Both

firs

lw Pll 01 -cm·cu i c\lH c• m;\tux ca leu! n t ('cl 111

met hod. , \ ·D 1. a pp li d on

data s1 ace> (RR ' PC K F ( D )) \\·hil • m t lw "'c>c·oud mc·t hod t hP S\'

1~ apphPd on

the error-e varian ·' matnx caknh1tc>d m th(' c·n~<>mhlc· :-.pace· (RRSPl'KF ( )).

lw

r duced-rank. squar -root mat nx is nsrcl to sPlPct t lH' most import aut sigma poir1t s
t hat can retain th

main statistical fc·HtnrPs of tlw ongmal s1gma pmnt s

Oh-;c•n'a-

tion simulation xperiment.· (o r prrft>d moclrl rxpNinwnt ~) arc• p ·r fonued nsing t hP

L r nz-96 mod l.

T o achieve th
RRSP

se

KF and t

nd objective. I 1 ropo. e a h~· brid-localization sch •me for th
t using th

Lor nz- 6 model (Lorc>nz and Emc-muf'l. 190 ) . The

exper iments a r e conclucte l in the 1 erfect model setup. ~ I an~· sensitivity experiments
are con d u cted with varying parameters to under tand how and why changes in the filter param ter s ( n

m l le size. covarian e lo a lization. a nd covariance inflation ) aff ct

t h e qu a lity of the anal:vsis.

The Lhird objective is r ealiz cl by implem ent ing t h

RRSP C KFs to a realist i · climate

m od el. The r ealistic climate model emp loye l in this st u dy is t h e LD 05 (LamontD h erty E a rth Obs rvat or y, v rsion 5) ver sion o f the Zebiak-Cane (Z ) mo d el. which
1s a n intermediat e-complexit y

oupled E

S O 1 redi c tion m o d el.

om e important

1 rop r t i sand t h e s LimaLion accuracy of t.h RRS P U KF a re explored . Th p roperties

17

of th RR P TK ~ ar also ompm d \nth c fnll-rank '- PC l\

1.6

and

nSR ~

Outlin

This dissrrtatwn addrP~. '!--a jHd( tical clpphc ell lOll of .' I K

dP\'doping two rPdmnl-

rank RPlroxunation: of ,' P l 'F <-llld mtwdndng alocalllrlliou ~dwnw for tlw SPK ·
Th reel ur 'cl-rank . Pl\F~ arP appliPd l o a l'Palist H <lima t

mod!•\. Thf' org<mizH l lOll

of this dissertation JS a. follow-.,.

hapte1 :2 co11taiw-. a brid tlH'Ol<'l Hal 1"<'\'J('\\. of dif!Pn·ut d<1l a-<-Js:-.uuilat ion 111<'1 lwcb
and the introduction to th modPb 11!--><'d m tlu .. twh·. In ,'('clwn 2.2 tlH· founnlations
of

n K . En RF am-i

En RF . t h

m t hod

P Tl\F are discus!-> 'd in a -.,plf coni airwd m;-muC'r. In 'uKF and
f gen rat ion of nsrmblC' ~iz ' i: not d('l Prmi11isl ic in a ngorons

tatistical ense. In contra. t.
ation.

P TK

prPsC'uts a dC>t rminist1c way of c·nsf'mblP gPrH'r-

c tion 2.3 contains c hri f introduction to th Lorrn z-0 (Lorenz . 1006: Lorenz

and Emanuel. 199 ) moclel as w 11 as the Z

modC'l (Zcbiak and

'ane. 19 7).

C ha pter 3 aclclresse · the first ol jectiYe and discuss s t h fmnmla t ion of t lw rl:'cl uc<'drauk sigma-p oint un ·ccut 'UK· hmm fi lt er · ancl a rcclucccl-Hmk sigma-p oint UllSCClltccl
K alma n filt er b asPd on thP data. pace' (RR SPPKF (D)) is proposC'd c:md d<'rivPd. R <mk
r du tion employs th TSVD to factorize the c variance matrix and re lu ·e iLs rank
through truncation. S ction 3.3 describes the th or etical f rmulation of re luce l-rank
igm -point un ented K alma n fi]t r based on the ensemble space (RR PlTKF (E)).
T SVD is mployed for rank redu ti n her as ,~·ell . Before going to the ar plication of
RRSPUKFs for a r alis tic climate m d el,

ec tion

A clesc rib es ih" data-assimilation

x p eriments with the Loren z-96 model in th e p rfe t m od el assumption to get a basi

1

i<i n of th pPtformanc' of both tlw HH .'P ' J\f..,
In 'hapt ('l'

.

I pwpo..,p and cll"i( \lSS t IH' b~·hrid-localizfl t iclll ..,c}H'llH' fm H nsrr I\

Tlus chapt •r addtPs"'c" "'1Hh is~IH'~ as iulnPPdlll~ hltPJ dl\<'IgPW<' ,md ~ puriou s C'or-

r<'latiOll and how to imprm·<' th JH'tfmnH1!H<' of lw fi!tPr h~ Plllplovi ng lo<<dizntwll
and inflc1 t 1011 :clwnH'S.

Tlw imp!PllH'lll (It ion dPt nls of coVilllc11H P loc;diz;JtJon <llld

c<)\'miancP inflatinu i~ d<'snil><•d in, c'c I ion..., J : ~<nul

. l. tPSJH'C ' t i\'C'l}

~,wet ion

1.5 pro-

pos<'s nud descnl><'" tll<' tlH'Ol'Ptical formnlntion of thc•lJ\l>IHI-lcH',dJ!cllion sdu•mc• fm
HI ,' PL'l\F ( ') .<'<lion J

contains tlw mmu•nc ·;d <'XJH'lllll<'IIIs \\'ith !IH· Lon·u; DG
ft C'l' dP~<Tipt ion~ of implPIIH'lll at iou. I lH'

modC'l m t lw pPdc •c t mod<'l <lS"illlllJ>I ion .

n•sults of mult 1plc• sc•nsJ! tnt\' <'XpPl'illH'Ill s ell<' hm\·11 ,mel <'<JlllJ>il!'Pcl.

Chaptci - adclr SS<'"' mv hnal ohjPctin•. i.<'. t llf' clpplwat 1011 of R H Pl"hFs in a I(' ThP n·ali:t ic modPl 11sPd in I hi~ st11dy is tlH· Z ' nu Hlc·l for

alist 1 chmat c mod Pl.

prediction. Thi. chaptn contcUll"i mtplc·nH•n!cltiou dP!ails of HH .' J>CI\F (I <llld
E ) mPt hods m t lw Z ' mode• 1 for
pcra t ur

anomaly (

T

I\.' 0 prPc bet ion by as~nnila 1ing '-,(';-\ "m f..wP I c·m-

) delta . The Z ' modd com p11 t Ps m1 omcd I<'~ of cl t mos plH'l ic

aml ou·e:mic fields. 1dativc to a sp<·cifi<·d uwnthl~· llH'ctll dimatologY. TllC'U' IS dC't i'likd
information about the modC'l. ex1wrinwnt al s<' t 11p and implc·nH'nt at ion dC't mls. Fom

clata-a.ssimila t ion met hods (RR PCKF(D). HR.'PCKF (E). , 'PC KF aud
impl ment cl for assimilating

T

n 'R F) ar<>

into ZC modrl . ~I11ltiplP SC'llsitivit \' C'xperimc•nts

a re condu ct ed by varying the number of s1gma points. • s timation acntrac.Y and tuan~·
import a nt p1 opertics of cliff rent m t h od. a r discussed.

In

hapt r G, I summari ~ my r rs ult s with a d e tailed disc11ssion and g ive thP conclu-

sion of the present r srarch . This chapt r also provid e's som e s11ggcs t ions for f11 t urP
work. Throughout Chapt ers 2 through

. I use tlw 1s t p e rson phtral to ncknov,.Jc•dgC'

19

the cantri l utwn.

f other. 1 to lh1. work .

1 us t he 1st person 1 lura! to acknow ledge t he cont ribut ions of my su pervisor aml co-~upcrvbor
to t his work
1

20

Chapter 2

Assimilation methods and Models used
in this study

2.1

Introduction

Th p urp o

of t hi Ch apter is t g1v a th oret ical revi w of data-assimilat ion meth-

d in thi dissertation a well as to giv a brief introduct ion to t h e model. nsc>d .
The theory and deriva t ion of a Kalman filt er. EnKF . En RF and SP l'KF ar g1ven
in this chapt r , and the ad vantages of SP UKF over En RF and EnKF are also discussed . Th theory and derivation of

n RF pre. ent d her is mainl.v based on the

works by Whitaker and Hamill (2002) and Tip I tt et al. (2 003) wherea · the t h em·~·
of SPUKF is m ainly 1 asecl on the works by J ulier t al. (1995). Julier and
(1997), J ulier (2002 ), Van d r I\ Ierw (2004) and Van cler

21

hlmann

Icrw ei al. (200--! ). The

inf rmatwn al out th

mod L lvcrlm this "'tndv 1" f!; IYPn in th

last srctwn of this

hapt er.

2.2

Th

r

ti

1r

.

l

he data-assin11latwn pwi>lPm i8 an P~tmmtion prohlPin : its o!Jjpc·tivc• 1s to PstimaiP
tlw statr ofoc<'ani j ntmo...,ph'll( fi·lds In· f11..,inp. th
model.

oh..;c•I\'cltillllS with a dvn ctlllH <d

up pose• ~\t 1~ t h forpcast st nt ' of c\ dn1nmical svst Plll of cliuwusiou L at t imP

t . and ) '1 is the )hsc•n·ation at tinw /.\\·,can IC'J>IP...,c·nt thP stat •-sp cl('f> Pqnations of
th dynamical . vstem a. hPlm\· (\dwreYPl po .... IhlP. notdtH>ll. an· bm,£>d on ldP PI al.

(1 7)):

(2 1)

( 2.2)

wher f() i the nonlinear m d 1 that takes . tat

X ?--- 1 to ..\ 16 and q1 1 is I hr random

model error foll wing a Gau .. ian eli. tril uti n v:ith zero mean and covariance Q1 1 .
Further h() is the nonlinear observation or measurement function . which maps th
model varialle to observational spac . "Csually the dimension of )(t is greater than
the dimension of Yt b cause th oh 'ervation i Sl arse and irregular in spac and tim .
Furth rmor , 'rt is the observati n noi. , which is also Gaussian with mean zero and
covarianc Rt.

If we assum that

xru is th true staLe of the system th n the forecast error and

22

for "-'c ast -err r CO\ 'Rl l<:l ll ·e can Le d 'llllPd a~

b

cl

\ ·lntf'

=-- • I

'

\ -b
I

,,·hr rr the (.) is t hr stat r:-,l rcnl rxpc>ctaticm op<'l rllor.

Tlw oh:-,Pl \'Hiicm Nro r and

obsen ·a t ion-PrTor cm·axrancc> can lw d 'not Pel 1)\·

(2.5)
(2. G)

Our ohj e tivP i. to fi nd t lw np rLltr'd. tn.tP (ra]]('(] anah·s i ) x ;I from t lH' fo r Pcas t s ta i r
X tb

and obs n ·ati n }/. \\. can \\Ti t

.\;1 as (he }mra r wei~ht r d mea n of x:~ a nd

} ~:

' ' H'"")
X Ia = •'-,.bt + I\ -(
1 1 1•"- t

(2. 7)

where Hi the linearized m as urem ent op ra lor and 1\ is thew ight. Anal,vsis error
and analysis-error covariance can I e represent cl as:

(2. )
(2. 9)

On olving equ ations (2.3). (2.4), (2.7) and (2. ). th equ a tions for anal)·sis rror and

23

anal~·si -

rror ovanancr an h r \\Tl t t rn as

E7 ~ c7 + l\." ( 1 t - H --~)
Pt - (!- l\ tH )P/'(I

whC're I is an id 'lltlty nwt n:x

(2. 10)

l\-1 H ) 1

f\-, H.l\/

(2.11)

lw opt nnal "'oluiH>n JS oht ainPd wlwn t lH' (nH'c' of

Pt is a mininmm ancl I hr opt mwl \\"f' lght [\. c nJlf'd t lw I\ cllmall gai n and i ~ d 'finrd
as:

(2. 12 )

On ·ub. tit u t ing t h \·al uc of ]\. in qun t ion ( 2.11) \\'P gc>t :

(2 13)

In the traditional Kalman filter th forecast- rror covariance is c akulatf'd using the
equation :

(2. 14)

where !II is the TLJ\1 oft h nonlin a r model. Th larg numb er of degrees of freedom
and nonlinearit y res trict th direc t implem entation of K alm an fi lt ers for G 'J\ I . Th
en. emble formul at ion of a K alman filter known as

ns mble K alman filter (EnKF )

pr po. d by Evensen (1994) i::; a practi cal implement ation of K alma n filt er for th<.->
nonlinear sysi m with large dimension.

24

Th fr n w r

2.2.

h fornmlat ion of ~ nK

1. hcl~ 'd on t h' id

a a. tocha. ti cliff r ntwl

qncltlClll.

h

a that 1ft lw fm ('( ast modrl is int Npl C'( c·d

fmt>Ul"t-etH 1

using ens mhlP intpgrations ( ~\'C'll:-.('11. 1. --1. 1
~ \'Cns n ( 1

I)

\\.Jth tlw

) and Hout kam 1 and ~I1tdH'll ( 1

gratrd [ rward 111 tim

·tcltlsiH'" Crlll b approxilnatPrl
n KF introdnrPd h.v

) . 'll"PlllhlP m ·ml H'rs are mt P-

and updcllPd (up dat •d <'llSPml>l(•: an· known as analysis rn-

s mh} , ) wh llC'\'C'f 11 W oh: n·ation.

X 1° =
J\'t

CUP clYclJlahlP. n<-Ulldy:

x:) l\.·~o~ HXt )

= P/>HT (HP/JHT

(2. 15)

R) I

(2. 1 )

P,b = E~(s:~- Xf )( X 1h - Xt )r]

(2. 17)

P1a = (I- XtH )P/>

(2.1 )

wh re Xf i th anal~·zed tate and H is the lin arized measun•mc·nt operator. Tlw
sup er ript b indicates the mod l-forecast stat . l\'1 is th

K alman gain. P/ i~ tlw

for cast- rror covarianc . E [.) repr . nt. th expectation valn e. the overbar denotes
the

nsemll

mean and Pt i the anah·sis-error covanance.

I is an identity ma-

trix.

En. emhlr I< (l,lman filtrr h a,.c; been widrly 11. rei in atm o. plwri r and ocr(tl1!C snrncr.
b caus

of its algorithmi

t al. 2012 · Tang
d at a-

. imrlicity and it is relativ l.Y easy to implement (Deng

t al. , 2014b). There are som

. imila1ion syst em . Th size

f th

lisadvant ages also f r an EnKF

nsemble used is ·:t d cisive factor in the

p erforma n e of th EnKF . A finit e ensemble size reduces th
of th st ati ti al mom nts wl1 r as a larg

nscmbl

c

ura )'of th

stimate

izc is omputati onall)· not afford-

al 1 .

h lim1 t d ensrml l , 1z ca usPs t h und 1 Pslmw I H n of I h
. _()lJl).

n isy corr lat ion (Ho nt kamc'r and :.Iltdwll. 1

nor 'O \ 'Cll wn <' and

o solvP tlus prohl 111. co-

variam:e mftatton cmd loca ltzatiou 1 ust'd 111 c-H lvmH cd Enl\F'"'. :.Im <' dist uss1o11 a bout
tlw covari rtncr mflc1.tinn (And rson and AndPt:--on L
and r-ditch ll. 2001) m >thod. can h found in

110tl1Pr disackan\agP of
( mhad an and

)

cUHllo< ahzcti 10ll (HoniPka nH'l

'h<lpl<'t 4

lS (}H' c\'-i'-\llll1ption of t}H' lillf'i-11" lllf'HSlll'<'lllPllt Cl!H'ralor

ang <'I al.. :...0 1 a ). Thr cwt Pill

an g. :...00

an implicit assumpt10n:
them an of th

111\

L

nK

<llgorit hm co11l mns

t hr fmP< ()<;( of tlw lllPH.' "'lll<'lll<'Ill function 1s nnhJas('(l or

forC'cast Pcpml to tllC' fon'Ca<;( of tlw m •an

f the n nlitlC'ar mc·asurcmPnt fmwtion in

~ n l\ ~

<llld

Tlw dnPct app lwalwn

11. 'HF imposPs au implicit

lin arization1 roc ss using 'lls mhlP nwmi>Pr.. ,,·luch is appr oximat<·lv hold trm• ou l.v

if:

(2.l!J)

D tailed d eriva tion can I e se n in T ang et a l. (201 4a) and thr rdc>n'ncc>!::i t hPrP in. In
m any real-world sit n at ion · the measurem nt fun ·tion is nonlin ar . For rxample the
observation d a t a a re at llit

r adia n ·es. but the variabl required for assimilation is

t mp erature. Linearizing th

m easurem ent function will result in estim a tion errors

and the improv m nt du

to the as imila tion of orres1 onding ob servatio n~ m ay b e

small (Hout eka m er and lit ch ell. 2001).

PKF aclclrcsscs this issue Gy using a cliffcrenl

approach for calculat ing th K alman gain and err r covarian ces (Amh a cla n and T ang.
2009 ; Tang et al. , 2014a).

2G

r

2.2.2

In t lw lassi

u l ti r

f

n

nK .. s hc>lllP" . t cHHlomh· 1 ' It urh d oh~c·n·a t ions ar ' nsnally llSC'd

to avmd tlw nnd 'r strmntion of th

ancdY"lS-Pitol

c·m·arimH'(' (B m gc•rs c•t al.. 1. 9 :

H ut kamer and )-Irt ch ll. 1(())

Prtn rhmg th' olJ~C'I\'H(J<Jll. lH>\\'('\''r. w1ll act as

a not h r s m CP of sam phng ('lTm cmd d 'C 1 'cl:--P t lw c-H·c·nracy of t lw ana l \"Sls- ' tTor ('CJvanancC' matrix (\\'hrt a kPr and Hamill . 20 l2) .
use of unp rturl Pel ohser,·ations. hut d\'OH l.

hns c-111 optimlllll approach tlwt makPs
""

l<'lllcltic ll1Hlc·r<'stimatwn of ;-mal.vsis-

rror cm·arianc . \\'a. clP\'Plop Pd 1)\· \\ 'l11t ak r and Hamill (2002). C'H llf'd the· c·n~PrublP
.~ q u ar0-root filt0r

(En. R )

For conveni nee. analysi.· stat rquation (_.l.J) can hP rc·\\·trttc·n as th' snm of uH·c-m
f

anal~·.i '

wher

' tate Xf and the dC'viation from th mean ~tatP x;~:

Xt - Xt

1\t (}I - H X {')

(2 .20)

.\f f = );f

1\t ( 't~' - H x:/)

(2.2 1)

f (t is the traditional Kalman gam and !1~1 rs th

gam used to npdat e thr

deviation from the mean . Forth traditional EnKF , J\ 1 = A-1 . Here 1 ~' is the rancloml_v
p rturl eel observation

EnSRF

rrors (\\'hitaker and Hamill. 2002: Burgers t al.. 199 ). In

quation (2.20) will l

X~

I

written a~:

= X/

I

--

1
-

I

..--

f{t H X 1b = (I - l\.-tH ).:'\.7

I

(2.22)

J(t i~ d w. ' ' ll ~n ch that it.s uc£initiou will sati ·fy t he t-m al)'~i '-l'rror covariance equation

27

(2.1 ).

In

h n th , olui10n w1ll l

qu al t o:

n R . ohsrn ·a t10ns rc n cllso h p roc·pssPd on· l-1 ( a lll lH'

R] 1

(2 .. )

lll tha t casP

H 1 /' H 1 a nd

R rrducr t o scala rs. R ' \\Titmg r qn n t ion ( ..... 22). 1f /\. 1 - n l\'1 . \\'lH·I r n is a cons t a nt.
on solving fm n. tlw s >lntion ob ta in •d 1. (\\'l11tak •r Hnd Ha nllll. 2002):

-,--R-R)

(2.2.J: )

1

H P/' H 7

2.2.3

In

Sign1.a point un

nKF . t h

nt d K li

n srml 1 m mh rs a r

us u a lly g<'nPrat Pd r an d o ml ~· nsm g !->Olll P v.:Pll-

d esign d r andom p erturb a tion s ·h e m e. (E,· nsPIL 200~ ) . In
en semble iz i u su ally d et ermined

pl xity of t h e

KF )

n filt r (

n KF and E n. 'RF . t h e

onsidPrin u t lw lim ita ti ons d ictat ed b~· the corn-

imila ti on s~·stem. or cal ulat i n cap acit y. or iu som e casrs b,v sen-

sitivity studies ( ~Iit ch ell

t al.. 2002 : Lor n c. 2003: Lia ng. 2007: 'u rn e t al.. 201 2).

v. hich is not d et rminis tic in a rigo rou s s t a tistical sen se. In contr a.s t.
d t erministically chosen s h em

FCKF u ses a

t o gen r at e ensembl m emb er , ( i. .. sigm a p oints) b~·

the seal d unsc nt ed tra nsforma tion formula ( Julier. 2002 ):

t

V(L + A)P a L . f r i = L 2 ..... L

\lJ 1 = X~ + [

1

\ll ~~ = X t - [ J (L + .\ )P,a ],- L , fori = L + 1. .... 2L

2

(2.2.5)

h rc is a, et of wpighL a. o 1at d w1th th "<' s1gma pnmts
,\

L

,\

. for i - 0 and

(2.2})
. for i- 1. ... 2L

wh r

tlw sn1 'rscTlpt.

lll

<'Oll<'sponcl to t lw lllP<lll. HPu' 2L

are g npraled ac ordmg to quatwn (2.'2·-1)

1 Pnsc'mblP llH'mhPrs

LIS tlH' dmH'nsion of (lllf.!,llH'll l<'d statPs

concat nHting the moclPl stat<'.. pm<Pss nois<' <1lld ohsc•rvatiou nois<'. [ j(L
denote,' thP z1h
\"=tnan

.A)P/'L

olmun of tlw \\''Ight •d lllcltllX !:'qllau·-wot of tlw annlvs1s-c•uor co-

matnx . ). -

2

(L + k)- L 1. c1 o...,calmg pcuamPtPr. n is a snwll posltJvP vahw

b hveen 0 and 1 and d nde. the spr >ad of t lw :ignw points around I lH' m<'an state

X f. It ·hould Ideally be , lllctll to nunillllht' t ht• higlw1-o1< lc·1 df<'c·t. wlH'll tlH· IWllllllarities are. tr ng. k is a control param<'t<'l that gwuantc•f's positin· sc·mi-ddinitC'llPSS
f th

covarian

matrix and is an important issue in implemC'nt mg . 'Pl'KF. TllC'

valu of k al. o d 1 ends on the valne of n . k c>nsun's that ,\is positive. othC'rwisr the
weight

f the zeroth sigma point. u·[;' < ()and the non-positiv' \\'C'ights can makP the

covariance non-positiv

midefinite ( Julier. 2002: \'an d('r ~lC'rwc. 200-J). {j is a non-

negativ weighting term introduc d to reduce the approximation error. If the stat
distribution i. Gaussian. 8 = 2 is an optimal choice (Van cler ~Ierwe. 200-J: Julier.

2002). In this th is th valu sofa= L k = 0 and 8 = 2 are used. whose rationales
and th

detailed derivations

an he found in Van der ~ I erwe (200.:1) and refer nces

therein.

SP K F mak s u. of the fo llowing r formulated equation to obtain the Kalman gain
f (t

and analysis-error covarianc

matrix Pt so that it d es not require a TLT\1 or

29

lin arize l mea, urem nt op rat r:

(2.27)
(2.2 )

P/"Y is t h noss-cm·ananr mat nx of t h ' st (1 t r (11H l oh:-.('1 \'<1 t ion pn•dJCt ion f'rror. I /111 rs
t h cm·arianr mat nx of t h o hspr \'<-1 t ron pr PclH t H m c•nor.
\ an cler I\Irn\·r (200.-J ) and

ml>cHlan <-llld

or ;.1 dc•t ni lPd clPll vat 1011 sc•p

(mg (_OOD) and H·fc·tC'llCf's thc·n·in.

equation (~.2 7 ) and stat s ar updat<'d 11smg, tlw followmg, c'qHation:

(2.20)

At thi .'tep one can gen rat n w sigma point . hy:

(2.30)

Th sigma-point vect r rs 1 r 1 agat d through equations (2. 1) and (2.2). Th nwan
and covariance ar

alculated u ing the following f rmulas :
l=2L

Xf =

L w;n X~~~

(2.3 1)

1=0
l=2L

Y. b I

-

L-t 'U?m,1 rbt ,7

'""'

(2.32)

7=0
l=2L

pbt = '""'
Xb/ ,1 _ X Ib) (-''Kb/ ,1 _ -')(b)T
L-t u{(
1
t
1=0

30

(2.33)

h

ross c van an

P/" 11 and proJec t H n m·ati ancP ? 11111 ar

n dlw l C'd hy t h P followmg

qu a t ions:

- ,,\ .h)(
'1 ·h
- fh)7
1
11
I

(2~ cl )

,r( '1 ·h _ ,1 ·h)( '1 ·h _ fh)I

( .. !) )

,c ( , .h

ll,
1

, \ 11

0

1= 2/

11,

11

1

11

I

lw \\' 'igh t s associa t rd \\' 1t h t lw cm·arian< · ' calcnlat ion is

(1

V ing th

')

n~

1) . fm I = [) and

PCKF . the S<' oncl ord <'r arcnrac\' can hP p t PSPf\'C'd m t hC' m Pan and C'O-

v n an c . Higher- rcl >r inform at ion of t h ' sys t Pm in t h cm·ari ancP can lH' iudndPd hy
u ' ing the scaling p a ra m eters o a nd i3 (.Juli r . ~ 002: .Julwr a nd -hlma nn . 200 1).

2.3

Models

In thi study. VI e us th Lor nz- 96 m odel (Lor enz. 1996: L renz a nd EmanueL 1 9 )
and the Zebiak-Cane m od l ( h en et al.. 2004). h r aft er ZC m od l). A bri f introdu ction to the models is given in the Subsections 2.3.1 a nd 2.3.2 .

m or

d et ailed

description a l out the Lorenz-96 model can 1 f und in Loren z (1996) and Lor nz a nd
Emanuel (199 ); more detail about Lh ZC mod 1 can b e found in Zebia k a nd Cane
(19 7).

31

2.3.1

Tl

h Lorrnz-

L r n z-

m

m o I '1 (Lor n z, 1

1

o n ' ll7 a nd ~ m a nuPl . 1

l )

1s a hi ~hly- nonlinr a r

model ft r nus d as a I s t-h 'd fm ll C'\\. d a t a-assuml a iHm ;.llgorithms.

his modd ha s

m a ny comm m asp r t s w1th a tmosph ric m od (+, ' \ 'C'll tho ng h it is n o t nwt 'omlogica lly
v r.\· realist 1

Th m odel qu ti on 1. ddi n ' d as.

1

~ / - .r, 1 ( .r 1 - r 2 ) - .r,
1

(2 ~37 )

F

1

wh re z = 1. - · ....Y.r . r 1 r sen t. clos d c n li e b o und ar\' concb t 1ons TlH' .r, ·s conld h P

1 os ly tr a t ed as a n a tmo,' ph ric va ri a bl a t d iscret r longitnd 'S a ronnd a con s! a ut
latitude

ir 1

u ch tha t :ro = J.'.Yr· 1'- 1 = .rsr- 1· .l'N:r

b e au e the dimen sion is 'm all a nd coulcl. h .,
similaritie in cha ti

1

= .rJ.

his sy t Pm is 11sPd

asily :caled and a lso h rca use of its

b eh aviour a nd tim s ·ale of predict a bilit y as found in realisti c

weather mod l . Th t erm F i. a n xt rna l fo r ing and h m odel lw h avc's ch aotically
when F

=

The Rung -Kutt a f urth- orcler ,'ch m e is used to solvE tlw mod el

equation with time st ep equal to 0.05. P erf rming d at a assimila ti on ever:v time st Pp
orr

I onds approxim at ely to 6-hours in a gl b a l weather m od 1 (Loren z and Ema nncl.

199 ). A mor d etailed discussion of the model a nd it · h a ract eristi ·scan 1
in Lorenz (1996) and Lor n z and Emanu 1 (199 ). The exp rim nt s a r

fmm l

conduct ed

in th p erfect model s t up: that i . a long-t rm integration fr om an a rl itr ar~· initial
condition is u s d as th

tru state and synth ti obs rvation i gener a t ed hy a dding

normal random errors to the true stat .

2

z

2. .2

h

l

k-

5 \'('fSiOn of th

modP} (ZPl IHk c\ll<l

Z

'cUH' . 1 ( I.

'}wn P(

al.. '()().J) is

used as the r a list IC rlnnat ' mod<'l in t lus -;t IICh· Tlw Z ' llH>< lc·l Is a couplc·d oc·c·mlat mosplwrr mod 1 and ha
in t ns1 t y
lat 1

lw '11 wJdPh· 11 ·pc[ for

'\'C'nt · for 1 ot h

f

Os(\\'·bstrrandialnH'r.l

nus d h~·

t h ' t iminf!; . phasr and

X]Wl im 11 t a 1 awl opPnl t ion al

purpos ·s since t lH'

-. l .cusp•ckand . ndc'rsmL20()7 ). Th1snwdPlhas

'hf'n Pt al. (:...00 ) for h snc · crs~..fnllcm g- tc · rm IPlwsppr·tivP ptPchctwn

of E:\' 0 for 1!

\'<'Hr. ( 1 ) - 200. )

Tlu.

Wrts t lw fi1 st

dvwunir a] r onplf'd nwdr' l

lus mode•! has C'Olll])()llPnh of hot b drs('harg('- rr·chargP

developed to pr diet
and d la~· d

JH Pdi<'t ing

sc1llator ph,\·:iC's of ~ 0:.'0 .

The Z

m d 1 i an anomaly modrl that computes anomali s of atmosphPriC' and

oceam

fi ld. r latin· to asp

dynami s follov,· Gill (1

ifi d monthly man dimatolog,v. Tlw atmosplwuc

0). with a ' teacl,v-statr. linPar. shallov:-walC'r Pqnations on

an quatorial beta plan . Th circulation is f rePel h.\ ' a hPating an nnaly that dPpends
on the

Tan mal~· and moi. ture conYerg nc paranwtc>rized in t rrn:-:. of surface wind

conYC'rgrncC'. ThC' modd domain is confinC'd to t hC' tropical Pacific (k an ( 101 .2.S 0 E
73.125°W, 29°S- 2 °1\} The grid r solution. for ocean lynamics is 2° longitude=' b:v 0 ..5°
1 titude and for S T physi ·,' and the atmosphere the model grid is 5.625° longitude by
2° latitude. Th ocean d~·namics use the reduced-gravity modC'l. The surface currents
ar generated 1 y spinning-up the model with monthly mean climatological \vinds ancl
is then u. d for t h

current anomaly calculation . Th

the SST anomaly a nd h eat-flux

tlwrmodynami s describe

h ange. The governing equation

f thermod~·namics

contain thr e-dimensional temp rat ur advection by th calculated anomalous current
and Lhe sp cifi d m ean

urr nt ( ane et al., 19 G: Zehiak and Cane. 19 7) . The

mod l1 ~ gi \ ' Cll b,·

gov rnmg qnation of thcrmocl)·namics m th Z

aT

-

~ = - [.

-

T - C.\l(T

-

nT

JI (W

T) - [J/(Il'

(2.:3 )

wh rc the harrccl and nnhanPd quantitH'S H'pt<'~Pnt 11H'a n and anomaliPs lC'SpC'ctl\Tl.v.
C and\\. rPptP~cnt hori?oulal smL·HT cmu'nt~ c~ud upwdling \·p]ocily H'~P('C'tivdv awl

n i a th rmnl damping odficiPnt. TlH' hmt?ontal cHlvc'ctwu tc'rlll~ mr n ' plPSP ntc'd
by th

first two trrm. on the right lwnd ~tdC' of tllC' al>OV<' Pcpwt1011

of anomal us ur w lling in th

The dfC'cts

prPsPncc' of thP llH'an vnt1ull lrmlH'ratnn' gradic·llt.

~~- ancl thr total upwelling in tlw pre's nc of thP mwmc-dous \'f'ttical temprratnrc
gradient. ~~. ar

rrprr. nt cl in the t h1rd and fonrt h t Pnns rc'srwct ivel.v.

lw la s t

term is a lin ar damping term that rcprPs nt~ the chcmg of,. 'T

dne to tlH' lH'a t

exchang 1 tween th ocean and atmosphc>re (Zrbiak and 'anC'. 1

7: Battisti. 19

h ng et al.. 2010). Th fun tion J£ (.1·) i. drfinrd hy :

0.

JI (.r) =

.r < 0

(2.39)

-

{

I.

I> 0

The function !IJ(x) rnsun's that surfr cr tPmprr().tnrr i. ().ffrctrd hy vrrtic().] ;vlvrction
only in th presence of upwelling (Zebiak and Can . 19 7).

The surface wind-stress anomaly is gen rat d from the background mean winds sp cified by the obs rvation and surfac

wind anomalies produced 1 y the atmospheric

model. The ocean dynamics time step is 10 clays. Further details about the Z
can b f und in

ane et al. (19 6) and Zel iak and

34

anc (19 7).

mod l

Chapter 3

RRSPUKFs:

Derivation and applica-

tion on the Lorenz-96 model

3.1

Introduction

Thi,

har ter con ·titutes :ome of the main thror tical contrilmtiun~ of thi~ chs~etta­

tion.

he ad\'antag s f th the

PCK

algorithm compmed to

uK

and En 'RF m

st ima ting t h

true m an and cm·ariancr ha\'e alrrad)· bern n•,·i ,,·pd in t lw prPYions

ch apters. In

PCKF. th sigma point. are generated clctt>rministicall.\· hy the s ·ah•d

un cent ecl transf rmati n f rnmla (Julier. ~002). The main limit<:1tion to apply thr

P KF to c hi gh r-dimensional syst m su ch as r\ \ ' P is t h rcqnirPm •nt of morr than
t wic

th

number of sigma 1 oints than t h

S)'St em stat s. If the S)'St em dimt'nswn

is L , thc:r is c requir ment of 2L + 1 sigma points for C'stimating the mran and co-

35

vanan

investigat

ne of th' 1 oss1hl solutwns IS to

<1111 utatwnall.v unf c.Ihl

. and that i.

if th numb r of s1gma pmnts an h lmvPI d hv takmg tlH' rPdnrrd-rank

appro ·imation f thr fnll,
rigina1 sigma pomts.

Cl\ . wluch cant tHin most of th' main f<•alluPs f th

hP possihilit \' of const lllCI lllg a rc>dncPd-umk approximation

is invrst igat rd in t h , c>ct ion :3.-.

3.2

R du

d-r nk

.

.

1

1

l1n n

filt r

(RRSPUKF)
Th r du · d-rank ap1 roxnnatwn is mainlv hasPd on the cone 'Pt that most largr-scal
gcoph)"ical phcnonH.'ll<:t can b · app1 uxima t'cl bv a hmt <' lllllni>el of d<'gH·t·s of i1 <'<'dom and th ir cl minant variahilit \' can he rxplainc>cl hv a limit Pd mnnl H'r of modPs
(Temam. 1991: Lermu ' iaux. 1 97: L rnmsianx and Robin.-on. 1

: Ambadan and

Tang. 2009). Icl ·1ll)' these m l · sh ulcl b all to dc>scribP th Pvolving dynamics
of the y t m. Comm n t

hnique' us<>cl to r ducc> thC' numb r of mode.· to a lim-

it d number that is st ill a lle t

repr sent th

dominant f at ur s of the system are

dynamical. ingular vector (P alm rand Zanna. 2013). mpirica1 orthogonal functions
(Lor nz. 1965), prin ·ipal oscillation and int ra tion 1 at terns (Penland . 19 9) and
radial functions and wavelets (Lermusiaux . 1997; L rmusiaux and Robinson. 1999).
Lermu iaux and Robin on (1999) pr pos d the ·oncept of error-sui space stati. tical
estimation (ESSE) to find the dominant rror subspa

. d scrib cl by err r-subspace

singular ve ors and values. In the following section a similar oncept is applied in
the case of SPUKF, assuming that most important errors oft he original sigma-point
space can be estimated using a fewer number of Lhe dominant . igma points (l\ Ianoj

3

t al.. 2014).

r

s can hr sr n in t hr quat ion lwlow . 1ll

T

h . for t hP com plPt

d 'SCript lOll of all

the> rrrors we Iwrd to nsr tlw numlwr f s1gma points dc>t<'lllllll d h~· tlw diniC'nsiou of
t h rrror-covarian r mat 11x. Pt:

( ~~ 1)

(L
>.)PnI

J I

[,

by finding the redu Tel-rank a.pJ roxinnJ1011 of

•

fo1 ; -

L

1. .... 2L

Pr tlw munlH'r of s1gma point s can hC'

reduced. Two methods are proposPd for tlw rank t Pdnction and hotb rln' hasPd on
th T VD. The f rmulation of both uH'thod. i. dPscrihPd in .'c'C'(Ion ~3.2.1.

3.2 .1

RRSP U KF (D ): T SVD 1n t h data pac

In the RR P

KF (D ) m et hod. T \ 'Dis aJ plied n the covariance matrix constructed

in the data space. so the filter will bed noted as redu ·ed-rank. sigm -point 1msc nt r d
Kalman filter (Data) (RRSP"CKF(D)). As can b

cen in equation (3.1) the numbC'r of

sigma points is d termined by the dim nsion of the error- ·ovariance matrix . Tlm.·. t h
central point of RRSPUK(D ) i , to effcctin~ly reduce the rank of th' enor-covariauc'
matrix, which will b

achieve l by the TSVD m thod. Th analysis-error ·cn·anance

Pt in (2.15) is symmetric and ·an be clecom1 o

l as:

(3.2)

37

wh r

D~ -= dwg (CJf. 1 ... a}d is a dwgonal matnx cons1stmg of thr

igf'l1\'Hlu

saL.

s rted in dcsrf'nciing ordPr. i.r .. al. > ot. ? 0 fori > .J. Er = [ 11· ... J't.d is thr
1

1

matrix consist mg of t hP corr spondmg igPll\'PC t m s c 1,,.

or dnce tlw mtmh ,r of sigma point s. a lPdllcC'd-tcmk. 'lTOl'-(m'ari nn c<' matrix. dP-

notrd hy P1°. can lw approxnnately ohtainC'd hY thP trllncatmn of Pqnntion (:5. 1).

pn- EaII Dn (Ell1,1 )r
I

1./

wher list h , tnmca tion1mmlwr The lH'W s1gma pomts a r P gc'lH'I at Pd a."i follows :

w1 .o = x,a

JU + A) ]CJt,l c for z - 1. 2 ... /
'-l! ;l = X il- [ JU A )]CJ~.~-trl.l-1 . for 1 = l + 1. ... .2!

w~, , = x,a + [

where o i th . quar root

1,1 .

(3.5)
(3. G)

f the eigen va ht e and c is tlw eigen \'ector.

Th new sigma point vector nO\\' b com es:

(3. 7)

Th numb r of igma- points generat d by equations (3. 4 to 3.6) a re 2l + 1.

3

R

3.2.2

K

Inth RR Pl' K ("')nwthod.
in t h c'nsc'mhl su hspac

v

( ):

\ '

ii

h

If' applJf'd on th

cc \'cHl;-Ul P nwlux consl rncl ed
hP HRS I l ' K .. ( )

( nsc'mhl '). de' not Pd a.., 1

can cfTe ti,· 1~· . 8\'P comp11tationallnu thro11gh r •d11cing th s 1g ma pollltf. hom '2L
to 2/ + 1.

h

sa\·ing 1s C'sp< cwll\' signifi ani \\·hc·n t l1C' C'llC>l C'O\'CU ir111c

1

P1n can h

cl!l roximat d ,,. ll by a fc'w lc,adin g mod '"' so that a ..,mall/ can haY<' a good lnmcalion
cccura ~·. HowC'\'Pr. a large' challcng in RH ,' J l'l\ ( ) 1s th · compnlahon of Pt 1lsPlf.
\\'hen th

state dim nsion is hug'. 1t IS cmuputalionallv cxp<'nsi\'C' to cornpnt<' such

a hig m atr ix and it is <'V<'n more eli Hie nl to . I en c'

n th r app roach to sc>C'k the mo.· t Important sigma points . which can avoid thr
af r m ntion d hall nge. is through
Tang et al.. 2014b ). \\ ~h n the stat
dim n ion n. i.

\ 'Din th rns •mblr spac (.:\Imwj 't al.. 2014;
dimension L.\1 is much grc>ater than the• Pnsc·mhlP

L ,\1 >> 11. it i · possibl

com1 ul

thr , '\ '

on thr nXn coYariancc

matrix (v n Star hand Hanno. cho k. 19 4: \\Tilk ·. 2011). which is com 1 utationall.v
affordethk for a. syst<'m with la.rgP clinwn. ion . Tlw computettimml COlllpl <'xit~· of thr
a lgorithm to find the covaricmrr matrix in th<' data spac<' is O(L~ 1 x n ); however.
the m at h 11atical complexity of t h
ensemlle subspace is O(n 2

a lgorithm to find t h

covaric n · matrix in t h

LAJ ). The cliff rene in computational compl exit)· b eh\·c n

t h two a lgorit hms is significant when L AI >> n , that i , usually t h
N~ P model. One of the main advantage of RR P

as for a r c- list ic

KF ( ) over RR PCKF (D ) i. that

in RRSPUKF (E ) we can calculate th analysi -error covaria n ce explicitly. which is a
requirement for gen rating th sigma points, even for a larg -climcnsionalm del.

In th

RRSPUKF( ) method a ll the ens mhl

ar lll dat l according to equation

(2.3) and h n from t h cs m mbers find the analysis-error cm·ari a n ce m at rix PtE in

39

the n mblc subs] ac accordmg l o:

(3. )

wlwre Xf is l he anal~·s1:-. m 'an.

h n , '\'D 1" 1wrfornwc l on tlw 11 X 11 anal~·sis-PlTOr

CO\'ariancr mal nx:

( 3. )

wher Ft is thr igc'm ·cctor:-. and ;;1 i:-. d diagonalmalnx of the rigPnvalnP~.

Th

ig m ·ec l or: of pta I:- and Pt m r diff('l rut. hut 1lw lrading 11 eigrllv(•C( ors of Pt can

b

mputed from the rig 11\' ctors Ft of PtE: (von

torch and Hamw~chock. 10 4:

\\'ilks. 2011) using :

(3.10)

where l = 1.. 11. The role of the denominator is to mak snre t hal the resulting E~
have unit length. The maximum value of l is l = (n- 1) /2. so that the total numb r
of ensemhle members is n for every time step. The new igma points are gcn ,rated
in the same way as in equation (3.4 to 3.6). i.e. the mean and (n - 1) /2 of positiv
and negative pairs of p rturbation to the mean.

40

Appli

3.3

ti n

nth

z- 6 m

r

l

In thi . eel ion. th 1 n'lcticalit)' of ll'-'ing tlw n dm d-n111k s1gma-pomt I alman filtc·rs

alg,ori t hm a::- an dl 'ell n' da t a-e~ss1mi la t 1011 llH'l hod for high!\· w mlin 'H r wo<h+, is
x1lor cL

hr highlv nonlinc•ar mod 'l us<'d in thl!-> stwl\' 1s tlw LmC'nZ ( 1

G) modc·l.

Th' modd quatwn 1s gl\·rn by:

d.rl
dt
- .rl 1 (.rl 1

,,·her

z =

F

c .11)

1. 2 . ....Y:r . Tbi. model simulate::- tlH' imC' Pvolution of an un rwcifiC'd ~calar

atmo 1h ri

variahl ..r. at .Y:r 'qnicli~tant grid pomts around a constant latitnclP

circle. \\rh n th
(Lor nz and

valu

of rxtPrnal forcing F

manueL 1

= . th ' systPm hC'h cwC's chaotically

) . Tlw fonrt h-ord ·r R nng -Kntt a sclH'lll(' i!-> nsed for t h('

time int gration of the mod l with time st p = 0.0.5 .

In th

fiPld of cl("ta a .. imilation , thi. modPL clong with thP Lorenz (19G3 ) thrC'c-

variable model. i. oft n u. d a
data-

similation m thod

a test-b d for examining the properties of varions

(~ Iiller

et al.. 1994:

et al. , 2012) not only because of it

mbadan and Tang. 2009: Kalna.v

smaller d gree. of freedom. allowing vanous

xp rim nts , 1 ut a lso because it includes many d)·namic features of realistic wee-It h -r
sy t m

(Lorenz 2006). For example, it has

those of full

rror growth characteristics similar to

\iVP model . A more det il d des Tipt ion al out l h model and it

ch ar acterist ics can b found in Lorenz an l Emanuel (199 ) .

On of the reason s for using t h Lor nz-96 model for data a:-;s imilation xperimcnts

is that , if new assimilation algorithm giv unfavourable results for a high!)· iclralizPd

41

g t r asona hl

mod l. it

ht.[?,hly unllk,ly fn the s(m

nH'th d to

r snlt s for a lugh-dum'n~wrwl 1 Palu.., t 1c weather model.

fnll-scal

m d l sn h as t h

L rcnz-

1.'

exprrim nt using a high-dim 'llS1011c11 mod lrs a time consnmm.[?, and vast 1111dPrt a king.
h

'lWrimcn t r snl t ~ fl om ~1mpl modC'b can g1 \'P ~omr dn P< 1ions on gc'nC'r al t rPnds

and possihl hcha\'lours.

3.4

Exp rim nt I

\\- perform t h

tup

rl.ata assimilation <'xprrimcut :-.; undf'r t lw ohservmg s.v:-.;1 <'lll simula-

tion exp rim nt: (

E). al. 'o known a~ thr rwtfC'ct uw ll c·xprnnH'nts. OSS""'s an'

g n rally d , ign d t

te, t th p rforme:mcr of liff r nt data assimilation algorithms.

Thy may a lso b

used to ass ss the potential im1 act of an obsC'rvation array to hP

deploy d u 'ing data-c " imilation mrthods (.~ l a~ntani Pt al.. 2010: Hontf'kanHT. 2012).
In an 0

. th m rl.cl trajector~· is created 1.v a long integration of thf' m )(lPl forward

in tim from a known state. Th mod l traj ctor~' is known as tlw .. trnth''. Synthetic
ob 'crvation , arc created by sampling the model tH"tjcctmy at asp ·cifi<'cllocatioll awl
interval using an ol s rvat ion forward op rator (for a rral model th

frecp1cn ~· an l

lo at ions of sampled observations . hould be irl.enti ·al to that of an a ·t ual observing sy tem). In th

next step. these s~·nthetic observation. a r

assimilated into the

model using th state of th art data-a similation method an I th r sultant product
is known as the an alysis. The assimilation p rformanc can b e evaluat d comparing
Lh a nalysis with th

"truth''.

42

n rati n

3.4.1

h

f trut

rv ti n

n

x1 ermwn tal set UJ 1s s1milar to that of Lor nz ( 00 ) . w h 'r .. \ ·r- 0 c nd F. the

magnit ucle of t h

for ing. 1s "''( to

. 111 which t hP svst m 1s chaot ir.

h

nalurr or

ohsrrYation sinmlat1 m nm1s cr'atc'd h\· mtc'giclting th' sys <'lll mTl 4000 tim stc'ps
by using th

fmrlh-orclrr Run gr-1\utta "dH'mr. The' intq:;n1tion ::-;1 pIS S<'l to 0 .05

(i. .. -hours). Th data from t h nat nr<' n m 111 t h tim ' st <'ps 1000 2000 a l'<' I akPn as
the truth. ThC' ohs<'n'Htional datc1 s'ls <ll'C' gPnPratPd b\· adding nmm:-dlv dis trihnt<'d
noi e }\ r(O.

3.4.2

2) t

A

the truth.

imil ti n m th d

nd pr

dur

\\T u e three l ta-, " imilation met h ds ( 'P ' KF . RR , 'PCKF (D). RR . 'Pl.KF( ) ) .
To implrmrnt thr , PeKF mrthod. thr tatr H'ctm i. n'drfinrrl ( s tlH' ('Onf'atrnation

f th

m del

tate . mod l err rs. and m asurement err rs as dis('ussrd in . 'ret ion

2.2.3. For a full-rank. sigma-point space. ther

is a total of 241 sigma points or 241

n. emble meml er . 'vVe assume that th model and observation errors arc uncorrelat eel
in both pac and tim and that the ob. ervation. of all m del states ar, ol sen· cl an l
assimilated every five tim
the mod l

st ps .

rror. the magnitud

Becau e th re i. no general method of set ting

of mod l error u ed in th

Kalman filt er i often

d termined experim nt ally 1 y trial and error or hy statistical m t hods su ·h as the
Monte

a rlo m tho l ( 1ill r et al., 1994). In our experim nts. the model errors are

intentionally designed in su ·h a way that the model would not drift too far from th
tru e state.

43

rr r

3.4.3

he quality of th analvsis ism asur<'d m t rms oft lw root lllC'an-sqnHn' rrror (Rl\ lS )
and time av raged rc>la t i,· R ~ L c>t ror ( R R ~1.

) lwt \\'P n ! h(• anal vsis ns mhlC' llH'clll

and th lru stale':

(:3 .12)

H r e X:r is number of th model dlln 11. '1011 (S r=.JO m this case> ):

wh r

Tma :r is the numb

r of a.·similation cycl and 11.11 2 is the L2 norm .

3.5

Results and discussion

3.5.1

State estimation exp erime nts with RRSP U KF (D ) and
RRSPUKF (E)

On important. elem nt in the ar p lication of the RRSPUKF method is the size of the
truncated m od es of TSVD . or efi'ect ive sigma points.

sm all ·iz.e f- il:s to ch ara<:icriz.e

important. error st atisti .· an llea ls top

r a ·simila lion analysis. whereas <-1 large size

is likely computationally unaffonlablc .

goo d st r a t egy for det cnuiuiug the siz.e oft he

44

trun atrc1 m d

i, s nsitl\'ity Xl eriments. which are oftpn us 'din .. nKF to xanune

the anal~·si.

rror as a function f the nsPmhlc srz (~ ht chcll C't al.. 002;

Liang. 2007:

nrn tal.. 2012).

m nc. 2003:

xpcllllH'llb cUl' condlldl·d llsmg dlff<'l<'llt t1tmcat<•d

moo s after th

\ "D tving hoth thr RR. r 'I\ (D) and RRSP TI\F( .. ) nwthods and

th rcsnlts ar

mpar c1 w1t h that of a fnll-r ank SI '1\ .

Figure .1 she \\'S th \'Hriation of RR.:\1.
used.

as a fnndwn of thP llllllliH'I of srgma points

s can be seen m Fignre .. 1. h low 21 ..,rgnw points t lw Psi uw1 t 1011 skill is \Try

poor for both s lwnw~ lmt as th' mtmlwr of sigma pomh llH 'H'H;.;('s th

C'stimatwn

err r de reasc~. \\'hen thC'r' arr :31 srgma pomts or more. fm tlwr unprovc'mPnt is
minimal . nggcsting that

1 sigma points conlcllw n~('(l for cl~~Jmilk-ttion. lmlanciug tbC'

accu ra ~·and computational cost.

Figur

.2(a) sh ws th

assimilation solution for thC' variablr Xl of thf' LorPnz-OG

mod 1 f r the full-rank

PCKF compared with the truth w1th time' stC'p f . In thC'

. e ond a e. the RR PCKF (D) is u

cl to r duce thP sigma points. In this casC'. ;31

igma point are . 1 cted that a ·c unt for mor than 3Vc oft he total varicmce (Figur

3.2(b)). The model state can be fairly wPll estimated by the RRSPC'KF(D).

although its R:f\1 E i not a

mall . the full

PKF (Figure 3.3). In the third case,

the RRSP KF (E ) algorithm is used to re luce th

number of sigma 1 oints.

s m

these ond ase, 31 sigma points are cho. n. Shown in Figure 3.2( c) is the simulation
of variable Xl from the RRSPUKF (E) algorithm comp a r d to th truth. Figure 3.3
compares the

stimation error (R 1SE) in the cases of (a) Full-rank SP TKF. (h)

RRSPUKF(D) with 31 sigma points. and (c) RRSPUKF (E) vvith 31 sigma points. In
the cas of both RRSP KF(D) and RRSPUKF (E), the Rl\ISE is Ycr.v cl se to that of
a full-rank SP KF. As the full-rank SP KF ontain more numb er of sigma p ints it
Lakes longer time t.o r a h th equilibrium vc lue. Comparison heh"'een Figures 3.3( c)

45

- - RRSPUKF(D)
RRSPUKF(E)

0.35

w

0.3

CJ)

:E
0::

Q)

....n:s> 0.25
Q)
loo..

Q)

0')

n:s
loo..

0.2

Q)

>

<(

0.15

··· ~~~~~--'I

0.1
10

20

I

I

1

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

40
50
60
70
30
Number of ensemble members

80

90

Figu re .1: Th tim ave rag cl r lati ve RT\I 'E a. a function of number of sigma points
( ensembl m embers) u ed for the Lor nz-96 1110 l l.
and 3.3(b) rev als that th

RRSP KF (E ) is ·o1111 ara hl

with the sam

error magnit u le, sugges ting a p ossible solution to ap-

approximat

plying SPUKF in high-dimensional systems .

vvith th

RR PCKF (D )

1\ Iore det ailed discussion a bo 1t the

RRSP KF (E ) is given in S ction 3.5.2.

Computation Lime for various m thods using a Linn PC with a 2.0GHz Int 1 P ent inm

Dual Core pro · sHor is shown in Table

.1. Th

·om put at ion time is for 1000 c~·des of

an alysis- forecast steps on th Loren z-96 model. This ·ompu tat ion timin g results from

46

Full-rank SPUKF
- - - True

(a)

.

X

-10

100

0

200

300

400

500

600

700

800

900

1000

1me steps

RRSPUKF(O)
--- True

(b)

10
5

. 0

X

-10

100

0

200

300

400

500

600

700

800

900

1000

1me steps

RRSPUKF(E)
--- True

(c)

10
\

5

I

,..
X

0-

-10 '------~-----~----------~

0

100

200

300

400

500

600

700

800

900

1000

Time steps

Figure 3 .2: C ompa rison h ehY<>en tlw t rue \·alne a nd a na l~·s i s for Ya ri ahlc> Xl wi t h tim <'
st ep for (a) the full-r a nk P U KF , (h ) RR SPl KF (D ) ,,·ith 31 sigm a p oints (('llsemhl('
m em1w rs) and (c) RR PUKF (E ) wit h 31 sigm a p oints.
the exp eriment s on Lor n z- 96 m od el giYP a gC'n era l trend of how RR SP U KF m et h ods
p erform compa red to the· full-ra nk SPUKF . RR SPUKF (E ) nncl (D ) nr<' m1wh fas t er
tha n the Full-R a nk SPUKF . The timing of RR ,' PUKF (E ) is s li ghtl~· h dkr tha n t h l'
RRSPUKF (D ). The timing r<>sults from both the RR , P

I\:Fs nrc' cmnp a ra hl< ' 1H'cn ust>

of the low-clim<>nHiona lit y of the' Loreu-9G mod('l. For a n 'nliHti c ntmosphcric or oe<'<llli ('

(/)

~ 4
2
0
0

100

200

300

400

500

600

700

BOO

900

1000

Time steps

10

I

B
w 6

(/)

~

a: 4
2
0
0

100

200

300

400

500

600

700

BOO

900

1000

600

700

BOO

900

1000

Time steps
(c) RRSPUKF(E)

10

B
w

(/)

~

a:

100

200

300

400

500
Time steps

Figure 3 .3: h e va ri a tion of the roo t m P<-m -squ a rr d error ovpr th e .J() va ri a ble's with
time s t 1 for (a) the full- ra nk PCKF . (h ) RR ' PCKF (D ) with. 1 sigm a p oint s ( P Hsembl e m emb ers). and (c) RR PCKF ( ~ ) with . 1 sigm a 1 oint s.
, ysi em with high-dim nsionali t~· and large> numb r of o bsc>rva t ions w can a ilo rd onl v
R P

KF ( ) m thod .

3.5.2

n itivit

r im

WI

F( )

h

m thod

In th ,en,iti,·it)· rxperim nt~ wr us diffe1011t t1mwatc'd moclP~ after the . VD. Figur

...1 cli.'pla)-.' th s0nsit i,·rty C'XI PrimPnt 10snlh of RR . 'PCKF ( ~) mPt hod . and t lH'

variation

f th timr mean c f th R:.I E wit h tl1C' diffPrPut nmnlwr of sigma point s

(i.e., enseml le 'ize).
cr a 'es fr m 21 t

s shov;n in Figure' :3 ...1 . wlwn the nmnhf'r of s1gma points in-

31. h R\'erage rror quickly dee r 'ases. whC'reas wlwn t hC' nnmiH'r

of sigma point. chang

from. 1 to ·-11. the im1 ro,·rmc'nt in th ' analysis is not VC'l.V sig-

nificant. This suggc ·t , that ;)1 ·igma pomt · aH' most likd~- sufficic11t t o clmradcrize
th error statistics in thi

·ase. Figur s 3 ..5 and 3.G shows tlw assimilation solution

for a different number of sigm<1 point. <1nd fn ll . Pl"KF in ('om pari son to t hr trnt h .
\A/h n the number of sigma points ar 3, .s and 7. the assimilation solution is far away
from the truth 1 ut as the number of igma points increas

the assimilation solutions

ar closer to the truth (Figure 3.5 and 3.6). \\'h n the numl er of sigma points are 1
or abov the assimilation solution is similar to that of a full-rank sigma-point K alman
filt r (Figur 3.6).

Figure 3.7( ) shows th assimila tion solution for the variation f the Rl\IS "' over thC'
40 varic hl s with tim step l for 1h e full-rank SPKF. Fnll-ran k

49

PK

can result in a

4.5

4

3.5
w

CJ)

:a:
a:

3

Q)
0')

m
m
2.s
>

<(

2

1.5

1
0.5 ~------~--------~------~------~~------~

0

20

10

30

40

50

Number of ensemble members

Figure 3.4: The time m an of th R1I E as a function of truncaiecl modes (ens ml >lc
size) for RRSP TKF (E ) method.
very sm 11 RMSE for the es timation of th mod l states after approxim ately· 120 t imr
st eps. The RI\1SE of the analyse.

om par d to th truth for (b) ..11 sigma points. (c)

31 sigma points (d) 21 sigma points, (e) 11 sigm a point s are also shown as a function

of tim step , furth r confirming that th ensemble siz of 31 i
imr rovem nt with a lditional points is only

uffi ient and t hat the

1htle.

Anothrr meLhod for Lh del rmination of the truncatcclmoclcs is based m th \'ari<- nee
:l

explained by t.he Lruncat.cd m drs with respect to the Lot al variance of a full-rank

20 ~

7-slgma points
True

-.,-

,...
X

-10

0

100

200

300

400

500

600

700

BOO

900

1000

nme steps
-

b

20

5-sigma pomts
True

,..
X

-10

0

100

200

300

400

500

600

700

BOO

900

1000

nme steps
-

20 - - - - - - - - - - - - - -

3-sigma points
True

,..
X

100

200

300

400

SOD
Time steps

600

700

BOO

900

1000

Figure 3.5: Comparison brtwePn the trnP nllm-' and anal~·sis of RR SPl KF (E) for
the variable Xl. The anal:vsis from tlw truncated nwcl('~ of :3. 5. and 7 an' shown in
Figures 3.5a :3.5 ·. indicating a poor Pstimation skill in all thn'e casPs .
covarian cP matrix Pt. For RR SPUKF (E). hmvew'r. the total Ynriance P/' is unknown
b ecause RR SPUKF ( ) onl:v calculates the coYariancc P1al:. of the ('nsemhh' spact>.
which is relatPcl to the mnnb er of initial p<'rt urbations. Therefon' . one challengp in
using RRSPUKF (E ) is cletennining thP tnmcat<'d modes. An npproxinwh' solntion is
to explore' the expla inC'cl vari<:mC'P with res1wd to the' total , ·c.- uiancP of / )1a£ (Fi gnrc :~ "' ).

51

---Full-rank SPUKF
--- True

ta'
,..
X

-10

0

100

200

300

400

500
T1me steps

600

700

800

900
-

lb

1000

41-sigma points

- - - True

10

,..
)(

-10
0

100

200

300

400

500
T1me steps

600

700

800

900
-

1000

31-slgma points

- - - True

,..
X

100

200

300

400

500
T1me steps

600

700

800

900
-

d

1000

21-s1gma pomts

- - - True

10

,..
X

-to ~-

0

100

200

300

400

500
T1me steps

600

700

800

900
-

tOOO

11-s1gma points

- - - True
10

,..
X

100

200

300

400

500
T1me steps

600

700

BOO

900

1000

Figure 3.6: Comparison b etween the true Yalue and anal~·sis of RRSP KF ( ) for the
vaiable Xl. The analysis from the full-rank SPUKF and tnmcated mode's of ~11. :31.
21 and 11 are shown in Figures 3. Ga 3. 6<' .
The vanance explai ned increaRes with thr ensemble size, hut the rate is not mn ·h
hi gh er, especially beyond tlw SlZ(' of 11. Howc'YPr. the assimilation C'XIWrinwnt with

11 ensemble memlwrs 1s poor , indicating that such a stratC'g~· ma.\' not ])(' usdul
(Figure :3 A) .

(a)

100

200

300

400

600

700

800

900

1000

600

700

800

900

1000

500
1me steps
(d)

600

700

800

900

1000

500
Time steps

600

500
Time steps
)

100

200

300

400

500
1me steps
(c)

100

100

200

200

300

300

400

400

-

700

BOO

21-sigma points

900

1000

Figure 3. 7: The variation of th RJ\1 E over th 40 variables with time step for the
different truncated modes (i.e .. cu Tmblc size) fur the RH PCKF (E ) method.

3.6

Summary

In this chapter, we eli ussed th

formulation of redu ed-rank ell proximations of

SPUKF. The redu eel-rank approximation was mainly based on the
m st large-seal g 01 hysical pr cesses can b

53

oncept that

appr ximated by a limited munl er of

86

-. 84
~
0

_..
Q)

0

~ 82

loo..

C'a

>

"'C

~ 80

C'a

a.
><

w 78

76

74 ~------~--------~------~--------~------~

0

10

20

30

40

50

Number of ensemble members

Figur 3. : The varianc xplain cl by the truncatrd mod ( nsemhl s) with respect
to the total variance for the RR PCKF (E) method.
d gr

s of fr eedom and their dominant Yariahilit:v patt rns can be cl scrib ed b:v a fi nit e

numb r of modes (Tem am , 1991 · Lermu. ia uc. 1997; Lermusiaux and Robinson. 1099:
Ambadan and Tang. 2009). Wear pli d a similar c nc 1 t as in error-subspa ·e statislical estimation (Lermusiaux nd Robin 'On, 199 ) for the redu eel-rank ar proximati n
of the SP KF , ssuming that the most imp ort a nt st a tistical features of the original sigma-1 oint sr ace can b e stimat d using ·1 f '"'er number f the l minant sigma
poinLs (Man j t al., 2014). Tw methods: (i) r duced-rank sigma-1 oint unscented

54

l alman filter (Data) (RR p
Kalman filt c1 (En Tmbl , ) ( RI

1

1 F(D)). and (ii) U'dllted-lallk ~igma-point \lllSC'llted
P TI\F ( ) ) \\TH' lll\T~t 1gat ed to find the red Heed-tank

apr roximation of th

PCK ~.

In RR P ' K ( ).

\ 'D wa~ apphrd on thr cm·ariRIH'<' mnltix P1n. construclf'rl in

the data sr ac .

he rrclnccd-Iank. ~quarr-wot matrix \\·a~ n~ed to ~Plc'C't lhr most

import ant sigma points t lwl can rrt am the main stat i...,t leal fpa tun's of t lw origmal
igma point.. Th' RI

I CKF(D) can dfrctin'ly san' computational time lhrcJilgh

approximating the rrror covariance P1n hv a fr,\· l<'ading mode's tlw1ehv rPdncing I he
numl er of , igma-p int s from 2L + 1 to 2/

1. \YhPn the state dinwnsion is large.

it i. computCLtionally cxprn.·in' to compnt

and cllfficnlt 1o st mr Pt itsPlf. That is

on of th large ·hallenges of RR P TKF (D) fort hr applJCal iou in a larg<'-dmwnsional
mod 1

In RR PCKF (E). T \ ' D v;as applird on the ·ovariancP matrix P1nf:_. constrnctPd in th
en, mble subspace. Th

comr utational complexity of th

RR 'PCKF (E) algorithm

(O (n 2 x LM )) i mu h smaller than that of the RR PCKF (D) a lgorithm (O(L~ 1 x n)).
The difference in computational complcxi ty l C't WC('ll the two algm it lnus is ~ignificcmt
when th

model dim nsion is much gr ater than the en. emile dimension. that is

u. ually the ·ase for a reali ti c N\YP model.

Data-assimilation exp riment s w re condu ·t ed usmg the LorPnz-96 moclPl.

Three

different data-assimilation methods (S PUKF , RRSP I<F (D ). RRSP KF (E)) \Yere
applied. The result. · of both the RRSPUKFs were com par d to each other and with
that of a full-r ank SP KF. Scn::;itiYity experiments were conJucl C'd using diHcn·nt
truncate l mod . aft. r the singular-valu decomr osition. A small siz of sigma point
failed to car t ur the main fpat urcs of th important error :-~tatistlcs ancllcad to poor
stimate, wh r as a large numb r of sigmc 1 oinls were computationally not feasible.

55

Th r snlt s showed t hFI t the 1 c>rf >rman c or rst imat 1011 . kill of hot b t h

H P rK

m thods were I oor wh n the numb r of ~ i gma I omt s \\' rP 21 or lwlow. hut wlwn
the numlwr >f sigma pmnt s incr a;;ed th ' rstnnatwn accnu-1cy abo mcH'asrd . \Vh rn
the nnmhPr of sigmFI point s was

1 or lll OH'. f11rl lwr nnp1 m·c•mpn( wn:-. Sill> t le Sltg;-

g;rsting that ' 1 sigma points conld he usrd for a:-.sm1ilation . balancing tlw accuracy
and com i utational cost.

ompnn~on lH'(\\'PPll

the• lC'sult s from RR ' P\.: KF (~) all(!

RR P ' K (D ) showed that RR ' PCI\F ( ) was comp<uahlc with thP RHSI CK _,( D )
\\'ith th samr HPl roximate <'rrm nwgnit tHIP. Th P rPsnlts of RRSPCKFs wPr<' comparecl with that of full-rank
hew

PCKF . showmg that. both 1lw H·ducrd-rank schPm<'S

naly is skill dos to that of C1 full-rank s1gma-point 1mscrulC'd Kalman filter

ew•n \\'ith minimum number of. igma points sug;gc>sting a poss1blP solution to applying PCKF in high-dimensional s~·stcms.

5G

Chapter 4

•

Localization 1n Sigma Point Kalman
Filter

4.1

Introduction

In this chapter th pot ntial problem s within th RRSP TKF (E) are inv stigated along
with their possible solutions so that R R PCKF (E) can be implement ed for r alist ic
atmospheric and oceanic mo lels. As w have alre ely se n in Charter 3. th compntatioual co!:lt limits th' si:tc of ·igma poiuhi we can afford. On the other hand. too small
a numl er of sigma points may cause problems such as spurious correlation, inhre cling
8.nd filtrr cbvrrgrncC' (Hamill ct. al., 2001; \iVhit akrr and HamilL ~002: Loren , 2003:
P tri , 2008 ; Deng et al., 2010, 2011. 2012; Tang et al., 201--!a). Thus , it is interesting

t.o xplor

Lhe 1 ossihili!.y f implementing a lo alization strategy for RRSPLTKF ( ).

57

ction .J.

In

w

Rr

p1

1 osmg t1 h~·hnd-locc hzatl n sch me for RR P rK ~ ( ) so

that thr problem. v,·itll in\nc 'chug, filter dn·ngclH C' ami sptuwus coudntwn wlll IH'
minimiz d.

Tumcri al xprrmwnt.' arr rondnct C'd usmg LorPnz-. > moclC'l with ( hrC'r

rl.iffrrrnt munhrr of~ atr va1 inhk

4.2

Inbr

.

Ill

filt r

.

IV

r

n

an

.

pur1ou

r lation
\\'hen the hackgrouu l- 'rror cm·arimH 'C' is nndPlP~timatC'd tlwrP is grPater c<·rtaint,v in
th

forcca.st pstimate and the iilt('l' will give' mon-' vwight to th' fon•c·ast compau•d

to oh.ervations. On th

oth r hand. if th

backgronncl co\·;.uiancc> is too large. the

filtrr will givr lrss wright to t hr forrrn,:-.t . l' ndrrsampling will lc'a.cl to a. plwnmUC'llOll
known as inbree ling, w hrre t h forecast -error covarianc is unclerest ima t eel ( Lon'llC'.
2003; P trie. 200 ). If tlw ha.ckgrmmcl covariance is too smalL th filtPr si m ts giving
maller weight to th observational data. Ignoring th<" observation information by the
n eml l memb rs in the successive assimilation )'des will lracl to t1 process known
as filt er divergenc (H outekamer and

Iit chelL 199 : Hamill et al.. 2001: Ehrenclorfer.

2007).

The 1 ackground ·ovanance g nerated nsing a limit ed number

f ensemble members

often produces sp uri usly large correla tions b et ween grid points, which arc at grrat
clistan es a part (Hout ka mrr and I\ Iitchcll, 199 ; Hamill t al., 2001) . Th( spurious
correlation bet ween indep endent. gr at l~· distant gricl points that arc not ph~·sicall.Y
rela t cl can cxaggerat e Lhr fore cas t -error covariance. may in (rocluce err )r in i he analysis. and can cRusr filter livergencr (Honirkanwr and lit dwll, 190 ; Hu d al., 2010).

5

Itha b

nalso bsrvdi~·Houl kamrand:\Iit hrll(1

) an 1 Hamill rl al. (2001)

that avoiding throbs rvation that ar at remotr locations fromthr analysis p;rid can
abo im] rov th analysis.

Thr errors in the ron1rianc

stimatPd f1om a finite numhn of c'nsf'mhl<'s can oftf'n

cHIS a biased mean Hncl in~uifiu nt vmwm f' in t hC' analysis . In the following fon'ca~t

st c>p.' . t hr chaot 1c d nun nics of tlw mo ld mav t akr the' c•nsc>mhl<' h1rl hrr away

from the tnH' stHlr of lh' svstc>m In thP snc'C' •ssi\'f' assilllilation cydPs. lhP forPcaslC'rror cm·ariancr 1s fnrther nnclrrPstmwtPd. and comparativC'l~· rvpn mon' disrPgard lbe
ol sen·ation mformation. resulting in dPgra led c>nsemhle of forecasts (Hcllnill 'I al..
2001). Tlw met hods to emu bat t llC' problems of nndc>rsampling indnde covariance
lo alization. cm·ariancC' inAation iUld ]oC"C1lizC1tion a.<., in tlw ]o('a] rn . C'llllJlr hausform

Kc lman filtPr (LETEF) (Ott et al.. 200.J. ~liyoshi. 2005: zunyogh c>t al.. 2005).

4 .3

Covariance localization

There are variou , methods developed to ov r om or lessen the unclersampling problem in EnKF. One good method i

ovariance localization pror oscd by Gaspari and

Cohn (1999) and by Houtekamer and I\Iitchell (2001). In this method. the backgroundcovariance matrix P1b is localized by applying a
multiplication) with a distance cle1 nclent

chur product (an element by elrmrnt

orrelation function p. The valu

of the

correlati n function varies from a valu of unit.Y at the observation lo aticm to 0.0 at
some pr leterminecl listances:

(.J.1)

59

h re. ult ant pr dn t iH als
an

rna t rix is f e

c

covanan

matrix (Horn. 1

0) awl t hP n finpJ co vari-

f. or has mmimal errors d n e to . spnri m s noise.

i udirs on

nI

con lu tccll .\' Hmnill t al. (200 1) ancl H utrkamcr ancl i\Iiichell (2001 ) su gg 'S teel that

CUlFI.ly. <'. wrr . moot hrr mul c losrr to t lw tr111 h wlwn t hr inflnrncrs of ohsrr a1 ions
hp~·o ncl a . JWrifi Pd ch. t a n cl' Wl'r

4.4

C vari n

mmt tl'd

infl ti n

On met hod for a dclrrs. ing the unclc>rcst imat ic n of t lw hackgr mmd nror-covariancc·
matrix i cm·ari n c

inflat ion ( n lPrson and

ndnson. 1

: Li et al.. 200 : D eng

et al.. 2010). Ther a re variou: methods to do that. One method a d ls a positi ve semidefinit matrix to t h coYari a n c matrix. Th r i no unique vva:v. h owever , of ch oosing
th

mi-d finit

inflation matrix t

add to th

coYaria nc

m atrix. This process, of

finding the ~ uit ab l p osit ive. mi-definite m atrix. is treat d as R p art of mod l tllning.
cluing a po ·itivc scmi-Jdinit c m atr ix to the backgiouncl-covcuiallcc m at1 ix i:-, also
one of the ways to account for the model rrors in EnKF (Hunt et a l.. 2007). Several
diffrrrnt mo lrl rrror schc>mrs h cwr hrrn propo. rd for t h r covarian r inflation (Li
et al., 2009 ). Th mo. t common . lwmf's <'Hf' additivf' inflation ( Corazza ei a l., 2007).
multiplicativ inflation (Anderson and

nd erson. 1999) a nd hybrid m thods (Li et al. .

2009). In the additive-inflation m ethod an id ntity matrix scaled by a small r eal
r nd m vector is added to the raw covari·::m · matrix P1~e·

clcling a random vector

can act as an additional ource of noise and destroy the correlation st ruct ur

of the

covariance matrix.

In t.hr nmltiplicfltivr-iuflflJion :·whrmr the raw covnriculCr matrix is nmltipliccl b~, a
sm< ll m unhf'r grf'< trr than onf'. This dirl'd inflation of hackgrmmrl-rrror cnYminn< ·r

GO

'qwva l 'll( to th<' inflati Hl of mod<'l stal<'s d tlH' llHHl<'l

1s

Jl OporiJOnal to th
I ackgronnd- fl()l

to lw

(Lr <'I al.. 200. ).

In tlns nl<'lhod. tlH' rmv

I/ IS lllfia {'d hv ct ia< I 01 ( 1

J). wlwr <' (J Js I h<' inti at ion

I ac+gwnnd
0\'allHlH' (

'l'l'Ols Hl<' asst llll<'d

11 H

factm abo call<'d as tlw t tlllclhl' pautm<'IC'r. ...,o tlw nP\\' h<1C'kground cm·arianc \:vill

h:

I I>
I

(1

o) P/',

( .2)

h sr;,' oft h' pau-unPt <'r c,J wav change' <H·c·ording to the' umulH'l of C'Ils<'mhlP llH'lllh 'rs (\\'llltak<'t and Hamill. '...00:2 )

\\'p

n C' t]u..., ...,cJtc'lll<' in ,' p('(JCJll LG alHl in

'hap-

r - . ).Iul!Iph 'cttiv inficttwn i. fim\'-cl JH'IHI<'nt c1JHI cc111 avmd tlH' di\'C'IP,<'IlCC' awl

drng pwblm of tlH' filt r of finite <'nsPmhlP

inbr

11<'

(.\nd<'l'-tOIL 2001. II mnill and

\Yhit aker. 200:: : Fnrn'r and Bc•ngtsson. 200/: l\arspc·ck and And<'tsou . 2007. )k<' C'l al..

2007).

4.5
Th

Localization in RRSPUKF(E)
locali zation . ch('l1H' ns('d in t lw EnKF. addrc'ssc•s t h(' computational dfi cic'IH'V

he main chall en g for implem nting the localization in the

PKF is tha t th s1gma

p oint s ar gc>n rm ted based on t lw globa l ana lysis-c>rror ovariance. As w

have al-

r ady disc ussed the glol al ana lysis wit hont locali zation for a largr- dimrnsional mod l
wonlrl

a usr t hr follm\'ing problems: (i) it would hr compnl a t iona1ly exp ensive Lo

upd Rle equa tion (3.7): and (ii ) tlw anal_ysis will h r drgrad ed h

anse of the spurious

correlation h rtwc'c'u the rc'mote grid points . In tlm; Secl.ion (4.") we propose a h_yhridlocalization S('hc>mC' for thr RRSP ' KF (E ). in whi ch mocld stale's are analyze lloca lly
at each modd grid bnl the sigma poiuls a rc g<'uera t cd globally.

The 8l m os1 h ric or ocP<lmc model slat' v<'ctor a t time I can be d cno lC>cl by X 8 , 1 ,
\Vh ere s is t lw location of t lw grid in t hr mmwrical mmlrls a nrl is nsually fixrcl etncl
clis r te.

In generaL s(lnt.lon . .::) bet:-; three compcnwnts. the geographi cal lati tude

(lo t ). longitude (!on) cmlll1C' altitude(.::). For simplicity we discnss the loce:1 lizat ion
in the case of a une-climensional model that can l>C' easil_y ex t ended to a l wo or three
dim nsionRl model. Therefore . in ou r cases has onl.v onr coordinate, sa_y s(lon). To
perform the analysis locally. first the local vector containing the information from the
localization r gion is definrcl (Patil ei aL 20()1 ; Ott et al.. 2004). If the analysis grid
p oints are centred at the grid point l' with the location coordinate s( lon) ancl the
lo calization radius is d. then the grid points inside the loca l region are X s10 ,

rt 1

to

X Stan rl• 1 . There is a total 2d + 1 gri l points centred a t t'. The eqn al ion for the local
an a lysis at grid point u is :

(-± .3)

where th subscript l ac in ibc <'qtwtion above denotes tlH' local snbspacC'.

s to,,l arC'

the local vecl ors from the model forecast . A toc,t is t be I\ aluwu gain en kula t cd in the

G2

loca l sn bspacr:

J\.to<
· .I -_ pry
( p!!Y ) - 1
/111 I
In< .t

( 4.4)

\\'lwrP P1 ~~. 1 and Pt~,;. t <-HP t lw cross cm·nrHmcP and projPct ion covariance in t hP local
snhspa C' respPcti\'d)'. and nllculat('cl <H'C'ou ling to tlw c>qnal Jmls given in SPction 2.2.3

of

hapter ~ ·

sinular opC'l n t ion 1:-. 1H'r fonu eel 011 ('<H'h llH H lei grid I' nsiug oul.v t hr ohsPrva t ions
local to 1' Hlld the mw}vsis x;~. / lS l'C'taillC'd .

he global-analysis VC'ctor X;' is t hC'

combination of all t hC' x;~ t ·

The 0aloha! anah·:-.Js-c
·m·mi;-mce mal nx iu the ('llSC'mblP spacC' is cons! rnct C'd nsing t hC'
.
cqm1tion:

(<± ..5)

On 1 rforming \ 'D on l he PtE:

PIaE -_ pnca(Fa)T
I 't
I

(4.G)

Then the leading f'igPm'c'ct ors are fmmd w-;ing the equation :

11

\o)T
* po1.1
I

\ '11 _
(. \ 1

•

(X;r

xr V * Ft1 l

( 1. 7)

The glol al sigma poi11ts nrc gen<'rnt<'d Hc·cordiug to the ('CJ1li1tl()llS P,l\'Cll m ~Pctiou
.2.2 of 'hap ter 3.

4.6

. nt
Num ric 1 xp r1m

rnnwriral C'xpenm 'll(s an' conducted to test tlw 1wrformancc of RRSP K ( ) with
a h)·hricl-lorahza t ion sclH'llH' on t 1H' Lorc'nzlllHllllC'L

1 9 )) nsiug (lll

)

model ( Loren ~\, 1

G; Lorenz and

'SE sdH'llH'. The sc·nsitivit .r C'XJH'rimcnls in this sPc--

lion ar<' motn·atPd hv Ott •t al. (200-1) .
Thr mod<'l c•quat ion 1s gl\'<'11 hy :

(1. )

4.6.1

E x p e rim ntal s tup

In our xperinwnt s. thE valne of the ext rrnal forc-ing F =

is nsrd and the fonrt h-

order Runge-Kntta scheme is nsrd for the time integration of the model with time
step=0.05. The experiments are condnc-ted for scvc•ral sPt lings of llH model h)· varying
the numb r of spatial cooHlinatPs (}\ rl = -±0. 0 an l 120) as in Ot l et al. (200-±). The
nature or observational nm is created by inlPgrating thC' llHHlPls m'cr 3 '0() time steps
l y using the fourth-order Rnnge-Kntta scheme'. The intC'gralion sl<'p is sct to 0.05
(i.e .. G-honrs). The lata from the model trajectory in th<' time stC'ps () 10()() for the
initial spin-up are cliscarcl('(l awl tlw cL-1ta from the tmH' st<'p 1000 ;:)'- ()()me t<lkt'n
as the truth. Th' ohsPrvahon vPctor (lt ench time ;-;[(·pis CT<'nh•d h)· ndcliug random
normal error to the truth , wlH'lC' the ohsetT<ltion <'rrm-<·m·mimlc<' ll1 <1tri:x R 1s an
iclrntiLy matrix. TlH' RRSPCI\:F(E) lll<'thod is Hscd to nssJmilntt• the uhs<•n·ntJous
(j 1

v ry I en time st ps ancl all the variables are observ d .

s the Loren z- G moclPl h as only on spatial clim nsion , the localization is carried out
as follows . If lh<' analys is Rrid is d enot 'cl by I ' ancl the localization raclins by d , the
g rid pomt s ccmsidPred in th<' local analysis (U(' t' - d . ... 11- 1. 11, u + L ..., v + d at an
incremcnt of one grid point in cit her direC' I ion .

In t lw fir ~ t :->Pt of PXJWrinwnt s, diffrrC'll( din)('n siona.l SIZC' of t }JC' s~·stC'lll (

1

.

40 . 0 cmd 1-0 ) a rc' consider ed a nd assnuilatC' lhP synthC'Iic obsnvations nsmg
RR , PCh.F (E ). Here no locali zati on is a ppli r d ancl tlw analysis is pc'rformed on the
g,l )hal dom a in us ing a \'arying numb er of sigm a point s. The n ew sigma point s are
g n erat ed follov: ing <' 1na tioni'> (:3...! ) to (3. 10). There' is no cova riance' inflatiou usrd
( i. .. 0 = 0). In t lw :-.rcoml set of exp erimc>nt s the localiza h on r a di ns is fi xed CI S d = G,

whi his th '-' d r fa nlt va ln r used for the Lorcn z- OG sys t<'m (Ott P( a L 2004 ) . The infi a tion p a ranwt cr is fix<'u t o zero (o = 0) and how m a n.v ensembles a rc required t o
rea ch th minimum RI\1 SE is ch ecked in the cases of sys t em s with differ ent dim n. ion

(N x = 40. 0 a nd 120). In the thiru se t of experime11t s. the iuflation parm11et er is set
t

c/J = 3% a nd the lo a lizcttion radius d = G. In the fourth schem e. the imp cd of th e

localization r a dius (d) on th assimila tion an alysis is explored . For this. the following
p a ram t er values a re u sed. N am el.v. the numb er of m o lel varia bles (

·.r

= .JO ). numb er

of ensemble m emb ers ( k = 7), c:m d inflati on pa r a m et er ( r:p = 3<;/(). As I h e LorC'n z- 96
model h as only one sp a tia l dimension . the lo ca l r egion is d cnotC'd h~· a single :-:~p ati a l
index (d ). For a localization radius (d ), th e observations from ... r/ + 1 grid p oint s ceutrecl at th ana lysis point a re used . Var ying d from 2 to 7 is u sC'd in thi s expninwnt.
In the fifth set of exp criment s. the va hlC's of d = G. <md k = 7 ctrC' kept C'O llsl<mt hnt
th e value oft h e infi nti on paramet er r/J \'<Hies from 0 I o 0.07.

Th

quaE t.y of 1he cu w lysis is llH'as urPd in ( C'l'lll:-1 of H ISE of I hv mwl.\·si:-1-c' n:-><'mhlc

G' )
r..

mean against the 1rue stat C' . Tl1 C' cle t a1·1s o r· t 1H-' specJ·1·1c experiments are summarized
lll

(\ hl

~1.1.

ahlc 4.1: Lornlllal 1011 C'XIWlimcnt cl<'1 mls. T 1.2 and T.J .. are rxpPrimrnt munlwrs
for 1lw Pxprrinwnts to grm'lHI<' Tahl<' 12 and T1hle .-J ..3 rrsprctively
~lode!
Localiznt ion Inflation
NmnlH'r of
~ xpt.
Assimila t i< m
\'1-HHli>J 'S
rmlins
par am 'I C'r ensrml)lC'S
n 1unl wr nwt hod
( ,\'7 )
(d)
(cj;)
(k)
()
1
HRSf>CJ\f(E) .-JO. 80 ;-md 120 ()
3 to 91
()
RRSPtTKF(E) -!0. &l c\lld 120 G
2
3 Lo 31
().()3
RHSPl , 1\F(E) ..!(). R() ;-wd 1:20 G
3
3 to 31
(J.(r)
2 t () 7
-!
RRSPCI\F (E) .-J()
7
r:
() t () 0.07
7
RHSPL'I\F (E) <-!()
G
f).(n
2 1() 7
3 to 31
RRSPCKF (E) .-J()
T ·1.2
() t () 0.07
3 to 31
1D
G
T -1 .3
RR~SPC I\F (E)
(

4.6 .2

R

In this

ecticm the lH'rformancc' of thC' byl>rid-localiz<ltion scheme' is assessed in the

ult and di c u

Ion

numl r of ensemhlc nwmlwrs ( k). iuAat ion factor (9). localization rHclius (d) and t hC'
numb r f model \'cHicl hlPs ( S 1 ) is chC'ckC'cl on the quality of I hC' analysis.
The time mean H IS error of Uw RHSPUKF (E) schenJC' as a fund ion of nnmlwr
of ·igma points fm diiferent sizP (.Vr) s~·s(C'lllS is shown in Figm(' 4.1. If the nwdPl

has more degrees of frePdom it reqnires a greater nmuhcr of ensC'mhks to clescrihc
t lw C'volntion of t 11<' nwan and coYarianc<'

From 1he U'snl t s of I he first <'X I><'l'llll<'ll t

(Figure 4.1) it can bC' sC'en hm\· t lw R lSE \ '<Hi<>s as <1 fnuct ion of <'nscmhl<' llH'mh<'rs
for diffPrPnt dinwnsional sizes of tlH' s~·s t<'m (Sr -

.J.O . 80 and 120). It rnn nlso lw

sC'en that the Rl\lSE cmlV<'rg<'s to approxJm<ll<'ly 0.00 irr<'SlH'<'tl\'<' of the diuH'llSJon of

the system. Syst('lllS of diffen'ut siz<'S llC'<'d diff<'ll'llt mmdH'rs of <'llS<'llthk nH'lllhl'rs to
(j(j

r a h th mimmmn valur of R L

. \\.hC'n the sys t m dimension is small (

.r

= 40 ,

the sohu line). a ~ mall '1 uuml){'t of siiSm a powt s i:-, JH'<'<kclto t each t h e 1 'vd ofi' valt1,

f R J\ 1

~.

s thr sys tC'm snr incn'asrs. morr sigma point s are re Jnired Lo r arh the

lrn'l off ,·nlnr of 11 ~1. E a:-. rvi dr ucrd in t hr ('<lsr of N r =

0 and

.r

-= 120 .

No localization
5.5

--Nx=40
- Nx=80
· · · · · · · Nx=120

5
4.5

\

4

'

.

'\

\
\

''

3.5
w

(/)

:IE

a:

\
\

3

\

'

\
\

2.5

\
\

''

2

\

' ....

1.5

\

1
0.5

0

20

80
60
40
Number of ensemble members

100

Figure 4.1: Thr lime mean R!\IS error of the RR SPC I\F (E) sdH'lllP ns <-1 fnnction of
number of sigma points (ensemhlr nwmh r rs). Thr results nr<' shcm-n for diH'neut size

(Nx) syst em .
In the second se t of <'XpPrinH'nls tlw locali.lalion is introdlH'<'<l. Th<' tmw llH'<lll R).IS
error of tlw RRSP 1\.F (E) sdH'llH' as a fmlC'liou of numlH'r of sigmn points (<'llSPmhl<'

ll1Pml ><' r.) w]H'll ]oca li ;m lion is IIS<'d for diff<'r<'lll :-.i7<' (Sr) sysl<'lllS JS shown in Fig(i7

s arC' d0snih rl. by th lh lo cal vc tors. Figur 4 .2 shows

ur 4.2 . HC'r th covarian

Localization
5.5
Nx=40
- Nx=80
·" "" Nx=120

5
\

4.5

'

\

'

\

'

',

\

4

'

·.

\

'

\

'

\

3.5

\
\

w

(/)

:2

\

3

\

a:

\
\

2.5

\
\
\

2

\

\
\

1.5

\
\

1
0.5

0

\

5

--

~ . ~ .~. ~ . ~ .~. ~ . ~ .·

25
20
15
10
Number of ensemble members

30

35

Figure 4 .2: Th e time mean RtiS error of the RRSPeKF (E) sch lll<' as a function of
number of sigma point s (ensemble mcmlwrs) when lo ca li zation is used . Th<' results
a re sh own for differf'nt size ( Sr) system.

t h at introducing the loca li zat ion signifi c<mtly rPdu c:cs th e numbn of sigm a points n'quirect to achiev€ 11H' minimum ('stimi:1tion <'rror . \\' lwn thC' s~·stcm siz(' .Y7

-

.JO

(sohd line), the wuuh er of sigma point s n ' qnir<'d to <lcbic\'(' the minimum I I\ISE 1s
aronnd 11 and it does not dwng<' with fnrt h<'r llHT<'<lSl'S in th<' nnmlwr of s1gma pomts .

\ h n the sy s t em sizes a r

1

.r -

0 and

x

= 120, the minimum numh er

f sigm a

point s required t o reach 1h minimmn an a lysis error is 15 and 21 , res pectively. Wh en
omp aring Fig11re 4 .2 with F igur 4.1. it can h e seen that as the r sult. of l calization

t lw nnmlwr of fligm n po int s 1 ('(Jll irr d t o nd1irYr t lH' minimnm R l\ISE is signifir c:mtly
redu c d .

h e rcsult s m d icate tha t tlw backgro und nncert aint y is low-dimen sional

a nd nm h sm a llc' r th c:m tlw m un lwr of d c'glC'C'S of fr re lom of th e sys trm . Another
a dYant age of this loralt za t ion sd wmc' fo r a pplicat wn in a

1 is 1ha t a na lysis can

b r d on e> in p arallel for each local n 'gicm in clep eml<>n1 of th e sys t Pm dimen sion , m aking
the m th d snit ahlc for p a ra lld comp ntation.

The tim e llH'<Ul R.:\I S en or of thc RH SP C KF ( ) sch em e as a function of the nmn h r r
of . igm a p oint s \\'he'll h ot h loca li 7ahon and inf-l ati on are 11S('d fo r diffc'r r nt sizr (N:r)
yst ms is sh 0\\'11 in Fignr P -L3, w hrn' b ot h locali z;1Jion and infhtion <tr r introd ucc>d .
Th R:i\1 E read w cl t h e minimu m \'a lur at a sigm a p oin ts size of scvc'n for alllhr 1hrPr
ca,

wit h diffe rent s)·stem d imen si n (1\ rJ = 40. 80 a nd 120) and d octl not ch a nge

furth er v, ith increase in t h e num b er of sigma p oint s uscd (Figure 4 .3). This result h as
imp ort ant implications v,·hen we use t he ' PKF on a high-dimension al sys tem . b ecc:w . e
it addr sse the imp orta nce of loca lization and coYa rian ce inflation . In t h r a bsen ce of
p ar all l com puta h n . t he 1im e reqnired for th e local analysis incr rases linearly hut if
w can use p a r a ll l algorit hms t he local analysis can b e d on e simult aneously. One of
the very import ant highlight s is the 1ime used fo r local a u alysis is indepen dent of t he
dim n sion of l he sys t em ..VJ. and the R l\ ISE is indep endent of sys t em dimension for
a simple m od el su ch as the Lorrnz-9G m udel. Therefore. S .r = 40- \·aria ble s~·s t cm is
used for furth r exp erimeuts.

The result s of (he four! h se t of exp erim ent s (Fig nrC' -1.4) slww h ()\\' R l\ [SE cbangPs wit h
localization radius when thC' nmuh cr of t' ll SC' lllhk ll H'lllh C'r s (k

7), nnd th e inll ntinn

Localization + Inflation
2.8
Nx=40
- Nx=80
· · · · · · · Nx=120

2.6
2.4
\

\

2.2

\

2

\
\

w

~ 1.8

\

a:

1.6
1.4
1.2
1
0.8

0

'

5

'

20
25
15
10
Number of ensemble members

30

35

Figure 4.3: The time mean RI\I error of the RR PCKF(E) sdwmc as a function of
uumLer of sigma point~ (eusemLle member~) when both louilizatiou c\llU iuflHtion clr<'
uscJ. The results arc shown for Jiitcrcnt size ( N.r) s.vst em.
pcramet r (cp = 3o/c ). and the mm1l>er of model variahlrs (}Yr - 40) arc fi>::cd

ThC'

minimum RI\1SE is ol>tainccl wlwn the localization radius is d - G. suggesting that it
may be a good choice for the furl hrr data-assimilation experinH'lll s with LmTnz-90
system (Figure 4.4). This result indica t C'S that the backgrouud llllC'<>rt nint)· can ht' wdl
approximated iu a low-dimensional (d - G) local space pr<)\'id<'d tlwt th ' \'HhH's \lS('d
for ext rna l forcing (F) . nmnlH'r of ('llS('lllhlt' llH'llllH'rs (k) and th<' inflntinllJMr;ull<'l<'r
( ~) arc

, 7, alHI :3. n·spcctivel.\'. \\\, h<W<' cml<ltH'I<'d mor<' S<'nsiti\·ity <'XP<'rinH'll!s

70

0.965 r-----.-----.---.----------r-----.--------,-----,

0.96

0.955

w

0.95

(/)

:2
0:

0.945

0.94

0.935
0.93

L _ __

2

____j _ _ _ _ . L _ _____J__ __ _ _ __ __ l __ _J . . __

3

4

6
5
Localization radius

7

8

__j

9

Figure .J ..J:: The time mean R~l C'rror of thP RRSPCKF(E) scbc'lllC' as a fnnction of
localization rnclius. Th(:' iufiotion parameter is ().():3.
wher the value of variancr inflation paramctC'r C/J is fixc>cl as 0.0:3. lmt the localizntion
radius (d) varic>cl from 2 9 an l the munb r of rnsc>mhle nH'mlwrs ( k) from 3 31 and
th

results are summarized in the THhle --!.2. Iu e~ll Ci:ls<'s. wlwn tlw size's of sigma

p oints ar

< 7 rC'sultecl in lcugc> Rl\ISE irrespC'ctivc> of the locnlih<llion radins . This

su ggests th8i i h state in the respect ivc' local sp<H'C' is higlwr-dimC'nsiouHl nnd n'qnires
more sigma points to approxima I C' I lH' lmckgronnd-cm'<l n<HH 'C' mn I nx ThC' minimum
Rl\'ISE is oht aiuC'd for <lll C'nscml>k sii';C' of 7 <llld lo<'alizn t ion mdnts of 6.

71

Tahlr 4.2: he drprndrncr of limP mran RM
on the localization ra li us ami. number
of sigma pomt s The inilat iou codlicicnt is' () 03
Locc liz at ion N'umlwr of sigma points
ra !i us .j,
1.5
7
3
11
27
19
23
31
2
2 ..5940 0.9579 0.9769 0.9904 0.9973 0.9996 1.0008 1.0007
3
2.3366 0.9"i62 0.9662 0.0912 1.0003 1.0016 0.9992 1.0048
--1
2.3..!00 01Jf=1--1 O.!J7f,9 0.9925 0.9967 1.0011 1.004.5 0.9999
2. ~1076
5
0.9..! 73 0.06--19 0.9879 0.9952 1.0020 1.00.58 1.0001
2.:3283 0.0300 0.0771 0.9892 0.9980 1.0008 1.0014 1.0003
6
7
2 3923 0 0-108 0.9708 0.0853 0.9060 1.0007 1.0006 1.0013
2.3220 0.9'-22 0.0665 O.!J902 0.9977 0.9996 0.9998 1.0002
8
2.37/9 0.9547 O.!J722 0.9903 0.9996 1.0000 1.0008 1.0033
9
°

In the lilt h ~d of ('XI H'l'i llH'llt ~ w h<'ll 1 h<' value~ of d = G. au cl k -= 7 arr k0pt constant
(Fignrl' 4.5 ) and thc inflrltion paranwrc'r varif's, tlH' miuimnm Rr-. ISE is ohtaincd <lt
th value of o is 0.0 or :Ylc. It can hr also sePn from Figure ..! .5 that whcn thr inflation
pannneler o = 0. R}.L E i~ large indicating 1lw necessity of inflation . I\ fore sensitivity
xpt>riments arc cotHluctecl to clwc k the influen cr of i11flation whC'll a differ 'nt number
of sigma points are usPd and the results arr s1m1marized in Table' L1.3. In this set of
exp riments. the localization radius d = 6 is kept as a constant and other pan1met ers

(dJ and k) are Yariecl. The inflation parametC'r ¢ is variC'cl from 0.0 to 0.07 at an
increment of 0.01. and nnmber of C'nseml k mrmbers (k ) from 3 31

The results

are summarized in Table --1: .3. \ \ 'lH'll the infle~tion parameter CfJ = 0 results in large

Rr-. ISE, ·w hich shows the importance of inflation. to cope with thr inbreeding nncl filter
divergence . It can he also sren from th e TahlC' --1: ..3 that as the mtmlwr of ensembles
in creases , t.hc , ·,due of 1he inflatioll < oc'Hicicut ll<'Cded 1o reach the miniunu11 H}.lSE

oC'crC'asC's . ThC' optimnm inflc tion codfici<'nr fm diff<'n'nt C'l1S<'mh](' si7c's may not
be the same (Table 4.3). For example, \\' he'll thr <'nsc'mhlr SIZ<' is I tbP minimum

R ISE obtai11ed whrn t lw inftnt ion paramct n is O.CU, wh<'n'n:-1 wh<'n t h<' numlH'r of
sigm a points is 11 the minimum R\TSE is oi>LlinC'd whe11 the m[Jntiuu parc\uH't<'r i~
0.01.
72

2.8
2.6

4~

2.4
2.2
2
I.J.J

en

~

1.8

cr

1.6
1.4
1.2
7

0

-

-

o.o3

o.o4

-

0.8

-1..

0.07
0.02

-

o.os

Inflation coefficient

4. 7

...

....-

o.o6

0.07

Sununary

I n Ihis chapter. " hylJrid-loc,]izat ion ·'<·heme """" in! rodllr·r·d for llJr• illl.c;p l "I\P(E:)

data-assimilation ll](•/ hort. comiJininR lor·aJ ilnaJ ·.sjs '"" J RloiJ,i] gc'll('tnt i(
1

Points.

The 1ocali>-Cli io11 sc]l< 'll)(• was IJ, "'"J u I/Jr, '"'"'llJ >

r
111

0

o[ .'>iglll a

11 . .11 1·
lllodeJ, Ihe sl <l/ e vr•ct ors of forPr·ast """''ria illiJr•s !J" h1 · ] , .. ,1 , J,,
, ..
OJJ
Ja 1. ",,. "
1 0

'

11

' '

~loi >a/

11 1
' ' ( (' 0 llliJc1I SJIJa//, '1'

Tabl --1.3: Th dcp ncl nee of timr m an R~I Eon the inilat.ion efiicicnL and numb r f sigma 1 in( s. h loc alization radius' is 6
Inflati n
Nmnlwr of sigma points
ocfiiC'i nt ..!- 3
11
7
27
15
19
23
31
4.3628 2.G7r:.s 1.1--1<-!6 0 .9831 0.9922 0.9412 0.9466 0.9543
0
0.01
3.1660 1.00<-!3 0.9266 0.9592 0.9732 0.9823 0.9908 0.9915
0.02
2 .3'-181 0~539 0.9699 0.9950 0.9990 1.0000 0.9991 1.0018() . 0~3
1.7910 [) .9:3()0 0.991.S 1.oo.r. 1 1. 0072 1.005.5 1.0061 1.0059
1.r:470 0.98<-!6 1.0097 1.0082 1.0103 1.0106 1.0089 1.0073
0.04
o .or:
1.-1610 1.0028 1.()1;3() 1.0170 1.0127 1.0101 1.0090 1.0084
1.3 105 1.0116 1.0220 1.ff201 1.0133 1.0111 1.0081 1.0039
0.06
1.2880 1.0132 1.0224 1.0233 1.0136 1.0100 1.0084 1.00'16
0.07

dimension than that of the full state vc'ctor in lhC' global clomain (Ott E'l aL 2004). The
localiza.tlon was importallt lwc <\ llSC' it makC's tlw fi.]tf'I mon' cnuqmtatimmlly fC':-tsihlC'
and mew mak the analysis estimate optimal by red ncing l he problems of spurious

COlT lation, inbrcC'cliug ancl fi ltf'I diYC'I geneT.

The localization v;a,· performed h.v updating the analysis at racb modd grid point
using the ob .rnttion within a predeiinr 1 local subspace cc'nlred at that grid point .
Combining the local analysis at each grid point the global-analysis vector is constru cted.

Th

global anal)·sis-covariance matrix was cons! ruct ed in the ensemble

space and the global sigma points were gencrat eel as in HR PCKF(E). CoYariancr
inflation wets etl. o etpplird for the' RRSPlrKF (E) tn oYerrome illC' nmlNC'stimation of
covariance mal rix beca nsc of th e l oo small m1mlwr of sigma points .

Numerical experiments of onr met hods nt ilizrd the LorC'nz- 96 model. ThE 1wrformnnce
of the hybrid-localization scheme was nssrsscd in the' presc'ncr of \'arying pnramdt'rs
1

su ·h c:ts thr ll1lllll wr of sigma points (k) , inDntioll fndor (~ l). locali za tion nHlius (d)
and th

m1mher of model varinhles (.V,) . \\' hen !hC' locah 7.nt ion \\'as impl('l1H'lltl'cl.

tlw 1mmlwr of sigma point s H' qniiTd !o aclll<'Y<' t lw minimum Til\lSE was s igmfi.ca nt]y

71

of m d l va ria bles ( :r).
introrl.uced , th

01 timal

\ Vhen b oth the' lo calization and in ilat ion sch em : :; wer e
s tima t

o h!.ainecl wa~ mu ch less t han the cas

wher

only

localizatiou was mtwduccd . . :_\.not hc'l iutclC'St ing findin g from t hC' same cxperi mcut
was th a t the acT nra t e ;.m a h ·sis o ht a inr d was indep nclr nt of t lw sys t em dimension of

X r in t h casf' of a simple' s.vst em hke Lor C'n z;- 9G . If t lw rpsnlts can h gen er a lized ,
thi s will h e a p ositivP s t Pp tc wards !h C' pot enli <-1 l a pplica tion of R RSPUKF (E ) for
a n ocranic or a lmosplwric

7

·~ I. Tlw sC'n si t 1vi l y expninwnt s a lso provid , suit a h lc>

p a r a nwt 'r ch oic<'s fo r the a.-;sn n ila t ion C'Xp <'rimPnl u sing t hC' Loren z- G modc>l. For
th e Lor en z X .r = -!0- \·ari a hlC' syst C'm . a ch oice of k = 7 . C/J - 0 .03 and d = G ga vC' the
minimum R~l . .

TlH' nnm N icnl c'xprrim r n tnl rrs ult s incli ca t r l t h at loca li ;m t ion a nd infl a ti on in
RR PeKF (E) wa!-. vP ry pffpctin' in imp1 m·i ng its p r rforn uu1 cP.

K e~·

pract ical as-

p ct s of the locaJizn tio n sch em e in RR . 'PCKF (E ) d a t a-assimil a ti on m c> thod were: (i)
a high dimen sion a l estima t e of t h e hackgronud covaria n ce was calcnla t eel in a local
subsp a e mu ch sm allr r t h an t h e glob a l d om a in u sing a sm Rll numb er of en semble; (ii )
the an al~·!::l is a t each local : :; ubsp ace can h e d on e indep enden tly m a king it suita ble for
p a r a llel comput a tion: (iii ) only low- lc>vel m ?tt ri.x op er a tions wC' re req u ired: a nd (iv)
th r was p ot enti a l t o obt a in t h e optima l es timat e at a very m o d es t os t comp a red to
Lhe case where there was no localizatiOn u sed . One of the limit a tion of this s tudy was
tha t . the Lo renz-9G m o d el. U!::led as t h e> a~simil a tio n pla tform , was a ·· t o.\·.. model: a fnll
a tmosr h eric or ocea ni c GC I is m u ch more' com p lrx. <:1 lld t h (' result s fr om Lorrnz-9G
m o d el can , at b es t . only indicat e som e general trends a nd p ossihk lw luwionrs.

Chapter 5

Application of RRSPUKF in an ENSO
Model

5 .1
This

Introduction
ha1 ter fo uses on the investigation of the pos~ihility of applying a reclucecl-rank

SPUKF to a realistic climate model. Th important pror <:'r!i ~ and th cstimHtion
accur cy of the recluc d-rank SPUKF are explored. Emphasis is plc-1cecl on the implementation of the reduced-rank SPUKF ancl on il~ com1 arison against a full-rank
SP KF using a realistic climate moclel. whi h has not h en rt>ported before . The
model usecl is the LDEO r: (Lamont-Doherty Earth Ohscrnrlor.v. version 5) version
of the

z biak- 'ane El- Niilo prediction model, wbi ch is nn in l<'rmedin t P c·ompl('xit y

model. The RR SPUKF rc'snlts Hrc' also comparc'cl with the' EuSRF n'sults .

7G

5.2

Mod 1 and m thod

5.2.

Z bi k-

an

M d 1 LDE 0 5 V r Ion

Thr rC'alisi ic mod l w<' nsc' m t hi:-, study is the LD "05 vC>rsion oft hf' Z ' modr>l ( 'h n
C't al.. 200-J). \\'hicb is an

', '() ]Hc>cliction nwdcl (Zc hiak and

'anc. 19 7).

NS

Is

the stronges sigwd in lh<' n1rialnlity of thf' global climate' system em the interannual
time-:t ales.

It occm:-, in t lw twpwal Pacific OcC'all in C'gnlarly a ! intnvab of 2- 7

y0ar. ( nrl. infinc'nc<'.' the global < linJatC' 1hrongb tclrconnC'ctions (\1\'ang <'1 al. , 1999).
In C'auadil. thr largrst internnnnal vmiation in \\"illt('l trrnpcratnr<' is inft11<'1lr<'d ·hy

E r 0. Dnring the \\·arm pha~e of

I'\ SO. most of

anada experiences cl hove 11ormal

wint r t mpC'ra t nrc's and bC'low normal summer precipitation (HsiC'h et al.. 1999).
Dery and \Y od (:...005) found a corrclat ion lJC'twr<'n large-scalE tc>lccmmC'ct ions snrh as

ENSO to the total annual freshv:at 'l' chschargr iu G4 rivrrs in northrrn CanacL·l.
tudi s condn ted h)· Ch n et al. (200-1) suggest that model-based E

T,

0 predict ion

depends more on the initial conclit ions than on unpredictable at mosphcric noise. In
this st ud)· we use the intermediate-complex ZC model as a represented ive of a realistic
climat model to assess the perfonnancf' of SPKF algorithms. The ZC modrl has b 'Cn
widely used for predicting the timing. pha.s e and intPnsity of ENSO events for both
experimental and operational purposes since the late 19 Os (\\'chstPr all< l Palm<'r.
1997; K a rspeck and AndPrson. 2007).

nother n1timwk for using tlH ZC nwdd is

that we can afford the full signw points rC'quirccl 1>.\· 1he full-rank SP1. 'I\.F met hod for
assimi lation ancl cmnpar their pcrformanc<' ,,·ith RHSPPI\J~ mcthocl.

The

z is an anomaly nw< kL wb ich com putt'S <lll<>Ul<llws of at nwsplH'ric and <W<'<lll ic
77

fic,lds, relative to a prescribed mcmt hly llH ''Ul dimat.ological In ·kgr nml. The dyuamics of thr atmosphc'ric com1 on 'nt of Z 'modrl arc i:>imilar to that of

ill (Hl 0), using

a stc'ad)·-state. linPnr. shallow-\\'a(c'r 'quat ion on an cqwtlorial b '(a plan .

he atmo-

sp lwri c rirculatiou i~ fmrNl hy n h<'al ing anomah· thnt dqwncls on th<' h<'at-flu x dn<'
to '.

anomah· and mcnst nr c·om·c'rg<'nce paramC't erizC'd in t c>nus of surface wind

cmn·C'rgPilCC'. Tlw at 1110~phnic modPl genf'ratc•s the wind fip)rJs which arc then conVC'rtC'd to stlTs~ cmomnlie's that force ' thC' oc·e'e:m mod '1. ThP dyuamics of thP ocean

compon 'lll of Z ' 11~<' t lw redu <'d-gnl\'ily nHHlC'l. The dynamics arC' modelled as a
single haroclinic mode• oft hC' s hallm\·-wa t <'r c•qua t ions acting 1H' lH'H t h a shallow snrfac<'
mixC'd let \'<'r ( Kars pc·ck awl Anderson . 2007). The snr facC' curn·n (s arc gC'nf'ra l cd b.Y
spinning-up l h<' modd with nwut hl~· m<'an dinwt ologinll wind .

h<' t lwnnodynmnic

equ a ti ons clc>scri lw the ~· 'T e:m omc-d y and lwRt -flux chan g<'. Th<' model tinH' slc'p
i:-; 10 clelys. Thl' wodd clonwin i:-; CO lliiiH'd to tlw tropical Pa('ihc o('('(lll (10l.:r 0 E-

7 .12

\\' . 29

'). The grid fen oct'clll dynamics is 2r longitmle h.v 0.5° latitude

and for ' T physi cs and the at mosphcre the mo lcl grid is 5.G25o longitude hv 2r
latitucl . Further details about the ZC model can h e fonnd in Section 2.3.2 .

5.2.2

Methods and experime ntal setup

\Ale apply 1he R R SPUKF (D and E) cia t a-assimilation met hods to assimilat <' SST
anomali

int 0 the ZC model. The ckt mlcd deri\·a lion of 1>ot h t lw clnt <1-a:-;similat ion

methods is xplained in Chapter 3 (RRSPl'l\F (D) 111 SC'ction :3.2. 1 <mel RRSPCI\:F (E)

in Se tion 3.2.2). Th<' C'stimalion skill of reducC'd-nmk met bods h <1\'P bC'en compnn•d
t.o a full-rank SPUKF (rc'\'iewed i11 Srdiou 2.2 .3) nncl EnSHF (s<'< ' S<•ction 2.2 .2).

with random vc r t ors ma.r act as a sourrc of a.dclitional nois and destroy the correlation

Lalancc. The mfla L('d covariance mat1ix 1t1:

pb
I

(1

(jJ) p Ib,P

(5. 1)

when~ P111 r is tlH' raw background-error cm·arian cc: ¢ i · th r inflation cordficient usually
ohtainc>cl h.Y tuning <>xpennwnts (also callC'd tunable param<'IC'r) . In this study. thr
vahw 9 is s<'l to 0.1.'>. wlllch \\'a~ fmmd to rPsnlt in th<' l> C's l analysis ba.s rd on a trial
and c>rror method (md the' literatnr<' (I\arspC'ck an d

5.2 .2 .3

o an an

ndC'rson. 2007 ).

1 ali zat i n

To avoid th possible impa ·t of spurious corrc>lation on the ana lysis , w a P}ly covari an c l calization in this stud~· (Ham ill C'l aL 2001: Hout cka uwr an d
The 1 ackground crror-cm·ariance matrix P111 is multipliecl ( l h r

Iit ch cll . 2001).

clmr product) by a

clist an e dependent localization correlat ion matrix p. an d t h<'ir prodn t will a lso lw
a covarianc matrix (Horn. 1990: Gaspari and Cohn. 1999). T h e m odified conlriCl iKe

matrix h as Leen defined in ection 4.j a:-. p o Pt

(5.2)

with the members of tlw correla tion matrix :

(.l.:3)

where p 1 a lo calize t ion funr11on that 1las a va 1ue of unity a t the analysis grid point
and 1t

,·alm' d

1 C'cl....,P:-.

a"' the di~t a ncr from t lw c'l llcl, 1.\sis
.. ·. gn·d p omt
· m
· creases, d rs
·

the decondatwn 1 ngth amt1 is tlw ch~tancP
hctwc""'ll
.· l pomt
· s 1 and j. The
·
' tl1r gr1c

l caliza t ion wdw~ used in t hi~ ~ t 1H h·
h 'C"lsuc
1t '"'"
• is 4."'
(
n~· fOltll(1 t 0 1)e a gooci value
based on th '-'Pll"'ltl\·it~· te·t: and Parlic>r studH'~ (.Jiang et a l.. 2012) .

~. in1i lc t i

. 2.2.

11

In our C'Xprrmwnt s \\T assunH' that t lw (lh~erYa t ion Prrors and model nrors are uncorrelated in "'PrlCP <liHl tim('.
assmulation domain l"' )4()

Tlw total IllllllhCT of spatial grid point s within Chc

\rhrn SPl-1\:f IS llllplC'lllC'lltc>cL the sla t e VC'C'(Or is reclc-

fined a. t h C' cone a Pna wn of lw lll<)(lPl stat<'~. modPl Prrors a nd rnPas11rrment Prrors,
thu ' the augmc nt <'d stat P dinwnswn \\·ill hC'com<' 3 x 540. Consequently. the number
of a full

PCEF ~ignw pnints i:-. (:2 x(3 Y ."'40)) + 1. In our study. to decrease the

numb r of sioma p oinh. rank reductions arc applied. In RRSPCKF(D) th TSVD is
applied t

the anal_,·sis error-conuiance mat nx constructed in the clat a sp ace.

Pr and

in RR PCKF (E) T YD is applied in the anal~·si~ error-covariance matrix con st ruct ed
in th en embl

pace. PtE. Th e total number of sigma points 2L + 1 is thus rcclucccl

to 2l + 1. The e timate accurac~· of the truncated covaricmce matrix depends on the
choice of l. The select ion of l should be chosen cautiously in such a way that it should
not be too small nor too large. If l is too small. some of the information from the
pta will be lost and a la r ge l '"'·ill create a computational burden.

The Nil-10-3.4 index.

defined as the average SST anomaly in the eqnatorial Pacific (.:JoS-.5oN. 120-170°\\'). is
u sed as an indicator to track the phase and intensity of the El'\SO ('\'C'llt s. Tlw riiio3.4 region (Figure 5.1 ) h as large variability 011 ENSO t imc scnks. The expnimeut
details are summa rized in Ta hle 5 .1.

Bl

30N
20N
10N
EQ

Nino 4

10S
20S
30S
L 120E

150E

180

150W

120W

90W

Fignrc 5. 1: ~rnphical dq>i ction of tlw :\iiio r q?, icm~ (from :'\ational C'PntcTs for ElwiromlH 'Il1 nl Pn,clidiou (:'\ ' _Jr )).
TahlP .. 1. £:'\'.' () stat<' <'stimation : Expc•riuwut S1llllllwr~·
. ssimila t ion I TimC'
0 hs<'rY<-\ I ional ltal.\ ·sis
nwthO<l
p<'riod I data
~pda!<'
1971 - 1haplcm <'tal. ( 199 )
ll
~Inltiplicativf'
1110111 1
2000 -L.i.i!A dat n
~· infiation

-rA

t

1971-

:2000
RR Pl KF (E )
EnSRF

I Kaplan ct al. (199 ~
ll
111011 1

'TA d a t a
1971 ~ K aplan C't al. ( 19D )
2000
S. 'TA d a t a__ 1
1971 al. ( 19D )
:20ClCl
1

• ',

t

Localization~
method
'ovariancC'
localization

~lnltiplicativ<' C'ovarimHT~

inftaticm
-+-localizati<>ll
ll
~lnltiplica tivC' Co,·arimH '<'
nont L~· I inftatiou
localizatiou
y

---r--K inltiplicativ<'

mont 111~·
·
. Hat ]()11
.
m

Covariance~

loca lizatio11

Forth(' sensiti,·it~· expPrim<'nt s with RRSPlTKF (D ) clifi<'r<'nl muul)('rs of sigma point s
ranging from 5 to 101 nre tc•st ed and tlw rC'snlts Hr<' ('Xplnin<'cl ill Sc'C'tl<>ll s .: ~ . l. To
furthPr explore ' the imp;H'i of tlw RRSPUKF (D ) assimilation method with diff'crcnt
f'llSPmhl(' siz<' on

:\SO prc •clictiou . the • SSTA hinckas ts of the d111ntion uf 12 months.

ini t ializ<'d frolll the first d<1y of each cnklH lm 111011 t h. an ' IH'r fonu <'d ht't W<'<'ll t l1<' p l'ri <)(I

1971 2000 . TlH' Kapl <m SSTA dntn set is wwd to \ T nf.\ ' tlw n1ocld ptTdictiou~ . The
r<'sults of the hiwlcnst C'X]H'l'illH'llt S nrc ' disc·nss< 'd iu Ill<' S<·dioll s.:{.:3

J

_J

cakulat cd as h lm , '''it h ach va ria ! l having t h sam n t ctti on as thaL in

hap t r

p Ia = £111,1 Dn
(Ent .L )T
t .L

.2.2.5

\ ,n rl(171 r fTJl -

(2:::'

at i ·t i al ·ignifi ·an

' t ' ·t

I

Inn
)
l
' · ' ! trnr·c(D 0
)
1, /.

(5.4)
X

lOOo/r

(5.5)

Th h oot strap lllP1lw d is n~ c'cl t o pc' rform Fl s t clt ist ical signif-icance trst for I I\ I E and
correla tion s bll. as follows: 1) or a given grid point . t lw a n alysis <-md obser ved sawplC'
of 1 he sa m e time> a n ' combin ed t o form a p a ir sample ovrr 1971 2000 ; 2) 95rX of the
p a ir . a mple'. arr r a nd o m]~ · ch nsrn to calcnl a t r t h r cnrr r la1ion cor ffi cir nt and R ~L 'E

h twc n a na lysis ancl uhse n ·ation: 3) procrss (2) is rep f'Cl 1ed 10000 times to prodncr

10000 correla tion s a nd R~I S Es. from which th r st a nda rd d evia li on is obt ain d : and
4) the 9.5o/r con fi d n ee int erval i, used as thf' thresh old valu f' of sampling rrors .

5.3

R esults and discussion

5 .3.1

Sens itivity exp erime nts with RRSPUKF(D)

In this sec1ion the p erfon mm c of RR SP UKF (D). as n fnu c t ion of the 1111ml wr of
tr uncat ed m od es of the SVD . is exa mi n< d. A comp arison among rliffl'rcnt t nnwJ t ('d
mo d es is shown in t erms of t oo t m c<lll-SC!llHrC'd c'rror (.RI\ISE) h C' twt'('ll tlH' mw l\'sis
valu sand the ohscrvt'd va lue's ( ignre G.:...) . \\'he'll t h P numh('l' nf sign w p omt s 111-

r ases fr m

to 101. th R 1

of

mo- .4

T

anal.ysis omparcd to observations

deer a!1e.. How v r. v,·h n the nmnl cr of sigma point s nsed is beyond 40 sigm a point s
(i.e., I = 20). th re is lit tl imr rovem cnt in t he analysis. ~'he n the number c f tnma t r d moctPs is lPss th an app roxi m atel_v 10 t hcr<> is considerable' increa,c.; in th RI\I ' E .

ne of t hr rPasons for higlwr R~l '

whC'n t hP rnsPm hlr nnmh er is sm all is du e to

unct erPstim a t ion of thr cm·miancc> mat rix h~· t hC' fi nit e> trun cat.C'd mode's.

0 . 25 ~----~------~----~------~----------~

•

Analysis

0.2

-

( .)
0

;

0.15

(J)

~

a:
or::t
M

g 0.1

z
0.05

O L_----~----~----~----~------~--~

0

20

40

60

80

100

120

Number of ensemble members

Figur .. 2: Th roo t mc>an-sqnar<'d ('rror (RI\ ISE) of SST (5°S-G I'\. U0-110°\\')
b tween analysis and trut h ns a function of m1mlH'r of ('llSC'lll hies w·wcl.
Thr R I\1. . . cmd cor rrla t ion skill for t lH' 1H'riod J!JI 1 '2000 O\'<'r t h(' <'11 tin' hn:-~m <ln'

sh wn in Figure 5.3 and Figur

s in Fignre 5.2, the analysis has a larg

when the ensemble size is sm c: 11, as indica t rd hy the high r RI\18

rror

and lower c rr -

latio11 co 'Hici<'nt . 1 his is a ca.:.;e of filter divn gcllcc. which wmally o Tun; whcu the
for ca. t co ariancr is sm a ll comparrd to t Jw mrasnn ' ment nn certaintirs and t lw fi lter
assigns sm a ll weight to the nwasurement (Furrrr and Brngtsson. 2007· Karspeck and
ncl rs n. 2007). \Ylwn tl1P "llSC'mhl<' mnnl)cr iucreasrs. the RI\1. E deer ases; hut
h .\'Ond 40 ensc'mhlc•s. thPre is bttlc 1mprovenwnt in the analysis.

his suggps l.s that.

a n ensrmbl sizr of -10 is sufficirnt to approxi mate them an an l sprc>acJ of th forecast
clist ri hut ion as wr ll as to char act crizc t hP coherC'nt st rnct ure hd WPC'n the mcasurenwn t
and th pri r statc>. Different ensemble sizc>s fro m 31 .J 1 ca n lC'acl to similar behaviour
in b oth R~I ' E and corn'lation. hut lwlmv 21 PnsPmhlPs the skill is vrry poor (Figurrs .
.3 anc1 5...!). This indicates that if the basic ch aracteris ti cs can b r captnred by tlw
minimum truncat 'd modes . thP sensiti\·it)' of the <-:tssimilalion pr rfonn ancc t o furthN
in reasing the tnmcat rd modes is not hi gh .

5.3 .2

Comparison of RRSPUKF(D) and SPUKF

Based on sensit ivity experiments of thr analysis error to the trnncation modt's /.
it a ppears that 4:1 sigma points (i.e., l

20) arc suffici<'nt to estim ate t h e llll' <tll

and covarian ·e. The resulting analysis is very good ancl comparable v:ith a fnll-rank
SPUKF as shown in Figure .s ..s and Figure .S.G. The vRric-lllC(' explained h~· the -11
sigm a p oints is more than

5o/c of the Iot al variance.

Figures 5.r:: (a) and (h) show th e RI\ IS ~ C'rror of tlw Nii'io-3. 1 indC'x (the H\ '('rag,e SST
anomaly in .) 0 8-5° , 120-1/() 0 \\T) IJC'(V\('('11 \hC' Hllell)'Sis HlH l\}H' ohSC'l'\'<1(ioll for both
thC' full-r ank SPL KF aml RR SPll\F (D) with -11 sigma poiuts . n'sp<'din'l)·.. \ :-. C<lll

5 Ensemb les

11 Ense mb les

::: [;o }- o0 2
>

2

o 2(o 2s(

SN

,.(",
-- v E J

2 I
J 1 'JQ 3 ) (

EO

L
r 7 f

55

'1

0 2

'
1OS

l OS

155
130[

150E

17 0 W

170£

!SOW

130W

!lOW

95W

~?

'-',.)2~>
0 2

)

o( ·)s

150[

(

~

'(

(__:

)

}01)1/:::-v
3 (0 s', '- 0 3 ...----0 ,
-0.25 -v--. J~c

.)

J

o 1",

170[

025

_.-_()/ ' >-

2 'r:O

\ J
0 / ~ 7')

Q ~ ,)

0.75 0 2
(0

o2 \ _\ 0 2 ~ j ,~ ~
o.~)o 2 cp5_j 3
· /

0 3

(

')

130E

1

025\.

0
, L '"'0 1 ')
1

\

t

0 2

') 15

)

l

'I) 25

::J

r ...-

f

l \o J "'-

!SOW

j

r

l

170W

-

E J

(l

11

2~
\j\
\ I
I

~ ') j

130W

11OW

95W

IJO W

11OW

95W

31 Ensembles

2 1 En semb les

( -

ISN
ION

EO
55
lOS
ISS
150E

130E

170W

170E

!SOW

130W

!lOW

95W

170[

150[

130[

ISN ,.,----;:----=:-;:-:~----------~..,..,
11
0 .\.>
12N 0 '
C'J---->

'---<c

I SN

55
l OS

!SOW

51 Ensembles

41 Ensembles

EO

I 70W

o·

9N

.....----...

6N

(v---)Q

3N

C 15

EO

•"

3S

('. \

0(~ 0 1

~0 is

~

I\-

)

)

o o~

0 05
0 0')

~0).\) 0"> ~

U 05

c 0 OS J')
0 85-)~
C 05 v
.) I'- / o o_5)',\::'(

.1 ? 1 / ~"0
/,I -

\ \I \0 1 0 ')')I
12S ~

15

IS~ 3~0~E~~~~50~E-~17~0~E--1~70-W--1-S~OW--13~0-W--1-10~W-~95W

:: ~~ c:_,; ' )
\

~

I SS

130E

J

150E

170E

170W

150W

130 W

11 OW

95W

(?

1 ')

Figure .5 .3: The RI\ISE of SSTA lwt ween the anal:vsis nncl l ruth <W<' ragcd over the
period 1971- 2000 forth(' RR SP KF (D ) when diUl'H'lll umnl H'r~ of ilm1 cnll'd mod l's
( l ) ar used . Thr numlH'r of cus<'mblC's is eqn a l t o (2/ + 1).

5 Ensembles

11 Ense mbl es

--15 N

15N

I ON

IO N

r::

5N

~ -0

EO

EO

55

55

105

105

155

155

r f

8

(\~Ns

l

0.8

150W

1 1OW

130 W

8

150E

17,0[

\

I ON

'

I

~ 95 "

EO

{I

1o5 ,_

l R

l

-

'

0<1

0 9

/1

\

() Y

(

55

I

. <

0 9)
J y

170W

I SOW

lJOW

~/

s

155 )

G8

11 OW

~ 9 ')

\

lOS ""> Q

{fl t'"
,

170E

') 95

1:;
)

!JOE

95W

150E

170E

)

130W

11 OW

95 W

130W

110W

95 W

(' 95

: ~ gs!()

Q.C,~

095'..)
0 95/ ~

3N

:: ~ Leo~',"

Eo
J5

~ ~ 0 95

o es

>' )

170W

150W

130W

110W

I.J

\

09 0- q )

/""'\..::

o ·:J:::. ~o 95

95

0 95
(

170[

I

65

155 {_ 9 5
150E

150W

-,

12N

JSN

130E

170W

5 1 En semble s
15N

lOS

95 W

\

EO

41 En semb le s

SN

0 7

5N

\(\

150E

JON

11 OW

ION

)

8':>

130E

13,0W

(

~/'rJq09'1
g
r
J

150W

15N

0 y~

!"\"0 -'\ '1

)

155

)

0 9

\.

(''l g

/ (0 f c,

O Cj r

(0 8

31 En sembles
-~

r

5N

55

_

0 9'1 J

170 W

ro 9

/

~--'>..'\.....-,JD::l....D-0~~-,--.1...._(~

2 1 En semble s
15N

0 9~

0 9"J

0 9'1
(

1JOE

95 W

0 95

/
G

1,----L....-,-

170 W

0 8

(._)

SN

0

130[

>~

0 g

g

125 '

95 W

155
130E

(

°" -,"'150E

170E

170W

150W

Figure 5 .4 : The correlat ion coeHicient of SSTA b et ween tlw ana lysis and t ru t h aYcragcd over the p eri od 107 1- 2000 for the RR SP VKF (D) vdH'll diil('l Cllt umn b c1:-. of
truncat ed m odes ( 1 ) a re nsrd . Th numb er of ensemhlrs is eqm"tli o (21+ 1).

be seen in Fig ure 5 .5. t he R l\ ISE of RR SP V KF (D ) is similar to th n1 of thr full-r ank
SP KF in m os t of th e assimilation \\'iwl ov:. This r e-m also br s<'rn in F igun' 5.G wlwn'
the Rl\1SE a nd correlation between the n m1l~·si s nncl ohser\'<ltion for b o th nH' t ho ds M<'
compa r d inlhe entire h as n1. Tlw b oo ts lrnp <'xp crinH'n ls nn ' conducl('d to 1w rformlhe
staJistic (l,l siguificauc-<' t<'s l for TI ~I SE aud corrdnt ion skill . F q.?,nr t>s G .)(' nnd 0 .Gf :-.how

7

(a)

0.12
-

RRSPUKF(D)

1995

2000

,..., 0.1

(.)
0

......

w 0.08

1/)

~

a: 0.06
~
C")

0 0.04 .

c:

z

0.02
1975

1980

1985
Years

1990

(b)

0.12

-Full-rank SPUKF

,..., 0.1

(.)
0

......

w 0.08

1/)

~

a: 0.06
~
C")

0 0.04

c:

z

0.02
1975

1980

1985
Years

1990

1995

2000

Figure 5.5: The Rl\ISE of the ino-3 .--1 index of the analysis against the ohsc'rYcd
counterpart. The panel (a) is for RRSP KF (D) whrn 41 C'nsc•mhles are nsed and
p anel (h) is for the full-rank SP KF .
the diff renee of Rl\ISE <md corrdation behn'en SPCKF and HR 'PC KF (D). with
t h e sta.tisticcd signiiir8ncc area. haded C\( thr couiiclC'ncc k\'<'1 of 957t. Thr correlation
skill of RRSPUl F(D ) lws 110 siguilicn ut diil('l'PllCP from tlwt or(\ fttll-nmk ' I L'KF f(ll
mos t of the dornHin of! he nssimila! ion (Figm(' S.Gi). TlH· dill('ll'IHT of lL\lSE \)('( ,, ('('11
TIR SP UKF (D) and SPl'KF. howr 'V<'l'. is signifi('all1 for almost tlw <'lllil<' as:-.imil,\11011

(a) RMSE SPUKF

ISN
12 N
9N
6N
JN

/~

0 0 5 r-.

(

l' ~)' o o<0 r
0 os

JS
6S

12S

12N

0')

9N
6N

ot[l~ o'1
0 0."' l

(0 0'>

EO

9S

IS N

-,

(/

...

0 JS

35

r

0 0')

65
{

(l)

95

0 0'1
ISOE

\70W

170E

1SOW

1JOW

II OW

9SW

(.'

12N

EO

nor,.,

35

I'

1S5
IJOE

r

01
1s

(

,-

(' 1

)

\

,. 01 ;\

---~~

0 1~
J 1

\ \

;(_

1

( 1

~

c J<

0 1' l
l )

f)'

6S

125

\

0 96

\_/

Gog

150E

ISOE

\70W

170E

IJOW

I SOW

110W

9SW

\

J

0 1 (Q 1
)

)

)\

0I

')

c 1 0 1I 0'

170E

170W

I SOW

-· 9'))

9N

(r

,J

0 '.l

12N

6N

0 C.fl
JN
EO
35
65

~ 9')

:\/
0 g)

('J')jl) (
, n 3~
I
0 90
\

95

J <)',

)

0 Q~
/

__,~o

1JOW

I lOW

95W

ISS
130E

9,

C-

f\

_/

0 99 ""

)'H)

J gg/
(

150E

170E

- 0 99

\

125

~

/

ISS
IJOE

ISN

l [,

0 1

0 ' 01
0 1

~ 0 1"

6N

l '

\

)"

) 1

95

g

(d) Correlalton

(c) RMSE
ISN

3N

\0 gg

\

125

I

ISS
!JOE

9N

f~ ':-J '\.J~

EO

l p os -o

I

0 O'J

JN

(b) Co rre latiOn Coef fi ctent SPUKF

170W

150W

130W

/ r-

0 CJ6
0 9~
110W

95W

12N
9N
6N
JN
EO
35
65
95

0 0.:'

I?S

Figure 5.G: The Rf\ISE ancl corrdaticm codiicicut of SSTA bet ween the cuwlysi~ aml
truth for full-rank SPUKF and RRSPCKF (D ) with --11 ensembles for ihe prriocll071
2000 ( a-o} Fignrr :J.G( r) ctnd (f) <trr t br cliffC'rrncr in , 'STA R ?\ISE nnd corrrlntion
coefficient of SPPI\F and RRSPUI\F(D) wbrrc the statis tically significant areas (at

957c confidence lPvc>l) are shadP(L
domain as shown in Fignre 5.Gc. This is mainly lwc<&ns<' tht' HI\ISE itself is vc'r.\· smnll

in both mel hods. leading to vC'r.v small cri t C'ria valn<'s . Tlms. it nm lw nrgHPd t hnl
the RRSPUKF could fairly wcll simula!c t lH' pcrfonwmce oft bt' fnll-nmk SPl_TI\F m

L rms of thP llH'asnrc' of coJor<'ln!ion skilL and <llso t'Hll g<'lH'r<l!<' small I }.lSE

2.5

T

T

T

2

..
+

1.5

~

++

u
0

~

~
Cll

0.5

E
0

c

Cll

0

t-

(/)
(/)

- 0.5
-1

:
+

- 1.5

1~70

-

t

+

\

+

1975

1980

1985
Years

1990

1995

+

••+

RRSPUKF(D)
Observation

2000

F1gure 5.1: One-t imP st<'p fon 'cas t fen t hC' riil.o-:3.--l rr'gion (-'> . •_;: c- :. 120-170 \\'). +
inclicatc>s th<' ' iil.o-:3 .-1 ilHl<'x of the ohsC'rvation and the' solid lmc' indicate's tlH' .l\iiio-3 .4
indc>x of a one month lc>cHl forPcast of RR.'Pl'KF (D ) \\'itb 11 C'llSC'lllhlC's.
Figure ;:_7 shm\·s the one step forecast (mH'-mouth lead forc>cas t ) of ' iiio-:3 .-1 ''TA
index from the RR 'Pl'KF (D ) '"·ith 41 sigma point s during the> time JWriod 1971 2000 .
Th solid line is the mode 1stat r bdorc assimilation ( i.C' .. t lw prC'dict ion ini t ializ<'d from
the analysis of last step analysis) and + dPnotcs the> ohsnnltmns. The RRS I l'KF (D )
fi ltrr has a good capability for f'Stllnating thr phase' and intf'nsit~· of all nm.im E::'\. ()
vents during the ent irP p c>riod (Fig nre 5.7). The' la rgc' nrors occurs at a f<'\\" t im<'
stc>ps when thPrC' 1s high frl'CJUC'llC.\' \'ariahility in SSTA . which is r<'Hsonahl<' gi\Tll tlwt
th observation used for assimih1tion can also contain 111H'<'ltnintiC's .

!)()

5.3.3

kill

r ca t

he qualit~· of t lw ~ :\'" , 0 hindcaf,t of 12-mont h lc>ad initializf'd from an PllSf'mblP of

analysis ohtaiw'd nsiug t lw RR . Pl. K (D ) is c'\'HlnatC'd in tC'rms of thP RI\L
corrdn tion skill of pn'diC't <'cl , ,

and

against t lw ohs<'ITC'd f'onnt c'rpart . Fig nrC'S. shows

(a)

0.8
,....
(.)
0

0.6

UJ
(/)

~

a: 0.4

-

~

C')

5 sigma points
11 sigma points
21 sigma points
31 sigma points
41 sigma points
51 sigma points

0

z 0.2c

0
0

---~

2

4

6
Lead time (months)

8

10

12

(b)

-

5sigma points
11 sigma points
21 sigma points
31 sigma points
41 sigma points
51 sigma points

c: 0.90

...m

-

~ 0.8r'-

-

0

(.)

~ 0.7 r-

C')

0

c:

z 0.6
0.5

0

2

4

6
Lead time (months)

8

10

12

ignre 5. : Th<' Ni1-10-:~ -1 forPcH st RI\~SE a11d <'<HT<'lclt.ion with 12-lllouth l<>nd t illH' fur
RR SPUKF(D) for tlw case's with diff< 'n'lll 111 1111l><'r of <'llS<'Jllhl<' l!H'lllh<•rs
tlH' RI\IS

and co 1 n ·lnt ion ski ll of :\u-tu :~. 1 SSTA. Hs n f'tuwt Jon of l<'<Hl tinl<' fur difr<'rDl

n t rnscmble siz s.

h

rrelat ion skill is 0.5

r higher for p r eli Lions

of np to 12 months irresprctive of the rnsemhl , size. Bot h the R I\1

t lead tim s

and correlation

skills elecT a;.;r \\'it h lPad time. as char act rrizrcl in a nonlinear or st ochasti · d ynami a l
syst rm such a.'> ~
an l

dw1 L 1

(.J in ct aL 1

; Pc>uland and

ar<lrshnmkh. 19 5; K irtman

'lwn Pt al.. 200L1). On thr other hand, the RI\IS ~ decrea.'>rs and

corrC'la t ion skill increase's as t h

C'llSC'mhlC' size increasrs.

ThC' predict ion skill sLar Ls Lo

lC'vrl off and haYr litt l improvC'm 'llt ''"h 11 the numhrT of C'llS mbles is approximately
ah n·e 20.
in

his resnlt 1s consistent with thos<' obtained in the sensitivity cxp ('rimrnts

C'ct ion r-.: .1.

Compari on

5.3.4

b tw

th

n

RRSPUKF(D)

and

RRSPUKF(E) approach s

An alternate method for finding the dominant sigma points i through TS \ rD in the
ensemble subspa · as introduced in Chapter 3. This method is compntationall.v feasibl

ven for a high-dimensional model.

analyzed

Tino-3.4: SST

Fignr

5.9 compares the Rt\ ISE skill of

index oht ained hy I he RRSP( KF(D) and RRSPl'KF (E )

methods, respectively. for the period from 1971 2000. against the ohservat ions. The
RRSPl:KF(D) is better than RRSP KF(E) when the C'ns mble size is small. inclicating thaJ thr lattrr is morr indficirnt in using small rnsc'mhlrs to capt1u·c' the'
main feat nres of the full covariance matrix. This is because the n RL PCI\F (D) dirrctly decomposes the full covariance> whereas the R RSPCKF(E) uses t hP <'nst>mhle
to a p proximHt.c> (he covariance uw t rix .

s the' <'nsemhlC' size increases. bm\'t' \ 'C'L the

d iffer 11 e hrtw<'('ll t lw t-wo mrt hods I>C"COllH'S \'('I'~' small . Figure !) .10 shows t lH' n ~IS E
(a an l c) ancl the cmrd<lliou (h nnd d) skill of mwl.\ 'Z.C'd SSTA using tlw two llH'thods

0.35 ,--,-----,----r---;=========;-1
e RRSPUKF(D)
+
· · ·+ · · RRSPUKF(E)
0.3

0.25
.-

+

u

-w 0.2
0

(/)

:2

a:

. 0.15

~

C"')

+

0

·-zc:
0.1

0.05

20

80
100
60
Number of ensemble members

40

120

Figure 5.9: The root mean-squared rrror (R}.l E) of S TA (5°8-5 N. 120-170°\\')
between analy,'is and truth as a function of the nmnbrr of c•nscmhlC's used for tho
RR PCKF(D) and RRSPCKF(E) m thocls.
for the period 1971 2000 v;hcn the cnsrmblr size is ~11 . Figures

5.10r and

.S.lOf

show the differences in R}.1SE and in corrolation value~ het w<>rn the twn methods .
Similar to Figure .S.G ancl as discussed in Section r: .3, thr hootstrnp mctlwd is nst'd
to perform th statistical s1gnil1cancr tt'st ton 9.) o/t conlidt lH'P h' \'t'l. Tht>u' is no signihcal11 diHcn·w·<· of em rdal iou awl H l\lSE skdl hd '''t't'll t lw t \\'u m<'l hot 1:-; fot <lilllost

th rntir domain of the a;.;similation. indicating that the skill of HR P KE ( ) 1s
entirely romparahl to the RR P TK ( ).
(a) RMSE RRSPUKF(E)

ISN
12N

(/

12N

9N

0 1

9N

6N
3N
EO
3S
6S

~n\

c: 1 -

Q 1r

J

I

(

~

1

0 1 °) (
( 1
0 1

(1'

(

('

/c , I

-o 1 I

3S
6S

_r,

9S

01

1

\>
)
'J g(;l 96

)

;.),

12S

ISS '
130[

170W

170E

II OW

130W

I SOW

95W

RM S E RRSPUKF(O)
15N

t:

12N
9N

1

{ .(

I

J 1 -

6N
3N
EO
3S

/"

)

"

)

-~ 0 1~

no~s
I\
0 1

6S

ISS
130E

(I

0 '
(

(

J 1::
<:..

'J ' 1

1,'-> 0 1 •

r

--'

'Cl1(01

I

1\'-

(d)

r

'

6N

{) <)')

")

I

(, '

,-

r '•£)
3N
E

\

0

65 > \

J

95

c" 1 0 1 J 1

170E

170W

150W

II OW

_)

r 99
I SOW

I tOW

130W

0 a:)
"-'r'" 9

_/

qc,_

-o gg

\

/c
_j
(

170[

150E

o'

~

'J CJ9'

.0 <J f)

(

95W

9SW

,G.93 Q g 'd

') ')9

Y'->

J

./0

y t)

130W

I

co gg

f

('1'=:!'~)
(
~
(1
'>

I

0

•

<

0 g'

' .

3S

170W

J9~)

9N

I\ I

l

C orreia t1on Coefflc1ent

Q T
I

)

0

') l f

12N

l) 1
(<
I

)

\

150E

1

<(
)

~-

125

r )
I

Gl

01

9S

0 1

170E

150E

ISN

-

f Y9

)
(

\

ISS
130E

(

{l

<?y

0 'l6

12S
150[

)0 99

'
96

;j)

3N
EO

Carrelat1on Coefficient RRSPUKF(E)

lo

6N

r0 '" ,,..

I

(

9S

/~(

/\_

J

(b)

ISN

170W

150W

0 q3
130W

llOW

95W

15N
12N

c

)

9N \
6N :1.

tJ 02
3N
l

I
\

I \
') 0

( J
0

J
!

'- /

tf J 4 ~ ~

EO '(;
35

6S , r ) 0 C' 2• r
95 )
125

k

'L~

,

0 OL

0 0

'

I

.;" ,-... '1 4

. , u5

lS~3~0~ELi~1~50~E---1~70~E~-1~7~
0WL-~15~0W----13~0-W~~~~IO~W~9~5W

Figure 5. 10 : The R:MSE and correlation coefficient of SST.-\ lwiween the analy!--i~ and
truth for RRSPlTKF (E) and RHSPL'KF (D) \\'ith .-J.1 PnsemhlPs (<Hl). e and f an·
the cliff rPnce in Rt\1SE and in COIT<'lation coefficient of SSTA lwtw<'<'ll RR SPl. I\:F (E)
and RRSPUKF (D). Tb s t ati~ticnll~· significant nrf'as (at 95~~~ <onfidenc<' l<'Y<' l) ar<'

shaclPd.
Figure 5.11 shows the explHiuP<l

\'Hri<llH' t'

HS

n

fnnc! ion of

<'llS<'mhl<·

Sll<'

for

bot.h RR SP KF (E) a1H l th e HHSPtTKF (D ). The <'xplnnwd \ 'Hri<llH'e 1s lmg<'r m

96
'''

94
92
''

'''
'''

'''

'''

''.

' ' ''

'''

'''

'''

'''

'''

~ 90

-u
0

Q)

c: 88

'''

'''

ro
lo..
ro
> 86

"C
Q)

c:

ro 84

a.
><
w 82

80
78
76

- - RRSPUKF(D)
RRSPUKF(E)

0

20

40

60

80

100

120

Number of ensemble members

Figure .5.11: The explained variancr (%) for the RRSPCKF (D and E ) met hods as n
fun ·tion of number of ensrmhles nsecl .
RRSPCKF (E ) than in the RRSPCK (D) (FigmP 5.11) . This is mtC'rrsting IH'cm1sc
the RRSP1_T KF (D ) usually has hettC'r pnfonmmcr than the RRSPCKF (E ) for tlw
same ensrmhlr size (FignrC' 5.9) . One possih!C' reason is that both lllC'thods lwvc' diffrrent total variances (dC'nominntors ), lll <-1king them incmnpnrahle . It shonld l>C' nntccl
that th RR SPUKF (E) only usc's lwlf of tlw c·nseml>l<' (i.e'., throwin g out the• uthC'r
half ) to caknlate the c·m·arimH·c' watnx to kc'c'p !h<' s;mH' C'llS<'lllhlC' silt' of:!./ 1,
givPn a presC'ril>C'd nmuhC'r of /. Fur! hc •nuon>, IH'< elliS<' t lw <'nsc'mh lc' siz<' 1s not lmgl'

.
r un d errs t 1ma
t1. n f or t }1e pre d iction co-

nough, t h rc IS lll c'l (' llr·a (- rn)
\I rrsrn (a 1.wn
variance. Th
lead t

\'

that is perform d on su ch an inarcnratc> covarian e matrix can

s ub-optunal sigma I oints. suggrs ting tha t explained varian

as the single rnt it)· to clcciclE' the numb r of sigma points t

5.3.5

C 1npar1 on b tw

n

th

cann L b taken

b , used .

RR PUKF(D)

and En-

RF

ssimil ation <'Xp('rimcnt s arC' a lso pf'rfornwd nsing the En.'RF algorithm with the
sam" expNinH•ntal sC'tnp as those in RR. ' P

KF (D ) . In all exp riment s both the

RR PCKF (D ) and En.'RF start from the s<:mH' initi<-11 conditions.

Shown in Fig-

ure .5.12(a) is the R~L E of the I'\iiio-3.4 incl x <:mRlys is hy RR .' PCKF (D ) a nd (b )
En RF . b oth ·with enscm1J le size of ell and agains t the obsNvation connt rrp a rt . Thr
a n aly is from two cliffrrcnt data-assimilation nwtlwds can diYergr from each ot h er
at cert ain as. ·imilatwn steps along ! h
a bilit. v

assimilat ion track.

It can 1w ba.">e l on the

f the assimila lion algorithm in cap turing the obse rvat ion informH tion and

mix with th

model in the transit ioning st ates

If one part icnlar cia t a-assimih t ion

method an approximate the rror covariance' hett cr t h;:lt met h od can h et t er est inw t e
the stat

at phas transition steps. Thus Figure G.l2 suggests that RRSPl'I\.F(D) is

probably hct t er than EnSRF in t be assimi lation of some ! nmsit ion states using noisy
observations. The Rl\ISE and corrc>lation over tlw cnt ire hnsi11 is <-llso compc-ucd in
Figure 5.13, showing that thr RR SPU KF (D) resu lt s in the smaller Hl\ISE and slightl.\·
high er correlation skill than EnSRF, although thrir diHc'rc'nce::-. arc nnt statistical!~· significant for most of the domain (Figure's !> .1:3e ;md .r; .nf). The hPtt<•r perform<HlC'l'
of RR.SP KF over the ~uSHF is pmlH1hly heccwsc' of its supC'nor nwthPlllil!tcnl prop-

G

timrs 1 ss than .J-'- months.

s thr lea d timr incrcas s thr RI\ 1 "' of RRSl

KF (D)

(E) methods cuc significantly low<'l than that of En 'RF method but

aw l RR 1 ' I

orrrlation ski ll arP still comparahl for all thrc•r methods. indicating the RI\1 E skill
is more sc•ns1tive to thr a.'-lsimilation mC'thod .

(a)

0.8

----

....-..

u
0

--0.6
UJ
(f)

:E
0:

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'l:t

RRSPUKF(D)
RRSPUKF(E)
-- - EnSRF

M

0

c:: 0.2

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0
0

2

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8

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Lead time (months)
(b)

1

RRSPUKF(D)
RRSPUKF(E)
- - - EnSRF

c::

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C'\l

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(1)

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u

. 0.7

'l:t
M

0

c:: 0.6

z

0.5

0

2

4

6

8

10

12

Lead time (months)

Figure 5.1-1: Niilo-3.4 index fon'casl RI\ISE and corrc>l<ltion w1th 12 mouth IP<HllmH'
for RRSPUKF(E). RRSPFKF(D) and EnSRF. Th<' \'t'rt icnl <'rror bars m<' the sampling l'rrors nt q;/;{ confidC'll<'r int C'r\'al ohtaiuc'd 1l!"i11)2, hnotst rap <'X]Wl i11H'1lb at <'<H'h

lead time.

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Figme ~.1:3: ''T \ H~ISE and coru·l<dioll codfin<'llt fm E11. 'HF ctll< l HHSl)l'l\F(D)
method for t lw p C'nod 1071 2000 whC'n 41 <'ll~<·mhlc·~ ;m• usPd (<H i ). <' and { ,m• the·
diffC'r nc<' in R~JSE awl corr<'lat)()n ccwftici<'nt of SST.\ of En.' RF and RRSP1'T\F(D)
where the stat istical lY "ignihcant ;uc•as (nt !l:i 1; ; con fid<' JH (' lc·\·c·l) HI<' ~h,td<'d
As in

i gmT f> . . Figure' 5.1 •1 shows tll<' hiudc<lS( <'XJH'IlllH'llls of ' mo-:~ 4 SST.\ llliti:d-

izPcl by the HRSPll\F(I ). "'nS HF <111 <1 HHSPl' I\F (E) Th<• hind<"nst S<'ttinns ,\l<' th<•
sanH' as thos<' m F1gnr<' f>.L

l3ootstmp <'XJH'rinH'lliS ;m• ]><'IfotllH'd to t<•o.;t tlw ::;t:lti~ti -

C'al si g nificanc <'for hot h H\fSE ,\llrl c oJ n·bt ion ..;kill. I'll<' <'XJH'IIIlH'JJ( :-.1't tin .~ jo.; "llllil.u

time's l('ss than Ll_r: months.

s tlw lC'ad timP iucr('asc's tlw R I. .. of RR P rKF ( )

and HH 'I 'KF(E) llH·thod, (Ut' signihcnntlv lm\Tl th;.Ul th at of En 1 F UH'thod but
cmrdation sklll mP still c·ompa rahlP for all till <' nwthods. indicatmg th(' fU.L' .. ski ll

mm ' s 'nsr t 1\ 'C' to the assrmrla t Hlll 111<'( hod.

1.

(a)

0.8

----

...........

u
._ 0.6
w
0

(J)

:E
a: 0.4

.

~
("')

-

0

c 0.2

·z

~

0
0

2

4

6

RRSPUKF(D)
RRSPUKF(E)
EnSRF

10

8

12

Lead time (months)
(b)

1

-

RRSPUKF(D)
RRSPUKF(E)
- - - EnSRF

c

-0

nl

0.9

Q)

I..
I..

0

0.8

u

. 0.7

~
("')

0

----

c 0.6

·-z

0.5

0

2

6

4

8

10

12

Lead time (months)

Hf\ISE nne! c·mr<'i<IIJ!lll \\·ith 1~ month l<•nd tillH'
,
,
, , ,
.
..
,
, ..
for RR SPL'KF(b). HHSJ>t' KF(D ) mHI b1SlU. Ihc· \'<'rtw,d !'II<n h,ll" ,ll< tlH ::>cllll.
<)~~ ·;
r· 1. c·c· intc·rYnl ClhtniJl<'d \ISIIlg lHHlfstrnp <'XJH'IIIll<'lll:-. at<',\( h
phng
C'ITOIS a( , ·>I< ( <>ll H <I 1
.

"' 1g nr e

.1 .

f-.1'-i.

r· ·
·1 1 · clc•x fon·c·lst
111<>-.1 · 111
'·

l<·ad t inw .

!l!l

5.

umm ry

ln tlus

'hnpter. both H'dun'd-rank stgmR-pomt llllSC<'ntcd J<alman filter (Data )

(HI ' 1 L' h~F ( D )) cllld u•ducC'd-umk sigllla-poiut unscc•utc•d Kalman ii.lter (EnsC'lllh1c•)
(Rl . r ' K ' ( ')) \\'{'1'(' applH'd 011 a l<'HhstH' ' I ' ino . 0lltlH'lll Oscillation (E rso) prP1

chctHHl mod ·1 (L E ) .J) and assululatc·d .'.

1

data. Fom mC'tlwds (RRS I

K ·(D) .

._' Pl'l\.F and En.' H ·) wc·r· Impl<·nH'ntc•d and tt'stcd on tlw Z

1

mod d .

. Pnsiti\·it\· <'X]H'llllH'llt \\'C'l'C ' cmHluC'tC'd usmg Rl SICK · (D) with varying nmni>C'r of
truncated nwdc •s aftc•t th<' .' \ 'D ou th<' c·m·ari;-llH'< matrix

Th<' tc'slllts showPd that

\\·lwn the' llllllliH'l of sigma pomt s llH'l'C'Hsc·s thf' R?\1. 'E of ' iil0-:3.-1 S 'T
c·om parcel to o hsPr\'Cl t tons dPnc·asc·s

\ \'lH·n the llllml H'l of sigma points wcr<' a hm·c•

10 . howf'\'<'l'. the' estnnat ion skill kvds off.
v.: hcn t h

C'llSC'lll hle 1111111 hc>r \\·as small

analysis

On<' of t lw rc'asou:-. for higll<'l 1\:\ISE

was clllP to nndc•n •s tmw t ion of the cm·aruuH'<'

matrix by t hc> fin it <' t iunc itt c>cl mod<•s. Th<' R -:\L E and coJ rclat ion skill of t lH' <'lll in•
basin were compared as a function of tlH' nmnher of signw points used . TlH' n 's ult s
showe l that the> c•stimatc had mon• C'rror whC'n the' llllllliH'l of sigma pomts '''<' n '
small as C'vidPnc<'d fwm the highc·r R~L E nnd lmn'r c·<m <'lnt ion codficiC'nt. This \\·ns
a

case of filt<'r cliv<'lgC'll<'<' ,,·hich nsmtlh· <H'( nrs \\'lH'Il t h<' 1>ackgrmmd cm·m i,nH'<' 1s

nnderestimat<'d lH•cnnsc' of tlH' snw ll lltlllllH'l of <'lls<'lllhl<'s

\\'ll<'n tlw l>nckgwund

covariance 1s nudC'r<'s t ima t P d t lH'r<' is g1<'cl t c•r c·<'rt a !Ill\' 111 t lH ' f< >1 <'('il!'\l <'Sl imn t <' :ut< I
the filt er will gi \'<' lllOlC' \Wight to tlw fmc•c;tst <Olll]>ilr< 'd to oh:-~<'t'\'il!H>ll:-.

ThP c•stinwtio 11 s kills of HHSPt ' J\:F (lJ ) ;uul HHSPCh.F (E) \\ <' t< ' tompan'd to <'cl< h
o (hC'l' . \VlH'llth<' (11!1lCCiti o tl n•adH':-1 i\ ti ad<-o fllH'I\Yt '!'ll <'I>S( 1'\.}H'IlS!' ,\lid 1':-;lillldli<lll

l()()

8' ura ~·. RR.' l , K ~ sw n-'ahlPtoanalyz th 1has andintrnsityofallm8jor ~

'0

\'Pnts from 1 71 2000 with comparal>lr Pstimation accuracy. For small r c·nsrmhlP
Sit

tlw R l ,'1 "KF (D) pC'rform •d hPtt<'r than HR P , Kl~ ( ~ ). suggrsting that thC' lat-

t r1 wa~ 11101 r> mdticiC'nt ll1 11 ing slllall rnsc'mhlc'~ to capt 1m' t hr main fC'atnrC'~ of t ll~
fnll cm·arimH'P llHltllx .

lns was lJC•causP the• HR.' l L'K I' (D) dirrctly decomposes thP

full cm·arimH ·c• nwtux wlH•n•a;.; thC' HH. 'P ·1\ ' ( ~) llSC's thP c•nsc•mhlC' to approximate·
t h' cm·anm1c ' lllcl t IJX

. ~ t lw llllllll H'r of ~tgma pomt s tucrc•asPcl. howpv 'r. t lH' chf-

fpr ' 11<" ' lwt wc•c•n t he• two lllC't hod~ 1JC 'C'HllH' llltmuwl. \ \ '11<'11 Lll sigma points wPn' usPcl
thc' l

was 110 Slt?,lltticant drffc'IC'll<"C' of coltC'lation ami R~l.

method~ for almo~t

skill lwtwc•cn t hC' two

thC' c•ntac• donwin of tlH' assimih1tion . indicatin g that tlw skill of

RR .' Pl"K • (·) \\·a~ c•utHC'lY compmal>lC' to thC' RRSPl"l\F (D ).

hP maiu <-Hh·anlcl gP of

RR PCKF ( ) llH'tlwd m·c·t RR . 'P "1\F (D) m<'thod i~ that tlH' formN mC'Ihod ~1gnificant }~ · rPdu cPs the· com put at Hlll cll cost and lllPmory st or age• and makc•s t hC' . '] L'KF
mC't hod ff'H:-il hlC' fm a lugh-dinwnsJowd syst C'lll .
The pstimation sk1lls of RR. 'Pl "KF wJth ~11 ~i gnw pomts \\'C'l<' \'C'ry close to that of a
full-rank . igma-pomt UllS( PntPcl Kalman tiltr>r (. PCKF ). ThP RR . P"CI\F could fauh·

well sim ula t (' t hP per formancP of t hC' fnll-rank . 'I l "I\F iu t C'rms of t lw llH'HSl m ' of
correlation skill. and al so can g<'lH'l'lliC' snwll R~ISE . ThC' IC'S lllt~ slH>\\'C'd tlwt both
the RR PCKF ~ \\'C' H ' mon' computnliOiwll_v dfi.cic•nt the1n the' full-r.lllk SPL 1\.F . iu
spite of losi11g ~ ollH ' c•st inw t ion acc·mH cy.
ThP estimatioll skill of C'llSC'lllhlc• sqwm•-wot hit C'r ( Eu~H F ) h;tcl also < mnp,u c·cl In
R R 'P K . ;.mel t h<' resnl t s shm\'C'd t l1<1 t H HSPl' l\F wns 111o1 <' robu st t h;m t lH' In t t <'r.

Jw JwttPl IH·dornwuc<' of HHSPL'J\F mTr tlH' EnSHF \\'< lS pwb;1bh lH'<<lllS<' of th e•
SllJH'rior 111 atl)('!llHtic;ll pwpc •!tic •s of' the• SPl\F ;llgmith111 ,ts <'X pl niuc •d 111 ('h ,t plc'l '-' 1
and 2.

HJll<'h . tlw HHSPl l(F C'i tll bc'tl<'r c•s tim nl<' ti\C' lir:-.1- nile! :-.c•cnwl-nrdc'l -..l; t-

I () I

trst1cc lmonH'nts nsmg thr ~ \'

techniqn to aPJ roximat

the known true statistical

mom nt~ m contta:-.t to c111 C'stimatC' of unknown tluc monH'nts in

n R .

s the

II .'1 ' KF nlgouthm can lH'tiPr Ps tunat<' thC' Prror cm·ariancC' compar d to ~ n 'J F .
it ha~ h •ttc•r nhilit\' m captmmg tll(' oh~c'I\'atrmimformalion Hll(lmix with thP modPL
tlwrPh)· pto\'ldmg hPtiPr state c·~tmwtC' . Fmthermore. m II Sl
PHOI

can lw cpr ant 1fwd hut in t lw c a ~C' of

J!L

the truncation

RF t lH' sc'lC'dion oft he mnnb 1 of <'ll-

:-.c'mhlc'. 1s allllt 1 <ll \ ' rmd t nH' prPcli ct ion c·m·anaiH'P IS 11 ~ ually unknown (I\lanoj Pt al..
2011 ).

10:2

Chapter 6

Conclusions and Future directions

6.1

Introduction

This clisscrt at ion primarily addu•ss<•d t h<' issue of dP\Tloping a practical sdH'llH' of
:igma-point h:almnn tilt r (. Ph~F) for its appl!ca t ion in a n•<llist I< dmwt (' nHHld . Th<'

PKF assinnlatiou system with Its clt'tenuinistic nnd robust mntlH·nwticalJHOJH'rtit>s
is rxpr ·ted to producC' n more' rf'liahle cstiuwtl• of thl' stat<' of tl](' oc<'<lllic/ ntnwsph<•nc
syst m.

To th<' lwst of lll\' knmd<'dgc·. tlus \\'clS 1hl' first trial in th<' <lSsimil<ltion

comnnmit~· to nppl.\· SI 1\F in n lPalistic climatP mode·!.

This <'h<lpt<'l pwnd<•s a

concise synopsis of the work JHTS<'Jlt<•d lll tl11s cliss<'llcllion. liliJHHt <mt coutnhutions
of this study me cllso clisC\lSSC'd Th<'r<' is cdso (\ S('('t ion Oil JH>Ssil>lc· rut llf<' dirP<'I lOllS
indmled at t lw <·ud.

6.2

In

n r

.

1 v rv1 w

'hapt<'r 2. l dls<IIss'd thr tlH'or<>tical fmmulntion of th' l alman filt r , Pnsrm-

IJ)p Kalmm1 filt<'l

(EnKF). ('llsc•mhl' :-.qtwn·-wot filt<'l (Eu 'HF) aud sigma-point tlll-

:-.ccntcd l 'a luwn hlt('l ( '1 l'l\.F). The· c·us<'lllhl<--lmsed 1 alumu Iilt<'lS s uch as Eu HF
luwe suCT'ssfHll\' ci<'monstullC'cl tlH'ir ah1llty m occ'cllll<' and atmosphc·ric data assimilatHJn (

·ng <'t al.. 2010. 2011). Til<' main disadvantage's of th sc• m •thods arC' tllC'

uncPrtaintv m constructmg thr c>usc•tuhlc•s and the r<'qnir<'llH'llt of th' grc•at<'r size
of C'llscmhlc's fo1 th<> cH·cmatc• appwxiuwtion of the lll<'Hil and variance' of the• prior
and post Prior cl!:-.t ul mt ion of llHHIPl stat Ps (Kim c•t al.. 2007:
~00

: . mhadan nnd

'handtasPkar <'t al..

ang. 200 ). The' , 'I l'KF is rC'lativc·h· lH'W iu atmosplH'ric and

oc anograplnc data assimilat iou. which dC'l c•tminist ically dwosPs c•nsc•mhlc• nH•mhC'rs
and thC'rc by rPducPs uw ·c•rtainty in p,C'nc•rating <'llsc•mhl<>s (.Julin . 2003).

!so. the

P TKF make's liS<' of a deri,·a ti,·c•-lC'ss optimal C'stimatiou tC'dmicpH' that nsc•s a mmimal nnmlwr of statistically \\'eight c•d sam piPs calle-d sigma points to calcnla t <' t h<' c•rrm
statistics. \\'hen t ransformPd t hro11gh t lH' 111mlinear mode•} t ll<',\' capt m <' t h<' true llH'<lll
ancl ·m·ariance accm i:l t <' ly 1o t lw secoud m clcr in t hP TH.d m sc•tws expnnsiou (.T nlH·r.

2002). The

P"CI\:F n~<·s tdmllmlatcd l'qllatious to fiud tlw I\: ahwu1 gmu all<l <'llOl

covarianc<> so t lu-tt t lH'H' is no lH't>d to calcnlat e• t lu• .lHcohian or t <lllgt•nt lilH'Hl lllcH!Pl
(TLI\ 1) as in t h<' (',tS<' of <'Xh'lHl<'cl J\alnwn tilt ('J' (EJ\F): and t lwrr i~ no IH'<'d to lm<'ariz<' th<'

uonlm<'ar m<'asm<'llH'nt function a~ in the EuKF-hnst•d filt< 'J:-. (Tnug nud

Ambadan. 200D: Taug <'! al.. 2011n).
ln

'haptcr :3. l dt·~ClJI><•d the• dc'll\'H(ioll of two l<'<iiH't'd-l<lllk llH'thod:-.. n•dun'd-l'Ctllk

sigma-pow( tlllS<'<'lll<'d 1\,dlllall 1ilkt ( U;l! i1) ( H HSP L' l\F ( D)) <llll I I l'dtl<'!'d-1 <\Il k sll!,lll,tpoint unscc'ute·d 1\:;dlllall filtc'l ( Eu~c'lllhk) (HHSI l Tl\F (E)) ))!'spit<- ;dl the• llH'Iit:-- ()f

l () 1

1

'P K . th nPcd fm t w1c th nnmhc'r of sigmCJ points than th s.ystcm stat s r mams
KF into rC'alisti oc an or at-

a maJor computational c·on '111 fm th 'applicat10n of 'I
mosph '1

modds . ;1\'C'll th<> computntional constraints of full-ordc>r Prror-covariancc

•stnnatlOn. tlH' data-assmnlation commtuJJt~· 1s \'Cry intC'rPsted in the dc>v lopmPnt of

a INhH·c·d mdn sdH'nH' fm tlH' statP c•stJmntJon.

lH' main ol ,]C'ctiv of this tlH'sis

was to dc'\'C'Iop cl l <'d llcC'd-!ctllk sJgllla-J><Hllt un sc·c'nl<•d 1\nllllall filtC'r , and to app ly
thc> rC'dtHC'd-Jank hltc•t to a n'a!Js(H· clnnct!<' modC'l

In the· JPclucPd-rank mPthnds,

t lw challc•nge' of t ll<' c·ompu t ;-\1 ional r·oust 1 aint fot Sl l TKF is succ·c•ssfnlly ov<'lTOllH'
hy

mplo\·mg n ll111H'C1tPd smgulm-\'nlu<' cle•c·omJH>sitHJll (

'VD) tc•clmicplC' for dinH'll-

swnal com Jn·c'ssHlll or rank tPdu ct tml of tlH' c'tTor-c·m·arianc·<' matrix . 1 applic·d both
lllt'thods on t hP Lmc•nz- G mod<'l (Lor c·nz cllHl "'llH\111H'L 1r 0 ) in t bC' pC'rfe'C't nwdC'l
.cc>mlno. Tht> n'sttlts slwwC'd thc~t thC' r<'cli!Ce'cl-lank m<'lhods s ign ificantly dc'C'lC'asc' thP
mnnlwr of sigma pomt s and t h<'il <'st im,ll ion ac·cuuH·y is clos<' to t bat of a fnll-rank

, PCKF .
In Chapte1 4. I clJscussNl t hC' common pwhlems in PllS<'mhl<'-hasc>d filt <'1S such a~
inhr

cling. filtc>r dJvC'rgC'nCC' and spnuons cmTc>lation.

I impkmC'utc>cl <l hybrid-

localization sdwmC' for RRSPCKF( ). In this wmk. I placC'd morC' C'mphasls 011

RR PCKF ( ) J>pcaus<' ,,·ith that lll('thod tlw anal~·sis <'lTor-cm'<HI<UH'<' mattix 1s <'stimat crl. in t hP <'llS('llll>l<' space. which ;woids t h<' est inw t 1on of a ua lvsis t'rror-e ·u,·arimH 't'
matrix of modPl stale' spaC'C' glo lwll~·. lu th<' localiz,ltion sclH'llH'. thC' <HWl.\·sJ:-- \\'<\:--p ' rformPd m a local sul>spac<' whose' dmH'llSH>ll lllil\' IH' gn'C\1 h· Jm,·e·r t h <lll t ll<' :---\ st l'lll

climC'nsion hll t t hC' signw poiut s an' f.!,('lH'r<l t l'd l>as<'d on t lw gloh;ll doma111. 1 hl' lll<lill
ndvantng<' of this sclH'llH' is t lint it not em!~· nwk<'s tIll' lilt <'l 111m<' <nmpnt :II ion:1 lh·
f'fficient , hut also !I lake's t lw anal~·sis <'st inull <' llH>l c' :we 111 :1 t <'h.' r <'d\l( ·ing tlu' "Jllllloll~
c orr ]at ion lH'f\\'C'C'll I<'!IIO(<' gnd JH>II!ts

.

·

In rm· <'XJH'llliH'll ls I 11'-'<'d difl<'(('lll systl'lll

all<l th<' l<'~ldts sh()\\'l'd th :tl . (() <H'hll'\l' thl' l>ptirnal
. ( ,1\~' .t - J() • tX() cIJH] 1'>0)
~

( l lllH' llS lOllS

stimat (m tc•nus of minuuum HI\lS ). th number of sigma 1 oints n 'E' led to incrras
a.· th :-.ystPm dmwnswn mcre•as sin tlH' ahsc'nc of localization. But wh nth locall-"ation wa:-. mtmdtH·c•cl. tlw munlH•r of s1gma points rPquirC'd to achi v th minimum
Prrm nwmh· dC'p<'IHb on IIH' local dmH'nsion and is mdC'p<'ndPnt of the climrnswn of
tlw glohal domain .

hC' te•:-.nlts of ll1tllH'ncal <' ·rwrimC'uls provide• a good trst-lH'd for

th appbcal1on of tlw ll\·lnHl-locab6alion sdH'lll' in an aetna! at mospheric 01 oce•cuJic

In 'lwpiC'r 0. I clpplH'd both th<' HR.SI rKF (D) and HR.SI l TKF ( ) sdH'llH's to H rc'alisllC" dnnatC' mode'! . I nsC'd tllC' LD '05 of IIH' Zehiak- 'anC' (:0 ')mode'! and assimilaiC'd
s•a surfacC' t<'lllj)<'rallll<' anomc-dv (S.'TA) data . The JH'Jfonllanc-P of rc•chwc•cl-rank
sch me:-. wa:-. compme•d w1th that of th<' fnll-umk .'P TKF . ThC' sc•nsiti\'ity c•xp<'nm nt: t<'stdts shm\·Pd that both the· HI SPCKF (D and ') :-.c!H·mc•s. c•vc•n \\' Ith a

lllinimum nmnl ><'l of sigma points as low as ~11. w<'J <' snffic i<'nt I o g<'l an analysis ll<'ar
to the full-rank 'PCKF analvsis. ThC' alnlit \'of R.RSPCKF (D and

) am! 'nS I . in

rms of R).I. E and conPlation coefucJC'llts was cmnpa1Pd. \\'hen tlw smn<' ntulllH'l
of ens mbles W<'l (' usPd. the pPrfonwmc<' of RR. PCKF \\'<lS s!Jght ly hPt t <'I than t }}('
n RF. which snggests tlwt thr HR.SPlTKF schrnw can ol>IHin more' accurnt<' <'stimates becansP of its bel t<'r algorithmic prop<'rties. ;-.r_v lC'snlts showC'd that whc·n I hc·

pns mble size 1s clS small as ..J.l. th<' pc•rfonumH'<' of both sdH'llH'S HHSI l ' KF (D <llld

6.3

Concluding urn

ry

lw aun of dntn nssilllilntion is to optitu.dlY <'stilll:ll<' th<· stcll<' (lr th<' g<'oplt\·si< ;d
sysl C'lll fro111 eli! il\'111' I<l 1>]<' 1"11 f<>IIIJ<ll ioll . wltwh :II"<' uwiul.\· t hv dywunic;d llllHlds .llld
1()(i

ohs rvat IOns ( alagrand. 1
a~ th

7).

h ontput of th data-assimilation procrss known

analvs1s ~~ ll~C'd c1s thC' uutial condition for tllC' fut ur

forrcast.

It can hr

abo 11~ d tog'( more' infmmatio11 clho nt thr arPa whC'l'<' tlw ohsc>rvation is sparse'
~uch n;.. tlH' dC'<'P layc•r. of t hP CH"C'Hll

Data as~Imilat Jon is an PssC'ntial comiHHH'nt

of tlw 11\llllPrical \\'PHllH'l prC'clJc t 1011 ( ' \\'P ) JHOC'('SS . v,·h ·r, rc•asoning mH!c>r nncntaintv Js a u•cp!ac'llH'Ill.
\'aluc\hl

).lc•thod~ hcl~c·d 011 u·cur~ivc> 13a.\'C'sian C'stimation provide':-. a

tool for t lH' S<'quc·nt 1al stat P-opt inuzat ion pwhiPms. Kalman filtPI:-. give' t h

h '~l liJH'Hl

nn1wt:-.c•cl \·tinwl<' (13Lr") for lnwm s~·stc•ms. howc•vc•r. its dirC'ct implf'-

nwntation 1~ not pl<H "lHal fm

T\\'P pwh]C'llls hPUlllS<' of Its large· dimC'nsimwlit~ · and

non-linPant , ..

Tlw <'Ibc mhlc· formnl<ltH>n of tllC' I\:ahnnn filtf'I. E11KF and ib dnivati\'C'S arc' widc·lv
us d in lltlllH'l ems gc•ophv~H al dat a-as~uuilat ion ptohlC'm:-. 1H'<"HIIS<' oft lwir algorithmic
simplicity and tlwy arC' C'as_,. to illlplC'11H'llt. The> main prohlc•m with

nKF and its

ach·anc ,d ,·arinnts such a:-. En. 'HF rs that tll<'rP is 110 we'll dc•sig11<'d tPch11ique to choose'
t h optimal c>nspmhl<' si;c'. TlH'Y also calllH>t prm·ide mformal ion about t h<' 1wu·c•nt ag<'
of the st inw t C'd prrm ,·aria11cc reb t Pd to t h<' t me em mt <'I pmt for a gi \'('11 c·n~<·mh 1<'
size. Th , P"CKF IS a dC'tPnuimstic. dc•tJ\'<ltJ\'<'-kss algonthm optu11nl for nonlnH'CH
st irnat ion prohkms. SPCKF mak(·~ us<' of a novd d('t Prmiuist IC' smnplmg st rat eg_,.
based on thP llllS<"Pllt<'cl tnmsfonw1tion . Tlw new Plls<'lllhks c;dlc·d th<' sigma potuts
can accurate!~· d<•sctiiH' tlH' statisticnlnu>lll<'llts of the• wmlitH',ll uwdPl

Iu "J>ll<' of

all tllc ];<'ndit s of SPl.Tl\.F. th<' t<'qttiH llWill of llH>l<' th:111 twit<' tlw Iltlllllwr of :-.ignw
points than th<' S\St<'lll stale's milk<'" it impractical for tii<'ll <tpplwntJmlm t':-.!lllt<ltion
pro l>kms in m<'! emol ogy or oc-<'<11 ll >gnt p hy. This rwn•ssi t <1 t <'S t lH' 1 <•q llll<'llH'll t of dt '\'<'I
oping a practic<llly uuplc•uwnt:tl>l<' <t pJ>I<>.·i tuCltion f<li SPl l\F Thi~ disst>r!Clti<>ll hns

1

lllatll .\' fO('l\S<'C

't lld ill\'('s tig<lt<'d (\\'ll nwtll!Hb l<> <lJl]>I<>Xilll<l(<' tlu• <'IT<>I
I (l ll tll <·tt issll<'
• •
'
•

·
1
1 '<II'Iltl· t]>]>I.<>'".·lllldti<lll <lllll ;d:--.u]>l!l]H>:--<'<1 <t 1<H '< tli;.nti<lll sl'!l!'llll' :--<>
covaru-HH'P >.V n•c 1w< - ' \ '
·'"
]()/

PI ' .. can hP ap1h l t a r absti high-dim nsiona l climate mo ll.

that

h

mam

contulntwn of tlns diss<'rtatwn IS dPscriiH'd in tlw fo llov;ing paragraphs.

I intloclu('(cl <1 11d in\'C'stigat 'd (\\'o emuputationall:v rfficiC'nt algorithms for sigma-point
I\:alman filt e' rs. makiu g t lH'm Ylahle candidates for a pphcat ion in 1 C'alistic at mosphC'l ic
and occ'anic modds

The' te'chtcC'd-rauk apptoximation is nwin ly hasPcl on t lw con-

<' <'pt thcll most l<llP,<' sea l<' geop hysiC'a ] plH'llOllH'llCl ccm lH' approximatC'd by a finit<'
mtml>Pt of dPg LC'Ps of frc•e•dom and tlH'ir dominant \'ariahil!ty C'Hll lw <'xplainPd l>y a
l11m t eel mnnl H' l' of nwdc':-.. • ssmmng that most important e'J rors of the original sigmapomt spac<' can he' PSI ima t C'd usin g a fe·wc'r nnmhn of the' clonnmmt sigma points .
two reduced-tank approximaiH>ll lll<'lhods Hr<' propose'd lll tlu s wmk. The' sdH'llH'S
nr<' IC'dtiCc'd-umk :-.tgma-pmnt uu...,cc'lll('(l I\a llllnll filtc'l (Data ) (RR .' P1 ' KF (D )) awl
I C'd 1J( Pcl-I ,m k sigma- point t msc-c' ll t Pel I\: aim, m fi I tc'I ( Ensc'Jll hie ) ( R R . P1 ' K F (E)) Bot h

nwt hods arc' ha:-.Pd em the t rtlllC<It c>d , 'VD to fact omw the> cm·;.uimH'<' matrix and n 'ducc its rank through trnucation . Iu th<' first tc'dmiqu<' . th<'

VD is a pplH'd 011 th<'

Prror cm·c-uianC'<' matrix calculat<'d Ill the' data space' (HHSPl' KF (D )) wlwrPa:-. Ill tlH·
sp ·ond t pchnicpH' the . '\ 'D is applied on t lw C'JTOl cm·arimH'P matrix cakula t <'d m t lH'
PnsPmhl spac<' (R RSPCKF ( ) ) . Tlw 1 ed ucPd-ra nk. square'- root m ,1t nx IS usC'd to S<'-

1'd th most 1mportaut sigma poiiLt s that can retain the' main statistical f<'nlun's of
t lw original sigma pmnt s.

ThC' RRSP"l'KF (D ) ca n C'ff<'c lt\'<'1~· S<I\T conlputaltonal tillH' tltmu gh JPdU<lllg tlH'
numh~r of s 1gma point s. Th<' sm· mg is <'sp<'C'lcdl~· signifi ccllll \\'h<'n tlH' <'lror C<l\'ari,nH <'

ill t h<' data spm·c'. P/1 c-;m hC' Hppl oxJm;d <'d wdl h~· a f< '\\' lc';lding modc's so t lw t t lt<'
f<'We'r nmnlH•r of signw pmut s c<lli ilpproxinwlc' tllC' stnti s ti c;d IIH>IlH'llls \\'ith <I good
truncation ;.H·ctunc ·y llm\'<'V<'r." lcugc• ch;dl<'ng<' 111 HHSPl ' h.l' (D ) is tlw <'O lllpllLiti<lll
of 1 a it sPlf
1

\\' lH 'll t IU' ,-;t;llc' dllltc 'l!Sion IS Lu·g<'. 11 is <'OlllpttLit ion,dh C''\P<'IlSIV<' to

•stin1at<.> thl' PllOl l'ovcuian ·e in thl' data spcH'l' l'V<'n diilicult t.o compute and store
such a hig mat11x . \\"hen th s tate duu ns1on LA! is much gr atC'r than thr C'llsrmblr
dun ' llSlOll n. i.(' L" >> 11. it IS 1 OSSihle t () C'O lllput

mat nx const n I<·t <'d in t hr

tlw ' D 011 t h

11

r

n covarianc

ns<'mhlP sn hsp<H"<' (von .'torch and Ila nnoschock . 1

~1 ;

\\"llks. 2011). whic ·h 1s co mputationally aflmclahiP for a syf' lf'l11 with larg<' dinH' llsion .

Th' computatiOnal coinpi<'Xl1 .\· of t hP algorithm to fiucl tlw covariance' ma1rix in thC'
d ata spac<' Is O(L~ 1

n). 110\\'(' \ T l. tlH' nwtlH'nwtical cm11plPxity of th<' algorithm to

find the' l 0\'clllcllH'<' llld tux Ill the' <'ll'-i<'lllbl<' S ll hspa('<' IS () ( 11 2

L ,\1). TlH' cliff<'r 'llCC' in

comput ational co u1p lC'xity hc't\\'PC'll the' two algorit hms is significan1 wlwn L ,\1 >> n .
that is usuallv tlH' casC' for <l r<'a hst Jc "\\"1 modd. lu thC'] RSP TKF (') lllC'thocl
\'

1s appliC'd on thP analYsis c•rror-c·m·arimH'P mat1ix ? 111 1• iu th<' <'llS<'mhl<' sub-

. p ac . Both the' lH'\\' algorithms <lr<' first kst c•d using t lH' Lorc'nz ~lG ( LorC'nz and
lll<lllt! Pl. 1

) mudd. which is a orH•-di llH'nswna l <l t mosph<'I ic to~· nwdC'l . as t <'st-

bccl . Th l'stmwtioll ski lls of both th<' rank-IPcluction mrthocls we're' llC'Hr to that of
full-rank ' PL'KF .
I proposed and tPst <'d a ll\·bricl-localizat ion schC'me for HI . 'I Cl\F ( ) nwt hod . Th<•
localization is \'<'r\' important in thC' case of RRSPl"Kl·s for Its piaC'tical appliC'ntion
in a glo hal ('ir('ula t ion modd ( ; ·~ I ) l> <'c a us<' \\'<' arC' a pproxima ting t hC' full- rnn k ('O\'ari a ncr w11h its tC'dncC'd-nmk apJn oxi nw ti on.

Lon1!Jwtion uwk<'-.. lh<' filtl'I uwn'

comput atimwll.\' fc'asih l<' and makc•s tlH' anah·sis c•stimnt<' uptmwl ln· u·clu('ing the'
pioblcn 1s of spUli<>liS condatwu. whi<'<'dlllg am l filt<'r dl\Tig<'IH'C' lH'< nns<' of -..JJl<dkr
PllS('llll>lP SlZ<'. Th<' uwiu dwll<'ll).!,C' for imph'llH'ntmg tlw loC'a]izatiou iu th<' ,' Pl\F Is
that th<' signw poiuts an• g<'ll<'mtc•d hnsc•cl ou the• gluh;d ;1nnh·s1s c·nm c·m·ailHll<'<'. In
this dissC'llatioll I dc•vc•lop<'cl aloc·il li;.;lti<lll sch<'lll<' in tlwt the' ;m,l h·srs is JH'rfoi!lH'd
locall~' aw l the• sigill il·J><HUts au' gc'll<'I<ltc'd gloh,dh

rlH' llllllH'ri<"<d <'XJH'rillH'llh rc'-

sult s from ( lH' LorC'llZ C)() uwdcl shm\'!'d t lull wlH'll t h<' Inc cllizat J(lll ,\lld inflat )()Jl ell('
1()!)

intr >duccd. thc> optimal Pstimatr ohtanwd is indq cnde'nt of thC' syst. m li mrnsi n.

'I h<' ~ell~1(1\ It V <'XP<'l'illl<'Ut l<'~ults ~uggc·~t<'d poss1bk wnys to llllJHOVC' the hlt<'l p '!formane '.

lw locahzat 1011 sch 'Ill<' gave' ·m·onraging r<'snlts making thr RH 'P rK ( ~)

method a smtahl ' cawbdatc' fm assnnilatiou in a lugh-dimc'usional systc·m.
I xplou'cl tlw c1pplication of .'Pl ' l\ · and 1 1 .'Pl"KF(D and ~ ) in a realistic clinH1( C'
modd of int '1lllC'dwt <' com ph xll ,. To t h<' l><'st of mv kw>wlc•dg<' . this was t lH' first
tmH'

IK

\\' H;..

apph •d to a rc'ah~tH · llHHld 111 th · HtniosplH'ric / oC'c>anic scic'1H'C's . The•

1C'ahst1 · modc'l ttsc>d m tlus stmly \\'< lS the LD ~ Y VCTSIOIJ of thC' Z 'modrl fm

I\. '0

pr<'< he t ion. • m pha.'·Ds \\'as pia ce'd on t h<' im pl<'llH'nt at ion of H HSP l' KF for the> LD ~

5

with thC' assimilHtion of .'.'T . Both the' HHSPCKFs cmtld analyze' \\'Cll thP pha,'-><'
and int ·nsity of all majm '"' ·. '0 P\'C'1Jts dmin g tlw st mlv !H'Jiod with comparable'
'stimation accuwc.\·.

mthrnnm<'. the' RH. ' I CKF is c·cmlpmwl against th<' '"' IL ' R • .

shm,·ing that t h<' m·c·t all anal.\ ·si~ ~k1ll of RH 'PC KF and E11SHF arP compmahlc• to
ach othrr lmt th • form<'r is more• robust than th<' lattC'r . ThC' sC'nsitivity PxpnmH'll(
r 'snlt shov;C'd that RH.'PCKF with 41 c•use'ml>ks can helVe' c•stimat1on Hccnracy close'
to that of a full-rank 'I 'KF. mal·ing it a potC'ntial cmHlidat<' for dc-1ta assnuilntion
in large-dimPnsional at nwsphf'nc or ocC'an : ').Is.

Th<' n.clYCtnt nf!_(' of R H. 'P'l ' KF (E) O\'C'I' R RSP'l TT\:F (D) Js in it s aflo1 dal >IC' cnn lput ,ttional
cost and low<'l llH' lllOI y nsag<' . Th<' lllain dwllc•nge of HHSPl' 1\:F (I ) Is to 1 <'PU'Sl'llt t lH·

full anal~·s i s-covm icmcc· nwt rix Pxp!IcJt 1~·. \\·hich is still com put at ion,t!Jy nnaft()Jcl.thlt•
for high chnH'llsicmalmodc'ls . lu tlH ' cas<' of HHSPl.l\F (l<..) tlH• c•nsc'mhh's me' llsC'd
to approximat<' th<' covarimH·c• matrix . cnlcnlatl'd 111 th<' n •dlwc•d clinH'nsiou (t'llsl'lllhll'
space). TlH' localizatioll sclwuH' Illakc's HHSPl' I\:F (E ) lll<>l<' c onlput.\1 ion,dh dll ci c'lll
IH·cmlse• Jt 11 w~· rPdnc ·c· t ll<' <lllHlvsJs <' I rms frcnu spmions c or1 <·lnt ion . inhll '<'d ing :llld

IiltC'f div<'tg<'lH "<' , nwklll g 1t <lll idcnl c<llHIHlnt<' fm iruplc·nwn t. ll ion on :1 J'< '. d \\C 'd t hc•r

l I0

modrl.

~h· go( l WH!-1 to dPvPlop a practical sdH'llH' forth

PK

dillH'JlSlOllcd . \'!-It PlllS

angP t al. (201 a) havC' shown th a t

lw tndH'sc·onduC'tc·d l>Y

for tbC'ir a pplication in high-

t lw I\: alman fi lt <'l' <'CJI Iat ion. nsc•d m , PKFs bavr lH't t <'l <'S( imat ion ac ura C)' wh ' 11
t h' m 'Hs lll <'lll<'llt fmlC't 1011 is uonlin<·a r c·ompnrPcl to t IH' mH' usC'cl in "' nK

and "' n-

'R ·s. The· clnPct application of tlH• nmthlH'H l llH'asur<'nH'll( function in ' nKF and
• n,' F 1111pospo..; <111 nupbc it liiH'<1lll<ll H>ll JHOCC'ss usmg <'llS<'mhlc' nH•mlH'rs. The• pn-

fornHUH '<' of HI . 'PCI\Fs \\'<'H' more' robust tlwu ' 11. 'HF whC'n a ppli<'d for assimila ting
' T

mto Z ' modd ().lallo.J Pt al.. 2(Jl·1). Iu HHSI rK •s thC' Pnsc'mhks arc gc'n-

<'ratPd d 't<'llllillJ"iiHcdh· ( ~lmw.i c·t a!.. 2()1 .1: Tang d a l. . 20 llh ) c-ompm Pd to tlH'
arllltrary natut<' of 111Jtwl 'llsPmhlc· g<'lH'ration in • nKFs (Lwu g. 2007:

'uru •t al..

2012). RR. PC1\:Fs c,tll g<'lH'Jat<' finit<' llllllliH' l of sigma points fm thP optnnal <'Stimate of prc·cliction-<' tTor cm·a nnncc ·(~ l ano.i l't al.. 2014: Tang <'t al., 201 ~1> ). TlH'
imll<'m r nt ation of lo calizatiou and infiatwn sell m s in RR . P"CKF(E) l C'sul tC'd in it s
h '( tcr p rfornHmcc>
m

The R~lS

was md< 'lH'lH lc•nt of systc•m clmH'nsiou for a si111plc'

1 1 , uch as t lw Lol<'!lz-OG mockl si mil;u to the finclmgs of Ott <'( ell. (200--1). Snllllctr

t o th LETKF clata-assnmlatiou s\·st<'lll (~ ll.\ ·oshi. 2005: Li <'I al.. 2009) HI .' PCI\:F ( ·)
has an advantage for parallel computHtiounwkmg it thP lH•st cnll<lldatr for apphcHt1o11
in sys t ems of high-dinH'nswmllity.

6.4

Futur

dir ction

couple of kc·v dwll<'llg<'S r<'bl<'d to th<' SP I\F d:lln i\ssnndaiHlll nwthnd~ h<l\<' IH'<'11
ickntific'd . 'll w !irst <h:tllc·ng<' is tlw J>Cll<'llllal nppli<':llion ()f HHSPl'h.F(l·-) 111 .111
. <'<'"' · c· ('('~ l Tlw S<'<"<lllcl isstH' < 'O II< ' <'Lll~ til<' Jll:thdit\' of <'ll::-.t'lllhh•
a t UlOS Jl 1l(' IIC ' Ol Cl " 11 1 ' "
·
I 1l

basc•d IiltE rs S\ 1('11 c'\!-,,

I l<".... .rl~ t o assum
. '1a ( <' as.y nc11rcmous ob::>c'lTat ions.

uot ht>r issu<' is

t hr ass1 llll pt 1011 of ; a nsswn dist ul m t ion for t h<' ohsprva t ion and background <'rrors in

PKF Iu the followm g sub.<'< t 1011s I bndl,v chsc 11":-. t 1l<'S<' I::>su-. and also ::;omc· po::;sib1e
ways t bat t lwv mm· I w addrPss<'d.

. .1

Itnpl

nt

n

fRR

KF (E) £raG M

In th1s chss<'Itation I <'xplm<'d puH'tiC'al a1gonthms fm SI Kl~ to makP It s uit a h1<· for
applicatiOn Ill an atmosp1H'nc or oc·Pmuc· ~ '~I. The RR SJ CKF("') m •thod with tlH'
h.Y hrid-1ocah;,atmn sdH'llH' propos<'d in t1us dissc'rlalion show<'cl c·stimalion cH·c· nuH·y
clos<' to that of a fnll-umk 'PC l\:F making it au idC'a1 c·c-llldidat' for impl 'nH'll(Htion mto a

; '). 1 In this stwh· HR~I l'KF (E) is app1wcl 011 Cl munlH'r of modds

with ,·;u~· iug complc'xit~· and tlH' I<'sults indicated that th <' method is Y<'I.V <'fl <'ctiY<'
in assimihltiug dnta . How('\'<'r. thC' lll<'thod ha!-1 some limitations as wdl for application into a

; '). 1 in an oprraticma1 sC'tling. \\'lwn \\'(' dC'vc•lop au optim ..ll n'<lhsiH '

RR PCKF (E) data-assi11ulation systc'm for an ocPan or almosplH'r<' . , '). 1. thC'r<' m<'
many important issues that IH'C'cl to IH' adclrPssc•cl. ThP moclC'l n'solution . munlH'l ()f

variables and t h(' o hsN\'H t ions to hP e~ssi llliln t C'd will h<> mnch lnrg<'r for a

;c;.. r 1h;m

what \\'C' hcWC' <'lH 'OlUlt PrC'd so fm . Dc'slgn ing I hP modd C'l ror . ol>sc'n·nt ion <'I rm nnd
initial pert nrhe~t icm will l>f' much hm dc•r in I he• cas<' of n GC).l. .\ not hn impm tcmt

fac·tor is tllC' dfici<'ll<Y of c·omputation 1Tnhk<' in ...,jmplc• Illo<lc•]., \W c,wnot ;dfoHIIo
('()ll(h! C( so llll\11,\' S<'llSlt i\' J()" ('X ]H'l'ill]('llt s \\' II I! \'C\1'\'lllg ]><H C\IIH'( ('l'S ( () ('h()(),'-i(' t IH' ()j)( 1-

mal param('(<'r s<'llings 111 th<' C'HS<' of'' C:C~ l The' n'sltlh fw111 tlw <'Xp<'rinwnts \\Ith
simpk modc·ls cm1 IH' us<'d ""'a gui dc •I IIH' for tlH' <lpplicnlillll <>1' HH~Pt ' h.F (E) inn
comp1c•x C:CI\1.

ll~

..2

l

rid

imil· ti n

up ling 3D- Var / 4D-V r +

h ' main adnmt agP of aclvmH'<'d PllS<'mhl<' hasc'd lllC'I hods such as , 'P 1 .. ov<'r v;.ma t imwl nwt lwei~ 1~ I lw I. in I lH' fmnH'l I lH' Pstima t <' of t lH' l>ackgronnd-cova rimH ·<'
matrix i:-. flow-dC'p<'ndc'nt 01 1t ':-.1mwl<'s I lw ··errors of I lw cia~·". On<' of tlH' primar y
ad,·antag' of 1 -\ 'm ,\ssumlaiH>ll lll('thod 1s 1ts ahtht~· to assimila te' asynchronou s
ohs<'lTaltolls.

lH' c·onplmg of ~C'(pl <'lllllll C'\'olntion of c·ovarimH'<' matric<'S with varia-

tional data-asstnulatiou lll<'thods is m1 active' a r<'a of n'sc·arch in tlw dclla-assimllation
comm1mitv (Zhang al}(l Zhang. 2012: Zha11g Pt al. 20n). Hc·c<'lll stndic·s lW\'C' shown
that thC' d e:l t <Hl'->Sllllilntioll svst<'lll c·cnnl>ining

nKF with .JD-\'m pC'rfornJPd hc·ttc•r

thannncoupl<'d lD-\ 'a r or "n KF (Zhang aw l Zlumg. ~01:2)

I\ lost oft hP st udiP~ on cmtpling <'llS<'ml>le-hasC'd d ata-assimi lation mC'thods \\·it 1J \'arin tionalmcthods ns<' "nl\

as a lC'pr<'S('nlali\'C' of thP c>nS('mhlc•-l>asC'd dHt <Ht'similaticm

mrth >cl. As discnssC'd in thi~ chssC'rtation .. 'PKF has Sll]WlH>r matlH' matical ptop<'ltiC's over EnKF and It will llC' int<'n's tmg to explor<' n hybrid data-clSsJmilatJoll ~.\·s tc•m
coupling the advall<·ed SP KF such as RHSI CKF ( .. ) \·n th th(' lD-\ 'nr lll<'lhod . Following Zhang <Hld Zlwng (20 12 ). t h<' h,1~ic idc·a of c·onplmg IS I lwt . I lH ' c'xplicit c·m·mi<11H '('
information from tbP sigma-pmut fm<'< ';lst::-; arc' us('cl m tlH' ulilllllll l<lt Joll of tlH' cost
fmlCticm of lD-\ 'm. Th(' lD-\ 'm illlC\1.\·si~ (I<l]C'C'(()l'\'lS llSC'cl clS tlH' nlC'illl of the· ,' Ph.r
mull~·sis . P osit IV(' aw lu('ga Ii\'<' p<1i1 of mwh·s1s pPrl 111 hilI ions nn · g('lH'nl t ('d ;H·c·ordiug

to thP ,' Pl\F sdH'Ill<' for tllC' ll< 'X I cyc ·lc • of forc•c ·Cis ts lu tlt( ' ll\ lHJ<I <l ppw;lch. hoth the•
, PKF a ncl lD- V;11 will nm iu p111 ;d]( 'l wl11l(' lH'ndit ing fwtu t lH' t'X('hnug(' of info1 ll\il1ion \\'It h ('ciC'h ot IH'I ell (liscr<'l (' I Jill<' int (') ,-,ds . 'ill<' lllcllll dJ ;dl('l\!.!,(' or ( IH!pl!ug ''P I\ F

with 1 -\'m will llC' tlw C'(lll l]>lll <llionnl C'ost.

n-

n

t ti ti

1n

PKF

Kalman filtc>1 della-a~ nmlation lll<'lhods gin' thP optimal PstimatC'. BL E. whPn thC'
modds a~ \\'Pll cls tlw llH'asm<'llH'llt opnatm that rdatc·s mode·! stale's to ohsc•rvations
me' lmC'ai and tlH' nssoc1nt 'd background ami ohsc'n·at ion C'rrors arC'

'anssian ( m-

hadan and Tang. 2011. \·au L<'<'U\\'C'll. 2012 .. \mhadan. 2()1:~). \\.hc'n \\'<' move• away
fmmtlw as....,umption of liiH'aritv
. and ;,111ssiamtv.
. that Js oftc·n tlH' case' with !llC' n'alwmld pwh!C'm~. the' opt1maht \' of the· c·st mw t C' oht nmC'd from the' Pnsc•mhlC' lH'C'OllH'S
qu 'Stionahlc' (v<lll LP<'ll\\'C'll. 2012) Til<' ass1m1ption of ;nussianit.v doC's not hold for

shmt tc'rlll (cJ..nlv 01 \\·c·c•klv) <l\'C'lage·s of mnny atmosplH•nc vmmhlC's ( 'nra all([ Haunachi. _()1-! ). R.c•cc'nt studi<'s hm·c· shown tlwt mnltiplH'ativP (statC'-dC'pC'ndc'ut) nmsc'

mod •Is can l>C' usC'd to accmmt fm thC' inte'rncll (dn1c-uniC'al and nunH'ncal) and <'Xtcrnal (ohsc•n·ation) C'Imrs and th<'ll associat 'd dc'\'i<1tions from ;aussmn probability
clistrilmtious (Amhadan and Tang. 2011: Sma and llmmachi . 2011) .

hP challC'ngc's

of nmltiplicatin• 1101se• can he sohTd \\'Jth <1 ll\·hrid d<lta-assimilat]()llllH'thod of a pcutiel filtC'r and

PI\F. knmn1 ns signw-pomt particle' filtc'r ( PPF) (\ '<m dC'1 ).lc•n\·e' .

200-!: Amhadan and Tang. 2011) .
. PPF makes usc' of t Jw important p1 <>JH'rt iC's of hot h t hC' part icl<' filtC'r and , PT\F
l-nlike tlw I\,llnwn tiltc•rs. th<' parti<l<' fiii<'J solv<'s tlw <ompl<'!<' nouliiH'<ll dntn-

assimilation pro!Jkm and Ill<' j>lOhcll>JlJt\·-dc•usJ(,V fllll<"(i()ll is I'llli\' j)!()pClgc1l<'d Ill lllll<'.
FnrthPnrHHC'. a particle' filt<'r do<'s not ll<'<'d mtih<ml t<'cluuqu<'s hkC' cm·mimH'<' Ill
fiat ion and locali:wtwu to oht aiu opt Jln,d e•:·d in1.1h' fm n hil-!,il -< lillH'llslont~l syst <'Ill

Th<' llH\lll di~<HlY<1llt age' of t lw J><ll! l<'l<' filt e'l' is hlt l'l dc'g< 11<'1', 1< ' \' t lll:-1 <'.Ill h<' ...,oh·l'd
b,v introclnclllg n good r<'S<11Uplmg stnll<'g\

(\'illl

L<'e'll\\<'ll. ~OI:Z). lu .'PPl· .. ' Ph.F

(('dlllolog.v is I!S('d foi se•qllt'Jltial illlp<Ht<lllt J('!-1(\llljlling (SIH) F()t clppli<<lllllll Ill <1

II I

hi g h-dnn 'nsional sys tc'm , ' PK . rcsamplmg tc ·hnology is not computationally fpasthlP.

lw dat a-a:->snuil atwn commmtit v ts vPry intPrPs tcd in d cv loping a hy hnd

pcutHlc filt '1. wludt (ctlll lS(' the RR

r TI\F (E ) sdH'lll(' fm the

I l.'"i

IR .

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)
11 C'\'aln,tt 1011 of thr' nonlm<'<U / noll- f?.<t llssian filtf'rs fc>r
~
H an. X . ,t 11 cl I 1. ' " ('1()0
thr scqu<'ntwl d <liH assimilation Rr molt Sr nsmq of Ent•!mnnu nl. 112 Ll:3J 111!1.
J

•

•"

.

•

, ,

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.JnliN . S. (2 00 :~)

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BC'aVC'rton.

nitC'd Stat<'s.

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