Mul$-­‐Scale  Data  Assimila$on  for  Fine-­‐Resolu$on  Models    

Zhijin  Li  Jet  Propulsion  Laboratory,  California  Ins$tute  of  Technology    

with    James  C.  McWilliams  (UCLA)  and  Kayo  Ide  (UMD)  

Workshop  on  Sensi-vity  Analysis  and  Data  Assimila-on  in  Meteorology  and  Oceanography  Roanoke,  WV,  1-­‐5  June,  2015  

Copyright  ©  All  rights  reserved    

Atmospheric  Data  Assimila$on  for    Cloud  Resolving  Models  

•  Regional  opera-onal  models  oGen  

have  a  resolu-on  of  higher  than  4  

km  to    resolve  cloud  systems  

•  Are  data  assimila-on  schemes  

based  on  op-mal  es-ma-on  

theory  are  suitable  for  cloud  

resolving  models?  

 Oceanic  Data  Assimila$on  for  Sub-­‐Mesoscale  Processes  

Surface  Water  and  Ocean  Topography  (SWOT)  Satellite  

Salinity  Processes  in  the  Upper  Ocean  Regional  Study  (SPURS)  (hOp://spurs.jpl.nasa.gov)  

•  Mesoscale  40km  –  400  km  •  Sub-­‐mesoscale  1  km  -­‐  40  km    

Conven$onal  Data  Assimila$on:  Op$mal  Es$ma$on  

 §     prescribed  B  §     op-miza-on  algorithm                                    

Varia-onal  methods  (3Dvar/4Dvar):  

Sequen-al  methods  (Kalman  filter/smoother)  

§   dynamically  evolved  B  §   analy-cal  solu-on  (matrix  manipula-ons)  

minxJ = 1

2(x − xb )T B−1(x − xb )+ 1

2(Hx − y)T R−1(Hx − y)

Forecast  

Analysis  

Observa$on  

Maximum  Likelihood  

Error  Covariance:  Spreading  and  Filtering  

2

2

2)( Dr

erc−

=

cn =D2π

exp −k2D2

2"

#$

%

&'

B = ΣCΣ

2D   2D  

Correla-

on  

Power  Spe

ctrum  

50 100 150 200−1.5

−1

−0.5

0

0.5

1

1.5

STAT

E DI

STRI

BUTI

ON

(a)

OBS: RMSE=0.150GUESS: RMSE=0.275

TRUEGUESSOBS

50 100 150 200−1.5

−1

−0.5

0

0.5

1

1.5

AB−DA: RMSE=0.064MS−DA: RMSE=0.061

(b)TRUEAB−DAMS−DA

50 100 150 200−1.5

−1

−0.5

0

0.5

1

1.5

GRID NUMBER

D=10: RMSE=0.074D=20: RMSE=0.122

(d)TRUESS−DA D=10SS−DA D=20

50 100 150 200−1.5

−1

−0.5

0

0.5

1

1.5

GRID NUMBER

STAT

E DI

STRI

BUTI

ON

D=5: RMSE=0.063D=35: RMSE=0.144

(c)TRUESS−DA D=5SS−DA D=35

50 100 150 200−1

−0.5

0

0.5

1

GRID NUMBER

(d)GUESSMS−DA LSMS−DA SS

50 100 150 200−1

−0.5

0

0.5

1

GRID NUMBER

ANAL

YSIS

INC

REM

ENT

(c)GUESSAB−DA LSAB−DA SS

50 100 150 200−1

−0.5

0

0.5

1

(b)GUESSSS−DA D=10SS−DA D=20

50 100 150 200−1

−0.5

0

0.5

1

ANAL

YSIS

INC

REM

ENT

(a)GUESSSS−DA D=5SS−DA D=35

2

2

2)( Dr

erc−

=

Filtering  Proper$es  An  Idealized  1D  Experiment  

With  a  large  correla$on  scale,  DA  corrects  the  large  scale  error  

1D  Problem  with  Homogeneous  and  Isotropic  Background  Error    

 For  both  global  and  regional  domains,  we  have                

BS ≈ FBFT =σ b2S

S = FCFT

F  –  Discrete  Fourier  Transform  (DFT)    

•   The  eigenvalues  of  C  are  its  Fourier  transform  coefficients,  namely,  values  of  the  spectral  density  func-on.  

•  The  vectors  that  define  the  discrete  Fourier  transform  are  eigenvectors  of  C.  

Filtering  Proper$es            

xa = xb +BHT HBHT + R( )−1y−Hxb( )

H = I

sa = Fxa

sd = F y− x f( )BS = FBF

T =σ b2S

RS = FRFT =σ o2I

sa = sb + S S +σo2

σ b2 I"

#$

%

&'

-1

sd

Filtering  Proper$es:  Analysis  Increment  Scales          

Conven$onal  data  assimila$on  is  unable  to  update  fine-­‐scale  informa$on  

sa = sb + S S +σo2

σ b2 I!

"#

$

%&

-1

sd

σ o

σ b =12

0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1

DISTANCE r (KM)

CORR

ELAT

ION

BACKGROUND ERROR

exp(−r2/2D2)

D=50KMD=10KM

200 100 50 40 30 25 200

0.2

0.4

0.6

0.8

1

WAVELENGTH L (KM)

STAN

DARI

ZED

POW

ER S

PECT

RUM

ANALYSIS INCREMENT

INC., 50KMINC., 10KM

2D  2D  

Meso-­‐  and  Small  Scale  Component  in  Background  Error  Covariance    

1.   Meso-­‐  and  small-­‐  scale  

systems  are  intensive,  but  

are  localized  and  

intermiOently  occur.    

2.   The  forecast/background  

error  covariance  is  primarily  

determined  by  large  scale  

systems  

3.   The  correla$on  scale  is  inevitable  to  be  large  scale    

Ques$ons  and  Challenges  

•  Do  we  need  to  update  fine-­‐scale  informa-on  in  data  assimila-on?    

                 maybe  not  ?  

•  If  yes,  how  can  we  update  fine  scale  informa-on?  

We  here  suggest  a  scheme  to  update  fine  scale  informa$on    

Mul$-­‐Scale  Data  Assimila$on:  Data  Assimila$on  Separately  for  Dis$nct  Scales    

SL

SL

eeexxx

+=

+=

( )

( ) )()(21

21min

)()(21

21min

11

11

yxHRHHByxHxBxJ

yxHRHHByxHxBxJ

ST

LT

SSSTSx

LT

ST

LLLTLx

S

L

δδδδδδ

δδδδδδ

δ

δ

−+−+=

−+−+=

−−

−−

Scale  decomposi$on  

Mul$-­‐scale  DA  

SL

TSL

BBB

ee

+=

= 0

bxxx −=δ (Li  et  al.,  2015,  MWR)  

Proper$es  of  Mul$-­‐Scale  Data  Assimila$on  

•  The  decomposed  cost  func-on  can  be  derived  by  maximizing  the  condi-onal  

probability  

•  Separate  es-mate  of  the  state  for  dis-nct  scales  using  decomposed  cost  func-ons  

•  Explicit  incorpora-on  of  mul-ple  decorrela-on  scales,  thus,    Mul$-­‐Scale  Data  

Assimila$on  

p(xL | y)p(xS | y)

Mul$-­‐Scale  Representa$veness  Errors  and  Aliasing  

( )

( ) )()(21

21min

)()(21

21min

11

11

yxHRHHByxHxBxJ

yxHRHHByxHxBxJ

ST

LT

SSSTSx

LT

ST

LLLTLx

S

L

δδδδδδ

δδδδδδ

δ

δ

−+−+=

−+−+=

−−

−−

sa = sb + S S +σo2

σ b2 I!

"#

$

%&

-1

sd

Scales  untangled  

Scales  tangled   Scale  aliasing/contamina$on  ?  

Mul$-­‐scale  representa$veness  error  

Assimila$on  of  Decomposed  Observa$ons  

Mul$-­‐scale  representa$veness  error  

δy = δyL +δyS

minδxL

J = 12δxL

TBL−1δxL +

12(HδxL −δyL )

T HBSHT + RL( )

−1(HδxL −δyL )

minδxS

J = 12δxS

TBS−1δxS +

12(HδxS −δyS )

T HBLHT + RS( )

−1(HδxS −δyS )

NEXRAD  Reflec$vity:  Meso-­‐Scale  Connec$ve  System  (MCS):  June  14,  2007  

UTC  00:00   UTC  03:00  

UTC  06:00   UTC  09:00  

Improved  Reflec$vity:  UTC  06,  14  June,  2007  

NO  DA  

DA  

Simulated  900  hPa    Reflec$vity  

(Li  et  al.,  2015,  JGR)  

Summary  

•  Due  to  the  filtering  proper-es,  conven-onal  data  assimila-on  can  not  

effec-vely  update    fine-­‐scale  informa-on.  

•  To  update  fine-­‐scale  informa-on,  it  is  suggested  that  the  fine  scale  should  be  

treated  separately  from  larger  scales.  

•  The  cost  func-on  is  mathema-cally  decomposed  for  formula-ng  a  MS-­‐DA  

scheme  .  

•  The  decomposed  cost  func-on  allows  for  the  background  error  covariance  to    

explicitly  incorporate  mul-ple  decorrela-on  scales.  

•   Experiments  show  promising  performance  of  the  MS-­‐DA  scheme  in  3Dvar  for  

both  oceanic  and  atmospheric  applica-ons