A method for the assimilation of Lagrangian data

C.K.R.T. Jones and L. Kuznetsov,

Lefschetz Center for Dynamical Systems , Brown University

K. Ide, Atmospheric Sciences, UCLA

Data assimilation

0 0Initial condition: ,x y

1 1 1Predicted state: , ,b b bx y P

1 1True state: ,t tx y

1 1State estimate: ,a ax y

1 1Measurement: o tx x

Discrete ocean model

t = i t

x = k x, k=1,n

y = l y, l=1,m

t = t i

kx

lyk

l

k

i

Model state vector x = {v, T, S, ,…}

x R (N = 5 n m)

i

N

Ocean model:

M – model’s dynamics operator

b b1 [ ]i i iM x x

Observations

True ocean:

o t

b b1

t

o

observation operator

observation error

state vector of the true ocean

typically

y [ ]

y

]

]

]

[

[

[

,

i i i i

i

i

i i i i

Ti i i

i

Li

Ti i i

N L

H

H

E

M

E

x x η

Q η η

x ε

ε

x

R

R ε ε

Covariance of the model residual:

Covariance of the observation error:

Extended Kalman Filter

b b t b t b b1

Keep track of the state vector error covariance matrix

using TLM:

linearized model operator

Combine model and observations into new state

[( )( ) ]

/ :

T Ti i i i i i i i i i

i i

E

M

P x x x x P M P M Q

M x

x

a b o b

a

b

a

b

1b

a

in a way that minimizes

: linearized observation function

: updated st

( )

tr :

/

i i i i i

i i

i i i i

i i i

T Ti i i i i i i

H

H

x x K d d y x

K P H H P H

P

H x

I K

R

P H P ate error covariance matrix

Ocean vs Atmosphere

Ocean:Drifters/floatsSlow (weeks) Horizontal coverage

Atmosphere:Balloons Fast (days) Vertical structure

Difference in scales and cost($drifters & floats>>$balloons)

Lagrangian information

y i-1,1

Lagrangian observations from drifters and floats do not give the data in terms of model variables. They are rarely used for assimilation into ocean models

Solution: Include drifter coordinates into the model

y i-1,2

y i,1

y i,2

Methodology

1 1

drifters "read" velocity information,

correlat

, ; [ ]; [ , ];

0

ions bet

0 0

wee n

F D F F F D D D Fi i i i i i i

FF FF DFi i i

i iDF DD DF DDi i i i

DF DFi

M M

M

M M

M

x x x x x x x x

P PM P

P P

P

x

1 1b b

( )= , = ,

and appear.

Observation of drifter po

sitions:DF

T T DDii i i i i i i i iDD

D F

i

H

PK P

x

x Hx H 0 I

H H P H R P RP

Point vortex systems

• Point vortex systems provide a rough model of 2d flows dominated by strong coherent vortices

• Simple dynamics (small number of degrees of freedom) makes them an attractive testing ground

• We consider flows due to N vortices. L tracer particles are observed. Tracers do not influence the flow.

b bb * b* b*

t tt

b*1,

* t * t * t *1

1

, 1

b

, 2 2

, , ,

, 2 2

State vector: vortices: tracers:

Th

e error:

N L

N Nj j

m m m m Ni j m im j m

N Nj j

m mi j m im j m j

j

i iz

z

i iz

z

z

z

z

z

x z ζ z C ζ C

x x

*

2* b* b* * * *

* *

t *1 2

1 2* *2 1

, , 22

, 0, ,

; TLM: ( )

jDFmj

m j

T T

FF

T TDF DD i

AFF T Tz

DF DD

T

A 0

ηη ηηA A

A QA 0 η η η η

A A

x x P x x P x x

P PP

P P

0

0

P AP AP Q

2

( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) , , 0x y x x y y x yi

0 I

I 0

η η η η η η η η η

*2

* * *

o t t

o oa

a a

= , 2

*b1 2

b b * b1 1 2 2

b

2 1

b

1 1 2

Observation of tracer positions:

Update:

,

y

y

y

T T

T T

i i i

i i i i i i i

i i

i ii ii

i i

εε εε 0 I0 IH R

I 00 I ε ε ε ε

00

x K K

I K HP K HP K

Hx ε ζ ε

x ζ

P HP KP

ζ

H

1 1

b*1

2b

b b1 1

2

a a

b* *

1

and are used as initial conditions to forecast and

;i iT Ti i i

i i

i

ii ii

K KK P H HP H R

K K

PP xx

P

Two point vortices

21,2 1 2

2

1/ 2

Deterministic model: vortices rotate around the origin:

( ) / 2, /(2 ), | |

Rescaling: 2 / , [( / 2 ) /( / 4)] 2 , 2

( / 2 ) , 2 /

When the noise is present

i tz t e z z

z z t t

t b 3/ 2 2

the system drifts away from the model:

| | where is shear at vortex location:

We take =0.01 0.05; deterministic model completely

loses track of the system in 1-3 motion periods.

z z t

Simple example: N=1, L=1

b b1

t t1

i i

i i

z z

z z

b t1 1i i

bi

oiy

* 21 1 2 =0i iz z

2

a b t b2 2i i i iz z z z

ai

ti

aiz

Two vortices, N=2, one tracer, L=1

1 2

1,2

1, 1

2

1

0.04

0.6 0.3

0 . 2 0

z

i

z

T

z 1 z 2

It works!

Two vortices, N=2, one tracer, L=1

1 2

1,2

1, 1

2

0.04

0.6 0.3

0

5

.

2

1

.0

i

z z

T

z 1 z 2

Or does it?

The overall performance of EKF is represented by tr P

Efficiency of tracking of individual vortices is measured by |z|

a

0.02 =0.02 1 0.05 =0.05 T

N=2, L=2

Assessment of method

• When does the assimilation works and when does it not?

• How does the filter fail?

• What is the role of Lagrangian structures

• Compare with the assimilation of velocity data directly

0, , , T

N=2, L=1 (Ne =100 noise realizations)

0.75 1.0 1.25 1.5 1.75 2.0

0 0 2 2 8 18

0.09 0.10 0.12 0.19 0.35 0.51

0 1 7 19 22 29

0.12 0.13 0.22 0.40 0.43 0.67

4 7 16 25 - -

0.20 0.29 0.51 0.78 - -

σ=0.02

ρ=0.02

σ=0.02

ρ=0.05

σ=0.03

ρ=0.02

T

fN

d

fN

d

fN

d

Tracer launch location: (0) 0.6 0.3 in all experimentsi

Ensembles of different system noise realizations, N=2, L=1

1 0.02 0.02 1.5 T T

2fN 0fN

Divergence of the filterCorotating reference frame

1 2

1,2

1, 1

2

0.04

0.6 0.3

0

5

.

2

1

.0

i

z z

T

Exponential separation of trajectories near the saddle causes the divergence of the filter

Ocean Atmosphere

Drifters/floats Balloons

Launching strategy Targeted observations

Dependence on initial tracer location

1 2 2 0.3-0.6i 1-0.6i 1-i 2.4-2.4i -1.75i

0.5 55.5 0.5 15 0

0.12 1.90 0.11 0.29 0.11

0ζ

(%)fN

d

0.02,

0.04,

1T

Vortices of different strength

1 23 3 0.02, 0.04, 1T

Four vortices, N=4

Vortices: (-4,-1), (-4,1), (4,-1), (4,1) (regular motion)

(5.24,3.61),(-2.12,0),(2.12,0),(0,-3) (chaotic motion)

Tracers: (-0.5,1) and (0.5,0), L =2

T=0.5, = 0.005, = 0.02

z 5

z 2 z 3

z 4

z 1

z 6

z 5z 2

z 3z 1

z 6

z 4

N=4, L=2

Regular Chaotic

Comparison with assimilation of velocity

o o1

1/ 2 21 1

Velocity from consecutive drifter observations:

( ) /

- error resulting from position observation error ,

model noise and finite

( )/ ( ) 2 /( )

i i i

i i i

T

T

T T T

v y y

R R

/ T Q

0.4 0.5 0.6 0.7 0.8

0 4 6 46 76

0.18 0.24 0.31 0.58 0.71

f

T

N

d

o o o

1 2Second order: (3 4 ) /(2 ) : gives some improvement

For same , our scheme worked up to 1.0

i i i iT

T

v y y y

σ=0.02

ρ=0.02

Conclusions

• Assimilation of the tracer data into the point vortex system affords successful tracking of the flow (N ~ L, both small)

• EKF fails for large T, , : TLM is not a good approximation

• Passage next to (Lagrangian) saddle can cause filter divergence – efficiency of the method depends on the launch location relative to the Lagrangian flow structures

• Basis for launch strategies of floats and/or drifters

• Extend to layered models

• Future work: analyze filter behavior near the saddle; move to gridded ocean models increase N use other methods for model error propagation