lagrangian data assimilation and observing system design ......observing system design for...

NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation

Lagrangian Data Assimilation and Observing System Design

from Dynamical Systems Perspective

Kayo Ide, UCLAChris Jones, Hayder Salman and Liyan Liu, UNC-CH

Project Sponsored by


Lagrangian Properties of the OceanCoherent structures

Commonly observedDescriptive physical pheonomenaare often in Lagrangian nature

Lagrangian trajectoriesDirectly related to dynamics

Maybe associated with Lagrangian structures

( , )d t dt d= +x v x η


Combining the Elements

Data AssimilationUsing Eulerian Models

Lagrangian ObservationsAlong the Paths

Lagrangian Data Assimilation

Observing System Design:Observing System Design:Optimal Deployment StrategiesOptimal Deployment Strategies

Dynamical Systems Theory


Eulerian Model and Observation

Eulerian model Eulerian observation at rs

( )( )

( )( )( ) ( )

a f o fF F F S S F

1f f oF FF S S F S S

f f t f tF F F F F

a f o fF F S S F S F

T T

TE

−

= + −

= +

⎡ ⎤= − −⎣ ⎦= = −

v

v v v

v v

x x K y H x

K P H H P H R

P x x x x

P P H R I K H PA

( ) ( )( ) ( )

f aF F F 1

f aF F F 1

k k

k k

t t

t t−

−

=

=

x m x

P PM

Forecast Analysis

Kalman Filter


Lagrangian Observations and Eulerian Model

Observation yok of the true positions yt

k subject to noise εtk

along the instrument path, k=1,…,K,

True drifter dynamics (may be stochastic)

( )( ) ( ) ( )

t t

t t t0 0

: dynamic noise tk

k t

d t dt d d

t t t dt d

= +

= + +∫

y v y η η

y y v y η

( )o t t t t 0k k k k kN= +y y ε ε R∼

Model forecast xD(tk)Simulated drifter dynamics (deterministic)

( )( ) ( ) ( )

f f fD F D

f f f fD D 1 F D1

tkk k tk

d t dt

t t t dt− −

=

= + ∫

v

v

x H x x

x x H x x


Lagrangian Data Assimilation (LaDA)

( ) ( )( ) ( )

F F F DFF FD F F F

DF DD D D DD F D D

T T

ET T

⎡ ⎤∆ ∆ ∆ ∆ ∆ −⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥≡ = ∆ ≡ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ∆ −⎝ ⎠ ⎝ ⎠ ⎝ ⎠∆ ∆ ∆ ∆⎣ ⎦

x x x xP P x x xP x

P P x x xx x x x

Augmented systemfor the ocean (xF) and drifter (xD) statesState vector x is N=NF + ND dimensional & dynamics is one-way interaction

Error covariance P is NxN=(NF + ND)(NF + ND ) dimensional

( )( )

F FF F F

D F DD D D

=

td Etdt

⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎡ ⎤≡ =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎝ ⎠

m xx x xx x

m x xx x x

( )( )

( )1f f oa f f f f

FD DD DDo o fF F FD F FD D D Da f f f f 1f f o

D D DD D D DD DD DD

−

−

⎛ ⎞+⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟= + − = + −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ +⎣ ⎦ ⎝ ⎠

P P Rx x K x xy H y x

x x K x x P P R

Update mechanism

Lagrangian observation operator HD =(0 I) extracts the drifter information

Correlation PFD enables to propagate (yoD -xf

D) into xaF as well as xf

D

FDD D D

DD

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

PH x x H P

P


Ensemble Kalman Filter (EnKF)

xF

xD

yD

( ) ( ) ( )a f o fj k j k k j k jt t= + −x x K y H x

( )( ) ( )( )( )( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

1f f o

f f f f f

1

f f f f f

1

f f

1

1 1

1 1

1

T T

k k k k k k k

Ne TT

k k j k k k j k k kj

Ne TT

k k k k j k k k k j k k kj

N

k j kje

t t

t t t t tNe

t t t t tNe

t tN

−

=

=

=

= +

⎡ ⎤ ⎡ ⎤≈ − −⎣ ⎦ ⎣ ⎦−

⎡ ⎤ ⎡ ⎤≈ − −⎣ ⎦ ⎣ ⎦−

=

∑

∑

K P H H P H R

P H x x H x H x

H P H H x H x H x H x

x x

e

∑

Analysis

( ) ( )f a1j k j kt t −=x m x ( )o t o

k k k k kt= +y H x ε

Ensemble Forecast Observation


Application to Shallow-Water Ocean Circulation

Can LaDA recover xt(t) after assimilating the drifter positions yo(tk)?How many drifters?How often?Where to deploy?

T=0(IC) ??

80 Ensemble membershave=550mhstd =50m

control runhave=500m


One Drifter at ν= 500m2s-1 & (∆T, Ne, rloc) =(1day, 80, ∞)

T=90days

T=0(IC)

Truth (Control run) Without DAWith DA


Performance VerificationParameters

Degree of turbulence νAssimilation time interval ∆TEnsemble number Ne

Localization length scale rloc

Performance validation by comparing True error norm

Predicted error and ensemble spread

( ) ( )

( ) ( ) ( ) ( )

, , ,2 2f f f f, , , ,

, , ,

, , ,2 2 2 2f f f f f f f, , , , , , , ,

, , ,

/

/

Nx Ny N Nx Nye

i j i j k e i ji j k i j

Nx Ny N Nx Nye

i j i j k i j i j k e i j i ji j k i j

h h h N h

KE u u v v N u v

= −

= − + − +

∑ ∑

∑ ∑

( ) ( )

( ) ( ) ( ) ( )

, ,2 2t f t t, , ,

, ,

, ,2 2 2 2t f t f t t t, , , , , ,

, ,

/

/

Nx Ny Nx Ny

i j i j i ji j i j

Nx Ny Nx Ny

i j i j i j i j i j i ji j i j

h h h h

KE u u v v u v

= −

= − + − +

∑ ∑

∑ ∑


Lagrangian Decorrelation Time Scale (TL) at ν= 500m2s-1

Region A: TL ~ O(10 days)

Region B: TL ~ O(100+ days)

Results for TL are similar at ν= 400m2s-1


Effect of Assimilation Time Interval ∆T

Lagrangian time scale is about TL~10days

∆T =1,2,4,5,8,10,15 and 20 days

(ν, rloc, Ne) =(500m2s-1, 300km, 80)

The method is stable if ∆T <TL

Height Error Kinetic Energy Error


Effect of Turbulence νReducing ν leads to turbulent flow

ν=500m2s-1 to 400m2s-1

(rloc , ∆T, Ne) =(∞, 1day, 80)

Convergence deteriorate relative to ν=500m2s-1

Predicted error does not match true error

Increasing to 36 drifters does not rectify the problem

Norm comparison Chaotic advection of xD


Effect of Ensemble Size Ne and Localization rloc

Error covariance is approximated by ensembles;Slow convergence ~ (Ne)-1/2

Noisy correlation between remote regions for small Ne

leading to deterioration of filtering

A remedy is to introduce localization of error correlationK= ρ ○(PHT) ( ρ○(HPHT) + R )-1

ρ ○() is the function of distance betweenGrid pointsGrid and drifterdrifters

denoting the Schur product (Hamill, 2001)rloc gives the radius of influence


Effect of Localization rloc for Turbulent Dynamics

Three localization radii investigated

rloc=150km, 300km, 600m

(ν,∆T, Ne) =(400m2s-1, 1days, 80) and 36 drifters

Better convergence using the localization

Optimal rloc=300km

Norm comparison Chaotic advection of xD


Drifter Update Examples

Along the jetLarge derivation: can be still successful

In the recirculating region

Can Can LaDALaDA handle chaotic drifter dynamics?handle chaotic drifter dynamics?

1

2

3

1

23

In the eddiesDetrainment process by the saddle

3


Dealing with Lagrangian Saddle:Estimation of Coherent Structures

Parameters (σ, ρ, ∆T)=(0.04, 0.02, 1.5)Ex: failuer case for I.C.no.3


Saddle Effect with the Eulerian Model

Large updates in drifter position occur near saddle pointPrior PDF distribution can be bimodal; Unimodal Gaussian distribution breaks down.

It can potentially produce large and spurious changes in the flow

Ensemble Spread of drifter Update mechanism of the mean


Saddle EffectSaddle effect occurs near the linearly hyperbolic region of velocity (λ: hyperbolicity given by the

positive local Lyapunov exponent)

∆xaF may be unreasonably large because

PFD may be approximated by too large with exponent λ:

Dragged by a large innovation (yoD-xo

D)

( ) ( )1a f f o o fD DD DD D D D

−∆ = + −x P P R y x

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

D D F D D D F D 0

DD D F DD DD D F DD D F DD 0 D F

+ T T

d t tdtd t tdt

= ≈ =

= ≈ =

x M x x x F x x

P M x P P M x P F x P F x

( ) ( )1a f f o o fF FD DD D D D

−∆ = + −x P P R y x

∆xaD can be updated correctly: PDD grows linearly in time with exponent 2λ:


Tracer Control: Sanity check for ∆xaF

CF ≡ Standard deviation of ∆xaF with respect to the expected error

PfFF is independent of xf

D or xtD: the flow has no knowledge of the drifters

Implementationif CF < δ: Update of xF

if CF ≥ δ: No update of xF (but xD is updated)

CF is computed for xD within rFD from xD (rFD <rloc)

Control is applied to entire xD within rloc from xD

( ) ( )( ) ( )

a aF F

1

1a f f o o fF FD DD D

Ff

FF

D D

TC

−

−

=

∆

∆

+ −

∆

=

P

P P y

x

R x

x

x

PfFF

∆xaF

cF


Tracer Control with Vortex System

∆T=1.5 and δ=3

EKFWithout TC

With TC

EnKFWithout TC

With TC


Dynamical Systems TheoryTwo-dimensional drifter dynamics

( ) ( ) ( )

( ) ( ) ( )

, , , , , ,

, , , , , ,

d x u x y t x y t a x y tdt yd y v x y t x y t a x y tdt x

∂= = − ψ ×

∂∂

= = ψ ×∂

Unsteady flow

Steady flow

• Velocity u(x,y,t), is tangent to the streamfunction Ψ(x,y,t)

• Any trajectory remains on the iso- Ψ(x,y) curve• Streamfunction field Ψ(x,y) completely describes the global flow geometry• Stable and unstable trajectories from the hyperbolic fixed point (saddle) define the global

template

( ) ( )( )( ) ( )( ) ( ) ( )( )0 0

, 0

, ,

D D

D D D D

d x t y tdt

x t y t x t y t

ψ =

ψ = ψ

• Stable and unstable invariant manifolds (material curve) from the hyperbolic trajectory (Lagrangian saddle) define the global template of the Lagrangian dynamics


Targeted Observing System Design

Centers (eddies)Hyperbolic trajectoriesMixed cases

CentersHyperbolic trajectories


Preliminary Results

( ) ( ), ,2 2

, , ,, ,

/Nx Ny Nx Ny

t ti j i j i j

i j i jh h h h= −∑ ∑ ( ) ( ) ( ) ( )

, ,2 2 2 2

, , , , , ,, ,

/Nx Ny Nx Ny

t t t ti j i j i j i j i j i j

i j i jKE u u v v u v= − + − +∑ ∑


Control

Mixed

Centers

Hyperbolic trajectories

T=365 daysTargeted Observation System


Control

Mixed

T=50T=25T=0


T=200T=100

Control

T=75

Mixed


Observing System Design for Transient Flow Dynamics: Finite Time Lyapunov Exponents


Future Directions: Atmospheric ApplicationsAtmospheric motion vector / Cloud wind vector

Global wind vectors are used for the initial value of Numerical Weather Prediction (NWP), via data assimilation.

Satellite-based observations provide wide coverage for the monitoring of atmospheric motion vectors by tracking the movement of clouds and interpolating the movement in time.

Idea: Use Lagrangian data assimilation method & assimilate the cloud feature positions directly into the morel, - without any interpolation in time- cloud height information is naturally taken care of


Summary and Work in Progress

Data AssimilationUsing Eulerian Models

Lagrangian ObservationsAlong the Paths

Lagrangian Data Assimilation

Observing System Design:Observing System Design:Optimal Deployment StrategiesOptimal Deployment Strategies

Dynamical Systems Theory

lagrangian data assimilation and observing system design ......observing system design for...

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