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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
One Drifter at ν= 500m2s-1 & (∆T, Ne, rloc) =(1day, 80, ∞)
T=90days
T=0(IC)
Truth (Control run) Without DAWith DA
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
= −
= − + − +
∑ ∑
∑ ∑
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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:
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
Tracer Control with Vortex System
∆T=1.5 and δ=3
EKFWithout TC
With TC
EnKFWithout TC
With TC
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
Targeted Observing System Design
Centers (eddies)Hyperbolic trajectoriesMixed cases
CentersHyperbolic trajectories
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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= − + − +∑ ∑
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
Control
Mixed
Centers
Hyperbolic trajectories
T=365 daysTargeted Observation System
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
Control
Mixed
T=50T=25T=0
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
T=200T=100
Control
T=75
Mixed
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
Observing System Design for Transient Flow Dynamics: Finite Time Lyapunov Exponents
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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
NOAA EMC Seminar June 30, 2005 K. Ide Observing System Design for Lagrangian Data Assimilation
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