Lagrangian Data Assimilation:
Method, Applications, and
Strategy for Optimal Drifter Deployment
Kayo Ide, UCLA
C.K.R.T. Jones, Guillaume Vernieres,UNC-CH
Hayder Salman, Cambridge U.
Liyan Liu, NCEP
Lagrangian Instruments in the Ocean: Drifters
Observations at sea surface
T
: Temperature
along (x(2D) )(tk)) at sea surface
Float Package
Temperature Sensor
Data available from
http://www.aoml.noaa.gov/phod/dac/dacdata.html
http://www.drifters.doe.gov/design.html
Lagrangian Instruments in the Ocean: Floats
Observation on the isopyncnal surface
(T,S )
(u,v)
along (x(2D) )(tk), p(x(2D) )(tk))
http://www.dosits.org/gallery/tech/ooct/rafos1.htm
http://www.whoi.edu/instruments/
Global Ocean Observing System by Drifters
Global observation network by drifters
1250 drifters to cover at the 5ox5o resolution
Drifters are used as the platform
Eulerian observations of T (SLP, Wind)
http://www.aoml.noaa.gov/phod/dac/gdp.html
Assimilation and Short-Range Forecast for Regional Ocean
Real-Time Regional ocean off the U.S. West Coast
Observations:
Remote-sensing
In situ
Model: Regional Ocean Modeling System
(ROMS)
One-way nested configuration with
increasing resolution for smaller domain
COAMPS forcing
Method: Incremental 3D-Var
Weak constraints by dynamic balance
Inhomogeneous / anistropic background error
covariance using Kronecker product
Li, Chao, McWilliams, Ide (2007a,b)
Ocean Observations: Remote-Sensing by Satellite
Sea Surface Temperature (SST)
Sea Surface Height (SSH)
Data available at
http://ourocean.jpl.nasa.gov/
http://nereids.jpl.nasa.gov/
Ocean Observations: In Situ by Stationary Platforms
Mooring
(T , S, p)
(u, v)
Data available at http://ourocean.jpl.nasa.gov
http://www.mbari.org
Ocean Observations: In Situ by Movable Platforms
Glider
At the surface: xG(2D)
In the water:
(T , S, p)
Ship
http://www.mbari.org
(T , S, p)
(u, v)
Data available at http://ourocean.jpl.nasa.gov
Ocean Observations
Currently available observations are inhomogeneous and sparse in space &
sporadic in time. Available observations are mostly
T and S
In the upper ocean
Routine Observation for ROMS 3D-Var system
Type
Platform
Number / day
SST
Remote-sensing
Satellite
O(102) - O(104)
SSH
Remote-sensing
Satellite
0 - O(102)
T&S
In situ
Mooring, ship,
glider, AUV
O(102)
Ocean observations are precious
New types of observations: SSS by Satellite, Coastal HF radar
New technology for cost effectiveness: Lagrangian data
Ocean Observation: Remote-Sensing by HF Radar
Coastal Oceans Currents Monitoring Program (COCMP)
http://www.cocmp.org/
http://www.cencoos.org/currents
Lagrangian Dynamics of Drifters
QuickTime™ and a
decompressor
are needed to see this picture.
Data available from http://www.aoml.noaa.gov/phod/dac/dacdata.html
Outline
Ocean observation for data assimilation systems
Lagrangian data assimilation (LaDA) method
Application I: Double-gyre circulation. “Proof of concept”
Application II: Gulf of Mexico. “Efficiency”
Design of optimal deployment strategy using dynamical systems
theory
Concluding remarks
Summary
Future Directions
Basic Elements of Lagrangian Data Assimilation System
Eulerian Model:
State xF

xF tk
 M
u t
 ijl k
  v ijl tk

 hijl tk

 M











Lagrangian Observation:
Location yD

yD, j tk
NF : 10
5-7

M
 r ( x) t
 D, j k
(y )
  rD, j tk

 rD,( p)j tk


M
Data Assimilation Method











LD  2 [or 3]
per drifter
Data Assimilation Method: Kalman-Filter Approach
Forecast from tk-1 to tk:
Observation at tk:
x ak 1  x kf 
 a  f
Pk 1  Pk 
 y ok 
 o
R k 

xkf  x kt ~ N 0,Pkf

 
 h x 
y ok  y kt ~ N 0,R ok
y kt
t
k

 I  K H P
k
x ak  x kf  K k y ok  y kf

x ak  x kt ~ N 0,Pka

Analysis at tk:
x k   x k 
 f   a
Pk  Pk 
f
a
Pka
k
k
 
 P H R

f
k
y kf  hk x kf
Kk
Hk 
f
T
o
k
k
k

h
x k
xf
k
 HkPkfHTk

1
Elements of Assimilating Lagrangian Data
Essence of analysis in data assimilation

K  P H R
y  h x 
xa  x f  K yo  y f
f
f
T
o

 H PfHT

1
f
Elements in hands
Forecast flow state
xF as xf
Lagrangian observation yD as yo
Missing elements
h that gives yfD from xfF , because nothing in xfF directly relates to yoD
Pf that gives K for optimal impact of yo on xa
Assimilation of Lagrangian Data: Conventional Method
Transform from Lagrangian data
  
yDo  yDo t j
to Eulerian (velocity) data
y oV 
  
yDo t j  yDo t j 1
t j  t j 1
Observation operator
y Vf  HV x f
Hv is linear spatial interpolation.
Feedback the mismatch of observation (innovation) into the model variable


R
xFa  xFf  K V y oV  y Vf
K V  P HV
f
T
o
V
 HVP HV
f
T

1
Carter (1989)
Kamachi, O’Brien (1995)
Tomassini, Kelly, Saunders (1999)
Assimilation of Lagrangian Data: Conventional Method
Transform from Lagrangian data
  
yDo  yDo t j
to Eulerian (velocity) data
y oV 
  
yDo t j  yDo t j 1
t j  t j 1
Observation operator
y Vf  HV x f
Hv is linear spatial interpolation.
Feedback the mismatch of observation (innovation) into the model variable


R
xFa  xFf  K V y oV  y Vf
K V  P HV
f
T
o
V
 HVP HV
f
T

1
Carter (1989)
Kamachi, O’Brien (1995)
Tomassini, Kelly, Saunders (1999)
Assimilation of Lagrangian Data: Conventional Method
Transform from Lagrangian data
  
yDo  yDo t j
to Eulerian (velocity) data
y oV 
  
yDo t j  yDo t j 1
t j  t j 1
Observation operator
y Vf  HV x f
Hv is linear spatial interpolation.
Feedback the mismatch of observation (innovation) into the model variable


R
xFa  xFf  K V y oV  y Vf
K V  P HV
f
T
o
V
 HVP HV
f
T

1
Carter (1989)
Kamachi, O’Brien (1995)
Tomassini, Kelly, Saunders (1999)
Assimilation of Lagrangian Data: Conventional Method
Transform from Lagrangian data
  
yDo  yDo t j
to Eulerian (velocity) data
y oV 
  
yDo t j  yDo t j 1
t j  t j 1
Observation operator
y Vf  HV x f
Hv is linear spatial interpolation.
Feedback the mismatch of observation (innovation) into the model variable


R
xFa  xFf  K V y oV  y Vf
K V  P HV
f
T
o
V
 HVP HV
f
T

1
Carter (1989)
Kamachi, O’Brien (1995)
Tomassini, Kelly, Saunders (1999)
Lagrangian Data Assimilation (LaDA) Method
Elements in hands
xF
Augmented state x and model m
Flow state xF and model mF
Drifter state xD and model mF
x 
x   F
x 
xD
D

  

    
 mF xF tk 1

 mD xF tk 1 ,xD tk 1
  
m x tk 1
Observation yD and operator h that relates yD to x
x 
 0 I  F
 xD 
  
h x
Missing elements
Pf that gives K for optimal impact of yo on xa
Ide, Jones, Kuznetsov (2002)
[Ide and Ghil (1997)]
Lagrangian Data Assimilation (LaDA) Method
Elements in hands
Augmented state x and model m
Flow state xF and model mF
Drifter state xD and model mF
x 
x   F
 xD 

  

    
 mF xF tk 1

 mD xF tk 1 ,xD tk 1
  
m x tk 1
 
dxD,j  uF xD, j dt
Observation yD and operator h that relates yD to x
x 
 0 I  F
 xD 
  
h x
Missing elements
Pf that gives K for optimal impact of yo on xa
Ide, Jones, Kuznetsov (2002)
[Ide and Ghil (1997)]
Lagrangian Data Assimilation (LaDA) Method
Elements in hands
Augmented state x and model m
Flow state xF and model mF
Drifter state xD and model mF
x 
x   F
 xD 

  

    
 mF xF tk 1

 mD xF tk 1 ,xD tk 1
  
m x tk 1
Observation yD and operator h that relates yD to x
x 
 0 I  F
 xD 
  
h x
Missing elements
Pf that gives K for optimal impact of yo on xa
Ide, Jones, Kuznetsov (2002)
[Ide and Ghil (1997)]
Ensemble-Based Data Assimilation


Use of ensemble X  x ,...,x
to represent the uncertainty of x
in particular, mean and covariance
1
Ne
mean
 xF 
1 Ne  xF,n 
x  
 
 xD  Ne n1  xD,n 
covariance
P
PFD 
P   FF
 PDF PDD 




 x x x x
1
F,n
F,n


Ne  1 n1  x  x x  x
 D,n
F,n
Ne
 
 
T
T




T
xF,n  x xD,n  x 

T
xD,n  x xD,n  x 
r 
xD   x 
 ry 
Ensemble-Based LaDA
xD
1. Ensemble forecast from tk-1 to tk
 
t  m x
f
a
xF,n
tk  mF xF,n
,
f
xD,n
k
D


tk 1
a
a
,
x
, tk 1
F,n
D,n
for n = 1,…, Ne
2. Ensemble update at tk to incorporate
  and
yDo tk

o
RDD
tk
xF
Analysis (dropping tk )
a


x
F,n
a
xn   a 
 xD,n 
f
 xF,n
  PFDf  f
o
  f    f  PDD  R DD
 xD,n   PDD 

 y
1
o
f



x
D
D,n
D,n
o

Salman, Kuznetsov,Jones, Ide (2006)
Salman, Ide, Jones (2007)
Ensemble-Based LaDA
xD
1. Ensemble forecast from tk-1 to tk
 
t  m x
f
a
xF,n
tk  mF xF,n
,
f
xD,n
k
D


tk 1
a
a
,
x
, tk 1
F,n
D,n
for n = 1,…, Ne
2. Ensemble update at tk to incorporate
  and
yDo tk

o
RDD
tk
xF
Analysis (dropping tk )
a


x
F,n
a
xn   a 
 xD,n 
f
 xF,n
  PFDf  f
o
  f    f  PDD  R DD
 xD,n   PDD 

 y
1
o
f



x
D
D,n
D,n
o

Ensemble-Based LaDA
xD
1. Ensemble forecast from tk-1 to tk
 
t  m x
f
a
xF,n
tk  mF xF,n
,
f
xD,n
k
D


tk 1
a
a
,
x
, tk 1
F,n
D,n
yD
for n = 1,…, Ne
2. Ensemble update at tk to incorporate
  and
yDo tk

o
RDD
tk
xF
Analysis (dropping tk )
a


x
F,n
a
xn   a 
 xD,n 
f
 xF,n
  PFDf  f
o
  f    f  PDD  R DD
 xD,n   PDD 

 y
1
o
f



x
D
D,n
D,n
o

Ensemble-Based LaDA
xD
1. Ensemble forecast from tk-1 to tk
 
t  m x
f
a
xF,n
tk  mF xF,n
,
f
xD,n
k
D


tk 1
a
a
,
x
, tk 1
F,n
D,n
yD
for n = 1,…, Ne
2. Ensemble update at tk to incorporate
  and
yDo tk

o
RDD
tk
xF
Analysis (dropping tk )
a


x
F,n
a
xn   a 
 xD,n 
f
 xF,n
  PFDf  f
o
  f    f  PDD  R DD
 xD,n   PDD 

 y
1
o
f



x
D
D,n
D,n
o

Ensemble-Based LaDA
xD
1. Ensemble forecast from tk-1 to tk
 
t  m x
f
a
xF,n
tk  mF xF,n
,
f
xD,n
k
D


tk 1
a
a
,
x
, tk 1
F,n
D,n
for n = 1,…, Ne
2. Ensemble update at tk to incorporate
  and
yDo tk

o
RDD
tk
xF
Analysis (dropping tk )
a


x
F,n
a
xn   a 
 xD,n 
f
 xF,n
  PFDf  f
o
  f    f  PDD  R DD
 xD,n   PDD 

 y
1
o
f



x
D
D,n
D,n
o

Mechanisms of Lagrangian Data Assimilation (LaDA)
Forecast from tk-1 to tk:
Observation at tk:
x ak 1  x kf 
 a  f
Pk 1  Pk 
 y ok 
 o
R k 
 
 mF xFa 
 xFf 
 x f    m x a , x a 
 D F D  k 1
D k

 PFFf
 Pf
DF

yDo tk


o
with RDD
tk
f
a

 PFFa PFD

PFD

M
f 
 Pa Pa 
PDD
k
DF
DD
k 1
Analysis at tk:
Other Methods
OI:
Molcard et al (2003)
4D-Var: Nodet (2006)
x k   x k 
 f   a
Pk  Pk 
f


f
f
o
 xFa 
 xFf   PFD PDD  R DD
 x a    x f    f
f
o
D
D
k
 PDD PDD  R DD
  y

 
a
1
1
o
D
 xD
f

 PFFa
 Pa
DF
k
a
f

 PFFf PFD

PFD

F
a 
 Pf Pf 
PDD
k
DF
DD
k
Application I. Mid-latitude Ocean Circulation:
“Proof of Concept”
nature run (simulated truth)
ht=500m
x1000km
x1000km
Ocean circulation
1-layer shallow-water model
Domain size: 2000km x 2000km
Wind-driven: =0.05 Ns-2
Salman, Kuznetsov,Jones, Ide (2006)
Perfect model scenario
Model spin-up for 12 yrs
- Nature run (truth) with H0=500m;
- Ensemble with (Hmean, Hstd)=(550m,50m)
Drifter released at the beginning of 13 yrs
observed every day
Ex.1: ν=500m2s-1, (∆T, LD )=(1day, 1), (Ne, rloc)=(80, ∞)
Truth
T=0
T=90
days
With LaDA
Without DA
Ex.1: ν=500m2s-1, (∆T, LD )=(1day, 1), (Ne, rloc)=(80, ∞)
Truth
T=0
T=90
days
With LaDA
Without DA
Application II. Gulf of Mexico
“Why is the LaDA Efficient?”
Ocean circulation:
Loop-current eddy
3 layer shallow-water model with the
structured curvilinear grid
Horizontal resolution: 5-13km
(average 8.3km)
Vertical resolution: 2 layers
at 200m, 800m, 2800m
Current forcing at 22.4Sv
Data assimilation system
Perfect model scenario
Ne =32-1028
LD =2-6
Initial perturbation in layer depth
only (velocity determined by
geostrophic balance)
QuickTime™ and a
decompressor
are needed to see this picture.
Vernieres, Ide, Jones, work in progress
Motivation for Eddy Tracking
Aug 28
Aug 28
NOAA GOM surface dynamics report for Katrina
http://www.aoml.noaa.gov/phod/altimetry/katrina1.pdf
Aug 31
Benchmark Case: (Ne, LD)=(1028, 6)
Analysis
Control
Time=0
Time=30
Time=50 days
Effect of Number of Drifters: LD (Ne=384)
QuickTime™ and a
decompressor
are needed to see this picture.
Analysis Mechanism: Representer
Analysis equation:
 xFa   xFf   PFDf  f
o


P

R
DD
 x a   x f   Pf  DD
D
D
DD

 y
1
f

x
D
D
o

 1 Ne f
f
f
f
x

x
x

x

f
 N  1 n1 F,n F D,n D


P
P fHT   FDf    e N
 PDD   1 e f
f
f
f
x

x
x

x

 N  1 n1 F,n F D,n D
e
Representer

PFDf (u,v,h), (xD ,y D )

ijk , l



T







T
f
 u f 
 uF,n
 
F,n
Ne 
1
 f 
 f   f
f

) l  (xDf ,y Df )l 
  vF,n    vF,n   (xD,n ,y D,n


Ne  1 n1  f
h 
hf  
 F,n  ijk  F,n  ijk 
rFDf (u,v,h), (xD ,y D )ijk , l  normalized PFDf (u,v,h), (xD ,y D )ijk , l
Convergence of rfFD (h1, xD) vs Ne
Ne=32
Ne=128
Ne=64
Ne=256
At Day 5, LD =2, No localization
Convergence of rfFD (h1, xD) vs Ne
Ne=384
Ne=640
Ne=512
Ne=1024
Convergence of rfFD (h1, xD) vs Ne : RMS over GoM
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decompressor
are needed to see this picture.
Vertical Impact: (Ne, LD)=(384, 2-4)
Volume of Influence: Lagrangian vs Eulerian
Green: ∂VL={(i,j,k) | rfFD ((u,v,h), (xD  yD))|ijk,l=1 =0.3}
Red: ∂VE={(i,j,k) | rfFE ((u,v,h), SSH)|ijk,l=1 =0.3}
Volume of Influence: Time Evolution
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decompressor
are needed to see this picture.
Volume of Influence: Vertical Structure
Remarks for Eddy Tracking in the GOM
LaDA can track the detaching eddy quite effectively
Efficiency can be explored using the representer
Lagrangian observation has large volume of influence than Eulerian
observation, both horizontally and vertically
Maximum impact may not necessarily at the location of the observation
For eddy tracking
Implicit hypothesis:
observations should be for the
drifters in the eddy
Implicit action: deployment of
the drifters in the eddy
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decompressor
are needed to see this picture.
Elements of Drifter Deployment: Lagrangian Tracers
Lagrangian coherent structures
i.e., ocean eddies (macroscopic)
Collection of tracers that evolve
and stay together much longer
than the Lagrangian
autocorrelation time scale
Drifters (microscopic)
Individually, tracers can be
entrained into or detrained from
the coherent structures across
the boundaries
Working Hypotheses for Optimal Drifter Deployment
Optimal deployment strategy should take into account of
Evolving Lagrangian coherent structures
(macroscopic view)
Moving observations by drifters {yoD,l(tk)}
(microscopic view)
Working hypotheses
O For eddy tracking
 Deploy drifters in the eddy
O For estimation of the large-scale flow
 Deploy drifters that spread quickly
and visit various regions of the
large-scale flow
O For balanced performance
 Use combination
 Without knowledge of the flow field
 Deploy drifters uniformly or
based on some intelligent guess,
and hope for the best
An Immediate Difficulty for Directed Deployment
Use of these hypotheses requires the evolving Lagrangian info.
How to obtain such information?
We have the data set of instantaneous Eulerian fields {xF(t)}
but Lagrangian trajectories don’t follow the instantaneous streamlines
We can simulate a bunch of drifter
trajectories {xD(t)}
but the spaghetti diagram does not
give cohesive information
We have the drifter observations {yD(tk)}
but they are too sparse to give the
complete Lagrangian flow information and
give no information for the future
Drifter Deployment Design: Dynamical Systems Theory
“Concept”
Dynamical systems theory: A tool to analyze Lagrangian dynamics
given a time sequence of the Eulerian flow fields
Stable and unstable manifolds = “material boundaries” of the distinct
Lagrangian flow regions
Instantaneous (Eulerian) field
Lagrangian flow template
Dynamical
Systems
Theory
Poje, Haller (1999)
Ide, Small, Wiggins (2002)
Mancho, Small, Wiggins, Ide (2003)
….
Immediate difficulty”

How to get Lagrangian flow template
Intermediate difficulty:
How to detect manifolds
Dynamical Systems Theory for Lagrangian Flow Template:
“Method for Detecting Manifolds”
Direct Lyapunov Exponents (Finite Time Lyapunov Exponents: FTLE)
Divergence of the nearby trajectory

 


x t0  T ; x0   x0 ,t0  x t0  T ; x0 ,t0   x t0  T ; x0 ,t0



FTLE max  x t  T ; x ,t  exp  t ; x ,t  x
0
 0 0 0
x
0 0
0
Day 0
Day 60
Day 110
Theory:
Haller (2001, 2002), …
Application to DA: Salman, Ide, Jones (2007)
Lagrangian Flow Template of the Double-Gyre Circulation
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
Hypothesis Testing Using the Lagrangian Flow Template
Goal: Given LD, design the “optimal” deployment strategy
Perfect model scenario using the shallow-water model
Nature run 12yr spin-up with H0=500m; Drifter released at year 13
Ensemble members with (Hmean, Hstd)=(550m,50m)
EnKF Parameters: (Ne, rloc)=(80, 600km)
LaDA Parameters: (∆T, LD)=(1day, 9)
Deployment strategies:
(a) Uniform
(b) Saddle
(c) Center
(d) Mixed
(3x3)
(3 saddles: 3 each)
(3 centers: 3 each)
(3 centers: 1 each;
2 saddle: 3 each)
Salman, Ide, Jones (2007) submitted
Distinctive Drifter Motion by the Deployment Strategies
Directed deployment
Convergence of the Basin-Scale Error Norms
Flow Estimation: Uniform Deployment
Day 25
Truth
Uniform
Day 100
Day 300
RMSE Spatial Pattern: Uniform Deployment
Day 25
h
KE
Day 100
Day 300
Flow Estimation: Center Deployment
Day 25
Truth
Center
Day 100
Day 300
RMSE Spatial Pattern: Center Deployment
Day 25
h
KE
Day 100
Day 300
Flow Estimation: Saddle Deployment
Day 25
Truth
Saddle
Day 100
Day 300
RMSE Spatial Pattern: Saddle Deployment
Day 25
h
KE
Day 100
Day 300
Flow Estimation: Mixed Deployment
Day 25
Truth
Mixed
Day 100
Day 300
RMSE Spatial Pattern: Mixed Deployment
Day 25
h
KE
Day 100
Day 300
Remarks on Deployment Strategy
Deployment strategy
It is “targeting” in the Lagrangian flow template hidden in a time sequence
of Eulerian flow field
It should most naturally be built on dynamical systems theory
Real Difficulty
Drifters are to be released in the real ocean {xtF (t)}, while the template is
build for the model flow field {xfF (t)}
FTLE computation requires i{xfF (t)} in the past and future, thus
predictability of both the Eulerian flow and Lagrangian dynamics must be
taken into account.
Predictability of drifters is doubly-penalized by
Uncertainty of the Lagrangian dynamics
Uncertainty of the Eulerian flow field
BUT detection of Lagrangian coherent structures is a robust procedure
Summary of LaDA
The Lagrangian data assimilation (LaDA) a natural and effective
method for the direct assimilation of Lagrangian observations such
as drifters
Advantage and efficacy of the LaDA are due to
Large volume of influence horizontally and vertically
Mobility
Optimal deployment strategy is intimately related to
Two aspects of Lagrangian tracers: macroscopic (evolution of fluid body
as Lagrangian coherent structures) and microscopic (dynamics of
individual tracers)
Dynamical systems theory, which offers an ideal vehicle for the optimal
deployment strategy = targeting in the LaDA
Future Direction I. Further Development LaDA Method
More realistic applications / situations.
Advancement of the optimal deployment strategy
+ building of Lagrangian analysis and forecasting system
Assimilation of float data (3D Lagrangian observations)
Assimilation of quasi-Lagrangian observation instruments, such as
gliders and Autonomous underwater vehicles (AUVs).
Deployment strategy: estimation/prediction problems  control problem
Theoretical study of observability of Lagrangian observation vs
Eulerian observation.
Future Direction II. Atmospheric Applications of LaDA
Vorecore balloons
http://www.southpoledudes.com/mcmurdo0506/
T
z
u
Hertzog, Basdevant, Viall,
Mechoso (2004)
Cloud feature tracking???
v
Hurricane tracking???
Future Direction III. Development of ROMS LETKF System
Model:
Regional Ocean Modeling
System (ROMS)
Method:
Local Ensemble
Transform Kalman Filter
(LETKF)