Lagrangian Data Assimilation: Method, Applications, and Strategy for Optimal Drifter Deployment

1
Lagrangian Data AssimilationMethod,
Applications, andStrategy for Optimal Drifter
Deployment

• Kayo Ide, UCLA

C.K.R.T. Jones, Guillaume Vernieres,UNC-CH Hayder
Salman, Cambridge U. Liyan Liu, NCEP
2
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
3
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.whoi.edu/instruments/
http//www.dosits.org/gallery/tech/ooct/rafos1.htm
4
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
5
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)
6
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)
7
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/
8
Ocean Observations In Situ by Stationary
Platforms

• Mooring

• (T , S, p)
• (u, v)

http//www.mbari.org
Data available at http//ourocean.jpl.nasa.gov
9
Ocean Observations In Situ by Movable Platforms

• Glider

• At the surface xG(2D)
• In the water
• (T , S, p)

• Ship

• (T , S, p)
• (u, v)

Data available at http//ourocean.jpl.nasa.gov
10
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

• Ocean observations are precious
• New types of observations SSS by Satellite,
  Coastal HF radar
• New technology for cost effectiveness Lagrangian
  data

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Ocean Observation Remote-Sensing by HF Radar

• Coastal Oceans Currents Monitoring Program (COCMP)

http//www.cocmp.org/
http//www.cencoos.org/currents
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Lagrangian Dynamics of Drifters
Data available from http//www.aoml.noaa.gov/phod/
dac/dacdata.html
13
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

14
Basic Elements of Lagrangian Data Assimilation
System
Eulerian Model State xF
Lagrangian Observation Location yD
Data Assimilation Method
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Data Assimilation Method Kalman-Filter Approach
Observation at tk
Forecast from tk-1 to tk
Analysis at tk
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Elements of Assimilating Lagrangian Data

• Essence of analysis in data assimilation

• 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

17
Assimilation of Lagrangian Data Conventional
Method

• Transform from Lagrangian data
• to Eulerian (velocity) data

• Observation operator

Hv is linear spatial interpolation.

• Feedback the mismatch of observation (innovation)
  into the model variable

Carter (1989) Kamachi, OBrien (1995) Tomassini,
Kelly, Saunders (1999)
18
Assimilation of Lagrangian Data Conventional
Method

• Transform from Lagrangian data
• to Eulerian (velocity) data

• Observation operator

Hv is linear spatial interpolation.

• Feedback the mismatch of observation (innovation)
  into the model variable

Carter (1989) Kamachi, OBrien (1995) Tomassini,
Kelly, Saunders (1999)
19
Assimilation of Lagrangian Data Conventional
Method

• Transform from Lagrangian data
• to Eulerian (velocity) data

• Observation operator

Hv is linear spatial interpolation.

• Feedback the mismatch of observation (innovation)
  into the model variable

Carter (1989) Kamachi, OBrien (1995) Tomassini,
Kelly, Saunders (1999)
20
Assimilation of Lagrangian Data Conventional
Method

• Transform from Lagrangian data
• to Eulerian (velocity) data

• Observation operator

Hv is linear spatial interpolation.

• Feedback the mismatch of observation (innovation)
  into the model variable

Carter (1989) Kamachi, OBrien (1995) Tomassini,
Kelly, Saunders (1999)
21
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

xF
xD

• Observation yD and operator h that relates yD to
  x

Ide, Jones, Kuznetsov (2002) Ide and Ghil
(1997)

• Missing elements
• Pf that gives K for optimal impact of yo on xa

22
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

• Observation yD and operator h that relates yD to
  x

Ide, Jones, Kuznetsov (2002) Ide and Ghil
(1997)

• Missing elements
• Pf that gives K for optimal impact of yo on xa

23
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

• Observation yD and operator h that relates yD to
  x

Ide, Jones, Kuznetsov (2002) Ide and Ghil
(1997)

• Missing elements
• Pf that gives K for optimal impact of yo on xa

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Ensemble-Based Data Assimilation

• Use of ensemble to
  represent the uncertainty of x
• in particular, mean and covariance

• mean

• covariance

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Ensemble-Based LaDA
xD

• Ensemble forecast from tk-1 to tk

for n 1,, Ne
2. Ensemble update at tk to incorporate
and
xF
Analysis (dropping tk )
Salman, Kuznetsov,Jones, Ide (2006) Salman, Ide,
Jones (2007)
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Ensemble-Based LaDA
xD

• Ensemble forecast from tk-1 to tk

for n 1,, Ne
2. Ensemble update at tk to incorporate
and
xF
Analysis (dropping tk )
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Ensemble-Based LaDA
xD

• Ensemble forecast from tk-1 to tk

yD
for n 1,, Ne
2. Ensemble update at tk to incorporate
and
xF
Analysis (dropping tk )
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Ensemble-Based LaDA
xD

• Ensemble forecast from tk-1 to tk

yD
for n 1,, Ne
2. Ensemble update at tk to incorporate
and
xF
Analysis (dropping tk )
29
Ensemble-Based LaDA
xD

• Ensemble forecast from tk-1 to tk

for n 1,, Ne
2. Ensemble update at tk to incorporate
and
xF
Analysis (dropping tk )
30
Mechanisms of Lagrangian Data Assimilation (LaDA)
Observation at tk
Forecast from tk-1 to tk
Analysis at tk
Other Methods OI Molcard et al (2003)
4D-Var Nodet (2006)
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Application I. Mid-latitude Ocean Circulation
Proof of Concept
nature run (simulated truth) ht500m
x1000km
x1000km

• Ocean circulation
• 1-layer shallow-water model
• Domain size 2000km x 2000km
• Wind-driven ?0.05 Ns-2

• Perfect model scenario
• Model spin-up for 12 yrs
• - Nature run (truth) with H0500m
• Ensemble with (Hmean, Hstd)(550m,50m)
• Drifter released at the beginning of 13 yrs
  observed every day

Salman, Kuznetsov,Jones, Ide (2006)
32
Ex.1 ?500m2s-1, (?T, LD )(1day, 1), (Ne,
rloc)(80, 8)
Without DA
With LaDA
Truth
T0
T90 days
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Ex.1 ?500m2s-1, (?T, LD )(1day, 1), (Ne,
rloc)(80, 8)
Without DA
With LaDA
Truth
T0
T90 days
34
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)

Vernieres, Ide, Jones, work in progress
35
Motivation for Eddy Tracking
Aug 28
Aug 28
Aug 31
NOAA GOM surface dynamics report for Katrina
http//www.aoml.noaa.gov/phod/altimetry/katrina1.p
df
36
Benchmark Case (Ne, LD)(1028, 6)
Control
Analysis
Time0
Time30
Time50 days
37
Effect of Number of Drifters LD (Ne384)
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Analysis Mechanism Representer

• Analysis equation

• Representer

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Convergence of rfFD (h1, xD) vs Ne

• Ne32

• Ne64

• Ne128

• Ne256

At Day 5, LD 2, No localization
40
Convergence of rfFD (h1, xD) vs Ne

• Ne384

• Ne512

• Ne1024

• Ne640

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Convergence of rfFD (h1, xD) vs Ne RMS over GoM
42
Vertical Impact (Ne, LD)(384, 2-4)
43
Volume of Influence Definition
?VL(i,j,k) rfFD ((u,v,h), (xD ? yD))ijk,l1
0.3
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Volume of Influence Lagrangian vs Eulerian
Green ?VL(i,j,k) rfFD ((u,v,h), (xD ?
yD))ijk,l1 0.3 Red ?VE(i,j,k) rfFE
((u,v,h), SSH)ijk,l1 0.3
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Volume of Influence Time Evolution
46
Volume of Influence Vertical Structure
47
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

48
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

49
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
• For eddy tracking
• ? Deploy drifters in the eddy
• For estimation of the large-scale flow
• ? Deploy drifters that spread quickly and visit
  various regions of the large-scale flow
• 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

O
O
O
?
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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 dont 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

51
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 ?
Intermediate difficulty How
to get Lagrangian flow template How to detect
manifolds
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Dynamical Systems Theory for Lagrangian Flow
Template Method for Detecting Manifolds

• Direct Lyapunov Exponents (Finite Time Lyapunov
  Exponents FTLE)
• Divergence of the nearby trajectory

• FTLE

Day 0
Day 60
Day 110
Theory Haller (2001, 2002),
Application to DA Salman, Ide, Jones (2007)
53
Lagrangian Flow Template of the Double-Gyre
Circulation
54
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 H0500m 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 (3x3) (b) Saddle (3 saddles 3
each) (c) Center (3 centers 3 each) (d) Mixed
(3 centers 1 each 2 saddle
3 each)
Salman, Ide, Jones (2007) submitted
55
Distinctive Drifter Motion by the Deployment
Strategies
Directed deployment
56
Convergence of the Basin-Scale Error Norms
57
Flow Estimation Uniform Deployment
Day 25
Day 100
Day 300
Truth
Uniform
58
RMSE Spatial Pattern Uniform Deployment
Day 25
Day 100
Day 300
h
KE
59
Flow Estimation Center Deployment
Day 25
Day 100
Day 300
Truth
Center
60
RMSE Spatial Pattern Center Deployment
Day 25
Day 100
Day 300
h
KE
61
Flow Estimation Saddle Deployment
Day 25
Day 100
Day 300
Truth
Saddle
62
RMSE Spatial Pattern Saddle Deployment
Day 25
Day 100
Day 300
h
KE
63
Flow Estimation Mixed Deployment
Day 25
Day 100
Day 300
Truth
Mixed
64
RMSE Spatial Pattern Mixed Deployment
Day 25
Day 100
Day 300
h
KE
65
RMSE Spatial Pattern in KE Uniform and Center
Strategies
Day 25
Day 100
Day 300
Uniform
Center
66
RMSE Spatial Pattern in KE Saddle and Mixed
Strategies
Day 25
Day 100
Day 300
Saddle
Mixed
67
RMSE Spatial Pattern in h Uniform and Center
Strategies
Day 25
Day 100
Day 300
Uniform
Center
68
RMSE Spatial Pattern in h Saddle and Mixed
Strategies
Day 25
Day 100
Day 300
Saddle
Mixed
69
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 ixfF (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

70
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

71
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.

72
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???

73
Future Direction III. Development of ROMS LETKF
System

• Model
• Regional Ocean Modeling System (ROMS)

• Method
• Local Ensemble Transform Kalman Filter (LETKF)