Lagrangian Data Assimilation and Observing System Design from Dynamical Systems Perspective

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Lagrangian Data Assimilation and Observing
System Design from Dynamical Systems Perspective

• Kayo Ide, UCLA
• Chris Jones Hayder Salman, UNC-CH

SAMSI DA Closing Workshop, October 5 2005
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Lagrangian Properties of the Ocean

• Coherent structures
• Commonly observed
• Descriptive physical phenomena are often in
  Lagrangian nature

• Lagrangian trajectories
• as spagetti diagram

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Lagrangian Properties of the Ocean

• Coherent structures
• Commonly observed
• Descriptive physical phenomena are often in
  Lagrangian nature

• Lagrangian trajectories
• Directly related to dynamics
• Maybe associated with Lagrangian structures

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Combining the Elements
Data Assimilation Using Eulerian Models
Lagrangian Observations Along the Paths
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Eulerian Model and Observation

• Eulerian model

• Eulerian observation at rs

Forecast
Analysis
Kalman Filter
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Lagrangian Data Assimilation (LaDA) Augmented
State Vector

• Observation Position observations yok along the
  path, k1,,K

• True drifter dynamics

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Ensemble Kalman Filter (EnKF)

• Ensembles xj are the samples from the probability
  density function of x

Ensemble Forecast
xD
xF
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Ensemble Kalman Filter (EnKF)
Ensemble Forecast
Observation
xD
yD
xF
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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?

• 80 Ensemble members
• have550m
• hstd 50m

• control run
• have500m

T0 (IC)
?
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One Drifter at ? 500m2s-1 (?T, Ne, rloc)
(1day, 80, 8)

• Truth (Control run)

T0 (IC)
T90 days
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Performance Verification

• Parameters
• Degree of turbulence ?
• Assimilation time interval ?T
• Ensemble number Ne
• Localization length scale rloc
• Performance validation by comparing
• True error norm

• Predicted error and ensemble spread

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Summary, So Far

• Lagrangian DA method by augmentation really
  works!
• Direct assimilation of Lagrangian observations
  along the path without velocity interpolation
  EKF, EnKF
• Sampling ?T can be as big as Lagrangian
  correlation time scale
• For turbulent flow (small ?)
• Using a small number of ensembles (Ne),
  correlation between the far fields can be noisy
  and spurious correlation in PHT and HPHT
• Localization radius rloc defines the radius of
  influence, which is about the order of the
  Lagrangian (coherent structure) length scale
• Saddle effects should be addressed correctly

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Drifter Update Examples

• Along the jet
• Large derivation can be still successful

• In the recirculating region

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Can LaDA handle chaotic drifter dynamics?
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Dynamical Systems Theory

• Two-dimensional drifter dynamics

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

• Steady flow

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

• Unsteady flow

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

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Reconstruction of Velocity Field
Poje, Toner, Kirwan Jones (2002)
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Targeted Observing System Design

• Centers (eddies)
• Hyperbolic trajectories
• Mixed cases
• Centers
• Hyperbolic trajectories

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Preliminary Results
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Targeted Observation System
T365 days

• Control

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• Control

• Mixed

T200
T100
T0

• Mixed

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Observing System Design for Transient Flow
Dynamics Finite Time Lyapunov Exponents
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Summary and Work in Progress
Data Assimilation Using Eulerian Models
Lagrangian Observations Along the Paths
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Work in Progress for Improving the Current

• Assimilation of Lagrangian data (drifter
  positions)