Better%20Data%20Assimilation%20through%20Gradient%20Descent

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Better Data Assimilation through Gradient Descent

• Leonard A. Smith, Kevin Judd and Hailiang Du
• Centre for the Analysis of Time Series
• London School of Economics

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Outline

• Perfect model scenario (PMS)
• GD method
• GD is NOT 4DVAR
• Results compared with Ensemble KF
• Imperfect model scenario (IPMS)
• GD method with stopping criteria
• GD is NOT WC4DVAR
• Results compared with Ensemble KF
• Conclusion Further discussion

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Experiment Design (PMS)
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Ensemble techniques

• Generate ensemble directly, e.g. Particle Filter,
  Ensemble Kalman Filter
• Generate ensemble from perturbations of a
  reference trajectory, e.g. SVD on 4DVAR

Gradient Descent (GD) Method
K Judd LA Smith (2001) Indistinguishable States
I The Perfect Model Scenario, Physica D 151
125-141.
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Gradient Descent (Shadowing Filter)
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Gradient Descent (Shadowing Filter)
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Gradient Descent (Shadowing Filter)
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Gradient Descent (Shadowing Filter)
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Gradient Descent (Shadowing Filter)
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GD is NOT 4DVAR

• Difference in cost function
• Noise model assumption
• Observational noise model 4DVAR cost
  function
• GD cost function not depend on noise model
• Assimilation window

• 4DVAR dilemma
• difficulties of locating the global minima with
  long assimilation window
• losing information of model dynamics and
  observations without long window

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Methodology
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Form ensemble
GD result
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Form ensemble

• Sample the local space
• Perturb observations and run GD

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Form ensemble
t0
Ensemble trajectory
Draw ensemble members according to likelihood
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Form ensemble
Ensemble trajectory
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Ensemble members in the state space

• Compare ensemble members generated by
  Gradient Descent method and Ensemble Adjustment
  Kalman Filter method in the state space.

Low dimensional example to visualize, higher
dimensional results later.
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Ikeda Map, Std of observational noise 0.05, 512
ensemble members
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Evaluate ensemble via Ignorance
Ensemble-gtp(.)

• The Ignorance Score is defined by
• where Y is the verification.

Ikeda Map and Lorenz96 System, the noise model
is N(0, 0.4) and N(0, 0.05) respectively.
Lower and Upper are the 90 percent bootstrap
resampling bounds of Ignorance score
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Imperfect Model Scenario
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Toy model-system pairs

• Ikeda system

Imperfect model is obtained by using the
truncated polynomial, i.e.
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Toy model-system pairs

• Lorenz96 system

Imperfect model
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Insight of Gradient Descent
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Insight of Gradient Descent
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Insight of Gradient Descent
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Insight of Gradient Descent
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Implied noise
Imperfection error
Distance from the truth
Statistics of the pseudo-orbit as a
function of the number of Gradient Descent
iterations for both higher dimension Lorenz96
system-model pair experiment (left) and low
dimension Ikeda system-model pair experiment
(right).
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GD with stopping criteria

• GD minimization with intermediate runs produces
  more consistent pseudo-orbits
• Certain criteria need to be defined in advance to
  decide when to stop or how to tune the number of
  iterations.
• The stopping criteria can be built by testing the
  consistency between implied noise and the noise
  model
• or by minimizing other relevant utility function

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Imperfection error vs model error
Obs Noise level 0.01
Model error
Imperfection error
Not accessible!
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Imperfection error vs model error
Obs Noise level 0.002
Obs Noise level 0.05
Imperfection error
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GD vs WC4DVAR
Model error assumption

• WC4DVAR

Model error estimates
GD
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Forming ensemble

• Apply the GD method on perturbed observations.
• Apply the GD method on perturbed pseudo-orbit.
• Apply the GD method on the results of other data
  assimilation methods.

Particle filter?
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Imperfect model experiment Ikeda
system-model pair, Std of observational noise
0.05, 1024 EnKF ensemble members, 64 GD ensemble
members
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Evaluate ensemble via Ignorance

• The Ignorance Score is defined by
• where Y is the verification.

Systems Ignorance Ignorance Lower Lower Upper Upper
Systems EnKF GD EnKF GD EnKF GD
Ikeda -2.67 -3.62 -2.77 -3.70 -2.52 -3.55
Lorenz96 -3.52 -4.13 -3.60 -4.18 -3.39 -4.08
Ikeda system-model pair and Lorenz96
system-model pair, the noise model is N(0, 0.5)
and N(0, 0.05) respectively. Lower and Upper are
the 90 percent bootstrap resampling bounds of
Ignorance score
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Conclusion

• Methodology of applying GD for data assimilation
  in PMS is demonstrated outperforms the 4DVAR and
  Ensemble Kalman filter methods
• Outside PMS, mmethodology of applying GD for data
  assimilation with a stopping criteria is
  introduced and shown to outperform the WC4DVAR
  and Ensemble Kalman filter methods.
• Applying the GD method with a stopping criteria
  also produces informative estimation of model
  error.

No data assimilation without dynamics.
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Thank you!

• H.L.Du_at_lse.ac.uk
• Centre for the Analysis of Time Series
• http//www2.lse.ac.uk/CATS/home.aspx

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