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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

      London Mathematical Society - EPSRC Durham Symposium
                 Mathematics of Data Assimilation
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
Experiment Design (PMS)
     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.
Gradient Descent (Shadowing Filter)
Gradient Descent (Shadowing Filter)




                                      s0
       F ( s5 )
s5



          s 4
Gradient Descent (Shadowing Filter)
Gradient Descent (Shadowing Filter)
Gradient Descent (Shadowing Filter)
     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
Methodology
Form ensemble
                             GD result
      Reference trajectory


                      Obs


                                         t=0
    Form ensemble




                                                         t=0
                                Candidate trajectories


   Sample the local space
   Perturb observations and run GD
Form ensemble




                                           t=0
                     Ensemble trajectory


     Draw ensemble members
       according to likelihood
Form ensemble



                Obs


                                       t=0
                 Ensemble trajectory
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.
Ikeda Map, Std of observational noise 0.05, 512 ensemble members
Evaluate ensemble via Ignorance
                                            Ensemble->p(.)
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
Imperfect Model Scenario
Toy model-system pairs

 Ikeda system:




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

 Lorenz96 system:




Imperfect model:
Insight of Gradient Descent




 Define the implied noise to be

 and the imperfection error to be
  Insight of Gradient Descent



                                s0
         f ( s5 )
   s5
 0        w

           s 4
Insight of Gradient Descent




        w
Insight of Gradient Descent




         w0

  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).
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
Imperfection error vs model error

                           Obs Noise level: 0.01




Model error              Imperfection error
Not accessible!
Imperfection error vs model error

Obs Noise level: 0.002                Obs Noise level: 0.05




                 Imperfection error
GD vs WC4DVAR

Model error
              WC4DVAR
assumption

              Model error
       GD
               estimates
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?
Imperfect model experiment: Ikeda system-model pair, Std of observational
 noise 0.05, 1024 EnKF ensemble members, 64 GD ensemble members
 Evaluate ensemble via Ignorance
 The Ignorance Score is defined by:


 where Y is the verification.

Systems        Ignorance              Lower               Upper
           EnKF        GD       EnKF          GD     EnKF         GD
 Ikeda     -2.67       -3.62    -2.77      -3.70     -2.52      -3.55

Lorenz9    -3.52       -4.13    -3.60      -4.18     -3.39      -4.08
   6
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
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.
Thank you!

         H.L.Du@lse.ac.uk

 Centre for the Analysis of Time Series:
 http://www2.lse.ac.uk/CATS/home.aspx

				
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posted:9/21/2012
language:English
pages:36