# du

Document Sample

```					Better Data Assimilation through

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

K Judd & LA Smith (2001) Indistinguishable States I: The Perfect Model Scenario, Physica D 151: 125-141.

s0
F ( s5 )
s5

s 4
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:

Define the implied noise to be

and the imperfection error to be

s0
f ( s5 )
s5
 0        w

s 4

        w

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

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 17 posted: 9/21/2012 language: English pages: 36