Better Data Assimilation through Gradient Descent Leonard A. Smith, Kevin Judd and Hailiang Du Centre for the Analysis of Time Series London School of.

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)

5s

0s

4s

)( 5sF

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

Obs

t=0

Reference trajectory

GD result

Form ensemble

t=0Candidate trajectories

Sample the local space

Perturb observations and run GD

Form ensemble

t=0Ensemble trajectory

Draw ensemble members according to likelihood

Form ensemble

Obs

t=0Ensemble 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

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

Ensemble->p(.)

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

5s0s

4s

)( 5sf

w0

Insight of Gradient Descent

w

Insight of Gradient Descent

0w

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).

Implied noise

Imperfection error

Distance from

the “truth”

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

Model error Imperfection error

Obs Noise level: 0.01

Not accessible!

Imperfection error vs model error

Imperfection error

Obs Noise level: 0.002 Obs Noise level: 0.05

GD vs WC4DVAR

WC4DVAR Model error

assumption

GDModel error

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.

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

Systems Ignorance Lower Upper

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

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