better data assimilation through gradient descent
DESCRIPTION
London Mathematical Society - EPSRC Durham Symposium Mathematics of Data Assimilation. 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. Outline. Perfect model scenario (PMS) - PowerPoint PPT PresentationTRANSCRIPT
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!
Centre for the Analysis of Time Series:http://www2.lse.ac.uk/CATS/home.aspx
