Dynamically constrained uncertainty for the Kalman filter covariance in the presence of model error

Speaker: 
Colin James Grudzien
Affiliation: 
NERSC
Seminar Date: 
4. April 2017 - 11:15 - 11:45
Location: 
Lecture room, Ground Floor, NERSC

The forecasting community has long understood the impact of dynamic instability on the uncertainty of predictions in physical systems and this has led to innovative filtering design to take advantage of the knowledge of process models. The advantages of this combined approach to filtering, including both a dynamic and statistical understanding, have included dimensional reductions and robust feature selection in the observational design of filters. In the context of perfect models we have shown that the uncertainty in prediction is damped along the directions of stability and the support of the uncertainty conforms to the dominant system instabilities. Our current work likewise demonstrates this constraint on the uncertainty for systems with model error, specifically:

  • we produce uniform analytical upper bounds on the uncertainty in the stable, backwards Lyapunov vectors in terms of the Lyapunov exponents and the amplitude of the additive noise,
  • we demonstrate a lower bound on the maximal uncertainty as an inverse relationship of the leading Lyapunov exponent and the observational certainty,
  • and we numerically estimate the evolution of model error in terms of the system's local Lyapunov exponents, which describes the local growth or dissipation of error on finite timescales.

The dynamic evolution of model error is strongly forced by the local Lyapunov exponents and in this way the physical process itself may amplify or abate modelling errors. Under generic assumptions we demonstrate that independently of filtering, the error in the stable, backwards Lyapunov vectors is uniformly bounded and we connect the observational design of filters to take advantage of the dynamic characteristics of error evolution.