Controlling Unpredictability with Observations in the Partially Observed Lorenz '96 Model

Controlling Unpredictability with Observations in the Partially Observed Lorenz '96 Model

K.J.H. Law, D. Sanz-Alonso, A. Shukla, A.M. StuartControlling Unpredictability with Observations in the Partially Observed Lorenz '96 Modelsubmited to arXiv:1411.3113v1, Nov. 2014.
K.J.H. Law, D. Sanz-Alonso, A. Shukla, A.M. Stuart
Controlling Unpredictability with Observations in the Partially Observed Lorenz '96 Model
2014
‚ÄčThe long-time behavior of filters for the partially observed Lorenz '96 model is studied. It is proven that for both discrete-time and continuous-time observations the 3DVAR filter can recover the signal within a neighborhood defined by the size of the observational noise, as long as a sufficiently large proportion of the state vector is observed; an explicit form for a sufficient constant observation operator is given. Furthermore, non-constant adaptive observation operators, with data incorporated by use of both the 3DVAR and the extended Kalman filter, are studied numerically. It is shown that for carefully chosen adaptive observations, the proportion of state coordinates necessary to accurately track the signal is significantly smaller than the proportion proved to be sufficient for constant observation operator. Indeed it is shown that the necessary number of observations may even be significantly smaller than the total number of positive Lyapunov exponents of the underlying system.