An Information Criterion for Choosing Observation Locations in Data Assimilation and Prediction

An information criterion is proposed for determining the observation locations based on maximizing the information gain in the posterior distribution from data assimilation. It is applied to developing an off-line strategy using the long-term statistics from Eulerian observations and an online ensemble strategy for determining the initial locations of Lagrangian tracers. Decompose the total information gain into a signal and a dispersion part, accounting for the posterior mean and posterior uncertainty, respectively. Despite the information criterion being a nonlinear function of the posterior estimates and the intrinsic nonlinearity in the Lagrangian data assimilation, the total information gain can be solved via closed analytic formulae. The signal part is given by the solution of a set of Sylvester equations, and the dispersion part is associated with a Riccati equation. Numerical experiments based on a multiscale compressible rotating shallow water equation show that the information gain using the optimal strategy and that using the random assignment increase as a linear and a logarithm function of the number of the Eulerian observations L, respectively, until L approaches the model degree of freedom, at which time the difference between the two information gains reaches the maximum. Afterwards, both the information gains are dominated by the dispersion part and increase as a function of In L. On the other hand, the optimal initial locations of the Lagrangian tracers resulting from the ensemble based strategy also succeed in improving the skill of recovering complex flow patterns and extreme events associated with single random realizations of the model.