  Deploying Lagrangian drifters that facilitate the state estimation of the
underlying flow field within a future time interval is practically important.
However, the uncertainty in estimating the flow field prevents using standard
deterministic approaches for designing strategies and applying trajectory-wise
skill scores to evaluate performance. In this paper an information measurement
is developed to quantitatively assess the information gain in the estimated
flow field by deploying an additional set of drifters. This information
measurement is derived by exploiting causal inference. It is characterized by
the inferred probability density function of the flow field, which naturally
considers the uncertainty. Although the information measurement is an ideal
theoretical metric, using it as the direct cost makes the optimization problem
computationally expensive. To this end, an effective surrogate cost function is
developed. It is highly efficient to compute while capturing the essential
features of the information measurement when solving the optimization problem.
Based upon these properties, a practical strategy for deploying drifter
observations to improve future state estimation is designed. Due to the
forecast uncertainty, the approach exploits the expected value of spatial maps
of the surrogate cost associated with different forecast realizations to seek
the optimal solution. Numerical experiments justify the effectiveness of the
surrogate cost. The proposed strategy significantly outperforms the method by
randomly deploying the drifters. It is also shown that, under certain
conditions, the drifters determined by the expected surrogate cost remain
skillful for the state estimation of a single forecast realization of the flow
field as in reality.
