Sparse Sensor Placement for Reducing Forecast Errors in Ensemble Kalman Filtering

Abstract

Designing efficient observation networks for reducing forecast errors is a fundamental challenge in numerical weather prediction. Data-driven sparse sensor placement (SSP) and ensemble-based data assimilation via the Ensemble Kalman Filter (EnKF) have each addressed this challenge independently, yet their mathematical connections have not been systematically formalized. This study presents a unified theoretical framework integrating SSP and EnKF through optimal experimental design, providing new theoretical and algorithmic results. While conventional SSP methods aim to reduce analysis errors, this study extends the SSP to target forecast error reduction by using a tangent linear model approximated by an ensemble forecast. We derive the Fisher information matrices in the ensemble and model spaces for the EnKF, and clarify the mathematical interpretations of A-, D-, and E-optimality in terms of forecast error reduction. A-optimality in the model space minimizes the mean forecast error variance; D-optimality is ill-defined in the model space due to rank deficiency and is therefore formulated in the ensemble space, where it maximizes the Shannon information content of assimilated observations; and E-optimality in the model space minimizes the worst-case forecast error variance. We further propose a fast greedy algorithm for selecting observation locations under A-optimality in the model space, avoiding matrix inversion at each greedy step and substantially reducing computational cost. Numerical experiments using the Lorenz-96 model support these theoretical findings. Among the three optimality criteria, A-optimality in the model space most consistently reduces forecast spread and root-mean-square error, and yields stable incremental improvements consistent with post-assimilation observation impact diagnostics.