PNNL

Abstract

The four-dimensional variational data assimilation methodology for assimilating noisy observations into a deterministic model has been the workhorse of the forecasting centers for over three decades. While this method provides a computationally efficient framework for dynamic data assimilation, it is largely silent on the important question concerning the minimum number and placement of observations. To leverage this question, we demonstrate the dual advantage of placing the observations where the square of the sensitivity of the model solution with respect to the unknown control variables, called forward sensitivities, attains its maximum. Hence, we can force the observability Gramian to be of full rank which in turn guarantees efficient recovery of the optimal values of the control variables which is the first of the two advantages of this strategy. We further show that the proposed strategy of placement of observations has another inherent optimality: square of the sensitivity of the optimal estimates of the control with respect to the observations (used to obtain these estimates) attains its minimum value -- a second advantage which is a direct consequence of the above strategy for placement of observations. Proof-of-concept is demonstrated using test cases representing linear and nonlinear dynamical systems. We support our numerical experiments with an analytical framework for assessing the observation placement strategy and the associated dual advantages.