Optimal placement of mobile sensors for data assimilations

Abstract

A B S T R A C TWe explore the theoretical framework as well as the associated algorithms for the problem of optimally placing mobile observation platforms to maximise the improvement of estimation accuracy. The approach in this study is based on the concept of observability, which is a quantitative measure of the information provided by sensor data and user-knowledge. To find the optimal sensor locations, the observability is maximised using a gradient projection method. The Burgers equation is used to verify this approach. To prove the optimality of the sensor locations, Monte Carlo experimentations are carried out using standard 4D-Var algorithms based on two sets of data, one from equally spaced sensors and the other from the optimal sensor locations. The results show that, relative to equally spaced sensors, the 4D-Var data assimilation achieves significantly improved estimation accuracy if the sensors are placed at the optimal locations. A robustness study is also carried out in which the error covariance matrix is varied by 50% and the sensor noise covariance is varied by 100%. In addition, both Gaussian and uniform probability distributions are used for the sensor noise and initial estimation errors. In all cases, the optimal sensor locations result in significantly improved estimation accuracy.

Observability for optimal sensor locations in data assimilation

“…While this may be surprising since BFGS was designed for smooth problems previous work by [10] in nonsmooth optimization via BFGS indicates that the conventional BFGS works well for nonsmooth, nonconvex problems so long as the appropriate line search method is used. We expect BFGS to converge linearly while the decent methods similar to the one used in [11] may fail to converge to optimal points. To find the conditions necessary for optimality let (σ, λ) be an eigenpair for the gramian G then…”

“…It is our intent to demonstrate that using optimal sensor locations results in smaller estimation error. For these experiments we incorporate the error covariance of the initial background and in our computations which was not previously done in [11]. We assume the background error covariance is unknown but may be approximated by a Laplacian correlation matrix (3) is approximated by L −1 c .…”

Comparison of actuator placement criteria for control of structures

“…Maximizing controllability (or observability) over a finite-dimensional subspace, for instance the first three modes, could be formulated so that consistent results were obtained as the approximation order increases. See [18] for a treatment of using observability over a subspace of the state-space as a criterion for the dual problem of sensor placement. However, since the actuator will be used with a controller, it makes sense to choose the actuator location to achieve the same objective as used in the controller design.…”

An optimization framework to improve 4D-Var data assimilation system performance

“…The approximate gradients (17) are used in the numerical calculations. This experiment is set up slightly different.…”