Optimal Sensor Placement and Movement in Data Assimilation

Abstract

Designing optimal locations for stationary sensors or optimal trajectories for moving sensors within a constrained sensor budget is crucial for Data Assimilation (DA) to reconstruct dynamical systems, such as ocean models or weather forecasts. Commonly used Lagrangian sensors for collecting observational data from the dynamical system may fail to capture important information due to their passive trajectory following the pathlines. This could lead to sensor clustering and undersampling of information-rich regions.

The first goal of this thesis research is to study the performance of Lagrangian sensors and potential improvements on a typical DA method called the Azouani-Olson-Titi (AOT) nudging algorithm. Two dynamical systems were used as testbeds: the one-dimensional Kuramoto-Sivashinsky equation (1D KSE) and the two-dimensional turbulent Navier-Stokes equations (2D NSE).

Computational experiments showed that, depending on the Stokes number of the sensors (e.g., from $St = 0$ for ideal Lagrangian sensors to $St = 1$ for realistic Lagrangian sensors with inertia), the clustering of the sensors degrades the performance of DA. However, introducing random perturbation to the ideal and realistic Lagrangian trajectories can achieve faster convergence for DA, and thus more effective reconstruction than their unperturbed counterparts. These observations suggest that ideal Lagrangian and inertia sensors may not be optimal, as even a simple random perturbation provides improvement. Therefore, it is reasonable to expect a better sensor movement strategy to achieve better convergence of DA.

The second goal is to propose an optimal sensor movement strategy that directs sensors toward information-rich regions of the flow. This is achieved by maximizing the convergence rate of the AOT algorithm, potentially yielding fast and effective reconstruction of a dynamical system. The thesis demonstrates that directed sensors can outperform both Lagrangian and inertia sensors based on AOT reconstruction, particularly in a sparse sensor scenario.

These findings can hopefully contribute to the DA community by showing that (i) ideal Lagrangian and inertia sensors are not necessarily good sensor movement strategies for reconstructing dynamical systems, and (ii) suggest a novel strategy to plan optimal sensor trajectories for moving sensors, which maximizes the convergence rate for the sequential DA.