Computational issues on observability and optimal sensor locations

In this paper we discuss computational issues related to optimal sensor placement in numerical weather prediction (NWP). Specifically we will discuss the application of observability as a metric for sensor placement to an atmospheric flow model and the arising optimization problem. Atmospheric data assimilation is the process of estimating the initial system state based on observations needed in NWP to produce a forecast of future weather conditions. Optimal placement of sensors for data assimilation leading to an improvement in the analysis of the data assimilation and improved forecast quality is of great interest. The traditional definition of observability is not necessarily suitable for NWP applications because of the high dimensions used in NWP. We use the concept of partial observability where the observability of a system is computed on a reduced subspace and is obtained using dynamic optimization. This definition allows for a characterization of the observability of complicated systems. Using partial observability for optimal sensor placement leads to a max-min problem. We use an empirical gramian to reduce this problem into one of eigenvalue optimization. Our focus will be to develop computational methods that are both efficient and scalable. We will leverage tools typically available in data assimilation and introduce tools used in nonsmooth optimization. We will use the shallow water equations as a testbed for our method of optimal sensor placement in four dimensional variational data assimilation.