Partial Observability for the Shallow Water Equations

In this paper we discuss a quantitative measure of partial observability for the shallow water equations. The quantity is consistent if approximated using well posed approximation schemes. A first order approximation of an unobservability index using empirical gramian is discussed. For linear systems with full state observability, the empirical gramian is equivalent to the observability gramian in control theory. We present algorithms for the computation of partial observability for the shallow water equations. These algorithms approximate the unobservability index using the empirical gramian via the nonlinear system and the linearized system given by the tangent linear model. This work has applications to optimal sensor placement for numerical weather prediction.