By tracking a GPS satellite with an antenna of receivers, it is possible to estimate the difference between the satellite
elevation angle and the actual arrival angle of the transmitted signal in the line of sight of the antenna. Those
measurements are assimilated through the use of a fast ray-tracing observation operator and its adjoint into a high
resolution version of the Weather Research and Forecast model. Such assimilation has the potential to improve the
description and prediction of the local refractivity field, through improved pressure, temperature and humidity, around
the antenna.
We introduce a formalism for computing the Cramer-Rao lower bound (CRLB) for a general dynamical system,
develop an approach to bounding the process noise for a general dynamical system, and discuss the application
of this formalism in the context of a prototypical forecasting model. This model consists of a simple transport
diffusion process with assimilation updates based on point source measurements. We investigate the use of Krylov
subspace techniques for efficient computation of two point correlation functions, and the use of this technique in
generating a coarse-grained state covariance.
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