Accurately Estimating the State of a Geophysical System with Sparse Observations: Predicting the Weather

Utilizing the information in observations of a complex system to make accurate predictions through a quantitative model when observations are completed at time $T$, requires an accurate estimate of the full state of the model at time $T$. When the number of measurements $L$ at each observation time within the observation window is larger than a sufficient minimum value $L_s$, the impediments in the estimation procedure are removed. As the number of available observations is typically such that $L \ll L_s$, additional information from the observations must be presented to the model. We show how, using the time delays of the measurements at each observation time, one can augment the information transferred from the data to the model, removing the impediments to accurate estimation and permitting dependable prediction. We do this in a core geophysical fluid dynamics model, the shallow water equations, at the heart of numerical weather prediction. The method is quite general, however, and can be utilized in the analysis of a broad spectrum of complex systems where measurements are sparse. When the model of the complex system has errors, the method still enables accurate estimation of the state of the model and thus evaluation of the model errors in a manner separated from uncertainties in the data assimilation procedure.