  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.
