The prediction degrades smoothly with time, remaining better than climatology after 60 days. The prediction experiment forms a cross-validation of the model physics, since the better the model, the less error is added to the forecast, and the longer the predictability time. There is a chaotic limit to predictability, because of the exponential divergence of nearby model trajectories, as well as possible growing modes from linear instabilities or local forcing. The most likely cause of the growth of errors in the example is advection and propagation of new features into the region of interest from outside.
The initialization shown in the figure is derived from 12 days of travel time data, optimally combined with the model constraints using an iterated sequence of linearized inversions. Fitting the model to 24 days (rather than 12 days) shows much greater sensitivity to initial conditions, and a more nonlinear fit.
DISCUSSION
If the model dynamics hold exactly, without error or forcing, then the only uncertainty is the initial state of the model. Elements representing uncertainty in other model parameters can be appended to the vector m of unknowns. For example, uncertainty in viscosity, friction, modal deformation radii or interaction coefficients can be included as parameters, requiring the model to be run to calculate the sensitivity of the calculated data to changes in these parameters. Unknown forcing or bottom topography can add many more parameters, although the uncertainty may be factorable into a reduced number of independent basis vectors. Finally, dynamical errors in the model, producing errors in the state vector forecast, can be included explicitly by adding an extra forcing term (representing the error) for each element of the state vector at each timestep [Bennett and Thorburn, 1992], although this may be handled more efficiently by a sequential algorithm like the Kalman smoother.
