One can get reasonable accuracy with a Kalman Filter even with a poorly hacked together model. As a naive dabbler in these matters, I learned recently that though the Kalman Filter states you observe might be filtered suitably, the hidden, or rather, unmeasured states may not necessarily yield sensible values . Also, increasing the model complexity doesn’t necessarily improve matters if your fundamental assumptions about the model are flawed to begin with.
The measurement and model covariance matrices don’t change their values over time. They remain at the initialized values. However, the pre and posterior error co-variance values constantly change. The variances of the states start at fairly low values. As time progresses, the variance values tend to increase and then stabilize at fairly high values. This is probably an artifact of the model selection being poor.
It was also observed that the corrected state at the end of a cycle, if propagated forward in time, yields the predicted state for the next cycle. (In retrospect this should have been obvious – but one has a lot to learn)