Kalman Filters track both an estimate x_hat and the variance of that estimate, P. There is also some true state, x_true. The definition of the variance of the estimate is E[(x_true - x_hat)^2], where E[.] is the expected value operator.
Using the example from the post, there is a known dynamics function that propagates the state one time step forward. That function is f(x, w) = x + velocity * Delta_t + w, where w is zero-mean Gaussian noise (meaning that E[w] = 0) with some known variance sigma_w^2 (meaning that E[w^2] = sigma_w^2). To update the true state, we do x_true_k+1 = f(x_true_k, w) = x_true_k + velocity * Delta_t + w.
So what should the filter's estimate of the state at time t_k+1 be? Well, if our estimator has been working up to this point, then x_hat_k = E[x_true_k] (this is just saying that our estimate is the "best guess" of the true state). We also want it to be true that x_hat_k+1 = E[x_true_k+1]. Plugging in the true dynamics from earlier, we get that x_hat_k+1 = E[x_true_k + velocity * Delta_t + w] = E[x_true_k] + E[velocity * Delta_t] + E[w] = x_hat_k + velocity * Delta_t + 0. Note that we are not adding noise to the estimate. The filter has no way of knowing the noise that enters the system, the filter will just be correct /on average/ for the noise that comes up.
It is also important to update the uncertainty in the estimate. State estimates are of little use without knowledge of the uncertainty. If the estimate is being fed into a self-driving car or something, "The car is here!" is pretty useless on its own. "The car is here, plus or minus 10 cm" means the car can drive as normal, but "The car is here, I think, or maybe in one of the surrounding states" means there's a problem.
We already know that the definition of the variance is P_k=E[(x_true_k - x_hat_k)^2]. Let's say that at time t_k, P_k = sigma^2. To get the variance at time t_k+1, we can just apply the definition. P_k+1 = E[(x_true_k+1 - x_hat_k+1)^2] = E[((x_true_k + velocity * Delta_t + w) - (x_hat_k + velocity * Delta_t))^2] = <skipping some steps> = E[(x_true_k - x_hat_k)^2] + E[w^2] = P_k + sigma_w^2 = sigma^2 + sigma_w^2. So the variance at the new time is equal to the old covariance sigma^2, plus some uncertainty accounting for the noise added in the dynamics sigma_w^2.
Sorry about the poor math formatting, but I hope that answers your question!
I enjoyed reading your article, but I want to note that the implementation is incorrect. The biggest error is that you are not propagating uncertainty forward in time, and therefore are underestimating the error. This shows up in your plots---the error bounds should cover the true state most of the time, but they do not in your results.
A couple more nits to pick:
"In essence, each of the 1000 passengers are doing thus: take the last known position (at the time before the present), add the velocity, and also, knowing that the wind and the water waves are going to slightly alter the course, add some random estimated fluctuations to it." The passengers don't add noise to their estimate. For the passengers, the expected value of the state at time k+1 given the state at time k will just be `position_k+1 = position_k + velocity * Delta_t`. The /true/ dynamics include noise, which is accounted for in the filter by adding to the estimated covariance. Your code doesn't break because you're taking a bunch of samples of the dynamics (by having 1000 passengers) and numerically taking the variance of the results. This is very different than what is usually done in practice.
It's a common misconception that GPS is affected by weather. It is not.
You are using a very non-standard definition of consistency. In estimation theory, having an estimator be consistent means that as you get more and more data, your estimator will converge to the true value.
Thank you for writing up the article and simplifying for a general audience, but there are a few misunderstandings here that seem to be causing problems. I am a graduate student in estimation theory and I would be happy to talk more about this if it would be helpful.
I just started a blog! I'm new to this kind of thing and so my first entry is just a short post on my preferred method for plotting covariance ellipses and ellipsoids.