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This paper uses a probabilistic approach to analyze the converge of anensemble Kalman filter solution to an exact Kalman filter solution in thesimplest possible setting, the scalar case, as it allows us to build upon arich literature of scalar probability distributions and non-elementaryfunctions. To this end we introduce the bare-bones Scalar Pedagogical EnsembleKalman Filter (SPEnKF). We show that in the asymptotic case of ensemble size,the expected value of both the analysis mean and variance estimate of theSPEnKF converges to that of the true Kalman filter, and that the variances ofboth tend towards zero, at each time moment. We also show that the ensembleconverges in probability in the complementary case, when the ensemble isfinite, and time is taken to infinity. Moreover, we show that in thefinite-ensemble, finite-time case, variance inflation and mean correction canbe leveraged to coerce the SPEnKF converge to its scalar Kalman filtercounterpart. We then apply this framework to analyze perturbed observations andexplain why perturbed observations ensemble Kalman filters underperform theirdeterministic counterparts.

An Explicit Probabilistic Derivation of Inflation in a Scalar Ensemble Kalman Filter for Finite Step, Finite Ensemble Convergence