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The perturbed-observation EnKF update preserves exchangeability

Statement

Fix the perturbed-observation update of Perturbed-observation ensemble Kalman filter. A row of random inputs is exchangeable if its joint law is invariant under every common permutation of the member indices. If the forecast states, forecast errors, observation perturbations, and any reference-member variables form an exchangeable input row, then the updated states together with those reference variables are exchangeable.

Facts & Assumptions

Given: The inputs, maps, and update rule in the Statement.

[A1]

The input row's joint law is invariant under common index permutations.

[F1]

The algorithm forms member predictions, empirical means and covariance blocks, one common gain from those empirical covariance blocks, and then updates each member using its own perturbed datum. Perturbed-observation ensemble Kalman filter

Proof

Given: The hypotheses in the Statement.

1.1A1F1givenalgebra

Fix a permutation $σ$ of ${1, \dots, N}$ . Applying it to the input states and errors permutes the forecast states and predicted observations by the same $σ$ . Their empirical means are unchanged, while the sums defining $C_{x z}$ and $C_{z z}$ are unchanged because they only reorder their summands. Therefore $S_{N}$ and the common gain $K_{N}$ are unchanged.

2.1A1F1step 1.1∎

With the same $K_{N}$ , the update at permuted index $i$ is exactly the original updated state at index $σ (i)$ , since its datum perturbation and reference variable are permuted along with that member. Thus the full updated tuple under $σ$ is a permutation of the original updated tuple. Its law is unchanged by [A1], which is joint exchangeability.

Depends on

Perturbed-observation ensemble Kalman filter

Used by

The perturbed-observation EnKF converges to the Kalman filter at fixed horizon Theorem

Dependency tree · two levels

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Sources

Jan Mandel, Loren Cobb, and Jonathan D. Beezley, On the Convergence of the Ensemble Kalman Filter, §2, Lemma 1 (standard reference, not scraped)
K. J. H. Law, A. M. Stuart, and K. C. Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 4 §4.2.3 (standard reference, not scraped)