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Scalar Gaussian 3DVAR returns the exact analysis mean

Example

Take a scalar Gaussian forecast with mean $m^{f} = 0$ and variance $B = 1$ , observation map $H = 1$ , error variance $Γ = 1$ , and observed value $y = 2$ . The 3DVAR objective is $J (u) = \frac{1}{2} u^{2} + \frac{1}{2} (2 - u)^{2}$ .

Facts & Assumptions

Given: AC and the one-dimensional Gaussian prior/error model, with $B = Γ = H = 1$ and $y = 2$ .

[A1]

AC is assumed for the Gaussian laws and conditional-law results. The Axiom of Choice

[F1]

The one-step linear-Gaussian model uses independent Gaussian initial and observation errors. Linear-Gaussian state-space models with independent noises

[F2]

The 3DVAR gain is $K = B H^{T} (H B H^{T} + Γ)^{- 1}$ and its analysis is $m^{f} + K (y - H m^{f})$ . Finite-dimensional 3DVAR analysis

[F3]

The 3DVAR objective minimizer is the Gaussian MAP point and coincides with the Kalman analysis mean under these hypotheses. The 3DVAR quadratic has the Gaussian MAP and Kalman analysis update

[F4]

The exact Kalman analysis covariance is $P^{a} = P^{f} - K S K^{T}$ when the innovation $S$ is positive definite. Exact linear-Gaussian Kalman filter

[F5]

Under AC the Kalman recursion is the exact Gaussian conditional law. The finite-horizon linear-Gaussian Kalman filter is exact

Verification

Given: The objects and observations in the Example.

1.1F2givenalgebra

Here $S = H B H^{T} + Γ = 2$ and $K = B H^{T} S^{- 1} = 1 / 2$ . Thus the analysis is $m^{a} = 0 + (1 / 2) (2 - 0) = 1$ .

2.1A1F1F3F4F5givenstep 1.1algebra∎

The independent Gaussian prior and observation-error model in the Example is the one-step model specified by [F1]. Completing the square gives $J (u) = \frac{1}{2} u^{2} + \frac{1}{2} (2 - u)^{2} = (u - 1)^{2} + 1$ , so its unique minimizer is $1$ , agreeing with step 1.1 and [F3]. The Kalman covariance is $P^{a} = 1 - (1 / 2) 2 (1 / 2) = 1 / 2$ by [F4]; [F5] therefore identifies the exact conditional law as $N (1, 1 / 2)$ . The 3DVAR output is its MAP point and supplies no covariance output.

Depends on

Used by

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Sources

K. J. H. Law, A. M. Stuart, and K. C. Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 4 §4.2.1 (standard reference, not scraped)
Geir Evensen, Femke C. Vossepoel, and Peter Jan van Leeuwen, Data Assimilation Fundamentals, Chapter 6 §6.2 (standard reference, not scraped)