The 3DVAR quadratic has the Gaussian MAP and Kalman analysis update

For $z\mathrm{\ne }0$, $z^{T}Sz=(H^{T}z{)}^{T}B(H^{T}z)+z^T\mathrm{\Gamma }z>0$, so $S\succ 0$ and $K$ is defined. The inverses $B^{-1}$ and ${\mathrm{\Gamma }}^{-1}$ are positive definite: for nonzero $v$, $v^T{B}^{-1}v=(B^{-1}v{)}^{T}B(B^{-1}v)>0$, and the same argument applies to $\mathrm{\Gamma }$. By [F3], choose Cholesky factors $B^{-1}=L_B{L}_B^T$ and ${\mathrm{\Gamma }}^{-1}=L_{\mathrm{\Gamma }}{L}_{\mathrm{\Gamma }}^T$. Define $Tu=(L_B^{T}u,L_{\mathrm{\Gamma }}^{T}Hu)$ and $b=(L_B^T{m}^f,L_{\mathrm{\Gamma }}^{T}y)$. Then $2J(u)=\mathrm{\parallel }Tu-b{\mathrm{\parallel }}^2$. The first block $L_B^T$ is invertible, hence $\ker T=\{0\}$.

The normal equation in [F2] is $B^{-1}(u-m^f)+H^T{\mathrm{\Gamma }}^{-1}(Hu-y)=0$. Let $\delta =y-H{m}^f$ and $u=m^f+B{H}^T{S}^{-1}\delta =m^a$. Then $B^{-1}(u-m^f)=H^T{S}^{-1}\delta$, while $y-Hu=(I-HB{H}^T{S}^{-1})\delta =\mathrm{\Gamma }{S}^{-1}\delta$. Substitution gives the normal equation. By [F2] this $u$ minimizes $J$, and because $\ker T=\{0\}$ it is the unique minimizer.

Under the Gaussian prior and independent Gaussian observation-error model, [F4] gives a prior density proportional to $\exp [-\frac{1}{2}(u-m^f{)}^T{B}^{-1}(u-m^f)]$ and a jointly continuous likelihood density proportional to $\exp [-\frac{1}{2}(y-Hu{)}^T{\mathrm{\Gamma }}^{-1}(y-Hu)]$. The likelihood is strictly positive and bounded above by its finite Gaussian normalizing constant; hence its prior integral is finite and positive. Bayes' formula [F5] makes the posterior density proportional to $e^{-J(u)}$; its unique maximum is therefore the unique minimizer from step 2.1.

Apply [F6] to the one-step model with forecast covariance $B$, identity transition, zero process covariance, and observation covariance $\mathrm{\Gamma }$. Its innovation covariance is $S\succ 0$ by step 1.1, so the output mean is $m^f+K(y-H{m}^f)=m^a$. By [F7] this is the mean of the exact conditional Gaussian law.