Finite-dimensional 3DVAR analysis

For integers $d,p\ge 1$, given a forecast mean $m^f\in {\mathbb{R}}^d$, fixed background covariance $B\in {\mathbb{R}}^{d\times d}$, linear observation matrix $H\in {\mathbb{R}}^{p\times d}$, datum $y\in {\mathbb{R}}^p$, and observation-error covariance $\mathrm{\Gamma }\in {\mathbb{R}}^{p\times p}$, require $B$ and $\mathrm{\Gamma }$ symmetric positive definite. Define the 3DVAR objective $J(u)=\frac{1}{2}(u-m^f{)}^T{B}^{-1}(u-m^f)+\frac{1}{2}(y-Hu{)}^T{\mathrm{\Gamma }}^{-1}(y-Hu)\mathrm{.}$ Form $S=HB{H}^T+\mathrm{\Gamma }$. It is positive definite because for every $z\mathrm{\ne }0$, $z^{T}Sz=(H^{T}z{)}^{T}B(H^{T}z)+z^T\mathrm{\Gamma }z>0.$ Factor $S=L{L}^T$ with the unique positive-diagonal Cholesky factor from A matrix admits a Cholesky factorisation with positive diagonal exactly when it is Hermitian positive definite, and that factor is unique. Solve $LW=HB$ and $L^T{K}^T=W$; thus $K^T=S^{-1}HB$ without explicitly forming $S^{-1}$. Return the analysis point $m^a=m^f+K(y-H{m}^f)\mathrm{.}$ For a sequential 3DVAR filter, first set $m_{j+1}^f={\mathrm{\Psi }}_j(m_j^a)$, then apply this update with the fixed matrices $B,H,\mathrm{\Gamma }$ and observation $y_{j+1}$. The routine returns a point estimate only. Its unique-minimizer and Gaussian-MAP properties are established separately; no analysis covariance is returned.