Finite-dimensional data assimilation, forecasts, analyses, filtering, and smoothing

Fix a finite horizon $T\ge 0$, state dimensions $d_t\ge 1$, observation dimensions $p_t\ge 1$ for $1\le t\le T$, and forecast maps ${\mathrm{\Psi }}_{t-1}:{\mathbb{R}}^{d_{t-1}}\to {\mathbb{R}}^{d_t}$ and observation maps $G_t:{\mathbb{R}}^{d_t}\to {\mathbb{R}}^{p_t}$ that are measurable for the Borel sigma-algebras: inverse images of Borel sets are Borel (A measurable function between measurable spaces, The Borel sigma-algebra of a topological space). A deterministic model trajectory satisfies $x_t={\mathrm{\Psi }}_{t-1}(x_{t-1})$; an additive-error model instead has $X_t={\mathrm{\Psi }}_{t-1}(X_{t-1})+q_t$. Observations have the form $Y_t=G_t(X_t)+r_t$. A forecast at time $t$ is information about $X_t$ formed from the model and observations through time $t-1$, before using $Y_t$; an analysis at time $t$ is information after using $Y_t$. In the state-space terminology of Filtering, predictive, and smoothing laws in a finite-horizon state-space model, a filtering law is the conditional law of $X_t$ given $Y_{1:t}$ and a predictive law is the conditional law of $X_t$ given $Y_{1:t-1}$. A fixed-interval smoothing law conditions a state, or the full state path, on observations through the terminal time, as in Fixed-interval smoothing law in a linear-Gaussian model. Under that page's standard-Borel hypotheses, the filtering, predictive, and full-path smoothing conditional kernels exist by Filtering and finite-horizon smoothing laws exist; conditional versions may differ on null observation histories.

For a symmetric positive-definite matrix $M$, write $\mathrm{\parallel }v{\mathrm{\parallel }}_{M^{-1}}^2:=v^T{M}^{-1}v$. In a finite-dimensional Gaussian 3DVAR update, the forecast mean is $m^f$, its fixed background covariance is $B\succ 0$, the linear observation map is $H$, the datum is $y$, and the observation-error covariance is $\mathrm{\Gamma }\succ 0$. The objective is $J_{3D}(u)=\frac{1}{2}\mathrm{\parallel }u-m^f{\mathrm{\parallel }}_{B^{-1}}^2+\frac{1}{2}\mathrm{\parallel }y-Hu{\mathrm{\parallel }}_{{\mathrm{\Gamma }}^{-1}}^2\mathrm{.}$ It returns one analysis point; the fixed background covariance is not an analysis posterior covariance.