Finite-modal fluid observation and noise model

Let $I\subseteq \mathbb{R}$ be a nonempty open interval and let $E$ be a three-dimensional oriented Euclidean affine space. Fix an orientation-preserving affine isometry $x:E\to {\mathbb{R}}^3$ as the Cartesian inertial chart on Newtonian spacetime $I\times E$, with absolute time $t$ in seconds and spatial position in metres. Let $D\subset {\mathbb{R}}^3$ be a bounded Lebesgue-measurable observation region with volume $0<\mathrm{\mid }D\mathrm{\mid }<\mathrm{\infty }$, and let $\mathcal{H}=L^2(D;{\mathbb{R}}^3)$ be the space of almost-everywhere equivalence classes of square-integrable vector fields, with norm $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathcal{H}}=({\int }_D\mathrm{\mid }v(x){\mathrm{\mid }}^2{d}^{3}x{)}^{1\mathrm{/}2}$. Write $\mathcal{B}(\mathcal{H})$ for the sigma-algebra generated by the norm-open sets. A velocity field has units m s${}^{-1}$.

Choose an integer observation horizon $T\ge 0$ and times $t_0<t_1<\cdots <t_T$ in $I$, where observations are taken at $t_1,\dots ,t_T$. For each $0\le k\le T$, let $m_k\ge 1$ and choose dimensionless modal fields ${\varphi }_{k,1},\dots ,{\varphi }_{k,m_k}\in \mathcal{H}$. Define the modal map ${\mathrm{\Phi }}_k:{\mathbb{R}}^{m_k}\to \mathcal{H}$ by ${\mathrm{\Phi }}_k(a)={\sum }_{j=1}^{m_k}{a}_j{\varphi }_{k,j}$. The coefficient vector $a_k\in {\mathbb{R}}^{m_k}$ has units m s${}^{-1}$, with initial mean ${\mu }_0$ in m s${}^{-1}$, and its represented velocity field is $U_k={\mathrm{\Phi }}_k(a_k)$. This finite-dimensional representation is an adopted approximation for the selected observation times; it does not assert that an untruncated fluid field lies in one of these finite-dimensional spaces.

For each $1\le k\le T$, choose $p_k\ge 1$ sensor weights ${\psi }_{k,1},\dots ,{\psi }_{k,p_k}\in \mathcal{H}$, with units m${}^{-3}$. The ideal sensor on a field $v\in \mathcal{H}$ is the weighted spatial average $S_{{\psi }_{k,i}}(v)={\int }_{D}v(x)\cdot {\psi }_{k,i}(x)d^{3}x$; it is not a point evaluation. Define $L_k:\mathrm{span}\{{\varphi }_{k,1},\dots ,{\varphi }_{k,m_k}\}\to {\mathbb{R}}^{p_k}$ by $(L_{k}v{)}_i=S_{{\psi }_{k,i}}(v)$ and define the dimensionless observation matrix by $[H_k{]}_{ij}=(L_k{\varphi }_{k,j}{)}_i$. Thus $L_k(U_k)=H_k{a}_k$, and each datum and observation-noise component has units m s${}^{-1}$.

On a common probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$, adopt the finite linear-Gaussian coefficient model $a_0\sim N_{m_0}({\mu }_0,P_0),a_k=F_k{a}_{k-1}+q_k,y_k=H_k{a}_k+r_k(1\le k\le T),$ where ${\mu }_0\in {\mathbb{R}}^{m_0}$, $P_0\in {\mathbb{R}}^{m_0\times m_0}$, $Q_k\in {\mathbb{R}}^{m_k\times m_k}$, and $R_k\in {\mathbb{R}}^{p_k\times p_k}$; $F_k\in {\mathbb{R}}^{m_k\times m_{k-1}}$ is dimensionless, $q_k\sim N_{m_k}(0,Q_k)$ has coefficient units m s${}^{-1}$, and $r_k\sim N_{p_k}(0,R_k)$ has observation units m s${}^{-1}$. The vectors $a_0,q_1,r_1,\dots ,q_T,r_T$ are mutually independent. Here $N_d(\mu ,C)$ denotes the $d$-dimensional Gaussian law with mean $\mu$ and covariance $C$, including singular covariance. A symmetric matrix $C$ is positive semidefinite, written $C⪰0$, if $z^{T}Cz\ge 0$ for every vector $z$ of the matching dimension. The covariance matrices $P_0,Q_k,R_k$ have this property, with entries carrying the corresponding squared units. Assume the Axiom of Choice (AC), meaning every family of nonempty sets has a choice function, for the Gaussian-law construction. The finite-dimensional Gaussian laws and their common probability-space realization use this assumption. For each $k$, define the field law by ${\mu }_k(B)=\mathbb{P}(U_k\in B)$ for $B\in \mathcal{B}(\mathcal{H})$; finite modal fields with finite-second-moment coefficients have finite-second-moment laws.

These linearity, Gaussianity, independence, covariance, finite-mode, and ideal-sensor conditions are adopted physical-model assumptions for this selected regime. The sensor weights and covariance matrices are model inputs; no calibration procedure is asserted. These conditions report no measurements and do not assert that an actual flow, observation system, or noise source satisfies them. Biased or temporally correlated errors, nonlinear observation maps, and untruncated fluid dynamics require separately stated models.

Classical frames, primitive quantities and SI units

Finite-second-moment random velocity sensor law

The L2 velocity sensor is well-defined

Finite modal velocity fields inherit finite second moments

Finite coefficient observation and innovation interface

The Axiom of Choice

Law, Stuart, and Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 2 §§2.1, 2.4 and Chapter 4 §4.1

Särkkä, Bayesian Filtering and Smoothing, Chapter 4 §§4.1–4.3