ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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One weighted sensor for one constant velocity mode

Physical scope

Symbolic one-mode calculation inside the adopted finite-modal, linear-Gaussian fluid observation model, with one ideal weighted spatial velocity sensor. The datum is a model variable, not a reported measurement; no claim about an actual flow or instrument is made.

Example

Assume the Axiom of Choice (AC), meaning every family of nonempty sets has a choice function, for the finite Gaussian-law construction in the adopted model. Let $D \subset R^{3}$ be bounded and Lebesgue measurable in SI spatial coordinates, with volume $V = ∣ D ∣ > 0$ measured in m $^{3}$ , and let $e \in R^{3}$ be a fixed unit vector. Define the dimensionless constant modal field $ϕ : D \to R^{3}$ and the sensor weight $ψ : D \to R^{3}$ by $ϕ (x) = e$ and $ψ (x) = e / V$ for $x \in D$ ; the weight has units m $^{- 3}$ . Define the ideal weighted spatial sensor $S_{ψ} : L^{2} (D; R^{3}) \to R$ by $S_{ψ} (v) : = \int_{D} v (x) \cdot ψ (x) d^{3} x .$ Then $ϕ, ψ \in L^{2} (D; R^{3})$ and $S_{ψ} (ϕ) = 1$ , a dimensionless response. Let the scalar modal coefficient have prior law $a_{0} \sim N (μ, P)$ , where $N (μ, P)$ denotes the scalar Gaussian law with mean $μ$ and variance $P$ , including $P = 0$ , with $μ$ in m s $^{- 1}$ and $P \geq 0$ in (m s $^{- 1}$ ) $^{2}$ . Set the dimensionless transition $F_{1} = 1$ , zero process covariance $Q_{1} = 0$ in (m s $^{- 1}$ ) $^{2}$ , and zero process noise $q_{1} = 0$ in m s $^{- 1}$ , so $a_{1} = F_{1} a_{0} + q_{1} = a_{0}$ . Take observation noise $r_{1} \sim N (0, R)$ with $R > 0$ in (m s $^{- 1}$ ) $^{2}$ , independent of $a_{0}$ and $q_{1}$ . The represented velocity is $U_{1} (x) = a_{1} e$ for $x \in D$ and the single observation is $y_{1} = a_{1} + r_{1}$ in m s $^{- 1}$ . The exact one-step filter outputs $m_{1 ∣ 1} = μ + \frac{P}{P + R} (y_{1} - μ), P_{1 ∣ 1} = \frac{P R}{P + R} .$ This is a symbolic model calculation, including the deterministic-prior case $P = 0$ ; it records no measured fluid data.

Example references

Finite-second-moment random velocity sensor law

The L2 velocity sensor is well-defined

Finite-modal fluid observation and noise model

Exact conditional law for the finite-modal fluid filter

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

Särkkä, Bayesian Filtering and Smoothing, Chapter 4 §4.3

The Axiom of Choice

Facts & Assumptions

Given: A finite-volume region $D$ of volume $V > 0$ , a unit vector $e$ , the one-mode field $ϕ (x) = e$ and sensor weight $ψ (x) = e / V$ for $x \in D$ , prior $a_{0} \sim N (μ, P)$ with $P \geq 0$ , transition $F_{1} = 1$ with $q_{1} = 0$ , and independent observation noise $r_{1} \sim N (0, R)$ with $R > 0$ .

[A1]

Assume AC, meaning every family of nonempty sets has a choice function, for the Gaussian-law construction. The Axiom of Choice

[F1]

A weighted sensor on $L^{2} (D; R^{3})$ is the integral $S_{ψ} (v) = \int_{D} v \cdot ψ d^{3} x$ , and it is not a point evaluation. Finite-second-moment random velocity sensor law

[F2]

The sensor pairing is finite, representative-independent, and linear for all $v, ψ \in L^{2} (D; R^{3})$ . The L2 velocity sensor is well-defined

[F3]

The adopted model defines the observation matrix entry as $H_{1, 11} = S_{ψ} (ϕ)$ ; the example chooses $F_{1} = 1$ and $q_{1} = 0$ . Finite-modal fluid observation and noise model

[F4]

Under these hypotheses and positive-definite innovation, the exact conditional field law is the pushforward of the coefficient Kalman law. Exact conditional law for the finite-modal fluid filter

[F5]

The fixed Cartesian inertial chart uses SI time and spatial units; fluid velocity has units m s $^{- 1}$ . Classical frames, primitive quantities and SI units

Verification

Given: The hypotheses and notation in the Example.

1.1F1F2F3F5givenalgebra

Since $∣ e ∣ = 1$ , $∥ ϕ ∥_{L^{2}}^{2} = \int_{D} ∣ e ∣^{2} d^{3} x = V < \infty$ and $∥ ψ ∥_{L^{2}}^{2} = \int_{D} ∣ e / V ∣^{2} d^{3} x = 1 / V < \infty$ . Linearity of the sensor integral gives $S_{ψ} (ϕ) = \int_{D} e \cdot (e / V) d^{3} x = V / V = 1,$ so the scalar observation matrix is $H_{1} = 1$ .

2.1F3givenstep 1.1algebra

The chosen state transition and process noise give forecast mean $m_{1 ∣ 0} = μ$ and variance $P_{1 ∣ 0} = P$ . Thus the innovation covariance is $S_{1} = H_{1} P_{1 ∣ 0} H_{1}^{T} + R = P + R > 0$ , the gain is $K_{1} = P / (P + R)$ , and the covariance update is $P_{1 ∣ 1} = P - \frac{P^{2}}{P + R} = \frac{P R}{P + R} .$ Substituting $v_{1} = y_{1} - μ$ into the mean update gives the displayed $m_{1 ∣ 1}$ . These formulas hold at $P = 0$ as well because $R > 0$ .

3.1A1F4step 2.1given∎

Apply [F4] to the same one-step model: the conditional field law is the pushforward of $N (m_{1 ∣ 1}, P_{1 ∣ 1})$ under $a \mapsto a e$ . Therefore the symbolic update in the Example is the exact conditional law within this finite model, not an empirical claim about a measured flow.

One weighted sensor for one constant velocity mode

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources