Combine input uncertainty through a measurement model.
Key knowledge
For y = f(x₁,…,xₙ), first-order propagation weights input variances and covariances by the model's sensitivity coefficients. Correlations must be retained when the input estimates are correlated.
Input/output estimates, standard uncertainties, sensitivity coefficients and covariances have the stated dimensions; each first-order variance contribution has squared output units.
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Agreement within this scope
Adequate first-order linearization propagates input covariance through model sensitivities.
Correlated inputs require covariance terms; strongly nonlinear cases need further evaluation.
Covariance C_ij = E[(X_i − E X_i)(X_j − E X_j)] describes joint second moments. A covariance matrix contains variances on its diagonal; correlation is r_ij = C_ij/(u_i u_j) when both standard deviations are positive.
Shared influences require joint input covariances in propagation when significant.
A measurement model relates quantities involved in a measurement. It may be implicit, multivariate or algorithmic; Y = f(X₁,…,X_n) is an explicit scalar special case.
The explicit smooth scalar case permits first-order propagation; general models are broader.
An uncertainty budget states the measurement model, input estimates, standard uncertainties, distributions, covariances, sensitivities, evaluation methods and degrees of freedom, together with how they combine and any coverage factor. It includes relevant corrections, reference standards, environment and sampling effects without counting the same contribution twice.
Budget components, sensitivities and dependencies are propagated through the specified model.
An illustrative temperature model y=x̄+C+δ has x̄=20.00 °C from four independent readings with s=0.20 K, C=+0.50 K with u_C=0.20 K, and independent zero-centered δ uniform on ±0.05 K. It gives y=20.50 °C and u_c=√[(0.20/√4)²+0.20²+(0.05/√3)²]=0.22546 K; choosing k=2 gives U=0.45092 K without automatically establishing 95% coverage.
The constructed additive temperature model has independent variance components and unit sensitivities.
For the valid linear calibration y=a+bx, an unknown input is estimated as x̂=(y₀−â)/b̂. Its uncertainty includes the future indication, fitted intercept/slope, their covariance, reference standards and applicable extra influences. First-order propagation uses gradient (1/b̂,−1/b̂,−x̂/b̂) with respect to (y₀,â,b̂).
The inverse calibration gradient propagates the joint future-indication and fitted-parameter covariance.
For a differentiable real model Y=f(X), first-order output covariance is Σ_Y≈JΣ_XJᵀ, J_ai=∂f_a/∂x_i at the input estimate; the rule is exact for an affine model with finite input covariance. For impedance inferred from voltage/current amplitudes and phase, R=V cosφ/I and X_r=V sinφ/I share the same uncertain inputs, so their covariance must be retained as well as their separate uncertainties.
The vector Jacobian generalizes the scalar local sensitivities while preserving shared-input output covariance.
Coulometry infers reacting amount from integrated charge and electron stoichiometry. Controlled-potential and controlled-current methods require attention to completion, background charge and which reactions carry the current. Current efficiency must be effectively complete or independently quantified before total measured charge is assigned to the analyte.
Current/time and correction uncertainty affect the inferred amount.
Photometric counts need a calibrated band response and zero point before they represent an astronomical flux or magnitude. For positive flux F with small standard uncertainty u(F), first-order propagation through m=−2.5log₁₀F+C gives u(m)≈[2.5/ln10]u(F)/F. Calibration and extinction uncertainties remain separate contributions.
The derivative propagates the stated small flux uncertainty.
For parent P decaying into daughter D at constant λ, normalization to a stable isotope S gives D/S=(D/S)₀+(P/S)(e^(λt)−1). Coeval samples with a common initial daughter ratio, differing parent ratios and closed subsequent evolution define an isochron with slope m=e^(λt)−1 and age t=ln(1+m)/λ. A fitted straight line by itself does not prove those geological assumptions.
Slope and decay-constant uncertainties affect an age inference.
For a planar shear-fault source with uniform shear modulus μ and consistently directed slip magnitude D, scalar moment is M₀=μAD̄, equivalently μ∫_A D dA. Moment magnitude uses Mw=(2/3)[log₁₀(M₀/(1N m))−9.1]. Source moment describes fault-slip size and is distinct from radiated energy or intensity at a site.
Uncertainty in inferred area, slip and modulus propagates into the source estimate.
A Landsat thermal-band TOA radianceL_λ>0 maps to product brightness temperature T_B=K₂/ln(K₁/L_λ+1), using its band-specific metadata constants. Brightness temperature is the equivalent blackbody temperature for the specified radiance/band convention; atmospheric effects and surface emissivity separate it from the physical temperature of the ground or sea surface.
For compatible near-infrared and red surface-reflectance bands, NDVI=(ρ_NIR−ρ_R)/(ρ_NIR+ρ_R). With nonnegative reflectances and positive sum it lies between−1 and1. The ratio can help characterize vegetation greenness, but does not uniquely measure biomass, plant health or a species. Common multiplicative scaling cancels; additive atmospheric effects do not generally cancel.
Errors near a small denominator can strongly affect the ratio.
For a monostatic pulse with corrected round-trip delayτ and constant propagation group speedv_g, one-way slant range is R=v_gτ/2. Platform position and beam direction map that range to a reflecting location. A bare-earth digital elevation model represents terrain heights relative to a specified horizontal/vertical reference; a measured canopy or building echo is not automatically bare ground.
Timing, position, medium and classification errors affect elevation.
Ensemble assimilation estimates forecast-error covariances from model realizations and updates the state using observations and their error model. Diagnose finite-ensemble spurious correlations and miscalibrated spread; covariance filtering or inflation requires method-specific justification. The analysis is conditioned on the observing system and dynamical model.
Conceptual association supported by the recorded research scope; this relation does not assert a proven derivation or mandatory prerequisite.
U–Pb discordance can reflect common lead, radiogenic lead loss, initial daughter disequilibrium or mixtures of differently aged growth zones. Correlated ratio errors and geological overdispersion must be retained; a discordia fit does not alone identify a unique geological event history.
Conceptual association supported by the recorded research scope; this relation does not assert a proven derivation or mandatory prerequisite.
Cosmological parameter inference combines data likelihoods with model priors and nuisance parameters for foregrounds, calibration and selection. Shared data/covariance prevent arbitrary multiplication of apparently independent probes. Constraints and discrepancies change when model freedom and systematics change.
Conceptual association supported by the recorded research scope; this relation does not assert a proven derivation or mandatory prerequisite.
For y = f(x₁,…,xₙ), first-order propagation weights input variances and covariances by the model's sensitivity coefficients. Correlations must be retained when the input estimates are correlated.
The uncertainty rule uses the joint covariance of input quantities. Finite covariance in the declared measurement model; retain shared dependencies.
For y = f(x₁,…,xₙ), first-order propagation weights input variances and covariances by the model's sensitivity coefficients. Correlations must be retained when the input estimates are correlated.
Sensitivity coefficients arise from the first-order multivariable Taylor linearization. Differentiable measurement function and adequate first-order approximation over the uncertainty region.