Error Propagation and Linearization

computed quantity

is a mathematical quantity obtained from one or more direct field measurements.

Error propagation

is the process of determining how errors in original measurements affect a computed quantity.

y = ax + b

x = x_t + dx

y_t = ax_t + b

y = a(x_t + dx) + b

y = ax_t + a dx + b

y = y_t + a dx

Therefore:

dy = a dx

Functional Substitution

y = ax + b

dy/dx = a

dy = a dx

Total Differentiation

linear function

functional substitution and total differentiation give the same propagated error.

nonlinear function

functional substitution and total differentiation do not give exactly the same result.

Total differentiation

is a linear approximation.

Taylor series

A nonlinear function can be represented using a ___ around a reference value x₀.

y ≈ y₀ + (dy/dx at x₀) Δx

Δx = x - x₀

y₀ = f(x₀)

linearized form of the function, if the second-order and higher-order terms are neglected

Linearization

is the approximation of a nonlinear function using its first-order Taylor series.

partial derivatives

When several measured quantities contain errors, ___ are used.

dy = (∂y/∂x₁)dx₁ + (∂y/∂x₂)dx₂ + ... + (∂y/∂xₙ)dxₙ

total differential for multiple variables

Jacobian matrix

contains the first partial derivatives of the output quantities with respect to the input quantities.

y ≈ y₀ + J ΔX

dy = J dX

J = Jacobian matrix

ΔX = vector of changes or errors in the input variables

dy = vector of propagated changes or errors in the output variables

For multiple input variables, the linearized equation can be written in matrix form:

The first-differential approach:

dy = Σ(∂y/∂xᵢ)dxᵢ

is commonly used when the errors are treated as systematic or deterministic changes.

random errors

the individual errors are generally not simply added. Instead, their variances are combined.

(σy)² = (∂y/∂x₁)²(σx₁)² + (∂y/∂x₂)²(σx₂)² + ... + (∂y/∂xₙ)²(σxₙ)²

σ = standard deviation

σ² = variance

For uncorrelated random errors:

Systematic/deterministic propagation

errors are directly combined using their signs.

Random/uncorrelated propagation

variances are combined, then the square root is taken to obtain the standard deviation.

ΣY = J ΣX Jᵀ

ΣY = covariance matrix of the output quantities

J = Jacobian matrix

ΣX = covariance matrix of the input measurements

Jᵀ = transpose of the Jacobian matrix

When input errors are correlated, the errors are not statistically independent.

The general matrix formula for covariance propagation is:

Linearization

is an approximation and is not always accurate.