Error Theory Advanced Concepts

What is the formula for propagating error through a function f(x,y)?

σf² = (∂f/∂x)²σx² + (∂f/∂y)²σy² + 2(∂f/∂x)(∂f/∂y)σxy For independent variables (σxy = 0): σf² = (∂f/∂x)²σx² + (∂f/∂y)²σy² — Ghilani & Wolf Ch. 6

What is the principle of least squares?

Minimize the sum of the squares of the weighted residuals: Σ(wᵢvᵢ²) = minimum where w = weight = 1/σ² and v = residual = adjusted - observed — Ghilani & Wolf Ch. 10

What does degrees of freedom (ν) represent?

Degrees of freedom = observations - unknowns ν = n - u • ν = 0: Unique solution, no adjustment • ν > 0: Overdetermined, adjustment possible • ν < 0: Cannot solve — Mikhail & Ackermann Ch. 4

How do you interpret the a posteriori variance factor?

σ₀² = Σwv²/ν Interpretation: • σ₀² ≈ 1.0: Weights properly assigned • σ₀² > 1.0: Observations worse than assumed • σ₀² < 1.0: Observations better than assumed — Ghilani & Wolf Ch. 11

What is the relationship between RMSE, bias, and σ?

RMSE² = bias² + σ² When bias = 0: RMSE = σ When bias ≠ 0: RMSE > σ Large RMSE with small σ indicates systematic error! — Ghilani & Wolf Ch. 2

How is position error affected by angular error in a traverse?

Position error from angular error: ε ≈ d × Δα (radians) Angular error causes position error proportional to distance from that angle. Example: 10" at 1000 ft: ε = 1000 × (10/206265) ≈ 0.048 ft — Ghilani Elementary Ch. 9

What does the covariance matrix tell us?

Diagonal: σx², σy², σz² (variance of each parameter) Off-diagonal: σxy, σxz, σyz (covariance between parameters) Full uncertainty description: • Precision of each coordinate • Correlations between coordinates • Basis for error ellipses — Ghilani & Wolf Ch. 11

What is 95% confidence level accuracy?

95% of true positions lie within the stated error circle. For 2D: RMSE_r × 1.7308 For 1D: σ × 1.96 This is the FGDC/NSSDA standard for reporting positional accuracy. — FGDC Standards Part 3