Closed-form variance growth for each error process, as a function of integration depth. q is the random-walk PSD, σ is white-noise standard deviation, σss is the Gauss-Markov steady-state σ, and τ is the GM correlation time.
| Model | Sensor σ(t) | Once-integrated σ(t) | Twice-integrated σ(t) |
|---|---|---|---|
| Constant bias b | 0 (deterministic) | |b|·t (linear) | |b|·t²/2 (quadratic) |
| White noise σ | σ (constant) | σ·√(Δt·t) | σ·√(Δt³·t³/3) |
| Random walk q | q·√t | q·√(t³/3) | q·√(t⁵/20) |
| Gauss-Markov σss, τ | σss (steady) | ≈ σss·√(2τt) for t≫τ | ≈ σss·√(2τt³/3) for t≫τ |
Random walk grows faster than white-noise integration: each integration adds another power of t under the radical. Gauss-Markov bridges the two — it behaves like white noise for t≪τ, like a slow random walk for t≪10τ, and saturates for very long horizons.