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📡 PPP EKF State Modeling and Noise Settings

This document describes the noise modeling assumptions and typical parameter values for an Extended Kalman Filter (EKF) implementation in a Precise Point Positioning (PPP) algorithm.

🧾 State Vector

The EKF considered in this project includes the following states:

  • position
  • clock_bias
  • ISB (Inter-System Bias)
  • iono (Ionosphere delay)
  • tropo (Troposphere wet delay)
  • ambiguity (float or fixed)
  • phase_bias

🎯 Process Models and Noise Parameters

State Model Φₖ Qₖ Recommended Parameters Units of σ
Position Random Walk 1 σ² ⋅ Δt σ ≈ 0.001–0.01 m/√s (static) or 0.1–1.0 m/√s (kinematic) meters / √s
Clock Bias Random Walk 1 σ² ⋅ Δt σ ≈ 1–3 m/√s → 3.3–10 ns/√s seconds / √s
ISB Random Walk 1 σ² ⋅ Δt σ ≈ 0.01–0.1 m/√s meters / √s
Ionosphere Random Walk 1 σ² ⋅ Δt σ ≈ 0.1–1.0 m/√s meters / √s
Ionosphere Gauss-Markov α = exp(-Δt/τ) σ² ⋅ (1 - α²) σ ≈ 2.0 m, τ ≈ 600–1800 s meters
Troposphere Random Walk 1 σ² ⋅ Δt σ ≈ 0.001–0.01 m/√s meters / √s
Troposphere Gauss-Markov α = exp(-Δt/τ) σ² ⋅ (1 - α²) σ ≈ 0.05–0.20 m, τ ≈ 3600–10800 s meters
Ambiguity Random Walk 1 σ² ⋅ Δt σ ≈ 0.1–10 cycles/√s (depends on ambiguity resolution) cycles / √s or meters / √s
Phase Bias Random Walk 1 σ² ⋅ Δt σ ≈ 0.001–0.01 m/√s meters / √s

📌 Detailed Notes per State

1. Position

  • Static user: use random walk with small noise (e.g., 1 cm/√s).
  • Kinematic user: use either random walk or a constant velocity model (e.g., include velocity states).

2. Clock Bias

  • GPS L1 time offset drift is about 10 ns/s ≈ 3 m/s.
  • Random walk modeling with σ_c = 1–3 m/√s is common.
  • You can optionally model clock drift as a separate state.

3. ISB (Inter-System Bias)

  • Slowly varying between constellations (e.g., GPS–Galileo).
  • Modeled as either a small random walk or a constant bias with low noise.

4. Ionosphere

  • Only needed if not using ionosphere-free combination.
  • Estimate one per satellite. Use random walk or Gauss-Markov (τ ≈ 1000–2000 s).
  • You may scale Q with satellite elevation angle to improve realism.

5. Troposphere

  • Zenith Wet Delay is modeled as a Gauss-Markov process.
  • Typical parameters: σ = 0.01 m, τ = 1800 s (30 minutes).
  • One state per receiver, not per satellite.

6. Ambiguities

  • Modeled as constant for continuous tracking.
  • When float, use small noise (e.g., σ = 0.01–1 m).
  • When fixed, set process noise to zero and apply integer fix.

7. Phase Bias (OSB)

  • Usually constant; can be initialized from IGS/CODE products.
  • Typically not time-evolving; use Φ = 1, Q = 0.

🔧 EKF Q Matrix Construction

Your Qₖ (process noise covariance) should be block-diagonal, with blocks like:

Qₖ = diag(
σ_p² ⋅ Δt ⋅ I, # position
σ_c² ⋅ Δt, # clock bias
σ_ISB² ⋅ Δt, # ISB
σ_iono² ⋅ Δt ⋅ I_N, # iono per sat (if modeled)
σ_tropo² ⋅ (1 - α²), # tropo (GM)
... # ambiguities, etc.
)

✅ Typical Sigma Values

State σ (Static User) σ (Kinematic User)
Position 0.01 m/√s 0.05–0.1 m/√s
Clock Bias 1–3 m/√s 1–3 m/√s
ISB 0.01 m/√s 0.02 m/√s
Ionosphere 0.02 m/√s 0.05 m/√s
Troposphere 0.01 m 0.01 m
Ambiguity (float) 0.01–1 m 0.05–1 m
Phase Bias 0 (usually fixed) 0

🔄 Process Noise Models in EKF

This section explains commonly used state evolution models in GNSS EKFs and how to construct their corresponding State Transition Matrix (Φₖ) and Process Noise Covariance (Qₖ) for time step Δt.

📌 1. Random Constant Model

Description:

  • Assumes the state does not change over time.
  • Used for static biases or fixed parameters (e.g., phase bias, fixed ambiguities).

State Equation:

xₖ = xₖ₋₁

STM:

Φₖ = 1

Process Noise:

Qₖ = 0  (or a very small value for numerical stability)

📌 2. White Noise Model

Description:

  • Each new state value is a sample from a zero-mean Gaussian distribution.
  • No temporal correlation between states.
  • Rarely used for states that evolve in time (too noisy), but can model sudden noise bursts.

State Equation:

xₖ = wₖ

STM:

Φₖ = 0

Process Noise (already in discrete time):

Qₖ = σ²

Discretization Notes:

  • If you're using a discrete-time white noise model directly, do not multiply by Δt. σ² above is in units unit²
  • If you're deriving from a continuous-time noise power spectral density q (in units of variance/sec), then discretize as:
Qₖ = q ⋅ Δt

This distinction is important for tuning based on physical noise characteristics.

Units and Physical Meaning:

Continuous-time white noise is defined via a power spectral density (PSD) q, with units:

[unit² / s]

When discretized over time step Δt, the process noise becomes:

Qₖ = q ⋅ Δt

and has units:

[unit²]

If you refer to the square root of the PSD (noise amplitude: square root of q), it has units:

[unit / √s]

This matches the rate parameter used in a random walk process noise:

σ_RW has units [unit / √s]

These unit relationships are important when tuning noise levels from physical models or sensor specs.

📌 3. Random Walk Model

Description:

  • Each state value is the sum of the previous state and a random perturbation.
  • Used when the state changes slowly over time (e.g., position, clock bias, ionosphere delay).

State Equation:

xₖ = xₖ₋₁ + wₖ

STM:

Φₖ = 1

Process Noise:

Qₖ = σ² ⋅ Δt

Where:

  • σ (continuous-time sigma) is the rate of change (in units/√s).
  • Δt is the time step (seconds).
  • wₖ has units [unit]
  • Qₖ has units [unit²]

Multidimensional Extension (e.g., 3D position):

Φₖ = Iₙ  
Qₖ = σ² ⋅ Δt ⋅ Iₙ

📌 4. Gauss-Markov Model (1st Order)

Description:

  • Captures exponential decay in state value over time.
  • Suitable for slowly varying quantities with mean-reverting behavior (e.g., troposphere, ionosphere).

State Equation:

xₖ = α ⋅ xₖ₋₁ + wₖ

Where:

  • α = exp(-Δt / τ)
  • τ is the correlation time constant (in seconds).

STM:

Φₖ = α

Process Noise:

Qₖ = σ² ⋅ (1 - α²)

Where σ is the steady-state standard deviation.

🧮 Summary Table

Model Equation Φₖ Qₖ Units of Sigma
Random Constant xₖ = xₖ₋₁ 1 0 or small ε -
White Noise (continuous-time) xₖ = wₖ 0 q ⋅ Δt q has units [unit² / s]
Random Walk xₖ = xₖ₋₁ + wₖ 1 σ² ⋅ Δt σ² has units [unit² / s]
Gauss-Markov xₖ = α ⋅ xₖ₋₁ + wₖ α = exp(-Δt/τ) σ² ⋅ (1 - α²) σ² has units [unit²]

✅ Guidelines

  • Use Random Walk for time-varying parameters like position, clock bias, and ionosphere delay.
  • Use Gauss-Markov for slowly drifting but mean-reverting quantities like troposphere delay.
  • Use Random Constant for fixed biases like ambiguities (after fixing) or phase biases.
  • Use White Noise only for modeling bursts or if the state resets every time step (rare in PPP/INS/GNSS).