Statistical Measures & Error Estimation

A. Covariance Matrices in Position Estimation

The **covariance matrix** stands as an indispensable mathematical construct for quantifying the uncertainty associated with GNSS position estimates. It offers a comprehensive perspective, encompassing not only the spread (variance) of individual coordinate components but also the interrelationships (covariances) between them.

For a three-dimensional position estimate (e.g., Latitude, Longitude, Altitude, or X, Y, Z in a local geodetic frame), it typically manifests as a 3x3 matrix. The **diagonal elements** of this matrix represent the variances (σ²) of the individual estimated coordinates. The **off-diagonal elements** quantify the covariances between different components, revealing the degree to which a change in one coordinate is related to a change in another.

B. Dilution of Precision (GDOP, PDOP, HDOP, VDOP, TDOP)

**Dilution of Precision (DOP)** is a geometric measure that describes how the spatial distribution of satellites affects the accuracy of a position fix. DOP is not an error source itself, but a multiplier that amplifies existing measurement errors. Lower DOP values indicate a better satellite geometry and, therefore, a more reliable position solution. A high DOP value indicates that small measurement errors will be significantly magnified into large position errors.

• GDOP (Geometric DOP): Overall 3D position and time accuracy.
• PDOP (Position DOP): 3D position accuracy only (latitude, longitude, and altitude).
• HDOP (Horizontal DOP): Horizontal accuracy (latitude and longitude).
• VDOP (Vertical DOP): Vertical accuracy (altitude).
• TDOP (Time DOP): Timing accuracy.

C. Error Ellipses & Confidence Regions

The covariance matrix provides the mathematical foundation for visually representing positional uncertainty.

• Error Ellipses: A two-dimensional graphical representation of horizontal position uncertainty. An elongated ellipse indicates a directional vulnerability, where errors are more likely to occur along its major axis.
• Error Ellipsoids: A three-dimensional volumetric confidence region that delineates where the true position is expected to lie .

D. Kalman Filtering (KF, EKF, UKF) for GNSS/INS Fusion

The **Kalman filter** is a powerful algorithm central to sensor fusion, especially for combining GNSS and Inertial Navigation Systems (INS). It provides an optimal estimate of the system's state (position, velocity, and attitude) by fusing noisy measurements over time.

The filter operates in two steps: a **predict step** and an **update step**. In the predict step, the filter uses a mathematical model to predict the next state of the system based on its current state. In the update step, it uses new measurements to correct its prediction.

• Kalman Filter (KF): The original filter, for linear systems.
• Extended Kalman Filter (EKF): An extension for non-linear systems, it linearizes the system model and measurement model around the current state estimate.
• Unscented Kalman Filter (UKF): A more advanced non-linear filter that avoids linearization and is generally more accurate and robust than the EKF.

E. Integrity Monitoring (RAIM, ARAIM, Fault Detection & Exclusion)

**Integrity** is the measure of trust that can be placed in the accuracy of the GNSS solution, and it is crucial for safety-critical applications.

• RAIM (Receiver Autonomous Integrity Monitoring): A technology that assesses the integrity of GNSS signals directly within the receiver, without external information.
• ARAIM (Advanced RAIM): A more sophisticated form of RAIM that leverages data from multiple GNSS constellations and frequencies, providing a more robust and reliable integrity assessment.
• Fault Detection and Exclusion (FDE): FDE is a key component of integrity monitoring, where the system not only detects a fault (e.g., an error in a satellite signal) but also excludes it from the positioning solution.

F. Measurement Residuals and Consistency Checks

**Measurement residuals** are the differences between the observed measurements and the measurements predicted by the navigation filter's solution. Analyzing these residuals is critical for diagnosing issues like multipath, jamming, or spoofing. Large or rapidly changing residuals can be a sign of a problem.

G. Position Accuracy Metrics: CEP, R95, 1σ, 2σ

Several statistical metrics are used to quantify the accuracy of a GNSS position solution:

• CEP (Circular Error Probable): The radius of a circle within which 50% of position fixes are expected to fall.
• R95: The radius of a circle within which 95% of position fixes are expected to fall.
• 1σ (One Sigma): In a normal distribution, 68.3% of the data falls within one standard deviation.
• 2σ (Two Sigma): 95.5% of the data falls within two standard deviations.

H. C/N₀ Statistics and Impact on Measurement Noise

**Carrier-to-Noise Density (C/N₀)** is a crucial metric that quantifies the signal strength of a GNSS satellite as received by the antenna. A higher C/N₀ value indicates a stronger, cleaner signal and directly correlates with lower measurement noise and improved positioning accuracy.