# Uncertainty Quantification of First Fix in a Time-Differenced Carrier Phase Observation Model

## TL;DR

This paper analyzes how errors in initial GNSS positioning affect displacement estimates in precise positioning systems.

## Contribution

The study introduces a theoretical framework to quantify displacement estimation errors caused by initial position errors in TDCP models.

## Key findings

- Displacement estimation error is linearly related to initial position error and time interval.
- A 10-meter first-fix error introduces at most a 1.3-millimeter displacement error for a 1-second interval.
- The misspecified Cramer-Rao bound provides a performance benchmark for displacement estimation.

## Abstract

This paper presents an uncertainty quantification analysis of the first fix in a time-differenced carrier phase (TDCP) observation model. TDCP is a widely used method in GNSS-based odometry for precise positioning and displacement estimation. A key point in the TDCP modeling is the assumption that the GNSS receiver’s initial position is perfectly known, which is never exactly the case in real-world applications. This study assesses the impact of initial position errors on estimated displacement by formulating a correct TDCP model and a misspecified one, where the first position is not correct. Theoretical derivations provide a generic framework of estimation under the misspecified model and its associated mean squared error (MSE), as well as estimation performance bounds through the misspecified Cramer Rao bound (MCRB) for the considered case. These theoretical considerations are then used to build an estimator of the receiver’s displacement, with comparisons to the MCRB for performance evaluation. Extensive simulations using realistic GNSS geometry assess the influence of a first-fix error under various conditions, including different time intervals, first-fix error norms, and first-fix error direction. As an example, it is shown that for the considered geometry, if a TDCP of t2−t1=1 s is built with an initial first fix error norm ∥Δr1∥=10 m, then it introduces an estimation of the displacement, with an error of norm equal to 1.3 mm, at most. The results indicate that the displacement estimation error is linearly related to the initial position error and the time interval between observations, highlighting the importance of accurate first-fix estimation for reliable TDCP-based odometry. The findings contribute to highlighting the order of magnitude of errors on solutions as a function of the error on parameters.