# Deriving a Dilution of Precision Indicator for GNSS Factor Graph Optimization Solutions +
## 1 Introduction  
The positioning algorithms implemented in GNSS receivers uses estimation techniques to combine GNSS observations, such as code pseudo-ranges or doppler offset measurements, to estimate the unknown position, velocity and clock bias of a GNSS receiver. Several algorithms are commonly used, such as the Weighted Least Squares estimation and the Kalman Filter [1, 2]. The Weighted Least Squares (WLS) estimation applied with code pseudo-ranges is called the Single Point Positioning (SPP) solution, and is the most basic solution provided by a GNSS receiver. Kalman filter-based techniques add a state transition model, which corresponds to a motion model for the position/velocity/clock states, in order to refine the estimate [3]. It can also be applied with phase pseudo-range observations, leading the well-known Precise Point Positioning [4].  
Those algorithms provide both the estimate and the covariance matrix of the state vector. The covariance matrix gives an indication of the uncertainty of the estimate, and its validity depends on the matching of the assumed models with the real conditions of the data collect. Among the usual quality indicators, the DOP (Dilution of Precision) factors are often encountered [1]. The DOP indicators are derived from the estimation covariance of the WLS solution, assuming that all code pseudo-ranges are affected by a Gaussian error that has the same variance and is independent from other pseudo-ranges. Given those assumptions, the DOP indicators only depend on the relative satellite positions with regards to the receiver. They quantify the impact of the satellite constellation geometryon the uncertainty of the WLS solution. In the context of this paper, satellite constellation geometry refers both to the satellite position relative to the receiver, but also to the possible masking of satellites by surrounding obstacles. DOP is used for assessment of solution quality, filtering bad position estimates or planning data collections.  
More recently, Factor Graph Optimization (FGO) has been applied to the GNSS positioning problem with great success [5,6]. FGO performs a batch WLS estimation over a large time window, as opposed to a snapshot algorithm as used in the SPP solution. In order to benefit from the optimization over a large time window, motion factors are introduced in addition to GNSS observations to provide constraints between states in adjacent epochs. Those motion factors are essentially similar to the state transition model in a Kalman filter. Therefore, all observations contribute to the accuracy of the solution at each epoch, through the correlation of the estimated states across epochs introduced by the motion factors.  
While the impact of the constellation geometry on the solution accuracy is well described for snapshot WLS solutions, such as the SPP solution, it is more difficult to determine how the constellation geometry will impact the solution accuracy in an FGO solution, due to the intricate interactions between the GNSS observations at each epochs and the propagation of the accuracy through the motion factors.  
This paper aims at deriving a DOP-like indicator for FGO solutions including a generic motion factor, that will provide a quality assessment of a FGO solution at each epoch, based on the constellation geometry and motion factor uncertainty only.