# Deriving a Dilution of Precision Indicator for GNSS Factor Graph Optimization Solutions +
## 4 Discussion  
This paper proposes a definition of the notion of DOP for batch solution including a generic motion factor, applicable to FGO solutions. The obtained DOP indicator mainly depends on the snapshot DOP, the FG length \(K\) and the motion to observation uncertainty ratio \(q/\sigma\). Depending on \(q/\sigma\), the batch DOP reaches a floor value for large FG length, meaning that increasing the FG length provides low marginal gains in terms of accuracy.  
A formula is derived for fast computation of batch DOP indicators. The approximation error comes from ignoring the correlated impact of observation errors on multiple states when computing the scalar uncertainty for each state. The error on HDOP is of the order of a few percents in open sky conditions, and rises to 22% in constrained conditions, when a lower number of satellites is available.  
Based on this formula, a future work could try to derive an optimal factor graph length depending on the satellite visibility conditions of consecutive epochs, in order to limit the computational complexity of large factor graph solution.  
The formula could also be improved by considering partial observability of the state vector, where the snapshot solution cannot be computed (e.g., when less than 4 satellites are available). Solutions to consider the covariance between the scalar state of the snapshot solution could also be investigated in order to reduce the approximation error.  
Finally, the proposed formula could be applied to other types of states, such as the float carrier ambiguities. This could provide insight to optimize factor graph length of a float solution, before trying to fix the carrier ambiguities.  
**Author Contributions:** Methodology, software, validation, writing--original draft preparation, P.T.; conceptualization, writing--review and editing, P.T., H.C. and J.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable  
**Informed Consent Statement:** Not applicable  
**Data Availability Statement:** The original data presented in the study are openly available in IGS repositories at [[https://igs.org/data-products-overview/](https://igs.org/data-products-overview/)]([https://igs.org/data-products-overview/](https://igs.org/data-products-overview/)), accessed on 29 April 2025 and on the Google Smartphone Decimeter Challenge 2021 Kaggle page at [[https://www.kaggle.com/competitions/google-smartphone-decimeter-challenge](https://www.kaggle.com/competitions/google-smartphone-decimeter-challenge)]([https://www.kaggle.com/competitions/google-smartphone-decimeter-challenge](https://www.kaggle.com/competitions/google-smartphone-decimeter-challenge)), accessed on 29 April 2025.  
**Acknowledgments:** The authors would like to thank [PERSON] and [PERSON], PhD students at ENAC, for the discussions leading to the mathematical formulas displayed in this paper.  
**Conflicts of Interest:** The authors declare no conflicts of interest.  
Figure 4: HDOP comparison for urban masking scenario.  
Appendix A Symbolic Expression of the First Few Diagonal Elements of \(\mathbf{P}_{j}^{(batch)}\)  
In the annexes, a simplified notation will be used by dropping the index \(j\) referring to a particular scalar state in the state vector: \(\mathbf{P}^{(batch)}\triangleq\mathbf{P}_{j}^{(batch)}\) and \(\sigma_{k}\triangleq\sigma_{j,k}\).  
It can be observed that the \(k\)-th diagonal element of \(\mathbf{P}^{(batch)}\) is a ratio of 2 expressions:  
\[(\mathbf{P}^{(batch)})_{k,k}=\frac{N_{k}(q,\sigma_{1},\dots,\sigma_{K})}{D(q, \sigma_{1},\dots,\sigma_{K})}\]  
where \(N_{k}\) and \(D\) are expressions depending on \(q,\sigma_{1},\dots,\sigma_{K}\)  
The first few diagonal elements of \(\mathbf{P}^{(batch)}\) are given in Table A1.