# Improving Real-Time Position Estimation Using Correlated Noise Models +
## Appendix A Augmented-State UKF Algorithm  
``` Initialise Set \(\mu_{0}\), \(\mathbf{K}_{0}\), \(\Sigma_{d}\) and \(\Sigma_{v}\) with a well-motivated estimate. for\(n=1...n_{t}-1\)do (1) Augment \(\mathbf{\mu}_{n-1}\) and \(\mathbf{K}_{n-1}\) with the process noise and measurement noise means (zeros) and their covariances, \(\mathbf{\Sigma}_{d}\), \(\mathbf{\Sigma}_{v}\): \(\mathbf{\mu}_{n}^{a}=(\mathbf{\mu}_{n-1},\mathbf{0},\mathbf{0})^{T}\), \(\mathbf{K}_{d,n}=\text{diag}\left(\mathbf{K}_{n-1},\mathbf{\Sigma}_{d},\mathbf{\Sigma}_{v}\right)\). (2) Calculate sigma points and weights \(\{\mathbf{x}^{(j)},W^{(j)}\}\), \(j=0\ldots N_{x}\), where \(N_{x}\) is the dimension of the augmented state vector, e.g., \(\mathbf{x}^{(0)}=\mathbf{\mu}_{at}^{a}\), \(\mathbf{x}^{(i)}=\mathbf{\mu}_{at}^{a}+\sqrt{(N_{x}+\lambda)\mathbf{K}_{a,n}}\), \(\mathbf{x}^{(i+



## References

* (1) [PERSON] Kalman-filtering with coloured measurement noise for deformation analysis. In Proceedings of the 11 th FIG Symposium on Deformation Measurements, Santorini, Greece, 25-28 May 2003. * (2) [PERSON]; [PERSON] Real-time deformation measurements using time-series of GPS coordinates processed by Kalman filter with shaping filter. _Surv. Rev._**2012**, _44_, 189-197. [CrossRef]
* (3) [PERSON]; [PERSON]; [PERSON] Interacting Multiple Model Filter-Based Sensor Fusion of GPS with In-Vehicle Sensors for Real-Time Vehicle Positioning. _IEEE Trans. Intell. Transp. Syst._**2012**, _13_, 329-343. [CrossRef]
* (4) [PERSON] A New Approach to Linear Filtering and Prediction Problems. _J. Fluids Eng._**1960**, _82_, 35-45. [CrossRef]
* (5) [PERSON]; [PERSON] Linear filtering for time-varying systems using measurements containing colored noise. _IEEE Trans. Autom. Control_**1965**, _10_, 4-10. [CrossRef]
* (6) [PERSON] Optimal filtering for correlated noise. _J. Math. Anal. Appl._**1967**, _20_, 1-8. [CrossRef]
* (7) [PERSON] Application of a colored noise Kalman filter to a radio-guided ascent mission. _[PERSON]. Spacerr. Rocket._**1970**, \(7\), 277-281. [CrossRef]
* (8) [PERSON]; [PERSON]; [PERSON] Enhanced noise models for GPS positioning. In Proceedings of the 6 th International Conference on Automation, Robotics and Applications (ICARA), Queenstown, New Zealand, 17-19 February 2015; pp. 28-33. [CrossRef]
* (9) [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] Consideration of time-correlated errors in a Kalman filter applicable to GNSS. _[PERSON]. Geod._**2009**, _83_, 51-56. [CrossRef]
* (10) [PERSON]; [PERSON]; [PERSON] Globally optimal Kalman filtering with finite-time correlated noises. In Proceedings of the 49 th IEEE Conference on Decision and Control (CDC), Atlanta, GA, USA, 15-17 December 2010; pp. 5007-5012. [CrossRef]
* (11) [PERSON]; [PERSON] On the Theory of the Brownian Motion. _Phys. Rev._**1930**, _36_, 823-841. [CrossRef]
* (12) [PERSON]; [PERSON] _Forecast Verification: A Practitioner's Guide in Atmospheric Science_; Wiley: Chichester, UK, 2003. * (13) [PERSON]; [PERSON] Scoring Probabilistic Forecasts: The Importance of Being Proper. _Weather Forecast_. **2007**, _22_, 382-388. [CrossRef]
* (14) [PERSON]; [PERSON]; [PERSON] Scoring rules and survey density forecasts. _Int. J. Forecast._**2011**, _27_, 379-393. [CrossRef]
* (15) [PERSON]; [PERSON] Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules. _J. Bus. Econ. Stat._**2011**, _29_, 411-422. [CrossRef]
* (16) [PERSON]; [PERSON]; [PERSON] Measuring the performance of sensors that report uncertainty. In Proceedings of the 21 st Electronics New Zealand Conference, Hamilton, New Zealand, 20-21 November 2014. [CrossRef]
* (17) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Real-time uncertainty quantification using correlated noise models for GNSS positioning. In Proceedings of the 2017 IEEE SENSORS, Glasgow, UK, 29 October-1 November 2017; pp. 1-3. [CrossRef]
* (18) [PERSON] A new look at the statistical model identification. _IEEE Trans. Autom. Control_**1974**, _19_, 716-723.