An Error Overbounding Method Based on a Gaussian Mixture Model with Uncertainty Estimation for a Dual-Frequency Ground-Based Augmentation System
<p>Block diagram of the Ifree filtering process.</p> "> Figure 2
<p>An equivalent diagram of the Ifree filtering process (<b>a</b>) and the overall corresponding overbounding process for an Ifree-based GBAS (<b>b</b>). The dotted-line block in (<b>a</b>) is the carrier smoothing module in (<b>b</b>).</p> "> Figure 3
<p>Flowchart of the MC simulation process for verifying the error types.</p> "> Figure 4
<p>NIG samples and overbounds with and without additive bias. These are plotted as CDFs with the <span class="html-italic">y</span>-axis scaled to better show the tails. OB is an abbreviation for “overbound“.</p> "> Figure 5
<p>(<b>a</b>) Top view of the Dongying Airport environment. (<b>b</b>) Locations of the experimental equipment deployed at the airport.</p> "> Figure 6
<p>CDFs of the Ifree-based range errors and overbounds for an elevation bin from 15° to 20°. The <span class="html-italic">y</span>-axis is scaled to show the tails.</p> "> Figure 7
<p>Motion trajectory of the land vehicle.</p> "> Figure 8
<p>Actual VPLs produced by the different overbounds against the observed errors on 25 January 2021.</p> ">
Abstract
:1. Introduction
2. Background
2.1. Previous Work on Single-Frequency Overbounds
2.2. GMM and GMM Overbounding Method
3. Materials and Methods
3.1. Overall Framework of the Ifree-Based GBAS Error Overbounding
3.2. Single-Frequency REDM Establishment
3.3. Single-Frequency REOM Establishment
3.4. Ifree-Based REOM Establishment
3.5. Ifree-Based Protection Level Calculation Utilizing GMM Overbounds
4. Results
5. Discussion
5.1. VPLs versus Alarm Limits
5.2. Computational Load
5.3. Limitation
6. Conclusions and Perspectives
- Our work proposes an overbounding framework for a dual-frequency GBAS.
- Our work redesigns a novel form of the GMM overbound to rigorously characterize heavy-tailed distributions. To determine whether the GMM parameter estimation approach is accurate, our work evaluates its point estimation accuracy and confidence interval estimation accuracy.
- To calculate the protection levels, our work derives Ifree-based protection levels by using the GMM overbound, which optimizes the availability without significantly increasing the computational load.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
ACEVS | Auto-covariance estimation of variable samples |
ARAIM | Advanced receiver autonomous integrity monitoring |
BDS | Beidou navigation satellite system |
CDF | Cumulative distribution function |
D-free | Divergence-free |
E | All potential errors |
EM | Expectation-Maximization |
EVT | Extreme value theory |
GAST | GBAS approach service type |
GBAS | Ground-based augmentation system |
GCET | Gaussian-core exponential-tail |
GLONASS | Russian global navigation satellite system |
GMM | Gaussian mixture model |
GNSS | Global navigation satellite system |
GPO | Gaussian-pareto overbound |
GPS | Global positioning system |
IFB | Inter-frequency bias |
I-free | Ionosphere-free |
NIG | Normal-inverse gamma distribution |
Probability density function | |
R | All common terms in code and carrier phase measurements |
REDM | Range error distribution model |
REOM | Range error overbounding model |
RTK | Real-time kinematic |
S | Projection matrix |
SBAS | Satellite-based augmentation system |
SUMD | Sum of difference |
Var | Variance of the variable |
VDB | Very high frequency data broadcast |
VPL | Vertical protection level |
PDF of Gaussian distribution | |
ζ | |
ɛ | Noise and multipath |
ρ | Range measurement |
ϕ | Carrier-phase measurement |
θ | GMM parameter set including the weight coefficient ω and the standard deviation σ |
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Parameters | Group | |||
---|---|---|---|---|
1 | 2 | 3 | 4 | |
0.85 | 0.95 | 0.975 | 0.50 | |
1.82 | 0.97 | 1.50 | 1.50 | |
0.75 | 0.11 | 0.30 | 0.50 |
Distribution | Parameters |
---|---|
NIG | = 0.65 = 0.65 = 0 = 0 |
GCET | = 0.8, El = 1.57 |
GPO | = 0.03 = 0.82 |
Stable | = 1.95 = 0.9 = 0 = 0 |
Group | 1 | 2 | 3 | 4 | |
---|---|---|---|---|---|
Absolute value bias | 0.001 | <0.001 | <0.001 | <0.001 | |
0.005 | 0.004 | 0.004 | 0.002 | ||
0.002 | 0.001 | <0.001 | <0.001 | ||
Absolute variance bias | 0.002 | 0.001 | <0.001 | 0.001 | |
<0.001 | 0.001 | 0.002 | 0.003 | ||
<0.001 | <0.001 | 0.001 | 0.002 | ||
Coverage probability | 95.6% | 94.2% | 95.1% | 96.2% | |
94.6% | 95.5% | 95.1% | 94.4% | ||
94.2% | 95.5% | 95.7% | 95.9% |
Group | 1 | 2 | 3 | 4 | |
---|---|---|---|---|---|
Absolute value bias | 0.047 | 0.028 | 0.011 | 0.002 | |
0.136 | 0.281 | 0.176 | 0.151 | ||
0.032 | 0.013 | 0.028 | 0.004 | ||
Absolute variance bias | 0.010 | 0.001 | <0.001 | <0.001 | |
0.023 | 0.020 | 0.148 | 0.134 | ||
0.006 | <0.001 | 0.006 | <0.001 | ||
Coverage probability | 55.6% | 97.8% | 82.7% | 81.6% | |
53.6% | 61.7% | 59.6% | 68.6% | ||
57.5% | 86.2% | 86.3% | 84.4% |
Parameters | Distribution | |||
---|---|---|---|---|
NIG | GCET | GPO | Stable | |
0.094 | 0 | 0.021 | 0.011 | |
0.276 | 0.088 | 0.271 | 0.281 | |
0.059 | 0.005 | 0.030 | 0.007 |
Distribution | Parameters | |
---|---|---|
GMM | = 2.98 | 0.0141 |
= 2.60 | ||
= 2.00 | ||
= 1.37 | ||
EVT-Gaussian [14] | = 2.2855 | 0.0835 |
GCET-Gaussian [25] | = 2.8308 | 0.1110 |
Distribution | Mean | Maximum | Standard Derivation |
---|---|---|---|
GMM | 26.74 | 31.29 | 1.49 |
EVT-Gaussian | 32.83 | 36.01 | 1.62 |
GCET-Gaussian | 43.03 | 47.39 | 2.25 |
Distribution | GMM | Stable | GPO |
---|---|---|---|
Relative time | 64 | 423 | ≈60,000 |
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Gao, Z.; Fang, K.; Wang, Z.; Guo, K.; Liu, Y. An Error Overbounding Method Based on a Gaussian Mixture Model with Uncertainty Estimation for a Dual-Frequency Ground-Based Augmentation System. Remote Sens. 2022, 14, 1111. https://doi.org/10.3390/rs14051111
Gao Z, Fang K, Wang Z, Guo K, Liu Y. An Error Overbounding Method Based on a Gaussian Mixture Model with Uncertainty Estimation for a Dual-Frequency Ground-Based Augmentation System. Remote Sensing. 2022; 14(5):1111. https://doi.org/10.3390/rs14051111
Chicago/Turabian StyleGao, Zhen, Kun Fang, Zhipeng Wang, Kai Guo, and Yuan Liu. 2022. "An Error Overbounding Method Based on a Gaussian Mixture Model with Uncertainty Estimation for a Dual-Frequency Ground-Based Augmentation System" Remote Sensing 14, no. 5: 1111. https://doi.org/10.3390/rs14051111