Long-Term Analysis of Carrier Phase Residual Variations 

Brian Breitsch

Jade Morton

Charles Rino

Using Geometry-Ionosphere-Free Combination of Triple-Frequency GPS Observations

ION GNSS 2017

Background and Motivation

Linear Estimation of GNSS Parameters

Geometry-Ionosphere-Free Combination

Application to Real GPS Data

Multi-Frequency GNSS

  • GPS Block IIF / Block III
  • Galileo (E1, E5a/b)
  • GLONASS-K2
Signal Frequency (GHz)
L1CA 1.57542
L2C 1.2276
L5 1.17645

triple-frequency GPS

12 satellites since 2016

GNSS Carrier Phase Observable

\Phi_i = r + c\Delta t + T + I_i + \lambda_i N_i + H_i + S_i + \epsilon_i
Φi=r+cΔt+T+Ii+λiNi+Hi+Si+ϵi \Phi_i = r + c\Delta t + T + I_i + \lambda_i N_i + H_i + S_i + \epsilon_i

HARDWARE BIASES

IONOSPHERE RANGE ERROR

CARRIER AMBIGUITY

SYSTEMATIC ERRORS / MULTIPATH

FREQUENCY INDEPENDENT EFFECTS

STOCHASTIC ERRORS

  • accumulated phase in meters
  • frequency \(f_i\)
I_i \approx -\frac{\kappa}{f_i^2} \text{TEC}
Ii≈−κfi2TECI_i \approx -\frac{\kappa}{f_i^2} \text{TEC}

Simplified Carrier Phase Model

\Phi_i = G + I_i + S_i + \epsilon_i
Φi=G+Ii+Si+ϵi\Phi_i = G + I_i + S_i + \epsilon_i

zero-mean

normally-

distributed

zero-mean

solve bias terms

Bias term accuracy can dominate error--

we address estimation precision, rather than accuracy

\Phi_i = r + c\Delta t + T + I_i + \lambda_i N_i + H_i + S_i + \epsilon_i
Φi=r+cΔt+T+Ii+λiNi+Hi+Si+ϵi \Phi_i = r + c\Delta t + T + I_i + \lambda_i N_i + H_i + S_i + \epsilon_i

"geometry" term

Ionosphere Range Error

consider first-order term in ionosphere refractive index

second and higher-order terms on the order of a few cm

I_i \approx -\frac{\kappa}{f_i^2} \displaystyle \int_\text{rx}^\text{tx} N_e \ ds = -\frac{\kappa}{f_i^2} \text{TEC}
Ii≈−κfi2∫rxtxNe ds=−κfi2TECI_i \approx -\frac{\kappa}{f_i^2} \displaystyle \int_\text{rx}^\text{tx} N_e \ ds = -\frac{\kappa}{f_i^2} \text{TEC}
\kappa = \frac{e^2}{8\pi^2\epsilon_0 m_e} \approx 40.301
κ=e28π2ϵ0me≈40.301\kappa = \frac{e^2}{8\pi^2\epsilon_0 m_e} \approx 40.301

TOTAL ELECTRON CONTENT

rx

tx

plasma / free electrons

units: \(\frac{\text{electrons}}{\text{m}^2}\)

f_i
fif_i

carrier frequency

Estimation of \(G\) and \(\text{TEC}\) Using Dual-Frequency GNSS

neglecting systematic and stochastic error terms, and after resolving bias terms:

\text{TEC} = \displaystyle -\frac{\Phi_1 - \Phi_2}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}
TEC=−Φ1−Φ2κ(1f12−1f22)\text{TEC} = \displaystyle -\frac{\Phi_1 - \Phi_2}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}
G = \displaystyle \frac{f_1^2 \Phi_1 - f_2^2 \Phi_2}{f_1^2 - f_2^2}
G=f12Φ1−f22Φ2f12−f22G = \displaystyle \frac{f_1^2 \Phi_1 - f_2^2 \Phi_2}{f_1^2 - f_2^2}

ionosphere-free combination

geometry-free combination

\Phi_i = G + I_i + S_i + \epsilon_i
Φi=G+Ii+Si+ϵi\Phi_i = G + I_i + S_i + \epsilon_i

Examples of Dual-Frequency TEC Estimates

Poker Flat, Alaska, 2016-01-02

We compute dual-frequency TEC estimates \(\text{TEC}_\text{L1,L2}\) and \(\text{TEC}_\text{L1,L5}\)

\(G_{\text{L1,L5}} - G_{\text{L1,L2}} = \text{TEC}_{\text{L1,L5}} - \text{TEC}_{\text{L1,L2}}\)

Poker Flat, Alaska, 2016-01-02

Can we characterize / find the source of these discrepancies?

Can we relate them to errors in range and TEC estimates?

Background and Motivation

Linear Estimation of GNSS Parameters

Geometry-Ionosphere-Free Combination

Application to Real GPS Data

model parameters

\mathbf{\Phi} = \left[\Phi_1, \cdots, \Phi_m\right]^T
Φ=[Φ1,⋯,Φm]T\mathbf{\Phi} = \left[\Phi_1, \cdots, \Phi_m\right]^T
\mathbf{m} = \left[G, \text{TEC}, S_1, \cdots, S_m \right]^T
m=[G,TEC,S1,⋯,Sm]T\mathbf{m} = \left[G, \text{TEC}, S_1, \cdots, S_m \right]^T
\mathbf{\Phi} = \mathbf{A} \mathbf{m} + \mathbf{\epsilon}
Φ=Am+ϵ\mathbf{\Phi} = \mathbf{A} \mathbf{m} + \mathbf{\epsilon}

Linear Inverse Problem

observations

\Phi_i = G + I_i + S_i + \epsilon_i
Φi=G+Ii+Si+ϵi\Phi_i = G + I_i + S_i + \epsilon_i
|G| , |I_i| \gg |S_i|
∣G∣,∣Ii∣≫∣Si∣|G| , |I_i| \gg |S_i|

a-priori information:

\mathbf{\hat{m}} \approx \mathbf{A^*} \mathbf{\Phi}
m^≈A∗Φ\mathbf{\hat{m}} \approx \mathbf{A^*} \mathbf{\Phi}

estimate:

\mathbf{A^*} =\begin{bmatrix} \mathbf{C}_G \\ \mathbf{C}_\text{TEC} \\ \mathbf{C}_{S_1} \\ \vdots \\ \mathbf{C}_{S_m} \end{bmatrix}
A∗=[CGCTECCS1⋮CSm]\mathbf{A^*} =\begin{bmatrix} \mathbf{C}_G \\ \mathbf{C}_\text{TEC} \\ \mathbf{C}_{S_1} \\ \vdots \\ \mathbf{C}_{S_m} \end{bmatrix}

geometry estimator

TEC estimator

systematic-error estimators

Linear Coefficient Constraints

\sum_i c_i = 0
∑ici=0\sum_i c_i = 0
\sum_i c_i = 1
∑ici=1\sum_i c_i = 1

geometry-free

geometry-estimator

\sum_i -\frac{\kappa}{f_i^2}c_i = 1
∑i−κfi2ci=1\sum_i -\frac{\kappa}{f_i^2}c_i = 1
\sum_i \frac{c_i}{f_i^2} = 0
∑icifi2=0\sum_i \frac{c_i}{f_i^2} = 0

TEC-estimator

ionosphere-free

\Phi_i = G + I_i + S_i + \epsilon_i
Φi=G+Ii+Si+ϵi\Phi_i = G + I_i + S_i + \epsilon_i

solutions lie along intersection of constraint hyperplanes

Estimators

\(\text{TEC}\) Estimator

TEC-estimator + geometry-free constraints

\(G\) Estimator

geometry-estimator + ionosphere-free constraints

\begin{aligned} & -\frac{\kappa}{f_1^2}c_1 - \frac{\kappa}{f_2^2}c_2 = 1 \\[1em] & c_1 + c_2 = 0 \Rightarrow c_1 = -c_2 \\[1em] \Rightarrow & -\kappa c_1 \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right) = 1 \end{aligned}
−κf12c1−κf22c2=1c1+c2=0⇒c1=−c2⇒−κc1(1f12−1f22)=1\begin{aligned} & -\frac{\kappa}{f_1^2}c_1 - \frac{\kappa}{f_2^2}c_2 = 1 \\[1em] & c_1 + c_2 = 0 \Rightarrow c_1 = -c_2 \\[1em] \Rightarrow & -\kappa c_1 \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right) = 1 \end{aligned}

recall:

\text{TEC} = \displaystyle -\frac{\Phi_1 - \Phi_2}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}
TEC=−Φ1−Φ2κ(1f12−1f22)\text{TEC} = \displaystyle -\frac{\Phi_1 - \Phi_2}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}

Geometry and TEC Estimators Using Triple-Frequency GPS

\begin{matrix} \text{Estimate} & c_1 & c_2 & c_3 & \sum_i c_i^2 \\[.5em] G_\text{L1,L2,L5} & 2.327 & -0.360 & -0.967 & 2.546 \\ G_\text{L1,L5} & 2.261 & 0 & -1.261 & 2.588 \\ G_\text{L1,L2} & 2.546 & -1.546 & 0 & 2.978 \\ G_\text{L2,L5} & 0 & 12.255 & -11.255 & 16.639 \end{matrix}
Estimatec1c2c3∑ici2GL1,L2,L52.327−0.360−0.9672.546GL1,L52.2610−1.2612.588GL1,L22.546−1.54602.978GL2,L5012.255−11.25516.639\begin{matrix} \text{Estimate} & c_1 & c_2 & c_3 & \sum_i c_i^2 \\[.5em] G_\text{L1,L2,L5} & 2.327 & -0.360 & -0.967 & 2.546 \\ G_\text{L1,L5} & 2.261 & 0 & -1.261 & 2.588 \\ G_\text{L1,L2} & 2.546 & -1.546 & 0 & 2.978 \\ G_\text{L2,L5} & 0 & 12.255 & -11.255 & 16.639 \end{matrix}
\begin{matrix} \text{Estimate} & c_1 & c_2 & c_3 & \sum_i c_i^2 \\[.5em] \text{TEC}_\text{L1,L2,L5} & 8.294 & -2.883 & -5.411 & 10.314 & \\ \text{TEC}_\text{L1,L5} & 7.762 & 0 & -7.762 & 10.977 & \\ \text{TEC}_\text{L1,L2} & 9.518 & -9.518 & 0 & 13.460 & \\ \text{TEC}_\text{L2,L5} & 0 & 42.080 & -42.080 & 59.510 & \end{matrix}
Estimatec1c2c3∑ici2TECL1,L2,L58.294−2.883−5.41110.314TECL1,L57.7620−7.76210.977TECL1,L29.518−9.518013.460TECL2,L5042.080−42.08059.510\begin{matrix} \text{Estimate} & c_1 & c_2 & c_3 & \sum_i c_i^2 \\[.5em] \text{TEC}_\text{L1,L2,L5} & 8.294 & -2.883 & -5.411 & 10.314 & \\ \text{TEC}_\text{L1,L5} & 7.762 & 0 & -7.762 & 10.977 & \\ \text{TEC}_\text{L1,L2} & 9.518 & -9.518 & 0 & 13.460 & \\ \text{TEC}_\text{L2,L5} & 0 & 42.080 & -42.080 & 59.510 & \end{matrix}

\(\text{L1,L2,L5}\)

\(\text{L1,L5}\)

\(\text{L1,L2}\)

\(\text{L2,L5}\)

Geometry and TEC Estimators Using Triple-Frequency GPS

Background and Motivation

Linear Estimation of GNSS Parameters

Geometry-Ionosphere-Free Combination

Application to Real GPS Data

Estimating Systematic Errors

apply both geometry-free and ionosphere-free constraints

For triple-frequency GNSS:

\begin{aligned} c_1 & = \ \ x \frac{\frac{1}{f_3^2} - \frac{1}{f_2^2}}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_2 & = - x \frac{\frac{1}{f_3^2} - \frac{1}{f_1^2}}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_3 & = \ \ x \end{aligned}
c1=  x1f32−1f221f22−1f12c2=−x1f32−1f121f22−1f12c3=  x\begin{aligned} c_1 & = \ \ x \frac{\frac{1}{f_3^2} - \frac{1}{f_2^2}}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_2 & = - x \frac{\frac{1}{f_3^2} - \frac{1}{f_1^2}}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_3 & = \ \ x \end{aligned}

system is linear subspace

there is "only one estimate" of systematic errors

note this requires \(m \ge 3\)

\Phi_i = G + I_i + S_i + \epsilon_i
Φi=G+Ii+Si+ϵi\Phi_i = G + I_i + S_i + \epsilon_i

Geometry-Ionosphere-Free Combination

\(\mathbf{C}_\text{GIFC} \perp \mathbf{C}_{G_{1,2,3}}\) and \(\mathbf{C}_\text{GIFC} \perp \mathbf{C}_{\text{TEC}_{1,2,3}}\)

\mathbf{C}_{\text{TEC}_{1,2,3}}
CTEC1,2,3\mathbf{C}_{\text{TEC}_{1,2,3}}
\mathbf{C}_\text{GIFC}
CGIFC\mathbf{C}_\text{GIFC}

information about systematic and stochastic errors present in GNSS carrier phase observables

\begin{aligned} \mathbf{C}_{\text{GIFC}_\text{L1,L2,L5}} & = \mathbf{C}_{\text{TEC}_{\text{L1,L5}}} - \mathbf{C}_{\text{TEC}_{\text{L1,L2}}} \\[1em] & = \left[-1.756, \ 9.520, \ -7.764\right] \end{aligned}
CGIFCL1,L2,L5=CTECL1,L5−CTECL1,L2=[−1.756, 9.520, −7.764]\begin{aligned} \mathbf{C}_{\text{GIFC}_\text{L1,L2,L5}} & = \mathbf{C}_{\text{TEC}_{\text{L1,L5}}} - \mathbf{C}_{\text{TEC}_{\text{L1,L2}}} \\[1em] & = \left[-1.756, \ 9.520, \ -7.764\right] \end{aligned}

We (arbitrarily) choose:

\Phi_i = G + I_i + S_i + \epsilon_i
Φi=G+Ii+Si+ϵi\Phi_i = G + I_i + S_i + \epsilon_i
\mathbf{C}_{G_{1,2,3}}
CG1,2,3\mathbf{C}_{G_{1,2,3}}

\( \text{GIFC} = \text{TEC}_{1,3} - \text{TEC}_{1,2} \)

Background and Motivation

Linear Estimation of GNSS Parameters

Geometry-Ionosphere-Free Combination

Application to Real GPS Data

Experiment Data

GPS Lab high-rate GNSS data collection network

  • Alaska, Hong Kong, Peru
  • 2013 - 2016
  • 1 Hz GPS L1/L2/L5 measurements
  • Septentrio PolarXs

GIFC Examples

PRN 24, Peru

PRN 24, Hong Kong

PRN 25, Peru

PRN 01, Alaska

GIFC Calendar

Hong Kong G24

Hong Kong G25

Alaska G24

Alaska G25

GIFC Calendar

Alaska G25

  • large, long-term GIFC trend likely due mostly to satellite thermal oscillations
  • studied by (Montebruck et. al. 2012) and (Li et. al. 2013)

\(\theta\)

GIFC Spectrum

Alaska G25

Hong Kong G25

Peru G25

Conclusions

  • GIFC is an indicator of the presence of systematic / stochastic errors
  • two main components to GIFC
    • large, low-frequency trend (likely due to satellite thermal oscillations)
    • multipath
  • this work is a step towards improved assessment of precision of GNSS phase-related estimates, particularly ionosphere TEC
  • small improvement of triple-frequency estimators over dual-frequency signal pair with widest frequency separation

References

O. Montenbruck, U. Hugentobler, R. Dach, P. Steigenberger, and A. Hauschild, “Apparent clock variations of the Block IIF-1 (SVN62) GPS satellite,” GPS Solutions, vol. 16, no. 3, pp. 303–313, 2012.

 

H. Li, X. Zhou, and B. Wu, “Fast estimation and analysis of the inter-frequency clock bias for Block IIF satellites,” GPS Solutions, vol. 17, no. 3, pp. 347–355, 2013.

 

B. Breitsch, "Optimal Linear Combinations of GNSS Phase Observables to Improve and Assess TEC Estimation Precision," Masters Thesis, Colorado State University

 

Acknowledgements

This research was supported by the Air Force Research Laboratory and NASA.

GIFC Histogram

Peru Example