Linear Combinations of GNSS Phase Observables

Brian Breitsch

Advisor:  Jade Morton

Committee:  Charles Rino, Anton Betten

to Improve and Assess TEC Estimation Precision

Background and Motivation

Linear Estimation of GNSS Parameters

TEC Estimate Error Residuals

Application to Real GPS Data

Earth's Ionosphere

J. Grobowsky / NASA GSFC

Ionosphere Effects on Electromagnetic Propagation

ionosphere = cold, collisionless, magnetized plasma

for L-band frequencies (1-2 GHz) refractive index given by:

n = 1 - \frac{1}{2} X \pm \frac{X|Y\cos\theta_B|}{2} - \frac{1}{8}X^2 \pm \mathcal{O}(\frac{1}{f^5})
n=1−12X±X∣YcosθB∣2−18X2±O(1f5)n = 1 - \frac{1}{2} X \pm \frac{X|Y\cos\theta_B|}{2} - \frac{1}{8}X^2 \pm \mathcal{O}(\frac{1}{f^5})

\(f\) = wave frequency

\(N_e\) = plasma density

\(\epsilon_0\) = permittivity of free space

\(e\) = fundamental charge

\(m\) = electron rest mass

\(B_0\) = ambient magnetic field strength

X = \frac{\omega_p^2}{\omega^2} \kern{1em} Y = \frac{\omega_H}{\omega} \kern{1em} \omega = 2 \pi f \kern{1em} \omega_p = \sqrt{\frac{N_e e^2}{\epsilon_0 m}} \kern{1em} \omega_H = \sqrt{\frac{B_0 |e|}{m}}
X=ωp2ω2Y=ωHωω=2πfωp=Nee2ϵ0mωH=B0∣e∣mX = \frac{\omega_p^2}{\omega^2} \kern{1em} Y = \frac{\omega_H}{\omega} \kern{1em} \omega = 2 \pi f \kern{1em} \omega_p = \sqrt{\frac{N_e e^2}{\epsilon_0 m}} \kern{1em} \omega_H = \sqrt{\frac{B_0 |e|}{m}}

radio source

ionosphere

phase shift / distortion

Global Navigation Satellite Systems (GNSS)

...a useful everyday radio source for geophysical remote-sensing!

GPS

GLONASS

Beidou

Galileio

...etc.

GPS - Global Positioning System

  • 32-satellite constellation
  • transmit dual-frequency BPSK-moduled signals
  • new Block-IIF and next-gen Block-III satellites transmitting triple-frequency signals
Signal Frequency (GHz)
L1CA 1.57542
L2C 1.2276
L5 1.17645

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 BIAS

IONOSPHERE RANGE ERROR

CARRIER AMBIGUITY

SYSTEMATIC ERRORS / MULTIPATH

FREQUENCY INDEPENDENT EFFECTS

STOCHASTIC ERRORS

accumulated phase (in meters) of demodulated GNSS signal at receiver for a particular satellite and signal carrier frequency \(f_i\)

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 = \displaystyle \int_\text{rx}^\text{tx} n - 1 \ ds \approx -\frac{\kappa}{f_i^2} \displaystyle \int_\text{rx}^\text{tx} N_e \ ds
Ii=∫rxtxn−1 ds≈−κfi2∫rxtxNe dsI_i = \displaystyle \int_\text{rx}^\text{tx} n - 1 \ ds \approx -\frac{\kappa}{f_i^2} \displaystyle \int_\text{rx}^\text{tx} N_e \ ds
n \approx 1 - \frac{1}{2} X = 1-\frac{\kappa}{f_i^2} N_e
n≈1−12X=1−κfi2Nen \approx 1 - \frac{1}{2} X = 1-\frac{\kappa}{f_i^2} N_e
\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: \(\frac{\text{electrons}}{\text{m}^2}\)

often measured in TEC units:

1 \text{TECu} = 10^{16} \frac{\text{electrons}}{\text{m}^2}
1TECu=1016electronsm21 \text{TECu} = 10^{16} \frac{\text{electrons}}{\text{m}^2}

Ionosphere Plasma Density

TEC and vertical TEC (vTEC) used to image plasma density structures

profile from CDAAC

image from Saito et al.

map from IGS

vertical distribution

horizontal distribution

travelling ionosphere disturbances (TIDs)

TEC

vTEC

TEC Estimation Using Dual-Frequency GNSS

\Phi_1 - \Phi_2 = \left(I_1 - I_2\right) + \left(\lambda_1 N_1 - \lambda_2 N_2\right) + \left(H_1 - H_2\right)
Φ1−Φ2=(I1−I2)+(λ1N1−λ2N2)+(H1−H2)\Phi_1 - \Phi_2 = \left(I_1 - I_2\right) + \left(\lambda_1 N_1 - \lambda_2 N_2\right) + \left(H_1 - H_2\right)
\approx -\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right) \text{TEC} + \left(\lambda_1 N_1 - \lambda_2 N_2\right) + \Delta H_{1,2}
≈−κ(1f12−1f22)TEC+(λ1N1−λ2N2)+ΔH1,2\approx -\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right) \text{TEC} + \left(\lambda_1 N_1 - \lambda_2 N_2\right) + \Delta H_{1,2}

satellite and receiver inter-frequency hardware biases

neglecting systematic and stochastic error terms:

after resolving bias terms:

\text{TEC} = \displaystyle \frac{\Phi_2 - \Phi_1}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}
TEC=Φ2−Φ1κ(1f12−1f22)\text{TEC} = \displaystyle \frac{\Phi_2 - \Phi_1}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}

carrier ambiguities

bias terms

Resolving Bias Terms

  • LAMBDA

  • code-carrier-levelling

carrier ambiguity resolution

hardware bias estimation

  • must apply ionosphere model
    • ​e.g. global ionosphere model using data assimilation and receiver networks
    • e.g. single receiver and linear 2D-gradient in vTEC (applied in Bourne 2016 [2])

Examples of Dual-Frequency TEC Estimates

Poker Flat, Alaska, 2016-01-02

Using methods similar to [2] and [3] to solve for bias terms, we compute dual-frequency TEC estimate \(\text{TEC}_\text{L1,L2}\) and \(\text{TEC}_\text{L1,L5}\)

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

Poker Flat, Alaska, 2016-01-02

Can we characterize / find the source of these discrepancies?

Can we relate them to errors in dual-frequency TEC estimates?

Objectives

Investigate the discrepancy in \(\text{TEC}_\text{L1,L5} - \text{TEC}_\text{L1,L2}\)

Derive optimal triple-frequency estimation of TEC

Provide a (partial) characterization of TEC estimate residual errors

Motivation

  • Push the boundaries of TID signature detection from earthquakes, explosions, etc.

  • Understand / address the errors in TEC estimates from low-elevation satellites

  • Improve user range error for precise positioning applications

Improve / understand TEC estimate precision

Approach

Apply framework to derive triple-frequency estimates of TEC and systematic errors

Develop framework for linear estimation of GNSS parameters

Relate to impact on TEC estimate error residuals

Background and Motivation

Linear Estimation of GNSS Parameters

TEC Estimate Error Residuals

Application to Real GPS Data

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

neglect bias terms

By neglecting bias terms, 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

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{A} = \begin{bmatrix} 1 & -\frac{\kappa}{f_1^2} & 1 & 0 & \cdots & 0 \\ 1 & -\frac{\kappa}{f_2^2} & 0 & 1 & \cdots & 0 \\ \vdots & & & & \ddots & \\ 1 & -\frac{\kappa}{f_m^2} & 0 & \cdots & & 1 \end{bmatrix}
A=[1−κf1210⋯01−κf2201⋯0⋮⋱1−κfm20⋯1]\mathbf{A} = \begin{bmatrix} 1 & -\frac{\kappa}{f_1^2} & 1 & 0 & \cdots & 0 \\ 1 & -\frac{\kappa}{f_2^2} & 0 & 1 & \cdots & 0 \\ \vdots & & & & \ddots & \\ 1 & -\frac{\kappa}{f_m^2} & 0 & \cdots & & 1 \end{bmatrix}
\mathbf{\epsilon} = [\epsilon_1, \cdots, \epsilon_m]^T
ϵ=[ϵ1,⋯,ϵm]T\mathbf{\epsilon} = [\epsilon_1, \cdots, \epsilon_m]^T
\mathbf{\Phi} = \mathbf{A} \mathbf{m} + \mathbf{\epsilon}
Φ=Am+ϵ\mathbf{\Phi} = \mathbf{A} \mathbf{m} + \mathbf{\epsilon}

Linear Inverse Problem

observations

stochastic error

forward model

Linear Estimation

\mathbf{\hat{m}}
m^\mathbf{\hat{m}}
\mathbf{A^*} = \ ?
A∗= ?\mathbf{A^*} = \ ?
\mathbf{\hat{m}} \approx \mathbf{A^*} \mathbf{\Phi}
m^≈A∗Φ\mathbf{\hat{m}} \approx \mathbf{A^*} \mathbf{\Phi}
\mathbf{A}^T \left(\mathbf{A}\mathbf{A}^T\right)^{-1}
AT(AAT)−1\mathbf{A}^T \left(\mathbf{A}\mathbf{A}^T\right)^{-1}

Poor results; treats each parameter with equal weight

We must apply a priori information about model parameters

model estimate

model estimator

A Priori Information

|G| \gg |I_i| \gg |S_i|
∣G∣≫∣Ii∣≫∣Si∣|G| \gg |I_i| \gg |S_i|

Under normal conditions, we know that:

G \sim
G∼G \sim
I \sim
I∼I \sim
S \sim
S∼S \sim

20,000 km

1 - 150 m

several cm

Using A Priori Information

We could apply \(|G| \gg |I_i| \gg |S_i|\) using Gaussian priors

Instead we derive each row separately:

\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

TECu estimator

systematic-error estimators

estimator

\mathbf{C} = \left[c_1, \cdots, c_m \right]^T \in \mathbb{R}^m
C=[c1,⋯,cm]T∈Rm\mathbf{C} = \left[c_1, \cdots, c_m \right]^T \in \mathbb{R}^m

(written as row vectors here)

How to Choose Optimal \(\mathbf{C}\)

Goals:

1. produce desired parameter with unity coefficient

2. remove / reduce all other terms

Linear combination \(E\) given by inner-product:

E = \langle \mathbf{C} | \mathbf{\Phi} \rangle
E=⟨C∣Φ⟩E = \langle \mathbf{C} | \mathbf{\Phi} \rangle

Approach:

First, constrain \(\mathbf{C}\) to satisfy Goal 1

Then, constrain / optimize \(\mathbf{C}\) to achieve Goal 2

Linear Coefficient Constraints

Use one or two of the following constraints to reduce search space for optimal estimator coefficients:

\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

Reduction of Error

\mathbf{C}^* = \ \displaystyle \arg\min_\mathbf{C} \mathbf{C}^T \Sigma_\epsilon \mathbf{C}
C∗= argminCCTΣϵC\mathbf{C}^* = \ \displaystyle \arg\min_\mathbf{C} \mathbf{C}^T \Sigma_\epsilon \mathbf{C}

Linear combination stochastic error variance:

\sigma_\epsilon^2 = \mathbf{C}^T \mathbf{\Sigma}_\epsilon \mathbf{C}
σϵ2=CTΣϵC\sigma_\epsilon^2 = \mathbf{C}^T \mathbf{\Sigma}_\epsilon \mathbf{C}

where \(\mathbf{\Sigma}_\epsilon\) is the covariance matrix between \(\mathbf{\epsilon}_i\)

Optimal \(\mathbf{C}\) for minimizing stochastic error variance:

\(\epsilon_i\) equal-amplitude and uncorrelated

\mathbf{C}^* = \ \displaystyle \arg\min_\mathbf{C} \sum_i c_i^2
C∗= argminC∑ici2\mathbf{C}^* = \ \displaystyle \arg\min_\mathbf{C} \sum_i c_i^2

TEC Estimator

1. apply TECu-estimator constraint

2. apply geometry-free constraint (since \(|G| \gg |I_i|\) )

Dual-Frequency Example

\Rightarrow c_1 = \displaystyle -\frac{1}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}
⇒c1=−1κ(1f12−1f22)\Rightarrow c_1 = \displaystyle -\frac{1}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}
\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}

TEC-estimator

geometry-free

recall:

\text{TEC} = \displaystyle \frac{\Phi_2 - \Phi_1}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}
TEC=Φ2−Φ1κ(1f12−1f22)\text{TEC} = \displaystyle \frac{\Phi_2 - \Phi_1}{\kappa \left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)}

Triple-Frequency TEC Estimator

Applying constraints yields following system of coefficients (with free parameter denoted \(x\):

\begin{aligned} c_1 &= \frac{\frac{1}{\kappa} + x \left( \frac{1}{f_3^2} - \frac{1}{f_2^2}\right)}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_2 &= \frac{-\frac{1}{\kappa} - x \left( \frac{1}{f_3^2} - \frac{1}{f_1^2}\right)}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_3 & = \ \ x \end{aligned}
c1=1κ+x(1f32−1f22)1f22−1f12c2=−1κ−x(1f32−1f12)1f22−1f12c3=  x\begin{aligned} c_1 &= \frac{\frac{1}{\kappa} + x \left( \frac{1}{f_3^2} - \frac{1}{f_2^2}\right)}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_2 &= \frac{-\frac{1}{\kappa} - x \left( \frac{1}{f_3^2} - \frac{1}{f_1^2}\right)}{\frac{1}{f_2^2} - \frac{1}{f_1^2}} \\[1em] c_3 & = \ \ x \end{aligned}
x^* = \frac{\frac{1}{\kappa}\left(\frac{2}{f_3^2} - \frac{1}{f_2^2} - \frac{1}{f_1^2}\right)}{\left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)^2 + \left(\frac{1}{f_2^2} - \frac{1}{f_3^2}\right)^2 + \left(\frac{1}{f_3^2} - \frac{1}{f_1^2}\right)^2}
x∗=1κ(2f32−1f22−1f12)(1f12−1f22)2+(1f22−1f32)2+(1f32−1f12)2x^* = \frac{\frac{1}{\kappa}\left(\frac{2}{f_3^2} - \frac{1}{f_2^2} - \frac{1}{f_1^2}\right)}{\left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)^2 + \left(\frac{1}{f_2^2} - \frac{1}{f_3^2}\right)^2 + \left(\frac{1}{f_3^2} - \frac{1}{f_1^2}\right)^2}

To satisfy \( \mathbf{C}^* = \ \displaystyle \arg\min_\mathbf{C} \sum_i c_i^2 \), choose

denote corresponding coefficient vector \(\mathbf{C}_{\text{TEC}_{1,2,3}}\) and its corresponding estimate \(\text{TEC}_{1,2,3}\)

TEC Estimator Using Triple-Frequency GPS

\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}
\mathbf{C}_{\text{TEC}_\text{L1,L2,L5}}
CTECL1,L2,L5\mathbf{C}_{\text{TEC}_\text{L1,L2,L5}}
\mathbf{C}_{\text{TEC}_\text{L1,L5}}
CTECL1,L5\mathbf{C}_{\text{TEC}_\text{L1,L5}}
\mathbf{C}_{\text{TEC}_\text{L1,L2}}
CTECL1,L2\mathbf{C}_{\text{TEC}_\text{L1,L2}}
\mathbf{C}_{\text{TEC}_\text{L2,L5}}
CTECL2,L5\mathbf{C}_{\text{TEC}_\text{L2,L5}}

Geometry Estimator

\begin{matrix} c_1 = & \frac{-\frac{1}{f_2^2} + x \left(\frac{1}{f_2^2} - \frac{1}{f_3^2}\right)}{\frac{1}{f_1^2} - \frac{1}{f_2^2}} \\ c_2 = & \frac{\frac{1}{f_1^2} - x \left(\frac{1}{f_1^2} - \frac{1}{f_3^2}\right)}{\frac{1}{f_1^2} - \frac{1}{f_2^2}} \\ c_3 = & x \end{matrix}
c1=−1f22+x(1f22−1f32)1f12−1f22c2=1f12−x(1f12−1f32)1f12−1f22c3=x\begin{matrix} c_1 = & \frac{-\frac{1}{f_2^2} + x \left(\frac{1}{f_2^2} - \frac{1}{f_3^2}\right)}{\frac{1}{f_1^2} - \frac{1}{f_2^2}} \\ c_2 = & \frac{\frac{1}{f_1^2} - x \left(\frac{1}{f_1^2} - \frac{1}{f_3^2}\right)}{\frac{1}{f_1^2} - \frac{1}{f_2^2}} \\ c_3 = & x \end{matrix}

For triple-frequency GNSS:

1. apply geometry-estimator constraint

2. apply ionosphere-free constraint since \(I_i\) are the next-largest terms

x^* = \frac{\frac{1}{\kappa}\left(\frac{2}{f_3^2} - \frac{1}{f_2^2} - \frac{1}{f_1^2}\right)}{\left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)^2 + \left(\frac{1}{f_2^2} - \frac{1}{f_3^2}\right)^2 + \left(\frac{1}{f_3^2} - \frac{1}{f_1^2}\right)^2}
x∗=1κ(2f32−1f22−1f12)(1f12−1f22)2+(1f22−1f32)2+(1f32−1f12)2x^* = \frac{\frac{1}{\kappa}\left(\frac{2}{f_3^2} - \frac{1}{f_2^2} - \frac{1}{f_1^2}\right)}{\left(\frac{1}{f_1^2} - \frac{1}{f_2^2}\right)^2 + \left(\frac{1}{f_2^2} - \frac{1}{f_3^2}\right)^2 + \left(\frac{1}{f_3^2} - \frac{1}{f_1^2}\right)^2}

To satisfy \( \mathbf{C}^* = \ \displaystyle \arg\min_\mathbf{C} \sum_i c_i^2 \),

We call this coefficient vector \(\mathbf{C}_{G_{1,2,3}}\) and its corresponding estimate \(G_{1,2,3}\)

the optimal "ionosphere-free combination"

Geometry Estimator 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}
\mathbf{C}_{G_\text{L1,L2,L5}}
CGL1,L2,L5\mathbf{C}_{G_\text{L1,L2,L5}}
\mathbf{C}_{G_\text{L1,L5}}
CGL1,L5\mathbf{C}_{G_\text{L1,L5}}
\mathbf{C}_{G_\text{L1,L2}}
CGL1,L2\mathbf{C}_{G_\text{L1,L2}}
\mathbf{C}_{G_\text{L2,L5}}
CGL2,L5\mathbf{C}_{G_\text{L2,L5}}

Systematic Error Estimator

Since \(|G| \gg |I_i| \gg |S_i|\), must 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 \(m \ge 3\)

Geometry-Ionosphere-Free Combination

We call the linear combination that applies both geometry-free and ionosphere-free constraints the geometry-ionosphere-free combination (GIFC)

FACT: The difference between any two TEC estimates produces some scaling of the GIFC

FACT: \(\mathbf{C}_\text{GIFC}\) and \(\mathbf{C}_{\text{TEC}_{1,2,3}}\) are perpendicular, i.e.

\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_{\text{TEC}_{1,2,3}} \rangle = 0
⟨CGIFC∣CTEC1,2,3⟩=0\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_{\text{TEC}_{1,2,3}} \rangle = 0

FACT:

\frac{\langle \mathbf{C}_\text{TEC} | \mathbf{C}_{\text{TEC}_{1,2,3}} \rangle}{|| \mathbf{C}_{\text{TEC}_{1,2,3}} ||} = || \mathbf{C}_{\text{TEC}_{1,2,3}} ||
⟨CTEC∣CTEC1,2,3⟩∣∣CTEC1,2,3∣∣=∣∣CTEC1,2,3∣∣\frac{\langle \mathbf{C}_\text{TEC} | \mathbf{C}_{\text{TEC}_{1,2,3}} \rangle}{|| \mathbf{C}_{\text{TEC}_{1,2,3}} ||} = || \mathbf{C}_{\text{TEC}_{1,2,3}} ||

i.e. \(\mathbf{C}_\text{TEC} \) projected onto direction  \(\mathbf{C}_{\text{TEC}_{1,2,3}}\) lands at \(\mathbf{C}_{\text{TEC}_{1,2,3}} \)

\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}

GIFC Triple-Frequency GPS

\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]^T \end{aligned}
CGIFCL1,L2,L5=CTECL1,L5−CTECL1,L2=[−1.756,9.520,−7.764]T\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]^T \end{aligned}

We (arbitrarily) choose:

Note: the triple-frequency GIFC does not have a well-defined unit.

GIFC in our results section have the scaling shown here.

Background and Motivation

Linear Estimation of GNSS Parameters

TEC Estimate Error Residuals

Application to Real GPS Data

Estimate Residual Error

Define the error residual vector \(\mathbf{R}\) with components:

The residual error impacting the TEC estimate is:

R_{\text{TEC}} = \langle \mathbf{C}_{\text{TEC}} | \mathbf{R} \rangle
RTEC=⟨CTEC∣R⟩R_{\text{TEC}} = \langle \mathbf{C}_{\text{TEC}} | \mathbf{R} \rangle
\text{GIFC} = \langle \mathbf{C}_{\text{GIFC}} | \mathbf{R} \rangle
GIFC=⟨CGIFC∣R⟩\text{GIFC} = \langle \mathbf{C}_{\text{GIFC}} | \mathbf{R} \rangle

Note that:

R_i = S_i + \epsilon_i
Ri=Si+ϵiR_i = S_i + \epsilon_i

A Convenient Basis

We transform \(\mathbf{R}\) using the orthonormal basis:

\mathbf{U}_1 = \frac{\mathbf{C}_{\text{TEC}_{1,2,3}}}{||\mathbf{C}_{\text{TEC}_{1,2,3}}||}
U1=CTEC1,2,3∣∣CTEC1,2,3∣∣\mathbf{U}_1 = \frac{\mathbf{C}_{\text{TEC}_{1,2,3}}}{||\mathbf{C}_{\text{TEC}_{1,2,3}}||}
\mathbf{U}_2 = \frac{\mathbf{C}_\text{GIFC}}{||\mathbf{C}_\text{GIFC}||}
U2=CGIFC∣∣CGIFC∣∣\mathbf{U}_2 = \frac{\mathbf{C}_\text{GIFC}}{||\mathbf{C}_\text{GIFC}||}
\mathbf{U}_3 = \mathbf{U}_1 \times \mathbf{U}_2
U3=U1×U2\mathbf{U}_3 = \mathbf{U}_1 \times \mathbf{U}_2
\mathbf{U} = \begin{bmatrix} \mathbf{U}_1 \\ \mathbf{U}_2 \\ \mathbf{U}_3 \end{bmatrix}
U=[U1U2U3]\mathbf{U} = \begin{bmatrix} \mathbf{U}_1 \\ \mathbf{U}_2 \\ \mathbf{U}_3 \end{bmatrix}
\mathbf{R}' = \mathbf{U} \mathbf{R}
R′=UR\mathbf{R}' = \mathbf{U} \mathbf{R}

Note that \(\mathbf{U}_3 \perp \mathbf{C}_\text{TEC} \) since \(\mathbf{U}_1\) and \(\mathbf{U}_2\) span the geometry-free plane

R_i' = \langle \mathbf{U}_i | \mathbf{R} \rangle
Ri′=⟨Ui∣R⟩R_i' = \langle \mathbf{U}_i | \mathbf{R} \rangle
R_1' = \frac{R_{\text{TEC}_{1,2,3}}}{||\mathbf{C}_{\text{TEC}_{1,2,3}}||}
R1′=RTEC1,2,3∣∣CTEC1,2,3∣∣R_1' = \frac{R_{\text{TEC}_{1,2,3}}}{||\mathbf{C}_{\text{TEC}_{1,2,3}}||}
R_2' = \frac{\text{GIFC}}{||\mathbf{C}_\text{GIFC}||}
R2′=GIFC∣∣CGIFC∣∣R_2' = \frac{\text{GIFC}}{||\mathbf{C}_\text{GIFC}||}

Note that:

TEC Estimate Residual Error

common TEC estimate residual error component

GIFC residual error component

\begin{aligned} R_\text{TEC} & = \langle \mathbf{U} \mathbf{C}_\text{TEC} | \mathbf{U} \mathbf{R} \rangle \\[1em] & = \langle \mathbf{U}_1 | \mathbf{C}_\text{TEC} \rangle R_1' + \langle \mathbf{U}_2 | \mathbf{C}_\text{TEC} \rangle R_2' \\[1em] & = R_{\text{TEC}_{1,2,3}} + \frac{\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_\text{TEC} \rangle}{||\mathbf{C}_\text{GIFC}||^2} R_\text{GIFC} \end{aligned}
RTEC=⟨UCTEC∣UR⟩=⟨U1∣CTEC⟩R1′+⟨U2∣CTEC⟩R2′=RTEC1,2,3+⟨CGIFC∣CTEC⟩∣∣CGIFC∣∣2RGIFC\begin{aligned} R_\text{TEC} & = \langle \mathbf{U} \mathbf{C}_\text{TEC} | \mathbf{U} \mathbf{R} \rangle \\[1em] & = \langle \mathbf{U}_1 | \mathbf{C}_\text{TEC} \rangle R_1' + \langle \mathbf{U}_2 | \mathbf{C}_\text{TEC} \rangle R_2' \\[1em] & = R_{\text{TEC}_{1,2,3}} + \frac{\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_\text{TEC} \rangle}{||\mathbf{C}_\text{GIFC}||^2} R_\text{GIFC} \end{aligned}

Express \(R_\text{TEC}\) as residual error components in transformed coordinate system:

\mathbf{U}_1 = \frac{\mathbf{C}_{\text{TEC}_{1,2,3}}}{||\mathbf{C}_{\text{TEC}_{1,2,3}}||}
U1=CTEC1,2,3∣∣CTEC1,2,3∣∣\mathbf{U}_1 = \frac{\mathbf{C}_{\text{TEC}_{1,2,3}}}{||\mathbf{C}_{\text{TEC}_{1,2,3}}||}
\mathbf{U}_2 = \frac{\mathbf{C}_\text{GIFC}}{||\mathbf{C}_\text{GIFC}||}
U2=CGIFC∣∣CGIFC∣∣\mathbf{U}_2 = \frac{\mathbf{C}_\text{GIFC}}{||\mathbf{C}_\text{GIFC}||}
\langle \mathbf{U}_3 | \mathbf{C}_\text{TEC} \rangle = 0
⟨U3∣CTEC⟩=0\langle \mathbf{U}_3 | \mathbf{C}_\text{TEC} \rangle = 0

TEC Estimate Residual Error Discussion

Term \( \frac{\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_\text{TEC} \rangle}{||\mathbf{C}_\text{GIFC}||^2} \) = amplitude of GIFC residual error component in TEC estimate

Term \(R_{\text{TEC}_{1,2,3}}\) = unobservable "TEC-like" residual error component

\(\text{TEC}_{1,2,3}\) is optimal in the sense that it completely removes the GIFC component of residual error

But can we say anything about the overall TEC estimate residual error?

Argument for Using GIFC to Assess Overall Residual Error

Assume \(\mathbf{R}\) has an overall distribution that is joint symmetric about the origin with distribution function \(f_R(x)\)

The distribution of a scaled version \(a \mathbf{R}\) for some scalar \(a\) is \(f_R (\frac{x}{a})\)

By definition, \(\mathbf{U}\mathbf{R} \sim \) symmetric with \(f_R(x) \) for any orthonormal transformation \(\mathbf{U}\)

\(R_i\) equal amplitude and uncorrelated

f_{R_\text{TEC}}(x) = f_{R}\left( \frac{x}{||\mathbf{C}_\text{TEC} ||} \right)
fRTEC(x)=fR(x∣∣CTEC∣∣)f_{R_\text{TEC}}(x) = f_{R}\left( \frac{x}{||\mathbf{C}_\text{TEC} ||} \right)
f_\text{GIFC}(x) = f_{R}\left( \frac{x}{||\mathbf{C}_\text{GIFC} ||} \right)
fGIFC(x)=fR(x∣∣CGIFC∣∣)f_\text{GIFC}(x) = f_{R}\left( \frac{x}{||\mathbf{C}_\text{GIFC} ||} \right)
f_{R_\text{TEC}} (x) = f_\text{GIFC}\left( \frac{||\mathbf{C}_\text{GIFC}||}{||\mathbf{C}_\text{TEC} ||} x \right)
fRTEC(x)=fGIFC(∣∣CGIFC∣∣∣∣CTEC∣∣x)f_{R_\text{TEC}} (x) = f_\text{GIFC}\left( \frac{||\mathbf{C}_\text{GIFC}||}{||\mathbf{C}_\text{TEC} ||} x \right)

Overall TEC Residual Error Discussion

The assumption that \(\mathbf{R}\) has joint symmetric distribution is wrong

We can do better by carefully assessing a priori knowledge about the error components in each \(\Phi_i\)

  • this is a lot of work
  • investigating GIFC is first-step in this process

\(f_{R_\text{TEC}}(x) = f_\text{GIFC} \left(\frac{||\mathbf{C}_\text{GIFC}||}{||\mathbf{C}_\text{TEC}||} x \right) \) is a coarse approximation

  • relates deviations as: \(\text{dev} R_\text{TEC} \approx \frac{||\mathbf{C}_\text{TEC}||}{||\mathbf{C}_\text{GIFC}||} \text{dev} \ \text{GIFC} \)
  • could be very wrong if \(R_{\text{TEC}_{1,2,3}} \gg GIFC \)

Relation Between GIFC and TEC Estimate Residual Errors

\begin{matrix} \text{Estimate} & \frac{\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_\text{TEC} \rangle}{||\mathbf{C}_\text{GIFC}||^2} & \frac{||\mathbf{C}_\text{TEC}||}{||\mathbf{C}_\text{GIFC}||} \\ \text{TEC}_\text{L1,L2,L5} & 0 & 0.831 \\ \text{TEC}_\text{L1,L5} & 0.303 & 0.885 \\ \text{TEC}_\text{L1,L2} & -0.697 & 1.085 \\ \text{TEC}_\text{L2,L5} & 4.723 & 4.796 \end{matrix}
Estimate⟨CGIFC∣CTEC⟩∣∣CGIFC∣∣2∣∣CTEC∣∣∣∣CGIFC∣∣TECL1,L2,L500.831TECL1,L50.3030.885TECL1,L2−0.6971.085TECL2,L54.7234.796\begin{matrix} \text{Estimate} & \frac{\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_\text{TEC} \rangle}{||\mathbf{C}_\text{GIFC}||^2} & \frac{||\mathbf{C}_\text{TEC}||}{||\mathbf{C}_\text{GIFC}||} \\ \text{TEC}_\text{L1,L2,L5} & 0 & 0.831 \\ \text{TEC}_\text{L1,L5} & 0.303 & 0.885 \\ \text{TEC}_\text{L1,L2} & -0.697 & 1.085 \\ \text{TEC}_\text{L2,L5} & 4.723 & 4.796 \end{matrix}

amplitude of GIFC error signal in TEC residual

relates deviation in GIFC and TEC residual

Background and Motivation

Linear Estimation of GNSS Parameters

TEC Estimate Error Residuals

Application to Real GPS Data

Experiment Data

GPS Lab high-rate GNSS data collection network

  • Alaska, Hong Kong, Peru
  • 2013, 2014, 2015, 2016
  • Septentrio PolarXs
  • 1 Hz GPS L1/L2/L5 measurements

Data Alignment

  • align data by sidereal day
    • 23h 56m 4.1s
  • level GIFC by one of two methods:
    • subtract overall GIFC mean
    • subtract GIFC mean near (5 minute window) highest-elevation epoch

GPS orbital period \(\approx\) 1/2 sidereal day

Examples

Method 1

Method 2

Data Jump Correction

  • jumps in carrier phase due to ionosphere activity, multipath, etc.
  • remove jumps in GIFC
    • ​detect large jumps in epoch-to-epoch GIFC difference
    • simultaneously estimate polynomial fit and jump amplitude

Example of significant GIFC jumps for which the method failed

GIFC Examples

Alaska

method 1 levelling

GIFC Examples

Hong Kong

method 1 levelling

GIFC Examples

Peru

method 1 levelling

GIFC Calendar

Alaska

method 2 levelling

GIFC Calendar

Hong Kong

method 2 levelling

GIFC Calendar

Peru

method 2 levelling

GIFC Heatmap

Alaska

method 1 levelling

GIFC Heatmap

Hong Kong

method 1 levelling

GIFC Heatmap

Peru

method 1 levelling

GIFC Deviations and TEC Residual Error Estimates

\begin{matrix} \text{Percentile} & \text{GIFC Deviation} \\ 50 & 0.11 \\ 75 & 0.19 \\ 90 & 0.21 \\ \end{matrix}
PercentileGIFC Deviation500.11750.19900.21\begin{matrix} \text{Percentile} & \text{GIFC Deviation} \\ 50 & 0.11 \\ 75 & 0.19 \\ 90 & 0.21 \\ \end{matrix}
\begin{matrix} \text{Percentile} & \kern{5em} & \text{TEC}_\text{L1,L5} & \text{TEC}_\text{L1,L2} & \text{TEC}_\text{L2,L5} \\ 50 & & 0.033 & 0.077 & 0.520 \\ 75 & & 0.058 & 0.132 & 0.897 \\ 90 & & 0.064 & 0.146 & 0.992 \end{matrix}
PercentileTECL1,L5TECL1,L2TECL2,L5500.0330.0770.520750.0580.1320.897900.0640.1460.992\begin{matrix} \text{Percentile} & \kern{5em} & \text{TEC}_\text{L1,L5} & \text{TEC}_\text{L1,L2} & \text{TEC}_\text{L2,L5} \\ 50 & & 0.033 & 0.077 & 0.520 \\ 75 & & 0.058 & 0.132 & 0.897 \\ 90 & & 0.064 & 0.146 & 0.992 \end{matrix}
\begin{matrix} \text{Percentile} & \text{TEC}_\text{L1,L2,L5} & \text{TEC}_\text{L1,L5} & \text{TEC}_\text{L1,L2} & \text{TEC}_\text{L2,L5} \\ 50 & 0.091 & 0.097 & 0.119 & 0.528 \\ 75 & 0.158 & 0.168 & 0.206 & 0.911 \\ 90 & 0.175 & 0.186 & 0.228 & 1.007 \end{matrix}
PercentileTECL1,L2,L5TECL1,L5TECL1,L2TECL2,L5500.0910.0970.1190.528750.1580.1680.2060.911900.1750.1860.2281.007\begin{matrix} \text{Percentile} & \text{TEC}_\text{L1,L2,L5} & \text{TEC}_\text{L1,L5} & \text{TEC}_\text{L1,L2} & \text{TEC}_\text{L2,L5} \\ 50 & 0.091 & 0.097 & 0.119 & 0.528 \\ 75 & 0.158 & 0.168 & 0.206 & 0.911 \\ 90 & 0.175 & 0.186 & 0.228 & 1.007 \end{matrix}
\frac{\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_\text{TEC} \rangle}{||\mathbf{C}_\text{GIFC}||^2}
⟨CGIFC∣CTEC⟩∣∣CGIFC∣∣2\frac{\langle \mathbf{C}_\text{GIFC} | \mathbf{C}_\text{TEC} \rangle}{||\mathbf{C}_\text{GIFC}||^2}
\frac{||\mathbf{C}_\text{TEC}||}{||\mathbf{C}_\text{GIFC}||}
∣∣CTEC∣∣∣∣CGIFC∣∣\frac{||\mathbf{C}_\text{TEC}||}{||\mathbf{C}_\text{GIFC}||}

GIFC deviation multiplied by scaling factor

Conclusions

  • introduced simplified GNSS carrier phase observation model
  • introduced methodology to choose linear combination coefficients for TEC and geometry estimators
  • derived GIFC for triple-frequency GNSS
  • characterized the GIFC
    • showed long-term calendar analysis
    • showed percentile deviations over all data
  • related deviations to TEC error residual estimates

Acknowledgements

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

References

[1] Saito A., S. Fukao, and S. Miyazaki, High resolution mapping of TEC perturbations with the GSI GPS network over Japan, Geophys. Res. Lett., 25, 3079-3082, 1998.

 

[2] Bourne, Harrison W. An algorithm for accurate ionospheric total electron content and receiver bias estimation using GPS measurements. Diss. Colorado State University. Libraries, 2016.

 

[3] Spits, Justine. Total Electron Content reconstruction using triple frequency GNSS signals. Diss. Université de Liège,​​ Belgique, 2012.