Breitsch Masters-thesis-presentation-corrected.pdf

Linear Combinations of GNSS Phase Observables to Improve and Assess TEC Estimation Precision Brian Breitsch Advisor: Jade Morton Committee: Charles Rino, Anton Betten

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Background and Motivation Linear Estimation of GNSS Parameters TEC Estimate Error Residuals Application to Real GPS Data

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Earth's Ionosphere

3 J. Grobowsky / NASA GSFC

Ionosphere Effects on Electromagnetic Propagation ionosphere = cold, collisionless, magnetized plasma

X=

ωp2 ω2

ω = 2πf

±

1 O( f 3 )

shift / phase ion distort

n=1−

1 2X

here ionosp

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

radio source

higher-order terms on the order of a few cm

ωp = √ Nϵ0eme

2

f = wave frequency

e = fundamental charge

Ne = plasma density

m = electron rest mass

ϵ0 = permittivity of free space

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Global Navigation Satellite Systems (GNSS)

“

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

GPS - Global Positioning System

GPS

GLONASS Beidou Galileio ...etc.

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

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GNSS Carrier Phase Observable accumulated phase (in meters) of demodulated GNSS signal at receiver for a particular satellite and signal carrier frequency fi FREQUENCY INDEPENDENT EFFECTS

IONOSPHERE RANGE ERROR

SYSTEMATIC ERRORS

Φi = r + cΔt + T + Ii + λi Ni + Hi + Si + ϵi CARRIER AMBIGUITY

HARDWARE BIAS

STOCHASTIC ERRORS

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Ionosphere Range Error consider first-order term in ionosphere refractive index

n≈1−

1 2X

Ii = ∫

tx

rx

=1−

κ N fi2 e

κ=

e2 8π 2 ϵ0 me

≈ 40.308

tx κ (n − 1) ds ≈ − 2 ∫ Ne ds fi rx TOTAL ELECTRON CONTENT

rx

tx plasma /

units:

electrons m2

often measured in TEC units:

1TECu = 1016 electrons m2

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vTEC

Ionosphere Plasma Density TE

C

v dis erti tri cal bu tio n

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

profile from CDAAC

tr ion avel dis osp ling tu rb here (TI anc Ds es ) ho dis rizo tri nta bu tio l n

map from IGS

image from Saito et al.

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TEC Estimation Using DualFrequency GNSS neglecting systematic and stochastic error terms:

Φ1 − Φ2 = (I1 − I2 ) + (λ1 N1 − λ2 N2 ) + (H1 − H2 ) ≈ −κ ( f12 −

1 ) TEC f22

1

+ (λ1 N1 − λ2 N2 ) + ΔH1,2 carrier ambiguities

after resolving bias terms:

TEC =

Φ2 − Φ1 κ ( f12 − 1

1 ) f22

satellite and receiver interfrequency hardware biases

bias terms

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Resolving Bias Terms carrier ambiguity resolution LAMBDA code-carrier-levelling [3] derives improved code-carrier leveling / ambiguity resolution using triple-frequency GNSS

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 (such as work by [2])

Example of L1/L2 TEC before and after codecarrier-levelling / ambiguity estimation, for satellite G01 and receiver at Poker Flat, Alaska.

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Examples of Dual-Frequency TEC Estimates Using methods similar to [2] and [3] to solve for bias terms, we compute dual-frequency TEC estimate TECL1,L2 and TECL1,L5

Poker Flat, Alaska, 2016-01-02

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TECL1,L5 − TECL1,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?

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Systematic Errors in GNSS Observations hardware bias drifts

Hi terms not constant

multipath reflected signals interfere with primary signal at receiver → causes fluctuations in phase / signal amplitude

ray-path bending r ≠ line-of-sight range

antenna phase effects relative displacement of satellite antenna phase centers changes as satellite moves / rotates

higher-order ionosphere terms need to consider orientation / strength of geomagnetic field

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Objectives Derive optimal triple-frequency estimation of TEC Investigate the discrepancy in TECL1,L5 − TECL1,L2 Provide a (partial) characterization of TEC estimate residual errors 14

Motivation Improve / understand TEC estimate precision

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 15

Approach Develop framework for linear estimation of GNSS parameters Apply framework to derive triple-frequency estimates of TEC and systematic errors Relate to impact on TEC estimate error residuals 16

Background and Motivation Linear Estimation of GNSS Parameters TEC Estimate Error Residuals Application to Real GPS Data

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Simplified Carrier Phase Model neglect bias terms By neglecting bias terms, we address estimation precision, rather than accuracy

Φi = r + cΔt + T + Ii + λi Ni + Hi + Si + ϵi "geometry" term

Φi = G + Ii + Si + ϵi zeromean

zero-mean normallydistributed

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Linear Inverse Problem Φ = Am + ϵ Φ = [Φ1 , ⋯ , Φm ]T

m = [G, TEC, S1 , ⋯ , Sm ]T

observations

⎡1 ⎢1 A=⎢ ⎢⋮ ⎣1

− fκ2 1 κ − f2 2

− fκ2

m

model parameters

1 0 0

0 1 ⋯

forward model

⋯ ⋯ ⋱

0⎤ 0⎥ ⎥ ⎥ 1⎦

ϵ = [ϵ1 , ⋯ , ϵm ]T stochastic error

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Linear Estimation ∗ ^ ≈ A Φ m

^ m

A∗ = ?

model estimate

model estimator

AT

−1 T (AA )

Poor results; treats each parameter with equal weight

We must apply a priori information about model parameters 20

A Priori Information Under normal conditions, we know that:

∣G∣ ≫ ∣Ii ∣ ≫ ∣Si ∣ G∼

20,000 km

I∼

1 - 150 m

S∼

several cm 21

Using A Priori Information We could apply ∣G∣ ≫ ∣Ii ∣ ≫ ∣Si ∣ using Gaussian priors Instead we derive each row separately:

C = [c1 , ⋯ , cm ]T ∈ Rm estimator

(written as row vectors here)

⎡ CG ⎤ CTEC ⎥ ⎢ ⎥ A∗ = ⎢ C S 1 ⎢ ⎥ ⎢ ⋮ ⎥ ⎣ CS ⎦ m

geometry estimator TECu estimator

systematic-error estimators

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How to Choose Optimal C Linear combination E given by inner-product:

E = ⟨C∣Φ⟩ Goals: 1. produce desired parameter with unity coefficient 2. remove / reduce all other terms Approach: First, constrain C to satisfy Goal 1 Then, constrain / optimize C to achieve Goal 2 23

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

Φi = G + Ii + Si + ϵi

∑i ci = 0

∑i ci = 1

geometry-free

geometry-estimator

ci ∑i f 2 i

=0

ionosphere-free

κ ∑i − f 2 ci i

=1

TEC-estimator

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Reduction of Error Linear combination stochastic error variance:

σϵ2 = CT Σϵ C where Σϵ is the covariance matrix between ϵi Optimal C for minimizing stochastic error variance:

C∗ = arg min CT Σϵ C C

ϵi equal-amplitude and uncorrelated

C∗ = arg min ∑ c2i C

i

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TEC Estimator 1. apply TECu-estimator constraint 2. apply geometry-free constraint (since ∣G∣ ≫ ∣Ii ∣ ) Dual-Frequency Example TEC-estimator geometry-free

−

κ c f12 1

−

κ c f22 2

=1

c1 + c2 = 0 ⇒ c1 = −c2 ⇒ − κc1 ( f12 − 1

⇒ c1 = −

1 ) f22

=1 TEC =

1 κ ( f12 1

−

recall:

1 ) f22

Φ2 − Φ1 κ ( f12 − 1

1 f22 )

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Triple-Frequency TEC Estimator Applying constraints yields following system of coefficients (with free parameter denoted x:

c1 = c2 =

( f12 − f12 )

1 κ +x

3

1 1 − f22 f12

− κ1 −x(

c3 = x

2

1 1 − f32 f12

1 − 1 f22 f12

)

To satisfy C∗ = arg min ∑ c2i , choose C

x∗ =

1 κ

(

1 − 1 f12 f22

2

i

( f22 − f12 − f12 )

) +(

3

2

1 − 1 f22 f32

1 2

) +(

1 − 1 f32 f12

2

)

denote corresponding coefficient vector CTEC1,2,3 and its corresponding estimate TEC1,2,3 27

TEC Estimator Using TripleFrequency GPS

CTECL1,L2,L5 CTECL1,L5 CTECL1,L2 CTECL2,L5

Estimate

c1

c2

c3

∑i c2i

TECL1,L2,L5 TECL1,L5 TECL1,L2 TECL2,L5

8.294 7.762 9.518 0

−2.883 0 −9.518 42.080

−5.411 −7.762 0 −42.080

10.314 10.977 13.460 59.510

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Geometry Estimator 1. apply geometry-estimator constraint 2. apply ionosphere-free constraint since Ii are the next-largest terms For triple-frequency GNSS:

c1 = c2 = c3 =

−

(

)

(

)

1 +x 1 − 1 f22 f22 f32 1 − 1 f12 f22

1 −x 12 − 12 2 f1 f1 f3 1 − 1 f12 f22

x

To satisfy C∗ = arg min ∑ c2i , C

x∗ =

1 κ 2

i

( f22 − f12 − f12 ) 3

2

1 2

2

( f12 − f12 ) +( f12 − f12 ) +( f12 − f12 ) 1

2

2

3

3

1

We call this coefficient vector CG1,2,3 and its corresponding estimate G1,2,3 the optimal "ionosphere-free combination"

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Geometry Estimator Using Triple-Frequency GPS

CGL1,L2,L5 CGL1,L5 CGL1,L2 CGL2,L5

Estimate

c1

c2

c3

∑i c2i

GL1,L2,L5 GL1,L5 GL1,L2 GL2,L5

2.327 2.261 2.546 0

−0.360 0 −1.546 12.255

−0.967 −1.261 0 −11.255

2.546 2.588 2.978 16.639

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Systematic Error Estimator Since ∣G∣ ≫ ∣Ii ∣ ≫ ∣Si ∣, must apply both geometryfree and ionosphere-free constraints note this requires m ≥ 3

For triple-frequency GNSS:

c1 = x

1 − 1 f32 f22 1 1 − f22 f12

c2 = −x c3 = x

1 − 1 f32 f12 1 − 1 f22 f12

system is linear subspace 31

Geometry-Ionosphere-Free Combination We call the linear combination that applies both geometry-free and ionospherefree constraints the geometry-ionosphere-free combination (GIFC)

FACT: The difference between any two TEC estimates produces some scaling of the GIFC FACT: CGIFC and CTEC1,2,3 are perpendicular, i.e.

CTEC1,2,3 CGIFC

⟨CGIFC ∣CTEC1,2,3 ⟩ = 0 FACT: ⟨CTEC ∣CTEC1,2,3 ⟩ ∣∣CTEC1,2,3 ∣∣

= ∣∣CTEC1,2,3 ∣∣

i.e. CTEC projected onto direction CTEC1,2,3 lands at CTEC1,2,3

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GIFC Triple-Frequency GPS We (arbitrarily) choose:

CGIFCL1,L2,L5 = CTECL1,L5 − CTECL1,L2 = [−1.756, 9.520, −7.764]T

Note: the triple-frequency GIFC does not have a welldefined unit. GIFC in our results section have the scaling shown here. 33

Background and Motivation Linear Estimation of GNSS Parameters TEC Estimate Error Residuals Application to Real GPS Data

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Estimate Residual Error Define the error residual vector R with components:

Ri = Si + ϵi The residual error impacting the TEC estimate is:

RTEC = ⟨CTEC ∣R⟩ Note that:

GIFC = ⟨CGIFC ∣R⟩ 35

A Convenient Basis Note that U3 ⊥ CTEC since

We transform R using the orthonormal basis:

U1 =

CTEC1,2,3 ∣∣CTEC1,2,3 ∣∣

U2 =

CGIFC ∣∣CGIFC ∣∣

U1 and U2 span the geometry-free plane

U3 = U1 × U2 ⎡U1 ⎤ U = U2 ⎣U3 ⎦

Note that:

R = UR

R1′

Ri′ = ⟨Ui ∣R⟩

R2′ =

′

=

RTEC1,2,3 ∣∣CTEC1,2,3 ∣∣ GIFC ∣∣CGIFC ∣∣

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TEC Estimate Residual Error Express RTEC as residual error components in transformed coordinate system:

U1 =

CTEC1,2,3 ∣∣CTEC1,2,3 ∣∣

U2 =

CGIFC ∣∣CGIFC ∣∣

⟨U3 ∣CTEC ⟩ = 0

RTEC = ⟨UCTEC ∣UR⟩ = ⟨U1 ∣CTEC ⟩R1′ + ⟨U2 ∣CTEC ⟩R2′ = RTEC1,2,3 +

⟨CGIFC ∣CTEC ⟩ ∣∣CGIFC ∣∣2 RGIFC

common TEC estimate residual error component

GIFC residual error component

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TEC Estimate Residual Error Discussion Term

⟨CGIFC ∣CTEC ⟩ ∣∣CGIFC ∣∣2

= amplitude of GIFC residual error

component in TEC estimate TEC1,2,3 is optimal in the sense that it completely removes the GIFC component of residual error Term RTEC1,2,3 = unobservable "TEC-like" residual error component But can we say anything about the overall TEC estimate residual error?

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Argument for Using GIFC to Assess Overall Residual Error Assume R has an overall distribution that is joint symmetric about the origin with distribution function fR (x)

Ri equal amplitude and uncorrelated

By definition, UR ∼ symmetric with fR (x) for any orthonormal transformation U The distribution of a scaled version aR for some scalar a is fR ( xa ) x fGIFC (x) = fR ( ∣∣CGIFC ∣∣ ) x fRTEC (x) = fR ( ∣∣CTEC ∣∣ )

GIFC ∣∣ fRTEC (x) = fGIFC ( ∣∣C ∣∣CTEC ∣∣ x)

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Overall TEC Residual Error Discussion The assumption that R has joint symmetric distribution is wrong We can do better by carefully assessing a priori knowledge about the error components in each Φi investigating GIFC is first-step in this process

GIFC ∣∣ fRTEC (x) = fGIFC ( ∣∣C ∣∣CTEC ∣∣ x) is a coarse approximation

relates deviations as: devRTEC ≈

∣∣CTEC ∣∣ ∣∣CGIFC ∣∣ dev

could be very wrong if RTEC1,2,3 ≫ GIF C

GIFC 40

Relation Between GIFC and TEC Estimate Residual Errors amplitude of GIFC error signal in TEC residual

Estimate TECL1,L2,L5 TECL1,L5 TECL1,L2 TECL2,L5

⟨CGIFC ∣CTEC ⟩ ∣∣CGIFC ∣∣2

0 0.303 −0.697 4.723

relates deviation in GIFC and TEC residual

∣∣CTEC ∣∣ ∣∣CGIFC ∣∣

0.831 0.885 1.085 4.796

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Background and Motivation Linear Estimation of GNSS Parameters TEC Estimate Error Residuals Application to Real GPS Data

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

GPS Lab high-rate GNSS data collection network

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Data Alignment and Correction align data by sidereal day = 23h 55m 54.2 s

GPS orbital period ≈ 1/2 sidereal day

must remove jumps in GIFC data due to ionosphere activity / multipath / interference

Outlier segments (∣GIFC∣ > 2) are removed from analysis

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GIFC Examples Alaska G01

G25

G24

G27

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GIFC Examples Hong Kong G01

G25

G24

G27

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GIFC Examples Peru G01

G25

G24

G27

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GIFC Calendar Alaska

G01

G25

G24

G27

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GIFC Calendar Hong Kong G01

G25

G24

G27

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GIFC Calendar Peru

G01

G25

G24

G27

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Satellite Antenna Phase Effects? angle cosine between Earth center, satellite, and Sun

antenna phase effects relative displacement of satellite antenna phase centers changes as satellite moves / rotates

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GIFC Heatmap Alaska

G01

G25

G24

G27

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GIFC Heatmap Hong Kong

G01

G25

G24

G27

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GIFC Heatmap Peru

G01

G25

G24

G27

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GIFC Deviations and TEC Residual Error Estimates GIFC percentile deviations computed over aggregate of all data

Percentile 50 75 90

Overall 0.11 0.19 0.21

GIFC deviation multiplied by scaling factor

∣∣CTEC ∣∣ ∣∣CGIFC ∣∣

Percentile 50 75 90

TECL1,L2,L5 0.091 0.158 0.175

TECL1,L5 0.033 0.058 0.064

TECL1,L2 0.077 0.132 0.146

TECL2,L5 0.520 0.897 0.992

TECL1,L5 0.097 0.168 0.186

TECL1,L2 0.119 0.206 0.228

TECL2,L5 0.528 0.911 1.007

[TECu]

Percentile 50 ⟨CGIFC ∣CTEC ⟩ 75 ∣∣CGIFC ∣∣2 90

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Recap simple phase observation model (ignore biases) methodology for choosing optimal linear estimators triple-frequency optima

l?

TEC1,2,3

GIFC characterize / relate to RTEC 56

Discussion TECL1,L2 residual error on order of 0.2 TECu includes large-scale trend → for TID detection, trend is removed and precision improves [4] cites 0.05 TECu fluctuations to be above noise for TID detection Improvement of TECL1,L2,L5 over TECL1,L5 seems minor:

∣∣CTECL1,L2,L5 ∣∣ = 10.314 ∣∣CTECL1,L5 ∣∣ = 10.977 ∣∣CTECL1,L2 ∣∣ = 13.460

...but it does eliminate GIFC component in TEC residual error

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Next Steps Use characterization of GIFC to address residual errors Is the GIFC trend variation due to satellite antenna phase effects? Can we obtain and apply better information on residual error components Ri ? Can we use GIFC to validate mitigation techniques for multipath, higher-order ionosphere terms, ray-path bending, antenna phase effects? → enable TEC estimation from low-elevation satellites

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

Thank you to my advisor, committee members, and all who provided me with feedback and criticism! 59

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. [4] M. Nishioka, A. Saito, and T. Tsugawa, “Occurrence characteristics of plasma bubble derived from global ground-based GPS receiver networks,” Journal of Geophysical

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