Linear Combinations of GNSS Phase Observables to Improve and Assess - PowerPoint PPT Presentation

Linear Combinations of GNSS Phase Observables to Improve and Assess TEC Estimation Precision Brian Breitsch Advisor: Jade Morton Committee: Charles Rino, Anton Betten 1

Background and Motivation Linear Estimation of GNSS Parameters TEC Estimate Error Residuals Application to Real GPS Data 2

Earth's Ionosphere J. Grobowsky / NASA GSFC 3

Ionosphere Effects on Electromagnetic Propagation ionosphere = cold, collisionless, magnetized plasma phase shift / ionosphere distortion radio for L-band frequencies (1-2 GHz) source refractive index given by: 1 1 n = 1 − X ± O ( ) f 3 higher-order terms on 2 the order of a few cm √ ϵ m 2 ω p e 2 X = ω = 2 π f ω = N e p ω 2 0 f = wave frequency e = fundamental charge N = plasma density m = electron rest mass e ϵ = permittivity of free space 4 0

Global Navigation Satellite Systems (GNSS) GPS “ ...a useful everyday radio source for GLONASS geophysical remote-sensing! Beidou Galileio GPS - Global Positioning System ...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 5 L5 1.17645

GNSS Carrier Phase Observable accumulated phase (in meters) of demodulated GNSS signal at receiver for a particular satellite and signal carrier frequency f i FREQUENCY IONOSPHERE INDEPENDENT RANGE SYSTEMATIC EFFECTS ERROR ERRORS Φ = r + c Δ t + T + I + λ N + H + S + ϵ i i i i i i i CARRIER HARDWARE STOCHASTIC AMBIGUITY BIAS ERRORS 6

Ionosphere Range Error consider fi rst-order term in ionosphere refractive index e 2 1 κ n ≈ 1 − X = 1 − κ = ≈ 40.308 N e 2 2 8 π ϵ m 2 0 f i e κ ∫ tx tx ∫ I = ( n − 1 ds ≈ − ) N ds i e 2 f i rx rx TOTAL ELECTRON CONTENT tx electrons units: m 2 rx often measured in TEC units: plasma / electrons 1TECu = 10 16 7 m 2

Ionosphere Plasma vertical distribution Density TEC and vertical TEC (vTEC) used to image plasma density vTEC structures TEC pro fi le from CDAAC travelling ionosphere disturbances (TIDs) horizontal distribution 8 image from Saito et al. map from IGS

TEC Estimation Using Dual- Frequency GNSS neglecting systematic and stochastic error terms: Φ − Φ = I − I ( 1 2 ) + λ N − λ N ( 2 ) + H − H ( 2 ) 1 2 1 1 2 1 1 ) ( f 1 1 ≈ − κ − TEC + λ N − λ N ( 2 ) + Δ H 1 1 2 1,2 2 2 f 2 carrier satellite and ambiguities receiver inter- frequency hardware biases after resolving bias terms: Φ − Φ 2 1 bias terms TEC = 1 ) ( f 1 1 − κ 2 2 f 2 9

Resolving Bias Terms carrier ambiguity resolution hardware bias estimation LAMBDA must apply ionosphere model code-carrier-levelling ​ e.g. global ionosphere model using data assimilation and [3] derives improved code-carrier receiver networks leveling / ambiguity resolution e.g. single receiver and linear using triple-frequency GNSS 2D-gradient in vTEC (such as work by [2]) Example of L1/L2 TEC before and after code- carrier-levelling / ambiguity estimation, for satellite G01 and receiver at Poker Flat, Alaska. 10

Examples of Dual-Frequency TEC Estimates Using methods similar to [2] and [3] to solve for bias terms, we compute dual-frequency TEC estimate TEC and TEC L1,L2 L1,L5 Poker Flat, Alaska, 2016-01-02 11

TEC − TEC L1,L5 L1,L2 Poker Flat, Alaska, 2016-01-02 Can we characterize / fi nd the source of these discrepancies? Can we relate them to errors in dual-frequency TEC estimates? 12

Systematic Errors in GNSS Observations hardware bias drifts H terms not constant i multipath re fl ected signals interfere with primary signal at receiver → antenna phase e ff ects causes fl uctuations in phase / relative displacement of signal amplitude satellite antenna phase centers changes as satellite moves / rotates higher-order ray-path bending ionosphere terms need to consider orientation / strength r ≠ line-of-sight range of geomagnetic fi eld 13

Objectives Derive optimal triple-frequency estimation of TEC Investigate the discrepancy in TEC − TEC L1,L5 L1,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 17

Simplified Carrier Phase Model neglect bias terms By neglecting bias terms, we address estimation precision , rather than accuracy Φ = r + c Δ t + T + I + λ N + H + S + ϵ i i i i i i i "geometry" term zero-mean Φ = G + I + S + ϵ normally- i i i i distributed zero- mean 18

Linear Inverse Problem Φ = Am + ϵ Φ = Φ , ⋯ , Φ m = G , TEC, S , ⋯ , S [ m ] T [ m ] T 1 1 observations model parameters ⎡ 1 ⎤ κ − f 1 1 0 ⋯ 0 2 ⎢ ⎥ ⎢ ⎥ 1 − f 2 0 1 ⋯ 0 κ ϵ = [ ϵ , ⋯ , ϵ ] m T ⎢ ⎥ 2 1 A = ⋮ ⋱ ⎣ 1 ⎦ stochastic error 1 − f m κ 0 ⋯ 2 19 forward model

Linear Estimation ∗ ≈ A Φ m ^ ∗ A = ? m ^ model estimate model estimator Poor results; treats T ( T ) − 1 A AA 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 ∣ ≫ ∣ I ∣ ≫ ∣ S ∣ i i G ∼ 20,000 km I ∼ 1 - 150 m S ∼ several cm 21

Using A Priori Information We could apply ∣ G ∣ ≫ ∣ I ∣ ≫ ∣ S ∣ using Gaussian priors i i Instead we derive each row separately: ∈ R C = c , ⋯ , c [ 1 m ] T m estimator (written as row vectors here) ⎡ C G ⎤ geometry estimator ⎢ ⎥ C TEC ⎢ ⎥ TECu estimator ⎢ ⎥ ∗ A = C S 1 ⎢ ⎥ ⋮ ⎣ C S m ⎦ systematic-error estimators 22

How to Choose Optimal C Linear combination E given by inner-product: E = ⟨ C ∣ Φ ⟩ Goals : 1 . produce desired parameter with unity coe ffi cient 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 Φ = G + I + S + ϵ i i i i optimal estimator coe ffi cients: c = 0 ∑ i c = 1 ∑ i i i geometry-free geometry-estimator κ c i = 0 − c = 1 ∑ i ∑ i f i i 2 2 f i ionosphere-free TEC-estimator 24

Reduction of Error Linear combination stochastic error variance: 2 σ = C Σ C T ϵ ϵ where Σ is the covariance matrix between ϵ ϵ i Optimal C for minimizing stochastic error variance: ∗ C = arg C Σ C T min ϵ C ϵ equal-amplitude and uncorrelated i ∑ 2 ∗ C = arg min c i C 25 i

TEC Estimator 1. apply TECu-estimator constraint 2. apply geometry-free constraint (since ∣ G ∣ ≫ ∣ I ∣ ) i Dual-Frequency Example κ κ − c − c = 1 TEC-estimator 1 2 2 2 f 1 f 2 geometry-free c + c = 0 ⇒ c = − c 1 2 1 2 1 ) 1 ( f 1 1 ⇒ − κ c − = 1 2 2 f 2 recall: Φ − Φ 2 1 TEC = 1 1 ) ( f 1 1 ⇒ c = − − κ 1 ) 2 2 1 f 2 ( f 1 1 − κ 2 2 f 2 26

Triple-Frequency TEC Estimator Applying constraints yields following system of coe ffi cients (with free parameter denoted x : 1 ) ( f 3 1 1 + x − ∑ 2 To satisfy C = arg min c , choose 2 2 κ ∗ f 2 = c 1 i C 1 1 − i 2 2 1 ) f 2 f 1 1 ( f 3 1 ) 2 1 − − ( f 3 2 2 2 κ ∗ f 2 f 1 x = 1 1 − − x − 1 ) 1 ) 1 ) 2 2 2 2 2 ( f 1 ( f 2 ( f 3 κ f 1 = 1 1 1 c 2 − + − + − 2 2 2 2 2 2 1 1 − f 2 f 3 f 1 2 2 f 2 f 1 = x c 3 denote corresponding coe ffi cient vector and its corresponding estimate TEC C 27 TEC 1,2,3 1,2,3

TEC Estimator Using Triple- Frequency GPS C TEC L1,L2,L5 C TEC L1,L5 2 Estimate ∑ i c 1 c 2 c 3 c i TEC L1,L2,L5 8.294 − 2.883 − 5.411 10.314 C TEC L1,L2 TEC L1,L5 7.762 0 − 7.762 10.977 TEC L1,L2 9.518 − 9.518 0 13.460 C TEC L2,L5 TEC L2,L5 0 42.080 − 42.080 59.510 28

Geometry Estimator 1. apply geometry-estimator constraint 2. apply ionosphere-free constraint since I are the i next-largest terms ∑ 2 For triple-frequency GNSS: To satisfy C = arg min c , ∗ i C 1 ) 1 ) i ( f 2 1 ( f 3 1 1 2 1 − + x − − − 2 2 2 2 2 2 κ ∗ f 2 f 3 f 2 f 1 c = x = 1 ) 1 ) 1 ) 1 1 1 2 2 2 ( f 1 ( f 2 ( f 3 − 2 2 1 1 1 − + − + − f 1 f 2 1 ) 2 2 2 2 2 2 ( f 1 f 2 f 3 f 1 1 1 − x − 2 2 2 f 1 f 3 c = 2 We call this coe ffi cient vector C 1 1 − 2 2 G 1,2,3 f 1 f 2 and its corresponding estimate G c = x 1,2,3 3 the optimal "ionosphere-free combination" 29