Ionospheric Data Processing and Analysis

Transcription

1 Ionospheric Data Processing and Analysis Dr. Charles Carrano 1 Dr. Keith Groves 2 1 Boston College, Institute for Scientific Research 2 Air Force Research Laboratory, Space Vehicles Directorate Workshop on Satellite Navigation Science and Technology for Africa The Abdus Salam ICTP, Trieste, Italy March 23 - April 9, 2009

2 Outline This talk is a tutorial on Total Electron Content (TEC) estimation using a GPS receiver Describe the GPS observables and the various linear combinations used to estimate TEC Demonstrate estimation and removal of the instrumental biases using several techniques A simple technique (setting the minimum value of TEC at night) A least squares approach (minimizing the variance of the vertical-equivalent TEC) Kalman filter estimation of total TEC Discuss the influence of ionospheric structure on the GPS TEC Discuss the influence of the plasmasphere on the GPS TEC A model for the plasmaspheric contribution to the total TEC Kalman filter estimation of ionospheric and plasmaspheric TEC Concluding remarks

3 The GPS Observables and TEC

4 The Pseudorange Observation Equations The Pseudorange Observation Equations: ( r s) P= ρ + c Δt Δ t + I + T+ b + b + m + ε P P P P 1 1 1r 1s 1 1 ( r s) P = ρ + c Δt Δ t + I + T+ b + b + m + ε P P P P 2 2 2r 2s 2 2 For each signal broadcast: 1 L1 ( x 10 6 Hz) 2 L2 ( x 10 6 Hz) Symbols: P Pseudorange (m) ρ Geometric range (m) I Ionospheric delay (m) T Tropospheric delay (m) b Instrumental bias for receiver and satellite Δt r Receiver clock error (s) Δt s Satellite clock error (s) m P Multipath (m) ε P thermal noise (m) 1 L1 ( x 10 6 Hz) 2 L2 ( x 10 6 Hz) Forming the difference P2-P1 and neglecting multipath and thermal noise gives: P P P P ( r r) + ( s s) P P= I I + b b b b = I I + b + b 2 1 P r P s (The geometric range, clock error, & tropospheric delay cancel)

5 The Carrier-Phase Observation Equations The Carrier-Phase Observation Equations: ( r s) Φ= ρ + c Δt Δ t + I + T+ b + b + λn + m + ε Φ Φ Φ Φ 1 1 1r 1s ( r s) Φ= ρ + c Δt Δ t + I + T+ b + b + λn + m + ε Φ Φ Φ Φ 2 2 2r 2s For each signal broadcast: 1 L1 ( x 10 6 Hz) 2 L2 ( x 10 6 Hz) Symbols: Φ Carrier-phase (m) ρ Geometric range (m) I Ionospheric delay (m) T Tropospheric delay (m) b Instrumental bias for receiver and satellite Δt r Receiver clock error (s) Δt s Satellite clock error (s) λ Wavelength (m) N Phase cycle-ambiguity m Φ Multipath (m) ε Φ thermal noise (m) Forming the difference P2-P1 and neglecting multipath and thermal noise gives: Φ Φ Φ Φ ( r r) + ( s s) ( λ λ ) Φ Φ r s ( λ λ ) Φ Φ = I I + b b b b + N N = I I + b + b + N N (The geometric range, clock error, & tropospheric delay cancel)

6 Derivation of the Pseudorange TEC Observable Ionospheric delay: I = f TEC f 2 f Signal frequency (Hz) I f Ionospheric delay (m) TEC total electron content (e - /m 2 ) Substituting ionospheric delay into the pseudorange observation equation gives: P P I I + b + b 1 1 TEC + b + b 1= 1 = f f P P P P r s r s Solving this for the TEC yields: TEC f f 1 2 = 2 2 = f f 1 2 P P ( 2 1) ( ) P P P P b + b ( P2 P1) ( b + b ) r s r s We define the pseudorange TEC without the bias terms in units of TECU: ( 2 1) TEC 9.52 P P where P 1 TECU = e - /m 2 unambiguous but noisy and therefore an imprecise observable

7 Derivation of the Carrier-Phase TEC Observable Ionospheric phase advance: I = f TEC f 2 f Signal frequency (Hz) I f Ionospheric delay (m) TEC total electron content (e - /m 2 ) Substituting phase advance into the carrier-phase observation equation gives: I I Φ Φ = TEC + b b N N r + s + Solving this for the TEC yields: TEC f f 1 2 = 2 2 Φ Φ r s f f 1 2 Φ Φ ( 1 2) ( b + b ) ( λ1n1 λ2n2 ) Φ Φ ( 1 2) ( b + b ) ( λ1n1 λ2n2 ) = Φ Φ We define the carrier phase TEC in units of TECU: f 1 r s ( λ λ ) Φ Φ f ( 1 2) TEC Φ 9.52 Φ Φ where 1 TECU = e - /m 2 Precise but ambiguous observable. Biases and ambiguities will be estimated using pseudoranges

8 Behavior of the Pseudorange and Carrier-Phase Measures of TEC Ancon, Peru ε < 20º ε 20º ε < 20º TECU TEC P Spread due to multipath & thermal noise TEC Φ cycle slip

9 Measurement of Cycle Slips Using Least Squares Approximate TEC Φ as a superposition of a quadratic polynomial and a Heaviside step function at the slip location Basis vectors: V 1 (t)=1 V 2 (t)=t V t = TEC j ij j V 3 (t)=t 2 V 4 (t)=h ts (t) Find solution to this over-determined system via least squares: T T ( V V) T = V Φ j Φ A cycle slip in TEC Φ TEC Φ (dots) Least-squares fit (open circles) Time Solve via singular-value decomposition Advantage over predictor methods uses data on both sides of slip, can Coefficient of the Heaviside step de-weight (or omit) data immediately function (V 4 ) is the size of the slip surrounding the slip, if noisy TEC Φ Slip not corrected if χ 2 > 1 TECU

10 Carrier-Phase TEC Corrected for Cycle-Slips Ancon, Peru ε < 20º ε 20º ε < 20º TECU TEC P TEC Φ post-slip gap (very common)

11 Leveling the Carrier-Phase TEC to the Pseudorange TEC Relative TEC (leveled phases): TEC R = TEC Φ + TEC P TEC Φ arc Offset Take difference between pseudorange TEC and phase TEC x i = TEC i P TEC i Φ Weighted average of x gives the offset: Offset = w x i w i i i i Summation is taken over all samples (i) in the same phase connected arc with ε i > 20º Weighted standard deviation, σ, provides estimate of the leveling error: 2 σ = [ ] [ ] w x w w x [ ] w w i i i i 2 i i 2 i i i i 2 i i 2 Weighting chosen is usually the sine of the satellite elevation, ε: w i = sin ( ε ) i Quality control: entire phase connected arc is discarded if σ > 5 TECU

12 Relative TEC and the Estimated Phase Leveling Error Ancon, Peru ε < 20º ε 20º ε < 20º TECU TEC P σ=2.0 TEC R = TEC Φ + <TEC P - TEC Φ > arc Offset=52.8 Only samples with ε 20º used. Arcs with σ 5 discarded

13 The Calibrated (Unbiased) Slant TEC Once the instrumental biases are known, they can be subtracted from relative TEC measurements to give the calibrated (unbiased) slant TEC b S =12.9, b R =33.8 Ancon, Peru ε < 20º ε 20º ε < 20º TECU TEC P -b S -b R TEC S TEC R -b S -b R

14 Estimation and Removal of the GPS Instrumental Biases

15 Estimation of GPS Instrumental Biases from the Measurements To estimate the instrumental biases from the measurements themselves, we must make assumptions about the (real) TEC we are trying to measure: TEC must be non-negative - and - The structure of the TEC is assumed to satisfy one or more of these We assume a value for the TEC attained at night Spatial gradients in TEC assumed negligible (at night) TEC well approximated by a polynomial of order N (higher order derivatives assumed negligible) TEC in the ionosphere well approximated by a polynomial and TEC in the plasmasphere by a model (generally structured according to dipole field-lines) These techniques work by exploiting the fact that slant TEC depends on elevation (since the path length through ionized region is longer) while the biases do not

16 A Useful Tool for Bias Estimation and Visualization: Computing the Vertical-Equivalent TEC Ionospheric Shell IPP RX η ε h TX The standard geometric mapping function, M, is the projection of slant distance onto zenith distance at the IPP: M ( ) s / d = sec η s η d R e R e The zenith angle at the IPP, η, can be expressed in terms of shell height, h, and Earth radius R: ( R + h) sin( η) = R sin( 90 + ε ) = R cos( ε ) e e e Application of mapping function to slant TEC gives the vertical-equivalent TEC: TEC V = TEC s M ( ε ) This gives the mapping function in terms of the satellite elevation, ε: M 1 Re ( ε ) = sec sin cos( ε ) R e + h

17 Calibrated Slant and Vertical-Equivalent TEC (All Satellites) Characteristics of well-calibrated TEC: TEC is non-negative Magnetic local time at the IPP Curves colored by magnetic latitude There are noteworthy Exceptions to this rule! TEC curves collapse well (especially at night and during post-sunrise ramp up)

18 Removing the Correct GPS Satellite Biases The technique used by the receiver to measure TEC P dictates the type of satellite instrumental biases that must be removed. Receiver Model Method used to measure the DPR Type of Satellite Bias to Remove Ashtech Z-12 L2(P2) - L1(P1) P1P2 bias Ashtech µz-cgrs L2(P2) - L1(P1) P1P2 bias NovAtel GSV 4004B L2(P2) - L1(CA) P1P2 bias minus the P1C1 bias Files containing monthly estimates for the P1P2 and P1C1 biases can be downloaded from These satellite differential codes bias are not absolute timing biases, instead they average to zero. The unknown offset is immaterial in that it will be lumped together and removed along with the receiver bias.

19 A Very Simple Technique for Approximate TEC Calibration Procedure: Assume the minimum TEC (generally attained during nighttime) is known, e.g. zero Download estimates of the satellite biases from CODE. Multiply the biases by TECU/ns to convert the reported biases from units of nanoseconds to TECU. Select the receiver bias to enforce that min(tecs) = TEC*: b R = min(tec R ) + b S TEC* TEC* = 0 TECU Compute the calibrated slant TEC: TEC S = TEC R b S -b R

20 A Better Technique for TEC Calibration (Performed Manually for Illustration) Procedure: If ionosphere is uniformly distributed in a thin slab (no spatial gradients) then the vertical-equivalent TEC estimates should the same for all satellites. Download estimates of the satellite biases from CODE. Multiply the biases by TECU/ns to convert the reported biases from units of nanoseconds to TECU. Manually change the assumed value of the receiver bias until the vertical-equivalent curves collapse most closely together (at least during nighttime hours)

21 Vertical-Equivalent TEC (no Biases Removed) Assumed receiver bias is too low

22 Vertical-Equivalent TEC (Satellite Biases Removed, b R = 0 TECU) Assumed receiver bias is too low

23 Vertical-Equivalent TEC (Satellite Biases Removed, b R = 10 TECU) Assumed receiver bias is too low

24 Vertical-Equivalent TEC (Satellite Biases Removed, b R = 20 TECU) Assumed receiver bias is approximately correct

25 Vertical-Equivalent TEC (Satellite Biases Removed, b R = 30 TECU) Assumed receiver bias is too high

26 Vertical-Equivalent TEC (Satellite Biases Removed, b R = 40 TECU) Assumed receiver bias is too high

27 Vertical-Equivalent TEC (Satellite Biases Removed, b R = 50 TECU) Assumed receiver bias is too high

28 Automated Receiver Bias Determination by Least Squares Assumption: In absence of spatio-temporal density gradients, the verticalized calibrated TEC measured by all satellites should be the same. Given the satellite biases and h, TEC V can be expressed as a function of b R : TEC V (b R ) = [ TEC R -b R + b S ] / M(ε, h) Single layer mapping function RMS (TECU) b R (TECU) We calculate the b R that minimizes Var(TEC V ) late at night when gradients are smallest

29 TEC Calibration by Least-Squares (Results) Final calibrated TEC result, using estimated value of 22.3 TECU for receiver bias

30 Variability of the Receiver Bias Estimates at Antofagasta Largest deviations from trend occur when TEC is structured late at night These nights often correlate with the occurrence of scintillation A closer look at two outliers: Late-night structure in TEC

31 Variability of the Bias Estimates at Kwajalein This station (Kwajalein) experienced weaker GPS scintillations than Antofagasta in 2005 Deviations in the receiver bias from the trend are correspondingly smaller Less structure at night generally means more accurate TEC calibration

32 Kalman Filter Estimation of Total TEC

33 The Kalman Filter Estimation Observation equation (for the i th GPS receiver-satellite pair): Measured slant TEC Bilinear fit to ionospheric TEC V Instrumental biases ( ) 0, 1, 2, TEC M a a d a d b b i i i i i i i i i RS = εrs R + R λrs + R ϕrs + R + S ε Elevation α Azimuth dλ Difference between MLT at ionospheric penetration point and station dϕ Difference between MLAT at ionospheric penetration point and station b R, b S Receiver and satellite instrumental biases Thin shell mapping function M R = Re + h e ( ε ) sec arcsin cos( ε ) Slide 33

34 The Kalman Filter Implementation Kalman state vector (unknowns) Ionospheric fit parameters Instrumental biases X k k k k k k k k R R R R S S S = a0, a1, a2, b b b b 1 2 N T Measurement vector (knowns) y Measured slant TEC k k k k RS RS RS = TEC TEC TEC 1 2 N T Kalman process to be estimated Identity state transition matrix X = Φ X + w k k, k 1 k 1 k 1 y = H X + v k k k k Zero-mean white Gaussian process noise w k and measurement noise v k Kalman updates performed every 60 seconds (each new data epoch) Slide 34

35 Kalman Filter Estimation of TEC in both the Ionosphere and Plasmasphere

36 Plasmaspheric Signatures in the Estimated TEC Gradients from the plasmasphere cause an apparent spread in the vertical-equivalent TEC which violates our assumption that spatial gradients are small. Moreover, the thin-shell approximation commonly used for the ionosphere is not a suitable representation for the plasmaspheric contribution to the TEC This effect is most evident during periods of very low solar activity such the one we are currently experiencing.

37 GPS Signal Paths through the Plasmasphere Electron Density in the Plasmasphere PTEC between 700 km and km TECU 60 Lat Increasing value of Kp Re TECU 30 Lat Plasmapause TECU 0 Lat Elevation Log (Ne) North Horizon Zenith South Horizon Plasmaspheric contribution to the TEC depends on location, azimuth, and elevation. Plasmapause location has strong influence on PTEC encountered at high to mid latitudes. Slide 37

38 Carpenter-Anderson Plasmasphere Model Carpenter and Anderson [1992] model for the electron density in the inner plasmasphere: i 2π logn = L cos e 365 Location of the plasmapause: Width of the plasmapause (Gallagher et al. [2000]), neglecting local time dependence: = 0.14 Electron density in the trough (Sheeley et al. [2001]) neglecting local time dependence: Regions spliced together using tanh step function L p ( L 2 ) ( d + 9) 4π ( d + 9) L w n ( 3/ ) t =124 L 4 e 0.075cos = Kp, max R e 1.5 Integration of electron density from 700 km to 20,200 km along signal path gives P(α, ε) Model very simple, but Kalman filter will scale the results to best fit the measurements Slide 38

39 A Numerical Experiment: Idealized Ionosphere Plus Plasmasphere Assume ionosphere is an idealized thin slab Construct slant TEC via thin-shell mapping fn Add slant TEC through model plasmasphere Verticalize the results

40 Comparing the Idealized and Estimated Total TEC TEC When Neglecting Plasmasphere Idealized Ionosphere and Plasmasphere Slide 40

41 The Kalman Filter Implementation (with Plasmasphere) Observation equation (for the i th GPS receiver-satellite pair): Measured slant TEC Bilinear fit to ionospheric TEC V Plasmaspheric slant TEC Instrumental biases ( ε ) ( ) 0, 1, λ 2, ϕ 3, α, ε i i i i i i i i i i i i RS = RS R + R RS + R RS + R RS RS + R + S TEC M a a d a d a P b b ε Elevation α Azimuth dλ Difference between MLT at ionospheric penetration point and station dϕ Difference between MLAT at ionospheric penetration point and station P(α, ε) PTEC from Carpenter-Anderson et. al [1992] (scaled to fit observations) b R, b S Receiver and satellite instrumental biases Thin shell mapping function (for the ionosphere only) M R = Re + h e ( ε ) sec arcsin cos( ε ) Slide 41

42 The Kalman Filter Implementation (with Plasmasphere) Kalman state vector (unknowns) Ionospheric fit parameters Plasmaspheric scaling Instrumental biases X k k k k k k k k k R R R R R S S S = a0, a1, a2, a3, b b b b 1 2 N T Measurement vector (knowns) y Measured slant TEC k k k k RS RS RS = TEC TEC TEC 1 2 N T Kalman process to be estimated Identity state transition matrix X = Φ X + w k k, k 1 k 1 k 1 y = H X + v k k k k Zero-mean white Gaussian process noise w k and measurement noise v k Kalman updates performed every 60 seconds (each new data epoch) Slide 42

43 Results at Greensboro (36, 280 ) on Nov 2007 Total Slant TEC Vertical-Equiv. Total TEC Ionospheric Slant TEC Vertical-Equiv. Ionospheric TEC Plasmaspheric Slant TEC Zenith Plasmaspheric TEC Combined Satellite + RX Biases Zenith Total and Ionospheric TEC TECU TECU TECU TECU TECU TECU TECU TECU MLT (hours) MLT (hours) Slide 43

44 Impacts of Ignoring the Plasmasphere when Estimating TEC Accounting for Plasmasphere Haystack (43, 288 ) Ignoring Plasmasphere Greensboro (36, 280 ) Roatan (16, 273 ) Neglecting the plasmasphere tends to cause overestimation of the total TEC at middle latitudes and underestimation at equatorial latitudes. Slide 44

45 Why the Estimated TEC Can Be Negative and What to Do About It If TEC is small and receiver bias is overestimated, negative TEC estimates can result TEC*=0 TECU When this happens, we fall back on the simple approach: choose the bias to enforce that min(tecs) = TEC* TEC*=2.7 TECU Now, however, we can make a more informed selection of TEC*. A reasonable value to use is the (zenith) plasmaspheric contribution according to the Carpenter-Anderson model.

46 Plasmaspheric Contribution to the GPS TEC According to the Carpenter-Anderson Plasmasphere Model Simulation conditions: 13 month average solar flux = 7.9; Kp=1; Day of year = 1 Local max due to offset dipole

47 Conclusions When estimating the GPS instrumental biases from the measurements we must make various assumptions about the structure of the ionized regions traversed by the signals Inaccuracies in estimation of the biases can be expected when these assumptions are violated. Phenomena that cause difficulty in estimating the biases include: Ionospheric structure and scintillation The contribution to the GPS TEC from the plasmasphere Neglecting the plasmasphere tends to cause overestimation of the total TEC at middle latitudes and underestimation at equatorial latitudes. Software to perform the calibrations using the Kalman filter approach (with and without the plasmasphere term) is available upon request. We will demonstrate this software during Wednesday s TEC calibration laboratory. A manuscript (draft) recently submitted to Radio Science describing the technique is also available upon request Slide 47