Turbulence Estimation Techniques for COSMIC Occultation Data

Transcription

1 Turbulence Estimation Techniques for COSMIC Occultation Data James Burkert, Penina Axelrad, Larry Cornman, Kent Goodrich, Scott Palo, Andrew Weekley Colorado Center for Astrodynamics Research University of Colorado, Boulder p. 1/20

2 Outline Objectives and Approach Model Review Estimators Half-Power Estimator, Maximum Likelihood Estimators Estimator Comparisons Simulated Data Real Data Future Work & Conclusions p. 2/20

3 Objectives & Approach Objectives Identify turbulence in COSMIC occultation data Estimate intensity and position Compare with in situ measurements Approach Select promising occultations Compute SNR power spectra or simulated data Estimate parameters by fitting model to computed spectrum Compare with aircraft measurements p. 3/20

4 Model Review Cornman, et. al. Model describes the amplitude, phase, and combined spectra in log-space Model varies with turbulence intensity C 2 n, size η, position η 1, and length scale L 0. p. 4/20

5 Model Review: C 2 n η C 2 n η is the combined intensity and length term Variation of Log Amplitude Spectrum with C n 2 η C 2 η=6e 015 n Log Amplitude Spectrum Increasing C n 2 η 2e 013 3e 011 f 8/3 Slope Frequency [Hz] p. 5/20

6 Model Review: η 1 η 1 /R is the distance of the turbulence η 1 from the GPS satellite normalized by the ray path length R Variation of Log Amplitude Spectrum with η 1 /R 10 5 Increasing η 1 /R (toward LEO) Log Amplitude Spectrum η /R= f 8/3 Slope Frequency [Hz] p. 6/20

7 Real Data Processing p. 7/20

8 Estimators Turbulence size and intensity, Cn η: 2 This parameter is very well-suited to the standard MLE approach, but requires an estimate of η 1 Cn η 2 affects the spectrum linearly MLE should be unbiased and Gaussian Position along ray path η 1 (or η 1 /R) Nonlinear effect on power spectrum C 2 n η is always estimated via Maximum Llikehood Estimator (MLE), given η 1, but we have three estimators for η 1 : Half-power, MLE 1 and MLE 2. p. 8/20

9 Estimators - Half-Power Cumulative Power over Frequency Frequency of -3 db Power Point Empirical Relationship η 1 (f) 1 Example Normalized Cumulative Power of Truth and Noisy Spectra Half-Power Estimator -3 db power point No initial guess needed for η 0 Normalized Cumulative Power Truth Noisy 1 Half Power Point 1 Noisy 2 Haf Power Point Frequency [Hz] p. 9/20

10 Estimators - MLE 1 Maximum Likelihood η 1, C 2 n η Initial estimate η 0 : From Half-Power Estimator C 2 n η: Single step MLE η 1 : Grid Search over Cost Function ln(l) Measured Ideal Correct Value MLE Cost Function η Error [km] (1) ln(l) = NX (ln( C 2 ˆ n η) + ln(φ(f, η 1 ))) + i=1 NX i=1 X i ˆ C 2 n η φ(f, η 1 ) (2) ˆ C 2 n η = NX i=1 X i N φ(f, η 1 ) p. 10/20

11 Estimators - MLE 2 Maximum Likelihood Initial estimate η 0 : Sample mean of η 1 values that give the lowest cost function, with η 1 values distributed across the atmosphere Cn η: 2 Standard MLE with η 1 estimate ln(l) MLE Cost Function Noisy Smooth Correct Value Sample Pts η Error [km] p. 11/20

12 MLE 1 : Simulated Data p. 12/20

13 MLE 2 : Simulated Data p. 13/20

14 Real Data: AF 447 Three occultations were identified by S. Sokolovskiy in the vicinity of Air France 447 s last known transmission Occultation Geometry Associated with the Air France 447 Distaster Latitude [deg] C G10 (06:32 06:34) C G28 (02:56 02:58) C G17 (04:23 04:25) Air France 447 Last TX (02:14) Longitude [deg] p. 14/20

15 Real Data: AF 447 MLE 1 MLE 2 η/r C n 2 η η/r C n 2 η e e e e e e e e e e Air France 447 Occultation 1 Results Frequency [Hz] p. 15/20

16 Conclusions C 2 n η estimation works well (about 10% error in simulated data) η 1 estimation is not reliable enough (about 2,000 km error in simulated data) Using the mean of the η 1 s that give the smallest cost function works about as well as perfect initial guess in simulated tests p. 16/20

17 Future Work Use phase data Case studies of particular occultations Process all COSMIC data Develop understanding of why η 1 is not well-determined from observations p. 17/20

18 Additional Slides p. 18/20

19 MLE 1 Statistical Tests χ χ 2 P value for χ 2 test χ 2 Test P Value Perfect η 0 η Error [km] x 10 6 MLE χ η Error [km] x 10 6 χ 2 Test P Value χ C n 2 η Error x 10 7 P value for χ 2 test C n 2 η Error x 10 7 p. 19/20

20 MLE 2 Statistical Tests χ χ 2 Perfect η MLE 2 η Error [km] x 10 6 P value for χ 2 test χ 2 Test P Value η Error [km] x 10 6 χ χ C n 2 η Error x 10 7 P value for χ 2 test χ 2 Test P Value C n 2 η Error x 10 7 p. 20/20