Bias and Uncertainty in Parameter Estimation in Infrasound Arrays

1. Bias and Uncertainty in Parameter Estimation in Infrasound Arrays John V. Olson, C. A. L. Szuberla C. R. Wilson, K. M. Arnoult and D. Osborne Infrasound Group, University of Alaska Fairbanks Presented at the Infrasound Technology Workshop Scripps Institute, UC San Diego October 2003

2. Infrasound Analysis:Detection and Estimation • Detection: the recognition that a signal of interest is present in the data • f-statistic (based upon beam power) • Correlation detection • Estimation: the calculation of parameters that describe the signal properties • k-space beamforming • Least-squares analysis • Minimum-variance, Maximum-likelihood, etc. Geophysical Institute, UAF

3. Classical Detection(Van Trees, H., Detection, Estimation and Modulation Theory, Wiley, 2001) Probability distribution functions Assuming the data can be represented as a sum of signal and noise (+clutter) classical detection methods are based upon the separation of signal and noise in some optimum sense. A threshold, CT, is set and the detected information is classified according to whether C > CT or C < CT. As a result, false alarms and missed events result. pdf noise signal missed events false alarms CT C Geophysical Institute, UAF

4. The UAF MCCM detector:(Mean of Cross-Correlation Maxima) For N sensors the N(N-1)/2 pair-wise cross-correlations are estimated and the maxima found. The mean of the cross-correlations is then taken as the detector output. A threshold is chosen (red line in top panel) and the values of MCCM that exceed the threshold are tagged. Note, the tagged values do not occur during the intervals with maximum amplitudes. Time (hrs) 0 24 Geophysical Institute, UAF

5. Estimation after MCCM detection Once the MCCM theshold is chosen, any values exceeding the threshold are tagged and the velocity and azimuth estimated. For the case shown at the left, the signals are high-velocity signals presumed to be associated with pulsating auroras. Geophysical Institute, UAF

6. Least Squares Estimationof azimuth and trace velocity τ= Xf + ε τ = (τ1τ2 … τN)T X = (r1 r2 … rN)T f = (k/V) ε = (ε1ε2 … εN)T Least squares solution: f* = (XTX)-1XTτ Giving V*=(|f*|)-1/2 Θ*=tan-1(f*2/f*1) N assume a plane wave front Θ ^ k Si rij ri Sj rj E Geophysical Institute, UAF

7. Variances in Velocity and Azimuth Estimates:Concentration Ellipsoid Geophysical Institute, UAF

8. Estimator Variances: I53US Azimuth uncertainty Trace Velocity uncertainty Geophysical Institute, UAF

9. Estimator Variance: I55US Azimuth uncertainty Trace Velocity uncertainty Geophysical Institute, UAF

10. Estimator VarianceCTBT 4 Element Prototype Trace Velocity uncertainty Azimuth uncertainty Geophysical Institute, UAF

11. Velocity and Azimuth Biases • When the least-squares estimator of velocity and azimuth is used to analyze signals from nearby sources, the wave fronts are not planar and the resulting estimates are biased. • Beyond 10 km the bias is less than 1%. Geophysical Institute, UAF

12. Velocity and Azimuth Bias for Nearby SourcesCTBT 4 –element array Velocity Bias Azimuth Bias Geophysical Institute, UAF

13. The UAF Arrays: Fairbanks, Alaska • In May 1999 a four-element, CTBT prototype, array was installed in Fairbanks near the Geophysical Institute on the campus of the University of Alaska. It has been in nearly continuous operation since that time. • In September 2002, the eight-element I53US array was brought on line. It is located in the same wooded location near the Geophysical Institute. Geophysical Institute, UAF

14. The Fairbanks Infrasound Arrays The CTBT I53US locations are shown in blue and are labeled H1…H8. The UAF array locations are shown in red and are labeled A,B,C,D. N E Geophysical Institute, UAF

15. Fairbanks Infrasound ArraysImpulse Responses -20<k<20 I53US: 8 sensors UAF: 4 sensors k = 2πf/c ≈ 19f so, k = 20 is equivalent to ≈ 1 Hz. Geophysical Institute, UAF

16. Mine Explosions near Fairbanks There are several gold mines located in the Fairbanks area. Located approximately 30 km North-East of Fairbanks are the True North and Fort Knox gold mines. The Fort Knox mine sets off a surface explosion each day near 3pm local time. Other surface mines set off explosions periodically. True North Fort Knox UAF True North: 32.5º, 23.8 km Fort Knox: 59.3º, 27.2 km Geophysical Institute, UAF

17. Mine Explosion WaveformsJanuary 9, 2003 23:51 UT Eight channels Velocity: 328 m/s Azimuth: 27.2º C: 0.90 sτ: 0.349*f8 = 2.56 Four channels Velocity: 339 m/s Azimuth 29.1º C: 0.85 sτ:: .0256*f4 = 2.56 Geophysical Institute, UAF

18. Detection and Estimation Jan 9, 2003 event Geophysical Institute, UAF

19. MCCM Mine Detections I53US missing sensors 2 missing (outer) 1 missing (outer) 2 missing (1 outer, 1 inner) 3 missing (2 outer, 1 inner) Geophysical Institute, UAF

20. Velocity-Azimuth Estimates71 events, Jan-July 2003 This figure shows the individual estimates of velocity and azimuth derived from the mine signals. One can see that the variance in the 4-element array estimates is larger than that of the 8-element array. Geophysical Institute, UAF

21. Velocity-Azimuth Estimates43 events, Jan-Aug 2003 True North Mine 32.5º UAF-4: 30.8o ± 6.3o 333.4 ± 48.8 m/s I53: 29.9o ± 2.7o 328.7 ± 4.7 m/s Fort Knox Mine: 59.3º UAF-4: 60.0o ± 6o 309.4 ± 42.7 m/s I53: 57.4o ± 1.2o 330.3 ± 9.8 m/s Geophysical Institute, UAF

22. Summary • Detection: Correlation detectors work well with infrasound data and are robust against the loss of sensors in an array • Estimation: For sources at distances more than 10x the array aperture, a plane-wave approximation and least square estimation of azimuth and trace velocity is accurate Geophysical Institute, UAF