Bias and Uncertainty in Parameter Estimation in Infrasound Arrays

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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
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Infrasound AnalysisDetection 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.

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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 gt CT or C lt
CT. As a result, false alarms and missed events
result.
pdf
noise
signal
missed events
false alarms
CT
C
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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
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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.
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Least Squares Estimation of azimuth and trace
velocity
t Xf e t (t1 t2 tN)T X (r1 r2
rN)T f (k/V) e (e1 e2 eN)T Least
squares solution f (XTX)-1XTt Giving V(f
)-1/2 Ttan-1(f2/f1)
N
assume a plane
wave front
T

k
Si
rij
ri
Sj
rj
E
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Variances in Velocity and Azimuth
EstimatesConcentration Ellipsoid
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Estimator Variances I53US
Azimuth uncertainty
Trace Velocity uncertainty
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Estimator Variance I55US
Azimuth uncertainty
Trace Velocity uncertainty
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Estimator VarianceCTBT 4 Element Prototype
Trace Velocity uncertainty
Azimuth uncertainty
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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.

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Velocity and Azimuth Bias for Nearby SourcesCTBT
4 element array
Velocity Bias
Azimuth Bias
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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.

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The Fairbanks Infrasound Arrays
The CTBT I53US locations are shown in blue and
are labeled H1H8. The UAF array locations are
shown in red and are labeled A,B,C,D.
N
E
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Fairbanks Infrasound ArraysImpulse Responses
-20ltklt20
I53US 8 sensors
UAF 4 sensors
k 2pf/c 19f so, k 20 is equivalent to 1
Hz.
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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
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Mine Explosion WaveformsJanuary 9, 2003 2351 UT
Eight channels Velocity 328 m/s Azimuth
27.2º C 0.90 st 0.349f8 2.56
Four channels Velocity 339 m/s Azimuth 29.1º C
0.85 st .0256f4 2.56
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Detection and Estimation Jan 9, 2003 event
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MCCM Mine Detections I53US missing sensors
2 missing (outer)
1 missing (outer)
2 missing (1 outer, 1 inner)
3 missing (2 outer, 1 inner)
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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.
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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
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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