The Pure-State Filter: Applications to Infrasound Data

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1 The Pure-State Filter: Applications to Infrasound Data John V Olson Geophysical Institute University of Alaska Fairbanks Presented at the US Infrasound Team Meeting Oxford, MS January 2009

2 The Pure-State Filter A musical preamble Pure states of information Process application Infrasound applications: wind noise reduction Summary US Infrasound Team Meeting, January

3 Paganini: Violin & Guitar: spectrum of recorded data US Infrasound Team Meeting, January

4 Paganini: Violin & Guitar: spectrum with white noise added US Infrasound Team Meeting, January

5 Paganini: Violin & Guitar: spectrum after Pure-State filter US Infrasound Team Meeting, January

6 Paganini: Violin & Guitar ~15 db US Infrasound Team Meeting, January

7 Pure States of Information Pure state: states of maximal information characterized by the existence of an experiment that gives a result with certainty (Fano, U., 1957) The spectral matrix, calculated from a multivariate data set, is viewed as a matrix representing the density of states present Operationally, a pure state is present if the spectral matrix has a single, non-zero eigenvalue US Infrasound Team Meeting, January

8 Stokes Vector Representation G. G. Stokes, 1852 Input: M time series, length N: Assemble M 2, trace-orthogonal matrices: Fourier transform the input: Expand the spectral matrix: Compute the spectral matrices: Choose U1 as the unit matrix (white-noise), And take the ratio of coherent to noise power The spectral matrix is MxM, hermitian and, by construction: US Infrasound Team Meeting, January

9 Statistics: PDF for 4 and 8 elements PDF s for 4 elements, 2, 4, 6, 10 dof PDF s for 8 elements, 2, 4, 6, 10 dof US Infrasound Team Meeting, January

10 Paganini: Violin & Guitar Returning to the music sample, we have plotted the values of P2 as a function of frequency. Note that the music harmonics have values of P2 that exceed 0.5 in most cases. US Infrasound Team Meeting, January

11 The Pure-State Filter Filter Development Raw, multivariate, data Input: x(t) The data are Fourier transformed: X(ω) Samson s P 2 parameter* is estimated at each frequency in the spectrum The data are filtered using P 2 (ω): X f (ω) = P 2 (ω) X(ω) The filtered data are transformed back to the time domain to obtain a filtered waveform: x f (t) The Pure-State filter is a data-adaptive filter that operates in the frequency domain to reduce or eliminate spectral components that represent noise in the data. The harmonic components of the signal of interest are not affected as they would be with a simple low-pass or high-pass filter. The result is a filtered waveform that preserves the coherent portions of the signal while reducing the spectral components of the ambient noise. Increases in signal-to-noise (SNR) levels of 20 db are commonly obtained. * 0 P 2 1, Olson, J. V., Inframatics, Sept 2004 at or contact J. Olson at jvo@gi.alaska.edu

12 The Pure-State Filter Process Application The following steps would be invoked by a data analyst to use the Pure-State filter to enhance the signal-to-noise, SNR, in a data sequence: The ordinary power spectral density (psd) function is estimated using data from an interval containing the signal of interest (SOI) The Pure-State filter is applied to the same interval and the power spectral density of the filtered data is plotted with the raw psd The analyst would then use the Pure-State psd to identify the frequency band that contains the principal information from the SOI Once determined, the analyst would band-pass filter the raw data to obtain the appropriate frequency band of interest Finally, the Pure-State filter is applied to the band-pass filtered data to achieve the coherent portion of the SOI in the chosen frequency band Increases in SNR of the order of 20 db are commonly achieved using the Pure-State filter. US Infrasound Team Meeting, January

13 Applications to Infrasound Data Primary noise source: wind-induced turbulent eddies Small scale eddies are relatively Incoherent over large distances: spectral matrices proportional to the unit matrix: traditional Pure-State filtering works well (Olson, 1982) Large scale eddies may appear partially coherent across smaller arrays US Infrasound Team Meeting, January

14 Kasatochi IMS Arrays (within ~6000 km) IMS Array Azimuth (o) GC Dist (km) I18 I I53 I10 I I30 K I56 I57 I I I I I59 I US Infrasound Team Meeting, January

15 Kasatochi Waveforms & Spectra Fairbanks Array, I53US ~60 db Infrasound amplitudes for the Kasatochi eruptions reached ~ 1 Pa at I53US and I56US US Infrasound Team Meeting, January

16 Spectrum of Microbaroms April 6, 2001 The microbarom spectrum is centered near 0.2 Hz and narrowband, with δf/f ~ 1. The narrow-band characteristic allows us to apply Hilbert transform techniques to identify wave packet boundaries Geophysical Instiute, UAF

17 Summary The Pure-State filter is a data-adaptive filter that uses an estimate of signal/noise derived from the spectral matrices at each frequency of the data transform In practice it can produce increases in signal/noise levels by as much as 20 db Infrasound noise, produced by wind turbulence, is easily suppressed with the filter US Infrasound Team Meeting, January

18 References Fano, U., Description of states in quantum mechanics by density matrix and operator techniques, Rev. Mod. Phys., 29, 74, 1957 Olson, J. V., Noise suppression using data-adaptive polarization filters: Applications to infrasonic array data, J. Acoust. Soc. Am., 72, 1456, 1982 Olson, J. V., The application of the pure-state filter to infrasound array data, Inframatics Newsletter ( Samson, J. C., Descriptions of the polarization states of vector processes: Applications to ULF magnetic fields, Geopys. J. R. Astr. Soc., 34, 403, 1973 US Infrasound Team Meeting, January

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