Abstract

Two experiments were conducted with a Fourier-transform infrared (FTIR) spectrometer. The purpose of the first experiment was to detect and identify Bacillus subtilis subsp. niger (BG) bioaerosol spores and kaolin dust in an open-air release for which the thermal contrast between the aerosol temperature and background brightness temperature is small. The second experiment estimated the concentration of a small amount of triethyl phosphate (TEP) vapor in a closed chamber in which an external blackbody radiation source was used and where the thermal contrast was large. The deduced BG (TEP) extinction spectrum (identification) showed an excellent match to the library BG (TEP) extinction spectrum. Analysis of the time sequence of the measurements coincided well with the presence (detection) of the BG during the measurements, and the estimated concentration of time-dependent TEP vapor was excellent. The data were analyzed with hyperspectral detection, identification, and estimation algorithms. The algorithms were based on radiative transfer theory and statistical signal-processing methods. A subspace orthogonal projection operator was used to statistically subtract the large thermal background contribution to the measurements, and a robust maximum-likelihood solution was used to deduce the target (aerosol or vapor cloud) spectrum and estimate its mass-column concentration. A Gaussian-mixture probability model for the deduced mass-column concentration was computed with an expectation-maximization algorithm to produce the detection threshold, the probability of detection, and the probability of false alarm. The results of this study are encouraging, as they suggest for the first time to the authors’ knowledge the feasibility of detecting biological aerosols with passive FTIR sensors.

© 2003 Optical Society of America

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References

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  1. P. L. Hanst, S. T. Hanst, “Gas measurement in the fundamental infrared region,” in Air Monitoring by Spectroscopic Techniques, M. W. Sigrist, ed. (Wiley, New York, 1994).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  12. G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (John Hopkins U. Press, Baltimore, Md., 1996).
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  14. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993), Chap. 4.
  15. C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, Oxford, 1995), Chap. 2.
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    [CrossRef]
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    [CrossRef] [PubMed]

2003 (1)

2002 (2)

1994 (1)

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

1987 (1)

1980 (1)

Acharya, P. K.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Adler-Golden, S. M.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Allred, C. L.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Anderson, G. P.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Ayres, S. D.

R. A. Sutherland, J. C. Thompson, S. D. Ayres, “Infrared scene modeling in emissive, absorptive, and multiple scattering atmospheres,” in Targets and Backgrounds VII: Characterization and Representation, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc.4370, 210–219 (2001).

Ben-David, A.

A. Ben-David, “Remote detection of biological aerosols at a distance of 3 km with passive Fourier transform infrared (FTIR) sensor,” Opt. Express 11, 418–429 (2003), http://www.opticsexpress.org .
[CrossRef] [PubMed]

A. Ben-David, A. Ifarraguerri, “Computation of a spectrum from a single-beam Fourier-transform infrared interferogram,” Appl. Opt. 41, 1181–1189 (2002).
[CrossRef] [PubMed]

A. Ben-David, “IR emission from aerosols: implications to passive remote sensing,” in International Symposium on Spectral Sensing (Deepak, Hampton, Va., 2001), pp. 141–151.

E. R. Schildkraut, R. Connors, A. Ben-David, “Initial test results from ultra-high sensitivity passive FTIR instrumentation (HISPEC),” in Proceedings of the International Symposium on Spectral Sensing Research, (Deepak, Hampton, Va., 2001), pp. 365–374.

Ben-Shalom, A.

Berk, A.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Bernstein, L. S.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Bishop, C. M.

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, Oxford, 1995), Chap. 2.

Brazilai, B.

Cabib, D.

Chang, C.-I.

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

Chetwynd, J. H.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Connors, R.

E. R. Schildkraut, R. Connors, A. Ben-David, “Initial test results from ultra-high sensitivity passive FTIR instrumentation (HISPEC),” in Proceedings of the International Symposium on Spectral Sensing Research, (Deepak, Hampton, Va., 2001), pp. 365–374.

Devir, A. D.

Dothe, H.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Gershenenfeld, N.

N. Gershenenfeld, The Nature of Mathematical Modeling (Cambridge U. Press, Cambridge, 1999), Chap. 10.

Gillespie, P. S.

Goldberg, S.

Golub, G.

G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (John Hopkins U. Press, Baltimore, Md., 1996).

Griffith, D. W. T.

D. W. T. Griffith, I. M. Jamie, “Fourier transform infrared spectrometry in atmospheric and trace gas analysis,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, Chichester, England, 2000).

Hanst, P. L.

P. L. Hanst, S. T. Hanst, “Gas measurement in the fundamental infrared region,” in Air Monitoring by Spectroscopic Techniques, M. W. Sigrist, ed. (Wiley, New York, 1994).

Hanst, S. T.

P. L. Hanst, S. T. Hanst, “Gas measurement in the fundamental infrared region,” in Air Monitoring by Spectroscopic Techniques, M. W. Sigrist, ed. (Wiley, New York, 1994).

Harsanyi, J.

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

Hoke, M. L.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Ifarraguerri, A.

Issacs, R. G.

Jamie, I. M.

D. W. T. Griffith, I. M. Jamie, “Fourier transform infrared spectrometry in atmospheric and trace gas analysis,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, Chichester, England, 2000).

Jeong, L. S.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993), Chap. 4.

Klett, J. D.

R. A. Sutherland, J. C. Thompson, J. D. Klett, “Effects of multiple scattering and thermal emission on target-background signatures sensed through obscuring atmospheres,” in Targets and Backgrounds VI: Characterization, Visualization, and the Detection Process, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc. SPIE4029, 300–309 (2000).

Ligon, D. A.

Lipson, S. G.

Matthew, M. W.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Oppenheim, U. P.

Pukall, B.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Richtsmeier, S. C.

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing, Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

Schaum, A.

A. Schaum, A. Stocker, “Subclutter target detection using sequences of thermal infrared multispectral imagery,” in Algorithms for Multispectral and Hyperspectral Imagery, S. S. Shen, M. R. Descour, eds., Proc. SPIE3701, 12–22 (1997).
[CrossRef]

Schildkraut, E. R.

E. R. Schildkraut, R. Connors, A. Ben-David, “Initial test results from ultra-high sensitivity passive FTIR instrumentation (HISPEC),” in Proceedings of the International Symposium on Spectral Sensing Research, (Deepak, Hampton, Va., 2001), pp. 365–374.

Stocker, A.

A. Schaum, A. Stocker, “Subclutter target detection using sequences of thermal infrared multispectral imagery,” in Algorithms for Multispectral and Hyperspectral Imagery, S. S. Shen, M. R. Descour, eds., Proc. SPIE3701, 12–22 (1997).
[CrossRef]

Sutherland, R. A.

R. A. Sutherland, J. C. Thompson, J. D. Klett, “Effects of multiple scattering and thermal emission on target-background signatures sensed through obscuring atmospheres,” in Targets and Backgrounds VI: Characterization, Visualization, and the Detection Process, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc. SPIE4029, 300–309 (2000).

R. A. Sutherland, J. C. Thompson, S. D. Ayres, “Infrared scene modeling in emissive, absorptive, and multiple scattering atmospheres,” in Targets and Backgrounds VII: Characterization and Representation, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc.4370, 210–219 (2001).

Thompson, J. C.

R. A. Sutherland, J. C. Thompson, S. D. Ayres, “Infrared scene modeling in emissive, absorptive, and multiple scattering atmospheres,” in Targets and Backgrounds VII: Characterization and Representation, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc.4370, 210–219 (2001).

R. A. Sutherland, J. C. Thompson, J. D. Klett, “Effects of multiple scattering and thermal emission on target-background signatures sensed through obscuring atmospheres,” in Targets and Backgrounds VI: Characterization, Visualization, and the Detection Process, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc. SPIE4029, 300–309 (2000).

Van Loan, C. F.

G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (John Hopkins U. Press, Baltimore, Md., 1996).

Wang, W.-C.

Wetmore, A. E.

Worsham, R. D.

Appl. Opt. (3)

IEEE Trans. Geosci. Remote Sens. (1)

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sens. 32, 779–785 (1994).
[CrossRef]

Opt. Express (2)

Other (13)

A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, M. L. Hoke, “modtran4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE3756, 348–353 (1999).
[CrossRef]

P. L. Hanst, S. T. Hanst, “Gas measurement in the fundamental infrared region,” in Air Monitoring by Spectroscopic Techniques, M. W. Sigrist, ed. (Wiley, New York, 1994).

D. W. T. Griffith, I. M. Jamie, “Fourier transform infrared spectrometry in atmospheric and trace gas analysis,” in Encyclopedia of Analytical Chemistry, R. A. Meyers, ed. (Wiley, Chichester, England, 2000).

R. A. Sutherland, J. C. Thompson, J. D. Klett, “Effects of multiple scattering and thermal emission on target-background signatures sensed through obscuring atmospheres,” in Targets and Backgrounds VI: Characterization, Visualization, and the Detection Process, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc. SPIE4029, 300–309 (2000).

R. A. Sutherland, J. C. Thompson, S. D. Ayres, “Infrared scene modeling in emissive, absorptive, and multiple scattering atmospheres,” in Targets and Backgrounds VII: Characterization and Representation, W. R. Watkins, D. Clement, R. R. Reynolds, eds., Proc.4370, 210–219 (2001).

A. Ben-David, “IR emission from aerosols: implications to passive remote sensing,” in International Symposium on Spectral Sensing (Deepak, Hampton, Va., 2001), pp. 141–151.

G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (John Hopkins U. Press, Baltimore, Md., 1996).

N. Gershenenfeld, The Nature of Mathematical Modeling (Cambridge U. Press, Cambridge, 1999), Chap. 10.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N.J., 1993), Chap. 4.

C. M. Bishop, Neural Networks for Pattern Recognition (Oxford U. Press, Oxford, 1995), Chap. 2.

L. L. Scharf, Statistical Signal Processing, Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

A. Schaum, A. Stocker, “Subclutter target detection using sequences of thermal infrared multispectral imagery,” in Algorithms for Multispectral and Hyperspectral Imagery, S. S. Shen, M. R. Descour, eds., Proc. SPIE3701, 12–22 (1997).
[CrossRef]

E. R. Schildkraut, R. Connors, A. Ben-David, “Initial test results from ultra-high sensitivity passive FTIR instrumentation (HISPEC),” in Proceedings of the International Symposium on Spectral Sensing Research, (Deepak, Hampton, Va., 2001), pp. 365–374.

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Figures (20)

Fig. 1
Fig. 1

Simplified geometry for sensor measurements of a cloud at distance z between cloud and sensor.

Fig. 2
Fig. 2

Atmospheric radiances M1 and M2 and transmissions t1 and t2 for distances z between source and detector of 1, 3, and 5 km. The order of the curves is marked: t1 (0–1 km) > t1(0–3 km) > t1(0–5 km) and M1(0–5 km) > M1 (0–3 km) > M1(0–1 km).

Fig. 3
Fig. 3

Mass-extinction coefficient of BG material.

Fig. 4
Fig. 4

Simulated measurements M(λ) (left-hand y axis) and the signal-information portion exp[-αρ]t1[M2 - B(Tc)] (right-hand y axis), which contain information on a bioaerosol cloud of size 100 m containing N particles/cm3. The bioaerosol particles are 5 K colder than the ambient atmosphere. The atmosphere (ideal case) is an isothermal atmosphere, M2 = B(T), represented by a blackbody at temperature T. The sensor LOS is horizontal, pointing to infinity, and the cloud is placed at a distance of 0 km from the sensor (t1 = 1, M1 = 0).

Fig. 5
Fig. 5

Same as Fig. 4: the cloud is at 0 km from the sensor (t1 = 1) but atmosphere M2 (behind the cloud) is simulated by the modtran radiative transfer program.

Fig. 6
Fig. 6

Same as Fig. 5 but with the cloud 1 km from the sensor (i.e., the atmospheric radiances M1 in front of the cloud and M2 behind the cloud as well as atmospheric transmission t1 to the cloud all simulated by the modtran radiative transfer program).

Fig. 7
Fig. 7

Block diagram of the identification process from which we estimate extinction spectrum α of the aerosol (or vapor) cloud.

Fig. 8
Fig. 8

Block diagram of the detection process for mass-column-density ρ̂(n) estimation and detection.

Fig. 9
Fig. 9

Block diagram of the detection process with a matched filter for target detection.

Fig. 10
Fig. 10

Spectral brightness temperature difference between the ambient atmosphere and the effective brightness temperature along the LOS. The normalized spectral mass-extinction coefficient of the bioaerosols is also shown. Negative thermal contrast ΔT indicates that the bioaerosol cloud was in an emission mode (ambient temperature higher than the brightness temperature along the LOS).

Fig. 11
Fig. 11

Estimated mass-extinction spectrum α̂(λ) of the bioaerosol cloud with the identification algorithm (Fig. 7) and BG library spectrum α(λ). The correlation coefficient between the spectra two is 0.97.

Fig. 12
Fig. 12

Estimated mass-extinction spectrum α̂(λ) of a kaolin cloud with the identification algorithm (Fig. 7) and the kaolin library spectrum α(λ). The correlation coefficient between the two spectra is 0.822.

Fig. 13
Fig. 13

Estimated mass-column density ρ̂(n) for the 2150 measurements taken during the bioaerosol experiment and detection threshold γ (computed with the EM algorithm). A 3-s moving average (i.e., 17 sequential measurements) procedure was preformed on the estimated ρ̂(n) to reduce noise.

Fig. 14
Fig. 14

pdf Model for ρ̂(n) (Fig. 13) computed with the EM algorithm, the data ρ̂(n) histogram, and the two pdfs for the two hypotheses, H0 and H1. The threshold of separation, γ = -14.87 mg m-2, between the two hypotheses is marked. Mass-column density ρ > γ belongs to hypothesis H1.

Fig. 15
Fig. 15

Matched filter score s(n) for the 2150 measurements of the bioaerosol experiment and detection threshold γ. A 3-s moving average (i.e., 17 sequential measurements) procedure was preformed on the estimated s(n) to reduce noise.

Fig. 16
Fig. 16

Same as Fig. 14 but for matched filter score s of Fig. 15. The threshold of separation, γ = 0.303, between the two hypotheses is marked. A score s > γ belongs to hypothesis H1.

Fig. 17
Fig. 17

Sensor uncalibrated measurements g(λ) with and without TEP in the chamber. The anticipated mass-column density [mg m-2] concentrations in the chamber (3-m path length) are shown. The difference among the five measurements is very small and is barely perceptible.

Fig. 18
Fig. 18

Deduced vapor spectrum for the TEP injections and a TEP library spectrum. The anticipated mass-column density [mg m-2] concentrations in the chamber (3-m path length) are shown.

Fig. 19
Fig. 19

Mass-column density ρ̂(n) for the 4000 measurements and detection threshold γ. A 2-s moving average (i.e., 25 sequential measurements) procedure was preformed on the estimated ρ̂(n) to reduce noise. Four injections of 0.25 ml each occurred at measurements 1, 1000, 2000, and 3000.

Fig. 20
Fig. 20

pdf model for ρ̂(n) (Fig. 19) computed with the EM algorithm, the data histogram, and five conditional pdfs (Gaussians); one for the hypothesis H0 and four for four hypotheses H1. The threshold of detection is the location of the intersection between adjacent Gaussians.

Tables (1)

Tables Icon

Table 1 Parameters of the pdf Modela of Fig. 20 and Threshold of Detection γ

Equations (24)

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Mλ=M1λ+1-exp-αλρBλ, Tct1λ+M2λexp-αλρt1λ+εbλBλ, Tbt2λexp-αλρt1λ,
M=M0-M2+εbBTbt2-BTt1αρ=M0-M0-M1-BTt1αρ,
M=M0-M0-BTαρ,
gλ=g0λ-g0λ-gbλ, Tαρ,
gbλ, Tg0λBλ, TBλ, Tb
OSPM=-OSPM0-Bαρ.
PU=I-UUTU-1UT
αˆλ, n=PUMλ, n
rn=corαˆn, α.
αˆ=Eαˆn|rn>r0.
Dλ, n0=-M0λ, n0-Bλ, Tαλ,
Tˆ=EB-1M0λ, n0,
PUDλ, n0ρn|n0-Mλ, n=0.
ρˆn|n0=PUDn0TPUDn0-1PUDn0TPUMn.
ρˆn=Eρˆn|n0.
pdfρ=w0 pdfρ|H0+1-w0pdfρ|H1,
pdfρ|H0=Nρ; μ0, σ02, pdfρ|H1=Nρ; μ1, σ12,
Nx; μ, σ2=2πσ2-0.5 exp-0.5x-μ2/σ2.
w0Nγ; μ0, σ02=1-w0Nγ; μ1, σ12.
PDγ=γpdfx|H1dx=2-11-erfγ-μ121/2σ1, PFAγ=γpdfx|H0dx=2-11-erfγ-μ021/2σ0,
erfx=12π0xexp-0.5t2dt.
sn=Mn-EM0TR-1α-EM0,
pdfρ=w0 pdfρ|H0+i=14 wi pdfρ|H1i,
i=04 wi=1,

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