## Abstract

Bio-aerosols containing *Bacillus subtilis* var. *niger* (BG) were detected at a distance of 3 *km* with a passive Fourier Transform InfraRed (FTIR) spectrometer in an open-air environment where the thermal contrast was low (~ 1 *K*). The measurements were analyzed with a new hyperspectral detection, identification and estimation algorithm based on radiative transfer theory and advanced signal processing techniques that statistically subtract the undesired background spectra. The results are encouraging as they suggest for the first time the feasibility of detecting biological aerosols with passive FTIR sensors. The number of detection events was small but statistically significant. We estimate the false alarm rate for this experiment to be 0.0095 and the probability of detection to be 0.61 when a threshold of detection that minimizes the sum of the probabilities of false alarm and of missed detection is chosen.

© 2003 Optical Society of America

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### Equations (19)

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(1)
$$M\left(\lambda \right)={M}_{0}\left(\lambda \right)-\left[{M}_{0}\left(\lambda \right)-B\left(\lambda ,T\right)\right]\alpha \left(\lambda \right)\rho $$
(2)
$$\mathit{OSP}\left(M\right)=-\mathit{OSP}\left(\left[{M}_{0}-B\right]\alpha \rho \right)$$
(3)
$$\mathit{pdf}\left(\rho \right)={w}_{0}\mathit{pdf}(\rho \mid {H}_{0})+\left(1-{w}_{0}\right)\mathit{pdf}(\rho \mid {H}_{1})$$
(4)
$$N(x;\mu ,{\sigma}^{2})=\frac{1}{\sqrt{2\pi {\sigma}^{2}}}\mathrm{exp}\left(-\frac{{\left(x-\mu \right)}^{2}}{2{\sigma}^{2}}\right)$$
(5)
$$\mathit{pdf}(\rho \mid {H}_{0})=N\left(\rho ;{\mu}_{0},{\sigma}_{0}^{2}\right)$$
(6)
$$\mathit{pdf}(\rho \mid {H}_{1})=N\left(\rho ;{\mu}_{1},{\sigma}_{1}^{2}\right)$$
(7)
$$\mathit{pdf}({H}_{0}\mid \rho )=\frac{\mathit{pdf}\left({H}_{0},\rho \right)}{\mathit{pdf}\left(\rho \right)}=\frac{{w}_{0}\mathit{pdf}(\rho \mid {H}_{0})}{\mathit{pdf}\left(\rho \right)}$$
(8)
$$\mathit{pdf}({H}_{1}\mid \rho )=\frac{\mathit{pdf}\left({H}_{1},\rho \right)}{\mathit{pdf}\left(\rho \right)}=\frac{\left(1-{w}_{0}\right)\mathit{pdf}(\rho \mid {H}_{1})}{\mathit{pdf}\left(\rho \right)}$$
(9)
$${w}_{0}=\int \mathit{pdf}({H}_{0}\mid \rho )d\rho \text{}\text{}$$
(10)
$${\mu}_{0}={w}_{0}^{-1}\int \mathit{pdf}({H}_{0}\mid \rho )\rho d\rho $$
(11)
$${\mu}_{1}={\left(1-{w}_{0}\right)}^{-1}\int \mathit{pdf}({H}_{1}\mid \rho )\rho d\rho $$
(12)
$${\sigma}_{0}^{2}={w}_{0}^{-1}\int \mathit{pdf}({H}_{0}\mid \rho ){\left(\rho -{\mu}_{0}\right)}^{2}d\rho \text{}$$
(13)
$${\sigma}_{1}^{2}={\left(1-{w}_{0}\right)}^{-1}\int \mathit{pdf}({H}_{1}\mid \rho ){\left(\rho -{\mu}_{1}\right)}^{2}d\rho $$
(14)
$${w}_{0}N(\gamma ;{\mu}_{0},{\sigma}_{0}^{2})=\left(1-{w}_{0}\right)N(\gamma ;{\mu}_{1},{\sigma}_{1}^{2})$$
(15)
$${n}_{\mathrm{det}\phantom{\rule{.2em}{0ex}}\mathit{ections}}=\left(n-{n}_{1}\right){P}_{\mathit{FA}}+{n}_{1}{P}_{D}$$
(16)
$${P}_{D}\left(\gamma \right)=\underset{\gamma}{\overset{\infty}{\int}}\mathit{pdf}(x\mid {H}_{1})\mathit{dx}={2}^{-1}\left[1-\mathit{erf}\left(\frac{\gamma -{\mu}_{1}}{{2}^{\frac{1}{2}}{\sigma}_{1}}\right)\right]=0.61$$
(17)
$${P}_{\mathit{FA}}\left(\gamma \right)=\underset{\gamma}{\overset{\infty}{\int}}\mathit{pdf}(x\mid {H}_{0})\mathit{dx}={2}^{-1}\left[1-\mathit{erf}\left(\frac{\gamma -{\mu}_{0}}{{2}^{\frac{1}{2}}{\sigma}_{0}}\right)\right]=0.0095$$
(18)
$$\mathit{erf}\left(x\right)=\frac{1}{\sqrt{2\pi}}\underset{0}{\overset{x}{\int}}\mathrm{exp}\left(-0.5{t}^{2}\right)\mathit{dt}$$
(19)
$${n}_{1}=\left({n}_{\mathrm{det}\phantom{\rule{.2em}{0ex}}\mathit{ections}}-n{P}_{\mathit{FA}}\right){\left({P}_{D}-{P}_{\mathit{FA}}\right)}^{-1}=240$$