## Abstract

Passive remote sensing of airborne chemicals at infrared wavelengths may be limited by temporal fluctuations in atmospheric brightness temperatures *δT*(*Δt*). Brightness temperatures in two infrared spectral bands were simultaneously measured on clear and cloudy days along three lines of sights. For time windows *Δt* < 3–5 *s*, *δT*(Δ*t*) remained constant at the sensor noise level and rapidly increased as Δ*t* increased. The fluctuation time scale for the cloudy day was longer than for the clear day. The long correlation time for *T*(*t*) limits the utility of signal averaging in improving detection signal-to-noise ratio (SNR). The simultaneous outputs of the two spectral channels during the clear day exhibited no spectral coherence at Δ*t* < 3 *s* and limited coherence at Δ*t* > 30 *s*. Measurements during the cloudy day were largely coherent. Consequently, band-by-band subtraction may have limited benefits.

© 2005 Optical Society of America

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### Equations (15)

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(1)
$${S}_{k}={A}_{k}+{B}_{k}{W}_{s}(T,{\lambda}_{k})-{C}_{k}{W}_{C}({T}_{C},{\lambda}_{k})$$
(2)
$${\left({\sigma}_{N}^{2}\right)}_{i}=\frac{1}{N}\sum _{j=1}^{N}{\left({T}_{j}-{\mu}_{i}\right)}^{2}$$
(3)
$${\mu}_{i}=\frac{1}{N}\underset{j=\left(i-1\right)\phantom{\rule{.2em}{0ex}}N+1}{\overset{\left(i-1\right)\phantom{\rule{.2em}{0ex}}N+N}{\sum {T}_{j}}}$$
(4)
$$E\left[{\sigma}_{N}^{2}\right]=\frac{1}{n}\sum _{i=1}^{n}{\left({\sigma}_{N}^{2}\right)}_{i}$$
(5)
$$E\left[{\sigma}_{N}^{2}\right]=\frac{1}{n}\sum _{i=1}^{n}\left[E\left({T}_{i}^{2}\right)-{\mu}_{i}^{2}\right]=\frac{1}{n}\sum _{i=1}^{n}E\left({T}_{i}^{2}\right)-\frac{1}{n}\sum _{i=1}^{n}{\mu}_{i}^{2}$$
(6)
$$\frac{1}{n}\sum _{i=1}^{n}\frac{{\left[N{\mu}_{i}-N\mu \right]}^{2}}{{N}^{2}}=\frac{{\sigma}^{2}}{N}\left[1+2\sum _{j=1}^{N-1}\left(1-\frac{j}{N}\right){\rho}_{j}\right]=\frac{1}{n}\sum _{i=1}^{n}{\mu}_{i}^{2}-{\mu}^{2}$$
(7)
$${\sigma}_{N}^{2}\cong {\sigma}^{2}-\frac{{\sigma}^{2}}{N}\left[1+2\sum _{j=1}^{N-1}\left(1-\frac{j}{N}\right){\rho}_{j}\right]$$
(8)
$${\rho}_{N}=\frac{\sum _{k=1}^{m-\mathit{1}}\left[{T}_{k}-\mu \right]\left[{T}_{k+N}-\mu \right]}{{\sigma}^{2}\left(m-N\right)}$$
(9)
$${\sigma}_{N}^{2}\cong {\sigma}^{2}-\frac{{\sigma}^{2}}{N}\left(1+\left(N-1\right){\rho}_{o}\right)$$
(10)
$$\left\{\begin{array}{c}{\rho}_{N-1}=\frac{{K}_{N}-\sum _{j=1}^{N-2}\left(1-\frac{j}{N}\right){\rho}_{j}}{1-\frac{N-1}{N}}\\ {K}_{N}=\left(\frac{{\sigma}^{2}-\delta {T}^{2}\left(\Delta {t}_{N}\right)}{{\sigma}^{2}}\right)\frac{N}{2}-\frac{1}{2}\end{array}\right\}\phantom{\rule{.5em}{0ex}}N>1$$
(11)
$$\mathrm{log}\left[\delta T\left(\Delta t\right)\right]=\sum _{i=0}^{4}{\phantom{\rule{.2em}{0ex}}\beta}_{i}\phantom{\rule{.2em}{0ex}}\mathrm{log}{\left(\Delta t\right)}^{i}$$
(12)
$$\phantom{\rule{18.2em}{0ex}}={\beta}_{0}+{\beta}_{1}\phantom{\rule{.2em}{0ex}}\mathrm{log}\left(\Delta t\right)+{\beta}_{2}\phantom{\rule{.2em}{0ex}}\mathrm{log}{\left(\Delta t\right)}^{2}+{\beta}_{3}\phantom{\rule{.2em}{0ex}}\mathrm{log}{\left(\Delta t\right)}^{3}+{\beta}_{4}\phantom{\rule{.2em}{0ex}}\mathrm{log}{\left(\Delta t\right)}^{4}$$
(13)
$${\sigma}_{\mu}^{2}\cong \frac{{\sigma}^{2}}{N}\left[1+2\sum _{j=1}^{N-1}\left(1-\frac{j}{N}\right){\rho}_{j}\right]$$
(14)
$${\rho}_{max}\ll \frac{1}{N{Q}^{2}}$$
(15)
$$\left\{\begin{array}{c}{\rho}_{N-1}=\frac{{K}_{N}-\sum _{j=1}^{N-2}\left(1-\frac{j}{N}\right){\rho}_{j}}{1-\frac{N-1}{N}}\\ {K}_{N}=\left(\frac{{\sigma}_{{\mu}_{N}}^{2}}{E\left({x}^{2}\right)}\right)\frac{N}{2}-\frac{1}{2}\end{array}\right\}N>1$$