## Abstract

We extend the probability model for 3-layer radiative transfer [Opt. Express **20, **10004 (2012)] to ideal gas conditions where a correlation exists between transmission and temperature of each of the 3 layers. The effect on the probability density function for the at-sensor radiances is surprisingly small, and thus the added complexity of addressing the correlation can be avoided. The small overall effect is due to (a) small perturbations by the correlation on variance population parameters and (b) cancelation of perturbation terms that appear with opposite signs in the model moment expressions.

© 2013 OSA

## 1. Introduction

In a recent paper [1,2] a general probability theory for a 3-layer radiative transfer model in the long wavelength infrared (LWIR) spectral region was presented. With the theory, one can characterize fluctuations in spectral radiance due to variations in the physical environment, which is of interest for evaluating sensor performance and developing detection algorithms for clutter-noise-limited scenarios. The key parameters that determine detection performance are the temperature and transmission of each layer in the atmosphere. In the model (under local thermodynamic equilibrium), each layer was characterized by a temperature, *T*, and a transmission, *t*, which were assumed to be independent and therefore uncorrelated. Layer transmission depends on the density of attenuating constituents, which in turn may depend on temperature. Therefore, the transmission, *t*, and Planck blackbody function, *B*(*T*), may actually be correlated, raising a question regarding the effect of neglecting correlation in [1]. We were not able to predict *a priori* whether neglecting correlation (which was done for convenience in [1]) would have a large impact. The ideal gas law [3] is used in many scenarios to model the behavior of gases in the atmosphere. This law describes the dependence between temperature and molecular density, and predicts that fluctuations in density will depend on fluctuations in temperature and partial pressure. In this paper we extend [1] to address the dependence (correlation) between density and temperature and study its effect on the statistics of the 3-layer remote sensing geometry.

The remainder of this paper is organized as follows. Section 2 presents the modifications to the model that result when the dependency between transmission and temperature is taken into account. Section 3 presents simulation results and discussion regarding the effect of correlation on the model. A summary follows in Section 4. Details regarding ideal gas law and the resulting correlation between transmission and temperature are in the Appendix.

## 2. Theory

The random variables for the 3 layers and the external source are shown in Fig. 1. The foreground layer is layer 1 with transmission ${t}_{1}$, temperature *T*_{1}, and blackbody radiance ${B}_{1}=B({T}_{1})$. The target cloud layer is layer 2 with ${t}_{2}$, *T*_{2}, and *B*_{2}. The background layer is layer 3 with ${t}_{3}$, *T*_{3}, and *B*_{3}, and the external source at the end of the line of sight (LOS) has radiance ${L}_{s}$.

Transmission terms are modeled as lognormal variates, $t~LN({\mu}_{t},{\sigma}_{t}^{2})$, whereas temperatures are considered to be normal, $T~N({\mu}_{T},{\sigma}_{T}^{2})$, where *μ* and *σ*^{2} are the population parameters of the distributions. Please note that the mean and variance of a lognormal variate functions of the population parameters (see Appendix A in [1]). In [1], it was shown that *B*(*T*) is very nearly lognormally distributed.

The total at-sensor radiance, *M* (see (1) and (3) in [1]), can be written as a function of seven terms, *x _{j}*,

*i*= 1, 2, 3) to capture the correlation between

*B*(

*T*) and ${t}_{i}$ caused by a dependence between density and temperature. Using the ideal gas law (see Appendix for details) transmission increases with increasing temperature, since density (and therefore optical depth) decreases with temperature. Thus, the correlation coefficient between ${t}_{i}$ and

_{i}*B*(

*T*) will be positive. Complete dependence (correlation coefficients ${\rho}_{1}={\rho}_{2}={\rho}_{3}=1$) is caused when the partial pressures of the attenuating gases are constant. Correlation values less than one can result due to fluctuations in partial pressure. We are interested in the behavior for the limiting case when ${\rho}_{1}={\rho}_{2}={\rho}_{3}=1$ (and therefore the effect of correlations will be maximized) but for generality and flexibility we formulate the equations with arbitrary coefficients (${\rho}_{1},{\rho}_{2},{\rho}_{3}$).

_{i}Equation (1) is the radiative transfer equation at local thermodynamic equilibrium (i.e., no change in the optical properties of the medium while photons are traveling). During the measurement time of the sensor, air currents, turbulence and other sources cause changes in the macroscopic state of the environment, resulting in temperature and density fluctuations. The relaxation time needed to restore equilibrium is so short compared to these changes that thermodynamic equilibrium is always maintained, and the ideal gas law (and the radiative transfer equation) is always valid. It is important not to confuse these fluctuations with the fluctuations predicted by statistical thermodynamics [4, Chapter 12]. In [4, Eq. (114).5] it is stated that in a closed system when fluctuations are small, the fluctuations between temperature and volume are uncorrelated (volume fluctuations are important because density is the ratio of number of molecules to volume: for a fixed number of molecules, fluctuations in density are determined by fluctuations in volume). The fluctuations in [4] are fluctuations *away* from thermodynamic equilibrium, which occur on time scales that are orders of magnitude smaller than those considered in our work. (For example, given a density of air molecules of ~10^{19} *cm*^{−3} moving on average at the speed of sound at standard temperature and pressure, the time per collision is on the order of 10^{−11} *s*. Usually only a few collisions are necessary to establish thermodynamic equilibrium, and thus a reasonable estimate of the relaxation time is on the order of nanoseconds or less.) Thus, the statement from [4] that temperature and volume fluctuations are uncorrelated does not apply to our physical scenario.

In our probability model [1] we compute the first four central moments of the radiance *M*, the mean, variance, skewness, and kurtosis (*E*, *V*, *S*, and *K*, respectively), which are used to fit a Johnson S_{U} *pdf*. *E*, *V*, *S*, and *K* are a function of the raw moments (E(*M ^{k}* ) for

*k*= 1, 2, 3, 4). In this paper, we modify the expressions for the first 4 raw moments to include the effect of within-layer correlations.

The *k*^{th} power of *M* can be computed with the multinomial theorem [5] as

*k*that sum to

_{j}*k*. Each of the

*x*’s is a lognormal variate or a product of lognormal variates, which [using (A3)] is also lognormally distributed. Thus, each product of

_{j}*x*’s appearing in the multinomial series is given as a lognormally distributed variate,

_{j}*μ*and

*σ*

^{2}are population parameters in absence of correlation (they are not a function of ${\rho}_{i}$), and

*f*is an adjustment on the population variance,

*σ*

^{2}, due to correlation. As there are (

*k*+ 6)!/(

*k*!6!) terms in the multinomial series, there are (

*k*+ 6)!/(

*k*!6!)

*μ*,

*σ*

^{2}, and

*f*parameters. The correlation is captured by a modification of the variance parameter for each term in the multinomial series. The modification factor,

*f*, is always positive (since all correlation coefficients are expected to be positive) and thus the effect of correlation is to increase the variance parameter of each term in the multinomial series: for each term of the form ${n}^{2}{\sigma}_{t}^{2}+{m}^{2}{\sigma}_{B}^{2}$, a term of the form $2\rho \text{\hspace{0.17em}}n\text{\hspace{0.17em}}m\text{\hspace{0.17em}}{\sigma}_{t}{\sigma}_{B}$ is added. Note that the range of ${n}_{i}$ and ${m}_{i}$ is from 0 to

*k*; for any positive integers

*n*and

*m*, it can be shown that ${n}^{2}{\sigma}_{t}^{2}+{m}^{2}{\sigma}_{B}^{2}\ge 2n\text{\hspace{0.17em}}m\text{\hspace{0.17em}}{\sigma}_{t}{\sigma}_{B}$ (equality occurs when $n=m$ and ${\sigma}_{t}={\sigma}_{B}$). Hence it can already be anticipated that the effect of correlation will be “small” (or at least not “too large”—at most increasing each variance parameter by a factor of 2). The effect is further limited in magnitude when ${m}_{s}^{2}{\sigma}_{s}^{2}$ (the leading term in

*σ*

^{2}) is large, since the perturbations in

*f*will be a smaller fraction of the value of

*σ*

^{2}. Note that all moments of a lognormal distribution have a dependence on the variance parameter,

*σ*

^{2}(see Appendix A in [1]), therefore perturbations of

*σ*

^{2}will affect

*E*,

*V*,

*S*, and

*K*. If the perturbation is small, then the expected effect on the moments will be small, too.

Computing the *k*^{th} moment of *M* involves taking the expectation of (2). Because the expectation operator distributes over a sum, $E({M}^{k})$ can be given by (2) where ${x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {x}_{7}^{{k}_{7}}$ is replaced by

*k*

^{th}moment gives

*E*), and would not affect (6) or the partial cancelation of terms due to opposite signs. Therefore, the effect of within-layer correlation is reduced by two considerations, (a) “small” perturbations of the variance parameters ${\sigma}^{2}$ and (b) cancelation of perturbation terms that appear with opposite signs.

Equations (1)–(6) are formulated for the “*H*_{1} scenario” when the target cloud is present within the field of view. The sensor measurements $M|{H}_{0}$for the null hypothesis or “*H*_{0} scenario” (target cloud is absent) may be obtained by letting ${t}_{2}\to 1$in (1), resulting in ${x}_{3}={x}_{4}$ and the disappearance of all terms that depend on *B*_{2}. Equations (2)–(6) may be used as written by forcing the exponents ${n}_{2}\to 0$, and ${m}_{2}\to 0$ (to remove ${B}_{2}$).

The thermal contrast, $\Delta T={B}^{-1}({L}_{in})-{T}_{2}$, is another important quantity in LWIR remote sensing, where ${L}_{in}$ is the radiance incident on the back of the 2nd layer. *L _{in}* may be obtained from (1) by letting ${t}_{1}\to 1$ and ${t}_{2}\to 1$. Equations (2)-(6) may be used to compute the moments of

*L*by forcing $({n}_{1},{n}_{2},{m}_{1},{m}_{2})\to 0$ in (3) to remove the presence of ${t}_{1}$, ${t}_{2}$,

_{in}*B*

_{1}, and

*B*

_{2}. Section 3.2.3 in [1] may then be used to determine the moments of Δ

*T*.

## 3. Results

We experimented with simulations for within-layer correlation coefficients ${\rho}_{1}={\rho}_{2}={\rho}_{3}=1$ for a triethyl phosphate (TEP) cloud observed at 1049 *cm*^{−1}. The transmission for the *i*^{th} layer is given by ${t}_{i}~LN(-{r}_{i}{Q}_{i}{\mu}_{Ti}^{-1},{({r}_{i}{Q}_{i})}^{2}{\mu}_{Ti}^{-\text{\hspace{0.17em}}4}{\sigma}_{Ti}^{2})$ where ${r}_{i}$ is the pathlength, ${\mu}_{Ti}$ and ${\sigma}_{Ti}^{2}$ are population parameters for normally-distributed temperature *T _{i}*, and

*Q*depends on the absorption coefficients and partial pressures for the absorbing species in the layer (see Appendix for details). For the cloud layer (layer 2), the mass-absorption coefficient of TEP is 8389

_{i}*cm*

^{2}/

*g*, the molecular weight is 182.16

*g*/

*mol*, and the partial pressure is computed from the desired concentration of TEP in

*ppm*by volume (in dry air). For the 1st and 3rd layers (ambient atmospheric layers with pathlengths ${r}_{1}$ and ${r}_{3}$

*km*, respectively), an atmospheric volume extinction coefficient of ${\alpha}_{v}=Q{\mu}_{T}^{-1}=0.1413$

*km*

^{−1}is taken from a MODTRAN [6] run using the 1976 US standard atmosphere model, and thus ${t}_{i}~LN(-0.1413{r}_{i},\text{\hspace{0.17em}}{(0.1413{r}_{i})}^{2}{\mu}_{Ti}^{-\text{\hspace{0.17em}}2}{\sigma}_{Ti}^{2})$ for $i=1$ and $i=3$. Simulations were run for ${\rho}_{1}={\rho}_{2}={\rho}_{3}=1$ (to observe the maximum effect of correlation) and compared to simulation results with ${\rho}_{1}={\rho}_{2}={\rho}_{3}=0$.

We experimented with different temperature population parameters $({\mu}_{T},{\sigma}_{T}^{2})$ and various pathlengths $({r}_{1},{r}_{2},{r}_{3})$of the 3 layers (which serve to scale the optical depth). We show typical results for a TEP cloud with 0.5 *ppm* concentration. The population parameters for temperature of the 3 layers (foreground, cloud, background) were chosen to be distinct (to accentuate the difference between the layers), ${\mu}_{Ti}=(288\text{\hspace{0.17em}}K,\text{\hspace{0.17em}}292\text{\hspace{0.17em}}K,\text{\hspace{0.17em}}296\text{\hspace{0.17em}}K)$ and ${\sigma}_{Ti}^{2}=(25\text{\hspace{0.17em}}{K}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}100\text{\hspace{0.17em}}{K}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}225\text{\hspace{0.17em}}{K}^{2})$; the pathlengths of the 3 layers are the same as used in [1], ${r}_{i}=(1\text{\hspace{0.17em}}km,\text{\hspace{0.17em}}0.1\text{\hspace{0.17em}}km,\text{\hspace{0.17em}}9\text{\hspace{0.17em}}km)$; and $({\mu}_{s},{\sigma}_{s}^{2})$ for the external source were taken from reference [1], Table 1 (physical simulation). The transmission of the 3 layers in this example are $E({t}_{i})=(0.868,\text{\hspace{0.17em}}0.727,\text{\hspace{0.17em}}0.281)$, i.e., a moderately thick cloud with 73% transmission on average, and standard deviations $\sqrt{V({t}_{i})}=(2.13\cdot {10}^{-3},\text{\hspace{0.17em}}7.94\cdot {10}^{-3},\text{\hspace{0.17em}}1.81\cdot {10}^{-2})$. The blackbodies for the 3 layers have mean radiances (in *W*/*cm*^{2}/*sr*/*cm*^{−1}) of $E({B}_{i})=(7.35\cdot {10}^{-6},\text{\hspace{0.17em}}7.99\cdot {10}^{-6},\text{\hspace{0.17em}}8.73\cdot {10}^{-6})$ and standard deviations $\sqrt{V({B}_{i})}=(6.74\cdot {10}^{-7},\text{\hspace{0.17em}}1.43\cdot {10}^{-6},\text{\hspace{0.17em}}2.31\cdot {10}^{-6})$. For the cloud layer, this simulation results in a mean density of 3.80·10^{−9} *g*/*cm*^{3} of TEP and a mean volume extinction coefficient of 3.19 *km*^{−1}.

Sampled raw moments in the presence and absence of correlation are tabulated in Table 1 for the radiance under *H*_{0}, the radiance under *H*_{1}, and the thermal contrast, Δ*T*. Percent differences are relatively small, showing that the effect of correlation is small. The largest percent differences occur for Δ*T*, possibly because fewer terms impact the computation of *L _{in}*, and thus there is a reduced likelihood of correlation effects cancelling. Table 2 shows central moments, where the standard deviation is displayed instead of the variance to facilitate comparison to the mean (in Table 1). Percent differences for the skewness are relatively large, however, skewness values are small: absolute differences in skewness of 0.1 to 0.2 units are not very significant (a standard deviation of sample skewness of 0.1 results from ~600 samples drawn from a normal random variable, as can be deduced from [7], p452, Exercise 12.9). Overall, the effect of correlation on the central moments is small.

Figure 2 shows *pdf*s for the radiance under *H*_{1}, *H*_{0}, and Δ*T* in the presence and absence of correlation. Histograms computed from sampled data are shown as dotted curves (red for ${\rho}_{i}=0$, green for ${\rho}_{i}=1$); theoretical *pdf*s from Johnson S_{U} fits of moments computed from (2–6) are also shown as solid curves (thick gray for${\rho}_{i}=0$, black for${\rho}_{i}=1$). All *pdf*s are very close to one another, which serves as a visual confirmation that the effects of within-layer correlations (even for ${\rho}_{i}=1$) are not very significant, and also shows that the theory approximates the truth well.

Differences between the uncorrelated and correlated cases may also be judged by the location of quantiles. For example, for the *M|H*_{0} radiance, there is a 0.03% difference in the location of the median and a 1.5% difference in the location of the 90th percentile. For the *M|H*_{1} radiance, there is about a 0.2% difference in the median whereas there is a 1.2% difference in the location of the 90th percentile. For Δ*T* there is a 0.2*K* difference in the location of the median versus a difference of 0.77*K* (a 4% difference) for the 90th percentile. This shows that the effects of correlation have a larger relative impact on the tails of the distributions, however the small values confirm the similarity of the *pdf*s and reinforces the small impact of correlation.

## 4. Summary

An extension of a probability theory model [1] for passive infrared remote sensing is given for the practical scenario where the transmission of each layer is correlated with temperature. Correlation is caused by the dependence of the transmission on the density of absorbing gases within the layer. In LWIR passive remote sensing, the temperatures of atmospheric layers and the target (a gaseous cloud) are key parameters that determines the performance of a sensor in detecting the presence of a target embedded in a cluttered environment. The effect of neglecting the correlation between temperature and transmission [1] had to be addressed, since *a priori* we could not predict its impact. In this paper we investigated the effects of correlation between density and temperature given by the ideal gas law. In the ideal gas law density is inversely proportional to temperature, which introduces a correlation between the Planck blackbody function, *B*(*T*), and the transmission function $t=\mathrm{exp}(-\tau )$ where the optical depth, *τ*, is proportional to density. On one hand the presence of correlation adds complications to the probability model, but on the other hand the dependence of transmission on temperature practically ensures $t\le 1$ without the need to use truncated lognormal distributions (as needed in [1]). We were initially surprised to find that the effect of within-layer correlation did not significantly affect the *pdf* of the sensor measurements. The weak sensitivity to the correlation can be seen in two ways: (a) inspection of (3) shows that the effect of the correlation is to modify the population parameter ${\sigma}^{2}$ by adding “perturbation” terms in the form of $2\rho \text{\hspace{0.17em}}n\text{\hspace{0.17em}}m\text{\hspace{0.17em}}{\sigma}_{t}{\sigma}_{B}$ to terms in the form of ${n}^{2}{\sigma}_{t}^{2}+{m}^{2}{\sigma}_{B}^{2}$, but since ${n}^{2}{\sigma}_{t}^{2}+{m}^{2}{\sigma}_{B}^{2}\ge 2n\text{\hspace{0.17em}}m\text{\hspace{0.17em}}{\sigma}_{t}{\sigma}_{B}$, the perturbations are “small”; and (b) inspection of (6) shows that the moments depend on a sum of perturbation terms that appear with opposite signs and thus will—at least partially—cancel out. In our simulations we found that the *pdf*s for sensor measurements $M|{H}_{1}$, $M|{H}_{0}$, and for the thermal contrast Δ*T* are very close to those that are produced in the absence of within-layer correlation, hence the theoretical derivations given in [1] can be used without the need to introduce within-layer correlations.

The dependence between the density of an attenuating vapor and temperature predicted by the ideal gas law depends on the partial pressure of the gas. The maximum correlation (dependence) between temperature and transmission occurs when the partial pressure is constant. Considering this case serves as an upper bound on the potential impact of correlation between temperature and transmission. Fluctuations in partial pressure or the presence of other components (e.g., aerosols) within a layer whose density does not follow the ideal gas law will reduce the correlation. The fact that the effect of correlation is insignificant – even in a scenario where its impact has been maximized $\rho =1$– means that the conclusion that the effect of correlation is small remains valid.

In principle we could also incorporate *between-layer* correlation coefficients $({\rho}_{12},{\rho}_{13},{\rho}_{23})$ to capture dependencies between the temperatures of the 3 layers. Including between-layer correlation coefficients may be necessary for long time-scale measurements (e.g., for diurnal cycles that may affect several layers in the same way). We are typically interested in much shorter time-scales and generally assume${\rho}_{ij}=0$, hence we did not generalize the model in this way. Note that due to the ideal gas law, correlation between temperatures *T _{i}* and

*T*would introduce a correlation between the densities of the components of layers

_{j}*i*and

*j*, and therefore would also introduce correlation between transmissions ${t}_{i}$ and ${t}_{j}$. If between-layer correlation coefficients were to be added, the effect would be to simply add the quantity $2{\displaystyle \sum {}_{i-1}^{2}{\displaystyle \sum {}_{j=i+1}^{3}}}{\rho}_{ij}{m}_{i}{m}_{j}{\sigma}_{{B}_{i}}{\sigma}_{{B}_{j}}$ to the definition of

*f*in (3).

## Appendix. Product of correlated lognormals, $t(T)\times B(T)$

In this Appendix we obtain the *pdf* of $z=t(T)\times B(T)$ by showing that both *t*(*T*) and *B*(*T*) are proportional to $\mathrm{exp}(const/T)$. It follows that the product $z=t(T)\times B(T)\propto \mathrm{exp}(const/T)$ is lognormally distributed if 1/*T* is normally distributed. Then, we show a more general case for *z* when *t* and *B* are not entirely dependent and thus have a correlation coefficient $\rho <1$. Finally, we give an equation regarding the product of an arbitrary product of correlated lognormal variates (each raised to an arbitrary power) that is useful in deriving (3).

In the ideal gas law [3], the density, *d*, of an absorbing species is inversely proportional to temperature, given by $d=q/T$. The proportionality, *q*, is given by $q=(P\text{\hspace{0.17em}}\times MW)/R$, where *P* is the partial pressure, *MW* is the molecular weight, and *R* is the universal gas constant. The optical depth for a layer with *n* absorbing constituents, each with mass extinction coefficient ${\alpha}_{s}$ (*s* = 1, 2, …, *n*), is $\tau ={\displaystyle \sum {\alpha}_{s}{d}_{s}r=}r\left({\displaystyle \sum {\alpha}_{s}{q}_{s}}\right)/T$. Note that the product ${\alpha}_{s}{d}_{s}$ is the volume extinction coefficient (units of reciprocal length) for species *s*. Letting $Q={\displaystyle \sum {\alpha}_{s}{q}_{s}}$, the transmission for the layer is given by $t(T)={e}^{-\tau}={e}^{-rQ/T}$. From [1], if the temperature is normally distributed, $T~N({\mu}_{T},{\sigma}_{T}^{2})$, a 1st order Taylor expansion for 1/*T* results in ${T}^{-1}~N({\mu}_{T}^{-1},\text{\hspace{0.17em}}{\mu}_{T}^{-\text{\hspace{0.17em}}4}{\sigma}_{T}^{2})$. If the partial pressures of the constituents are constant, then the transmission is soley dependent on *T* and is distributed as $t~LN(-rQ{\mu}_{T}^{-1},\text{\hspace{0.17em}}{(rQ)}^{2}{\mu}_{T}^{-\text{\hspace{0.17em}}4}{\sigma}_{T}^{2})$. Note that $Q{\mu}_{T}^{-1}$ is in units of reciprocal length and may be interpreted as the total volume extinction coefficient for the layer, ${\alpha}_{v}$. Thus, alternatively, the transmission may be expressed as $t~LN(-{\alpha}_{v}r,\text{\hspace{0.17em}}{({\alpha}_{v}r)}^{2}{\mu}_{T}^{-\text{\hspace{0.17em}}2}{\sigma}_{T}^{2})$. Note also that because ${\mu}_{T}>>{\sigma}_{T}$ (e.g., ${\mu}_{T}\cong 300$*K*, ${\sigma}_{T}<50$*K*), the constraint $t(T)={e}^{-\alpha qr/T}\le 1$ is naturally enforced without introducing truncated lognormal distributions for transmissions as was necessary in [1]. For example, for ${\mu}_{T}/{\sigma}_{T}=5$, the probability for *T* < 0 —and therefore *t* > 1 —is negligible (less than 3×10^{−7}).

The Planck blackbody function is given by $B(T)={k}_{1}{({e}^{{k}_{2}/T}-1)}^{-1}$, where *k*_{1} and *k*_{2} are constants that depend on wavelength (in the Appendix, *k*_{1}, *k*_{2}, and later, *k*_{3} and *k*_{4}, are constants not to be confused with the indices *k _{j}* that appear in the body of the paper). In [1] (section 3.1.1), we used a lognormal approximation for the term $x={e}^{{k}_{2}/T}-1$ to show that

*B*(

*T*) is lognormally distributed. The approximation for

*x*as a lognormal variate involves using the first-order Taylor for 1/T and finding population parameters ${\mu}_{x}$ and ${\sigma}_{x}$ such that the correct mean and variance are produced. The mean and variance of

*x*are $E=\text{E}({e}^{{k}_{2}/T})-1=\mathrm{exp}({k}_{2}{\mu}_{T}^{-1}+{\scriptscriptstyle \frac{1}{2}}{k}_{2}^{2}{\mu}_{T}^{-4}{\sigma}_{T}^{2})-1$, $V=\mathrm{var}({e}^{{k}_{2}/T})=[\mathrm{exp}({k}_{2}^{2}{\mu}_{T}^{-4}{\sigma}_{T}^{2})-1]{(E+1)}^{2}$, respectively. Using the moment matching conditions (property (ii) in Appendix A of [1]) results in ${\sigma}_{x}^{2}=\mathrm{ln}(1+V{E}^{-2})$, ${\mu}_{x}=\mathrm{ln}E-{\scriptscriptstyle \frac{1}{2}}{\sigma}_{x}^{2}$.

Since $x~LN({\mu}_{x},{\sigma}_{x}^{2})$, there exists an associated variable $u=\mathrm{ln}x~N({\mu}_{x},{\sigma}_{x}^{2})$ that is a linear transformation of 1/T, $u={\scriptscriptstyle \frac{{k}_{3}}{T}}-{k}_{4}$, such that $E({\scriptscriptstyle \frac{{k}_{3}}{T}}-{k}_{4})={\mu}_{x}$ and $\mathrm{var}({\scriptscriptstyle \frac{{k}_{3}}{T}}-{k}_{4})={\sigma}_{x}^{2}$. Thus, the lognormal approximation for *x* is consistent with the functional approximation $x={e}^{{k}_{2}/T}-1\cong {e}^{({k}_{3}/T)-{k}_{4}}$ with ${k}_{3}={\sigma}_{x}{\sigma}_{T}^{-1}{\mu}_{T}^{2}$ and ${k}_{4}=-{\mu}_{x}+{\sigma}_{x}{\sigma}_{T}^{-1}{\mu}_{T}$. As a result,

*B*(

*T*) share the same functional dependence on

*T*, and the product $z=t(T)\times B(T)\cong {k}_{1}{e}^{-({k}_{3}+rQ)/T+{k}_{4}}$ is lognormally distributed,

In this work we are also interested in the effect of partial dependence between *t* and *B*. It is well-known that $z={y}_{1}{y}_{2}~LN({\mu}_{1}+{\mu}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma}_{1}^{2}+{\sigma}_{2}^{2}+2\rho {\sigma}_{1}{\sigma}_{2})$ where ${y}_{1}~LN({\mu}_{1},{\sigma}_{1}^{2})$, ${y}_{2}~LN({\mu}_{2},{\sigma}_{2}^{2})$, and $\rho $ is the correlation coefficient between *associated normal* variables ${u}_{1}=\mathrm{ln}{y}_{1}~N({\mu}_{1},{\sigma}_{1}^{2})$ and ${u}_{2}=\mathrm{ln}{y}_{2}~N({\mu}_{2},{\sigma}_{2}^{2})$. For the special case where ${y}_{1}=B(T)$, ${y}_{2}=t(T)$, and$\rho \to 1$, this result reproduces (A2). Note that we always give the correlation coefficient in terms of the associated normal parameters ${u}_{1}$ and ${u}_{2}$ (i.e., in the “normal space”), however, if needed the correlation coefficient between lognormal variates ${y}_{1}$ and ${y}_{2}$ is given as ${\rho}_{{y}_{1}{y}_{2}}=cor({y}_{1},{y}_{2})=({e}^{\rho {\sigma}_{1}{\sigma}_{2}}-1){[({e}^{{\sigma}_{1}^{2}}-1)({e}^{{\sigma}_{2}^{2}}-1)]}^{-1/2}$.

Combined with the lognormal property that ${y}^{n}~LN(n\mu ,{n}^{2}{\sigma}^{2})$, it follows that a product of *k* correlated lognormal variates (raised to arbitrary powers) can be given by

*ρ*are set to zero if

_{ij}*y*and

_{i}*y*are not the respective blackbody radiance and transmission for the same layer (e.g., ${y}_{i}={t}_{2}$ and ${y}_{j}={B}_{2}$).

_{j}## Acknowledgments

We thank Prof. Seth Lichter of Northwestern University for stimulating discussions about thermodynamics.

## References and links

**1. **A. Ben-David and C. E. Davidson, “Probability theory for 3-layer remote sensing radiative transfer model: univariate case,” Opt. Express **20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

**2. **A. Ben-David and C. E. Davidson, “Probability theory for 3-layer remote sensing radiative transfer model: errata,” Opt. Express **21**(10), 11852 (2013), doi:. [CrossRef] [PubMed]

**3. **M. L. Salby, *Fundamentals of Atmospheric Physics* (Academic, 1996).

**4. **L. D. Landau and E. M. Lifshitz, *Statistical Physics* (Pergamon, 1969).

**5. **National Institute of Standards and Technology, (2010). “NIST Digital Library of Mathematical Functions”. Section 26.4.9. http://dlmf.nist.gov/26.4#ii

**6. **MODerate resolution atmospheric TRANsmission (MODTRAN), atmospheric radiative transfer model software. http://modtran5.com

**7. **A. Stuart and K. Ord, *Kendall’s Advanced Theory of Statistics, Volume I* (Hodder Arnold, 1994).