Probability theory for 3-layer remote sensing in ideal gas law environment

Avishai Ben-David; Charles E. Davidson

doi:10.1364/OE.21.019768

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 T₁, and blackbody radiance $B_{1} = B (T_{1})$ . The target cloud layer is layer 2 with $t_{2}$ , T₂, and B₂. The background layer is layer 3 with $t_{3}$ , T₃, and B₃, and the external source at the end of the line of sight (LOS) has radiance $L_{s}$ .

Fig. 1 Geometry and random variables for each layer of the 3-layer model.

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Transmission terms are modeled as lognormal variates, $t ~ L N (μ_{t}, σ_{t}^{2})$ , whereas temperatures are considered to be normal, $T ~ N (μ_{T}, σ_{T}^{2})$ , where μ and σ² 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,

{\begin{cases} M = x_{1} - x_{2} + x_{3} - x_{4} + x_{5} - x_{6} + x_{7} \\ x_{1} = B_{1}; x_{2} = t_{1} B_{1}; x_{3} = t_{1} B_{2}; x_{4} = t_{1} t_{2} B_{2} \\ x_{5} = t_{1} t_{2} B_{3}; x_{6} = t_{1} t_{2} t_{3} B_{3}; x_{7} = t_{1} t_{2} t_{3} L_{s} \end{cases}} .

We introduce 3 within-layer correlation coefficients into the radiative transfer model (one per layer:

ρ_{i}

for i = 1, 2, 3) to capture the correlation between B(T_i) 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 B(T_i) will be positive. Complete dependence (correlation coefficients

ρ_{1} = ρ_{2} = ρ_{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

ρ_{1} = ρ_{2} = ρ_{3} = 1

(and therefore the effect of correlations will be maximized) but for generality and flexibility we formulate the equations with arbitrary coefficients (

ρ_{1}, ρ_{2}, ρ_{3}

).

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 ~10¹⁹ 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

{\begin{cases} M^{k} = \sum_{} (\begin{array}{l} k \\ k_{1} k_{2} k_{3} k_{4} k_{5} k_{6} k_{7} \end{array}) (^{- 1) k_{2} + k_{4} + k_{6}} x_{1}^{k_{1}} x_{2}^{k_{2}} x_{3}^{k_{3}} x_{4}^{k_{4}} x_{5}^{k_{5}} x_{6}^{k_{6}} x_{7}^{k_{7}} \\ (\begin{array}{l} k \\ k_{1} k_{2} k_{3} k_{4} k_{5} k_{6} k_{7} \end{array}) = \frac{k!}{k_{1}! k_{2}! k_{3}! k_{4}! k_{5}! k_{6}! k_{7}!} \end{cases}}

where the sum is over all combinations of 7 nonnegative integers k_j that sum to k. Each of the x_j’s is a lognormal variate or a product of lognormal variates, which [using (A3)] is also lognormally distributed. Thus, each product of x_j’s appearing in the multinomial series is given as a lognormally distributed variate,

{\begin{cases} x_{1}^{k_{1}} x_{2}^{k_{2}} x_{3}^{k_{3}} x_{4}^{k_{4}} x_{5}^{k_{5}} x_{6}^{k_{6}} x_{7}^{k_{7}} = t_{1}^{n_{1}} B_{1}^{m_{1}} t_{2}^{n_{2}} B_{2}^{m_{2}} t_{3}^{n_{3}} B_{3}^{m_{3}} L_{s}^{m_{s}} ~ L N (μ, σ^{2} + f (ρ_{1}, ρ_{2}, ρ_{3})) \\ n_{i} = \sum_{j = 2 i}^{7} k_{j}, m_{i} = k_{2 i - 1} + k_{2 i}, m_{s} = k_{7} \\ μ = m_{s} μ_{s} + \sum_{i = 1}^{3} (n_{i} μ_{t i} + m_{i} μ_{B i}) \\ σ^{2} = m_{s}^{2} σ_{s}^{2} + \sum_{i = 1}^{3} (n_{i}^{2} σ_{t i}^{2} + m_{i}^{2} σ_{B i}^{2}) \\ f (ρ_{1}, ρ_{2}, ρ_{3}) = 2 \sum_{i = 1}^{3} ρ_{i} n_{i} m_{i} σ_{t i} σ_{B i} \end{cases}}

where μ and σ² are population parameters in absence of correlation (they are not a function of

ρ_{i}

), and f is an adjustment on the population variance, σ², due to correlation. As there are (k + 6)!/(k!6!) terms in the multinomial series, there are (k + 6)!/(k!6!) μ, σ², 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} σ_{t}^{2} + m^{2} σ_{B}^{2}

, a term of the form

2 ρ n m σ_{t} σ_{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} σ_{t}^{2} + m^{2} σ_{B}^{2} \geq 2 n m σ_{t} σ_{B}

(equality occurs when

n = m

and

σ_{t} = σ_{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} σ_{s}^{2}

(the leading term in σ²) is large, since the perturbations in f will be a smaller fraction of the value of σ². Note that all moments of a lognormal distribution have a dependence on the variance parameter, σ² (see Appendix A in [1]), therefore perturbations of σ² 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 $x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{7}^{k_{7}}$ is replaced by

\begin{matrix} E (x_{1}^{k_{1}} x_{2}^{k_{2}} x_{3}^{k_{3}} x_{4}^{k_{4}} x_{5}^{k_{5}} x_{6}^{k_{6}} x_{7}^{k_{7}}) = \exp (μ + \frac{1}{2} (σ^{2} + f)) \\ = E (x_{1}^{k_{1}} x_{2}^{k_{2}} x_{3}^{k_{3}} x_{4}^{k_{4}} x_{5}^{k_{5}} x_{6}^{k_{6}} x_{7}^{k_{7}} | ρ_{i} = 0) \times \exp (\frac{f}{2}) \end{matrix}

where the first line of (4) is obtained from the moment expression of a lognormal distribution (see property (i) of Appendix A in [1]), and

E (x_{1}^{k_{1}} x_{2}^{k_{2}} \dots x_{7}^{k_{7}} | ρ_{i} = 0)

is the moment in absence of correlation (

ρ_{1} = ρ_{2} = ρ_{3} = 0

). Typically standard deviation parameters

σ_{t i}

and

σ_{B i}

are <<1 and a 1st order Taylor expansion can be used to linearize

\exp (\frac{f}{2}) ≅ 1 + \frac{f}{2}

, thus,

{\begin{cases} E (x_{1}^{k_{1}} x_{2}^{k_{2}} x_{3}^{k_{3}} x_{4}^{k_{4}} x_{5}^{k_{5}} x_{6}^{k_{6}} x_{7}^{k_{7}}) = E (x_{1}^{k_{1}} x_{2}^{k_{2}} x_{3}^{k_{3}} x_{4}^{k_{4}} x_{5}^{k_{5}} x_{6}^{k_{6}} x_{7}^{k_{7}} | ρ_{i} = 0) + Δ E \\ Δ E = \frac{1}{2} E (x_{1}^{k_{1}} x_{2}^{k_{2}} x_{3}^{k_{3}} x_{4}^{k_{4}} x_{5}^{k_{5}} x_{6}^{k_{6}} x_{7}^{k_{7}} | ρ_{i} = 0) \times f \end{cases}}

where

Δ E \geq 0

is the perturbation on the expected value of each term in the multinomial series due to the presence of correlation. Inserting

Δ E

into the expression for the k^th moment gives

E (M^{k}) = E (M^{k} | ρ_{i} = 0) + \sum (\begin{array}{l} k \\ k_{1} k_{2} k_{3} k_{4} k_{5} k_{6} k_{7} \end{array}) {(- 1)}^{k_{2} + k_{4} + k_{6}} Δ E

and shows that the effect of correlation is a series of positive contributions with alternating signs (due to factor

{(- 1)}^{k_{2} + k_{4} + k_{6}}

) that will partially cancel the effect of correlation on the moments. As a result the pdf due to the presence of correlation is similar to the one obtained without taking correlation into account. Note that the first order Taylor expansion used to obtain (5) could be performed to higher order if necessary (simply modifying the definition of ΔE), 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

σ^{2}

and (b) cancelation of perturbation terms that appear with opposite signs.

Equations (1)–(6) are formulated for the “H₁ scenario” when the target cloud is present within the field of view. The sensor measurements $M | H_{0}$ for the null hypothesis or “H₀ 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 B₂. 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, $Δ T = B^{- 1} (L_{i n}) - T_{2}$ , is another important quantity in LWIR remote sensing, where $L_{i n}$ 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_in by forcing $(n_{1}, n_{2}, m_{1}, m_{2}) \to 0$ in (3) to remove the presence of $t_{1}$ , $t_{2}$ , B₁, and B₂. Section 3.2.3 in [1] may then be used to determine the moments of ΔT.

3. Results

We experimented with simulations for within-layer correlation coefficients $ρ_{1} = ρ_{2} = ρ_{3} = 1$ for a triethyl phosphate (TEP) cloud observed at 1049 cm⁻¹. The transmission for the i^th layer is given by $t_{i} ~ L N (- r_{i} Q_{i} μ_{T i}^{- 1}, {(r_{i} Q_{i})}^{2} μ_{T i}^{- 4} σ_{T i}^{2})$ where $r_{i}$ is the pathlength, $μ_{T i}$ and $σ_{T i}^{2}$ are population parameters for normally-distributed temperature T_i, and Q_i 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 cm²/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 $α_{v} = Q μ_{T}^{- 1} = 0.1413$ km⁻¹ is taken from a MODTRAN [6] run using the 1976 US standard atmosphere model, and thus $t_{i} ~ L N (- 0.1413 r_{i}, {(0.1413 r_{i})}^{2} μ_{T i}^{- 2} σ_{T i}^{2})$ for $i = 1$ and $i = 3$ . Simulations were run for $ρ_{1} = ρ_{2} = ρ_{3} = 1$ (to observe the maximum effect of correlation) and compared to simulation results with $ρ_{1} = ρ_{2} = ρ_{3} = 0$ .

We experimented with different temperature population parameters $(μ_{T}, σ_{T}^{2})$ 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), $μ_{T i} = (288 K, 292 K, 296 K)$ and $σ_{T i}^{2} = (25 K^{2}, 100 K^{2}, 225 K^{2})$ ; the pathlengths of the 3 layers are the same as used in [1], $r_{i} = (1 k m, 0.1 k m, 9 k m)$ ; and $(μ_{s}, σ_{s}^{2})$ for the external source were taken from reference [1], Table 1 (physical simulation). The transmission of the 3 layers in this example are $E (t_{i}) = (0.868, 0.727, 0.281)$ , i.e., a moderately thick cloud with 73% transmission on average, and standard deviations $\sqrt{V (t_{i})} = (2.13 \cdot 10^{- 3}, 7.94 \cdot 10^{- 3}, 1.81 \cdot 10^{- 2})$ . The blackbodies for the 3 layers have mean radiances (in W/cm²/sr/cm⁻¹) of $E (B_{i}) = (7.35 \cdot 10^{- 6}, 7.99 \cdot 10^{- 6}, 8.73 \cdot 10^{- 6})$ and standard deviations $\sqrt{V (B_{i})} = (6.74 \cdot 10^{- 7}, 1.43 \cdot 10^{- 6}, 2.31 \cdot 10^{- 6})$ . For the cloud layer, this simulation results in a mean density of 3.80·10⁻⁹ g/cm³ of TEP and a mean volume extinction coefficient of 3.19 km⁻¹.

Table 1. Comparison of raw moments for correlated and uncorrelated cases.

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Sampled raw moments in the presence and absence of correlation are tabulated in Table 1 for the radiance under H₀, the radiance under H₁, and the thermal contrast, ΔT. Percent differences are relatively small, showing that the effect of correlation is small. The largest percent differences occur for ΔT, 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.

Table 2. Comparison of central moments for correlated and uncorrelated cases.

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Figure 2 shows pdfs for the radiance under H₁, H₀, and ΔT in the presence and absence of correlation. Histograms computed from sampled data are shown as dotted curves (red for $ρ_{i} = 0$ , green for $ρ_{i} = 1$ ); theoretical pdfs from Johnson S_U fits of moments computed from (2–6) are also shown as solid curves (thick gray for $ρ_{i} = 0$ , black for $ρ_{i} = 1$ ). All pdfs are very close to one another, which serves as a visual confirmation that the effects of within-layer correlations (even for $ρ_{i} = 1$ ) are not very significant, and also shows that the theory approximates the truth well.

Fig. 2 The effect of within layer correlation on the pdf of the radiance for H₀ (top), H₁ (middle), and thermal contrast ΔT (bottom). Histograms from sampled data (dotted curves) and theoretical pdfs (solid curves) are shown. The figure shows that the pdfs computed in the presence of within-layer correlation, $ρ_{i} = 1$ , are very close to the pdfs in the absence of correlation, $ρ_{i} = 0$ . Theoretical pdfs and histograms are also very close to one another.

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Differences between the uncorrelated and correlated cases may also be judged by the location of quantiles. For example, for the M|H₀ 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|H₁ 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 ΔT there is a 0.2K difference in the location of the median versus a difference of 0.77K (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 pdfs 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 $t = \exp (- τ)$ 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 $t \leq 1$ 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 $σ^{2}$ by adding “perturbation” terms in the form of $2 ρ n m σ_{t} σ_{B}$ to terms in the form of $n^{2} σ_{t}^{2} + m^{2} σ_{B}^{2}$ , but since $n^{2} σ_{t}^{2} + m^{2} σ_{B}^{2} \geq 2 n m σ_{t} σ_{B}$ , 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 pdfs for sensor measurements $M | H_{1}$ , $M | H_{0}$ , and for the thermal contrast ΔT 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 $ρ = 1$ – means that the conclusion that the effect of correlation is small remains valid.

In principle we could also incorporate between-layer correlation coefficients $(ρ_{12}, ρ_{13}, ρ_{23})$ 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 $ρ_{i j} = 0$ , 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_j would introduce a correlation between the densities of the components of layers 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 $2 \sum_{i - 1}^{2} \sum_{j = i + 1}^{3} ρ_{i j} m_{i} m_{j} σ_{B_{i}} σ_{B_{j}}$ 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 $\exp (c o n s t / T)$ . It follows that the product $z = t (T) \times B (T) \propto \exp (c o n s t / T)$ 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 $ρ < 1$ . 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 $q = (P \times M W) / R$ , 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 $α_{s}$ (s = 1, 2, …, n), is $τ = \sum α_{s} d_{s} r = r (\sum α_{s} q_{s}) / T$ . Note that the product $α_{s} d_{s}$ is the volume extinction coefficient (units of reciprocal length) for species s. Letting $Q = \sum α_{s} q_{s}$ , the transmission for the layer is given by $t (T) = e^{- τ} = e^{- r Q / T}$ . From [1], if the temperature is normally distributed, $T ~ N (μ_{T}, σ_{T}^{2})$ , a 1st order Taylor expansion for 1/T results in $T^{- 1} ~ N (μ_{T}^{- 1}, μ_{T}^{- 4} σ_{T}^{2})$ . If the partial pressures of the constituents are constant, then the transmission is soley dependent on T and is distributed as $t ~ L N (- r Q μ_{T}^{- 1}, {(r Q)}^{2} μ_{T}^{- 4} σ_{T}^{2})$ . Note that $Q μ_{T}^{- 1}$ is in units of reciprocal length and may be interpreted as the total volume extinction coefficient for the layer, $α_{v}$ . Thus, alternatively, the transmission may be expressed as $t ~ L N (- α_{v} r, {(α_{v} r)}^{2} μ_{T}^{- 2} σ_{T}^{2})$ . Note also that because $μ_{T} > > σ_{T}$ (e.g., $μ_{T} ≅ 300$ K, $σ_{T} < 50$ K), the constraint $t (T) = e^{- α q r / T} \leq 1$ is naturally enforced without introducing truncated lognormal distributions for transmissions as was necessary in [1]. For example, for $μ_{T} / σ_{T} = 5$ , 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 k₁ and k₂ are constants that depend on wavelength (in the Appendix, k₁, k₂, and later, k₃ and k₄, 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 $μ_{x}$ and $σ_{x}$ such that the correct mean and variance are produced. The mean and variance of x are $E = E (e^{k_{2} / T}) - 1 = \exp (k_{2} μ_{T}^{- 1} + \frac{1}{2} k_{2}^{2} μ_{T}^{- 4} σ_{T}^{2}) - 1$ , $V = var (e^{k_{2} / T}) = [\exp (k_{2}^{2} μ_{T}^{- 4} σ_{T}^{2}) - 1] {(E + 1)}^{2}$ , respectively. Using the moment matching conditions (property (ii) in Appendix A of [1]) results in $σ_{x}^{2} = \ln (1 + V E^{- 2})$ , $μ_{x} = \ln E - \frac{1}{2} σ_{x}^{2}$ .

Since $x ~ L N (μ_{x}, σ_{x}^{2})$ , there exists an associated variable $u = \ln x ~ N (μ_{x}, σ_{x}^{2})$ that is a linear transformation of 1/T, $u = \frac{k_{3}}{T} - k_{4}$ , such that $E (\frac{k_{3}}{T} - k_{4}) = μ_{x}$ and $var (\frac{k_{3}}{T} - k_{4}) = σ_{x}^{2}$ . Thus, the lognormal approximation for x is consistent with the functional approximation $x = e^{k_{2} / T} - 1 ≅ e^{(k_{3} / T) - k_{4}}$ with $k_{3} = σ_{x} σ_{T}^{- 1} μ_{T}^{2}$ and $k_{4} = - μ_{x} + σ_{x} σ_{T}^{- 1} μ_{T}$ . As a result,

B (T) = k_{1} x^{- 1} ≅ k_{1} e^{- (k_{3} / T) + k_{4}}

and

B (T) ~ L N (μ_{B}, σ_{B}^{2})

with

μ_{B} = \ln k_{1} - μ_{x} = \ln k_{1} - k_{3} μ_{T}^{- 1} + k_{4}

and

σ_{B}^{2} = σ_{x}^{2} = k_{3}^{2} μ_{T}^{- 4} σ_{T}^{2}

. It is easy to see that

t (T) = e^{- r Q / T}

and B(T) share the same functional dependence on T, and the product

z = t (T) \times B (T) ≅ k_{1} e^{- (k_{3} + r Q) / T + k_{4}}

is lognormally distributed,

z ~ L N (μ_{B} - r Q μ_{T}^{- 1}, {(σ_{B}^{} + r Q μ_{T}^{- 2} σ_{T}^{})}^{2}) .

In this work we are also interested in the effect of partial dependence between t and B. It is well-known that $z = y_{1} y_{2} ~ L N (μ_{1} + μ_{2}, σ_{1}^{2} + σ_{2}^{2} + 2 ρ σ_{1} σ_{2})$ where $y_{1} ~ L N (μ_{1}, σ_{1}^{2})$ , $y_{2} ~ L N (μ_{2}, σ_{2}^{2})$ , and $ρ$ is the correlation coefficient between associated normal variables $u_{1} = \ln y_{1} ~ N (μ_{1}, σ_{1}^{2})$ and $u_{2} = \ln y_{2} ~ N (μ_{2}, σ_{2}^{2})$ . For the special case where $y_{1} = B (T)$ , $y_{2} = t (T)$ , and $ρ \to 1$ , 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 $ρ_{y_{1} y_{2}} = c o r (y_{1}, y_{2}) = (e^{ρ σ_{1} σ_{2}} - 1) {[(e^{σ_{1}^{2}} - 1) (e^{σ_{2}^{2}} - 1)]}^{- 1 / 2}$ .

Combined with the lognormal property that $y^{n} ~ L N (n μ, n^{2} σ^{2})$ , it follows that a product of k correlated lognormal variates (raised to arbitrary powers) can be given by

{\begin{cases} z = \prod_{i = 1}^{k} y_{i}^{n_{i}} ~ L N (\sum_{i = 1}^{k} n_{i} μ_{i}, \sum_{i = 1}^{k} n_{i}^{2} σ_{i}^{2} + 2 \sum_{i = 1}^{k - 1} \sum_{j = i + 1}^{k} ρ_{i j} n_{i} n_{j}) \\ y_{i} ~ L N (μ_{i}, σ_{i}^{2}) \\ ρ_{i j} = c o r (e^{y_{i}}, e^{y_{j}}) \end{cases}}

where

ρ_{i j}

is again defined in the “normal” space. Equation (A3) is used to determine the distribution for terms such as

t_{1}^{n_{1}} t_{2}^{n_{2}} t_{3}^{n_{3}} B_{1}^{m_{1}} B_{2}^{m_{2}} B_{3}^{m_{3}}

that appear in (2-3). Correlation coefficients ρ_ij are set to zero if y_i and y_j are not the respective blackbody radiance and transmission for the same layer (e.g.,

y_{i} = t_{2}

and

y_{j} = B_{2}

).

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).

		$E (x)$	$E (x^{2})$	$E (x^{3})$	$E (x^{4})$
$x = M \| H_{0}$	$ρ_{i} = 0$	7.60 × 10⁻⁶	5.96 × 10⁻¹¹	4.83 × 10⁻¹⁶	4.05 × 10⁻²¹
	$ρ_{i} = 1$	7.56 × 10⁻⁶	5.89 × 10⁻¹¹	4.73 × 10⁻¹⁶	3.90 × 10⁻²¹
	*% diff.*	-0.470	-1.20	-2.23	-3.60
$x = M \| H_{1}$	$ρ_{i} = 0$	7.67 × 10⁻⁶	5.99 × 10⁻¹¹	4.77 × 10⁻¹⁶	3.88 × 10⁻²¹
	$ρ_{i} = 1$	7.63 × 10⁻⁶	5.93 × 10⁻¹¹	4.67 × 10⁻¹⁶	3.77 × 10⁻²¹
	*% diff.*	-0.467	-1.07	-1.83	-2.76
$x = Δ T$	$ρ_{i} = 0$	−2.40	239	−1550	1.69 × 10⁵
	$ρ_{i} = 1$	−2.64	231	−1830	1.60 × 10⁵
	*% diff.*	9.73	-3.41	18.3	-5.53

		$\sqrt{V (x)}$	$S (x)$	$K (x)$
$x = M \| H_{0}$	$ρ_{i} = 0$	1.39 × 10⁻⁶	0.476	3.33
	$ρ_{i} = 1$	1.32 × 10⁻⁶	0.341	3.14
	*% diff.*	-4.64	-28.5	-5.77
$x = M \| H_{1}$	$ρ_{i} = 0$	1.06 × 10⁻⁶	0.415	3.26
	$ρ_{i} = 1$	1.02 × 10⁻⁶	0.311	3.12
	*% diff.*	-4.25	-25.0	-4.32
$x = Δ T$	$ρ_{i} = 0$	15.3	4.17 × 10⁻²	2.99
	$ρ_{i} = 1$	15.0	−1.19 × 10⁻²	3.00
	*% diff.*	-2.02	-129	0.185

		$E (x)$	$E (x^{2})$	$E (x^{3})$	$E (x^{4})$
$x = M \| H_{0}$	$ρ_{i} = 0$	7.60 × 10⁻⁶	5.96 × 10⁻¹¹	4.83 × 10⁻¹⁶	4.05 × 10⁻²¹
	$ρ_{i} = 1$	7.56 × 10⁻⁶	5.89 × 10⁻¹¹	4.73 × 10⁻¹⁶	3.90 × 10⁻²¹
	*% diff.*	-0.470	-1.20	-2.23	-3.60
$x = M \| H_{1}$	$ρ_{i} = 0$	7.67 × 10⁻⁶	5.99 × 10⁻¹¹	4.77 × 10⁻¹⁶	3.88 × 10⁻²¹
	$ρ_{i} = 1$	7.63 × 10⁻⁶	5.93 × 10⁻¹¹	4.67 × 10⁻¹⁶	3.77 × 10⁻²¹
	*% diff.*	-0.467	-1.07	-1.83	-2.76
$x = Δ T$	$ρ_{i} = 0$	−2.40	239	−1550	1.69 × 10⁵
	$ρ_{i} = 1$	−2.64	231	−1830	1.60 × 10⁵
	*% diff.*	9.73	-3.41	18.3	-5.53

		$\sqrt{V (x)}$	$S (x)$	$K (x)$
$x = M \| H_{0}$	$ρ_{i} = 0$	1.39 × 10⁻⁶	0.476	3.33
	$ρ_{i} = 1$	1.32 × 10⁻⁶	0.341	3.14
	*% diff.*	-4.64	-28.5	-5.77
$x = M \| H_{1}$	$ρ_{i} = 0$	1.06 × 10⁻⁶	0.415	3.26
	$ρ_{i} = 1$	1.02 × 10⁻⁶	0.311	3.12
	*% diff.*	-4.25	-25.0	-4.32
$x = Δ T$	$ρ_{i} = 0$	15.3	4.17 × 10⁻²	2.99
	$ρ_{i} = 1$	15.0	−1.19 × 10⁻²	3.00
	*% diff.*	-2.02	-129	0.185

Probability theory for 3-layer remote sensing in ideal gas law environment

Abstract

1. Introduction

2. Theory

3. Results

4. Summary

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

Acknowledgments

References and links

Cited By

Figures (2)

Tables (2)

Equations (9)

Optics Express

Abstract

1. Introduction

2. Theory

3. Results

4. Summary

Appendix. Product of correlated lognormals, t(T)×B(T)

Acknowledgments

References and links

Cited By

Figures (2)

Tables (2)

Equations (9)

Optics Express

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