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

Optics Express, Vol. 21, Issue 17, pp. 19768-19777 (2013)

http://dx.doi.org/10.1364/OE.21.019768

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### 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.

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## 1. Introduction

## 2. Theory

*T*

_{1}, and blackbody radiance

*T*

_{2}, and

*B*

_{2}. The background layer is layer 3 with

*T*

_{3}, and

*B*

_{3}, and the external source at the end of the line of sight (LOS) has radiance

*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

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]

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]

*B*(

*T*) is very nearly lognormally distributed.

*M*(see (1) and (3) in [1

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]

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

_{j}*i*= 1, 2, 3) to capture the correlation between

*B*(

*T*) and

_{i}*B*(

*T*) will be positive. Complete dependence (correlation coefficients

_{i}**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*M*, the mean, variance, skewness, and kurtosis (

*E*,

*V*,

*S*, and

*K*, respectively), which are used to fit a Johnson S

_{U}

*E*,

*V*,

*S*, and

*K*are a function of the raw moments (E(

*M*) for

^{k}*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.

*k*

^{th}power of

*M*can be computed with the multinomial theorem [5

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

*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

*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

*k*; for any positive integers

*n*and

*m*, it can be shown that

*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

**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*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.

*k*

^{th}moment of

*M*involves taking the expectation of (2). Because the expectation operator distributes over a sum,

**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*k*

^{th}moment givesand shows that the effect of correlation is a series of positive contributions with alternating signs (due to factor

*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

*H*

_{1}scenario” when the target cloud is present within the field of view. The sensor measurements

*H*

_{0}scenario” (target cloud is absent) may be obtained by letting

*B*

_{2}. Equations (2)–(6) may be used as written by forcing the exponents

*L*may be obtained from (1) by letting

_{in}*L*by forcing

_{in}*B*

_{1}, and

*B*

_{2}. Section 3.2.3 in [1

**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*T*.

## 3. Results

*cm*

^{−1}. The transmission for the

*i*

^{th}layer is given by

*T*, and

_{i}*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

*km*, respectively), an atmospheric volume extinction coefficient of

*km*

^{−1}is taken from a MODTRAN [6

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

*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

_{U}fits of moments computed from (2–6) are also shown as solid curves (thick gray for

## 4. Summary

*between-layer*correlation coefficients

*T*and

_{i}*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

*f*in (3).

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

*d*, of an absorbing species is inversely proportional to temperature, given by

*q*, is given by

*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

*s*= 1, 2, …,

*n*), is

*s*. Letting

**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*T*results in

*T*and is distributed as

*K*,

*K*), the constraint

**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*T*< 0 —and therefore

*t*> 1 —is negligible (less than 3×10

^{−7}).

*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*that appear in the body of the paper). In [1

_{j}**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*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

*x*are

**20**(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]

*t*and

*B*. It is well-known that

*associated normal*variables

## Acknowledgments

## 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 |

2. | A. Ben-David and C. E. Davidson, “Probability theory for 3-layer remote sensing radiative transfer model: errata,” Opt. Express |

3. | M. L. Salby, |

4. | L. D. Landau and E. M. Lifshitz, |

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, |

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(010.1320) Atmospheric and oceanic optics : Atmospheric transmittance

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(030.5620) Coherence and statistical optics : Radiative transfer

(300.6340) Spectroscopy : Spectroscopy, infrared

(280.4991) Remote sensing and sensors : Passive remote sensing

(290.6815) Scattering : Thermal emission

**ToC Category:**

Remote Sensing

**History**

Original Manuscript: May 10, 2013

Revised Manuscript: July 8, 2013

Manuscript Accepted: July 9, 2013

Published: August 15, 2013

**Citation**

Avishai Ben-David and Charles E. Davidson, "Probability theory for 3-layer remote sensing in ideal gas law environment," Opt. Express **21**, 19768-19777 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19768

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### References

- A. Ben-David and C. E. Davidson, “Probability theory for 3-layer remote sensing radiative transfer model: univariate case,” Opt. Express20(9), 10004–10033 (2012), doi:. [CrossRef] [PubMed]
- A. Ben-David and C. E. Davidson, “Probability theory for 3-layer remote sensing radiative transfer model: errata,” Opt. Express21(10), 11852 (2013), doi:. [CrossRef] [PubMed]
- M. L. Salby, Fundamentals of Atmospheric Physics (Academic, 1996).
- L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, 1969).
- National Institute of Standards and Technology, (2010). “NIST Digital Library of Mathematical Functions”. Section 26.4.9. http://dlmf.nist.gov/26.4#ii
- MODerate resolution atmospheric TRANsmission (MODTRAN), atmospheric radiative transfer model software. http://modtran5.com
- A. Stuart and K. Ord, Kendall’s Advanced Theory of Statistics, Volume I (Hodder Arnold, 1994).

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