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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References

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  1. 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]
  2. 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]
  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).

2013 (1)

2012 (1)

Opt. Express (2)

Other (5)

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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Figures (2)

Fig. 1
Fig. 1

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

Fig. 2
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.

Tables (2)

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

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

Equations (9)

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{ 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 }.
{ M k = ( k k 1 k 2 k 3 k 4 k 5 k 6 k 7 ) (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 ( k k 1 k 2 k 3 k 4 k 5 k 6 k 7 )= k! k 1 ! k 2 ! k 3 ! k 4 ! k 5 ! k 6 ! k 7 ! }
{ 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 ~LN( μ, σ 2 +f( ρ 1 , ρ 2 , ρ 3 ) ) n i = j=2i 7 k j , m i = k 2i1 + k 2i , m s = k 7 μ= m s μ s + i=1 3 ( n i μ ti + m i μ Bi ) σ 2 = m s 2 σ s 2 + i=1 3 ( n i 2 σ ti 2 + m i 2 σ Bi 2 ) f( ρ 1 , ρ 2 , ρ 3 )=2 i=1 3 ρ i n i m i σ ti σ Bi }
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+ 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 )×exp( f 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 )=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 )+ΔE ΔE= 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 )×f }
E( M k )=E( M k | ρ i =0 )+ ( k k 1 k 2 k 3 k 4 k 5 k 6 k 7 ) (1) k 2 + k 4 + k 6 ΔE
B(T)= k 1 x 1 k 1 e ( k 3 /T)+ k 4
z~LN( μ B rQ μ T 1 , ( σ B +rQ μ T 2 σ T ) 2 ).
{ z= i=1 k y i n i ~LN( i=1 k n i μ i , i=1 k n i 2 σ i 2 +2 i=1 k1 j=i+1 k ρ ij n i n j ) y i ~LN( μ i , σ i 2 ) ρ ij =cor( e y i , e y j ) }

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