Abstract

A probability model for a 3-layer radiative transfer model (foreground layer, cloud layer, background layer, and an external source at the end of line of sight) has been developed. The 3-layer model is fundamentally important as the primary physical model in passive infrared remote sensing. The probability model is described by the Johnson family of distributions that are used as a fit for theoretically computed moments of the radiative transfer model. From the Johnson family we use the SU distribution that can address a wide range of skewness and kurtosis values (in addition to addressing the first two moments, mean and variance). In the limit, SU can also describe lognormal and normal distributions. With the probability model one can evaluate the potential for detecting a target (vapor cloud layer), the probability of observing thermal contrast, and evaluate performance (receiver operating characteristics curves) in clutter-noise limited scenarios. This is (to our knowledge) the first probability model for the 3-layer remote sensing geometry that treats all parameters as random variables and includes higher-order statistics.

© 2012 OSA

Full Article  |  PDF Article

Errata

Avishai Ben-David and Charles E. Davidson, "Probability theory for 3-layer remote sensing radiative transfer model: errata," Opt. Express 21, 11852-11852 (2013)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-21-10-11852

References

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2009 (1)

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402, 733402-16 (2009).
[CrossRef]

2008 (3)

2007 (1)

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wirel. Comm. 6(7), 2690–2699 (2007).
[CrossRef]

2005 (3)

J. Kerekes and J. Baum, “Full spectrum spectral imaging system analytical model,” IEEE Trans. Geosci. Rem. Sens. 43(3), 571–580 (2005).
[CrossRef]

A. Ben-David, S. K. Holland, G. Laufer, and J. D. Baker, “Measurements of atmospheric brightness temperature fluctuations and their implications on passive remote sensing,” Opt. Express 13(22), 8781–8800 (2005).
[CrossRef] [PubMed]

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

2004 (1)

D. E. Lane-Veron and C. J. Somerville, “Stochastic theory of radiative transfer through generalized cloud fields,” J. Geophys. Res. 109(D18), D18113 (2004).
[CrossRef]

2003 (3)

2001 (1)

E. Limpert, W. A. Stahel, and M. Abbt, “Lognormal distributions across the sciences: keys and clues,” Bioscience 51(5), 341–352 (2001).
[CrossRef]

1996 (1)

1995 (1)

1991 (1)

G. L. Stephens, P. M. Gabriel, and S.-C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20(2-3), 139–175 (1991).
[CrossRef]

1980 (1)

R. E. Wheeler, “Quantile estimators of Johnson curve parameters,” Biometrika 67(3), 725–728 (1980).
[CrossRef]

1976 (1)

I. D. Hill, R. Hill, and R. L. Holder, “Algorithm AS99: fitting Johnson curves by moments,” J. R. Stat. Soc., Ser. C, Appl. Stat. 25(2), 180–189 (1976).

1960 (1)

L. F. Fenton, “The sum of lognormal probability distributions in scatter transmission systems,” IRE Trans. Commun. Syst. CS-8, 57–67 (1960).

1949 (1)

N. L. Johnson, “Systems of frequency curves generated by methods of translation,” Biometrika 36(1-2), 149–176 (1949).
[CrossRef] [PubMed]

Abbt, M.

E. Limpert, W. A. Stahel, and M. Abbt, “Lognormal distributions across the sciences: keys and clues,” Bioscience 51(5), 341–352 (2001).
[CrossRef]

Acharya, P. K.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Adler-Golden, S. M.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Anderson, G. P.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Baker, J. D.

Baum, J.

J. Kerekes and J. Baum, “Full spectrum spectral imaging system analytical model,” IEEE Trans. Geosci. Rem. Sens. 43(3), 571–580 (2005).
[CrossRef]

Ben-David, A.

Berk, A.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Bernstein, L. S.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Chervet, P.

C. Quang, F. Dalaudier, A. Roblin, V. Rialland, and P. Chervet, “Statistical model for atmospheric limb radiance structure: application to airborne infrared surveillance systems,” Proc. SPIE 7108, 710805, 710805-9 (2008).
[CrossRef]

Chetwynd, J. H.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Cooley, T.

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402, 733402-16 (2009).
[CrossRef]

Cooley, T. W.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Dalaudier, F.

C. Quang, F. Dalaudier, A. Roblin, V. Rialland, and P. Chervet, “Statistical model for atmospheric limb radiance structure: application to airborne infrared surveillance systems,” Proc. SPIE 7108, 710805, 710805-9 (2008).
[CrossRef]

Davidson, C. E.

Embury, J. F.

Fenton, L. F.

L. F. Fenton, “The sum of lognormal probability distributions in scatter transmission systems,” IRE Trans. Commun. Syst. CS-8, 57–67 (1960).

Flanigan, D. F.

Fox, M. J.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Gabriel, P. M.

G. L. Stephens, P. M. Gabriel, and S.-C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20(2-3), 139–175 (1991).
[CrossRef]

Gardner, J. A.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Hall, J. L.

Herr, K. C.

Hill, I. D.

I. D. Hill, R. Hill, and R. L. Holder, “Algorithm AS99: fitting Johnson curves by moments,” J. R. Stat. Soc., Ser. C, Appl. Stat. 25(2), 180–189 (1976).

Hill, R.

I. D. Hill, R. Hill, and R. L. Holder, “Algorithm AS99: fitting Johnson curves by moments,” J. R. Stat. Soc., Ser. C, Appl. Stat. 25(2), 180–189 (1976).

Hoke, M. L.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Holder, R. L.

I. D. Hill, R. Hill, and R. L. Holder, “Algorithm AS99: fitting Johnson curves by moments,” J. R. Stat. Soc., Ser. C, Appl. Stat. 25(2), 180–189 (1976).

Holland, S. K.

Ifarraguerri, A.

Jacobson, J.

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402, 733402-16 (2009).
[CrossRef]

Johnson, N. L.

N. L. Johnson, “Systems of frequency curves generated by methods of translation,” Biometrika 36(1-2), 149–176 (1949).
[CrossRef] [PubMed]

Kerekes, J.

J. Kerekes and J. Baum, “Full spectrum spectral imaging system analytical model,” IEEE Trans. Geosci. Rem. Sens. 43(3), 571–580 (2005).
[CrossRef]

Lane-Veron, D. E.

D. E. Lane-Veron and C. J. Somerville, “Stochastic theory of radiative transfer through generalized cloud fields,” J. Geophys. Res. 109(D18), D18113 (2004).
[CrossRef]

Laufer, G.

Lee, J.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Limpert, E.

E. Limpert, W. A. Stahel, and M. Abbt, “Lognormal distributions across the sciences: keys and clues,” Bioscience 51(5), 341–352 (2001).
[CrossRef]

Lockwood, R.

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402, 733402-16 (2009).
[CrossRef]

Lockwood, R. B.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Manolakis, D.

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402, 733402-16 (2009).
[CrossRef]

D. Manolakis, D. Marden, and G. A. Shaw, “Hyperspectral image processing for automatic target detection applications,” Lincoln Lab. J. 14, 79–116 (2003).

Marden, D.

D. Manolakis, D. Marden, and G. A. Shaw, “Hyperspectral image processing for automatic target detection applications,” Lincoln Lab. J. 14, 79–116 (2003).

Mehta, N. B.

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wirel. Comm. 6(7), 2690–2699 (2007).
[CrossRef]

Molisch, A. F.

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wirel. Comm. 6(7), 2690–2699 (2007).
[CrossRef]

Muratov, L.

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

Polak, M. L.

Quang, C.

C. Quang, F. Dalaudier, A. Roblin, V. Rialland, and P. Chervet, “Statistical model for atmospheric limb radiance structure: application to airborne infrared surveillance systems,” Proc. SPIE 7108, 710805, 710805-9 (2008).
[CrossRef]

Ren, H.

Rialland, V.

C. Quang, F. Dalaudier, A. Roblin, V. Rialland, and P. Chervet, “Statistical model for atmospheric limb radiance structure: application to airborne infrared surveillance systems,” Proc. SPIE 7108, 710805, 710805-9 (2008).
[CrossRef]

Roblin, A.

C. Quang, F. Dalaudier, A. Roblin, V. Rialland, and P. Chervet, “Statistical model for atmospheric limb radiance structure: application to airborne infrared surveillance systems,” Proc. SPIE 7108, 710805, 710805-9 (2008).
[CrossRef]

Shaw, G. A.

D. Manolakis, D. Marden, and G. A. Shaw, “Hyperspectral image processing for automatic target detection applications,” Lincoln Lab. J. 14, 79–116 (2003).

Somerville, C. J.

D. E. Lane-Veron and C. J. Somerville, “Stochastic theory of radiative transfer through generalized cloud fields,” J. Geophys. Res. 109(D18), D18113 (2004).
[CrossRef]

Stahel, W. A.

E. Limpert, W. A. Stahel, and M. Abbt, “Lognormal distributions across the sciences: keys and clues,” Bioscience 51(5), 341–352 (2001).
[CrossRef]

Stephens, G. L.

G. L. Stephens, P. M. Gabriel, and S.-C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20(2-3), 139–175 (1991).
[CrossRef]

Tsay, S.-C.

G. L. Stephens, P. M. Gabriel, and S.-C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20(2-3), 139–175 (1991).
[CrossRef]

Wheeler, R. E.

R. E. Wheeler, “Quantile estimators of Johnson curve parameters,” Biometrika 67(3), 725–728 (1980).
[CrossRef]

Wu, J.

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wirel. Comm. 6(7), 2690–2699 (2007).
[CrossRef]

Zhang, J.

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wirel. Comm. 6(7), 2690–2699 (2007).
[CrossRef]

Appl. Opt. (4)

Biometrika (2)

N. L. Johnson, “Systems of frequency curves generated by methods of translation,” Biometrika 36(1-2), 149–176 (1949).
[CrossRef] [PubMed]

R. E. Wheeler, “Quantile estimators of Johnson curve parameters,” Biometrika 67(3), 725–728 (1980).
[CrossRef]

Bioscience (1)

E. Limpert, W. A. Stahel, and M. Abbt, “Lognormal distributions across the sciences: keys and clues,” Bioscience 51(5), 341–352 (2001).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

J. Kerekes and J. Baum, “Full spectrum spectral imaging system analytical model,” IEEE Trans. Geosci. Rem. Sens. 43(3), 571–580 (2005).
[CrossRef]

IEEE Trans. Wirel. Comm. (1)

N. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, “Approximating a sum of random variables with a lognormal,” IEEE Trans. Wirel. Comm. 6(7), 2690–2699 (2007).
[CrossRef]

IRE Trans. Commun. Syst. (1)

L. F. Fenton, “The sum of lognormal probability distributions in scatter transmission systems,” IRE Trans. Commun. Syst. CS-8, 57–67 (1960).

J. Geophys. Res. (1)

D. E. Lane-Veron and C. J. Somerville, “Stochastic theory of radiative transfer through generalized cloud fields,” J. Geophys. Res. 109(D18), D18113 (2004).
[CrossRef]

J. R. Stat. Soc., Ser. C, Appl. Stat. (1)

I. D. Hill, R. Hill, and R. L. Holder, “Algorithm AS99: fitting Johnson curves by moments,” J. R. Stat. Soc., Ser. C, Appl. Stat. 25(2), 180–189 (1976).

Lincoln Lab. J. (1)

D. Manolakis, D. Marden, and G. A. Shaw, “Hyperspectral image processing for automatic target detection applications,” Lincoln Lab. J. 14, 79–116 (2003).

Opt. Express (3)

Proc. SPIE (3)

C. Quang, F. Dalaudier, A. Roblin, V. Rialland, and P. Chervet, “Statistical model for atmospheric limb radiance structure: application to airborne infrared surveillance systems,” Proc. SPIE 7108, 710805, 710805-9 (2008).
[CrossRef]

A. Berk, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. J. Fox, S. M. Adler-Golden, J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, T. W. Cooley, and J. A. Gardner, “MODTRAN5: A Reformulated Atmospheric Band Model with Auxiliary Species and Practical Multiple Scattering Options,” Proc. SPIE 5655, 88–95 (2005).
[CrossRef]

D. Manolakis, R. Lockwood, T. Cooley, and J. Jacobson, “Is there a best hyperspectral detection algorithm?” Proc. SPIE 7334, 733402, 733402-16 (2009).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry and radiative transfer quantities for a 3-layer radiative transfer model for horizontal line of sight (LOS1) and up-looking slant path (LOS2). Transmissions and temperatures for the different layers are denoted by t i and T i , respectively. For LOS1 the foreground and background layer radiances ( L f , L b ) are given as emissions (1 t i )B( T i ) from a blackbody B( T i ) . The external source radiance, L s , at the end of LOS can be a function of temperature T s and emissivity ε s , i.e., L s = ε s B( T s ) .

Fig. 2
Fig. 2

Standard β 1 , β 1 chart, where the y-axis is reversed (by common convention). The lognormal curve (corresponding to both LN and SL distributions) is given by Eq. (A1) and is shown in red. The normal distribution is a point in blue at coordinates (0,3). The area “below” the lognormal curve contains all SU distributions. The area above the lognormal curve corresponds to the SB region. All probability distributions must lie below the shaded gray area in the upper right corner of the plot.

Fig. 3
Fig. 3

Comparison of data histogram (solid red) and model-predicted pdfs (thick dotted green) for several intermediate model outputs for the physical simulation. The L and M radiances (in units of W/cm2/sr/cm−1) are modeled with the Johnson SU distribution, and B( T c ) is modeled with a lognormal. Top row: emission radiances for the foreground and background layers, L f = M f , L b , and cloud layer emission radiance L c =(1 t c )B( T c ) . Middle row: At-sensor radiances for the external source, M s (0) , and the background layer, M b (0) , under the H0 scenario and the at-sensor radiance from the cloud layer, M c . Bottom row: At-sensor radiances for the external source, M s (1) , and background layer, M b (1) , for the H1 scenario and the Planck blackbody radiance for the cloud, B( T c ) .The KL distance between theory and sampled data for all plots is very small (<0.002), hence excellent agreement between the data and the model.

Fig. 4
Fig. 4

Comparison of data histograms (solid lines) and model pdfs (thick dotted lines) for the primary model outputs for the physical simulation. Top left: at-sensor radiance under H0 and H1 hypotheses, M (0) and M (1) , respectively. Top right: Differential radiance, ΔM= M (1) M (0) . Bottom left: Thermal contrast (in Kelvin), ΔT= B 1 ( L in ) T c , computed using Eq. (8). Bottom right: Receiver operating characteristics where the line of equal probability of false alarm and detection is shown in gray. The agreement between model predications and the data is excellent (KL distances for M (0) , M (1) , ΔM , and ΔT of 0.0014, 0.0015, 0.0014, 0.0016, respectively).

Fig. 5
Fig. 5

Same as Fig. 3 but for Simulation 1. The agreement between the model and the data for L and M radiances is poor (large KL values of ~1.3) with the exception of M s (0) and M s (1) for which the fit is good and the KL distance is small (0.005 and 0.008 respectively).

Fig. 8
Fig. 8

Same as Fig. 4 but for Simulation 2. The KL distance between theory and sampled data for M (0) , M (1) , and ΔM is very small (0.002, 0.0014, and 0.0013, respectively) indicating good agreement between the model and data. For ΔT the fit is reasonably good with a KL of 0.007. Detection performance for this scenario is reduced in comparison to Simulation 1: although the optical depth of the target cloud is much greater in Simulation 2, the thermal contrast is reduced and there is greater attenuation from the foreground layer.

Fig. 6
Fig. 6

Same as Fig. 4 but for Simulation 1. The KL distance between theory and sampled data for M (0) , M (1) , ΔM , and ΔT is very small (0.0014, 0.0013, 0.0013, 0.003, respectively), hence excellent agreement between the model and the data. Although the optical depth is similar and the thermal contrast larger for Simulation 1 in comparison to the physical scenario, the performance is reduced for Simulation 1 due to larger fluctuations.

Fig. 7
Fig. 7

Same as Fig. 3 but for Simulation 2. The KL distance for the L radiances are small (0.03-0.06) and are larger (~0.75) for M b (0) and M c . The agreement for M s (0) , M s (1) , and M b (1) is poor (KL of 2.7, 6, and 12, respectively).

Tables (2)

Tables Icon

Table 1 Parameters for Simulations

Tables Icon

Table 2 Parameters for Johnson Distributions, x~ S U (a,b,c,d)

Equations (31)

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M ( 1 ) = L f + t f (1 t c )B( T c )+ t f t c L b + t f t c t b L s .
M ( 0 ) = L f + t f L b + t f t b L s .
{ L f =(1 t f )B( T f ) L b =(1 t b )B( T b ) }.
{ ΔM= M ( 1 ) M (0) = t f (1 t c )ΔT ΔT= L in B( T c ) L in = L b + t b L s }
ΔT= B 1 ( L in ) T c
B(T)= k 1 e k 2 /T 1
B 1 ( L )= k 2 ln( k 1 /L +1 ) .
B 1 ( L )= B 1 ( L 0 )+( L L 0 ) B 1 L | L= L 0 +O( (L L 0 ) 2 ).
α 1 = B 1 L | L= L 0 , α 0 = B 1 ( L 0 )( B 1 L | L= L 0 ) L 0 ,
B 1 L | L= L 0 = B 1 ( L 0 ) L 0 ( k 1 + L 0 )ln( k1 L 0 +1 ) .
p x (x)= (2π σ 2 ) 1/2 exp( 1 2 (xμ) 2 σ 2 ).
p y (y)= (2π σ 2 y 2 ) 1/2 exp( 1 2 (ln(y)μ) 2 σ 2 ).
E( y n )=exp(nμ+ 1 2 n 2 σ 2 )
E(y)=exp(μ+ 1 2 σ 2 ), V(y)=[exp( σ 2 )1]E (y) 2 , S(y)=[exp( σ 2 )+2] exp( σ 2 )1, K(y)=exp(4 σ 2 )+2exp(3 σ 2 )+3exp(2 σ 2 )3.
{ β 2 = β 1 +exp(4 σ 2 )+exp(3 σ 2 )+1 β 1 =exp(3 σ 2 )+3exp(2 σ 2 )4 }.
{ σ 2 =ln( 1+ V(x) E (x) 2 ) μ=ln[ E(x) ] 1 2 σ 2 }.
{ σ 2 =ln( 1 2 + median (x) 2 +4V(x) 2median(x) ) μ=ln[ median(x) ] }.
z= i y i ~LN( μ z , σ z 2 ), μ z =ln( i E( y i ) ) 1 2 σ z 2 , σ z 2 =ln( 1+ i V( y i ) ( i E( y i ) ) 2 ).
ρ 12 =corr( y 1, y 2 )= E( y 1 y 2 )E( y 1 )E( y 2 ) V( y 1 )V( y 2 ) ,
z= y 1 + y 2 + y 3 ~LN( μ z , σ z 2 ), E(z)=E( y 1 )+E( y 2 )+E( y 3 ),
V(z)=V( y 1 )+V( y 2 )+V( y 3 )+2 ρ 12 V( y 1 )V( y 2 ) +2 ρ 13 V( y 1 )V( y 3 ) +2 ρ 23 V( y 2 )V( y 3 ) .
E( y n )= erfc( μ+n σ 2 2 σ 2 ) erfc( μ 2 σ 2 ) exp(nμ+ 1 2 n 2 σ 2 )
erfc( μ+n σ 2 2 σ 2 ) erfc( μ 2 σ 2 ) 1,
y=a+bsinh( xc d )
p y ( y )= dexp{ 1 2 [c+d sinh 1 ( ya b )] 2 } b 2π[ ( ya b ) 2 +1] .
E[ (yθ) n ]= j=0 n ( n j ) ( b 2 ) j ( aθ ) nj r=0 j ( j r ) ( 1 ) r e (j2r)Ω ω (j2r) 2
{ E=abωsinh(Ω), V= 1 2 b 2 ( ω 2 1)(1+ ω 2 cosh2Ω), S=ω ( ω 2 1 2 ) 1/2 ω 2 ( ω 2 +2)sinh3Ω+3sinhΩ ( 1+ ω 2 cosh2Ω ) 3/2 , K= 1 2 ω 4 ( ω 8 +2 ω 6 +3 ω 4 3)cosh4Ω+4 ω 4 ( ω 2 +2)cosh2Ω+3(2 ω 2 +1) ( 1+ ω 2 cosh2Ω ) 2 }.
E(z), V(z)=E( [ zE(z) ] 2 )=E( z 2 )E(z), S(z)= E( [zE(z)] 3 ) V (z) 3/2 = E( z 3 )3E( z 2 )E(z)+2E (z) 3 V (z) 3/2 , K(z)= E( [zE(z)] 4 ) V (z) 2 = E( z 4 )4E( z 3 )E(z)+6E( z 2 )E (z) 2 3E (z) 4 V (z) 2 ,
E[f(x)]= f(x) p x (x)dx
{ E(z)=E(x)±E(y) E( z 2 )=E( x 2 )±2E(x)E(y)+E( y 2 ) E( z 3 )=E( x 3 )±3E( x 2 )E(y)+3E(x)E( y 2 )±E( y 3 ) E( z 4 )=E( x 4 )±4E( x 3 )E(y)+6E( x 2 )E( y 2 )±4E(x)E( y 3 )+E( y 4 ) }.
{ E(z)=E(x)E(y) E( z 2 )=E( x 2 )E( y 2 ) E( z 3 )=E( x 3 )E( y 3 ) E( z 4 )=E( x 4 )E( y 4 ) }.

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