Abstract

Monte Carlo radiative transfer simulations are used to study the atmospheric spread function appropriate to satellite-based sensing of the earth's surface. The parameters which are explored include the nadir angle of view, the size distribution of the atmospheric aerosol, and the aerosol vertical profile.

© 1986 Optical Society of America

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References

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  1. J. Otterman, R. S. Fraser, “Adjacency Effect on Imaging by Surface Reflection and Atmospheric Scattering: Cross Radiance to Zenith,” Appl. Opt. 18, 2852 (1979).
    [CrossRef] [PubMed]
  2. A. P. Odell, J. A. Weinman, “The Effect of Atmospheric Haze on Images of the Earth's Surface,” J. Geophys. Res. 80, 5035 (1975).
    [CrossRef]
  3. Y. J. Kaufman, “Effect of the Earth's Atmosphere on the Contrast for Zenith Observations,” J. Geophys. Res. 84, 3165 (1979).
    [CrossRef]
  4. S. Ueno, Y. Haba, K. Kawata, T. Kusaka, Y. Terashita, Remote Sensing of the Atmosphere: Inversion Methods and Application, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, NY, 305 (1979).
  5. B. A. Kargin, S. V. Kuznetsov, G. A. Mikhaylov, “Monte Carlo Estimates of Brightness Contrast Transfer Function in a Light Scattering Medium,” Izv. Atmos. Oceanic Phys. 15, 717 (1979).
  6. J. V. Dave, “Effect of Atmospheric Conditions on Remote Sensing of a Surface Non-Homogeneity,” Photogramm. Eng. Remote Sensing 46, 1173 (1980).
  7. Y. Mekler, Y. J. Kaufman, “The Effect of Earth's Atmosphere on Contrast Reduction for a Nonuniform Surface Albedo and ‘Two-Halves’ Field,” J. Geophys. Res 85, 4067 (1980).
    [CrossRef]
  8. Y. J. Kaufman, R. S. Fraser, “Different Atmsopheric Effects on Remote Sensing of Uniform and Non-Uniform Surfaces,” Adv. Space Res. 2, 145 (1983).
  9. I. L. Katsev, “Modulation Transfer Function and Spread Function for a Medium with Strongly Anisotropic Scattering,” Izv. Atmos. Oceanic Phys. 17, 348 (1981).
  10. I. M. Levin, Izv. Atmos. Oceanic Phys. “Backscattering Noise and Contrast in Artificial Underwater Illumination,” 18, 149 (1982).
  11. V. V. Belov, V. E. Zuev, G. M. Krekov, Izv. “Visibility of Distant Objects in Scattering Media,” Atmos. Oceanic Phys. 18, 742 (1982).
  12. V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).
  13. D. Tanre, M. Herman, P. Y. Deschamps, A. de Leffe, “Atmospheric Modeling for Space Measurements of Ground Reflectances, Including Bidirectional Properties,” Appl. Opt. 18, 3587 (1979).
    [CrossRef] [PubMed]
  14. D. Tanre, M. Herman, P. Y. Deschamps, “Influence of the Background Contribution upon Space Measurements of Ground Reflectance,” Appl. Opt. 20, 3676 (1980).
    [CrossRef]
  15. D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground—I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97 (1984).
    [CrossRef]
  16. D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground. II. Computational Considerations and Results,” J. Quant. Spectrosc. Radiat. Transfer. 32, 279 (1984).
    [CrossRef]
  17. W. A. Pearce, A Study of the Effects of the Atmosphere on Thematic Mapper Observations, Report 004-77, contract NAS 5-23639, EG&G Washington Analytical Services Center, Inc., Riverdale, MD (1977).
  18. W. A. Pearce, CTRANS A Monte Carlo Program for Radiative Transfer in Plane Parallel Atmospheres with Imbedded Finite Clouds: Development, Testing and User's Guide, EG&G Washington Analytical Services Center, Inc., Applied Systems Department Report 006-76 (1976).
  19. R. S. Fraser, Goddard Space Flight Center, Greenbelt, MD; private communication.
  20. U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration, and United States Air Force, Washington, D.C. (1976).
  21. R. W. L. Thomas, “The Characterization of Atmospheric Spread Functions Affecting Satellite Remote Sensing of the Earth's Surface,” Adv. Space Res. 2, 157 (1983).
    [CrossRef]
  22. D. J. Diner, Jet Propulsion Laboratory, Pasadena, CA; private communication.

1984 (2)

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground—I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97 (1984).
[CrossRef]

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground. II. Computational Considerations and Results,” J. Quant. Spectrosc. Radiat. Transfer. 32, 279 (1984).
[CrossRef]

1983 (2)

R. W. L. Thomas, “The Characterization of Atmospheric Spread Functions Affecting Satellite Remote Sensing of the Earth's Surface,” Adv. Space Res. 2, 157 (1983).
[CrossRef]

Y. J. Kaufman, R. S. Fraser, “Different Atmsopheric Effects on Remote Sensing of Uniform and Non-Uniform Surfaces,” Adv. Space Res. 2, 145 (1983).

1982 (3)

I. M. Levin, Izv. Atmos. Oceanic Phys. “Backscattering Noise and Contrast in Artificial Underwater Illumination,” 18, 149 (1982).

V. V. Belov, V. E. Zuev, G. M. Krekov, Izv. “Visibility of Distant Objects in Scattering Media,” Atmos. Oceanic Phys. 18, 742 (1982).

V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).

1981 (1)

I. L. Katsev, “Modulation Transfer Function and Spread Function for a Medium with Strongly Anisotropic Scattering,” Izv. Atmos. Oceanic Phys. 17, 348 (1981).

1980 (3)

J. V. Dave, “Effect of Atmospheric Conditions on Remote Sensing of a Surface Non-Homogeneity,” Photogramm. Eng. Remote Sensing 46, 1173 (1980).

Y. Mekler, Y. J. Kaufman, “The Effect of Earth's Atmosphere on Contrast Reduction for a Nonuniform Surface Albedo and ‘Two-Halves’ Field,” J. Geophys. Res 85, 4067 (1980).
[CrossRef]

D. Tanre, M. Herman, P. Y. Deschamps, “Influence of the Background Contribution upon Space Measurements of Ground Reflectance,” Appl. Opt. 20, 3676 (1980).
[CrossRef]

1979 (4)

D. Tanre, M. Herman, P. Y. Deschamps, A. de Leffe, “Atmospheric Modeling for Space Measurements of Ground Reflectances, Including Bidirectional Properties,” Appl. Opt. 18, 3587 (1979).
[CrossRef] [PubMed]

J. Otterman, R. S. Fraser, “Adjacency Effect on Imaging by Surface Reflection and Atmospheric Scattering: Cross Radiance to Zenith,” Appl. Opt. 18, 2852 (1979).
[CrossRef] [PubMed]

Y. J. Kaufman, “Effect of the Earth's Atmosphere on the Contrast for Zenith Observations,” J. Geophys. Res. 84, 3165 (1979).
[CrossRef]

B. A. Kargin, S. V. Kuznetsov, G. A. Mikhaylov, “Monte Carlo Estimates of Brightness Contrast Transfer Function in a Light Scattering Medium,” Izv. Atmos. Oceanic Phys. 15, 717 (1979).

1975 (1)

A. P. Odell, J. A. Weinman, “The Effect of Atmospheric Haze on Images of the Earth's Surface,” J. Geophys. Res. 80, 5035 (1975).
[CrossRef]

Belov, V. V.

V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).

V. V. Belov, V. E. Zuev, G. M. Krekov, Izv. “Visibility of Distant Objects in Scattering Media,” Atmos. Oceanic Phys. 18, 742 (1982).

Borisov, B. D.

V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).

Dave, J. V.

J. V. Dave, “Effect of Atmospheric Conditions on Remote Sensing of a Surface Non-Homogeneity,” Photogramm. Eng. Remote Sensing 46, 1173 (1980).

de Leffe, A.

Deschamps, P. Y.

Diner, D. J.

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground—I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97 (1984).
[CrossRef]

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground. II. Computational Considerations and Results,” J. Quant. Spectrosc. Radiat. Transfer. 32, 279 (1984).
[CrossRef]

D. J. Diner, Jet Propulsion Laboratory, Pasadena, CA; private communication.

Fraser, R. S.

Y. J. Kaufman, R. S. Fraser, “Different Atmsopheric Effects on Remote Sensing of Uniform and Non-Uniform Surfaces,” Adv. Space Res. 2, 145 (1983).

J. Otterman, R. S. Fraser, “Adjacency Effect on Imaging by Surface Reflection and Atmospheric Scattering: Cross Radiance to Zenith,” Appl. Opt. 18, 2852 (1979).
[CrossRef] [PubMed]

R. S. Fraser, Goddard Space Flight Center, Greenbelt, MD; private communication.

Gemin, V. N.

V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).

Haba, Y.

S. Ueno, Y. Haba, K. Kawata, T. Kusaka, Y. Terashita, Remote Sensing of the Atmosphere: Inversion Methods and Application, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, NY, 305 (1979).

Herman, M.

Kabanov, M. V.

V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).

Kargin, B. A.

B. A. Kargin, S. V. Kuznetsov, G. A. Mikhaylov, “Monte Carlo Estimates of Brightness Contrast Transfer Function in a Light Scattering Medium,” Izv. Atmos. Oceanic Phys. 15, 717 (1979).

Katsev, I. L.

I. L. Katsev, “Modulation Transfer Function and Spread Function for a Medium with Strongly Anisotropic Scattering,” Izv. Atmos. Oceanic Phys. 17, 348 (1981).

Kaufman, Y. J.

Y. J. Kaufman, R. S. Fraser, “Different Atmsopheric Effects on Remote Sensing of Uniform and Non-Uniform Surfaces,” Adv. Space Res. 2, 145 (1983).

Y. Mekler, Y. J. Kaufman, “The Effect of Earth's Atmosphere on Contrast Reduction for a Nonuniform Surface Albedo and ‘Two-Halves’ Field,” J. Geophys. Res 85, 4067 (1980).
[CrossRef]

Y. J. Kaufman, “Effect of the Earth's Atmosphere on the Contrast for Zenith Observations,” J. Geophys. Res. 84, 3165 (1979).
[CrossRef]

Kawata, K.

S. Ueno, Y. Haba, K. Kawata, T. Kusaka, Y. Terashita, Remote Sensing of the Atmosphere: Inversion Methods and Application, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, NY, 305 (1979).

Krekov, G. M.

V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).

V. V. Belov, V. E. Zuev, G. M. Krekov, Izv. “Visibility of Distant Objects in Scattering Media,” Atmos. Oceanic Phys. 18, 742 (1982).

Kusaka, T.

S. Ueno, Y. Haba, K. Kawata, T. Kusaka, Y. Terashita, Remote Sensing of the Atmosphere: Inversion Methods and Application, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, NY, 305 (1979).

Kuznetsov, S. V.

B. A. Kargin, S. V. Kuznetsov, G. A. Mikhaylov, “Monte Carlo Estimates of Brightness Contrast Transfer Function in a Light Scattering Medium,” Izv. Atmos. Oceanic Phys. 15, 717 (1979).

Levin, I. M.

I. M. Levin, Izv. Atmos. Oceanic Phys. “Backscattering Noise and Contrast in Artificial Underwater Illumination,” 18, 149 (1982).

Martonchik, J. V.

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground—I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97 (1984).
[CrossRef]

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground. II. Computational Considerations and Results,” J. Quant. Spectrosc. Radiat. Transfer. 32, 279 (1984).
[CrossRef]

Mekler, Y.

Y. Mekler, Y. J. Kaufman, “The Effect of Earth's Atmosphere on Contrast Reduction for a Nonuniform Surface Albedo and ‘Two-Halves’ Field,” J. Geophys. Res 85, 4067 (1980).
[CrossRef]

Mikhaylov, G. A.

B. A. Kargin, S. V. Kuznetsov, G. A. Mikhaylov, “Monte Carlo Estimates of Brightness Contrast Transfer Function in a Light Scattering Medium,” Izv. Atmos. Oceanic Phys. 15, 717 (1979).

Odell, A. P.

A. P. Odell, J. A. Weinman, “The Effect of Atmospheric Haze on Images of the Earth's Surface,” J. Geophys. Res. 80, 5035 (1975).
[CrossRef]

Otterman, J.

Pearce, W. A.

W. A. Pearce, A Study of the Effects of the Atmosphere on Thematic Mapper Observations, Report 004-77, contract NAS 5-23639, EG&G Washington Analytical Services Center, Inc., Riverdale, MD (1977).

W. A. Pearce, CTRANS A Monte Carlo Program for Radiative Transfer in Plane Parallel Atmospheres with Imbedded Finite Clouds: Development, Testing and User's Guide, EG&G Washington Analytical Services Center, Inc., Applied Systems Department Report 006-76 (1976).

Tanre, D.

Terashita, Y.

S. Ueno, Y. Haba, K. Kawata, T. Kusaka, Y. Terashita, Remote Sensing of the Atmosphere: Inversion Methods and Application, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, NY, 305 (1979).

Thomas, R. W. L.

R. W. L. Thomas, “The Characterization of Atmospheric Spread Functions Affecting Satellite Remote Sensing of the Earth's Surface,” Adv. Space Res. 2, 157 (1983).
[CrossRef]

Ueno, S.

S. Ueno, Y. Haba, K. Kawata, T. Kusaka, Y. Terashita, Remote Sensing of the Atmosphere: Inversion Methods and Application, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, NY, 305 (1979).

Weinman, J. A.

A. P. Odell, J. A. Weinman, “The Effect of Atmospheric Haze on Images of the Earth's Surface,” J. Geophys. Res. 80, 5035 (1975).
[CrossRef]

Zuev, V. E.

V. V. Belov, V. E. Zuev, G. M. Krekov, Izv. “Visibility of Distant Objects in Scattering Media,” Atmos. Oceanic Phys. 18, 742 (1982).

Adv. Space Res. (2)

Y. J. Kaufman, R. S. Fraser, “Different Atmsopheric Effects on Remote Sensing of Uniform and Non-Uniform Surfaces,” Adv. Space Res. 2, 145 (1983).

R. W. L. Thomas, “The Characterization of Atmospheric Spread Functions Affecting Satellite Remote Sensing of the Earth's Surface,” Adv. Space Res. 2, 157 (1983).
[CrossRef]

Appl. Opt. (3)

Atmos. Oceanic Phys. (1)

V. V. Belov, V. E. Zuev, G. M. Krekov, Izv. “Visibility of Distant Objects in Scattering Media,” Atmos. Oceanic Phys. 18, 742 (1982).

Izv. Atmos. Oceanic Phys. (4)

V. V. Belov, B. D. Borisov, V. N. Gemin, M. V. Kabanov, G. M. Krekov, “Experimental and Mathematical Modeling of the Condition for Seeing Objects Through a Layer of a Turbid Medium,” Izv. Atmos. Oceanic Phys. 18, 1042 (1982).

I. L. Katsev, “Modulation Transfer Function and Spread Function for a Medium with Strongly Anisotropic Scattering,” Izv. Atmos. Oceanic Phys. 17, 348 (1981).

I. M. Levin, Izv. Atmos. Oceanic Phys. “Backscattering Noise and Contrast in Artificial Underwater Illumination,” 18, 149 (1982).

B. A. Kargin, S. V. Kuznetsov, G. A. Mikhaylov, “Monte Carlo Estimates of Brightness Contrast Transfer Function in a Light Scattering Medium,” Izv. Atmos. Oceanic Phys. 15, 717 (1979).

J. Geophys. Res (1)

Y. Mekler, Y. J. Kaufman, “The Effect of Earth's Atmosphere on Contrast Reduction for a Nonuniform Surface Albedo and ‘Two-Halves’ Field,” J. Geophys. Res 85, 4067 (1980).
[CrossRef]

J. Geophys. Res. (2)

A. P. Odell, J. A. Weinman, “The Effect of Atmospheric Haze on Images of the Earth's Surface,” J. Geophys. Res. 80, 5035 (1975).
[CrossRef]

Y. J. Kaufman, “Effect of the Earth's Atmosphere on the Contrast for Zenith Observations,” J. Geophys. Res. 84, 3165 (1979).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground—I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 31, 97 (1984).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer. (1)

D. J. Diner, J. V. Martonchik, “Atmospheric Transfer of Radiation Above an Inhomogeneous Non-Lambertian Reflective Ground. II. Computational Considerations and Results,” J. Quant. Spectrosc. Radiat. Transfer. 32, 279 (1984).
[CrossRef]

Photogramm. Eng. Remote Sensing (1)

J. V. Dave, “Effect of Atmospheric Conditions on Remote Sensing of a Surface Non-Homogeneity,” Photogramm. Eng. Remote Sensing 46, 1173 (1980).

Other (6)

S. Ueno, Y. Haba, K. Kawata, T. Kusaka, Y. Terashita, Remote Sensing of the Atmosphere: Inversion Methods and Application, A. L. Fymat, V. E. Zuev, Eds. (Elsevier, NY, 305 (1979).

W. A. Pearce, A Study of the Effects of the Atmosphere on Thematic Mapper Observations, Report 004-77, contract NAS 5-23639, EG&G Washington Analytical Services Center, Inc., Riverdale, MD (1977).

W. A. Pearce, CTRANS A Monte Carlo Program for Radiative Transfer in Plane Parallel Atmospheres with Imbedded Finite Clouds: Development, Testing and User's Guide, EG&G Washington Analytical Services Center, Inc., Applied Systems Department Report 006-76 (1976).

R. S. Fraser, Goddard Space Flight Center, Greenbelt, MD; private communication.

U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration, and United States Air Force, Washington, D.C. (1976).

D. J. Diner, Jet Propulsion Laboratory, Pasadena, CA; private communication.

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Figures (16)

Fig. 1
Fig. 1

Scattering phase functions for the aerosol models. Solid line, small model; dashed line, intermediate model; dotted line, large model.

Fig. 2
Fig. 2

Diffuse ground intensity crossing the albedo boundary of a semi-infinite Lambertian plane; albedo = 0.04, 0.4.

Fig. 3
Fig. 3

Line spread functions for the three lognormal aerosol models.

Fig. 4
Fig. 4

Modulation transfer functions for the three lognormal aerosol models.

Fig. 5
Fig. 5

Intensity components for observations of sinusoidal albedo patterns employed in computing the MTF for the small-size distribution model. Solid lines, direct ground intensity D (the upper curve is D = D+, the lower curve is D = 0); long-dashed lines, diffuse ground intensities, M+ and M; short-dashed line, atmospheric radiance I0.

Fig. 6
Fig. 6

Schematic illustrating the geometry: (A) detector azimuth = 0°; (B) detector azimuth = 180°. As shown here the detector is viewing the dark side of an albedo boundary 3 km from the border between the light and dark regions. The point of view here is −3.0 km.

Fig. 7
Fig. 7

Diffuse ground intensity component detected crossing an albedo boundary. Detector azimuth = 0°. Solid, dashed, dot-dashed, and short-dashed lines are for receiver nadir angles of 0, 20, 40, and 50°.

Fig. 8
Fig. 8

As Fig. 7 but with detector azimuth = 180°.

Fig. 9
Fig. 9

Normalized cumulative ground diffuse intensities crossing an albedo boundary (0.04,0.4). Detector azimuth = 0° for the solid, dashed, dot-dashed, and dotted curves corresponding to 0, 20, 40, and 50° receiver nadir angle. The short-dashed curve is for receiver azimuth 180°, nadir angle 50°.

Fig. 10
Fig. 10

Normalized line spread functions obtained from Fig. 9: (a) 0° detector nadir angle; (b) 40° detector nadir angle; (c) 50° detector nadir angle, 0° azimuth (small dots) and 180° azimuth (large dots).

Fig. 11
Fig. 11

Log–log plot of the cumulative diffuse intensities. The solid line shows the 0° receiver nadir angle results. Dotted curves are for 50° nadir angle and 0° (small dots) and 180° (large dots) azimuth.

Fig. 12
Fig. 12

Scatter plot correlating I0 with the single-scattering phase function S11. The parenthetic values denote (nadir angle, azimuth). The values in square brackets are the corresponding single-scattering angles.

Fig. 13
Fig. 13

Total intensities crossing the albedo boundary: solid line, 0° nadir angle; dotted curves show 50° nadir angle with 0° (small dots) and 180° (solid dots) receiver azimuth, respectively.

Fig. 14
Fig. 14

Vertical profiles of scattering optical density showing the Rayleigh component as dot-dashed, normal aerosol as solid, middle aerosol as dotted, and high aerosol as dashed lines.

Fig. 15
Fig. 15

Ground diffuse intensity components crossing an albedo boundary (0.04, 0.4) for the three aerosol profiles: Solid, dashed, and dotted lines are for the normal, middle, and high-profile models.

Fig. 16
Fig. 16

Log–log plots of the ground diffuse intensity on the high-albedo side of the boundary: (a) comparison of normal model with middle model horizontally scaled by a factor of 1.4545; (b) comparison of normal model with high model horizontally scaled by a factor of 8.0.

Tables (3)

Tables Icon

Table I Log Normal Size Distribution Parameters

Tables Icon

Table II Optical Thickness Estimated from the Intensity Jump at a Boundary

Tables Icon

Table III Measures of Mean Optical Altitude

Equations (34)

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I = α R S i s + α R g g α g s i s + α R g g α g g g α g s i s + ,
I ( x , Ω ̂ , Ω ̂ s ) = I 0 ( Ω ̂ , Ω ̂ s ) + α ( x , x , Ω ̂ , Ω ̂ r ) g ( Ω ̂ r , Ω ̂ i , x ) × Ĩ ( x , Ω ̂ i , Ω ̂ s ) d Ω ̂ r d Ω ̂ i d x .
Ĩ ( x , Ω ̂ i , Ω ̂ s ) = α g s i s + α g g ( x , x , Ω ̂ i , Ω ̂ j ) g ( Ω ̂ j , Ω ̂ l , x ) Ĩ ( x , Ω ̂ l , Ω ̂ s ) d Ω ̂ j d Ω ̂ l d x ,
I ( x , Ω ̂ , Ω ̂ s ) = I 0 + exp ( τ / μ ) g ( Ω ̂ , Ω ̂ i , x ) Ĩ ( x , Ω ̂ i , Ω ̂ s ) d Ω ̂ i + α ( d ) ( x , x , Ω ̂ r ) g ( Ω ̂ r Ω ̂ i , x ) Ĩ ( x , Ω ̂ i Ω ̂ s ) d Ω ̂ r d Ω ̂ r d Ω ̂ i d x ,
I = I 0 + A ( x ) exp ( τ / μ ) F ( x ) + α ( d ) ( x x ) A ( x ) F ( x ) d x
A ( x ) = A 0 , x < x 0 , = A 1 x x 0 ,
d I d x ( A 1 A 0 ) δ ( x x 0 ) exp ( τ / μ ) F ̅ + ( A 1 A 0 ) α L ( d ) ( x x 0 F ̅ ,
A ± = A 0 ± A 1 cos 2 π ν x
i ± = i 0 + α L ( x x ) { A 0 ± A 1 2 × [ exp ( 2 π i ν x ) + exp ( 2 π i ν x ) ] } F ̅ d x .
i + i = A 1 [ a ( ν ) + a ( ν ) ] F ̅ ,
a ( ν ) = α L ( x ) exp ( 2 π i ν x ) d x
i + + i 2 i 0 = α ( x x ) 2 A 0 F ̅ d x = 2 A 0 a ( 0 ) F ̅
MTF = i + i i + + i 2 i 0 = A 1 A 0 a ( ν ) + a ( ν ) 2 a ( 0 ) .
A ( x ) = B 0 ± B 1 sin 2 π ν x
J ± = i 0 + α L ( x x ) { B 0 ± B 1 2 i × [ exp ( 2 π i ν x ) exp ( 2 π i ν x ) ] } F ̅ d x
J + J = i B 1 F ̅ [ a ( ν ) a ( ν ) ]
( i + + i ) i ( J + J ) i + + i 2 i 0 = a ( ν ) / a ( 0 ) ,
( i + + i ) + i ( J + + J ) i + + i 2 i 0 = a ( ν ) / a ( 0 ) ,
τ = ln ρ .
I = S I R j S j exp ( t j ) W W k ,
n ( d ) = 0 d 0.06 μ m , = c 0.06 < d 0.2 μ m , = c d 4 0.2 < d 16.16 μ m , = 0.0 d > 16.16 μ m ,
n ( D ) = A ( D C ) π exp ( { A ln [ ( D C ) / ( B C ) ] } 2 )
S 11 ( θ , k ) = 2 k 2 [ J 1 ( k sin θ ) / sin θ ] 2 ,
J 1 ( k sin θ ) / sin θ = ½ k j = 0 ( ¼ k 2 sin 2 θ ) j / [ ( j + 1 ) ! j ! ]
S 11 ( 0 , k ) = ½ k 4 .
S 11 ( 0 , k ) n ( D ) d D = ½ L H C ( A / D ) 4 k 4 d D = C 2 ( A π / λ ) 4 ( H L ) .
n ( D ) D ( 5 + ) , > 0 .
MTF = D + ( M + M ) D + ( M + + M ) .
I cum = [ I ( x ) I ( ) ] / [ I ( ) I ( ) ] .
I c x p ,
α L = c x ( p 1 )
τ = ln [ Δ I ( θ 1 ) / Δ I ( θ 2 ) ] ( 1 / cos θ 2 1 / cos θ 1 ] 1 .
Δ τ ( 1 / cos θ 2 1 / cos θ 1 ) 1 { [ Δ M ( θ 1 ) / Δ I ( θ 1 ) Δ M ( θ 2 ) / Δ I ( θ 2 ) ] + 1 / Δ I ( θ 1 ) + 2 / Δ I ( θ 2 ) } .
α L ( i ) K i x 0.91 i = 1 , 2 , 3 ,

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