Abstract

The singly scattered irradiance (SSI) model is an approximate radiative transfer model designed to describe optically shallow stratified waters. It is intended to be a tool to aid interpretation of remote (satellite or aircraft) observations of water color. The SSI model was derived in analogy to single-scattering radiance models. As it is based on irradiance (rather than radiance), multiple-scattering events are included implicitly, yielding a model which is expressed in terms of readily measurable parameters and which preserves mathematical simplicity without sacrificing many important subtleties of the radiative transfer process. The success of the SSI model relies on the underwater radiance distribution being nearly independent of sun angle, cloud cover and depth, and the resulting quasi-inherent property of the irradiance attenuation coefficient, irradiance reflectance, and distribution functions. A derivation of the model is presented, along with evidence for the qualitative and quantitative correctness of the model.

© 1987 Optical Society of America

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References

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  1. H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery—A Review (Springer-Verlag, New York, 1983).
  2. H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton Pigments from the Nimbus-7 Coastal Zone Color Scanner; Comparisons with Surface Measurements,” Science 210, 63 (1980).
    [CrossRef] [PubMed]
  3. W. D. Philpot, V. Klemas, “Remote Detection of Ocean Wastes,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 189 (1979).
  4. D. R. Lyzenga, “Passive Remote Sensing Techniques for Mapping Water Depth and Bottom Features,” Appl. Opt. 17, 379 (1978).
    [CrossRef] [PubMed]
  5. J. M. Paredes, R. E. Spero, “Water Depth Mapping from Passive Remote Sensing Data under a Generalized Ratio Assumption,” Appl. Opt. 22, 1134 (1983).
    [CrossRef] [PubMed]
  6. R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
    [CrossRef]
  7. A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
    [CrossRef]
  8. R. W. Preisendorfer, Hydrologic Optics [U.S. Department of Commerce, NOAA, Environmental Research Laboratories, Pacific Marine Environmental Laboratory, U.S. GPO 76-677-811-36, Region 8, 6 Volumes (1976)].
  9. R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
    [CrossRef]
  10. R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
    [CrossRef]
  11. K. S. Baker, R. C. Smith, “Quasi-inherent Characteristics of the Diffuse Attenuation Coefficient for Irradiance,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).
  12. J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).
  13. J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon & Breach, New York, 1970).
  14. W. R. McCluney, “Ocean Color Spectrum Calculations,” Appl. Opt. 13, 2422 (1974).
    [CrossRef] [PubMed]
  15. H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed Relationships Between the Inherent and Apparent Optical Properties of a Flat Homogeneous Ocean,” Appl. Opt. 14, 417 (1975).
    [CrossRef] [PubMed]
  16. A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  17. L. Prieur, A. Morel, “Relations theoriques entre le facteur de reflexion diffuse de l'eau de mer a diverses protondeurs et les caracteristiques optiques (absorption, diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).
  18. S. G. Ackleson, “The Verification of a Radiative Transfer Model for a Multiple Layer Ocean and its Application to Remotely Sensed Imagery,” Master's Thesis, College of Marine Studies, U. Delaware, Newark (1981).
  19. H. R. Gordon, O. B. Brown, “Diffuse Reflectance of the Ocean: Some Effects of Vertical Structure,” Appl. Opt. 14, 2892 (1975).
    [CrossRef] [PubMed]
  20. B. N. Plass, G. W. Kattawar, J. A. Guinn, “Radiative Transfer in the Earth's Atmosphere and Ocean: Influence of Ocean Waves,” Appl. Opt. 14, 1924 (1975).
    [CrossRef] [PubMed]
  21. J. I. Gordon, “Directional Radiance (luminance) of the Sea Surface,” SIO Ref. 69-20, Visibility Laboratory, Scripps Institute of Oceanography, San Diego, CA (1969).
  22. W. D. Philpot, “Experimental Verification of a Radiative Transfer Model for Assessment of Water Quality,” NSF Final Report, grant CEE 82-04579 (1985).
  23. W. D. Philpot, “Experimental Test of the Singly-Scattered Irradiance Model,” in Ocean Optics: VIII, Proc. Soc. Photo-Opt. Instrum. Eng. 637, 72 (1986).

1986

W. D. Philpot, “Experimental Test of the Singly-Scattered Irradiance Model,” in Ocean Optics: VIII, Proc. Soc. Photo-Opt. Instrum. Eng. 637, 72 (1986).

1984

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

1983

1982

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

1980

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton Pigments from the Nimbus-7 Coastal Zone Color Scanner; Comparisons with Surface Measurements,” Science 210, 63 (1980).
[CrossRef] [PubMed]

1979

W. D. Philpot, V. Klemas, “Remote Detection of Ocean Wastes,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 189 (1979).

K. S. Baker, R. C. Smith, “Quasi-inherent Characteristics of the Diffuse Attenuation Coefficient for Irradiance,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

1978

D. R. Lyzenga, “Passive Remote Sensing Techniques for Mapping Water Depth and Bottom Features,” Appl. Opt. 17, 379 (1978).
[CrossRef] [PubMed]

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

1977

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

1975

1974

1960

J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).

Ackleson, S. G.

S. G. Ackleson, “The Verification of a Radiative Transfer Model for a Multiple Layer Ocean and its Application to Remotely Sensed Imagery,” Master's Thesis, College of Marine Studies, U. Delaware, Newark (1981).

Baker, K. S.

K. S. Baker, R. C. Smith, “Quasi-inherent Characteristics of the Diffuse Attenuation Coefficient for Irradiance,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

Brown, O. B.

Clark, D. K.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton Pigments from the Nimbus-7 Coastal Zone Color Scanner; Comparisons with Surface Measurements,” Science 210, 63 (1980).
[CrossRef] [PubMed]

Gordon, H. R.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton Pigments from the Nimbus-7 Coastal Zone Color Scanner; Comparisons with Surface Measurements,” Science 210, 63 (1980).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, “Diffuse Reflectance of the Ocean: Some Effects of Vertical Structure,” Appl. Opt. 14, 2892 (1975).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed Relationships Between the Inherent and Apparent Optical Properties of a Flat Homogeneous Ocean,” Appl. Opt. 14, 417 (1975).
[CrossRef] [PubMed]

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery—A Review (Springer-Verlag, New York, 1983).

Gordon, J. I.

J. I. Gordon, “Directional Radiance (luminance) of the Sea Surface,” SIO Ref. 69-20, Visibility Laboratory, Scripps Institute of Oceanography, San Diego, CA (1969).

Guinn, J. A.

Hovis, W. A.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton Pigments from the Nimbus-7 Coastal Zone Color Scanner; Comparisons with Surface Measurements,” Science 210, 63 (1980).
[CrossRef] [PubMed]

Jacobs, M. M.

Kattawar, G. W.

Klemas, V.

W. D. Philpot, V. Klemas, “Remote Detection of Ocean Wastes,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 189 (1979).

Lyzenga, D. R.

McCluney, W. R.

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

Morel, A.

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

L. Prieur, A. Morel, “Relations theoriques entre le facteur de reflexion diffuse de l'eau de mer a diverses protondeurs et les caracteristiques optiques (absorption, diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).

Morel, A. Y.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery—A Review (Springer-Verlag, New York, 1983).

Mueller, J. L.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton Pigments from the Nimbus-7 Coastal Zone Color Scanner; Comparisons with Surface Measurements,” Science 210, 63 (1980).
[CrossRef] [PubMed]

Paredes, J. M.

Philpot, W. D.

W. D. Philpot, “Experimental Test of the Singly-Scattered Irradiance Model,” in Ocean Optics: VIII, Proc. Soc. Photo-Opt. Instrum. Eng. 637, 72 (1986).

W. D. Philpot, V. Klemas, “Remote Detection of Ocean Wastes,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 189 (1979).

W. D. Philpot, “Experimental Verification of a Radiative Transfer Model for Assessment of Water Quality,” NSF Final Report, grant CEE 82-04579 (1985).

Plass, B. N.

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

R. W. Preisendorfer, Hydrologic Optics [U.S. Department of Commerce, NOAA, Environmental Research Laboratories, Pacific Marine Environmental Laboratory, U.S. GPO 76-677-811-36, Region 8, 6 Volumes (1976)].

Prieur, L.

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

L. Prieur, A. Morel, “Relations theoriques entre le facteur de reflexion diffuse de l'eau de mer a diverses protondeurs et les caracteristiques optiques (absorption, diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).

Smith, R. C.

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

K. S. Baker, R. C. Smith, “Quasi-inherent Characteristics of the Diffuse Attenuation Coefficient for Irradiance,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
[CrossRef]

J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon & Breach, New York, 1970).

Spero, R. E.

Tyler, J. E.

J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).

J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon & Breach, New York, 1970).

Appl. Opt.

Bull. Scripps Inst. Oceanogr.

J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).

IAPSO-IGGU XVI General Assembly (Grenoble)

L. Prieur, A. Morel, “Relations theoriques entre le facteur de reflexion diffuse de l'eau de mer a diverses protondeurs et les caracteristiques optiques (absorption, diffusion),” IAPSO-IGGU XVI General Assembly (Grenoble) (1975).

Limnol. Oceanogr.

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

Mar. Geod.

A. Morel, R. C. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1982).
[CrossRef]

Ocean Optics: VIII, Proc. Soc. Photo-Opt. Instrum. Eng.

W. D. Philpot, “Experimental Test of the Singly-Scattered Irradiance Model,” in Ocean Optics: VIII, Proc. Soc. Photo-Opt. Instrum. Eng. 637, 72 (1986).

Proc. Soc. Photo-Opt. Instrum. Eng.

W. D. Philpot, V. Klemas, “Remote Detection of Ocean Wastes,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 189 (1979).

K. S. Baker, R. C. Smith, “Quasi-inherent Characteristics of the Diffuse Attenuation Coefficient for Irradiance,” Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

Science

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton Pigments from the Nimbus-7 Coastal Zone Color Scanner; Comparisons with Surface Measurements,” Science 210, 63 (1980).
[CrossRef] [PubMed]

Other

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery—A Review (Springer-Verlag, New York, 1983).

R. W. Preisendorfer, Hydrologic Optics [U.S. Department of Commerce, NOAA, Environmental Research Laboratories, Pacific Marine Environmental Laboratory, U.S. GPO 76-677-811-36, Region 8, 6 Volumes (1976)].

J. E. Tyler, R. C. Smith, Measurements of Spectral Irradiance Underwater (Gordon & Breach, New York, 1970).

J. I. Gordon, “Directional Radiance (luminance) of the Sea Surface,” SIO Ref. 69-20, Visibility Laboratory, Scripps Institute of Oceanography, San Diego, CA (1969).

W. D. Philpot, “Experimental Verification of a Radiative Transfer Model for Assessment of Water Quality,” NSF Final Report, grant CEE 82-04579 (1985).

S. G. Ackleson, “The Verification of a Radiative Transfer Model for a Multiple Layer Ocean and its Application to Remotely Sensed Imagery,” Master's Thesis, College of Marine Studies, U. Delaware, Newark (1981).

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Figures (8)

Fig. 1
Fig. 1

Basic geometry of the SSI model: z1 = depth of observation; d = depth of water; z = 0+ (air side of air–water interface); z = 0− (water side of air–water interface).

Fig.2
Fig.2

Geometry of the two-layer form of the SSI model. The boundary between the two layers is at depth z = h. The observation is made at depth z1.

Fig. 3
Fig. 3

Spectral observations and model predictions at the reflectance Ad of the tank bottom (after Ackleson18).

Fig. 4
Fig. 4

Comparison of SSI predictions (solid lines) and Monte Carlo calculations19 (circles) for the case of a two-layer optically deep water mass.

Fig. 5
Fig. 5

Irradiance measurements. Symbols represent observations; dotted lines represent polynomial fits to the data: (a) 8–9 June: illumination varied from very hazy to completely overcast; no wind, clear waters; (b) 21 June: very clear skies, solar zenith angle ranged from 21 to 34°; light breeze, clear water; (c) 22 June: generally clear skies, a few high clouds, solar zenith angle ranged from 19 to 29°; little or no breeze, water with dye.

Fig. 6
Fig. 6

Derived optical parameters, 21 June. Parameters are derived from interpolated irradiance values.

Fig. 7
Fig. 7

Derived optical parameters, 22 June. Parameters are derived from interpolated irradiance values.

Fig. 8
Fig. 8

Observed (symbols) and predicted (dotted lines) values of upwelling irradiance Eu(z) for three study dates.

Tables (3)

Tables Icon

Table I Values of Optical Parameters Needed to Evaluate the SSI Model for Comparison with the Two-Layer Monte Carlo Model

Tables Icon

Table II Irradiance reflectance R(0+) just above the Water Surface of a Two-Layer System where the Optical Properties and Thickness of the Upper Layer is Varying; for the Lower Layer, ω0 = 0.4, a, = 0.062, and bb = 0.0544

Tables Icon

Table III Derived Parameters Used to Characterize the Homogeneous Water Column In the Tank Experiment

Equations (48)

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k d ( z ) = dE d ( z ) E d ( z ) dz ,
D d ( z ) = E od ( z ) E d ( z )
D u ( z ) = E ou ( z ) E u ( z )
R ( z ) = E u ( z ) E d ( z ) .
a ( z ) = dE ( z ) E 0 ( z ) dz ,
E ( z ) = E d ( z ) E u ( z ) ,
E 0 ( z ) = E od ( z ) + E ou ( z ) .
B ss ( z ) = dE uw ( z ) E d ( z ) dz .
E d ( z ) = E d ( 0 ) exp [ 0 z k ( z ) dz ] .
dE uw ( z 1 ) = dE uw ( z ) exp [ z z 1 k ( z ) dz ] ,
k ( z ) = a ( z ) D u ( z ) .
dE uw ( z 1 ) = E d ( 0 ) B ss ( z ) × exp ( 0 z kdz ) exp ( z 1 z k dz ) dz ,
E uw ( z 1 ) = E d ( z 1 ) z 1 d B ss ( z ) × exp ( z 1 z kdz ) exp ( z 1 z k dz ) dz ,
E d ( z 1 ) = E d ( 0 ) exp ( 0 z 1 kdz ) .
E ub ( z 1 ) = E d ( z 1 ) A d exp [ z 1 d ( k + k ) dz ] .
R ( z 1 ) = E uw ( z 1 ) + E ub ( z 1 ) E d ( z 1 ) = z 1 d B ss ( z ) exp [ z 1 z ( k + k ) dz ] dz + A d exp [ z 1 d ( k + k ) d z ] .
R ( z 1 ) = z 1 h B ss ( z ) exp [ z 1 z ( k + k ) dz ] dz + h d B ss ( z ) exp [ z 1 h ( k + k ) dz ] exp [ h z ( k + k ) dz ] dz + A d exp [ z 1 h ( k + k ) dz ] exp [ h d ( k + k ) dz ] dz .
R ( z 1 ) = B 1 z 1 h exp [ ( k 1 + k 1 ) ( z z 1 ) ] dz + B 2 exp [ ( k 1 + k 1 ) ( h z 1 ) ] × h d exp [ ( k 2 + k 2 ) ( z h ) ] dz + A d exp [ ( k 1 + k 1 ( h z 1 ) ] exp [ ( k 2 + k 2 ) ( d h ) ] ,
R ( z 1 ) = B 1 K 1 { 1 exp [ K 1 ( h z 1 ) ] } + B 2 K 2 exp [ K 1 ( h z 1 ) ] { 1 exp [ K 2 ( d h ) ] } + A d exp [ K 1 ( h z 1 ) ] exp [ K 2 ( d h ) ] ,
R ( 0 ) = i = 1 n B i K i exp ( j = 1 i 1 K j Δ h j ) [ 1 exp ( K i Δ h i ) ] + A d exp ( j = 1 n ) K i Δ h i ,
k d E d = dE d dz = aD d E d + b d E d b u E u ,
k u E u = dE u dz = aD u E u b u E u + b d E d ,
B ss E d = b d E d b u E u .
k d E d = dE d dz = ( aD d + B ss ) E d ,
k u E u = dE u dz = aD u E u + B ss E d .
B ss = b d b u R
( dE u Eu dz ) absorption = aD u = k ,
( dE u E d dz ) scattering = B ss .
B ss = k d aD d .
R ( z 1 ) = B 1 K 1 = B ss a ( D u + D d ) + B ss ,
k d = aD d + B ss ,
k u = aD u B ss / R .
k d k u = 0 = a ( D d + D u ) + B ss ( a 1 R ) ,
R = B ss a ( D d + D u ) + B ss ,
R = B ss ( 4.1 ) a + B ss .
R = n = 1 3 r n X n ,
R = 0.32 b b a + b b = b b 3.1 ( a + b b ) .
R = 0.33 b b a .
R ( 0 ) = B 1 K 1 [ 1 exp ( K 1 d ) ] + A d exp ( K 1 d ) .
R ( 0 ) A d exp ( K 1 d ) .
K 1 = ln ( R / R ) d d .
R ( 0 ) = R 1 [ 1 exp ( K 1 h ) ] + R 2 exp ( K 1 h ) ,
R = 0.0001 + 0.3244 X + 0.1425 X 2 + 0.1308 X 3 ,
X = ω 0 b ̅ b 1 ω 0 b ̅ f ,
k = k d ( 1 ω 0 b ̅ f ) cD d ,
c = a 1 ω 0 .
R ( 0 + ) = 1 S w 1 S a { R 1 [ 1 exp ( K 1 d ) ] + R 2 exp ( K 1 d ) } ,
E u ( z 1 ) = E d ( 0 ) exp ( kz 1 ) × ( B ss k + k { 1 exp [ ( k + k ) ( d z 1 ) ] } + A d exp [ ( k + k ) ( d z 1 ) ] ) .

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