Abstract

A statistical sea surface specular BRDF (bidirectional reflectance distribution function) model is developed that includes mutual shadowing by waves, wave facet hiding, and projection weighting. The integral form of the model is reduced to an analytical form by making minor and justifiable approximations. The new form of the BRDF thus allows one to compute sea reflected radiance more than 100 times faster than the traditional numerical solutions. The repercussions of the approximations used in the model are discussed. Using the analytical form of the BRDF, an analytical approximation is also obtained for the reflected sun radiance that is always good to within 1% of the numerical solution for sun elevations of more than 10° above the horizon. The model is validated against measured sea radiances found in the literature and is shown to be in very good agreement.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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  32. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
    [CrossRef]
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    [CrossRef] [PubMed]

2004

V. Ross, D. Dion, “Assessment of sea slope statistical models using a detailed micro-facet BRDF and upwelling radiance measurements,” in Proc. SPIE 5572, 112–122 (2004).
[CrossRef]

2003

B. Henderson, J. Theiler, P. V. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte-Carlo model,” Remote Sens. Environ. 88, 453–467 (2003).
[CrossRef]

W. J. Plant, “A new interpretation of sea-surface slope probability density functions,” J. Geophys. Res. 108, 3295–3298 (2003).
[CrossRef]

2002

Z. Jin, T. Charlock, K. Rutledge, “Analysis of Broadband Solar Radiation and Albedo over the Ocean Surface at COVE,” Bull. Am. Meteorol. Soc. 19, 1585–1601 (2002).

W. Su, T. Charlock, K. Rutledge, “Observations of reflectance distribution around sunglint from a coastal ocean platform,” Appl. Opt. 41, 7369–7383 (2002).
[CrossRef] [PubMed]

C. Bourlier, G. Berginc, “Microwave analytical backscattering models from randomly rough anisotropic sea surface—comparison with experimental data in C and Ku bands,” Electromagn. Waves 37, 31–78 (2002).
[CrossRef]

2000

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing function,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

Y. Liu, M.-Y. Su, X.-H. Yan, W. T. Liu, “The mean-square slope of ocean surface waves and its effects on radar backscatter,” J. Atmos. Ocean. Technol. 17, 1092–1105 (2000).
[CrossRef]

1999

P. K. Acharya, A. Berk, G. P. Anderson, N. F. Larsen, S.-C. Tsay, K. H. Stamnes, “MODTRAN4: Multiple Scattering and Bi-Directional Reflectance Distribution Function (BRDF) Upgrades to MODTRAN,” in Proc. SPIE 3756, 19–21 (1999).

1997

1995

1994

M. Mermelstein, E. Shettle, E. Takken, R. Priest, “Infrared radiance and solar glint at the ocean-sky horizon,” Appl. Opt. 33, 6022–6034 (1994).
[CrossRef] [PubMed]

J. R. Apel, “An improved model of the ocean surface wave vector spectrum and its effects on radar backscatter,” J. Geophys. Res. 99, 269–16 291 (1994).

1993

1990

J. Wu, “Mean squared slopes of the wind-disturbed water surface, their magnitude, directionality, and composition,” Radio Sci. 25, 37–48 (1990).
[CrossRef]

1988

P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13,903–13,912 (1988).
[CrossRef]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

1987

M. A. Donelan, W. J. Pierson, “Radar scattering and equilibrium range in wind-generated waves with application to scatterometry,” J. Geophys. Res. 92, 4971–5029 (1987).
[CrossRef]

1986

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

1975

1972

J. Wu, “Sea-surface slope and equilibrium wind-wave spectra,” Phys. Fluids 13, 741–747 (1972).
[CrossRef]

1969

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
[CrossRef]

1967

B. G. Smith, “Lunar surface roughness, shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

1966

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

1954

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Acharya, P. K.

P. K. Acharya, A. Berk, G. P. Anderson, N. F. Larsen, S.-C. Tsay, K. H. Stamnes, “MODTRAN4: Multiple Scattering and Bi-Directional Reflectance Distribution Function (BRDF) Upgrades to MODTRAN,” in Proc. SPIE 3756, 19–21 (1999).

Anderson, G. P.

P. K. Acharya, A. Berk, G. P. Anderson, N. F. Larsen, S.-C. Tsay, K. H. Stamnes, “MODTRAN4: Multiple Scattering and Bi-Directional Reflectance Distribution Function (BRDF) Upgrades to MODTRAN,” in Proc. SPIE 3756, 19–21 (1999).

Apel, J. R.

J. R. Apel, “An improved model of the ocean surface wave vector spectrum and its effects on radar backscatter,” J. Geophys. Res. 99, 269–16 291 (1994).

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Berginc, G.

C. Bourlier, G. Berginc, “Microwave analytical backscattering models from randomly rough anisotropic sea surface—comparison with experimental data in C and Ku bands,” Electromagn. Waves 37, 31–78 (2002).
[CrossRef]

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing function,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

Berk, A.

P. K. Acharya, A. Berk, G. P. Anderson, N. F. Larsen, S.-C. Tsay, K. H. Stamnes, “MODTRAN4: Multiple Scattering and Bi-Directional Reflectance Distribution Function (BRDF) Upgrades to MODTRAN,” in Proc. SPIE 3756, 19–21 (1999).

A. Berk, L. S. Bernstein, D. C. Robertson, “MODTRAN: A Moderate Resolution Model for LOWTRAN7,” Technical Report GL-TR-89-0122 (Air Force Geophysics Laboratory, Bedford, Mass., 1989).

Bernstein, L. S.

A. Berk, L. S. Bernstein, D. C. Robertson, “MODTRAN: A Moderate Resolution Model for LOWTRAN7,” Technical Report GL-TR-89-0122 (Air Force Geophysics Laboratory, Bedford, Mass., 1989).

Bourlier, C.

C. Bourlier, G. Berginc, “Microwave analytical backscattering models from randomly rough anisotropic sea surface—comparison with experimental data in C and Ku bands,” Electromagn. Waves 37, 31–78 (2002).
[CrossRef]

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing function,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Charlock, T.

W. Su, T. Charlock, K. Rutledge, “Observations of reflectance distribution around sunglint from a coastal ocean platform,” Appl. Opt. 41, 7369–7383 (2002).
[CrossRef] [PubMed]

Z. Jin, T. Charlock, K. Rutledge, “Analysis of Broadband Solar Radiation and Albedo over the Ocean Surface at COVE,” Bull. Am. Meteorol. Soc. 19, 1585–1601 (2002).

D. Rutan, F. Rose, N. Smith, T. Charlock, “Validation data set for CERES surface and atmospheric radiation budget (SARB),” World Climate Research Programme Global Energy and Water Cycle Experiment (WCRP/GEWEX) Newsletter 11, 11–12 (International GEWEX Project Office, Silver Spring, Maryland, 2001).

Churnside, J. H.

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Cox, C.

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

Daigle, S.

D. Dion, L. Gardenal, J. L. Forand, M. Duffy, G. Potvin, S. Daigle, “IR Boundary Layer Effects Model (IRBLEM), IRBLEM5.1 documentation, DRDC-RDDC Valcartier, Quebec, Canada (2004). (Available via e-mail by submitting request to the authors.)

Dion, D.

V. Ross, D. Dion, “Assessment of sea slope statistical models using a detailed micro-facet BRDF and upwelling radiance measurements,” in Proc. SPIE 5572, 112–122 (2004).
[CrossRef]

D. Dion, L. Gardenal, J. L. Forand, M. Duffy, G. Potvin, S. Daigle, “IR Boundary Layer Effects Model (IRBLEM), IRBLEM5.1 documentation, DRDC-RDDC Valcartier, Quebec, Canada (2004). (Available via e-mail by submitting request to the authors.)

Donelan, M. A.

M. A. Donelan, W. J. Pierson, “Radar scattering and equilibrium range in wind-generated waves with application to scatterometry,” J. Geophys. Res. 92, 4971–5029 (1987).
[CrossRef]

Duffy, M.

D. Dion, L. Gardenal, J. L. Forand, M. Duffy, G. Potvin, S. Daigle, “IR Boundary Layer Effects Model (IRBLEM), IRBLEM5.1 documentation, DRDC-RDDC Valcartier, Quebec, Canada (2004). (Available via e-mail by submitting request to the authors.)

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Forand, J. L.

D. Dion, L. Gardenal, J. L. Forand, M. Duffy, G. Potvin, S. Daigle, “IR Boundary Layer Effects Model (IRBLEM), IRBLEM5.1 documentation, DRDC-RDDC Valcartier, Quebec, Canada (2004). (Available via e-mail by submitting request to the authors.)

J. L. Forand, “The L(W)WKD Marine Boundary Layer Model - Version 7.09,” Technical Report 1999-099 (Defence Research Establishment of Valcartier (DREV), Valcartier, Quebec, Canada, 1999).

Gardenal, L.

D. Dion, L. Gardenal, J. L. Forand, M. Duffy, G. Potvin, S. Daigle, “IR Boundary Layer Effects Model (IRBLEM), IRBLEM5.1 documentation, DRDC-RDDC Valcartier, Quebec, Canada (2004). (Available via e-mail by submitting request to the authors.)

Gentili, B.

Gordon, H. R.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Guinn, J.

Henderson, B.

B. Henderson, J. Theiler, P. V. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte-Carlo model,” Remote Sens. Environ. 88, 453–467 (2003).
[CrossRef]

Hwang, P. A.

P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13,903–13,912 (1988).
[CrossRef]

Jin, Z.

Z. Jin, T. Charlock, K. Rutledge, “Analysis of Broadband Solar Radiation and Albedo over the Ocean Surface at COVE,” Bull. Am. Meteorol. Soc. 19, 1585–1601 (2002).

Kattawar, G.

Larsen, N. F.

P. K. Acharya, A. Berk, G. P. Anderson, N. F. Larsen, S.-C. Tsay, K. H. Stamnes, “MODTRAN4: Multiple Scattering and Bi-Directional Reflectance Distribution Function (BRDF) Upgrades to MODTRAN,” in Proc. SPIE 3756, 19–21 (1999).

Liu, W. T.

Y. Liu, M.-Y. Su, X.-H. Yan, W. T. Liu, “The mean-square slope of ocean surface waves and its effects on radar backscatter,” J. Atmos. Ocean. Technol. 17, 1092–1105 (2000).
[CrossRef]

Liu, Y.

Y. Liu, M.-Y. Su, X.-H. Yan, W. T. Liu, “The mean-square slope of ocean surface waves and its effects on radar backscatter,” J. Atmos. Ocean. Technol. 17, 1092–1105 (2000).
[CrossRef]

Mermelstein, M.

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Morel, A.

Munk, W.

C. Cox, W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198–227 (1954).

C. Cox, W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun glitter,” J. Opt. Soc. Am. 44, 838–850 (1954).
[CrossRef]

Pierson, W. J.

M. A. Donelan, W. J. Pierson, “Radar scattering and equilibrium range in wind-generated waves with application to scatterometry,” J. Geophys. Res. 92, 4971–5029 (1987).
[CrossRef]

Plant, W. J.

W. J. Plant, “A new interpretation of sea-surface slope probability density functions,” J. Geophys. Res. 108, 3295–3298 (2003).
[CrossRef]

Plass, G.

Potvin, G.

D. Dion, L. Gardenal, J. L. Forand, M. Duffy, G. Potvin, S. Daigle, “IR Boundary Layer Effects Model (IRBLEM), IRBLEM5.1 documentation, DRDC-RDDC Valcartier, Quebec, Canada (2004). (Available via e-mail by submitting request to the authors.)

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Albedos and glitter patterns of a wind-roughened sea surface,” J. Phys. Oceanogr. 16, 1293–1316 (1986).
[CrossRef]

Priest, R.

Robertson, D. C.

A. Berk, L. S. Bernstein, D. C. Robertson, “MODTRAN: A Moderate Resolution Model for LOWTRAN7,” Technical Report GL-TR-89-0122 (Air Force Geophysics Laboratory, Bedford, Mass., 1989).

Rose, F.

D. Rutan, F. Rose, N. Smith, T. Charlock, “Validation data set for CERES surface and atmospheric radiation budget (SARB),” World Climate Research Programme Global Energy and Water Cycle Experiment (WCRP/GEWEX) Newsletter 11, 11–12 (International GEWEX Project Office, Silver Spring, Maryland, 2001).

Ross, V.

V. Ross, D. Dion, “Assessment of sea slope statistical models using a detailed micro-facet BRDF and upwelling radiance measurements,” in Proc. SPIE 5572, 112–122 (2004).
[CrossRef]

Rutan, D.

D. Rutan, F. Rose, N. Smith, T. Charlock, “Validation data set for CERES surface and atmospheric radiation budget (SARB),” World Climate Research Programme Global Energy and Water Cycle Experiment (WCRP/GEWEX) Newsletter 11, 11–12 (International GEWEX Project Office, Silver Spring, Maryland, 2001).

Rutledge, K.

Z. Jin, T. Charlock, K. Rutledge, “Analysis of Broadband Solar Radiation and Albedo over the Ocean Surface at COVE,” Bull. Am. Meteorol. Soc. 19, 1585–1601 (2002).

W. Su, T. Charlock, K. Rutledge, “Observations of reflectance distribution around sunglint from a coastal ocean platform,” Appl. Opt. 41, 7369–7383 (2002).
[CrossRef] [PubMed]

Saillard, J.

C. Bourlier, J. Saillard, G. Berginc, “Effect of correlation between shadowing and shadowed points on the Wagner and Smith monostatic one-dimensional shadowing function,” IEEE Trans. Antennas Propag. 48, 437–446 (2000).
[CrossRef]

Sancer, M. I.

M. I. Sancer, “Shadow-corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. 17, 577–585 (1969).
[CrossRef]

Shaw, J. A.

Shemdin, O. H.

P. A. Hwang, O. H. Shemdin, “The dependence of sea surface slope on atmospheric stability and swell conditions,” J. Geophys. Res. 93, 13,903–13,912 (1988).
[CrossRef]

Shettle, E.

Smith, B. G.

B. G. Smith, “Lunar surface roughness, shadowing and thermal emission,” J. Geophys. Res. 72, 4059–4067 (1967).
[CrossRef]

B. G. Smith, “Geometrical shadowing of a random rough surface,” IEEE Trans. Antennas Propag. 15, 668–671 (1967).
[CrossRef]

Smith, N.

D. Rutan, F. Rose, N. Smith, T. Charlock, “Validation data set for CERES surface and atmospheric radiation budget (SARB),” World Climate Research Programme Global Energy and Water Cycle Experiment (WCRP/GEWEX) Newsletter 11, 11–12 (International GEWEX Project Office, Silver Spring, Maryland, 2001).

Smith, R. C.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Stamnes, K. H.

P. K. Acharya, A. Berk, G. P. Anderson, N. F. Larsen, S.-C. Tsay, K. H. Stamnes, “MODTRAN4: Multiple Scattering and Bi-Directional Reflectance Distribution Function (BRDF) Upgrades to MODTRAN,” in Proc. SPIE 3756, 19–21 (1999).

Su, M.-Y.

Y. Liu, M.-Y. Su, X.-H. Yan, W. T. Liu, “The mean-square slope of ocean surface waves and its effects on radar backscatter,” J. Atmos. Ocean. Technol. 17, 1092–1105 (2000).
[CrossRef]

Su, W.

Takken, E.

Theiler, J.

B. Henderson, J. Theiler, P. V. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte-Carlo model,” Remote Sens. Environ. 88, 453–467 (2003).
[CrossRef]

Tsay, S.-C.

P. K. Acharya, A. Berk, G. P. Anderson, N. F. Larsen, S.-C. Tsay, K. H. Stamnes, “MODTRAN4: Multiple Scattering and Bi-Directional Reflectance Distribution Function (BRDF) Upgrades to MODTRAN,” in Proc. SPIE 3756, 19–21 (1999).

Vaitekunas, D.

D. Vaitekunas, “Technical manual for SHIPIR/NTCS (2.9),” Document A912-002 (Davis Engineering Limited, Ottawa, Ontario, Canada, 2002).

Villeneuve, P. V.

B. Henderson, J. Theiler, P. V. Villeneuve, “The polarized emissivity of a wind-roughened sea surface: a Monte-Carlo model,” Remote Sens. Environ. 88, 453–467 (2003).
[CrossRef]

Wagner, R. J.

R. J. Wagner, “Shadowing of randomly rough surfaces,” J. Opt. Soc. Am. 41, 138–147 (1966).

Wu, J.

J. Wu, “Mean squared slopes of the wind-disturbed water surface, their magnitude, directionality, and composition,” Radio Sci. 25, 37–48 (1990).
[CrossRef]

J. Wu, “Sea-surface slope and equilibrium wind-wave spectra,” Phys. Fluids 13, 741–747 (1972).
[CrossRef]

Yan, X.-H.

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Figures (6)

Fig. 1
Fig. 1

Coordinate system and representations of relevant quantities: U n , facet Cartesian normal unit vector; U r , receiver Cartesian unit vector; etc. U s , source Cartesian unit vector; ω, reflection angle; θ i , zenith angle of U i vector ( i = n , r , s ) ; ϕ i , azimuth angle of U i vector ( i = n , r , s ) counterclockwise from upwind direction ( W ) ζ x , facet slope in the upwind ( W ) direction ζ y , facet slope in the crosswind direction.

Fig. 2
Fig. 2

Difference in the normalizing factor from the complete solution with the Gram–Charlier PDF at different wind speeds resulting from use of the Gaussian PDF to obtain an analytical solution.

Fig. 3
Fig. 3

Point source (80° zenith angle) reflectance for different wind speed and receiver nadir angles. The dashed curve is the analytical solution with the Gaussian PDF in both the numerator and the normalization factor, the plus signs ( + ) are the exact numerical solution with the Gram–Charlier approximation, and the solid curve is the analytical solution with the Gaussian PDF in the normalizing factor only.

Fig. 4
Fig. 4

Variations in the reflectance of a point source located at 80° zenith angle for a detector sweeping across the source azimuth at 80° nadir angle. The wind speed is 5 m s blowing toward the source. Variances used to obtain these results are those of Cox and Munk, Cox and Munk with its standard errors ( ± 0.004 ) , and Cox and Munk multiplied by two.

Fig. 5
Fig. 5

Validity domain for the analytical approximation for sun glint radiance according to maximum acceptable error.

Fig. 6
Fig. 6

(a),(b),(c) Measured and (d),(e),(f) simulated radiances averaged over 30 min on January 6, 2001, starting at 8:00 am (first column), 8:30 am (second column), and 12:00 noon (third column). Plots on the last row (g),(h),(i) are measured (solid curve) and simulated (dashed curve) azimuthal radiance distribution at 9.1°, 13.6° and 30.6° below the horizon, respectively.

Tables (2)

Tables Icon

Table 1 Speed Comparison between Analytical and Numerical Solution at Equal Accuracy

Tables Icon

Table 2 Environmental Conditions during the Data Measurements

Equations (82)

Equations on this page are rendered with MathJax. Learn more.

( x i y i z i ) = ( sin θ i cos ϕ i sin θ i sin ϕ i cos θ i ) ,
θ i = cos 1 ( z i ) ,
ϕ i = tan 1 ( y i x i ) .
( x n y n z n ) = 1 1 + ζ x 2 + ζ y 2 ( ζ x ζ y 1 ) ,
ζ x = x n z n ,
ζ y = y n z n .
U s = 2 cos ( ω ) U n U r ,
ω = cos 1 ( U r U n ) ,
U s = 2 ( U r U n ) U n U r ,
ζ x = ( x r + x s ) ( z r + z s ) ,
ζ y = ( y r + y s ) ( z r + z s ) .
p 0 ( ζ U ) = 1 2 π σ x σ y exp { 1 2 ( ζ x 2 σ x 2 + ζ y 2 σ y 2 ) }
p p 0 { 1 1 2 c 21 ( Y 2 1 ) X 1 6 c 03 ( X 3 3 X ) + 1 24 c 40 ( Y 4 6 Y 2 + 3 ) + 1 4 c 22 ( Y 2 1 ) ( X 2 1 ) + 1 24 c 04 ( X 4 6 X 2 + 3 ) } ,
X = ζ x σ x , Y = ζ y σ y .
σ x 2 = 0.003 16 U ± 0.004 ,
σ y 2 = 0.003 + 0.001 92 U ± 0.004 .
c 21 = 0.01 0.0086 U , c 03 = 0.04 0.033 U ,
c 40 = 0.40 , c 22 = 0.12 , c 04 = 0.23 .
σ s c 2 σ c m 2 = { 1.42 2.80 R i , 0.23 < R i < 0.27 0.65 , R i 0.27 } .
R i = g Δ T a w z T w U z 2 ,
P p ( ζ U ) d 2 ζ .
p ( ζ U ) d 2 ζ = 1 .
F ¯ = F ( ζ ) p ( ζ U ) d 2 ζ ,
Q q ( ζ Ψ ) d 2 ζ
Q v q v ( ζ Ψ r , Ψ s ) d 2 ζ
W ( ζ , Ψ r ) = cos ω cos θ n = U n U r z n ,
H ζ ( ζ , Ψ r ) = Υ ( U n U r ) ,
Υ ( x ) = { 0 , x 0 1 , x > 0 } .
S ζ ( ζ , Ψ s ) = Υ ( U n U s ) Υ ( π 2 θ s ) .
H w ( h , Ψ r ) = [ 1 1 2 erfc ( h ) ] Λ ( v r ) ,
Λ ( v r ) = exp ( v r 2 ) v r π erfc ( v r ) 2 v r π ,
v r = cot ( θ r ) 2 σ ( ϕ r ) ,
σ 2 ( ϕ r ) = σ x 2 cos 2 ϕ r + σ y 2 sin 2 ϕ r .
S w ( h , Ψ s ) = [ 1 1 2 erfc ( h ) ] Λ ( v s ) ,
Λ ( v s ) = exp ( v s 2 ) v s π erfc ( v s ) 2 v s π ,
v s = cot ( θ s ) 2 σ ( ϕ s ) .
q v ( ζ , h Ψ r , Ψ s ) = q v ( ζ Ψ r ) q v ( h Ψ r , Ψ s ) ,
q v ( ζ Ψ r ) = [ p ( ζ ) W ( ζ , Ψ r ) H ζ ( ζ , Ψ r ) ] ,
q v ( h Ψ r , Ψ s ) = [ p ( h ) S w ( h , Ψ s ) H w ( h , Ψ r ) ] ,
p ( h ) = 1 σ h 2 π exp ( h 2 2 σ h 2 ) .
q v ( ζ Ψ r , Ψ s ) = q v ( ζ Ψ r ) q v ( h Ψ r , Ψ s ) .
q v n ( ζ Ψ r , Ψ s ) = q v ( ζ Ψ r ) q v ( ζ Ψ r ) d 2 ζ × q v ( h Ψ r , Ψ s ) d h p ( h ) H w ( h , Ψ r ) d h .
p ( h ) H w ( h , Ψ r ) d h = p ( h ) [ 1 1 2 erfc ( h ) ] Λ ( v r ) d h = 1 1 + Λ ( v r ) .
q v ( h Ψ r , Ψ s ) d h = p ( h ) S w ( h , Ψ s ) H w ( h , Ψ r ) d h = p ( h ) [ 1 1 2 erfc ( h ) ] Λ ( v r ) + Λ ( v s ) d h = 1 1 + Λ ( v r ) + Λ ( v s ) .
q v ( ζ Ψ r ) d 2 ζ p 0 ( ζ ) W ( ζ , Ψ r ) H ζ ( ζ , Ψ r ) d 2 ζ = [ 1 + Λ ( v r ) ] cos ( θ r ) ,
q v n ( ζ Ψ r , Ψ s ) = [ 1 + Λ ( v r ) 1 + Λ ( v r ) + Λ ( v s ) ] p ( ζ ) W ( ζ , Ψ r ) H ζ ( ζ , Ψ r ) [ 1 + Λ ( v r ) ] cos θ r ,
= p ( ζ ) W ( ζ , Ψ r ) H ζ ( ζ , Ψ r ) [ 1 + Λ ( v r ) + Λ ( v s ) ] cos θ r ,
= p ( ζ ) W ( ζ , Ψ r ) H ζ ( ζ , Ψ r ) [ 1 + Λ ( v r ) ] cos θ r .
d 2 ζ = sec ω sec 3 θ n sin θ s d 2 Ψ s 4 = sin θ s d 2 Ψ s 4 z n 3 ( U n U r ) .
R ¯ sun ( Ψ r ) = L sun 4 sun disk r q v n sin θ s d 2 Ψ s z n 3 ( U n U r )
R ¯ sun ( Ψ r ) π ϵ 2 L sun r q v n 4 z n 3 ( U n U r ) ,
U ( z ) = u * k ln ( z z 0 )
I = p 0 ( ζ ) W ( ζ , Ψ r ) H ζ ( ζ , Ψ r ) d 2 ζ ,
p 0 ( ζ ) = 1 2 π σ x σ y exp { ζ x 2 2 σ x 2 ζ y 2 2 σ y 2 } ,
W ( ζ , Ψ r ) = ζ x x r ζ y y r + z r ,
H ζ ( ζ , Ψ r ) = Υ ( z n ζ x x r z n ζ y y r + z n z r ) .
ζ = ζ x cos ϕ r + ζ y sin ϕ r ,
ζ = ζ x sin ϕ r + ζ y cos ϕ r ,
ζ x = ζ cos ϕ r ζ sin ϕ r ,
ζ y = ζ sin ϕ r + ζ cos ϕ r .
p 0 ( ζ ) = 1 2 π σ x σ y exp { a 2 ζ 2 b ζ ζ c 2 ζ 2 } ,
W ( ζ , Ψ r ) = ζ sin θ r + cos θ r ,
H ζ ( ζ , Ψ r ) = Υ ( [ ζ sin θ r + cos θ r ] cos θ n ) = Υ ( ζ sin θ r + cos θ r ) ,
a = cos 2 ϕ r σ x 2 + sin 2 ϕ r σ y 2 ,
b = sin ϕ r cos ϕ r ( 1 σ y 2 1 σ x 2 ) ,
c = sin 2 ϕ r σ x 2 + cos 2 ϕ r σ y 2 .
I = I 1 + I 2 = sin θ r ζ p 0 ( ζ ) H ζ ( ζ , Ψ r ) d 2 ζ + cos θ r p 0 ( ζ ) H ζ ( ζ , Ψ r ) d 2 ζ .
p 0 ( ζ ) ζ = { a ζ + b ζ } p 0 ( ζ ) ,
p 0 ( ζ ) ζ = { b ζ c ζ } p 0 ( ζ ) ,
ζ p 0 ( ζ ) = β p 0 ( ζ ) ζ α p 0 ( ζ ) ζ ,
α = c c a b 2 , β = b c a b 2 ,
α = σ 2 ( ϕ r ) ,
β ζ ( H ζ ( ζ , Ψ r ) p 0 ( ζ ) ) β p 0 ( ζ ) H ζ ( ζ , Ψ r ) ζ α ζ ( H ζ ( ζ , Ψ r ) p 0 ( ζ ) ) + α p 0 ( ζ ) H ζ ( ζ , Ψ r ) ζ .
p 0 ( ζ ) H ζ ( ζ , Ψ r ) ζ d ζ = sin θ r p 0 ( ζ ) δ ( sin θ r ζ + cos θ r ) d ζ = p 0 ( cot θ r , ζ ) .
p 0 ( ζ ) = 1 2 π σ x σ y exp [ ζ 2 2 σ 2 ( ϕ r ) ] exp [ 1 2 { c ζ + b ζ c } 2 ] .
I 1 = σ 2 ( ϕ r ) sin θ r 2 π c σ x σ y exp [ cot 2 θ r 2 σ 2 ( ϕ r ) ] = σ ( ϕ r ) sin θ r 2 π exp [ cot 2 θ r 2 σ 2 ( ϕ r ) ] .
p 0 ( ζ ) H ζ ( ζ , Ψ r ) d 2 ζ = d ζ cot θ r p 0 ( ζ ) d ζ .
d ζ cot θ r p 0 ( ζ ) d ζ
= cot θ r 1 2 π σ ( ϕ r ) exp [ ζ 2 2 σ 2 ( ϕ r ) ] d ζ ,
I 2 = [ 1 1 2 erfc ( cot θ r 2 σ ( ϕ r ) ) ] cos θ r ,
I = σ ( ϕ r ) sin θ r 2 π exp [ cot 2 θ r 2 σ 2 ( ϕ r ) ] + [ 1 1 2 erfc ( cot θ r 2 σ ( ϕ r ) ) ] cos θ r .
I = [ 1 + Λ ( v r ) ] cos θ r .

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