Abstract

A bathymetric lidar survey is the most cost efficient method of producing bathymetric maps in near shore areas where the ocean bottom is both highly variable and of greatest importance for shipping and recreation. So far, not much attention has been paid to the influence of bottom materials on the lidar signals. This study addresses this issue using a Monte Carlo modeling technique. The Monte Carlo simulation includes a plane parallel water body and a flat bottom with or without seagrass. The seagrass canopy structure is adopted from Zimmerman (2003). Both the surface of the seagrass leaves and the bottom are assumed to be Lambertian. Convolution with the lidar pulse function followed by the median operator is used to reduce the variance of the resultant lidar waveform. Two seagrass orientation arrangements are modeled: seagrass in still water with random leaf orientation and seagrass with a uniform orientation as would be expected when under the influence of a water current. For each case, two maximum canopy heights, 0.5 m and 1 m, three shoot densities, 100, 500, and 1000, and three bending angles, 5, 25, and 45 degrees, are considered. The seagrass is found to induce a depth bias that is proportional to an effective leaf area index (eLAI) and the contrast in reflectance between the seagrass and the bottom material.

©2011 Optical Society of America

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References

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  1. G. C. Guenther, “Airborne lidar bathymetry,” in Digital Elevation Model Technologies and Applications: The DEM Users Manual, D. F. Maune, ed. (ASPRS, 2001)
  2. G. C. Guenther, A. G. Cunningham, P. E. LaRocque, and D. J. Reid, “Meeting the accuracy challenge in airborne lidar bathymetry,” EARSeL eProceedings (2001), Vol. 1, pp. 1–27.
  3. G. C. Guenther, R. W. L. Thomas, and P. E. LaRocque, “Design considerations for achieving high accuracy with the shoals bathymetric lidar system,” Proc. SPIE 2964, 54–71 (1996).
    [Crossref]
  4. M. Lee, “Benthic mapping of coastal waters using data fusion of hyperspectral imagery and airborne laser bathymetry,” PhD dissertation (University of Florida, 2003).
  5. C.-K. Wang and W. D. Philpot, “Using airborne bathymetric lidar to detect bottom type variation in shallow waters,” Remote Sens. Environ. 106(1), 123–135 (2007).
    [Crossref]
  6. L. R. Bissonnette, G. Roy, L. Poutier, S. G. Cober, and G. A. Isaac, “Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements,” Appl. Opt. 41(30), 6307–6324 (2002).
    [Crossref] [PubMed]
  7. R. M. Measures, Laser Remote Sensing (Krieger, 1992)
  8. L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60(4), 315–323 (1995).
    [Crossref]
  9. C. Flesia and P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the LIDAR equation,” Appl. Phys. B 60(4), 331–334 (1995).
    [Crossref]
  10. A. V. Starkov, M. Noormohammadian, and U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B 60(4), 335–340 (1995).
    [Crossref]
  11. E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Analytical solution to LIDAR return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60(4), 345–353 (1995).
    [Crossref]
  12. P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Monte-Carlo calculations of LIDAR returns: procedures and results,” Appl. Phys. B 60(4), 325–329 (1995).
    [Crossref]
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    [Crossref] [PubMed]
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  15. L. R. Poole, “Radiative transfer model for airborne laser fluorosensors: inclusion of water Raman scattering,” Appl. Opt. 21(17), 3063–3065 (1982).
    [Crossref] [PubMed]
  16. D. M. Winker and L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space based LIDARS,” Appl. Phys. B 60(4), 341–344 (1995).
    [Crossref]
  17. J. Heslin, W. J. Lillycrop, and R. Pope, “CHARTS: an evolution in airborne lidar hydrography,” presented at U.S. Hydro Conference, Biloxi, Missippi, 24–27 March 2003.
  18. G. Cunningham, Marine Survey Division, Optech Inc., 100 Wildcat Road, Toronto, Ontario M3J 2Z9, Canada (personal communication, 2004).
  19. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  20. G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics, (Springer-Verlag, 1980)
  21. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, 1972).
  22. R. Barbini, F. Colao, E. Cupini, N. Ferrari, G. Ferro, and A. Palucci, “Marine code for modelling range resolved oceanographic lidar fluorosensor measurements,” EARSeL eProceedings (2001), Vol. 1, pp. 77–87.
  23. R. C. Zimmerman, “A biooptical model of irradiance distribution and photosynthesis in seagrass canopies,” Limnol. Oceanogr. 48(1_part_2), 568–585 (2003).
    [Crossref]
  24. R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, 1981)
  25. Y. M. Govaerts, S. Jacquemoud, M. M. Verstraete, and S. L. Ustin, “Three-dimensional radiation transfer modeling in a dicotyledon leaf,” Appl. Opt. 35(33), 6585–6598 (1996).
    [Crossref] [PubMed]
  26. T. W. Brakke, J. A. Smith, and J. M. Harnden, “Bidirectional scattering of light from tree leaves,” Remote Sens. Environ. 29(2), 175–183 (1989).
    [Crossref]
  27. R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25(4), 045010 (2009), doi:.
    [Crossref]
  28. E. J. Hochberg, M. J. Atkinson, and S. Andrefouet, “Spectral reflectance of coral reef bottom-types worldwide and implications for coral reef remote sensing,” Remote Sens. Environ. 85(2), 159–173 (2003).
    [Crossref]

2009 (1)

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25(4), 045010 (2009), doi:.
[Crossref]

2007 (1)

C.-K. Wang and W. D. Philpot, “Using airborne bathymetric lidar to detect bottom type variation in shallow waters,” Remote Sens. Environ. 106(1), 123–135 (2007).
[Crossref]

2003 (2)

R. C. Zimmerman, “A biooptical model of irradiance distribution and photosynthesis in seagrass canopies,” Limnol. Oceanogr. 48(1_part_2), 568–585 (2003).
[Crossref]

E. J. Hochberg, M. J. Atkinson, and S. Andrefouet, “Spectral reflectance of coral reef bottom-types worldwide and implications for coral reef remote sensing,” Remote Sens. Environ. 85(2), 159–173 (2003).
[Crossref]

2002 (1)

1996 (2)

G. C. Guenther, R. W. L. Thomas, and P. E. LaRocque, “Design considerations for achieving high accuracy with the shoals bathymetric lidar system,” Proc. SPIE 2964, 54–71 (1996).
[Crossref]

Y. M. Govaerts, S. Jacquemoud, M. M. Verstraete, and S. L. Ustin, “Three-dimensional radiation transfer modeling in a dicotyledon leaf,” Appl. Opt. 35(33), 6585–6598 (1996).
[Crossref] [PubMed]

1995 (6)

D. M. Winker and L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space based LIDARS,” Appl. Phys. B 60(4), 341–344 (1995).
[Crossref]

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60(4), 315–323 (1995).
[Crossref]

C. Flesia and P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the LIDAR equation,” Appl. Phys. B 60(4), 331–334 (1995).
[Crossref]

A. V. Starkov, M. Noormohammadian, and U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B 60(4), 335–340 (1995).
[Crossref]

E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Analytical solution to LIDAR return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60(4), 345–353 (1995).
[Crossref]

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Monte-Carlo calculations of LIDAR returns: procedures and results,” Appl. Phys. B 60(4), 325–329 (1995).
[Crossref]

1989 (1)

T. W. Brakke, J. A. Smith, and J. M. Harnden, “Bidirectional scattering of light from tree leaves,” Remote Sens. Environ. 29(2), 175–183 (1989).
[Crossref]

1982 (2)

1972 (1)

Andrefouet, S.

E. J. Hochberg, M. J. Atkinson, and S. Andrefouet, “Spectral reflectance of coral reef bottom-types worldwide and implications for coral reef remote sensing,” Remote Sens. Environ. 85(2), 159–173 (2003).
[Crossref]

Atkinson, M. J.

E. J. Hochberg, M. J. Atkinson, and S. Andrefouet, “Spectral reflectance of coral reef bottom-types worldwide and implications for coral reef remote sensing,” Remote Sens. Environ. 85(2), 159–173 (2003).
[Crossref]

Bertero, M.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25(4), 045010 (2009), doi:.
[Crossref]

Bissonnette, L. R.

Boccacci, P.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25(4), 045010 (2009), doi:.
[Crossref]

Brakke, T. W.

T. W. Brakke, J. A. Smith, and J. M. Harnden, “Bidirectional scattering of light from tree leaves,” Remote Sens. Environ. 29(2), 175–183 (1989).
[Crossref]

Bruscaglioni, P.

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Monte-Carlo calculations of LIDAR returns: procedures and results,” Appl. Phys. B 60(4), 325–329 (1995).
[Crossref]

Cober, S. G.

Flesia, C.

C. Flesia and P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the LIDAR equation,” Appl. Phys. B 60(4), 331–334 (1995).
[Crossref]

Gordon, H. R.

Govaerts, Y. M.

Guenther, G. C.

G. C. Guenther, R. W. L. Thomas, and P. E. LaRocque, “Design considerations for achieving high accuracy with the shoals bathymetric lidar system,” Proc. SPIE 2964, 54–71 (1996).
[Crossref]

Harnden, J. M.

T. W. Brakke, J. A. Smith, and J. M. Harnden, “Bidirectional scattering of light from tree leaves,” Remote Sens. Environ. 29(2), 175–183 (1989).
[Crossref]

Hochberg, E. J.

E. J. Hochberg, M. J. Atkinson, and S. Andrefouet, “Spectral reflectance of coral reef bottom-types worldwide and implications for coral reef remote sensing,” Remote Sens. Environ. 85(2), 159–173 (2003).
[Crossref]

Isaac, G. A.

Ismaelli, A.

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Monte-Carlo calculations of LIDAR returns: procedures and results,” Appl. Phys. B 60(4), 325–329 (1995).
[Crossref]

Jacquemoud, S.

Katsev, I. L.

E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Analytical solution to LIDAR return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60(4), 345–353 (1995).
[Crossref]

Kattawar, G. W.

LaRocque, P. E.

G. C. Guenther, R. W. L. Thomas, and P. E. LaRocque, “Design considerations for achieving high accuracy with the shoals bathymetric lidar system,” Proc. SPIE 2964, 54–71 (1996).
[Crossref]

Noormohammadian, M.

A. V. Starkov, M. Noormohammadian, and U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B 60(4), 335–340 (1995).
[Crossref]

Oppel, U. G.

A. V. Starkov, M. Noormohammadian, and U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B 60(4), 335–340 (1995).
[Crossref]

Philpot, W. D.

C.-K. Wang and W. D. Philpot, “Using airborne bathymetric lidar to detect bottom type variation in shallow waters,” Remote Sens. Environ. 106(1), 123–135 (2007).
[Crossref]

Plass, G. N.

Polonsky, I. N.

E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Analytical solution to LIDAR return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60(4), 345–353 (1995).
[Crossref]

Poole, L. R.

D. M. Winker and L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space based LIDARS,” Appl. Phys. B 60(4), 341–344 (1995).
[Crossref]

L. R. Poole, “Radiative transfer model for airborne laser fluorosensors: inclusion of water Raman scattering,” Appl. Opt. 21(17), 3063–3065 (1982).
[Crossref] [PubMed]

Poutier, L.

Roy, G.

Schwendimann, P.

C. Flesia and P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the LIDAR equation,” Appl. Phys. B 60(4), 331–334 (1995).
[Crossref]

Smith, J. A.

T. W. Brakke, J. A. Smith, and J. M. Harnden, “Bidirectional scattering of light from tree leaves,” Remote Sens. Environ. 29(2), 175–183 (1989).
[Crossref]

Starkov, A. V.

A. V. Starkov, M. Noormohammadian, and U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B 60(4), 335–340 (1995).
[Crossref]

Thomas, R. W. L.

G. C. Guenther, R. W. L. Thomas, and P. E. LaRocque, “Design considerations for achieving high accuracy with the shoals bathymetric lidar system,” Proc. SPIE 2964, 54–71 (1996).
[Crossref]

Ustin, S. L.

Verstraete, M. M.

Wang, C.-K.

C.-K. Wang and W. D. Philpot, “Using airborne bathymetric lidar to detect bottom type variation in shallow waters,” Remote Sens. Environ. 106(1), 123–135 (2007).
[Crossref]

Winker, D. M.

D. M. Winker and L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space based LIDARS,” Appl. Phys. B 60(4), 341–344 (1995).
[Crossref]

Zaccanti, G.

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Monte-Carlo calculations of LIDAR returns: procedures and results,” Appl. Phys. B 60(4), 325–329 (1995).
[Crossref]

Zanella, R.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25(4), 045010 (2009), doi:.
[Crossref]

Zanni, L.

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25(4), 045010 (2009), doi:.
[Crossref]

Zege, E. P.

E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Analytical solution to LIDAR return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60(4), 345–353 (1995).
[Crossref]

Zimmerman, R. C.

R. C. Zimmerman, “A biooptical model of irradiance distribution and photosynthesis in seagrass canopies,” Limnol. Oceanogr. 48(1_part_2), 568–585 (2003).
[Crossref]

Appl. Opt. (5)

Appl. Phys. B (6)

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60(4), 315–323 (1995).
[Crossref]

C. Flesia and P. Schwendimann, “Analytical multiple-scattering extension of the Mie theory: the LIDAR equation,” Appl. Phys. B 60(4), 331–334 (1995).
[Crossref]

A. V. Starkov, M. Noormohammadian, and U. G. Oppel, “A stochastic model and a variance-reduction Monte-Carlo method for the calculation of light transport,” Appl. Phys. B 60(4), 335–340 (1995).
[Crossref]

E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Analytical solution to LIDAR return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60(4), 345–353 (1995).
[Crossref]

P. Bruscaglioni, A. Ismaelli, and G. Zaccanti, “Monte-Carlo calculations of LIDAR returns: procedures and results,” Appl. Phys. B 60(4), 325–329 (1995).
[Crossref]

D. M. Winker and L. R. Poole, “Monte-Carlo calculations of cloud returns for ground-based and space based LIDARS,” Appl. Phys. B 60(4), 341–344 (1995).
[Crossref]

Inverse Probl. (1)

R. Zanella, P. Boccacci, L. Zanni, and M. Bertero, “Efficient gradient projection methods for edge-preserving removal of Poisson noise,” Inverse Probl. 25(4), 045010 (2009), doi:.
[Crossref]

Limnol. Oceanogr. (1)

R. C. Zimmerman, “A biooptical model of irradiance distribution and photosynthesis in seagrass canopies,” Limnol. Oceanogr. 48(1_part_2), 568–585 (2003).
[Crossref]

Proc. SPIE (1)

G. C. Guenther, R. W. L. Thomas, and P. E. LaRocque, “Design considerations for achieving high accuracy with the shoals bathymetric lidar system,” Proc. SPIE 2964, 54–71 (1996).
[Crossref]

Remote Sens. Environ. (3)

C.-K. Wang and W. D. Philpot, “Using airborne bathymetric lidar to detect bottom type variation in shallow waters,” Remote Sens. Environ. 106(1), 123–135 (2007).
[Crossref]

E. J. Hochberg, M. J. Atkinson, and S. Andrefouet, “Spectral reflectance of coral reef bottom-types worldwide and implications for coral reef remote sensing,” Remote Sens. Environ. 85(2), 159–173 (2003).
[Crossref]

T. W. Brakke, J. A. Smith, and J. M. Harnden, “Bidirectional scattering of light from tree leaves,” Remote Sens. Environ. 29(2), 175–183 (1989).
[Crossref]

Other (11)

R. M. Measures, Laser Remote Sensing (Krieger, 1992)

R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, 1981)

M. Lee, “Benthic mapping of coastal waters using data fusion of hyperspectral imagery and airborne laser bathymetry,” PhD dissertation (University of Florida, 2003).

G. C. Guenther, “Airborne lidar bathymetry,” in Digital Elevation Model Technologies and Applications: The DEM Users Manual, D. F. Maune, ed. (ASPRS, 2001)

G. C. Guenther, A. G. Cunningham, P. E. LaRocque, and D. J. Reid, “Meeting the accuracy challenge in airborne lidar bathymetry,” EARSeL eProceedings (2001), Vol. 1, pp. 1–27.

J. Heslin, W. J. Lillycrop, and R. Pope, “CHARTS: an evolution in airborne lidar hydrography,” presented at U.S. Hydro Conference, Biloxi, Missippi, 24–27 March 2003.

G. Cunningham, Marine Survey Division, Optech Inc., 100 Wildcat Road, Toronto, Ontario M3J 2Z9, Canada (personal communication, 2004).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics, (Springer-Verlag, 1980)

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78 (Scripps Institution of Oceanography, 1972).

R. Barbini, F. Colao, E. Cupini, N. Ferrari, G. Ferro, and A. Palucci, “Marine code for modelling range resolved oceanographic lidar fluorosensor measurements,” EARSeL eProceedings (2001), Vol. 1, pp. 77–87.

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Figures (5)

Fig. 1
Fig. 1 The geometric relationship between the photon direction and the lidar receiver. The receiver collects a fraction of (a) scattered energy by sea water, (b) reflected energy by seagrass leaf, and (c) transmitted energy by seagrass leaf. [See the text for the explanation of the notations.]

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Fig. 2
Fig. 2 Comparisons of normal simulation (thin lines) and convolution of median waveform (thick lines) for beam attenuation coefficients of 0.25 and 0.45 m−1. The bottom is 20 m deep.

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Fig. 3
Fig. 3 Examples of seagrass beds (1 m x 1 m) with 1 m maximum canopy height and a shape factor of 4.75. Shoot density of 100 shoots/m2 with bending angles of (A) 5° (B) 25° (C) 45°, when the seagrass is in still water. Shoot density of 1000 shoots/m2 with bending angles of (D) 5° (E) 25° (F) 45°, when the seagrass is in still water. Shoot density of 1000 shoots/m2 and azimuth angles of (G) 0° and (H) 90°, when the seagrass is under the influence of a water current. The arrows denote the moving direction of the simulated airborne lidar.

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Fig. 4
Fig. 4 A subset of the results showing the depth error induced by turtlegrass with leaf azimuth angle 0° and canopy height of 1 m. The open and solid symbols denote sand and mud bottoms, respectively; the triangles, circles, and squares denote the bending angles of 5°, 25°, and 45°, respectively; red, blue, and green denote the shoot density of 100, 500, and 1000 shoots/m2, respectively.

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Fig. 5
Fig. 5 The depth errors induced by seagrass. (a) and (b) shows exactly the same data but are color- and symbol- coded differently according to parameters setup in Table 2. (a) The open and solid symbols denote sand and mud bottoms, respectively; the squares and circles denote eelgrass and turtlegrass, respectively; red denotes the seagrass in still water and green, blue, and cyan denote the seagrass affected by water current of 0°, 90°, and 180°, respectively. (b) The open and solid symbols denote the seagrass height of 0.5 m and 1 m, respectively; the triangles, circles, and rectangles denote the bending angle of 5°, 25°, and 45°, respectively; the red, blue, and green denote shoot density of 100, 500, and 1000 shoots/m2, respectively.

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Tables (3)

Tables Icon

Table 1 1 Optical Specifications of SHOALS1000 [18]

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Table 2 Optical Properties of Seagrass and Bottom Materials

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Table 3 Summary of eLAI and Depth Error for Different Seagrass Parameters

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Equations (25)

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

w p ′ = w p ( 1 - r F ) .
l = − ln ( Λ ) ,
w p ' = w p ⋅ ω 0 ,
ω 0 = b c .
2 π ∫ 0 π β ˜ ( θ ) sin θ d θ = 1.
β ˜ w ( θ ) = 3 16 π ( 1 + cos 2 θ ) .
β ˜ ( θ ) = β ˜ w ( θ ) ⋅ b w + β ˜ p ( θ ) ⋅ b p b w + b p ,
f θ ( θ ) = 2 π β ˜ ( θ ) sin θ ,           0 ≤ θ ≤ π ,
F θ ( θ ) = ∫ 0 θ 2 π β ˜ ( θ ′ ) sin θ ′ d θ ′ ,           0 ≤ θ ≤ π .
θ = F θ − 1 ( Λ ) .
w w = w P ′ β ˜ ( θ T , W ) × Δ Ω × ( 1 - r F ′ ) × ( T G ) N × exp ( - c d T ) ,
w p ′ = w p ⋅ R ,
F L ( θ L ) = 2 ∫ 0 θ L cos θ ′ sin θ ′ d θ ′ ,           0 ≤ θ ≤ π 2 .
θ L = F L − 1 ( Λ ) ,
w B = w P ′ cos θ T , B π × Δ Ω × ( 1 - r F ′ ) × ( T G ) N × exp ( - c d T ) ,
B ( z ) = ψ 1 + [ z I ] s ,
ψ = 2.51 h c − 0.79
I = 0.588 ( 1 − exp ( − 1.12 h c ) ) ,
h c = h m cos γ .
B ( Λ 1 ⋅ h c ) ≥ ψ ⋅ Λ 2 ,
w G = w P × R G cos θ T , G π × Δ Ω × ( 1 - r F ′ ) × ( T G ) N × exp ( - c d T ) ,
w G = w P × T G cos θ T , G π × Δ Ω × ( 1 - r F ′ ) × ( T G ) N × exp ( - c d T ) .
w P ′ = w P × ( 1 - A G ) ,
eLAI = A G × cos θ i , g A B × cos θ i ,
Δ D = t G − t B .

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