Abstract

Detecting objects hidden beneath forest canopies is a difficult task for optical remote sensing systems. Rather than relying upon the existence of gaps between leaves, as other researchers have done, our ultimate goal is to use light scattered by leaves to image through dense foliage. Herein we describe the development of a Monte Carlo model for simulating the scattering of light as it propagates through the leaves of an extended tree canopy. We measured several parameters, including the gap fraction and maximum leaf-area density, of a nearby sugar maple tree grove and applied them to our model. We report the results of our simulation in both the ground and the receiver planes for an assumed illumination angle of 80°. To validate our model, we then illuminated the sugar maple tree grove at 80° and collected data both on the canopy floor and at our monostatic receiver aperture. Experimental results were found to correlate well with our simulated expectations.

© 2009 Optical Society of America

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References

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  1. R. M. Marino and W. R. Davis Jr., “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J. 15, 23-36 (2005).
  2. B. W. Schilling, D. N. Barr, G. C. Templeton, L. J. Mizerka, and C. W. Trussell, “Multiple-return laser radar for three-dimensional imaging through obscurations,” Appl. Opt. 41, 2791-2799 (2002).
    [CrossRef]
  3. J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
    [CrossRef]
  4. J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
    [CrossRef]
  5. S. Jacquemoud and S. Ustin, “Leaf optical properties: a state of the art,” in 8th International Symposium Physical Measurements and Signatures in Remote Sensing (CNES, 2001), pp. 223-232.
  6. T. W. Brakke, “Specular and diffuse components of radiation scattered by leaves,” Agric. Forest Meteorol. 71, 283-295(1994).
    [CrossRef]
  7. E. A. Bucher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391-2400 (1973).
    [CrossRef]
  8. C. Kittel and H. Kroemer, “Collision cross sections and mean free paths,” in Thermal Physics (Freeman, 1980), pp. 395-397.
  9. E. F. Gilman and D. G. Watson, “Acer saccharum sugar maple,” Fact Sheet ST-51, Environmental Horticultural Department, Florida Cooperative Extension Service, Institute of Food and Agricultural Sciences (1993).
  10. R. B. Myneni, J. Ross, and G. Asrar, “A review of the theory of photon transport in leaf canopies,” Agric. Forest Meteorol. 45, 1-153 (1989).
    [CrossRef]
  11. M. A. Karam and A. K. Fung, “A canopy scattering model and its application to a deciduous forest,” in International Geoscience and Remote Sensing Symposium (IEEE, 1990), pp. 137-140.
  12. D. S. Falster and M. Westoby, “Leaf size and angle vary widely across species: what consequences for light interception?,” New Phytol. 158, 509-525 (2003).
    [CrossRef]
  13. M. Abramowitz and I. A. Stegun, “Laplace transforms,” in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), pp. 1020-1021.
  14. J. Ross, V. Ross, and A. Koppel, “Estimation of leaf area and its vertical distribution during growth period,” Agric. Forest Meteorol. 101, 237-246 (2000).
    [CrossRef]
  15. B. Lalic and D. T. Mihailovic, “An empirical relation describing leaf-area density inside the forest for environmental modeling,” J. Appl. Meteorol. 43, 641-645 (2004).
    [CrossRef]
  16. M. Greiner, B. Duncan, and M. Dierking, “Bidirectional scattering distribution functions of maple and cottonwood leaves,” Appl. Opt. 46, 6485-6494 (2007).
    [CrossRef]
  17. T. W. Brakke, “Goniometric measurements of light scattered in the principle plane from leaves,” in International Geoscience and Remote Sensing Symposium (IEEE, 1992), pp. 508-510.
  18. J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inversion of a soil bidirectional reflectance model for use with vegetation reflectance models,” J. Geophys. Res. 100, 25497-25508 (1995).
    [CrossRef]
  19. S. F. Ray, “The fisheye lens and immersed optics,” in Applied Photographic Optics (Focal, 2002), pp. 327-332.
  20. L. Claustres, Y. Boucher, and M. Paulin, “Wavelet-based modeling of spectral bidirectional reflectance distribution function data,” Opt. Eng. 43, 2327-2339 (2004).
    [CrossRef]
  21. S. L. Miller and D. Childers, “Operations on a single random variable,” in Probability and Random Processes (Elsevier, 2004), pp. 100-107.

2007 (1)

2005 (1)

R. M. Marino and W. R. Davis Jr., “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J. 15, 23-36 (2005).

2004 (3)

B. Lalic and D. T. Mihailovic, “An empirical relation describing leaf-area density inside the forest for environmental modeling,” J. Appl. Meteorol. 43, 641-645 (2004).
[CrossRef]

J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
[CrossRef]

L. Claustres, Y. Boucher, and M. Paulin, “Wavelet-based modeling of spectral bidirectional reflectance distribution function data,” Opt. Eng. 43, 2327-2339 (2004).
[CrossRef]

2003 (2)

D. S. Falster and M. Westoby, “Leaf size and angle vary widely across species: what consequences for light interception?,” New Phytol. 158, 509-525 (2003).
[CrossRef]

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

2002 (1)

2000 (1)

J. Ross, V. Ross, and A. Koppel, “Estimation of leaf area and its vertical distribution during growth period,” Agric. Forest Meteorol. 101, 237-246 (2000).
[CrossRef]

1995 (1)

J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inversion of a soil bidirectional reflectance model for use with vegetation reflectance models,” J. Geophys. Res. 100, 25497-25508 (1995).
[CrossRef]

1994 (1)

T. W. Brakke, “Specular and diffuse components of radiation scattered by leaves,” Agric. Forest Meteorol. 71, 283-295(1994).
[CrossRef]

1989 (1)

R. B. Myneni, J. Ross, and G. Asrar, “A review of the theory of photon transport in leaf canopies,” Agric. Forest Meteorol. 45, 1-153 (1989).
[CrossRef]

1973 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, “Laplace transforms,” in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), pp. 1020-1021.

Asrar, G.

R. B. Myneni, J. Ross, and G. Asrar, “A review of the theory of photon transport in leaf canopies,” Agric. Forest Meteorol. 45, 1-153 (1989).
[CrossRef]

Austin, W.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Barr, D. N.

Boucher, Y.

L. Claustres, Y. Boucher, and M. Paulin, “Wavelet-based modeling of spectral bidirectional reflectance distribution function data,” Opt. Eng. 43, 2327-2339 (2004).
[CrossRef]

Brakke, T. W.

T. W. Brakke, “Specular and diffuse components of radiation scattered by leaves,” Agric. Forest Meteorol. 71, 283-295(1994).
[CrossRef]

T. W. Brakke, “Goniometric measurements of light scattered in the principle plane from leaves,” in International Geoscience and Remote Sensing Symposium (IEEE, 1992), pp. 508-510.

Bridges, R.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Bucher, E. A.

Childers, D.

S. L. Miller and D. Childers, “Operations on a single random variable,” in Probability and Random Processes (Elsevier, 2004), pp. 100-107.

Claustres, L.

L. Claustres, Y. Boucher, and M. Paulin, “Wavelet-based modeling of spectral bidirectional reflectance distribution function data,” Opt. Eng. 43, 2327-2339 (2004).
[CrossRef]

Davis, R. E.

J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
[CrossRef]

Davis, W. R.

R. M. Marino and W. R. Davis Jr., “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J. 15, 23-36 (2005).

Dierking, M.

Duncan, B.

Emery, W. J.

J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inversion of a soil bidirectional reflectance model for use with vegetation reflectance models,” J. Geophys. Res. 100, 25497-25508 (1995).
[CrossRef]

Falster, D. S.

D. S. Falster and M. Westoby, “Leaf size and angle vary widely across species: what consequences for light interception?,” New Phytol. 158, 509-525 (2003).
[CrossRef]

Fung, A. K.

M. A. Karam and A. K. Fung, “A canopy scattering model and its application to a deciduous forest,” in International Geoscience and Remote Sensing Symposium (IEEE, 1990), pp. 137-140.

Gilman, E. F.

E. F. Gilman and D. G. Watson, “Acer saccharum sugar maple,” Fact Sheet ST-51, Environmental Horticultural Department, Florida Cooperative Extension Service, Institute of Food and Agricultural Sciences (1993).

Greiner, M.

Jacquemoud, S.

S. Jacquemoud and S. Ustin, “Leaf optical properties: a state of the art,” in 8th International Symposium Physical Measurements and Signatures in Remote Sensing (CNES, 2001), pp. 223-232.

Karam, M. A.

M. A. Karam and A. K. Fung, “A canopy scattering model and its application to a deciduous forest,” in International Geoscience and Remote Sensing Symposium (IEEE, 1990), pp. 137-140.

Kittel, C.

C. Kittel and H. Kroemer, “Collision cross sections and mean free paths,” in Thermal Physics (Freeman, 1980), pp. 395-397.

Koppel, A.

J. Ross, V. Ross, and A. Koppel, “Estimation of leaf area and its vertical distribution during growth period,” Agric. Forest Meteorol. 101, 237-246 (2000).
[CrossRef]

Kroemer, H.

C. Kittel and H. Kroemer, “Collision cross sections and mean free paths,” in Thermal Physics (Freeman, 1980), pp. 395-397.

Lalic, B.

B. Lalic and D. T. Mihailovic, “An empirical relation describing leaf-area density inside the forest for environmental modeling,” J. Appl. Meteorol. 43, 641-645 (2004).
[CrossRef]

Liu, J.

J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
[CrossRef]

Marino, R. M.

R. M. Marino and W. R. Davis Jr., “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J. 15, 23-36 (2005).

Melloh, R. A.

J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
[CrossRef]

Mihailovic, D. T.

B. Lalic and D. T. Mihailovic, “An empirical relation describing leaf-area density inside the forest for environmental modeling,” J. Appl. Meteorol. 43, 641-645 (2004).
[CrossRef]

Miller, S. L.

S. L. Miller and D. Childers, “Operations on a single random variable,” in Probability and Random Processes (Elsevier, 2004), pp. 100-107.

Mizerka, L. J.

Moran, S. E.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Murray, J. T.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Myneni, R. B.

J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inversion of a soil bidirectional reflectance model for use with vegetation reflectance models,” J. Geophys. Res. 100, 25497-25508 (1995).
[CrossRef]

R. B. Myneni, J. Ross, and G. Asrar, “A review of the theory of photon transport in leaf canopies,” Agric. Forest Meteorol. 45, 1-153 (1989).
[CrossRef]

Ochs, E. S.

J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
[CrossRef]

Paulin, M.

L. Claustres, Y. Boucher, and M. Paulin, “Wavelet-based modeling of spectral bidirectional reflectance distribution function data,” Opt. Eng. 43, 2327-2339 (2004).
[CrossRef]

Pinty, B.

J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inversion of a soil bidirectional reflectance model for use with vegetation reflectance models,” J. Geophys. Res. 100, 25497-25508 (1995).
[CrossRef]

Privette, J. L.

J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inversion of a soil bidirectional reflectance model for use with vegetation reflectance models,” J. Geophys. Res. 100, 25497-25508 (1995).
[CrossRef]

Ray, S. F.

S. F. Ray, “The fisheye lens and immersed optics,” in Applied Photographic Optics (Focal, 2002), pp. 327-332.

Roddier, N.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Ross, J.

J. Ross, V. Ross, and A. Koppel, “Estimation of leaf area and its vertical distribution during growth period,” Agric. Forest Meteorol. 101, 237-246 (2000).
[CrossRef]

R. B. Myneni, J. Ross, and G. Asrar, “A review of the theory of photon transport in leaf canopies,” Agric. Forest Meteorol. 45, 1-153 (1989).
[CrossRef]

Ross, V.

J. Ross, V. Ross, and A. Koppel, “Estimation of leaf area and its vertical distribution during growth period,” Agric. Forest Meteorol. 101, 237-246 (2000).
[CrossRef]

Schilling, B. W.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, “Laplace transforms,” in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), pp. 1020-1021.

Templeton, G. C.

Trussell, C. W.

Ustin, S.

S. Jacquemoud and S. Ustin, “Leaf optical properties: a state of the art,” in 8th International Symposium Physical Measurements and Signatures in Remote Sensing (CNES, 2001), pp. 223-232.

Vercillo, R.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Watson, D. G.

E. F. Gilman and D. G. Watson, “Acer saccharum sugar maple,” Fact Sheet ST-51, Environmental Horticultural Department, Florida Cooperative Extension Service, Institute of Food and Agricultural Sciences (1993).

Westoby, M.

D. S. Falster and M. Westoby, “Leaf size and angle vary widely across species: what consequences for light interception?,” New Phytol. 158, 509-525 (2003).
[CrossRef]

Woodcock, C. E.

J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
[CrossRef]

Agric. Forest Meteorol. (3)

R. B. Myneni, J. Ross, and G. Asrar, “A review of the theory of photon transport in leaf canopies,” Agric. Forest Meteorol. 45, 1-153 (1989).
[CrossRef]

J. Ross, V. Ross, and A. Koppel, “Estimation of leaf area and its vertical distribution during growth period,” Agric. Forest Meteorol. 101, 237-246 (2000).
[CrossRef]

T. W. Brakke, “Specular and diffuse components of radiation scattered by leaves,” Agric. Forest Meteorol. 71, 283-295(1994).
[CrossRef]

Appl. Opt. (3)

Hydrol. Process. (1)

J. Liu, R. A. Melloh, C. E. Woodcock, R. E. Davis, and E. S. Ochs, “The effect of viewing geometry and topography on gap fractions through forest canopies,” Hydrol. Process. 18, 3595-3607 (2004).
[CrossRef]

J. Appl. Meteorol. (1)

B. Lalic and D. T. Mihailovic, “An empirical relation describing leaf-area density inside the forest for environmental modeling,” J. Appl. Meteorol. 43, 641-645 (2004).
[CrossRef]

J. Geophys. Res. (1)

J. L. Privette, R. B. Myneni, W. J. Emery, and B. Pinty, “Inversion of a soil bidirectional reflectance model for use with vegetation reflectance models,” J. Geophys. Res. 100, 25497-25508 (1995).
[CrossRef]

Lincoln Lab. J. (1)

R. M. Marino and W. R. Davis Jr., “Jigsaw: a foliage-penetrating 3D imaging laser radar system,” Lincoln Lab. J. 15, 23-36 (2005).

New Phytol. (1)

D. S. Falster and M. Westoby, “Leaf size and angle vary widely across species: what consequences for light interception?,” New Phytol. 158, 509-525 (2003).
[CrossRef]

Opt. Eng. (1)

L. Claustres, Y. Boucher, and M. Paulin, “Wavelet-based modeling of spectral bidirectional reflectance distribution function data,” Opt. Eng. 43, 2327-2339 (2004).
[CrossRef]

Proc. SPIE (1)

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, “Advanced 3D polarimetric flash ladar imaging through foliage,” Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Other (8)

M. Abramowitz and I. A. Stegun, “Laplace transforms,” in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, 1972), pp. 1020-1021.

M. A. Karam and A. K. Fung, “A canopy scattering model and its application to a deciduous forest,” in International Geoscience and Remote Sensing Symposium (IEEE, 1990), pp. 137-140.

C. Kittel and H. Kroemer, “Collision cross sections and mean free paths,” in Thermal Physics (Freeman, 1980), pp. 395-397.

E. F. Gilman and D. G. Watson, “Acer saccharum sugar maple,” Fact Sheet ST-51, Environmental Horticultural Department, Florida Cooperative Extension Service, Institute of Food and Agricultural Sciences (1993).

S. L. Miller and D. Childers, “Operations on a single random variable,” in Probability and Random Processes (Elsevier, 2004), pp. 100-107.

S. F. Ray, “The fisheye lens and immersed optics,” in Applied Photographic Optics (Focal, 2002), pp. 327-332.

S. Jacquemoud and S. Ustin, “Leaf optical properties: a state of the art,” in 8th International Symposium Physical Measurements and Signatures in Remote Sensing (CNES, 2001), pp. 223-232.

T. W. Brakke, “Goniometric measurements of light scattered in the principle plane from leaves,” in International Geoscience and Remote Sensing Symposium (IEEE, 1992), pp. 508-510.

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Figures (19)

Fig. 1
Fig. 1

Example hemispheric image captured looking upward within a grove of sugar maple trees near Dayton, Ohio.

Fig. 2
Fig. 2

Canopy illuminated by a monostatic ladar system at an angle θ INC . The primary target is on the ground at the geometric center of the canopy, while a secondary target is located directly below the point where the incident beam enters the canopy.

Fig. 3
Fig. 3

Monte Carlo algorithm flow chart describing photon propagation through a tree canopy.

Fig. 4
Fig. 4

(a) Leaf orientation angles are defined in the ( x , y , z ) canopy coordinate system, while (b) leaf scattering angles are defined in the leaf coordinate system.

Fig. 5
Fig. 5

Representative maple leaf-area density as a function of canopy height.

Fig. 6
Fig. 6

Illustration of region-to-region propagation in a vertically segmented tree canopy. If the random propagation distance places a photon outside the current region, then we propagate the photon to the edge of the next region and select a new propagation distance. If the random propagation distance places the photon within the current region, then the photon is assumed to have experienced a leaf interaction.

Fig. 7
Fig. 7

Example BSDF for a sugar maple leaf illuminated at an incidence zenith angle of 110 ° . The BSDF is uniform in azimuth.

Fig. 8
Fig. 8

Mean gap fraction as a function of zenith angle for our nearby grove of sugar maple trees.

Fig. 9
Fig. 9

Probability of an unscattered photon reaching the ground as a function of L m for a zenith illumination angle of 80 ° . The arrow depicts the location where the probability of an unscattered photon is equal to the measured gap fraction of the canopy.

Fig. 10
Fig. 10

Characteristic temporal waveform based upon those simulated photons that strike the virtual canopy floor.

Fig. 11
Fig. 11

Simulated ground plane beam footprint for an illumination angle of 80 ° .

Fig. 12
Fig. 12

Simulated beam footprint cross sections and best- fit Gaussian distributions in the (a) range and (b) cross-range dimensions.

Fig. 13
Fig. 13

Spatial distribution of simulated RMS pulse widths on the virtual canopy floor.

Fig. 14
Fig. 14

Simulated 1-D temporal PDF measured in pupil plane of the virtual detector.

Fig. 15
Fig. 15

Characteristic temporal waveform measured by an upward-looking APD placed on the canopy floor.

Fig. 16
Fig. 16

Measured ground plane beam footprint for an illumination angle of 80 ° .

Fig. 17
Fig. 17

Measured beam footprint cross-sectional data and best-fit Gaussian curves in the (a) range and (b) cross-range dimensions.

Fig. 18
Fig. 18

Spatial distribution of actual RMS pulse widths measured on the canopy floor.

Fig. 19
Fig. 19

Actual 1-D temporal PDF measured with a range-gated intensified CCD camera located in the tower.

Equations (22)

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

[ x S y S z S ] = [ sin θ S cos ϕ S sin θ S sin ϕ S cos θ S ] .
[ x y z ] = [ cos θ L cos ϕ L sin ϕ L sin θ L cos ϕ L cos θ L sin ϕ L cos ϕ L sin θ L sin ϕ L sin θ L 0 cos θ L ] [ sin θ S cos ϕ S sin θ S sin ϕ S cos θ S ] .
θ = tan 1 ( x 2 + y 2 z ) , ϕ = tan 1 ( y x ) .
p d ( d ) = 1 D exp [ d D ] ,
D ( z , θ ) = 1 A ¯ p ( θ ) · N ( z ) ,
p = x ^ sin θ + z ^ cos θ .
n = x ^ sin θ L cos ϕ L + y ^ sin θ L sin ϕ L + z ^ cos θ L ,
A p = A 0 n p ,
A p = A 0 [ sin θ L cos ϕ L sin θ + cos ϕ L cos θ ] .
A p ¯ ( θ ) = A p ( θ L , ϕ L : θ ) p θ L ( θ L ) p ϕ L ( ϕ L ) d ϕ L d θ L .
p Φ L ( ϕ L ) = 1 2 π [ u ( ϕ L ) u ( ϕ L 2 π ) ] ,
p Θ L ( θ L ) = [ u ( θ L ) u ( θ L π 2 ) ] cos θ L .
A p ¯ ( θ ) = d 2 4 [ sin θ + π 2 4 cos θ ] .
L ( z ) = L m ( h z m h z ) n exp [ n ( 1 h z m h z ) ] ,
n = { 6 0 z z m 1 / 2 z m z h .
N ( z ) = L ( z ) A 0 = 4 π d L 2 L m ( h z m h z ) n exp [ n ( 1 h z m h z ) ] .
D ( z , θ ) = π [ sin θ + π 2 4 cos θ ] · L m ( h z m h z ) n exp [ n ( 1 h z m h z ) ] .
p Θ S ( θ S ) = ( 1 F s ) cos θ S + F S ( π 2 θ s ) ( 2 π θ Inc ) 2 exp ( ( π 2 θ s ) 2 2 ( 2 π θ Inc ) 2 ) ,
P { z k 1 < z < z k } = a k 1 a k 1 D k exp ( x D k ) d x a k 1 1 D k exp ( x D k ) d x ,
P { 0 < z < h ¯ } = k = 1 K [ 1 a k 1 a k 1 D k exp ( x D k ) d x a k 1 1 D k exp ( x D k ) d x ] .
P { 0 < x < h ¯ : L m } | 80 ° = exp ( 9.0490 · L m ) .
y = f ( x ) = F Y 1 ( x ) ,

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