Abstract

We model the effects of the leaves of mature broadleaf (deciduous) trees on air-to-ground free-space optical communication systems operating through the leaf canopy. The concept of leaf area index (LAI) is reviewed and related to a probabilistic model of foliage consisting of obscuring leaves randomly distributed throughout a treetop layer. Individual leaves are opaque. The expected fractional unobscured area statistic is derived as well as the variance around the expected value. Monte Carlo simulation results confirm the predictions of this probabilistic model. To verify the predictions of the statistical model experimentally, a passive optical technique has been used to make measurements of observed sky illumination in a mature broadleaf environment. The results of the measurements, as a function of zenith angle, provide strong evidence for the applicability of the model, and a single parameter fit to the data reinforces a natural connection to LAI. Specific simulations of signal-to-noise ratio degradation as a function of zenith angle in a specific ground-to-unmanned aerial vehicle communication situation have demonstrated the effect of obscuration on performance.

© 2006 Optical Society of America

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References

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  1. H. Willebrand and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today's Networks (Sams Publications, 2002).
  2. I. Kim, B. McArthur, and E. Korevaar, "Comparison of laser beam propagation at 785 nm in fog and haze for optical wireless communications," http://www.opticalaccess.com.
  3. M. Achour, "Free-space optics wavelength selection: 10 µm versus shorter wavelength," J. Opt. Network. 2, 127-143 (2003).
  4. C. C. Davis and I. I. Smolyaninov, "The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system," in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 4489, 126-137 (2001).
    [CrossRef]
  5. J. M. Vose, N. H. Sullivan, B. D. Clinton, and P. V. Bolstad, "Vertical leaf area distribution, light transmittance, and application of the Beer-Lambert law in four mature hardwood stands in the Southern Appalachians," Can. J. Forest Res. 25, 1036-1043 (1995).
    [CrossRef]
  6. S. N. Martens, S. L. Ustin, and R. A. Rousseau, "Estimation of tree canopy leaf area index by gap fraction analysis," Forest Ecol. Management 61, 91-108 (1993).
    [CrossRef]
  7. W. A. Lefsky,W. B. Cohen, G. G. Parker, and D. J. Harding, "Lidar remote sensing for ecosystem studies," BioScience 52, 19-30 (2002).
    [CrossRef]

2003 (1)

M. Achour, "Free-space optics wavelength selection: 10 µm versus shorter wavelength," J. Opt. Network. 2, 127-143 (2003).

2002 (2)

H. Willebrand and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today's Networks (Sams Publications, 2002).

W. A. Lefsky,W. B. Cohen, G. G. Parker, and D. J. Harding, "Lidar remote sensing for ecosystem studies," BioScience 52, 19-30 (2002).
[CrossRef]

2001 (1)

C. C. Davis and I. I. Smolyaninov, "The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system," in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 4489, 126-137 (2001).
[CrossRef]

1995 (1)

J. M. Vose, N. H. Sullivan, B. D. Clinton, and P. V. Bolstad, "Vertical leaf area distribution, light transmittance, and application of the Beer-Lambert law in four mature hardwood stands in the Southern Appalachians," Can. J. Forest Res. 25, 1036-1043 (1995).
[CrossRef]

1993 (1)

S. N. Martens, S. L. Ustin, and R. A. Rousseau, "Estimation of tree canopy leaf area index by gap fraction analysis," Forest Ecol. Management 61, 91-108 (1993).
[CrossRef]

Achour, M.

M. Achour, "Free-space optics wavelength selection: 10 µm versus shorter wavelength," J. Opt. Network. 2, 127-143 (2003).

Bolstad, P. V.

J. M. Vose, N. H. Sullivan, B. D. Clinton, and P. V. Bolstad, "Vertical leaf area distribution, light transmittance, and application of the Beer-Lambert law in four mature hardwood stands in the Southern Appalachians," Can. J. Forest Res. 25, 1036-1043 (1995).
[CrossRef]

Clinton, B. D.

J. M. Vose, N. H. Sullivan, B. D. Clinton, and P. V. Bolstad, "Vertical leaf area distribution, light transmittance, and application of the Beer-Lambert law in four mature hardwood stands in the Southern Appalachians," Can. J. Forest Res. 25, 1036-1043 (1995).
[CrossRef]

Cohen, W. B.

W. A. Lefsky,W. B. Cohen, G. G. Parker, and D. J. Harding, "Lidar remote sensing for ecosystem studies," BioScience 52, 19-30 (2002).
[CrossRef]

Davis, C. C.

C. C. Davis and I. I. Smolyaninov, "The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system," in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 4489, 126-137 (2001).
[CrossRef]

Ghuman, B. S.

H. Willebrand and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today's Networks (Sams Publications, 2002).

Harding, D. J.

W. A. Lefsky,W. B. Cohen, G. G. Parker, and D. J. Harding, "Lidar remote sensing for ecosystem studies," BioScience 52, 19-30 (2002).
[CrossRef]

Kim, I.

I. Kim, B. McArthur, and E. Korevaar, "Comparison of laser beam propagation at 785 nm in fog and haze for optical wireless communications," http://www.opticalaccess.com.

Korevaar, E.

I. Kim, B. McArthur, and E. Korevaar, "Comparison of laser beam propagation at 785 nm in fog and haze for optical wireless communications," http://www.opticalaccess.com.

Lefsky, W. A.

W. A. Lefsky,W. B. Cohen, G. G. Parker, and D. J. Harding, "Lidar remote sensing for ecosystem studies," BioScience 52, 19-30 (2002).
[CrossRef]

Martens, S. N.

S. N. Martens, S. L. Ustin, and R. A. Rousseau, "Estimation of tree canopy leaf area index by gap fraction analysis," Forest Ecol. Management 61, 91-108 (1993).
[CrossRef]

McArthur, B.

I. Kim, B. McArthur, and E. Korevaar, "Comparison of laser beam propagation at 785 nm in fog and haze for optical wireless communications," http://www.opticalaccess.com.

Parker, G. G.

W. A. Lefsky,W. B. Cohen, G. G. Parker, and D. J. Harding, "Lidar remote sensing for ecosystem studies," BioScience 52, 19-30 (2002).
[CrossRef]

Rousseau, R. A.

S. N. Martens, S. L. Ustin, and R. A. Rousseau, "Estimation of tree canopy leaf area index by gap fraction analysis," Forest Ecol. Management 61, 91-108 (1993).
[CrossRef]

Smolyaninov, I. I.

C. C. Davis and I. I. Smolyaninov, "The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system," in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 4489, 126-137 (2001).
[CrossRef]

Sullivan, N. H.

J. M. Vose, N. H. Sullivan, B. D. Clinton, and P. V. Bolstad, "Vertical leaf area distribution, light transmittance, and application of the Beer-Lambert law in four mature hardwood stands in the Southern Appalachians," Can. J. Forest Res. 25, 1036-1043 (1995).
[CrossRef]

Ustin, S. L.

S. N. Martens, S. L. Ustin, and R. A. Rousseau, "Estimation of tree canopy leaf area index by gap fraction analysis," Forest Ecol. Management 61, 91-108 (1993).
[CrossRef]

Vose, J. M.

J. M. Vose, N. H. Sullivan, B. D. Clinton, and P. V. Bolstad, "Vertical leaf area distribution, light transmittance, and application of the Beer-Lambert law in four mature hardwood stands in the Southern Appalachians," Can. J. Forest Res. 25, 1036-1043 (1995).
[CrossRef]

Willebrand, H.

H. Willebrand and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today's Networks (Sams Publications, 2002).

BioScience (1)

W. A. Lefsky,W. B. Cohen, G. G. Parker, and D. J. Harding, "Lidar remote sensing for ecosystem studies," BioScience 52, 19-30 (2002).
[CrossRef]

Can. J. Forest Res. (1)

J. M. Vose, N. H. Sullivan, B. D. Clinton, and P. V. Bolstad, "Vertical leaf area distribution, light transmittance, and application of the Beer-Lambert law in four mature hardwood stands in the Southern Appalachians," Can. J. Forest Res. 25, 1036-1043 (1995).
[CrossRef]

Forest Ecol. Management (1)

S. N. Martens, S. L. Ustin, and R. A. Rousseau, "Estimation of tree canopy leaf area index by gap fraction analysis," Forest Ecol. Management 61, 91-108 (1993).
[CrossRef]

J. Opt. Network. (1)

M. Achour, "Free-space optics wavelength selection: 10 µm versus shorter wavelength," J. Opt. Network. 2, 127-143 (2003).

Proc. SPIE (1)

C. C. Davis and I. I. Smolyaninov, "The effect of atmospheric turbulence on bit-error-rate in an on-off-keyed optical wireless system," in Free-Space Laser Communication and Laser Imaging, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 4489, 126-137 (2001).
[CrossRef]

Other (2)

H. Willebrand and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today's Networks (Sams Publications, 2002).

I. Kim, B. McArthur, and E. Korevaar, "Comparison of laser beam propagation at 785 nm in fog and haze for optical wireless communications," http://www.opticalaccess.com.

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Figures (13)

Fig. 1
Fig. 1

(Color online) Illustration of an air-to-ground optical communication link operating through partially obscuring foliage.

Fig. 2
Fig. 2

(Color online) SLICER uses lidar to develop a digitized waveform of foliage. (Picture courtesy of NASA Goddard Space Flight Center.)

Fig. 3
Fig. 3

(Color online) Canopy region divided into n cells as m leaves are uniformly distributed.

Fig. 4
Fig. 4

(Color online) Results of Monte Carlo simulation with different leaf sizes. The curves represent different sizes of ellipses. The ellipse major/minor axial lengths for the blue, green, red, magenta, and black curves are 3 cm∕2 cm, 5 cm∕4 cm, 7 cm∕5 cm, 8 cm∕6 cm, and 9 cm∕7 cm, respectively.

Fig. 5
Fig. 5

Schematic of the apparatus and the canopy.

Fig. 6
Fig. 6

FOV of the camera looking straight up through the canopy.

Fig. 7
Fig. 7

Tilted camera with a symmetry argument to recreate conical regions.

Fig. 8
Fig. 8

(Color online) (a) Sample picture that has been taken of foliage and (b) histogram of each pixel in (a).

Fig. 9
Fig. 9

(Color online) (a)–(d) Measurements versus fitted model and minimizing error curve for each site in Greenbelt National Park.

Fig. 10
Fig. 10

UAV flying over a receiver.

Fig. 11
Fig. 11

(Color online) (a) SNR versus time and (b) zenith angle versus time.

Fig. 12
Fig. 12

(Color online) Fraction signal drop-out from flyover with increasingly larger aperture sizes.

Fig. 13
Fig. 13

(Color online) SNR during flyover with a 2 cm aperture diameter.

Tables (2)

Tables Icon

Table 1 Leaf Transmittance Values

Tables Icon

Table 2 N ′ and LAI Values for Measured Data

Equations (37)

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P R = P T F ( θ B ) 4 π R 2 L ( θ ) A R ,
F ( θ B ) = 2 1 cos θ B ,
Pr { a   leaf   is   placed   in the   i th  cell } = Pr { i } ,
Pr { i } = 1 n .
Pr { i i + 1 } = Pr { i } Pr { i + 1 } = ( 1 n ) 2 .
Pr { two leaves in the   i th   cell } = Pr { i } Pr { i } = ( 1 n ) 2 .
Pr { m  leaves in specified   cells } = ( 1 n ) m .
Pr { X i | one   leaf   placed } = ( 1 1 n ) .
Pr { X i | m   leaves   placed } = ( 1 1 n ) m .
Pr { X i } = ( 1 1 n ) e - m / n .
Pr { y   cells   unobscured } = P { Y = y } ,
Pr { Y = y } = ( m y ) ( e - m / n ) y ( 1 - e - m / n ) m y .
E { Y } = n e - m / n .
fractional   unobscured   area   = E { Y } n = e - m / n .
var { Y } = n e - m / n ( 1 - e - m / n ) .
e x 1 x .
number   of   leaves = m = ρ canopy V canopy .
V canopy = 1 3 π ( L 3 D 3 ) tan ( θ ) .
V ( φ = 0 ) = 1 3 π ( L 3 D 3 ) tan ( θ ) .
V tilt = 1 3 π [ ( L ) 3 ( D ) 3 ] tan 2 ( θ ) ,
L = L cos ( φ )   and   D = D cos ( φ ) .
V ( φ ) = 1 3 π { [ L cos ( φ ) ] 3 [ D cos ( φ ) ] 3 } tan 2 ( θ ) = 1 3 π ( L 3 D 3 ) tan 2 ( θ ) [ 1 cos 3 ( φ ) ] ,
V ( φ ) = V ( 0 ) cos 3 ( φ ) .
FA = exp [ ρ V ( 0 ) n cos 3 ( φ ) ] .
N = n ρ V ( 0 ) ,
FA = exp [ 1 N cos 3 ( φ ) ] .
LAI = i = 1 m ( projected   area   of   leaf   X i ) total   area   in   region .
I = I 0 exp ( k   LAI ) ,
I I 0 = exp ( k   LAI ) .
FA = exp ( - m n ) ,
LAI = m n .
LAI = 1 N .
Pr { i th   cell   is   unobscured } = exp ( m n ) = exp ( LAI ) ,
e LAI = LAI cos ( φ ) .
Pr { i th   cell   is   unobscured } = exp ( e LAI )
Pr { i th   cell   is   unobscured } = exp [ LAI cos ( φ ) ] .
Pr { i th   cell   is   unobscured } = exp [ LAI cos ( φ ) ] .

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