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Geometric-Optical Modeling of Forests as Remotely-Sensed Scenes Composed of Three-Dimensional, Discrete Objects

  • A. H. Strahler
  • D. L. B. Jupp

Abstract

Mathematical modeling of the interaction of electromagnetic radiation with vegetation canopies is a research field that has been highly active in recent years. Modeling a canopy as a “turbid medium” of leaf and foliage elements in a slab geometry, the general equations of radiative transfer (Chandrasekhar 1950) have been approximated and solved in various ways and with varying types of description of the canopy. For example, both Suits (1972) and Verhoef (1984) use a four-stream approximation of the radiation flow and its interaction with canopy elements that has its origin in work by Schuster (1905) and Schwartzchild (1914). Cooper et al. (1982) and Kimes (1984) also use multistream approximations which, along with a variety of models for photon transport, are comprehensively reviewed in Myneni et al. (1989). In these models, the canopy elements are parameterized by such variables as the reflectance and transmittance of the leaf, leaf area index, and the leaf angle distribution. Many of these models are one-dimensional, in that the canopies vary only with height above the soil or ground surface.

Keywords

Component Radiance Leaf Canopy Bidirectional Reflectance Canopy Element Leaf Angle Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Symbols

B, B’

random sets of points within a space of arbitrary dimension

b

vertical radius of spheroid

Covf(d)

punctual covariance of scene

d

distance vector between two background points

dl,d2

intermediate variables in calculation of disk overlap function

ds-v

distance between centers of illumination and viewing projections

dv

elementary volume element for space of arbitrary dimension

f(x)

spatial binary (0,1) function returning 1 if point x lies in sunlit background, 0 else

G(θ)

projected area of a crown at angle θ

G(θ, z)

projected area of leaf at angle θ, depth z

G1

area of a single canopy envelope within a section z + dz

H

total depth of canopy

h

height of spheroid above ground plane

Ic

radiance of sunlit crown or leaf

IG

radiance of sunlit background

IS

radiance of scene

Is

radiance of an individual pixel

IT

radiance of shaded crown or leaf

IZ

radiance of shaded background

IPOV

Instantaneous Field of View of a sensor

JP(d)

convolution of point spread function with its conjugate at distance d

KG, KZ, KC, KT

proportion of scene in scene component, spheroid canopy model

kG, kZ, kC, kT

proportion of scene component within a individual pixel

LG, LZ, LC, LT

proportion of scene in scene component, leaf canopy model

N(z)

number of leaf centers in a section of the canopy between z and z + dz

Ō(θs, θV, ϕ)

average overlap of illumination and viewing shadows on the plane for each of a collection of objects above the plane

O(θs, θV, ϕ, r)

overlap area of illumination and viewing shadows on the background plane for a spheroid-on-a-stick of radius r

O(θs, θV, ϕ, z - t)

overlap of illumination and viewing shadows for a leaf at distance z - t

O*(d)

overlap function for compound figure of illumination and viewing projections for a discrete object on a plane

P

subscript denoting pixel

PSF

Point Spread Function

p(dv)

probability that a point falls within elementary volume element dv

p(r)

probability density function for r

p(θ)

gap fraction at angle θ

p(θ, z)

gap fraction at angle θ, depth z for leaf canopy

R2

domain of two-space containing the scene

r

horizontal radius of spheroid

rs, rv

radius of disc with area equal to elliptical projection of spheroid onto background area

ss, sv

intermediate variables in computation of disc overlap function

t

position vector variable of integration; variable of integration for canopy depth

ts, tv

intermediate variables in computation of disc overlap function

V(KG)

global (punctual) variance of KG

V(kG)

variance of kG observed within pixels of an image

w

intermediate depth within canopy

x

point (coordinate vector) in two-space

z

depth within canopy, 0 at top to H at ground surface

z’

variable of integration for canopy depth

λ

density of object centers on the plane

λ(z)

volume density of leaf centers at depth z

λ(d)

punctual variogram at distance d

γp(d)

variogram at distance d regularized over a pixel P

θs, θV

illumination (Is) or viewing angle (v), measured from nadir

θ’s, θ’v

nadir angle of direction vector associated with spheroid and illumination or viewing angle

σ2p(kG)

parametric variance of KG within pixels

ϕ

azimuth angle between illumination and viewing positions

χ

intermediate variable (angle) in disc overlap function

ψ

phase angle between illumination and viewing direction vectors for spheroid

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • A. H. Strahler
  • D. L. B. Jupp

There are no affiliations available

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