Photon-Vegetation Interactions pp 415-440 | Cite as

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

## 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

Biomass Anisotropy Attenuation Covariance Autocorrelation### Symbols

- B, B’
random sets of points within a space of arbitrary dimension

- b
vertical radius of spheroid

- Cov
_{f}(d) punctual covariance of scene

- d
distance vector between two background points

- d
_{l},d_{2} intermediate variables in calculation of disk overlap function

- d
_{s-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

- G
_{1} area of a single canopy envelope within a section z + dz

- H
total depth of canopy

- h
height of spheroid above ground plane

- I
_{c} radiance of sunlit crown or leaf

- I
_{G} radiance of sunlit background

- I
_{S} radiance of scene

- I
_{s} radiance of an individual pixel

- I
_{T} radiance of shaded crown or leaf

- I
_{Z} radiance of shaded background

- IPOV
Instantaneous Field of View of a sensor

- J
_{P}(d) convolution of point spread function with its conjugate at distance d

- K
_{G}, K_{Z}, K_{C}, K_{T} proportion of scene in scene component, spheroid canopy model

- k
_{G}, k_{Z}, k_{C}, k_{T} proportion of scene component within a individual pixel

- L
_{G}, L_{Z}, L_{C}, L_{T} 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(θ
_{s}, θ_{V}, ϕ) average overlap of illumination and viewing shadows on the plane for each of a collection of objects above the plane

- O(θ
_{s}, θ_{V}, ϕ, r) overlap area of illumination and viewing shadows on the background plane for a spheroid-on-a-stick of radius r

- O(θ
_{s}, θ_{V}, 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

- R
^{2} domain of two-space containing the scene

- r
horizontal radius of spheroid

- r
_{s}, r_{v} radius of disc with area equal to elliptical projection of spheroid onto background area

- s
_{s}, s_{v} intermediate variables in computation of disc overlap function

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

- t
_{s}, t_{v} intermediate variables in computation of disc overlap function

- V(K
_{G}) global (punctual) variance of K

_{G}- V(k
_{G}) variance of k

_{G}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

- γ
_{p}(d) variogram at distance d regularized over a pixel P

- θ
_{s}, θ_{V} illumination (I

_{s}) or viewing angle (v), measured from nadir- θ’
_{s}, θ’_{v} nadir angle of direction vector associated with spheroid and illumination or viewing angle

- σ
^{2}_{p}(k_{G}) parametric variance of K

_{G}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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