Geometric optical (GO) modeling of radiative transfer in plant canopy

Xin Xi

Classic radiative transfer theory in planetary atmospherethe Scattering Phase Function is used to define the

directional dependence of scattered radiation. The scattering source function is:

, ' ' '

4

,4scatJ P I d

' , cosP P

For randomly oriented spherical particles (air molecules, aerosols, cloud drops):

More complex in clouds with ice crystals and non-spherical aerosols…

Rayleigh scatteringMie-theory

Radiative transfer in plant canopyFor homogeneous agricultural crop and grasslands,

it’s an acceptable approximation to treat the plant canopy as plane-parallel layers.

Have to determine the optical properties (scattering phase function, single scattering albedo, etc) and structure-related properties (leaf size, shape, LAI, leaf angle distribution)

But for the discrete canopies (forest, savanna, row crops) with complex 3-D architecture (leaf with different size, shape and orientation; leaf/shoots/branch/crown/canopy) and spatial distribution pattern, the phase function is difficult to specify, not just a function of the scattering angle. The reflection has “bi-directionality”.

The Bi-directional Reflectance Distribution Function (BRDF) is used to describe the dependence of reflected radiation on the incident (i) and outgoing (v) directions (Nicodemus, 1977).

How surface structure affect BRDF ?

Volume scatteringeffect

Shading (single scattering) effect

Specular/glint effect

Ross-thick kernel

Li-sparse- reciprocal kernel

Sun behind observer

Sun opposite observer

Hotspot

Chen & Leblanc, 1997four-scale BRDF model

Ideas of GO modeling…

Early geometric optical model by Li&Strahler,1985

Assumptions:Parallel-ray geometry;Crowns as randomly distributed solid cones casting

shadow on a contrasting background (crowns/ shadows may overlap);

Cone size (height) distribution: lognormal (cone apex angle is constant) ;

Scene component model: total reflectance is a linear sum of the respective reflectance of each component (illuminated/shadowed crown, illuminated/shadowed ground ), weighted by their areas in the scene;

nadir view

Geometric effect on bi-directional reflectance:

Approaching the hybrid geometric-optical radiative transfer model

Mutual shadowing: Li&Strahler 1992

Inter-crown and within-crown gap prabability: Li&Strahler 1986, 1988

Multiple scattering (classic RT): Li&Strahler 1995

Gap probability is a key link of geometric optical and (classic) radiative transfer models for 3-D discrete plant canopies.

How gap probability works?

Gap probability: an incident photon passes directly through the canopy without being intercepted.

vertical distribution of sunlit crown surface, and single scattering source, which produces the hotspot effect;

openness: sky diffuse radiation into canopy and ground; ground reflection to the sky.

multiple scattering (determined by the pathlength distribution) within crowns and between crown and ground.

Within-crown gap probability as a function of the within-crown pathlength s:

, sk Dgapp s e Illumination direction Foliage area volume density, FAVD;

constant within crowns, zero outside

Knowing the distribution of pathlength s for a single crown, the mean gap probability over area A is:

At given height h, the distribution of radiance is calculated from the probability density distribution of pathlength s at h, P(s|h,θi).

Knowing the distribution of radiance with height, we can obtain the distribution of bean radiation with height that collides with the canopy. The unabsorbed proportion is further scattered (multiple scattering).

How to calculate P(s|h,θi) ?

gives the probability that a point at height h receives solar beam radiation without hitting any crown.

ViP n 0 | h, e

V can be integrated from V(h)

gives the areal proportion where solar beam radiation first enters the canopy and starts the scattering process within the thin layer

i i iP s 0 | h, =P n 0 | h, P n 0 | h ,h

h

also gives the vertical distribution of sunlit canopy

Knowing P(s=0|h), the pathlength distribution P(s|h) of solar beam radiation after entering the canopy and reaching the height h can be obtained.

A nested convolution process is used. Details are available in Li&Strahler 1988.

gives the vertical distribution of radiance.

It applied to points within and outside crowns. But scattering happens only inside canopy. So the within-crown proportion is scaled by a factor fB.

0 i iP | h, =P s | h,I

Openness for diffuse radiation

gives the proportion of sky diffuse radiation that enters the thin layer h-∆h to h.

is the proportion of upward scattered radiation exiting the canopy without further scattering.

( ) /(1 ( ))open openK h K h

( 0)openK hdescribe how much diffuse skylight reaches the ground directly without pass through the crowns

Multiple scattering

Assuming half of total single scattering goes upward and the other half downward at height h:

On the ground, only the upward radiation:

Knowing total single scattering flux at height h and assuming scattering sources are randomly spread without further linkage, the radiation reaching another height z is:

The contribution of J(h) in a thin layer centered at h to the next order scattering is

Contribution of J(h) to the next order of scattering on the ground (z=0):

Knowing the initial source distribution J(1)(h) generated by direct sunlight, the next order scattering source distribution is:

Accumulated scattering source at height h: