Radiative transfer of leaves

J Gómez-Dans (UCL & NCEO)


What is radiative transfer theory?

  • The properties of the leaves and the canopy are combined
  • For analysis, untangle response of the leaves and canopy.
  • We want to understand the processes and properties that control sunlight being reflected by an e.g. canopy.
  • RT tries to derive a physical description of the fate of photons as they are scattered and absorbed within a canopy.
  • Let's start with leaves
  • ... then we'll embed them in a canopy.

Modelling leaf optical properties

  • Photon hits a leaf, it can be
    • absorbed ($A$),
    • scattered ($R$), or
    • transmitted ($T$).
  • From energy conservation $A+R+T=1$
  • Implicit dependence on wavelength ($\lambda$)

How does a leaf look like?


monocot leave


dicot leave

Internal leaf structure diagram

Leaf reflectance: directional aspects

diffuse and specular

BRDF of a laurel leaf (Bousquet et al 2005)

  • The wax cuticle is bright, and scatters incoming radiation with a strong directional signal in the specular direction, $R_s$.
  • Specular response broadened by the leaf's surface microtopography.
  • At interfaces between different materials, photons are scattered.
  • Internal contribution $R_d$ is Lambertian (isotropic), the result of photons interacting with internal leaf structure and pigments.

Leaf reflectance: spectral aspects

Main spectral regions:

  1. VIS $[400-700nm]$: absorption of radiation in green leaves. Chlorophyll, carotenoids, ..
  2. NIR $[700-1100nm]$: multiple scattering due to air-cell wall interfaces,
  3. SWIR $[>1100nm]$: mostly water absorption, note peaks at 1450, 1950 and 2500 nm.
  • We will focus on the Lambertian effect
  • This is a consequence of internal photon scattering
  • therefore, it should tell us something about the internal structure and composition of the leaf.
  • Although specular component is important, it is often neglected!

A simple leaf model

  • A monocot leaf looks like a slab

  • What's weird about this photo? ;-)
  • We can model it by considering the internal reflections of a bean inside a plate:

maize leaf

  • Solution to plate model given Airy in 1833, applied to leaves by Allen and others in 1970s
  • We can write a fairly compact expression for the reflectance and transmittance of the plate that only depends on
    • The index of reflection of the plate, $n$, and
    • The absorption coefficient $k$.
  • The model works surprisingly well for monocots, but fails for more complex leaves or senescent leaves.

A multilayer leaf model

  • Numerous photon scatter events take place at the air-cell boundaries within the leaf mesophyll
  • So extend model to have a stack of plates separated by air interfaces to account for these interactions
  • Problem solved by Stokes in 1862.
  • Can calculate leaf reflectance and transmittance of $N$ layers as a function of a single layer ($R(1)$ and $T(1)$):
$$ \frac{R(N)}{b^{N}-b^{-N}}=\frac{T(N)}{a-a^{-1}}=\frac{1}{ab^{N}-a^{-1}b^{-N}}, $$


$$ \begin{align} a &=\frac{1+R^{2}(1)-T^{2}(1) + \Delta}{2R(1)}\\ b &=\frac{1-R^{2}(1)+T^{2}(1) + \Delta}{2T(1)}\\ \Delta &= \sqrt { (T^{2}(1)-R^{2}(1)-1)-4R^{2}(1)}. \end{align} $$
  • Can make number of layers a real number not an integer (e.g. 1.3 layers)
  • $\Rightarrow$ PROSPECT
  • PROSPECT predicts leaf reflectance and transmittance as a function of
    • $N$ (the number of layers $\sim \left[1.5-2.5\right]$, a sort of internal complexity of the leaf parameter)
    • Calculates $k$ from the concentrations of
      • Chlorophyll a+b
      • Leaf water
      • "Brown pigment"
      • "Dry matter"
      • Carotenoids

Chlorophyll a+b specific absorption

Cab sp abs

Carotenoids specfic absorption


Dry matter specific absorption


Water specific absorption


  • Cellulose and lignine as well as proteins have particular absorption fingerprints
  • However, absorption coincides with water absorption
  • Hence these responses lumped into "Dry matter".


  • Leaf BRDF
    • Directional component due to surface specular reflection
    • Lambertian component due to internal interactions
  • Leaves can be modelled as stacks of plates
  • Parsimonious model with a handful of parameters
    • N layers ($N\in R$)
    • Chlorophyll concentration $(\mu gcm^{-2})$
    • Carotenoids $(\mu gcm^{-2})$
    • Equivalent water thickness $(cm)$)
    • Dry matter $(gcm^{-2})$