Beer's law
Plants absorb, transmit, and reflect shortwave radiation; On both the downwards and upwards pass of radiation through the canopy, the canopy transmits a fraction per wavelength band of
\[f_{\rm{trans}} = e^{(-K(\theta_s) LAI \Omega)}.\\\]
The total fraction of the incident radiation on the land surface absorbed by the canopy is
\[f_{\rm{abs, canopy}} = (1 - \alpha_{\rm{leaf}})(1 - f_{\rm{trans}})(1+\alpha_{\rm{ground}}f_{\rm{trans}_\lambda}),\\\]
while the total absorbed by the ground is
\[f_{\rm{abs, ground}} = (1 - \alpha_{\rm{ground}})f_{\rm{trans}}.\\\]
The upwelling radiation from the land is
\[f_{\rm{refl}} = 1 - f_{\rm{abs, canopy}} - f_{\rm{trans}} * (1-\alpha_{\rm{ground}}).\]
where α_{leaf,λ} is the albedo of the leaves in that wavelength band, K is the extinction coefficient, θ_s is the zenith angle, LAI is the leaf area index, Ω is the clumping index, and α_{ground} is the ground albedo. Each of these is a function of wavelength.
The extinction coefficient is defined as
\[K = l_d/\max{(\cos{(\theta_s)}, \epsilon)}\]
where $l_d$ is the leaf distribution factor, and where the denominator is structured so that at night, when 3π/2 > $θ_s$ > π/2, $K$ is large (lots of extinction) and non-negative. The small value ε prevents dividing by zero.
The model has the following variables:
| Output | Symbol | Unit | Range |
|---|---|---|---|
| Absorbed fraction of radiative flux per band | fabsλ | W m⁻² | 0–1 |
| Reflected fraction of radiative flux per band | freflλ | W m⁻² | 0–1 |
| Transmitted fraction of radiative flux per band | ftransλ | W m⁻² | 0–1 |
| Input | Symbol | Unit | Range |
|---|---|---|---|
| Leaf reflectance (albedo) | $α\_{\rm{leaf},λ}$ | - | 0–1 |
| Extinction coefficient | $K$ | - | K>0 |
| Clumping index | $Ω$ | - | 0–1 |
| Zenith angle | $θ_s$ | rad | 0–π |
| Leaf Area Index | LAI | m² m⁻² | 0–10 |
| Leaf angle distribution | $l_d$ | - | 0–1 |
| Ground albedo | $α\_{\rm{ground},λ}$ | - | 0–1 |
Beer's law as implemented has the following simplifications:
- No leaf transmittance: Assumes leaves only reflect ($α_{\rm{leaf}}$) or absorb. Does not account for light transmitted through leaves ($τ_{\rm{leaf}}$).
- Single ground reflection: Accounts for one reflection from ground back through canopy, but not infinite cascade.
- No within-canopy scattering: Light scattered by leaves is not tracked within the canopy volume.
For vegetation with significant leaf transmittance (especially NIR where $τ_{\rm{leaf}}$ ~ 0.4-0.5) or applications requiring accurate multiple scattering, use the TwoStreamModel instead.