Beer's law
Plants absorb, transmit, and reflect shortwave radiation; the fraction of downwelling radiation partitioned into each of these categories under Beer's law is given by per wavelength band as
\[\rm{abs}_\lambda = (1 - \alpha_{\rm{leaf}, \lambda})(1 - e^{(-K(\theta_s) LAI \Omega)})\\ \rm{trans}_\lambda = e^{(-K(\theta_s) LAI \Omega)}\\ \rm{refl}_\lambda = 1 - \rm{abs}_\lambda - trans_\lambda * (1-\alpha_{\rm{ground}})\]
where λ reflects the wavelength of light, SW_d is the downwelling radiative flux, 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,λ} is the ground albedo.
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 | abs_λ | W m⁻² | 0–1 |
| Reflected fraction of radiative flux per band | refl_λ | W m⁻² | 0–1 |
| Transmitted fraction of radiative flux per band | trans_λ | 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 |