Absorbed lumped irradiance ========================== Absorption of lumped irradiance can be calculated following two models, respectively Beer's law **(Monsi and Saeki, 1953)** and **de Pury et al. (1997)**. Beer's law ---------- **Monsi and Saeki (1953)** were probably the first to use Beer-Lambert's law in order to simulate irradiance transfer through crop canopies. Following this approach, leaves are assumed to form a continuous turbid medium that intercepts the incident irradiance :math:`I_{inc} \ [W \cdot m^{-2}_{ground}]` and the ability of the canopy to transfer irradiance is represented by the so-called so-called extinction coefficient property :math:`k_{lumped} \ [m^2_{ground} \cdot m^{-2}_{leaf}]`. .. _fig_absorption_lumped: .. figure:: figs/absorption_lumped.png :align: center *Lumped* irradiance absoption. :math:`L \ [m^2_{leaf} \cdot m^{-2}_{ground}]` is downward-cumulative leaf area index, :math:`I_{inc} \ [W \cdot m^{-2}_{ground}]` is the incident global irradiance, :math:`I_{trans} \ [W \cdot m^{-2}_{ground}]` is the transmitted global irradiance below :math:`L`, :math:`I_{abs, \ lumped} \ [W \cdot m^{-2}_{ground}]` is the absorbed *lumped* irradiance by leaves throughout :math:`L`. Below a given depth :math:`L \ [m^2_{leaf} \cdot m^{-2}_{ground}]` inside the canopy, the flux density of the transmitted irradiance :math:`I_{trans} \ [W \cdot m^{-2}_{ground}]` is calculated as: .. math:: :label: lumped_beer_transmitted I_{trans} = I_{inc} \cdot \exp \left( -k_{lumped} \cdot L \right) Given a finite leaf layer of a thickness :math:`dL \ [m^2_{leaf} \cdot m^{-2}_{ground}]` at depth :math:`L` , the absorbed *lumped* irradiance :math:`d I_{abs} \ [W \cdot m^{-2}_{ground}]` can be calculated as: .. math:: :label: lumped_beer_absorbed_finite_layer \frac{d I_{abs, \ lumped}}{d L} =& - \frac{d I_{trans}}{d L} \\ =& k_{lumped} \cdot I_{inc} \cdot \exp(-k_{lumped} \cdot L) Layered canopies ++++++++++++++++ .. _fig_absorption_lumped_layered: .. figure:: figs/absorption_lumped_layered.png :align: center *Lumped* irradiance absoption by a leaf layer spanning between upper :math:`L_u [m^2_{leaf} \cdot m^{-2}_{ground}]` and lower :math:`L_l [m^2_{leaf} \cdot m^{-2}_{ground}]` depths. The rate of the absorbed irradiance of a leaf layer that spands between an upper depth :math:`L_u \ [m^2_{leaf} \cdot m^{-2}_{ground}]` and a lower depth :math:`L_l \ [m^2_{leaf} \cdot m^{-2}_{ground}]` (:numref:`fig_absorption_lumped_layered`) is obtained from :eq:`lumped_beer_absorbed_finite_layer` as: .. math:: :label: lumped_absorbed_layer_integral I_{abs, \ lumped} = \int_{L_u}^{L_l} {k_{lumped} \cdot I_{inc} \cdot \exp(-k_{lumped} \cdot L) \ dL} which yields: .. math:: :label: lumped_absorbed_layered I_{abs, \ lumped} = I_{inc} \cdot \left[ \exp(-k_{lumped} \cdot L_u) - exp(-k_{lumped} \cdot L_l) \right] Bigleaf canopies ++++++++++++++++ .. _fig_absorption_lumped_bigleaf: .. figure:: figs/absorption_lumped_bigleaf.png :align: center *Lumped* irradiance absoption by a *bigleaf* canopy. :math:`L_t [m^2_{leaf} \cdot m^{-2}_{ground}]` is the total leaf area index. Irradiance absorption by a *bigleaf* canopy (:numref:`fig_absorption_lumped_bigleaf`) is simply derived from :eq:`lumped_absorbed_layered` by replacing :math:`L_u` and :math:`L_l` by 0 and the total leaf area index :math:`L_{t} \ [m^2_{leaf} \cdot m^{-2}_{ground}]`, respectively, which yields: .. math:: :label: lumped_absorbed_big_leaf I_{abs, \ lumped} = I_{inc} \cdot \left[1 - \exp(-k_{lumped} \cdot L_t) \right] de Pury and Farquhar (1997) --------------------------- This model is simplified from Goudriaan models **(Goudriaan, 1977; 1988; 1994; 2016)** by disregarding the scattering effect of leaves. **de Pury and Farquhar (1997)** calculated the absorbed lumped irradiance as the sum of absorbed direct and diffuse irradiance rates, respectively :math:`I_{abs, \ direct}` and :math:`I_{abs, \ diffuse} \ [W \cdot m^{-2}_{ground}]`: .. math:: :label: lumped_de_pury I_{abs, \ lumped} = I_{abs, \ direct} + I_{abs, \ diffuse} Layered canopies ++++++++++++++++ On a ground area basis, the absorbed lumped irradiance by a leaf layer spanning between depths :math:`L_u` and :math:`L_l \ [m^2_{leaf} \cdot m^{-2}_{ground}]` writes: .. math:: :label: lumped_layered_de_pury I_{abs, \ lumped} &= I_{inc, \ direct} \cdot (1 - \rho_{direct}) \cdot \left( \exp(-k_{direct} \cdot L_u) - \exp(-k_{direct} \cdot L_l) \right) \\ &+ I_{inc, \ diffuse} \cdot (1 - \rho_{diffuse}) \cdot \left( \exp(-k_{diffuse} \cdot L_u) - \exp(-k_{diffuse} \cdot L_l) \right) where :math:`I_{inc, \ direct} \ [W \cdot m^{-2}_{ground}]` is the incident direct (beam) irradiance, :math:`I_{inc, \ diffuse} \ [W \cdot m^{-2}_{ground}]` is the incident sky-diffused irradiance, :math:`\rho_{direct} \ [-]` is canopy reflectance to direct irradiance, :math:`\rho_{diffuse} \ [-]` is canopy reflectance to diffuse irradiance, :math:`k_{direct} \ [m^2_{ground} \cdot m^{-2}_{leaf}]` is the extinction coefficient of the direct irradiance, and :math:`k_{diffuse} \ [m^2_{ground} \cdot m^{-2}_{leaf}]` is the extinction coefficient of the diffuse irradiance. These variables are thoroughly described in :doc:`absorbed_sunlit_shaded`. Bigleaf canopies ++++++++++++++++ For a bigleaf canopy, equation :eq:`lumped_layered_de_pury` become: .. math:: :label: lumped_big_leaf_de_pury I_{abs, \ lumped} &= I_{inc, \ direct} \cdot (1 - \rho_{direct}) \cdot \left( 1 - \exp(-k_{direct} \cdot L_t) \right) \\ &+ I_{inc, \ diffuse} \cdot (1 - \rho_{diffuse}) \cdot \left( 1 - \exp(-k_{diffuse} \cdot L_t) \right)