1.1. Optical Properties¶

ADELM derives the optical properties and geometric factors required by the two-stream solver for Shortwave Radiation. The main outputs are the single-scattering albedo \(\omega_\Lambda\), the upscatter parameters \(\beta_\Lambda^\mu\) (betad) and \(\beta_\Lambda^{\mathrm{d}}\) (betai), the direct-beam extinction coefficient \(K_{\mathrm{dir}}\), the diffuse optical-depth scale \(\bar{\mu}\), and the ground albedo \(\alpha_{\mathrm{g},\Lambda}\).

1. Vegetation area index and element fractions¶

The vegetation area index is

(1.1.1)¶\[\mathrm{VAI} = \mathrm{LAI} + \mathrm{SAI}\]

where \(\mathrm{LAI}\) (lai [m2 m-2; Driver: leaf area index]) is leaf area index and \(\mathrm{SAI}\) (stem_area_index [m2 m-2; PFT-based parameter: one-sided stem area index]) is stem area index.

The leaf and stem fractions used to weight optical properties are

(1.1.2)¶\[f_{\mathrm{leaf}} = \frac{\mathrm{LAI}}{\mathrm{VAI}}, \qquad f_{\mathrm{stem}} = \frac{\mathrm{SAI}}{\mathrm{VAI}}\]

where \(f_{\mathrm{leaf}}\) and \(f_{\mathrm{stem}}\) denote the leaf and stem fractions of \(\mathrm{VAI}\).

2. Single-scattering albedo¶

The single-scattering albedo is

(1.1.3)¶\[\omega_\Lambda = \rho_\Lambda + \tau_\Lambda\]

where \(\rho_\Lambda\) and \(\tau_\Lambda\) are canopy reflectance and transmittance in band \(\Lambda\). They are computed as leaf/stem-weighted means:

(1.1.4)¶\[\begin{split}\rho_\Lambda &= f_{\mathrm{leaf}}\,\rho_\Lambda^{\mathrm{leaf}} + f_{\mathrm{stem}}\,\rho_\Lambda^{\mathrm{stem}} \\ \tau_\Lambda &= f_{\mathrm{leaf}}\,\tau_\Lambda^{\mathrm{leaf}} + f_{\mathrm{stem}}\,\tau_\Lambda^{\mathrm{stem}}\end{split}\]

A larger \(\omega_\Lambda\) means that a greater fraction of intercepted radiation is redistributed by scattering, whereas a smaller \(\omega_\Lambda\) means that more is absorbed by vegetation. \(\rho_\Lambda\), \(\tau_\Lambda\), and \(\omega_\Lambda\) are clipped to \((0, 1)\) to ensure physical admissibility.

Symbol	ADELM variables
\(\omega_{\mathrm{vis}}\)	`canopy_single_scattering_albedo_vis` [Diagnostic: single-scattering albedo of the canopy in the visible band]
\(\omega_{\mathrm{nir}}\)	`canopy_single_scattering_albedo_nir` [Diagnostic: single-scattering albedo of the canopy in the near-infrared band]
\(\rho_{\mathrm{vis}}^{\mathrm{leaf}}\)	`leaf_vis_reflectance` [PFT-based parameter: leaf reflectance in the visible band]
\(\rho_{\mathrm{vis}}^{\mathrm{stem}}\)	`stem_vis_reflectance` [PFT-based parameter: stem reflectance in the visible band]
\(\tau_{\mathrm{vis}}^{\mathrm{leaf}}\)	`leaf_vis_transmittance` [PFT-based parameter: leaf transmittance in the visible band]
\(\tau_{\mathrm{vis}}^{\mathrm{stem}}\)	`stem_vis_transmittance` [PFT-based parameter: stem transmittance in the visible band]
\(\rho_{\mathrm{nir}}^{\mathrm{leaf}}\)	`leaf_nir_reflectance` [PFT-based parameter: leaf reflectance in the near-infrared band]
\(\rho_{\mathrm{nir}}^{\mathrm{stem}}\)	`stem_nir_reflectance` [PFT-based parameter: stem reflectance in the near-infrared band]
\(\tau_{\mathrm{nir}}^{\mathrm{leaf}}\)	`leaf_nir_transmittance` [PFT-based parameter: leaf transmittance in the near-infrared band]
\(\tau_{\mathrm{nir}}^{\mathrm{stem}}\)	`stem_nir_transmittance` [PFT-based parameter: stem transmittance in the near-infrared band]

3. Two-stream geometry¶

The Ross-Goudriaan coefficients are

(1.1.5)¶\[\phi_1 = 0.5 - 0.633\,\chi - 0.330\,\chi^2, \qquad \phi_2 = 0.877\,(1 - 2\phi_1)\]

where \(\chi\) (leaf_angle_dist_index [PFT-based parameter: leaf angle distribution index, ranging from -1 (vertical leaves) to +1 (horizontal leaves)]) is the leaf-angle distribution index, clamped to \([-0.4,\,0.6]\).

The mean projected vegetation area in the solar direction is

(1.1.6)¶\[G(\mu_s) = \phi_1 + \phi_2\,\mu_s\]

where \(\mu_s = \cos\theta_z\) (cosine_solar_zenith [Driver: cosine of the solar zenith angle]) is the cosine of the solar zenith angle.

See also

model.utils.solar_geometry.compute_cosine_solar_zenith_series()

The direct-beam extinction coefficient per unit VAI is

(1.1.7)¶\[K_{\mathrm{dir}} = \frac{G(\mu_s)}{\mu_s}\]

and the diffuse optical-depth scale is

(1.1.8)¶\[\bar{\mu} = \frac{1}{\phi_2}\left(1 - \frac{\phi_1}{\phi_2}\ln\frac{\phi_1+\phi_2}{\phi_1}\right)\]

Physically, \(K_{\mathrm{dir}}\) sets how fast the direct beam is attenuated through the canopy; \(\bar{\mu}\) plays the equivalent role for diffuse streams. A more erectophile canopy (\(\chi < 0\)) presents less projected area to a high sun, giving lower \(K_{\mathrm{dir}}\).

The direct-beam upscatter parameter \(\beta_\Lambda^\mu\) (betad) is

(1.1.9)¶\[\beta_\Lambda^\mu = \frac{1 + \bar{\mu}\,K_{\mathrm{dir}}}{\bar{\mu}\,K_{\mathrm{dir}}} \cdot a_{s,\Lambda}\]

where \(a_{s,\Lambda}\) is the single-scattering upward fraction for the direct beam:

(1.1.10)¶\[a_{s,\Lambda} = \frac{\omega_\Lambda}{2} \cdot \frac{G(\mu_s)}{G(\mu_s) + \phi_2\mu_s} \cdot \left(1 - \frac{\phi_1\mu_s}{G(\mu_s)+\phi_2\mu_s}\ln\frac{\phi_1\mu_s + G(\mu_s)+\phi_2\mu_s}{\phi_1\mu_s}\right)\]

The diffuse upscatter parameter \(\beta_\Lambda^{\mathrm{d}}\) (betai) is

(1.1.11)¶\[\beta_\Lambda^{\mathrm{d}} = \frac{1}{2\,\omega_\Lambda} \left[\omega_\Lambda + (\rho_\Lambda - \tau_\Lambda)\left(\frac{1+\chi}{2}\right)^2\right]\]

The upscatter parameters \(\beta_\Lambda^\mu\) and \(\beta_\Lambda^{\mathrm{d}}\) describe what fraction of scattered radiation is redirected upward. The \((\rho_\Lambda - \tau_\Lambda)\) term reflects that leaves which reflect more than they transmit scatter more radiation upward. The \(((1+\chi)/2)^2\) factor modulates this with leaf angle: horizontal leaves (\(\chi > 0\)) intercept more diffuse radiation from above and scatter a larger fraction back upward. \(\beta_\Lambda^\mu\) and \(\beta_\Lambda^{\mathrm{d}}\) are clipped to \([0, 1]\).

Symbol	ADELM variables
\(\beta_{\mathrm{vis}}^\mu\)	`canopy_direct_beam_upscatter_parameter_vis` [Diagnostic: upscatter parameter for direct beam radiation in the visible band]
\(\beta_{\mathrm{nir}}^\mu\)	`canopy_direct_beam_upscatter_parameter_nir` [Diagnostic: upscatter parameter for direct beam radiation in the near-infrared band]
\(\beta_{\mathrm{vis}}^{\mathrm{d}}\)	`canopy_diffuse_upscatter_parameter_vis` [Diagnostic: upscatter parameter for diffuse radiation in the visible band]
\(\beta_{\mathrm{nir}}^{\mathrm{d}}\)	`canopy_diffuse_upscatter_parameter_nir` [Diagnostic: upscatter parameter for diffuse radiation in the near-infrared band]

4. Ground albedo¶

The ground albedo is

(1.1.12)¶\[\alpha_{\mathrm{g},\Lambda} = (1 - f_{\mathrm{snow}})\,\alpha_{\Lambda}^{\mathrm{soil}} + f_{\mathrm{snow}}\,\alpha_{\Lambda}^{\mathrm{snow}}\]

where \(\alpha_{\Lambda}^{\mathrm{soil}}\) and \(\alpha_{\Lambda}^{\mathrm{snow}}\) are soil and snow albedo in band \(\Lambda\), and \(f_{\mathrm{snow}}\) is snow cover fraction.

Symbol	ADELM variable
\(\alpha_{\mathrm{vis}}^{\mathrm{soil}}\)	`soil_vis_albedo` [default: 0.15; Fixed parameter: snow-free soil albedo in the visible band]
\(\alpha_{\mathrm{nir}}^{\mathrm{soil}}\)	`soil_nir_albedo` [default: 0.29; Fixed parameter: snow-free soil albedo in the near-infrared band]
\(\alpha_{\mathrm{vis}}^{\mathrm{snow}}\)	`snow_vis_albedo` [default: 0.95; Fixed parameter: snow-covered surface albedo in the visible band]
\(\alpha_{\mathrm{nir}}^{\mathrm{snow}}\)	`snow_nir_albedo` [default: 0.65; Fixed parameter: snow-covered surface albedo in the near-infrared band]

The snow cover fraction is

(1.1.13)¶\[f_{\mathrm{snow}} = \frac{S_{\mathrm{snow}}}{S_{\mathrm{snow}} + S_{\mathrm{snow},0}}\]

where \(S_{\mathrm{snow}}\) (snow_water_storage [mm; State: snow water equivalent stored in the surface snowpack]) is snow water storage and \(S_{\mathrm{snow},0}\) is snow_cover_scale [default: 25 mm; Fixed parameter: characteristic snow water equivalent at which half the ground surface is snow-covered].

Note

Direct-beam and diffuse ground albedos are set equal in ADELM, so \(\alpha_{\mathrm{g},\Lambda}^{\mu} = \alpha_{\mathrm{g},\Lambda}^{\mathrm{d}} = \alpha_{\mathrm{g},\Lambda}\).