crt1d.leaf_angle#

Parameterizations of the impact of leaf angles on canopy RT.

Leaf angle factor \(G\) and the black leaf extinction coeff \(K_b\) have the following relationship:

\[K_b = G / \cos(\psi)\]

where \(\psi\) is the solar zenith angle and \(K_b = K_b(\psi), G = G(\psi)\).

\(G\) is the mean relative projection of leaf area in the direction \(\psi\).

\(g(\theta_l)\) is the PDF of leaf inclination angle \(\theta_l\) (relative to the horizontal plane). \(G(\psi)\) functions are derived from these distributions. The azimuth angle is usually assumed to have a uniform distribution and so does not have an impact.

Module Contents#

Functions#

G_ellipsoidal(psi, x)

\(G(\psi)\) for the ellipsoidal leaf angle distribution

G_ellipsoidal_approx(psi, x)

Campbell \(G\) approximate form.

G_ellipsoidal_approx_bonan(psi, xl)

Campbell \(G\) approximate form -- Bonan version.

G_horizontal(psi)

\(G(\psi)\) for horizontal leaves.

G_spherical(psi)

\(G(\psi)\) for the spherical leaf inclination angle distribution.

G_vertical(psi)

\(G(\psi)\) for vertical leaves.

g_ellipsoidal(theta_l, x)

PDF of \(\theta_l\) for the ellipsoidal distribution

g_erectophile(theta_l)

PDF of \(\theta_l\) for a mostly vertical distribution.

g_plagiophile(theta_l)

PDF of \(\theta_l\) for a distribution between horizontal and vertical.

g_planophile(theta_l)

PDF of \(\theta_l\) for a mostly horizontal distribution.

g_spherical(theta_l)

PDF of \(\theta_l\) for the spherical distribution.

g_uniform(theta_l)

PDF of \(\theta_l\) for a uniform distribution.

mla_from_g(g_fn)

Calculate (estimate) the mean leaf inclination angle (deg.)

mla_to_x_approx(mla)

Convert mean leaf angle (deg.) to x

mla_to_x_integ(mla)

Convert mean leaf angle (deg.) to x

x_to_mla_approx(x)

Convert x to mean leaf angle (deg.)

x_to_mla_integ(x)

Convert x to mean leaf angle (deg.)

xl_from_g(g_fn)

Compute \(\chi_l\), an index which quantifies the departure of the
crt1d.leaf_angle.G_ellipsoidal(psi, x)#

\(G(\psi)\) for the ellipsoidal leaf angle distribution with parameter x.

Reference: Campbell (1986) eqs. 5, 6 [Cam86]

Parameters:

  • psi (float) – Solar zenith angle in radians.

  • x (float) – b/a – the ratio of ellipse horizontal semixaxis length to vertical, s.t. x > 1 indicates an oblate spheroid.

crt1d.leaf_angle.G_ellipsoidal_approx(psi, x)#

Campbell \(G\) approximate form.

References

  • area ratio term: Campbell (1990) eq. 14 [Cam90]

  • exact formula: Campbell & Norman eq. 15.4 [CN12]

crt1d.leaf_angle.G_ellipsoidal_approx_bonan(psi, xl)#

Campbell \(G\) approximate form – Bonan version.

This uses \(\chi_l\), an index which quantifies the departure of the leaf angle distribution from spherical.

Warning

xl is not the same parameter as the x used elsewhere in this module. xl=0 gives spherical, whereas x=1 gives spherical.

crt1d.leaf_angle.G_horizontal(psi)#

\(G(\psi)\) for horizontal leaves.

crt1d.leaf_angle.G_spherical(psi)#

\(G(\psi)\) for the spherical leaf inclination angle distribution.

crt1d.leaf_angle.G_vertical(psi)#

\(G(\psi)\) for vertical leaves.

crt1d.leaf_angle.g_ellipsoidal(theta_l, x)#

PDF of \(\theta_l\) for the ellipsoidal distribution with parameter x. Following Bonan [Bon19] (p. 30, eqs. 2.11–14).

crt1d.leaf_angle.g_erectophile(theta_l)#

PDF of \(\theta_l\) for a mostly vertical distribution.

crt1d.leaf_angle.g_plagiophile(theta_l)#

PDF of \(\theta_l\) for a distribution between horizontal and vertical.

crt1d.leaf_angle.g_planophile(theta_l)#

PDF of \(\theta_l\) for a mostly horizontal distribution.

crt1d.leaf_angle.g_spherical(theta_l)#

PDF of \(\theta_l\) for the spherical distribution. Vertical leaves are favored, but not as much so as for erectophile.

crt1d.leaf_angle.g_uniform(theta_l)#

PDF of \(\theta_l\) for a uniform distribution.

crt1d.leaf_angle.mla_from_g(g_fn)#

Calculate (estimate) the mean leaf inclination angle (deg.) by numerically integrating the distribution’s PDF: \(g(\psi)\).

crt1d.leaf_angle.mla_to_x_approx(mla)#

Convert mean leaf angle (deg.) to x for the ellipsoidal leaf angle distribution. Using Campbell (1990) [Cam90] eq. 16 inverted.

crt1d.leaf_angle.mla_to_x_integ(mla)#

Convert mean leaf angle (deg.) to x for the ellipsoidal leaf angle distribution by optimization.

crt1d.leaf_angle.x_to_mla_approx(x)#

Convert x to mean leaf angle (deg.) for the ellipsoidal leaf angle distribution. Using Campbell (1990) [Cam90] eq. 16.

crt1d.leaf_angle.x_to_mla_integ(x)#

Convert x to mean leaf angle (deg.) for the ellipsoidal leaf angle distribution by numerically integrating the leaf angle PDF.

crt1d.leaf_angle.xl_from_g(g_fn)#

Compute \(\chi_l\), an index which quantifies the departure of the leaf angle distribution from spherical.

Vertical leaves have \(\chi_l = -1\) and horizontal leaves \(\chi_l = +1\).

Bonan [Bon19] eq. 2.16