Leaf distribution options
Leaf Area Density Distribution (LADD)
Relative height
Options for the marginal distribution function of relative height dTypeLADDh are:
| Distribution | dTypeLADDh | Function Definition |
|---|---|---|
| uniform | 'uniform' | $f(x) = 1$ |
| beta | 'beta' | $f(x,\alpha,\beta) = (x^{\alpha-1}(1-x)^{\beta-1})/\mathrm{B}(\alpha,\beta)$ |
| truncated Weibull | 'weibull' | $f(x,k,\lambda) = \frac{k}{\lambda} \left( \frac{x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k}$ |
| polynomial | 'polynomial' | $f(x,p_0,\ldots,p_n) = p_n x^n + \ldots + p_1 x + a_0$ |
Here $\mathrm{B}(\alpha,\beta) = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} dx$ is the beta function.
The possible parameter values pLADDh are:
| Distribution | pLADDh | Parameter Values |
|---|---|---|
| uniform | - | - |
| beta | [a b] | a, b $> 0$ |
| truncated Weibull | [k l] | k, l $> 0$ |
| polynomial | [p0 ... pN] | p0, …, pN $\in \mathbb{R}$ |
Relative branch distance / relative distance from stem
Options for the marginal distribution function of relative branch distance / relative distance from stem dTypeLADDd are:
| Distribution | dTypeLADDd | Function Definition |
|---|---|---|
| uniform | 'uniform' | $f(x) = 1$ |
| beta | 'beta' | $f(x,\alpha,\beta) = (x^{\alpha-1}(1-x)^{\beta-1})/\mathrm{B}(\alpha,\beta)$ |
| truncated Weibull | 'weibull' | $f(x,k,\lambda) = \frac{k}{\lambda} \left( \frac{x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k}$ |
| polynomial | 'polynomial' | $f(x,p_0,\ldots,p_n) = p_n x^n + \ldots + p_1 x + a_0$ |
Here $\mathrm{B}(\alpha,\beta) = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} dx$ is the beta function.
The possible parameter values pLADDd are:
| Distribution | pLADDd | Parameter Values |
|---|---|---|
| uniform | - | - |
| beta | [a b] | a, b $> 0$ |
| truncated Weibull | [k l] | k, l $> 0$ |
| polynomial | [p0 ... pN] | p0, …, pN $\in \mathbb{R}$ |
Compass direction
Options for the marginal distribution function of compass direction dTypeLADDc are:
| Distribution | dTypeLADDc | Function Definition |
|---|---|---|
| uniform | 'uniform' | $f(x) = 1/2\pi$ |
| von Mises | 'vonmises' | $f(x,\mu,\kappa) = e^{\kappa \cos(x-\mu)}/(2 \pi \mathrm{I}_0(\kappa))$ |
Here $I_0(\kappa) = \frac{1}{2\pi} \int_0^{2\pi} e^{\kappa \cos(x)} dx$ is the modified Bessel function of the first kind of order 0.
The possible parameter values pLADDc are:
| Distribution | pLADDc | Parameter Values |
|---|---|---|
| uniform | - | - |
| von Mises | [m k] | m $\in [0,2\pi]$, k $> 0$ |
Mixture models
For all LADD marginal distributions, it is also possible to define the distributions as mixture models of two distributions of the same type. This allows for a larger variety of marginal distribution shapes, like multimodal distributions. The definition of a mixture model requires setting the parameters of each distribution separately and defining a weighting factor among the distributions. The weighting factor w is given a value between 0 and 1, which determines the weights of the distributions to be w for the first distribution and 1-w for the second distribution. Mixture models can be defined for the following distributions:
| Distribution | pLADDh/pLADDd/pLADDc | Parameter Values |
|---|---|---|
| beta | [a1 b1 a2 b2 w] | a1, b1, a2, b2 $> 0$, w $\in [0,1]$ |
| truncated Weibull | [k1 l1 k2 l2 w] | k1, l1, k2, l2$> 0$, w $\in [0,1]$ |
| von Mises | [m1 k1 m2 k2 w] | m1, m2 $\in [0,2\pi]$, k1, k2 $> 0$, w $\in [0,1]$ |
Leaf Orientation Distribution (LOD)
Inclination angle distribution
Options for the marginal distribution function of leaf inclination angle dTypeLODinc are:
| Distribution | dTypeLODinc | Function Definition |
|---|---|---|
| uniform | 'uniform' | $f(\theta) = 2/\pi$ |
| spherical | 'spherical' | $f(\theta) = \sin(\theta)$ |
| generalized de Wit’s | 'dewit' | $f(\theta;a,b) = (1 + a \cos(b\theta))/(\frac{\pi}{2} + \frac{a}{b} \sin(b\frac{\pi}{2}))$ |
| beta | 'beta' | $f(\theta,\alpha,\beta) = (\theta^{\alpha-1}(1-\theta)^{\beta-1})/\mathrm{B}(\alpha,\beta)$ |
| constant value | 'constant' | - |
Here $\mathrm{B}(\alpha,\beta) = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1} dx$ is the beta function.
The possible parameter values for the function fun_pLODinc are:
| Distribution | fun_pLODinc | Parameter Values |
|---|---|---|
| uniform | - | - |
| spherical | - | - |
| generalized de Wit’s | [a b] | a $\in [-1,1]$, b $\in [2,4]$ |
| beta | [a b] | a, b $> 0$ |
| constant value | c | c $\in [0,\frac{\pi}{2}]$ |
Azimuth angle distribution
Options for the marginal distribution function of leaf azimuth angle dTypeLODaz are:
| Distribution | dTypeLODaz | Function Definition |
|---|---|---|
| uniform | 'uniform' | $f(\phi) = 1/2\pi$ |
| von Mises | 'vonmises' | $f(\phi,\mu,\kappa) = e^{\kappa \cos(\phi-\mu)}/(2 \pi \mathrm{I}_0(\kappa))$ |
| constant value | 'constant' | - |
Here $I_0(\kappa) = \frac{1}{2\pi} \int_0^{2\pi} e^{\kappa \cos(x)} dx$ is the modified Bessel function of the first kind of order 0.
The possible parameter values for the function fun_pLODaz are:
| Distribution | fun_pLODaz | Parameter Values |
|---|---|---|
| uniform | - | - |
| von Mises | [m k] | m $\in [0,2\pi]$, k $> 0$ |
| constant value | c | c $\in [0,2\pi]$ |
Leaf Size Distribution (LSD)
Options for the distribution function of leaf size distribution dTypeLSD are:
| Distribution | dTypeLSD | Function Definition |
|---|---|---|
| uniform | 'uniform' | $f(x,a,b) = (x-a)/(b-a)$ |
| normal | 'normal' | $f(x,\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}$ |
| constant value | 'constant' | - |
The possible parameter values for the function fun_pLSD are:
| Distribution | fun_pLSD | Parameter Values |
|---|---|---|
| uniform | [a b] | a, b $> 0$ |
| normal | [m v] | m, v $> 0$ |
| constant value | c | c $> 0$ |