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'	$f(x,p_0,\ldots,p_n) = p_n x^n + \ldots + p_1 x + a_0$
QSM-based*	'qsm'	Nonparametric distribution

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'	$f(x,p_0,\ldots,p_n) = p_n x^n + \ldots + p_1 x + a_0$
QSM-based*	'qsm'	Nonparametric distribution

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'	$f(x,\mu,\kappa) = e^{\kappa \cos(x-\mu)}/(2 \pi \mathrm{I}_0(\kappa))$
QSM-based*	'qsm'	Nonparametric distribution

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 $\in [0,1]$

* The QSM-based LADD

The QSM-based approach distributes the leaves evenly throughout the branch cylinders, emphasizing the leaves towards the tips of the branches, and can be considered as some kind of “automatic” leaf positioning for the QSM. This is useful if the user wants to generate a somewhat realistic foliage and the specific shape of LADD is not important. When using QSM-based approach the dTypeLADD field of each of the structural variables have to be set to 'qsm'. Also, the QSM-based approach is obviously available only when generating foliage on QSMs.

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 $> 0$