21-AllometricModels.Rmd

# (PART) Static modules {-}


# Allometric models {#allometricmodels}

The purpose of this chapter is to describe how certain plant structural attributes are estimated in **medfate** using allometric relationships. These static models are used to initialize vegetation input parameters from forest plot data before simulations (in functions `spwbInput()` and `growthInput()`), to update vegetation structure during growth simulations or to calculate foliar biomass for fuel assessments. 

## Input data

### Forest plot data

As explained in section \@ref(vegetationinput), **medfate** has been specially designed to work with forest inventory plots with data in form of plant cohorts. The tree/shrub cohort attributes required to apply allometric models are the following:

| Symbol | Units | R  | Description                        | trees | shrubs |
|--------|-------|----| ------------------------------------|-----|-----|
| $SP_i$  |  | `Species` | Species identity |  Y  |  Y  |
| $H_i$  | $cm$  | `Height` | Average tree or shrub height |  Y  |  Y  |
| $N_i$ | $ind · ha^{-1}$ | `N` | Density of tree individuals | Y | N |
| $DBH_i$ | $cm$ | `DBH` | Tree diameter at breast height | Y | N |
| $Cover_i$ | % | `Cover` | Shrub percent cover | N | Y |

Most allometric functions require data in form of a `forest` object (see section \@ref(forestobjects)).

### Species parameters

The following table shows the allometric coefficients needed for the calculations detailed in this chapter.

| Symbol       | Units | R param       | Description                      |
|--------------|-------|---------------|----------------------------------|
| $a_{fbt}$, $b_{fbt}$, $c_{fbt}$ | | `a_fbt`, `b_fbt`, `c_fbt` | Coefficients to calculate foliar biomass of an individual tree  |
| $a_{ash}$, $b_{ash}$ | | `a_ash`, `b_ash` | Coefficients relating the square of shrub height with shrub area |
| $a_{bsh}$, $b_{bsh}$ | | `a_bsh`, `b_bsh` | Coefficients relating crown volume with dry weight of shrub individuals |
| $a_{cr}$, $b_{1cr}$, $b_{2cr}$, $b_{3cr}$, $c_{1cr}$, $c_{2cr}$ | | `a_cr`, `b_1cr`, `b_2cr`, `b_3cr`, `c_1cr`, `c_2cr` | Coefficients to calculate crown ratio of trees |
| $cr$ | [0-1] | `cr` | Ratio between crown length and total height for shrubs |
| $r_{6.35}$ | | `r635` | Ratio between the weight of leaves plus branches and the weight of leaves alone for branches of 6.35 mm |
| $SLA$  | $m^2 \cdot kg^{-1}$ | `SLA` | Specific leaf area |

## Allometric relationships

### Leaf biomass 

Leaf biomass calculations are done differently for trees and shrub cohorts. They are calculated from `forest` objects using function `plant_foliarBiomass()`.

**Tree cohorts**

Foliar biomass for a single tree of cohort $i$ ($FB_{tree,i}$; in $kg$) is calculated using:
\begin{equation}
FB_{tree,i} = a_{fbt} \cdot DBH_{i}^{b_{fbt}}\cdot e^{c_{fbt}\cdot BAL_i} \cdot e^{-0.0001\cdot N_i}
(\#eq:onetreefoliarbiomass)
\end{equation}
where $DBH_{i}$ is the diameter of the tree (in $cm$), $BAL_i$ is the cummulative basal area ($m^2\cdot ha^{-1}$) of trees having an equal or larger diameter (including the target tree), $N_i$ is the density of the cohort, and $a_{fbt}$, $b_{fbt}$ and $c_{fbt}$ are species-specific regression coefficients. The first two determine the relationship between tree diameter and its foliar biomass, whereas the third one leads to reductions of foliar biomass because of over-shading and self-shading. Factor $e^{-0.001\cdot N_i}$ reduces foliar biomass for cohorts with very high density (such as very dense regeneration after a fire). The foliar biomass of the whole tree cohort ($FB_{i}$; in $kg\cdot m^{-2}$) is obtained multiplying tree foliar biomass by tree density  ($N_{i}$; in $ind.\cdot ha^{-1}$):
\begin{equation}
FB_{i} = FB_{tree,i}\cdot (N_{i}/10000)
(\#eq:treefoliarbiomass)
\end{equation}

**Shrub cohorts**

To calculate the leaf biomass of a shrub cohort, we first determine $A_{sh,i}$, the area (in $cm^2$) occupied by one average individual of height $H_{i}$ (in $cm$), using the relationship: 
\begin{equation}
A_{sh,i} = a_{ash} \cdot H_{i}^{b_{ash}}
\end{equation}
where $a_{ash}$ and $b_{ash}$ are species-specific parameters. The model then estimates the dry weight of leaves and branches up to 6.35mm in diameter ($B_{sh,i}$, in kg) of this average individual (i.e. fine fuel biomass), using an allometric relationship with shrub crown phytovolume assuming a cylinder (in $cm^3$):
\begin{equation}
B_{sh,i} = a_{bsh} \cdot (A_{sh,i}\cdot H_{i})^{b_{bsh}} \cdot e^{-0.235\cdot LAI^{live}_{trees}} 
(\#eq:shrubindividualfuel)
\end{equation}
where $a_{bsh}$ and $b_{bsh}$ are species-specific parameters. Here $LAI^{live}_{trees}$ ($m^2\cdot m^{-2}$) is the sum of leaf area index values of tree cohorts, which induces a reduction of $B_{sh,i}$ similar to the one described for tree foliar biomass by $BAL_i$. This reduction is important for biomass and leaf area estimation of shrubs in forests, since it has been observed that light availability affects the growth, architecture and bulk density of the understory [@pimont_simple_2018].

Shrub density ($N_{i}$; in $ind.\cdot m^{-2}$) can be grossly estimated from percent cover ($C_{i}$, in percent) and $A_{sh,i}$ (in $cm^{-2}$):
\begin{equation}
N_{i} = \frac{C_{i}/100}{A_{sh,i}/10000}
\end{equation}
The fine fuel biomass of a shrub cohort ($W_{i}$, in $kg \cdot m^{-2}$) is simply the product of $B_{sh,i}$ ($kg$ of dry weight) and $N_{i}$:
\begin{equation}
W_{i} =  B_{sh,i} \cdot N_{i}
(\#eq:shrubloading)
\end{equation}
Foliar biomass (in $kg \cdot m^{-2}$) can be obtained using the species-specific ratio $r_{6.35,i}$:
\begin{equation}
FB_{i} =  W_{i}/r_{6.35, i}
(\#eq:shrubfoliarbiomass)
\end{equation}
If not known, $r_{6.35,i}$ can be set to a default value of 2 (equivalent to 50% of weight corresponding to leaves).

**Herbaceous layer**

Foliar biomass of the herbaceous layer (which is equal to its fine fuel loading) is estimated from herbaceous cover ($Cover_{herb}$; in %) and height ($H_{herb}$; in cm) assuming a constant bulk density coefficient:

\begin{equation}
FB_{herb} = 0.014 \cdot Cover_{herb} \cdot (H_{herb}/100.0) \cdot e^{-0.235\cdot LAI^{live}_{woody}} 
\end{equation}

where the factor $e^{-0.235\cdot LAI^{live}_{woody}}$ is analogous to that of shrubs (eq. \@ref(eq:shrubindividualfuel)), but using the overall leaf area index of woody cohorts as driving variable. 


### Leaf area and LAI
Leaf area index ($LAI$ in $m^2 \cdot m^{-2}$) of a given cohort $i$ can be calculated from its foliar biomass (in $kg \cdot m^{-2}$) by using a *specific leaf area* coefficient ($SLA$, in $m^2 \cdot kg^{-1}$) that in **medfate** is species-specific:
\begin{equation}
LAI_{i} =  FB_{i} \cdot SLA
\end{equation}
The leaf area ($LA$ in $m^2$) of an individual of the cohort is found by dividing it by plant density ($N_i$, in $ind.\cdot ha^{-1}$) and multiplying by 10000 $m^2 \cdot ha^{-1}$:
\begin{equation}
LA_{i} =  10000 \cdot LAI_{i} / N_{i}
\end{equation}

LAI values of woody cohorts are calculated from `forest` objects using function `plant_LAI()`.

Leaf area index of the herbaceous layer ($LAI_{herb}$) is estimated from $FB_{herb}$ by applying a $SLA = 9$ value, is limited to $LAI_{herb} = 2\,\,m^2\cdot m^{-2}$, and can be calculated from `forest` objects using function `herb_LAI()`.

### Crown vertical dimensions {#crownbaseheight}


Crown base height, i.e. the height were the first living branch of trees or shrubs occurs, is an important parameter to know the vertical distribution of leaves (see section \@ref(leafdistribution)). Crown base height of a plant cohort $i$ (i.e., $H_{crown,i}$, in $cm$) is related to the total height of the cohort ($H_i$, in $cm$) and its crown ratio ($CR_i$), which is the ratio between crown length and plant height:
\begin{equation}
H_{crown,i} = (1 - CR_i) \cdot H_i
(\#eq:CrownHeight)
\end{equation}
In the case of shrubs the crown ratio $CR_i$ is normally an input (species-specific) parameter ($cr$). In the case of trees, the crown ratio is modelled as a function of tree size and stand competition, following a modification of the logistic equation of @Hasenauer1996 :
\begin{equation}
CR_i = \frac{1}{1+e^{-(a_{cr}+b_{1cr}\cdot HD_i +b_{2cr} \cdot (H_i/100)+b_{3cr} \cdot DBH_i^2+c_{1cr} \cdot BAL_i + c_{2cr} \cdot ln(CCF_i))}}
(\#eq:treecrownratio)
\end{equation}
where $HD_i = H_i/(100\cdot DBH_i)$ is the height to diameter ratio (in $m\cdot cm^{-1}$), $H_i$ is the tree height, $DBH_i$ is the diameter, $CCF_i$ is the crown competition factor and $a_{cr}$, $b_{1cr}$, $b_{2cr}$, $b_{3cr}$, $c_{1cr}$ and $c_{2cr}$ are species-specific parameters. The crown competition factor is in turn calculated using [@Krajicek1961]:
\begin{equation}
CCF_i = \sum_{i}{N_i \cdot \pi \cdot (CW_i/2)^2/100}
\end{equation}
where $N_i$ is the tree density and $CW_i$ is the crown width (in $m$) assuming an open-grown tree, estimated from an allometric relationship with tree diameter:
\begin{equation}
CW_i = a_{cw}\cdot DBH_i^{b_{cw}} 
\end{equation}
where again $a_{cw}$ and $b_{cw}$ are species-specific parameters. 

Crown ratios can be calculated from forest objects using function `plant_crownRatio()`. Similar calculations are performed to estimate crown length and crown base height, with functions `plant_crownLength()` and `plant_crownBaseHeight()`, respectively.