Geometry

• Height - Stem Diameter - Relationship
  • Polynomial approach
  • Saturation approach
  • Power-law approach
• Crown length - Height - Relationship
  • Linear approach (most frequently used)
  • Saturation approach
  • Polynomial approach
• Crown diameter - Stem diameter - Relationship
  • Exponential approach I
  • Exponential approach II
  • Polynomial approach
  • Linear approach
  • Saturation approach
  • Power-law approach (most frequently used)
• Crown area - Crown diameter - Relationship
• Aboveground biomass - Stem diameter - Relationship
  • Geometrical approach (most frequently used)
  • Power-law approach
  • Logarithmic approach
• Leaf area index - Stem diameter - Relationship
  • Power-law approach (most frequently used)
  • Linear approach
• Maximum Values

Although individual trees in real forests do not necessarily have identical shapes, we model each tree by a cylindrical stem and a cylindrical crown (see figure below). The geometry of an individual can be described completely by the following size characteristics: stem diameter ($D$), height ($H$), crown diameter ($C_\text{D}$), crown length ($C_\text{L}$) and crown projection area ($C_\text{A}$) as shown in the figure below.

Figure: Geometrical representation of a single tree . The following abbreviations describe size characteristics of the modelled tree geometry: $D$ -stem diameter, $H$ - height, $C_\text{D}$ - crown diameter, $C_\text{L}$ - crown length, $C_\text{A}$ crown projection area.

These size characteristics are functionally related to each other. In the following, we describe the functional relationships used. Parameters of the described relationships can vary between different tree types. Some graphical examples are given in figure at the end of the page.

Height - Stem Diameter - Relationship

The height $H\,[\text{m}]$ of an tree relates to its stem diameter $D\,[\text{m}]$ by several approaches. $h_\text{0}$, $h_\text{1}$ and $h_\text{2}$ are type-specific parameters.

Polynomial approach

\begin{equation} H=h_0+h_1 \cdot D+h_2 \cdot D^2 \end{equation}

Saturation approach

\begin{equation} H=\frac{D}{\frac{1}{h_0}+\frac{D}{h_1}} \end{equation}

Power-law approach

\begin{equation} H=h_0 \cdot D^{h_1} \end{equation}

Crown length - Height - Relationship

The crown length $ C_L \, [\text{m}]$ of a tree is modelled as a fraction of its height $ H\, [\text{m}]$. $c_\text{l0}$, $c_\text{l1}$ and $c_\text{l2}$ are type-specific parameters.

Linear approach (most frequently used)

\begin{equation} C_L=c_{l0} \cdot H \end{equation}

Saturation approach

\begin{equation} C_L=(-\frac{c_{l0} \cdot H \cdot c_{l1}}{c_{l0}\cdot H + c_{l1}} + c_{l2}) \cdot H \end{equation}

Polynomial approach

\begin{equation} C_L=(c_{l0} + c_{l1} \cdot H + c_{l2} \cdot H^2) \cdot H \end{equation}

Crown diameter - Stem diameter - Relationship

The second dimension of the cylindrical crown, i.e. the crown diameter $C_D\,[\text{m}]$ of a tree relates to its stem diameter $D\,[\text{m}]$ by several approaches. $c_\text{d0}$, $c_\text{d1}$ and $c_\text{d2}$ are type-specific parameters.

Exponential approach I

\begin{equation} C_D = D \cdot \left(c_{d0} + c_{d1} \cdot exp\left(-c_{d2} \cdot D\right) \right) \end{equation}

Exponential approach II

\begin{equation} C_D = c_{d0} \cdot D + c_{d1} \cdot exp\left(-c_{d2} \cdot D\right) \end{equation}

Polynomial approach

\begin{equation} C_D = c_{d0} + c_{d1} \cdot D + c_{d2} \cdot D^2 + c_{d3} \cdot D^3 \end{equation}

Linear approach

\begin{equation} C_D = c_{d0} \cdot D \end{equation}

Saturation approach

\begin{equation} C_D = \frac{D}{\frac{1}{c_{d0}}+\frac{D}{c_{d1}}} \end{equation}

Power-law approach (most frequently used)

\begin{equation} C_D = c_{d0} \cdot D^{c_{d1}} - c_{d2} \end{equation}

Crown area - Crown diameter - Relationship

The crown projection area $C_A \, [\text{m}^2]$ of a tree is simply the ground area of the modelled cylindrical crown: \begin{equation} C_A=\frac{\pi}{4} \cdot C_D^2 \end{equation}

Aboveground biomass - Stem diameter - Relationship

The aboveground volume of a tree captures biomass (i.e. organic dry matter). The following different ways of modelling the aboveground biomass are included in FORMIND.

Geometrical approach (most frequently used)

Aboveground biomass $B\,[t_\text{odm}]$ of a tree is calculated in relation to its stem diameter $D\,[\text{m}]$ and height $H\,[\text{m}]$ by: \begin{equation} B = \frac{\pi}{4} \cdot D^2 \cdot H \cdot f \cdot \frac{\rho}{\sigma} \end{equation}

whereby the calculation simply represents the volume of the stem (according to its geometry) multiplied by three factors, which describe the biomass content more concisely.

Firstly, $f\,[\text{-}]$ denotes a type-specific form factor, which accounts for deviations of the stem from a cylindrical shape. Secondly, the parameter $\rho\,[t_\text{odm}/\text{m}^3]$ represents the wood density, which describes how much organic dry matter per unit of volume the stem contains. Thirdly, the division by the parameter $\sigma\,[-]$, which represents the fraction of total aboveground biomass attributed to the stem, results then in the total aboveground biomass $B$.

In contrast to the constant parameters $\rho$ and $\sigma$, the form factor $f$ can change during the growth of a tree with respect to its stem diameter $D\,[\text{m}]$ via

\begin{equation} f=f_0 \cdot exp\left(f_1 \cdot D^{f_2}\right) \end{equation}

or

\begin{equation} f=f_0 \cdot D^{f_1} \end{equation}

$f_\text{0}$, $f_\text{1}$ and $f_\text{2}$ are type-specific parameters.

Power-law approach

Aboveground biomass $B\,[t_\text{odm}]$ of a tree can also be modelled in relation to its stem diameter $D\,[\text{m}]$ by:

\begin{equation} B =b_{0} \cdot D^{b_1} \end{equation}

whereby $b_0$ and $b_1$ are type-specific parameters.

Logarithmic approach

Aboveground biomass $B\,[\text{t}_{odm}]$ of a tree can also be modelled in relation to its stem diameter $D\,[m]$ by:

\begin{equation} B =exp \left(b_{0} \cdot (log(D)-b_{2}) \cdot \frac{2 \cdot b_{1}+(log(D)-b_{2})}{b_{1} +(log(D)-b_{2})}\right) \end{equation}

whereby $b_0$, $b_1$ and $b_2$ are type-specific parameters.

Leaf area index - Stem diameter - Relationship

In general, aboveground biomass is divided between woody biomass captured in the stem and green biomass captured in the crown leaves. Important for the photosynthetic production of a tree is the green biomass captured in crown leaves. As leaves absorb radiation for photosynthesis, the total amount of one-sided leaf area per unit of crown projection area (i.e. the individual's leaf area index) is of main interest. The leaf area index $LAI\,[\text{m}^2/{m}^2]$ of a tree relates functionally to its stem diameter $D\,[m]$ by:

Power-law approach (most frequently used)

\begin{equation} LAI =l_0 \cdot D^{l_1} \end{equation} whereby $l_0$ and $l_1$ are type-specific parameters.

Linear approach

\begin{equation} LAI =l_0 + l_1 \cdot D \end{equation} whereby $l_0$ and $l_1$ are type-specific parameters.

The figure below shows all modeled functional relationships with exemplary parameters.

Figure: Illustration of the modelled functional relationships, which are used to describe the geometry of a single tree. The approaches are in all cases the most frequently used ones. As parameters here we use the mean values of the parameter range, documented in table below.

Table: Summary of the parameter range based on tropical parameterizations.

Maximum Values

The trees cannot grow indefinitely in FORMIND. We introduce the following maximum values for a plausible geometry of a mature individual:

• maximum stem diameter $D_{\text{max}} [\text{m}]$
• maximum height $H_{\text{max}} [\text{m}]$

Either the maximum stem diameter or the maximum height is given as a type-specific input parameter. The missing maximum value and the corresponding maximum biomass $B\,[t_\text{odm}]$ are then derived using the functional relationships mentioned in section h-d-relationship and section b-d-relationship. The maximum values are used in section growth.