Growth
Modelling
The Modelling Approach
The Tyfiant Coed approach is based on aspects of a mechanistic stand growth model developed by Wenk (et al., 1990, 1994), which are applied to a spatially explicit individual tree approach. The model combines a mathematical mechanistic approach with statistical functions whose parameters are found through regression analyses. Statistical models whose parameters have been found through regression methods have the problem that they are valid only for a defined population. These models cannot be easily transferred to different populations in a different area. However, by combining the two different approaches, it may be possible to establish a model which is able to cope with the difficult data situation we are in at present. One appealing property of this modelling approach is that its functions work on the basis of generalised allometric relations and so-called growth multipliers (see eq. 5, 6-6.2) and thus have a comparatively small data demand and a relatively high degree of generalisation. The function parameters, especially the growth parameter c1 (eq. 1) and the parameter c2 (eq. 3) are interpretable and reflect the vitality of a tree. A preliminary framework of the model functions has been developed by Pommerening and Wenk (2002). This paper is also presented in the appendix so that the main model functions are only briefly repeated at this stage.
The
Preliminary Model Frame Work (Pommerening and Wenk, 2002)
The growth parameter c1 of a tree is given by its initial DBH, a competition index (in this case a DBH ratio) and a parameter x (eq. 1).
(1) 
where
i index of tree under study
e base of natural logarithm
b tree species specific coefficient
DBHratio ratio of DBHs of the tree
under study and its corresponding primary competitor
t current forecast year.
The number 0.19 in the numerator is
basically a variable as well and will in future applications be treated as such.
It corresponds with the growth parameter c1 of primary competitors
and open grown trees, which exist with almost no competition. The value of this
asymptote differs with tree species and site conditions.
The variable x is derived from the
relationship
(2) 
where
a tree species specific coefficient.
According to an approach developed by Wenk (1996, p. 5) each growth parameter estimated after equation (1) is additionally modified using a Weibull-distributed random number, in order to reflect biological variations in tree populations.
The parameter c2 is estimated from c1 via a linear function (3).
(3) 
where
c, d tree species specific coefficients.
The parameter 
accounts for
variability of juvenile growth, and usually loses its influence beyond age
40-50.
Now the relative volume increment can be calculated for the next ten year period with formula (4) according to Wenk (1969).
(4) 
where
ti’ transformed tree age[1]
refers to the final
value at the end of the forecasting step. In terms of the observed data from
research plots it could be expressed as
(4.1) 
where
V tree volume.
The primary focus in the model approach is on the volume, as this is an important variable in forestry, which can also act as an estimator of the tree’s biomass. Unlike other growth models volume is the first variable to be estimated in order to keep the effects of error propagation low. It is possible to use other functions describing the relative volume increment instead of eq. (4) without a need to fundamentally alter the model. Therefore it would also be possible to use a more physiologically based estimator instead.
As found in many biological studies (e.g. Bertalanffy, 1951), age is an important quantity in determining the growth of trees and stands. In uneven aged forests there is generally no unique overall age of a forest stand so that individual tree age will be estimated dependent on tree diameter and spatial tree position in the stand. The derivation of this algorithm is part of the current project.
From the estimated relative volume increment for a ten-year period, a one-year volume increment multiplier is derived following the method developed by Gerold and Römisch (1977). The general formula for the calculation of growth multipliers is as follows:
(5) 
where
Y arbitrary growth quantity (e.g. volume, height, diameter).
The increase in an arbitrary growth quantity Y is given by equation (6).
(6) 
According to Wenk et al. (1990, p. 95) equation (6) is based on the two statements:
- The increment is the difference in the growth quantity at different times.
- The increment is the product of growth quantity and relative increment.
By setting the two corresponding equations equal, the following equation (6.1) is obtained:
(6.1) 
,
from which by transforming the equation we can derive
(6.2) 
,
which corresponds with equation (6).
Allometric relationships are used in this approach to derive height and diameter increment from volume increment. Allometry is based on the fundamental finding that the relative increment of one growth quantity is proportional to another growth quantity of one and the same organism (see Pienaar and Turnbull, 1973) and was being used by 19th century foresters like Preßler and Schneider to estimate the growth of one part of a tree by the growth of a different part of the same tree. The allometric coefficient describing the height growth depending on volume growth of each tree can be derived from the growth parameter c1 by applying a Chapman-Richards growth function (Pienaar and Turnbull, 1973):
(7) 
where
g, k tree species specific coefficients.
Subsequently, the allometric coefficient will be estimated assuming its dependence on the spatial arrangement of the trees under study. This ensures that effects of competition on a tree’s allometrics are taken into account. The results are modified according to the approach of Wenk (1996, p. 5).
The multipliers for volume increment and height growth are related through the allometric equation. Thus the growth multiplier for tree height is derived from the volume multiplier using equation (8).
(8) 
where
h tree height
v tree volume.
Finally, the diameter multiplier is also obtained through allometric relationships from the form factor, the height multiplier and the allometric coefficient according to Wenk (1990, p. 109) using equation (9).
(9) 
where
d tree diameter at breast height
f form factor.
In Pommerening and Wenk (2002) this approach was applied to a Norway
spruce time series from
