Forest-Management Modelling - Environmental Modelling: Finding Simplicity in Complexity - page 370

form (Garcıa, 2005b). The most common model forms
include the Gompertz (1825), Bertalanffy (1949), and
Richards (1959) equations. Although these theoretical
models offer some biological interpretability (e.g. Zeide,
2004), it has been shown that well formulated empirical
equations can be just as accurate or even more accurate
for a wide range of data (e.g. Martin and Ek, 1984).
The dependent variables for updating individual
tree DBH have included diameter increment (Hann
et al ., 2006; Weiskittel et al ., 2007), diameter-inside-
bark-squared (Cole and Stage, 1972), relative-diameter
increment (Yue et al ., 2008), and inside-bark-basal-area
increment (Monserud and Sterba, 1996). The optimal
dependent variable has been debated, as West (1980)
found no difference between using diameter or basal
area to predict short-term increment (1 to 6 years) in
Eucalyptus . Two general conceptual approaches to model
formulation have been used to predict diameter incre-
ment: (i) a maximum potential increment multiplied
by a modifier and (ii) a unified equation that predicts
realized increment directly. Although the differences
between the two are mostly semantic as they both can
give reasonable behaviour (Wykoff and Monserud,
1988), they do illustrate a key philosophical decision
in modelling increment. The potential-times-modifier
approach to modelling diameter increment has long been
used in the past, but suffers from the inability to estimate
parameters simultaneously and estimating a potential
increment change can be challenging. Consequently,
empirical model forms that predict realized diameter
increment have become more common and differ
primarily in the covariates considered. The majority of
equations include two expressions of DBH to induce
a peaking behaviour (BAL), a measure of two-sided
competition and site index.
Modelling height increment is generally much more
difficult than diameter increment because of higher
within-stand variability, a more limited number of remea-
surements, and a closer connection to environmental
factors rather than stand-level ones. Like diameter incre-
ment, a variety of approaches have been used to model
height increment and the most common are of two types:
(i) potential times modifier and (ii) realized. One alterna-
tive to a height increment equation is to predict diameter
increment and use a static allometric height to diame-
ter equation to estimate the change in tree height. In
contrast to diameter increment modelling, the potential-
times-modifier approach is commonly used for predicting
height increment (Hegyi, 1974; Arney, 1985; Burkhart
et al ., 1987; Wensel et al ., 1987; Hann and Ritchie, 1988;

Hann and Hanus, 2002; Hann et al ., 2003; Weiskittel
et al ., 2007; Nunifu, 2009). One reason for this is that
dominant height equations can be easily rearranged to
give good estimates of potential height growth rather than
having to fit a separate equation or select a subjective max-
imum as was the case for potential diameter growth. The
prediction of height increment with a realized approach
has paralleled the approaches used for estimating diam-
eter increment directly. For example, Hasenauer and
Monserud (1997) used a height-increment-model form
similar to the diameter-increment equation of Monserud
and Sterba (1996), except tree height-squared was used
instead of DBH 2 .

23.3.3 Mortalityequations

Tree mortality is a rare yet important event in forest
stand development and has significant implications for
long-term growth-and-yield model projections (Gert-
ner, 1989). Of all of the attributes predicted in growth
models, mortality remains one of the most difficult due
to its stochastic nature and infrequent occurrence. For
modelling purposes, it is important to note the type
of mortality, which is generally described as regular or
irregular . Regular mortality can also be expressed as
density-dependent and is caused by competition-induced
suppression. Irregular or catastrophic mortality is inde-
pendent of stand density and is due to external factors
such as disease, fire, or wind. Previous reviews on mod-
elling mortality have concluded that there is no best
way to model it for all applications (Hawkes, 2000).
Nearly all of the tree-level mortality equations use logis-
tic regression to estimate the probability of a tree dying
(Hamilton, 1986; Monserud and Sterba, 1999; Hann
et al ., 2003). Thus, the primary differences between indi-
vidual tree-mortality equations that have been developed
are: (i) the type of data used; (ii) the statistical meth-
ods for estimating parameters; (iii) the length of the
prediction period; (iv) usage of additional equations to
constrain predictions; and (v) the tree and stand variables
utilized for predictions. Like allometric and increment
equations, DBH has been the primary variable in most
individual tree-mortality equations. DBH growth has also
been used as a covariate in mortality equations (Mon-
serud, 1976; Buchman et al ., 1983; Hamilton, 1986; Yao
et al ., 2001). Although data intensive and often explain-
ing a limited amount of variation, empirical equations
of mortality tend to perform better than theoretical
(Bigler and Bugmann, 2004) and mechanistic approaches
(Hawkes, 2000).