Forest-Management Modelling - Environmental Modelling: Finding Simplicity in Complexity - page 369

estimate of maximum crown width (MCW) for all trees in
a stand. Like defining the live crown base of an individual
tree, multiple definitions of crown width exist. Maximum
crown width generally refers to the width reached by an
open-grown tree, while largest crown width (LCW) is
the width of a stand-grown tree. Crown profile is the
change in crown width within an individual tree. Maxi-
mum (Ek, 1974; Paine and Hann, 1982; Hasenauer, 1997)
and largest (Moeur, 1981; Hann, 1997; Bechtold, 2003)
crown-width equations exist for several species. Gener-
ally, DBH is effective at capturing most of the variation in
both MCW (Paine and Hann, 1982) and LCW (Gill et al .,
2000). Two primary approaches have been used to model
crown profile: (i) direct and (ii) indirect characterization.
Direct characterization uses deterministic or stochastic
models to predict crown width (radius or area) from tree
attributes, whereas indirect characterization predicts the
attributes of individual branches and computes crown
width based on trigonometric relationships. The direct
characterization has been the predominant form of pre-
dicting crown profile (Nepal et al ., 1996; Baldwin and
Peterson, 1997; Biging and Gill, 1997; Hann, 1999; Mar-
shall et al ., 2003), but the indirect approach has also been
used for several species (Cluzeau et al ., 1994; Deleuze
et al ., 1996; Roeh and Maguire, 1997).
Stem form and volume are the two most important tree
attributes for determining value and the primary interest
of most growth-model users. A variety of approaches
for determining both attributes exist, even for a single
geographic region (e.g. Hann, 1994). The current trend
has been to move away from stem-volume equations
and rely more on stem-taper equations, which predict
changes in stem diameter from tree tip to base. Taper
equations have become preferred because they depict
stem form, provide predictions of total volume, and
can be used to determine merchantable volume to any
height or diameter specification. Limitations of taper
equations are that they are often overly complex, which
may limit their ability to extrapolate beyond the dataset
from which they were developed, and they are not opti-
mized to give volume predictions. Similar to volume
equations, most stem-taper equations are a function of
only DBH and total tree height, and a variety of model
forms exist. Taper equations are of three primary types,
namely: (i) single (Thomas and Parresol, 1991); (ii) seg-
mented (Max and Burkhart, 1976); and (iii) variable-form
(Kozak, 1988). Goodwin (2009) gives a list of criteria for
an ideal taper equation, but most of the widely used
forms do not meet all the criteria, which is important to
recognize.

Like stem volume, thousands of biomass equations
have been developed around the world. For example,
Jenkins et al . (2004) reported 2640 biomass equations
from 177 studies in North America. Other extensive
reviews have been done for Europe (Zianis et al ., 2005),
North America (Ter-Mikaelian and Korzukhin, 1997),
and Australia (Eamus et al ., 2000; Keith et al ., 2000),
which highlight the vast amount of work that has been
done on this topic. However, most biomass equations
are simplistic with parameters determined from relatively
small sample sizes. Zianis et al . (2005) found that more
than two-thirds of the equations they examined were a
function of just DBH and more than 75% of the stud-
ies that reported a sample size had less than 50 trees.
As a result of using simple model forms fitted to small
data sets, the application of the resulting equations to
other populations can produce large predictions errors
(e.g. Wang et al ., 2002). In addition, the development
of universal (Pilli et al ., 2006) and generalized (Muukko-
nen, 2007) allometric equations ignores significant species
variability and complex relationships, particularly when
the goal is to estimate regional and national biomass
(Zianis and Mancuccini, 2004). Efforts to localize allo-
metric biomass equations without requiring destructive
sampling by accounting for the relationship between tree
height and DBH as well as wood density (Ketterings et al .,
2001) or the DBH distribution (Zianis, 2008) have been
proposed. The most widely used biomass equations in
North America are reported in Jenkins et al . (2003).

23.3.2 Increment equations

Growth is the increase in dimensions of each individual
in a forest stand through time, while increment is the
rate of the change in a specified period of time. Although
growth occurs throughout a tree, foresters are primarily
concerned with changes in both tree DBH and height
because of their ease of measurement and strong corre-
lation with total tree volume. Tree growth has multiple
inter- and intra-annual stages that must be considered by
tree-list models. For example, a cumulative growth curve
of height over age shows three primary stages: (i) juvenile
period where growth is rapid and often exponential; (ii) a
long period of maturation where the trend is nearly linear;
and (iii) old age where growth is nearly asymptomatic. A
diameter growth curve would show much the same trend,
except there is a tendency toward curvilinearity during the
period of maturity. Various theoretical model forms have
been used to predict growth in forestry (Zeide, 1993), but
most of them can be generalized with a single equation