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An Adjustable Predictor of Crown Profile for Stand-Grown Douglas-Fir Trees

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This study developed a method for predicting the crown profile of stand-grown trees that can be adjusted to other populations of the same species by using either measurements or predictions of the largest crown width (LCW) for trees in the alternative population. The method should be of particular interest for tree species such as Douglas-fir that have exhibited variation in crown attributes across both their geographic range and genotypes. To model crown profile, the crown was divided into two segments: the portion of the crown above the point where LCW occurs, and the portion below that point. The equation for the upper portion predicted a crown profile that ranged in shape from nearly conic to parabolic, depending on position within the crown and the social status of the tree, as indicated by the ratio of total height divided by diameter at breast height for the tree. The equation for the lower portion predicted a crown profile with a cylindrical shape. This method explained nearly 94% of the variation in crown width when used with the felled tree measurement of LCW, 87% with a measurement of LCW taken while the tree was standing, and 83% with the use of a value for LCW that was predicted from an equation developed from an independent data set. For. Sci. 45(2):217-225.
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An Adjustable Predictor of
Crown Profile for Stand-Grown
Douglas-Fir Trees
David W. Hann
ABSTRACT. This study developed a method for predicting the crown profile of stand-grown trees that
can be adjusted to other populations of the same species by using either measurements or predictions
of the largest crown width (LCW) for trees in the alternative population. The method should be of
particular interest for tree species such as Douglas-fir that have exhibited variation in crown attributes
across both their geographic range and genotypes. To model crown profile, the crown was divided into
two segments: the portion of the crown above the point where LCWoccurs, and the portion below that
point. The equation for the upper portion predicted a crown profile that ranged in shape from nearly
conic to parabolic, depending on position within the crown and the social status of the tree, as indicated
by the ratio of total height divided by diameter at breast height for the tree. The equation for the lower
portion predicted a crown profile with a cylindrical shape. This method explained nearly 94% of the
variation in crown width when used with the felled tree measurement of LCW, 87%with a measurement
of LCWtaken while the tree was standing, and 83% with the use of a value for LCWthatwas predicted
from an equation developed from an independent data set. FoR. Sc•. 45(2):217-225.
Additional Key Words: Crown width, crown area, crown shape, crown architecture, crown modeling.
T HE ABILITY TO PREDICT CROWN PROFILE (i.e., crown
width at any point in the crown) for stand-grown
trees is becoming increasingly important in forest
management. Crown profile equations can be used to charac-
terize the crown closure, crown-area profile, crown volume,
and crown porosity of a stand (Dubrasich et al. 1997). These
measures of canopy architecture have been used to character-
ize stand structure and past disturbance history and their
effect on wildlife breeding densities of birds (Sturman 1968,
Mannan and Meslow 1984, Medman and Binkley 1988,
McComb et al. 1993). This type of information is useful for
making multiresource decisions.
Crown closure of the stand at various heights on a tree has
been used to predict individual-tree height growth (Wensel et
al. 1987, Mann and Ritchie 1988, Ritchie and Mann 1990,
Biging and Dobbertin 1995), basal-area growth (Wensel et
al. 1987, Biging and Dobbertin 1995), and probability of
mortality (Mann and Wang 1990) in stand-development
models, such as CACTOS (Wensel et al. 1986) and ORGA-
NON (Mann et al. 1995), which are used to make manage-
ment decisions. Predictions of the largest crown area of each
tree can be used to simulate the spatial arrangement of the
trees in naturally regenerated stands (Manus et al. 1998).
Crown profile equations are used to characterize the size and
shape of each tree' s crown as it is drawn on the screen of a PC
in stand-visualization programs, such as VIZ4ST (Hanus and
Harm 1997). These programs are used to display the results of
management actions.
Past approaches to characterizing crown profile can be
classed as using either an indirect or a direct method for
predicting crown width. The studies that have used the
indirect method first develop equations for predicting
branch attributes, such as branch length (BL), branch angle
(VA), and branch diameter (BD), through regression analy-
sis and then indirectly computing crown width from ap-
propriate trigonometric relationships. Studies that have
David w. Hann is Professor, Department of Forest Resources, Oregon State University, Cowallis, OR 97331--Phone: (541) 737-4673, Fax: (541)
737-3049; E-maih hannd@ccmeil.orst.edu.
Acknowledgments: The author wishes to thank Debbie Johnson, Janet Ross, and Joe Mortzheim for their assistance in field sampling and data
preparation. This is Paper 3239 of the Forest Research Laboratory, Oregon State University.
Manuscript received December 15, 1997. Accepted December 8, 1998. Copyright ¸ 1999 by the Society of Amedcan Foresters
Forest Sctence 45(2) 1999 217
used this approach include the recent work of Cluzeau et
al. (1994), Deleuze et al. (1996), Kurth and Sloboda
(1997), and Roeh and Maguire (1997).
The alternative approach has been to use regression
analysis to develop equations that directly predict crown
width (or radius or area) from various tree attributes. The
choice of tree attributes used in the equations has differed
among studies. Mitchell (1975) and Ottorini (1991) used
only depth into the crown from the tip of the tree (DINC)
as a predictor variable in their equations of Douglas-fir
(Pseudotsuga rnenziesii [Mirb.] Franco) crown radius.
Raulier et al. (1996) divided the crown of black spruce
(Picea mariana [Mill.]) into an upper "light crown," in
which they predicted crown radius as a function of DINC,
and a lower "shade crown," in which crown radius was
characterized as being constant. Honer (1971) included
total height (HI) and DINC in his equations for balsam fir
(Abies balsamea [L.] Mill.) and black spruce. Ritchie and
Hann (1985) used relative position within the crown (RPh,
where RPh = h/CL, h = HT- DINC, and CL = crown length)
and an estimator of the maximum crown width (MCW) of
open-grown trees as predictors of crown width for five
coniferous tree species, including Douglas-fir. Biging and
Wensel (1990) used RPh, DINC, HT, crown ratio (CR), and
diameter of the tree at breast height (DBH) as predictors of
crown area for six coniferous tree species, including Dou-
glas-fir. Finally, Baldwin and Peterson (1997) used RPh,
CR, DBH, and tree age as predictors of crown radius for
loblolly pine (Pinus taeda L.).
Data to model crown profile has usually come from
detailed measurements taken on felled (Mitchell 1975, Ritchie
and Hann 1985, Biging and Wensel 1990, Raulier et al. 1996)
or standing trees (Biging and Gill 1997, Roeh and Maguire
1997). As a result, the modeling data sets have been relatively
small in size and narrow in geographic range. For Douglas-
fir in western North America, crown profile has been mod-
eled with data collected on 26 trees from Vancouver Island
(Mitchell 1975), 43 trees from northern Idaho and western
Montana (Ritchie and Hann 1985), 53 trees in western Wash-
ington (Roeh and Maguire 1997), and 5 trees in northern
California (Biging and Gill 1997). Cumulative crown vol-
ume above crown base and the associated crown area profile
have been modeled with data from 115 trees in northern
California (Biging and Wensel 1990).
Past studies have shown that crown attributes often vary
across the range of Douglas-fir. MCW and largest crown
width (LCW) of stand-grown trees have been found to vary
with geographic location (Smith 1966, Paine and Hann 1982,
Hann 1997); the height-to-crown-base (HCB) equation of
southwestern Oregon (Ritchie and Hann 1987) predicts sub-
stantially longer CL than the equation for northwestern Or-
egon (Zumrawi and Hann 1989). Significant variation in CR
and relative crown-width ratio has also been found among
families of Douglas-fir (St. Clair 1994).
For this study, the direct method for predicting crown
profile was chosen for two reasons. First, past experience
with the application of VIZ4ST has shown that user judg-
ments about the success of the rendering ofa tree's crown in
218 Forest Sctence 45(2) 1999
such visualization programs was most often based on •ts
shape. Because the shape of a tree's crown is determined
predominantly by its profile, it was felt that the use of direct
profile equations could be more easily evaluated and con-
trolled for the appropriateness of the resulting crown shape.
Second, it was felt that the development of a predictor of
crown profile that could be adjusted to other populations
would be more easily accomplished by using the direct
approach. Given the sparsity of existing crown width data,
the effort and cost of collecting new data, and the likelihood
of geographic variation in crown attributes, an adjustable
crown-profile model could be very useful in the local calibra-
tion of computer models such as ORGANON and VIZAST.
As in Ritchie and Hann (1985), Raulier et al. (1996), and
Roeh and Maguire (1997), crown profile is defined herein as
a two-dimensional relationship that, for each whorl in the
crown, describes the horizontal distance between opposite
branch tips for the dominant branches in the whorl. This two-
dimensional relationship involves two absolute quantities,
CL and LCW, which define the maximum extent of the crown
in each dimension. A relative descriptor of the shape of the
crown profile can be formed by dividing crown width at a
given height (CWh) by LCW, and the reference height of
CWh (h) by CL. Therefore, variation among crown profiles
may be due to either variation in CL, LCW, or relative shape.
A number of past studies involving crowns have used
geometric forms to characterize or assess the shape of the
crown (e.g., Hamilton 1969, Mawson et al. 1976, Takeshim
1985, Koop 1989). An examination of the potential applica-
tion of simple geometric forms (i.e., cylinder, cone, parabola,
and neiloid) for characterizing relative crown shape (i.e.,
CWh/LCW) indicated that all of the geometric forms could be
generalized into the following relationship:
CW___2_h = (1)
LCW
The particular relative shape taken by Equation (1) de-
pends on the value of k: a value of 1.5 would indicate a crown
with a neloidic shape, a value of 1.0 a conic shape, a value of
0.5 a parabolic shape, and a value of 0 a cylindrical shape.
However, the value ofk does not have to be constant between
or within the tree. For example, k could change depending on
the social position of the tree within the stand, or it could take
on one value (and therefore crown shape) near the top of the
crown and another near the base of the crown.
Equation (1) can be modified to predict CWh by multiply-
ing it by LCW:
CWh = ICW' Reh k (2)
The advantage of this formulation for characterizing
crown profile is that it explicitly expresses the relationship
as a function of relative crown shape, which is then
expanded to an estimate of CWh through the multiplication
by LCW. As a result, the method allows the direct input of
a measure of LCW into the estimation process. The first
objective of this study, therefore, was to collect and then
use crown profile data from a set of felled trees to develop
a new modeting approach for directly characterizing crown
profile in Douglas-fir. The model would be based on
Equation (2), which explicitly includes the felled tree
measurement of LCW (LCWf) as one of its predictor
variables.
Because LCWfwill seldom be available in application of the
model developed under the first objective, direct measurement
of LCWwhile the tree was standing (LCWs) or a predicted LCW
(LCWt•) from equations such as those developed for Douglas-fir
by Smith and Bailey (1964), Smith (1966), Moeur (1981),
Warbington and Levitan (1993), Dubrasich et al. (1997), and
Harm (1997) would probably be used instead. However, these
substitutions can introduce measurement error (Germer 1991)
into the prediction process. The result can be biased predictions
and/or loss of precision (Kmenta 1986, Gerther 1991). The
second objective, therefore, is to conduct an empirical evalua-
Uon of the effects of substituting eitherLCW s orLCWt• for LCWf
on the accuracy and precision of predicting CWn.
Variables are defined and their abbreviations given at first
mention in the text. For easy reference, they are also summa-
nzed in Table 1.
Table 1. Variables.
Data
All suitable Douglas-fir stands on the 4,714 ha McDonald-
Dunn Research Forest located near Corvallis, Oregon, were
classified into nine strata based on the average diameter and
average crown ratio of the dominant trees in each stand.
Three stands were then randomly chosen from each stratum,
then four undamaged trees with crowns that from ground
inspection appeared symmetrical in length were chosen from
each stand for destructive sampling.
The range of diameters found in the management inven-
tory for the stand (Marshall et al. 1997) was divided into
quartiles, and one tree was selected for destructive sampling
from each quartile. Trees in each quartile that met the criteria
for selection and that could be safely felled with minimal
damage to the crowns were noted during a thorough field
inspection of the stand. Final tree selection in each quartile
was done in a manner that distributed the four trees across the
stand, thereby minimizing the spatial dependence between
the sample trees.
The following attributes were measured before falling
each tree: DBH (cm), HT (m), HCB (m), LCW (m) along a
BD
BL/
BL•
CL
CR
CWA•,
CWB•
DACB/
DA CB •,
DBH
DBTn•
DIBn•
HB
HCB
HLCW
HT
h
LCW
LCW ,
MCW
RPAn
RPB•
Diameter of the branch outside bark at approximately one branch diameter from the bole
Length of the main stem branch measured while the whorl was prone
Straight line distance between branch base and branch tip measured while the whorl was stood upright
Crown length
Crown ratio
Crown width at a height of h above the ground
Crown width at a height of h above the ground for the portion of the crown above the height where largest
crown width occurs
Crown width at a height of'h' above the ground for the portion of the crown below the height where
largest crown width occurs
Distance above crown base to where largest crown width occurs as measured on the felled tree
Distance above crown base to where largest crown width occurs as predicted from an appropriate equation
Diameter at breast height
Double bark thickness of the main bole of the tree at the height above the ground where the base of the
branch occurs
Diameter inside bark of the main bole of the tree at the height above the ground where the base of the
branch occurs
Height above the ground to the base of the branch
Height-to-crown-base
Height above the ground to the largest crown width of a stand-grown tree
Total height
Height above the ground to the crown width of interest
Largest crown width of a stand-grown tree
Largest crown width of a stand-grown tree measured after the tree is felled
Largest crown width of a stand-grown tree predicted from an appropriate equation
Largest crown width of a stand-grown tree measured while the tree is standing
Maximum crown width of an open-grown tree
Relative position of the crown width within the length of the crown
Relative position of the crown width within the length of the crown occurring above the position of the
largest crown width
Relative position of the crown width within the length of the crown occurring below the position of the
largest crown width
Angle between the bole, the base of the branch and the lower 30 cm of the branch while the whorl was
prone
Angle between the bole, the base of the branch and the branch tip while the whorl was stood upright
Forest Sctence 45(2) 1999 219
random horizontal axis centered through the bole of the tree,
and LCW(m) on the horizontal axis centered through the bole
of the tree and perpendicular to the first axis. After falling, HT
and HCB were remeasured along the bole of the felled tree.
Ten whorls were then selected at approximately equal spac-
ings between crown base and tip of the terminal leader. On
each chosen whorl, the largest undamaged, accessible branch
was selected and the following measurements taken: (1) the
angie between the bole, branch base, and the lower 30 cm of
the branch (VAf) in degrees; (2) the height (m) of the
branch (HB) measured from the ground to the base of the
branch; (3) the diameter (cm) of the branch outside bark
(BD) at approximately one branch diameter from the bole
to avoid basal swelling (Maguire et al. 1991); and (4) the
length (m) of the main stem of the branch (BLf). For trees
with lower crown damage due to falling, measurements
began at the first whorl above the crown base that con-
tained at least one undamaged branch.
The branch angie of a prone, felled tree can differ from
the branch angle of a standing tree; branches also usually
exhibit some degree of curvature, which can cause the
length of the branch itself to be longer than the straight-
line distance from the base of the branch to the branch tip
(Honer 1971, Remphrey and Powell 1984). To minimize
these problems, 251 measured whorls were cut from the
boles of a subset of trees and each whorl was stood upright
for additional measurements. For branches in this subset,
branch angle (VAs) in degrees was remeasured as the angle
between the bole, branch base, and the branch tip. Branch
length (BLs) in meters was also remeasured as the straight
line distance between branch base and branch tip. These
data were then used to develop the following equations for
predicting the attributes on branches that were not directly
measured:
BL• = BLf e ['-ø'ø3582ø'(3'28ø8'•/)(ø'794281 - 0.399676,BP)]
ß (3)
VA• =80
ß [1- ae]O.,
(4)
where
BP = the relative position of a branch within the crown
(HT- HB)/(HT- HCB)
The adjusted coefficient of determination ( •2 ) for Equa-
tion (3) was 0.3652 and •2 for Equation (4) was 0.9713.
Given VAs, BLs, and HB, h and CWh were calculated by:
h = HB + cos(VA s). BL s
CW•= 2.(sin(VAs).BLs) + OOBnB
where
DOBHB
DIBHB
= DIBHB + DBTHB
= diameter inside bark of the bole, in meters, atHB
predicted from the Walters and Harm (1986)
bole taper equation for Douglas-fir
DBTHB = double bark thickness of the bole, in meters, at
HB predicted from the Maguire and Harm (1990)
bark thickness taper equation for Douglas-fir
The resulting CWh describes only the longest, dominant
branches in a whorl. This is in contrast to the studies of Biging
and Wensel (1990) and Biging and Gill (1997), wherein
measurements were also taken in the internodes in order to
characterize the reduction in crown width that can occur
between whorls.
Examination of the resulting CWh data set indicated that
felling damage in the lower part of the crown prevented
determining the true position of LCW for six of the trees
Those trees were eliminated from the modeling data set, but
they were reserved for use as a verification data set. The
examination also found one suppressed tree with a dead
terminal leader; that tree was eliminated from both the
modeling and the verification data sets.
For the remaining 101 trees in the modeling data set, the
largest value of CWh (LCWf) was determined and the distance
(m) above crown base (DACBf) where LCW[did occur was
calculated for each tree. The height from the ground to LCWf
(HLCW) was then computed by adding DACB[to HCB. The
geometric mean of the two LCW measurements taken before
felling was used to calculate an average LCW (LCWs) for
each tree in both data sets (LCWs was not measured on one
tree in the modeling data set). A summary of the tree and
crown attributes for the modeling and the verification data
sets are found in Tables 2 and 3, respectively.
Analysis Methods
Crown Profile
Examination of the CWh data for Douglas-fir indicated
that LCW did not always occur at the crown base. Therefore,
one equation was developed for predicting CWh above HLCW
(CWAh) and a second equation for estimating CWh below
HLCW(CWBh). The two equations were then structured in a
Table 2. Tree and crown attributes for the crown profile modelin9
data set.
No. of
Attribute observations Mean Min Max
DBH (cm) 101 42.4 5.1 110.0
HT (m) 101 29.4 4.7 50.9
(m) 101 13.4 3.5 26.9
HT/DBH 101 0.76 0.44 1.40
DACB r (m) 101 0.8 0.0 6.4
LCW s (m) 100 8.0 3.0 17.3
LCW/(m) 101 7.9 3.2 17.1
CWh (m) 1,008 4.4 0.1 17.1
CWAh (m) 859 3.9 O. 1 15.5
CWB• (m) 48 5.8 1.8 11.8
220 Forest Sctence 45(2) 1999
Table 3. Tree end crown attributes for the crown profile verifica-
tion data set.
No. of
Attribute observations Mean Min Max
DBH (cm) 6 80.0 40.9 120.7
HT (m) 6 47.9 33.1 54.7
CL(m) 6 14.3 10.0 17.4
HT/DBH 6 0.64 0.46 0.82
LCW, (m) 6 10.8 5.8 15.4
CWh (m) 53 3.9 0.5 11.3
manner such that alternative estimators of LCW could be
easily inserted into the two-component profile equations.
To predict CWAh, Equation (2) was modified to the
following:
where
CWAh = LC%' ReA (5)
RPA h = (HT- h) / (HT - HLCW)
In this formulation, k is a constant for all trees and within
the tree's crown. Ford (1985), however, found that crown
shape depended on the position of the tree within the stand.
Oliver and Larson (1996) reported that Douglas-fir exhibits
a high degree of epinastic control when in the overstory, and
weak control in the understory, and that one characteristic of
trees with weak epinastic control is a wide, "umbrella-like"
appearance of the crown. A rounding of crown shape with the
loss of dominance has also been reported by Raulier et al.
(1996) for black spruce (Picea rnariana [Mill.]). Therefore,
k m Equation (5) should change with social status. Because
trees under competition stress will allocate more of their
photosynthate production to height growth rather than to
diameter growth (Oliver and Larson 1996), a small HT?DBH
value will usually indicate a dominant tree in the overstory,
and alarge value will usually indicate a suppressed tree in the
understory. Therefore, Equation (5) was further modified to:
CWA h = LCWf . RP4 aø+a"{ ...... ] (6)
Nonlinear regression was used to fit this equation to the
CWAh data; the residuals were then plotted across DBH, HT,
HT/DBH, CL, CR, LCWf, and RPAh to check for unexplained
trends. These plots indicated that the variance of the residuals
increased with LCWf and that a trend existed across RPAh,
indicating that relative crown shape differed based on the
position of the whorl within the crown. Several transforma-
tions of RPAh were tried and the following formulation was
selected as best for characterizing crown profile:
CWA = LCW/ ' (7)
The parameters of nonlinear equations such as Equation
(7) can be estimated by using ordinary nonlinear least squares
regression, if the variance of the residuals is homogeneous
(Seber and Wild 1989), an assumption which is not met in this
application. To mitigate this situation, the parameters of
Equation (7) were predicted through weighted nonlinear least
squares regression, a special case of the generalized least
squares model (Seber and Wild 1989). Given the behavior of
the residuals, the appropriate weight to use was judged to be
1.0/LCWf 2 . The parameters were then estimated by using
weighted nonlinear least squares regression, and the success
of the weighing scheme was evaluated by plotting the weighted
residuals across predicted, weighted CW h (Figure 1). The
weighted residuals appeared homogeneous.
The possible presence of serial correlation between re-
siduals within each profile was evaluated by fitting Equation
(5) to each tree using weighted nonlinear regression and
computing the Durbin-Watson statistic (Kmenta 1986). Of
the 101 trees in the modeling data set, 10 had Durbin-Watson
statistics that were significant (ix = 0.01), and 37 had tests that
were inconclusive. Raulier et al. (1996) reported 1 inconclu-
sive and 3 significant tests for their 32 trees. The higher
proportion of inconclusive tests in this study may be partly
due to the fact that fewer crown width measurements were
taken on each tree [ 10 measurements versus 18 for Raulier et
al. (1996)]. From this analysis it was concluded that serial
correlation was not a serious problem and could therefore be
ignored.
The following relative position variable was formed to
characterize CWBh:
RPB h = (h - HCB) / DACBf
When CWB h was plotted against RPBh, DBH, HT, CL, CR,
HT/DBH, DA CBf, and LCWf, a simple linear equation through
the origin on LCWfassociated with a residual that increased
with LCWfemerged as the most plausible and parsimonious
model. Although the application of ordinary linear least
squares regression to data with heterogeneous variance does
produce unbiased parameter estimates, the parameter esti-
mates are inefficient (i.e., they are not "best linear unbiased
estimators") and the estimates of the parameter variances are
biased (Kmenta 1986). Analogous to the nonlinear case,
Kmenta (1986) recommends the usage of weighted linear
least squares regression when the variance is heterogeneous.
Therefore, weighted linear least square regression (with a
weight of 1.0/LCW• 2) was used to estimate the parameter in
the following equation:
CWBh : ß œCW/ (8)
0A
ß .......•.', ..'e%'.".4',['.-•-..-".
.. , :, ß
'•.•t '' ''.'" '" '- •.".'' "". '.-
o o.• o,= o,a 0.4
Predicted, Weighted CWA h
Figure 1. Weighted residuals for Equation (7) plotted across
predicted, weighted CWA h.
Forest Sctence 45(2) 1999 221
Graphs of weighted residuals across DBH, HT, HT/DBH,
CR, and predicted, weighted CWB h suggested homogeneous
variance and no discernible trends.
These two-component profile equations [(7) and (8)] can
be combined into the following estimator of CWh:
CW h = Io CWA h + (1.0 - I) o CWB h (9)
where
I= 1.0 ifh _> HLCW
= 0.0 if h < HLCW
This combined equation assumes that DACBf and LCWf
are available as predictor variables. Unfortunately, DACBfis
difficult to determine and .measure while the tree is standing.
An estimator of DACBfwas used, along with both an estima-
tor and a standing tree measurement of LCWf, to explore their
impact on the accuracy and precision of predicting CWh.
To develop an estimator of DACBf, the variable was
plotted over DBH, HT, and CL. The plots indicated that
DACBf was best characterized as a simple linear equation
through the origin, with CL as the independent variable, and
that the conditional variance in DACBf increased with CL.
Weighted linear least squares regression (with a weight of
1.0/CL 2) was used to estimate the parameter in the following
equation:
OACBp = q o CL (10)
Graphs of weighted residuals across DBH, HT, HT/DBH,
CR, and predicted, weighted DACBp suggested homoge-
neous variance without trends.
Predicted Largest Crown Width
The following equation of Harm (1997) was selected as an
appropriate predictor of LCW:
LCWp = MCW ø cg (0'0143149CL + 0.0722402-•--•) (11)
where
MCW = 1.4081 + 0.22111 ß DBH- O.00053438.DBH 2
This equation was developed from an independent data set of
925 Douglas-fir trees collected on the McDonald-Dunn Re-
search Forest. The presence of CL and its associated param-
eter in this equation resulted in a decrease in LCWp as CL
increased. With CR fixed, the larger the CL, the taller the tree;
a tall tree has a greater chance of abrasion damage to its crown
due to the effect of wind whipping (Oliver and Larson 1996).
The presence of DBH/HTand its associated parameter in the
equation also resulted in a decrease in L CW for an increase in
DBH/HT. Dominant trees will therefore exhibit a greater
reduction in LCWp as CR decreases than will suppressed
trees. As a result, suppressed trees will exhibit relatively
broader crowns than dominants.
Residuals
Four sets of unweighted CWh residuals were used to
evaluate the impact on prediction accuracy and precision of
substituting alternative estimators into Equation (9). The sets
Table 4. Parameter estimates and associated asymptotic stan-
dard errors of the estimates for Equations (7), (8), and (10).
Parameter Asymptotic
Equation Parameter estimate standard error
(7) a 0 0.929973 0.0437554
a• -0.135212 0.0509390
a 2 -0.131316 0.0405132
(8) b, 0.896794 0.0163025
(10) ct 0.062000 0.0101440
of residuals were calculated by using the 100 trees in the
modeling data set with measurements of both LCWs and
LCWf and the following combinations of LCW and DACB
(1) LCWfand DACBf, (2) LCWfand DACBp, (3) LCWs and
DACBp, and (4) LCWp and DACBp. For the verification data
set, two sets of unweighted CWh residuals were calculated
with the following combinations of LCW and DACB: (1)
LCWs and DACBp, and (2) LCWp and DACBp. The average
residual, mean square error (MSE) of residuals, and the •2
were then computed for each residual data set. •2 was
computed by using the method recommended by Kvfilseth
(1985).
Results and Discussion
The strategy of dividing the crown into two compo-
nents at the point where LCW occurs in order to model
crown width has also been used by Badoux (1939) and
Raulier et al. (1996). Raulier et al. (1996) called the upper
portion the "light" crown, because it is usually exposed to
direct sunlight, and the lower portion the "shade" crown.
Equation (7), therefore, is a model of the light portion of
the crown and Equation (8) is a model of the shade portion
The parameter estimates and associated asymptotic stan-
dard error of the parameters for both models can be found
in Table 4. The fit to Equation (7) was very good, with a
weighted MSE of 0.008650 and a weighted, •2 of 0.8685
The shade crown was more ragged (and therefore less
predictable) than the light crown, producing a larger
weighted MSE of 0.012757.
...... 0,4S
0,4.
ta m
0,2 o.4 0,6
Relative Postlfion in the Crown (RPA•)
Figure 2. Relative crown width (CWAh/LCW •) plotted across
relative position within the crown (RPA h) for a dominant tree
with a HT/DBH ratio of 0.48 and a suppressed tree with a value
of 1.92.
222 Forest Sctence 45(2) 1999
40
30
20
10
i I I
10 15 2o 25 3o
Widths (m)
Figure 3. Predicted crown profiles for a dominant tree with a DBH
of 81.3 cm, e HTof 39.6 m, and a CRof 0.5, end a suppressed tree
with e DBH of 5.1 cm, e HT of 7.6 m, and a CR of 0.2.
For Douglas-fir, the shape of the light crown [Equation
(7)] depends on both the social status of the tree, as indicated
by its HT/DBH ratio, and the position of the whorl in the
crown (Figure 2). For a dominant tree with a HT/DBH ratio
of 0.48, the value of the power term on RPh ranges from 0.87
at the tip of the tree to 0.73 at HLCW, whereas an understory
tree with a HT/DBH ratio of 1.92 has a range of 0.68 at the
tip to 0.54 at HLCW. Therefore, a dominant Douglas-fir has
a more conic shape and an understory tree has a more
parabolic shape (Figure 3), confirming the previous observa-
tion of Oliver and Larson (1996).
LCW occurred above HCB in only 33 of the 101 trees
measured in this study. When this happened, the shade crown
[Equation (8)] was best described as having a cylindrical
shape. Raulier et al. (1996) also used a cylinder to character-
ize the shade crown of black spruce. It is likely that measure-
ment of more whorls on a tree could result in a higher
frequency where LCW occurred above HCB, particularly on
trees with long CL.
The profile model developed in this study used only
tree attributes as predictor variables. Ritchie and Hann
(1987), Zumrawi and Hann (1989), Larocque and Marshall
(1994), and others have shown that crown attributes can be
affected by stand density. Roeh and Maguire (1997),
however, found that the effect of density on Douglas-fir
crown width was adequately reflected in the DBH, HT, and
Table 5. Unweighted residual statistics for Equation (4} using
various measures of LCWand DACB and the modeling data set.
CW h predicted Unweighted residuals
using Mean Variance MSE Adjusted R 2
LCW/ and DACB/ +0.0292
LCW/ and DACB p -0.1745
LCW, and DACBp -0.1756
LCWp and DACB p -0.3759
0.528135 0.530053 0.9377
0.613219 0.645632 0.9241
1.067648 1.101823 0.8705
1.416244 1.562420 0.8336
CL of the tree. Curtis and Reukema (1970) found that
Douglas-fir "... trees of given DBH or total height but in
different spacings are quite similar in other stem and
crown dimensions" (p. 290). This suggests that measures
of density may be unnecessary in crown profile models for
Douglas-fir that already include DBH, HT, and CL.
When Equations (7) and (8) are combined through Equa-
tion (9), almost 94% of the variation in unweighted CW h for
the modeling data set can be explained if LCWfand DACBf
are known (Table 5). If DACBfis predicted by Equation (10)
and its parameter estimate in Table 4, the amount of ex-
plained variation is reduced by 1.4% with a concomitant
underprediction bias of 0.17 m, or 4% of the mean of CW h.
Using a measurement of LCW while the tree is standing
(LCWs), instead of LCWj• further reduces the amount of
explained variation by 5.4% with unchanged bias. Finally,
over 83% of the variation in CW h is explained when both
LCWand DACB are predicted, a reduction of 3.7% over using
LCW s, but the underprediction bias is increased to 0.38 m
(8% of the mean of CWh).
Results of the residual analysis were similar for the six
trees in the verification data set (Table 6). Equation (9)
explained 93% of the variation in CW h when using DACB•,
and LCW s and almost 83% when using DACB I, and LCWt,.
Instead of an underprediction bias, however, both methods
over predicted CW h by 0.18 m (or 5%) for DACBt, and LCW s,
and 0.06 m (or 1%) for DACB I, and LCWt,.
Although usage of DACB•,did introduce an underprediction
bias in the modeling data set, its effect on increasing the MSE
of the residuals was much smaller than using either LCWs or
LCW•, in place of LCWf Using LCWs measured in the field
instead of a prediction did improve the ability to predict CW h.
Therefore, any mensurational effort that improves the mea-
surement accuracy and precision of LCWs should improve
the ability to predict crown profile.
These findings indicate that either a measured or predicted
value of LCW can be used to adjust Equation (9) to other
populations, thereby producing reasonably accurate and pre-
cise estimates of crown profile between populations, if the
major variation in crown profile was due mainly to a variation
in LCWrather than relative crown shape. Final confirmation
Table 6. Unweighted residual statistics for Equetion (4) using
verious measures of LCWand DACB and the verification data set.
Unweighted residuals
CWh predicted using Mean Variance MSE Adjusted R 2
LCW s andDACBt, +0.1780 0.465771 0.528563 0.9307
LCWp and DACBp +0.0581 1.242239 1.321948 0.8266
Forest Sctence 45(2) 1999 223
of this assumption must await validation testing on other
populations of Douglas-fir. However, if this assumption does
hold for a species, then development of crown profile equa-
tions for that species can be accomplished in a more efficient
manner by minimizing the number of trees sampled for the
detailed and costly measurements needed to characterize
relative crown shape.
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Forest Sctence 45(2) 1999 225
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... It has been of interest for forest modelers to better understand this phenomenon, particularly on the basis of statistical modeling and analysis . The traditional crown profile modeling methods have been used to deal with the autocorrelation and heteroscedasticity in the crown profile equations, they are mainly direct variance-covariance modelling (Hann, 1999;Crecente-Campo et al., 2009;Crecente-Campo et al., 2013), mixed-effects modelling (Fu et al., 2013;Sharma et al., 2016;Fu et al., 2017;Sharma et al., 2017;Sun et al., 2017;Jia and Chen, 2019;Wang et al., 2019;Chen et al., 2021;Di Salvatore et al., 2021), and nonlinear marginal modeling (McCulloch and Searle, 2001;Lejeune et al., 2009;de-Miguel et al., 2012;Chen et al., 2022). With the rapid development of machine learning artificial intelligence, some machine learning algorithms have the characteristics of high accuracy and good robustness for the data with nonlinear features (Singh et al., 2016;Dong et al., 2021), which has subsequently been applied to crown profile modeling. ...
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