Forest Ecology and Management 326 (2014) 125–132
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Forest Ecology and Management
journal homepage: www.elsevier.com/locate/foreco
An allometric model for estimating DBH of isolated and clustered
Eucalyptus trees from measurements of crown projection area
Niva Kiran Verma a,b, David W. Lamb a,b,⇑, Nick Reid a,c, Brian Wilson a,c
a
Cooperative Research Centre for Spatial Information (CRCSI), University of New England, Armidale, NSW 2351, Australia
Precision Agriculture Research Group and School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
c
School of Environmental and Rural Science, University of New England, Armidale, NSW 2351, Australia
b
a r t i c l e
i n f o
Article history:
Received 26 March 2014
Received in revised form 3 April 2014
Accepted 4 April 2014
Keywords:
Diameter at breast height
Crown projection area
Scattered trees
Tree clusters
Eucalyptus
Farm land
a b s t r a c t
Owing to its relevance to remotely-sensed imagery of landscapes, this paper investigates the ability to
infer diameter at breast height (DBH) for five species of Australian native Eucalyptus from measurements
of tree height and crown projection area. In this study regression models were developed for both single
trees and clusters from 2 to 27 stems (maximum density 536 stems per ha) of Eucalyptus bridgesiana,
Eucalyptus caliginosa, Eucalyptus blakelyi, Eucalyptus viminalis, and Eucalyptus melliodora. Crown projection
area and tree height were strongly correlated for single trees, and the log-transformed crown projection
area explained the most variance in DBH (R2 = 0.68, mean prediction error ±16 cm). Including tree height
as a descriptor did not significantly alter the model performance and is a viable alternative to using crown
projection area. The total crown projection area of tree clusters explained only 34% of the variance in the
total (sum of) the DBH within the clusters. However average crown projection area per stem of entire tree
clusters explained 67% of the variance in the average (per stem) DBH of the constituent trees with a mean
prediction error ±8 cm. Both the single tree and tree cluster models were statistically similar and a combined model to predict average stem DBH yielded R2 = 0.71 with a mean prediction error (average DBH
per stem) of ±13 cm within the range of 0.28–0.84 m. A single model to infer DBH for both single trees
and clusters comprising up to 27 stems offers a pathway for using remote sensing to infer DBH provided
a means of determining the number of stems within cluster boundaries is included.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
Much of eastern Australia is characterised by diverse farming
landscapes, or ‘farmscapes’ containing a range of land-use systems
including crops, native and sown pasture, remnant vegetation and
trees at various densities and configurations. Native eucalyptus
trees, both individual and in clusters are an important feature of
Australian farmscapes, and provide shade and shelter for livestock
(Bird et al., 1992). They also have considerable value through their
influence on soils (Wilson, 2002; Goh et al., 1996; Barnes et al.,
2011a), biodiversity (Oliver et al., 2006) and their indirect role in
pasture quantity and quality (Barnes et al., 2011b). Scattered trees
outside the forest (STOF) contribute significantly to above and
below-ground carbon stocks in these landscapes (Baffetta et al.,
2011; Soto-Pinto et al., 2010). A summary of the role of scattered
⇑ Corresponding author at: Cooperative Research Centre for Spatial Information
(CRCSI), University of New England, Armidale, NSW 2351, Australia. Tel.: +61 2
6773 3565.
E-mail address: dlamb@une.edu.au (D.W. Lamb).
http://dx.doi.org/10.1016/j.foreco.2014.04.003
0378-1127/Ó 2014 Elsevier B.V. All rights reserved.
trees in landscapes is given in Manning et al. (2006). The
assessment of biomass in eucalypt systems has, to date, been
largely restricted to plantation forestry systems. However there
is also a growing need to assess carbon and biomass stocks across
our farmscapes in order to fully quantify carbon storage change in
response to management and provide evidence-based support for
carbon inventory and ultimately carbon trading. Such assessments
must also include scattered native trees.
Destructive sampling is considered the most reliable method
for determining biomass in grass and shrub vegetation but it is
rarely used for agroforest ecosystems (Snowdon et al., 2002).
The biomass of large vegetative structures like forests and open
woodland is usually estimated by applying ratio or regression
methods using empirical equations and allometric models
(Snowdon et al., 2002; Houghton, 2005; Makungwa et al.,
2013). Trunk diameter, more formally known as diameter at
breast height (DBH), is observed to be closely related to tree
biomass and carbon stocks (Ter-Mikaelian and Korzukhin, 1997;
Snowdon et al., 2002; Sanquetta et al., 2011a; Kuyah et al.,
2012; Beets et al., 2012).
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N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132
DBH is an important tree characteristic in its own right and is
straight forward to measure on the ground. The relationships
between tree canopy characteristics such as diameter, projection
area and coverage, and DBH is of considerable interest as DBH
can then be used to infer canopy attributes, for example to quantify
canopy competition in response to planting (stem) density, growth
potential or habitat modelling (Bella, 1971; Cade 1997; Grote
2003). Of course, large-scale collection of DBH data can be time
consuming and now satellite or airborne remote sensing, including
LIDAR can be used to infer canopy characteristics such as crown
projection area (e.g. Franklin and Strahler, 1988; Leckie et al.,
2003; Popescu and Wynne, 2004; Lee and Lucas, 2007).
Measuring the canopy diameter or crown projection area from
the ground involves measuring the crown projection across different angular segments of the canopy. These angular segments can
either be based on fixed (pre-set) directions (Röhle and Huber,
1985), or in the case of highly asymmetric canopies the directions
can be adapted to adequately characterise the actual tree dimensions (Hemery et al., 2005; Fleck et al., 2011). Based on an investigation involving 161 trees in an old-growth forest (approx. density
392 stems per hectare), Fleck et al. (2011) recommend the 8-point,
flexible approach be used to estimate crown projection area in
order to minimise errors due to deviations from canopy symmetry.
Other workers have concluded that two, orthogonal diameter measurements (4 radii) are suitable for computing crown diameter
(Hemery et al., 2005), from which crown projection area can be
subsequently derived.
The relationship between DBH and crown diameter or projected
area has been the subject of numerous papers in recent years, of
which a few exemplars will be discussed here. Unsurprisingly,
the exact form of the relationship is driven by competition with
neighbouring trees and understorey and much of this driven by
factors such as shade tolerance. Many of the relationships are
‘almost’ linear, with observed departures from linearity, for example logarithmic or square-root dependence, often at smaller DBH
(Hall et al., 1989; Hemery et al., 2005). Arzai and Aliyu (2010)
found statistically significant linear relationships between crown
diameter and DBH in some Eucalypt and other tree species from
the savana zone in Nigeria (R2 0.23–0.82), as did Sanquetta et al.
(2011b) for Araucaria (Araucaria angustifolia), Imbuia (Ocotea
porosa) and Canelas (Nectandra grandiflora) in the mid southern
Parana State of Brazil (R2 0.47–0.78). On the other hand, O’Brien
et al. (1995) observed the good species-dependent predictions
using log–log transformed data (R2 > 0.86), and Sanquetta et al.
(2011b) also improved their prediction, but only for the combined
species dataset, by a log–log transformation (R2 from 0.23 to 0.52).
Smith (2008) observed a strong linear relationship between canopy
area and trunk cross-sectional area (proportional to DBH2) for
native pecan trees grown in managed groves (Smith, 2008) and
Gill et al. (2000) observed DBH and DBH2, when used as a sole
independent variable, to be reasonably good at predicting canopy
radius in forest planted conifer trees (R2 > 0.45). Ultimately, models
relating crown projection area and DBH are genus dependant, and
often species dependent owing to the range of tree architecture
between species. Moreover, little is known about the transferability of models derived for single trees (for example in open
woodlands) as compared to clusters of trees (i.e., multiple stems)
in light of possible competition in growth (e.g. Biging and
Dobbertin, 1992; Biging and Dobbertin, 1995; Moeur 1997).
Crown projection area could therefore be used to infer DBH.
This offers an important pathway to deploy remote sensing tools
for the large-scale assessment of above ground or biomass stocks
across entire landscapes. From a remote sensing perspective,
crown projection area is more easily measured than canopy
diameter for the simple reason that the extracted crown diameter
parameter is influenced by the direction of the measurement
vector. Assuming image pixels, or objects are correctly classified
as a particular tree crown, and the tree crown is at the nadir viewpoint (or close to it), it remains to sum the pixels (of known dimensions) within that crown to determine the projected area. This is
not to say, however that crown diameter is irrelevant to the remote
estimation of DBH. Numerous workers have found strong relationships between crown diameter and DBH (for example Hall et al.,
1989; Gering and May, 1995) and given established sampling
regimes for determining crown diameter (or canopy extent) are
aimed at minimising the effect of crown asymmetry (Fleck et al.,
2011), crown diameter could potentially be derived from the remotely sensed canopy projection area measurements by applying
appropriate shape parameters such as discussed in the various
work of Nelson (1997); Grote (2003) and Fleck et al. (2011).
The present study aims to develop an allometric model for
predicting DBH of single trees and trees in clusters using the
tree/cluster characteristics of crown projection area and tree
height. The overall context of this study is remnant, native vegetation that exists in typical Australian farming landscapes. In this
study, we examine trees from the genus Eucalyptus, owing to its
overall significance in Australian landscapes (Commonwealth of
Australia, 1999) and in particular its prevalence in Australian farmscapes (Attiwill and Adams, 1996). In this study we hypothesize
that DBH can be predicted using field measurements of tree height
and crown projection area for both single trees as well as clusters
of trees. We also wish to test the hypothesis that the same model is
applicable for both single trees and isolated clusters of trees. The
definition of a cluster can be somewhat arbitrary, and proximity
of trees to one another will influence more than just competition
for growth. For example Barnes et al. (2011b) demonstrated the
extent to which the litterfall from an isolated tree will influence
soil nutrients beyond the immediate canopy envelope. Moreover,
given the context of this particular work in relation to the use of
large scale remote sensing tools to possibly infer DBH, consideration should be given to the spatial resolution of such large scale
sensing systems. To this end we define a cluster as a group of trees
with canopy envelopes within 3 m proximity to each other.
2. Materials and methods
2.1. Study area
The study site was located within the University of New England’s, ‘Newholme-Kirby SMART farm’, Armidale, New South
Wales, Australia (longitude 151°350 4000 E to 151°370 1200 E and latitude 30°260 0900 S to 30°250 120 0 S) (Fig. 1). The 662 ha site comprises
large tracts of natural eucalypt woodland and pasture cover.
Approximately one third of the study area is dense eucalypt woodland, one third open woodland, and the remainder native pasture.
Most of the study site is unimproved pasture grazed by sheep. The
soils within the study site vary from brown and yellow chromosols,
and the mean annual rainfall is 780 mm. It has a cool temperate climate with the majority of rain falling in the summer months
(National Parks and Wildlife Service, 2003; BoM, 2014).
2.2. Field measurements
High spatial resolution (15 cm), colour infrared (CIR) airborne
imagery of the study area was first used to identify single trees
and tree clusters in the field. A total of 52 tree clusters and 172
individual trees were identified (Fig. 2). Within the tree clusters
the number of stems ranged from 2 to 27 with a density ranging
from 38 to 536 stems per ha (SD 4.2 trees per cluster, 105 stems
per ha). The tree/cluster locations were extracted from the orthorectified, georeferenced imagery for subsequent field visitation,
N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132
127
Fig. 1. Location map of the study site in north eastern NSW, Australia. Source: open access data.
Fig. 2. Tree and tree cluster locations (white circles) overlaid on a grey-scale aerial
image of the field site.
aided by a DGPS (GPS PathfinderÒ Pro XRS receiver, Ranger TSC2
model, Trimble, California). The horizontal accuracy of GPS was
0.5 m allowing unambiguous identification of target trees/clusters on the ground. On-ground visitation and measurement of
selected trees and tree clusters was conducted during the period
September–December 2012.
The DBH of individual trees and those in clusters was derived
from the measured trunk circumference at 1.3 m above local
ground level. In the case of individual trees (as delineated by a single root-ball) with multiple stems at 1.3 m above the ground, the
DBH of each stem was measured and tree DBH calculated using:
DBH ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d1 þ d2 þ d3 For individual trees, the height (Ht) was measured using a
combination of laser rangefinder and clinometer (MDL LaserAce
300, Measurement Devices Ltd. Scotland, UK). Cluster height was
determined by first measuring the height of each tree within the
cluster, and the average height (Htav) calculated. In those clusters
where the individual tree crowns could not be delineated, Htav
was calculated from six measurements of the entire canopy
envelope acquired from different azimuthal viewer positions
relative to the cluster.
Canopy cover is defined as the proportion of ground covered by
the vertical projection of the tree crowns (Jennings et al., 1999). In
order to measure crown projection area (CA) of both single trees
and tree clusters, we adopted a hybrid version of the 8-point crown
projection method recommended by Fleck et al. (2011) and that
investigated by Röhle and Huber (1985). Firstly a pair of vertical
range poles were placed in the ground to delineate the edges of
the tree/cluster canopy along a cardinal direction, for example
N–S, passing through the tree stem or the estimated centre of
the cluster. The crown periphery for locating the pole positions
was located using a clinometer set to ‘vertical view’, in effect acting
as a crown mirror. The crown diameter (d) along this direction was
then measured using a laser range finder positioned at right-angles
to, and a known distance well back from, the line between the
poles. This measurement avoided errors that would otherwise be
incurred by using tapes to measure the straight-line distance
between the poles with tree stems in between. This measurement
was undertaken for six cardinal directions namely N, ENE, ESE, S,
WSW and WNW, respectively and the average diameter, which is
effectively the average crown spread, d, calculated (Sumida and
Komiyama, 1997). The crown projection area was then calculated
using:
2
ð1Þ
where d1, d2, d3 were the diameters of each stem (Pretzsch, 2009).
The DBH parameters investigated for tree clusters were the average
DBH of the individual trees within a cluster (DBHav) and the sum of
the DBH values (DBHsum).
CA ¼ pd =4
ð2Þ
The crown projection area parameters investigated for tree
clusters were the average of the individual trees within a cluster
(CAav) and the total area of the cluster (CAsum).
The information about tree species for each tree/cluster was
noted as was the number of tree stems in a cluster. The dimensions
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N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132
Table 1
The allometric models tested for crown projection area (CA, m2), tree height (Ht, m)
and DBH (m) for single trees and tree clusters. The subscript ‘av’ denotes the average
value per stem within a cluster. The subscript ‘sum’ as applied to DBH is the sum of
each value for the individual stems and when applied to CA is the size of the canopy
envelope enclosing all the trees within the cluster.
Regression models
Individual trees
Tree clusters
DBH = a0 + a1(Ht,CA)
ln(DBH) = a0 + a1ln(Ht, CA)
ln(DBH) = a0 + a1ln(Ht) + a2ln(CA)
DBHav = b0 + b1(Ht, CA)av
ln(DBHav) = b0 + b1ln(Ht, CA)av
ln(DBHav) = b0 + b1ln(Htav) + b2ln(CAav)
DBHsum = b0 + b1CAsum
ln(DBHsum) = b0 + b1ln(CAsum)
Table 2
Summary statistics for single trees; n is the number of trees used in both the model
development and validation.
Tree characteristics/species
n
Min
Max
Mean
SD
DBH
Apple box
Red gum
Stringy bark
White gum
Yellow box
23
5
51
28
65
0.34
0.5
0.36
0.36
0.28
1.65
0.83
1.33
1.92
1.47
0.840
0.690
0.839
0.911
0.807
0.310
0.151
0.221
0.391
0.244
Crown projection area (CA)
Apple box
Red gum
Stringy bark
White gum
Yellow box
26.26
70.85
42.41
36.83
9.34
750
167
413
732
683
212
136
195
230
222
183.5
41.6
100.4
164.1
132.3
Tree height (Ht)
Apple box
Red gum
Stringy bark
White gum
Yellow box
9.1
12.7
13.5
11.8
9.5
40.5
23.6
30.4
44.6
42.1
17.6
18.2
21.4
20.7
21.9
7.28
4.12
4.59
8.05
6.42
value per stem within the cluster. Similarly the subscript ‘sum’, as
applied to DBHsum is the sum of each value for the individual stems
and the parameter CAsum is the total size of the canopy envelope
enclosing all the trees within the cluster. In the case of non-normal
data distributions (Shapiro–Wilk Test), a logarithmic transformation was first carried out and the transformed data tested for species-specific variations. Linear regression models were evaluated
using the statistical software R (Studio Version 0.97.318). In all
the models, the interaction terms between the parameters were
also tested.
The data were tested for normality using Shapiro–Wilk test and
influential outliers, if any, were detected by means of Cooks distance
statistics of the residuals. Any data that had a residual Cook’s
distance value of P2 was cross-checked with the original dataset
to validate its precision and impact on the model. To satisfy the
assumptions of linear regression analysis, scatter plots of residuals
were checked for linearity, homoscedasticity and normality. The
strength of the underlying relationship of the predictor and
response variables was evaluated by analyzing the regression coefficients of the fitted models. The coefficient of determination (R2)
was used to evaluate the level of variance in DBH explained by the
variables. For each model, half of the samples, namely 86 for single
trees and 26 for tree clusters, respectively, were withheld from the
initial calibration for subsequent validation of the model. The prediction performance of each model was quantified using a mean
prediction error (MPE) given by MPE ¼ jDBHpredicted DBHactual j.
3. Results and discussions
3.1. Single trees
of five different Eucalyptus species, namely Apple Box (AB, Eucalyptus bridgesiana), Stringy Bark (SB, Eucalyptus caliginosa), Red Gum
(RG, Eucalyptus blakelyi), White Gum (WG, Eucalyptus viminalis)
and Yellow Box (YB, Eucalyptus melliodora) were sampled in this
way.
2.3. Model development and validation
1.6
1.6
1.4
1.4
1.2
1.2
DBH (m)
DBH (m)
The allometric models tested for crown projection area (CA,
m2), tree height (Ht, m) and DBH (m) for single trees and tree clusters are listed in Table 1 and follow the forms reviewed and listed
in Hall et al. (1989). The subscript ‘av’ denotes measurements for
tree clusters where DBHav, CAav and Htav are effectively the average
Table 2 lists the descriptive statistics for the single trees. With
all the species combined, the dataset was found to be non-normal,
in particular the subset comprising White Gum and Yellow
Box (Shapiro–Wilk test W = 0.97 p = 0.004). A logarithm transformation was sufficient to normalize the data (W = 0.992, p = 0.389).
Scatter plots of DBH versus individual tree height (Ht) and
crown projection area (CA) parameters are given in Fig. 3, along
with regression curves of the log-transformed models. The regression statistics are summarised in Table 3. When the models were
validated against the 86 trees retained from the data for this purpose, the log-transformed models yielded a MPE of 16 cm. A
multi-linear regression model involving both log-transformed Ht
and CA parameters combined gave a slightly improved prediction
of DBH with a MPE of 14 cm.
Interestingly, while log-transformed crown projection area on
its own explains significantly more variance in log-transformed
DBH than tree height (R2 = 0.68 compared to 0.31), both tree height
and crown projection area predict DBH with a similar MPE (16 cm).
This is a reflection of the strong inter-relationship between crown
1
0.8
0.6
0.4
0.6
0.4
R2 = 0.31
MPE= 0.16 m
0.2
1
0.8
R2 = 0.68
MPE= 0.16 m
0.2
0
0
0
10
20
30
Ht (m)
(a)
40
50
0
200
400
600
800
CA (sq m)
(b)
Fig. 3. Scatter plots of DBH versus (a) tree height (Ht) and (b) crown projection area (CA) for single trees (all species, n = 172). The regression curves are the log-transformed
models for each individual parameter (Table 5).
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N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132
Table 3
Derived regression parameters (95% confidence intervals) for single trees (n = 86). The
MPE values are derived from a separate validation dataset (n = 86).
Table 5
Derived regression parameters (95% confidence intervals) for tree clusters (n = 26).
The MPE values are derived from a separate validation dataset (n = 26).
Equation
R2
Fstat
p
MPE
(m)
Equation
R2
F-stat
p
MPE
(m)
ln(DBH) = 2.10229 + 0.61742 ln(Ht)
ln(DBH) = 2.40568 + 0.42616 ln(CA)
ln(DBH) = 2.64742 + 0.15142
ln(Ht) + 0.38002 ln(CA)
0.31
0.68
0.59
37.4
181.6
60.1
<0.0001
<0.0001
<0.0001
0.16
0.16
0.14
ln(DBHav) = 1.85617 + 0.42221 ln(Htav)
ln(DBHav) = 2.13471 + 0.335344 ln(CAav)
ln(DBHav) = 2.5813 + 0.18246
ln(Htav) + 0.31712 ln(CAav)
ln(DBHsum) = 2.17648 + 0.523622
ln(CAsum)
0.14
0.67
0.69
3.8
46.2
24.7
0.07
<0.0001
<0.0001
0.10
0.08
0.07
0.34
11.9
<0.01
0.66
800
Apple Box
R2 = 0.30
10
Red Gum
600
Stringy Bark
500
White Gum
9
R2 = 0.34
8
MPE= 0.66 m
Yellow Box
400
7
DBHsum (m)
CA (sq m)
700
300
200
100
6
5
4
3
0
5
15
25
35
45
2
Ht (m)
1
0
Fig. 4. Scatter plot of crown projection area (CA) against tree height (Ht) for single
trees (n = 172, 5 species). The solid regression line (and R2) is based on a linear
regression between the parameters.
Table 4
Summary statistics for tree clusters (all species); n is the total number of trees used
for model development and validation.
Tree characteristics
n
Min
Max
Mean
SD
DBHav (m)
Htav (m)
CAav (m2)
Number of stems
Stem density (/ha)
52
0.28
9.12
18.66
2
37.91
1.10
26.50
443.45
27
535.91
0.56
18.23
111.75
5.71
150.71
0.17
4.10
78.30
4.20
105.11
projection area and tree height. The scatter plot of crown projection area versus tree height for all species (Fig. 4) illustrates this,
with tree height explaining 30% of the variance in crown projection
area.
It can be concluded at this point that either Ht or CA could be
used to infer DBH for single trees, although based on the level of
variance explained in the DBH by CA, this latter parameter is likely
to provide better precision in predicting DBH on a tree by tree basis.
3.2. Tree clusters
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0
R2 = 0.14
MPE= 0.10 m
5.0
10.0
15.0
20.0
25.0
30.0
400
600
800
1000
1200
CAsum (sq m)
Fig. 6. Scatter plot of DBHsum versus the cluster crown projection area (CAsum) for
tree clusters (n = 52). The regression curve is the log-transformed model (Table 5).
average tree height (Htav) and average crown projection area (CAav)
are given in Fig. 5, along with the curves based on the derived
regression models. Regression statistics are summarised in Table 5.
An assessment of the datasets showed that, again DBHav was
not normally distributed (Shapiro–Wilk W = 0.953, p = 0.03147).
Once log-transformed, the parameter CAav explained a significantly
larger proportion of the variance observed in the average DBH per
stem in each cluster as compared to the average tree height within
the cluster (R2 = 0.67 compared to 0.14). Unlike the single trees,
Htav and CAav were not strongly correlated (R2 = 0.04, data not
shown), which suggests that a combination of both parameters
in a multi-linear regression model (log-transformed inputs) may
be desirable, even though this yields only modest gains in the level
of variance explained in DBHav (R2 = 0.69 compared to 0.67) and
precision in predicting values of DBHav (0.07 m compared to
0.08 m). The CAav parameter alone was found to yield a MPE of
0.08 m when predicting DBHav in the range from 0.28 to 0.84 m
(28.5% and 9.5% error, respectively)
The regression statistics for the sum of the DBH values within
the clusters (DBHsum) and the total crown projected area (CAsum)
of the clusters are also listed in Table 5 and a scatter plot of the
DBHav (m)
DBHav (m)
Table 4 lists the descriptive statistics for the tree clusters. For
the tree clusters, scatter plots of average DBH (DBHav) versus
200
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
R2 = 0.67
MPE= 0.08 m
0
50
100
150
Ht av (m)
CA av (sq m)
(a)
(b)
200
250
300
Fig. 5. Scatter plots of DBHav versus (a) average tree height (Htav) and (b) crown projection area (CAav) for tree clusters (n = 52). The regression curves are the log-transformed
models for each individual parameter (Table 5).
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the number of stems (R = 0.55). At the same time we observe the
stem density (stems/ha) – CAsum relationship (Fig. 7) to take the
form:
800
single tree envelope
Stem density (/ha)
600
stem density ¼ 6007:1 ðCAsum Þ0:652
stem density = 6007.1 x (CAsum)-0.652
R² = 0.35
400
200
0
0
200
400
600
800
1000
1200
CAsum (sq m)
Fig. 7. Scatter plot of stem density (/ha) versus total crown projection area (CAsum,
m2) for tree clusters. The dashed curve is the envelope for the single tree data,
calculated from the crown projection area (CA). The solid curve is represents the
fitted power curve to the cluster data.
Table 6
Derived regression parameters (95% confidence intervals) for a combined individual
tree–tree cluster model. The parameters DBH and CA incorporate data for both
single trees (DBH, CA) and the average values of each tree within the measured
clusters (DBHav, CAav). The parameters DBH+ and CA+ incorporate data for both single
trees (DBH, CA) and the sum of each tree within the measured clusters (DBHsum,
CAsum); n = 112. The MPE values are derived from a separate validation dataset
(n = 112). The percentages in brackets indicate the mean relative prediction error.
Equation
R2
F-stat
p
MPE (m)
ln(DBH) = 2.46441 + 0.426425
ln(CA)
ln(DBH+) = 3.73606 + 0.70211
ln(CA+)
0.71
265.0
<0.0001
0.59
157.0
<0.0001
0.13
(17%)
0.43
(31%)
of DBHsum versus CAsum is given in Fig. 6. Again a log transformation provided the best regression model, although the total crown
projected area only explained 34% of the variance in the total DBH
(R2 = 0.34) with a MPE of 0.66 m where the DBHsum ranged from
1.06 to 8.91 m (63% and 7.4% error, respectively).
The fact that the net crown projected area for a cluster is only
weakly related to the sum of DBH values within that cluster is
evidence of competition effects (Bella, 1971; White 1981). Fig. 6
exhibits increasing scatter at higher CAsum. In the clusters investigated in this work, a higher CAsum is correlated with an increase in
The larger tree clusters in our farmscape are made up of fewer,
older trees, consistent with the self-thinning rule discussed by
White (1981). Superimposed on Fig. 7 is the curve generated by
the single tree data; the so-called ‘single tree envelope’. Here the
stem density (/ha) for the single trees was calculated using the
ratio 10,000 m2/CA, effectively assuming the inter-stem distance
is limited to the unperturbed canopy envelope of the individual
trees themselves; namely zero influence overlap (Bella, 1971). It
is evident that in our scattered tree clusters, the self-thinning
mechanism is resulting in trees expressing similar canopy dimensions as isolated trees. With these older trees there is expected to
be an increase in variability in canopy extent due to effects of
weathering, pests and disease (for example Landsberg and
Ohmart, 1989; Elliott et al., 1993; Neumann, 1993; Köstner et al.,
2002) and this may also explain the increasing variability observed
at higher values of CAsum. Summing the DBH in Fig. 6 effectively
accumulates the effects of individual tree competition, and the
resulting departure from the behaviour of single, isolated trees.
The act of taking average (per stem) crown projected area and
average DBH in a cluster most likely reduces this accumulating
error.
3.3. Combining both single trees and tree cluster datasets
A comparison of the single tree and tree cluster regression models based on crown projected area alone (DBH = f(CA); Tables 3 and
5) suggests the parameters derived for single trees, and those
derived on a per-tree basis from tree clusters (stem numbers
ranging from 2 to 27), are similar. A comparison of the derived
regression equations for the individual trees and tree clusters
shows the interaction terms to be non-significant (p = 0.79). This
implies that the slopes and the intercepts of the two regression
models do not differ significantly; statistically the two equations
generate similar estimates of DBH.
The log-transformed regression models derived using the
combined datasets are given in Table 6 and scatter plots of the
model-predicted versus actual DBH are given in Fig. 8. Here the
models are based on the log-transformed DBH (DBH, DBH+) and
the log-transformed crown projection area (CA, CA+) values. The
superscript ‘’ denotes the fact that each parameter involves data
for both single trees (DBH, CA), and the average values of each tree
1.6
4
single trees
tree clusters
single trees
tree clusters
3.5
DBH+ predicted (m)
1.4
DBH* predicted (m)
ð3Þ
1.2
1
0.8
0.6
0.4
R2 = 0.71
MPE= 0.13 m
0.2
3
2.5
2
1.5
1
R2 = 0.59
MPE= 0.43 m
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
0
0.5
1
1.5
2
2.5
DBH* actual (m)
DBH+ actual (m)
(a)
(b)
3
3.5
4
Fig. 8. Scatter plots of (a) measured versus predicted DBH⁄ (comprising DBH of single trees and DBHav of tree clusters) using both single tree CA and tree cluster CAav data in
the regression model, and (b) measured versus predicted DBH+ (comprising both DBH of single trees and DBHsum of tree clusters) using both single tree CA and tree cluster
CAsum data in the regression model (n = 112). Solid lines indicate 1:1 equivalence between predicted and actual values.
N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132
within the measured clusters (DBHav, CAav). The superscipt ‘+’
denotes the fact that each parameter involves data for both single
trees (DBH, CA), and the net cluster projected canopy area and sum
of DBH of each tree within the measured clusters (DBHsum, CAsum).
Both models were created using a random selection of half the single and cluster data for calibration (n = 112) and the remaining
data withheld for subsequent validation (n = 112).
The data for the single tree and tree cluster datasets are shown
as separate symbols. The effect of the error in the DBHsum versus
CAsum, discussed earlier is again evident in Fig. 8(b), and it is clear
that a single model for both individual trees and tree clusters
breaks down when the DBHsum values are included in DBH+. In
the DBH – CA model that incorporates both DBHav and CAav values of clusters with the data for single trees, the model performs
well; it is noteworthy to observe the tree cluster data to be distributed amongst the single tree data points. The MPE in DBH for the
tree-averaged data is 0.13 m. While encouraging, this equates to
a mean relative prediction error approximately 17% of the DBH
values encountered in the sampling. An investigation of the relative prediction error on a sample by sample basis did not show
any systematic trend towards increasing prediction error with
increasing DBH except for values exceeding 1.2 m.
4. Conclusions
Simple regression models involving crown projection area of
Eucalyptus trees (six species), both isolated and in clusters of up
to 27 stems ranging from 38 to 536 stems per ha), explained
67% and 68%, respectively of the variance in stem DBH. A single
model involving both single trees and the tree clusters to predict
average stem DBH had similar explanatory power (R2 = 0.71) and
yielded a mean prediction error in average DBH per stem of
±13 cm. We conclude that it is sufficient to use crown projection
area to infer DBH for these species (and these stem densities and
stem numbers). While the results appear encouraging, we
acknowledge that the landscape investigated here was only
662 ha, and while encapsulating considerable variation in soils,
elevation and aspect (for example as reported in Garraway and
Lamb, 2011), it would be expected that the robustness and precision of the model could potentially be enhanced by including
other landscape parameters. This is the subject of further work.
Nevertheless, the use of larger scale data sources such as airborne
or satellite imagery to infer DBH from derived values of crown
projection area for both single Eucalypt trees and clusters up to
27 stems (between 22 and 536 stems per ha) appears feasible
and worthy of further investigation. Admittedly, the single model
developed here does require knowledge of the number of stems
within a given tree cluster. However the increasing use of other
sensing technologies like LIDAR to both infer total crown projection area and delineate and count the number of stems within a
canopy (for example Popescu et al., 2004) potentially offers the
means to achieve this.
Acknowledgments
This work was partially funded by the CRC for Spatial Information (CRCSI), established and supported under the Australian
Government Cooperative Research Centres Programme. One of
the authors (NKV) wishes to acknowledge the receipt of a
Postgraduate ‘Top-up’ Scholarship from the CRCSI. We would like
to thank Ashley Saint and Derek Schneider (UNE-PARG) for their
assistance in conducting the field work and Drs Gregory Falzon
(UNE-C4D), Robin Dobos (NSW DPI) and Jackie Reid (UNE) for their
helpful comments on the statistical analysis.
131
References
Arzai, A.H., Aliyu, B.S., 2010. The relationship between canopy width, height and
trunk size in some tree species growing in the savana zone of Nigeria. Bajopas 3
(1), 260–263.
Attiwill, P.M., Adams, M.A. (Eds.), 1996. Nutrition of Eucalypts. CSIRO Publishing,
pp. 448.
Baffetta, F., Corona, P., Fattorini, L., 2011. Assessing the attributes of scattered trees
outside the forest by a multi-phase sampling strategy. Forest 84 (3), 315–325.
Barnes, P., Wilson, B.R., Trotter, M.G., Lamb, D.W., Reid, N., Koen, T., 2011a. The
patterns of grazed pasture associated with scattered trees across an Australian
temperate landscape: an investigation of pasture quantity and quality. Rangel. J.
33, 121–130.
Barnes, P., Wilson, B.R., Reid, N., Koen, T.B., Lockwood, P., Lamb, D.W., 2011b.
Litterfall and associated nutrient pools extend beyond the canopy of scattered
eucalypt trees in temperate pastures. Plant Soil. 345, 339–352.
Beets, P.N., Kimberley, M.O., Oliver, G.R., Pearce, S.H., Graham, J.D., Brandon, A.,
2012. Allometric equations for estimating carbon stocks in natural forest in
New Zealand. Forests 3, 818–839.
Bella, I.E., 1971. A new competition model for individual trees. Forest Sci. 17, 364–
372.
Biging, G.S., Dobbertin, M., 1992. A comparison of distance-dependent competition
measures for height and basal area growth of individual conifer trees. Forest Sci.
38 (3), 695–720.
Biging, G.S., Dobbertin, M., 1995. Evaluation of competition indices in individual
tree growth models. Forest Sci. 41 (2), 360–377.
Bird, P., Bicknell, D., Bulman, P., Burke, S., Leys, J., Parker, J., 1992. The role of shelter
in Australia for protecting soils, plants and livestock. Agroforest Syst. 20, 59–86.
BoM., 2014. Bureau of Meteorology Climate Statistics for Australian Locations:
Armidale
airport.
<http://www.bom.gov.au/climate/averages/tables/
cw_056238.s html> (accessed 03.03.14).
Cade, B.S., 1997. Comparison of tree basal area and canopy cover in habitat models:
subalpine forest. J. Wildlife Manage. 61 (2), 326–335.
Commonwealth of Australia, 1999. Comprehensive Regional Assessment, World
Heritage Sub-theme: Eucalypt-dominated vegetation. Report of the Expert
Workshop, Canberra, 8 & 9 March 1999. (Commonwealth of Australia), pp. 105.
Elliott, H.J., Bashford, R., Greener, A., 1993. Effects of defoliation by the leaf beetle,
chrysophtharta bimaculata, on growth of Eucalyptus regnans plantations in
Tasmania. Aust. Forest. 56, 22–26.
Fleck, S., Mölder, I., Jacob, M., Gebauer, T., Jungkunst, H.F., Leuschner, C., 2011.
Comparison of conventional eight-point crown projections with LIDAR-based
virtual crown projections in a temperate old-growth forest. Annal. Forest Sci.
68, 1173–1185.
Franklin, J., Strahler, A.H., 1988. Invertible canopy reflectance modeling of
vegetation structure in semi-arid woodland. IEEE Trans. Geosci. Rem. Sens.
26, 809–825.
Garraway, E., Lamb, D.W., 2011. Delineating spatial variations in soil carbon using
remotely sensed data: a New England case study. In: Patterson, R., (Ed.).
Proceedings of the 2011 Regional Convention, Newcastle Division, Engineers
Australia, (Newcastle Division Engineers Australia: Newcastle West, Australia),
University of New England, Armidale. NSW 16–18th September 2011, pp. 45–
52, (ISBN 978-0-85825-870-9).
Gering, L.R., May, D.M., 1995. The relationship of diameter at breast height and
crown diameter for four species groups in Hardin county. Tennessee. Southern J.
Appl. Forest 19 (4), 177–181.
Gill, S.J., Biging, G.S., Murphy, E.C., 2000. Modeling conifer tree crown radius and
estimating canopy cover. Forest Ecol. Manage. 126, 405–416.
Goh, K.M., Mansur, I., Mead, D.J., Sweet, G.B., 1996. Biological nitrogen fixing
capacity and biomass production of different understorey pastures in a Pinus
radiata-pasture agroforestry system in New Zealand. Agroforest. Syst. 34, 33–
49.
Grote, R., 2003. Estimation of crown radii and crown projection area from stem size
and tree position. Annal. Forest Sci. 60, 393–402.
Hall, R.J., Morton, R.T., Nesby, R.N., 1989. A Comparison of existing models for DBH
estimation for large–scale photos. Forest Chron., 114–116.
Hemery, G.E., Savill, P.S., Pryor, S.N., 2005. Applications of the crown diameter-stem
diameter relationship for different species of broadleaved trees. Forest Ecol.
Manage. 215, 285–294.
Houghton, R.A., 2005. Aboveground forest biomass and the global carbon balance.
Global Change Biol. 11, 945–958.
Jennings, S.B., Brown, N.D., Sheil, D., 1999. Assessing forest canopies and
understorey illumination : canopy closure, canopy cover and other measures.
Forestry 72 (1), 59–73.
Köstner, B., Falge, E., Tenhunen, J.D., 2002. Age-related effects on leaf area/
sapwood area relationships, canopy transpiration and carbon gain of Norway
spruce stands (Picea abies) in the Fichtelgebirge. Germany Tree Physiol. 22,
567–574.
Kuyah, S., Dietz, J., Muthuri, C., Jamnadass, R., Mwangi, P., Coe, R., Neufeldt, H., 2012.
Allometric equations for estimating biomass in agricultural landscapes: I.
Aboveground biomass. Ag. Ecosys. Environ. 158, 216–224.
Landsberg, J., Ohmart, C., 1989. Levels of insect defoliation in forests: patterns and
concepts. Trends Res. Ecol. Evol. 4, 96–100.
Leckie, D.G., Gougeon, F., Walsworth, N., Paradine, D., 2003. Stand delineation and
composition estimation using semi automated individual tree+ crown analysis.
Rem. Sens. Environ. 85, 355–369.
132
N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132
Lee, A.C., Lucas, R.M., 2007. A Lidar-derived canopy density model for tree stem and
crownmapping in Australian forests. Rem. Sens. Environ. 111 (4), 493–518.
Makungwa, S.D., Chittock, A., Skole, D.L., Kanyama-Phiri, G.Y., Woodhouse, I.H.,
2013. Allometry for biomass estimation in jatropha trees planted as boundary
hedge in farmers’ fields. Forests 4, 218–233.
Manning, A.D., Fischer, J., Lindenmayer, D.B., 2006. Scattered trees are keystone
structures. Implications conserv. Biol. Cons. 132, 311–321.
Moeur, M., 1997. Spatial models of competition and gap dynamics in old growth
Tsuga heterophylla/Thuja plicata forests. Forest Ecol. Manage. 94, 175–186.
National Parks and Wildlife Service, 2003. Duval Nature Reserve, Plan of
Management. (Commonwealth of Australia) ISBN 0 7313 6600 X, pp. 17.
Nelson, R., 1997. Modelling forest canopy heights. The effects of canopy shape. Rem.
Sens. Environ. 60, 327–334.
Neumann, F.G., 1993. Insect pest problems of eucalypt plantations in Australia. 3.
Victoria. Aust. Forest 56, 370–374.
O’Brien, S.T., Hubbell, S.B., Spiro, P., Condit, R., Foster, R.B., 1995. Diameter, height,
crown and age relationships in eight neotropical tree species. Ecol. 76 (6),
1926–1939.
Oliver, I., Pearce, S., Greenslade, P.J.M., Britton, D.R., 2006. Contribution of paddock
trees to the conservation of terrestrial invertebrate biodiversity within grazed
native pastures. Australian Ecol. 31, 1–12.
Popescu, S.C., Wynne, R.H., 2004. Seeing the trees in the forest: using lidar and
multispectral data fusion with local filtering and variable window size for
estimating tree height. PERS 70 (5), 589–604.
Popescu, S.C., Wynne, R.H., Scrivani, J.A., 2004. Fusion of small-footprint Lidar and
multispectral data to estimate plot-level volume and biomass in deciduous and
pine forests in Virginia. USA Forest Sci. 50 (4), 551–564.
Pretzsch, H., 2009. Forest Dynamics, Growth and Yield: From Measurement to
Model. Springer, pp. 664.
Röhle, H., Huber, W., 1985. Untersuchungen zur Methode der Ablotung von
Kronenradien und der Berechnung von Kronengrundflächen. Forstarchiv 56,
238–243.
Sanquetta, C.R., Dalla Corte, A.P., Silva, F.S., 2011a. Biomass expansion factor and
root-to-shoot ratio for pinus in Brazil. Carbon Bal. Manage. 6 (6), 1–8.
Sanquetta, C.R., Dalla Corte, A.P., Jacon, A.D., 2011b. Crown area and trunk diameter
relationship for tree species at a mixed-araucaria natural forest in the midsouthern Parana state, Brazil. Floresta 41 (1), 63–72.
Smith, M.W., 2008. Relationship of trunk size to selected canopy size parameters for
native pecan trees. HortScience 43 (3), 784–786.
Snowdon, P., Raison, J., Keith, Heather., Ritson, P., Grierson, P., Adams, M., Montagu,
K., Bi, Hui-quan., Burrows, W., Eamus, D., 2002. Protocol for Sampling Tree and
Stand Biomass. National Carbon Accounting System, Technical Report No. 31
(Australian Greenhouse Office).
Soto-Pinto, L., Anzueto, M., Mendoza, J., Ferrer, G.J., de Jong, B., 2010. Carbon
sequestration through agroforestry in indigenous communities of Chiapas.
Mexico. Agroforest Syst. 78, 39–51.
Sumida, A., Komiyama, A., 1997. Crown spread patterns for five deciduous broadleaved woody species: ecological significance of the retention patterns of larger
branches. Annal. Biol. 80, 759–766.
Ter-Mikaelian, M.T., Korzukhin, M.D., 1997. Biomass equations for sixty-five North
American tree species. Forest Ecol. Manage. 97, 1–24.
White, J., 1981. The allometric interpretation of the self-thinning rule. J. Theoret.
Biol. 89, 475–500.
Wilson, B.R., 2002. Influence of scattered paddock trees on surface soil properties: a
case study of the Northern Tablelands of NSW. Ecol. Manage. Restor. 3, 211–
219.