Forest Ecology and Management 326 (2014) 125–132

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Forest Ecology and Management

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An allometric model for estimating DBH of isolated and clusteredEucalyptus trees from measurements of crown projection area

http://dx.doi.org/10.1016/j.foreco.2014.04.0030378-1127/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: Cooperative Research Centre for Spatial Information(CRCSI), University of New England, Armidale, NSW 2351, Australia. Tel.: +61 26773 3565.

E-mail address: dlamb@une.edu.au (D.W. Lamb).

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, Australiab Precision Agriculture Research Group and School of Science and Technology, University of New England, Armidale, NSW 2351, Australiac School of Environmental and Rural Science, University of New England, Armidale, NSW 2351, Australia

a r t i c l e i n f o

Article history:Received 26 March 2014Received in revised form 3 April 2014Accepted 4 April 2014

Keywords:Diameter at breast heightCrown projection areaScattered treesTree clustersEucalyptusFarm land

a b s t r a c t

Owing to its relevance to remotely-sensed imagery of landscapes, this paper investigates the ability toinfer diameter at breast height (DBH) for five species of Australian native Eucalyptus from measurementsof tree height and crown projection area. In this study regression models were developed for both singletrees 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 projectionarea and tree height were strongly correlated for single trees, and the log-transformed crown projectionarea explained the most variance in DBH (R2 = 0.68, mean prediction error ±16 cm). Including tree heightas a descriptor did not significantly alter the model performance and is a viable alternative to using crownprojection area. The total crown projection area of tree clusters explained only 34% of the variance in thetotal (sum of) the DBH within the clusters. However average crown projection area per stem of entire treeclusters explained 67% of the variance in the average (per stem) DBH of the constituent trees with a meanprediction error ±8 cm. Both the single tree and tree cluster models were statistically similar and a com-bined model to predict average stem DBH yielded R2 = 0.71 with a mean prediction error (average DBHper stem) of ±13 cm within the range of 0.28–0.84 m. A single model to infer DBH for both single treesand clusters comprising up to 27 stems offers a pathway for using remote sensing to infer DBH provideda means of determining the number of stems within cluster boundaries is included.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Much of eastern Australia is characterised by diverse farminglandscapes, or ‘farmscapes’ containing a range of land-use systemsincluding crops, native and sown pasture, remnant vegetation andtrees at various densities and configurations. Native eucalyptustrees, both individual and in clusters are an important feature ofAustralian farmscapes, and provide shade and shelter for livestock(Bird et al., 1992). They also have considerable value through theirinfluence on soils (Wilson, 2002; Goh et al., 1996; Barnes et al.,2011a), biodiversity (Oliver et al., 2006) and their indirect role inpasture quantity and quality (Barnes et al., 2011b). Scattered treesoutside the forest (STOF) contribute significantly to above andbelow-ground carbon stocks in these landscapes (Baffetta et al.,2011; Soto-Pinto et al., 2010). A summary of the role of scattered

trees in landscapes is given in Manning et al. (2006). Theassessment of biomass in eucalypt systems has, to date, beenlargely restricted to plantation forestry systems. However thereis also a growing need to assess carbon and biomass stocks acrossour farmscapes in order to fully quantify carbon storage change inresponse to management and provide evidence-based support forcarbon inventory and ultimately carbon trading. Such assessmentsmust also include scattered native trees.

Destructive sampling is considered the most reliable methodfor determining biomass in grass and shrub vegetation but it israrely used for agroforest ecosystems (Snowdon et al., 2002).The biomass of large vegetative structures like forests and openwoodland is usually estimated by applying ratio or regressionmethods using empirical equations and allometric models(Snowdon et al., 2002; Houghton, 2005; Makungwa et al.,2013). Trunk diameter, more formally known as diameter atbreast height (DBH), is observed to be closely related to treebiomass and carbon stocks (Ter-Mikaelian and Korzukhin, 1997;Snowdon et al., 2002; Sanquetta et al., 2011a; Kuyah et al.,2012; Beets et al., 2012).

126 N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132

DBH is an important tree characteristic in its own right and isstraight forward to measure on the ground. The relationshipsbetween tree canopy characteristics such as diameter, projectionarea and coverage, and DBH is of considerable interest as DBHcan then be used to infer canopy attributes, for example to quantifycanopy competition in response to planting (stem) density, growthpotential or habitat modelling (Bella, 1971; Cade 1997; Grote2003). Of course, large-scale collection of DBH data can be timeconsuming and now satellite or airborne remote sensing, includingLIDAR can be used to infer canopy characteristics such as crownprojection 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 fromthe ground involves measuring the crown projection across differ-ent angular segments of the canopy. These angular segments caneither be based on fixed (pre-set) directions (Röhle and Huber,1985), or in the case of highly asymmetric canopies the directionscan be adapted to adequately characterise the actual tree dimen-sions (Hemery et al., 2005; Fleck et al., 2011). Based on an investi-gation involving 161 trees in an old-growth forest (approx. density392 stems per hectare), Fleck et al. (2011) recommend the 8-point,flexible approach be used to estimate crown projection area inorder to minimise errors due to deviations from canopy symmetry.Other workers have concluded that two, orthogonal diameter mea-surements (4 radii) are suitable for computing crown diameter(Hemery et al., 2005), from which crown projection area can besubsequently derived.

The relationship between DBH and crown diameter or projectedarea has been the subject of numerous papers in recent years, ofwhich a few exemplars will be discussed here. Unsurprisingly,the exact form of the relationship is driven by competition withneighbouring trees and understorey and much of this driven byfactors such as shade tolerance. Many of the relationships are‘almost’ linear, with observed departures from linearity, for exam-ple 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 crowndiameter and DBH in some Eucalypt and other tree species fromthe savana zone in Nigeria (R2 0.23–0.82), as did Sanquetta et al.(2011b) for Araucaria (Araucaria angustifolia), Imbuia (Ocoteaporosa) and Canelas (Nectandra grandiflora) in the mid southernParana State of Brazil (R2 0.47–0.78). On the other hand, O’Brienet al. (1995) observed the good species-dependent predictionsusing log–log transformed data (R2 > 0.86), and Sanquetta et al.(2011b) also improved their prediction, but only for the combinedspecies dataset, by a log–log transformation (R2 from 0.23 to 0.52).Smith (2008) observed a strong linear relationship between canopyarea and trunk cross-sectional area (proportional to DBH2) fornative pecan trees grown in managed groves (Smith, 2008) andGill et al. (2000) observed DBH and DBH2, when used as a soleindependent variable, to be reasonably good at predicting canopyradius in forest planted conifer trees (R2 > 0.45). Ultimately, modelsrelating crown projection area and DBH are genus dependant, andoften species dependent owing to the range of tree architecturebetween species. Moreover, little is known about the transferabil-ity of models derived for single trees (for example in openwoodlands) as compared to clusters of trees (i.e., multiple stems)in light of possible competition in growth (e.g. Biging andDobbertin, 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 toolsfor the large-scale assessment of above ground or biomass stocksacross entire landscapes. From a remote sensing perspective,crown projection area is more easily measured than canopydiameter for the simple reason that the extracted crown diameterparameter is influenced by the direction of the measurement

vector. Assuming image pixels, or objects are correctly classifiedas a particular tree crown, and the tree crown is at the nadir view-point (or close to it), it remains to sum the pixels (of known dimen-sions) within that crown to determine the projected area. This isnot to say, however that crown diameter is irrelevant to the remoteestimation of DBH. Numerous workers have found strong relation-ships between crown diameter and DBH (for example Hall et al.,1989; Gering and May, 1995) and given established samplingregimes for determining crown diameter (or canopy extent) areaimed at minimising the effect of crown asymmetry (Fleck et al.,2011), crown diameter could potentially be derived from the remo-tely sensed canopy projection area measurements by applyingappropriate shape parameters such as discussed in the variouswork of Nelson (1997); Grote (2003) and Fleck et al. (2011).

The present study aims to develop an allometric model forpredicting DBH of single trees and trees in clusters using thetree/cluster characteristics of crown projection area and treeheight. The overall context of this study is remnant, native vegeta-tion that exists in typical Australian farming landscapes. In thisstudy, we examine trees from the genus Eucalyptus, owing to itsoverall significance in Australian landscapes (Commonwealth ofAustralia, 1999) and in particular its prevalence in Australian farm-scapes (Attiwill and Adams, 1996). In this study we hypothesizethat DBH can be predicted using field measurements of tree heightand crown projection area for both single trees as well as clustersof trees. We also wish to test the hypothesis that the same model isapplicable for both single trees and isolated clusters of trees. Thedefinition of a cluster can be somewhat arbitrary, and proximityof trees to one another will influence more than just competitionfor growth. For example Barnes et al. (2011b) demonstrated theextent to which the litterfall from an isolated tree will influencesoil nutrients beyond the immediate canopy envelope. Moreover,given the context of this particular work in relation to the use oflarge scale remote sensing tools to possibly infer DBH, consider-ation should be given to the spatial resolution of such large scalesensing systems. To this end we define a cluster as a group of treeswith 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 Eng-land’s, ‘Newholme-Kirby SMART farm’, Armidale, New SouthWales, Australia (longitude 151�3504000E to 151�3701200E and lati-tude 30�260090 0S to 30�250120 0S) (Fig. 1). The 662 ha site compriseslarge tracts of natural eucalypt woodland and pasture cover.Approximately one third of the study area is dense eucalypt wood-land, one third open woodland, and the remainder native pasture.Most of the study site is unimproved pasture grazed by sheep. Thesoils within the study site vary from brown and yellow chromosols,and the mean annual rainfall is 780 mm. It has a cool temperate cli-mate 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) airborneimagery of the study area was first used to identify single treesand tree clusters in the field. A total of 52 tree clusters and 172individual trees were identified (Fig. 2). Within the tree clustersthe number of stems ranged from 2 to 27 with a density rangingfrom 38 to 536 stems per ha (SD 4.2 trees per cluster, 105 stemsper ha). The tree/cluster locations were extracted from the ortho-rectified, georeferenced imagery for subsequent field visitation,

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 aerialimage of the field site.

N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132 127

aided by a DGPS (GPS Pathfinder� Pro XRS receiver, Ranger TSC2model, Trimble, California). The horizontal accuracy of GPS was�0.5 m allowing unambiguous identification of target trees/clus-ters on the ground. On-ground visitation and measurement ofselected trees and tree clusters was conducted during the periodSeptember–December 2012.

The DBH of individual trees and those in clusters was derivedfrom the measured trunk circumference at 1.3 m above localground level. In the case of individual trees (as delineated by a sin-gle root-ball) with multiple stems at 1.3 m above the ground, theDBH of each stem was measured and tree DBH calculated using:

DBH ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid1 þ d2 þ d3 � � �

pð1Þ

where d1, d2, d3 were the diameters of each stem (Pretzsch, 2009).The DBH parameters investigated for tree clusters were the averageDBH of the individual trees within a cluster (DBHav) and the sum ofthe DBH values (DBHsum).

For individual trees, the height (Ht) was measured using acombination of laser rangefinder and clinometer (MDL LaserAce300, Measurement Devices Ltd. Scotland, UK). Cluster height wasdetermined by first measuring the height of each tree within thecluster, and the average height (Htav) calculated. In those clusterswhere the individual tree crowns could not be delineated, Htav

was calculated from six measurements of the entire canopyenvelope acquired from different azimuthal viewer positionsrelative to the cluster.

Canopy cover is defined as the proportion of ground covered bythe vertical projection of the tree crowns (Jennings et al., 1999). Inorder to measure crown projection area (CA) of both single treesand tree clusters, we adopted a hybrid version of the 8-point crownprojection method recommended by Fleck et al. (2011) and thatinvestigated by Röhle and Huber (1985). Firstly a pair of verticalrange poles were placed in the ground to delineate the edges ofthe tree/cluster canopy along a cardinal direction, for exampleN–S, passing through the tree stem or the estimated centre ofthe cluster. The crown periphery for locating the pole positionswas located using a clinometer set to ‘vertical view’, in effect actingas a crown mirror. The crown diameter (d) along this direction wasthen measured using a laser range finder positioned at right-anglesto, and a known distance well back from, the line between thepoles. This measurement avoided errors that would otherwise beincurred by using tapes to measure the straight-line distancebetween the poles with tree stems in between. This measurementwas undertaken for six cardinal directions namely N, ENE, ESE, S,WSW and WNW, respectively and the average diameter, which iseffectively the average crown spread, d, calculated (Sumida andKomiyama, 1997). The crown projection area was then calculatedusing:

CA ¼ pd2=4 ð2Þ

The crown projection area parameters investigated for treeclusters 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 wasnoted as was the number of tree stems in a cluster. The dimensions

Table 1The 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 averagevalue per stem within a cluster. The subscript ‘sum’ as applied to DBH is the sum ofeach value for the individual stems and when applied to CA is the size of the canopyenvelope enclosing all the trees within the cluster.

Regression models

Individual trees Tree clusters

DBH = a0 + a1�(Ht,CA) DBHav = b0 + b1�(Ht, CA)av

ln(DBH) = a0 + a1�ln(Ht, CA) ln(DBHav) = b0 + b1�ln(Ht, CA)av

ln(DBH) = a0 + a1�ln(Ht) + a2�ln(CA) ln(DBHav) = b0 + b1�ln(Htav) + b2�ln(CAav)DBHsum = b0 + b1�CAsum

ln(DBHsum) = b0 + b1�ln(CAsum)

Table 2Summary statistics for single trees; n is the number of trees used in both the modeldevelopment and validation.

Tree characteristics/species n Min Max Mean SD

DBHApple box 23 0.34 1.65 0.840 0.310Red gum 5 0.5 0.83 0.690 0.151Stringy bark 51 0.36 1.33 0.839 0.221White gum 28 0.36 1.92 0.911 0.391Yellow box 65 0.28 1.47 0.807 0.244

Crown projection area (CA)Apple box 26.26 750 212 183.5Red gum 70.85 167 136 41.6Stringy bark 42.41 413 195 100.4White gum 36.83 732 230 164.1Yellow box 9.34 683 222 132.3

Tree height (Ht)Apple box 9.1 40.5 17.6 7.28Red gum 12.7 23.6 18.2 4.12Stringy bark 13.5 30.4 21.4 4.59White gum 11.8 44.6 20.7 8.05Yellow box 9.5 42.1 21.9 6.42

128 N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132

of five different Eucalyptus species, namely Apple Box (AB, Eucalyp-tus 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 thisway.

2.3. Model development and validation

The allometric models tested for crown projection area (CA,m2), tree height (Ht, m) and DBH (m) for single trees and tree clus-ters are listed in Table 1 and follow the forms reviewed and listedin Hall et al. (1989). The subscript ‘av’ denotes measurements fortree clusters where DBHav, CAav and Htav are effectively the average

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 10 20 30 40 50

DB

H (

m)

Ht (m)

R2 = 0.31MPE= 0.16 m

Fig. 3. Scatter plots of DBH versus (a) tree height (Ht) and (b) crown projection area (CA)models for each individual parameter (Table 5).

value per stem within the cluster. Similarly the subscript ‘sum’, asapplied to DBHsum is the sum of each value for the individual stemsand the parameter CAsum is the total size of the canopy envelopeenclosing all the trees within the cluster. In the case of non-normaldata distributions (Shapiro–Wilk Test), a logarithmic transforma-tion was first carried out and the transformed data tested for spe-cies-specific variations. Linear regression models were evaluatedusing the statistical software R (Studio Version 0.97.318). In allthe models, the interaction terms between the parameters werealso tested.

The data were tested for normality using Shapiro–Wilk test andinfluential outliers, if any, were detected by means of Cooks distancestatistics of the residuals. Any data that had a residual Cook’sdistance value of P2 was cross-checked with the original datasetto validate its precision and impact on the model. To satisfy theassumptions of linear regression analysis, scatter plots of residualswere checked for linearity, homoscedasticity and normality. Thestrength of the underlying relationship of the predictor andresponse variables was evaluated by analyzing the regression coef-ficients of the fitted models. The coefficient of determination (R2)was used to evaluate the level of variance in DBH explained by thevariables. For each model, half of the samples, namely 86 for singletrees and 26 for tree clusters, respectively, were withheld from theinitial calibration for subsequent validation of the model. The pre-diction performance of each model was quantified using a meanprediction error (MPE) given by MPE ¼ jDBHpredicted � DBHactualj.

3. Results and discussions

3.1. Single trees

Table 2 lists the descriptive statistics for the single trees. Withall the species combined, the dataset was found to be non-normal,in particular the subset comprising White Gum and YellowBox (Shapiro–Wilk test W = 0.97 p = 0.004). A logarithm transfor-mation was sufficient to normalize the data (W = 0.992, p = 0.389).

Scatter plots of DBH versus individual tree height (Ht) andcrown projection area (CA) parameters are given in Fig. 3, alongwith regression curves of the log-transformed models. The regres-sion statistics are summarised in Table 3. When the models werevalidated against the 86 trees retained from the data for this pur-pose, the log-transformed models yielded a MPE of 16 cm. Amulti-linear regression model involving both log-transformed Htand CA parameters combined gave a slightly improved predictionof DBH with a MPE of 14 cm.

Interestingly, while log-transformed crown projection area onits own explains significantly more variance in log-transformedDBH than tree height (R2 = 0.68 compared to 0.31), both tree heightand crown projection area predict DBH with a similar MPE (16 cm).This is a reflection of the strong inter-relationship between crown

(b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 200 400 600 800

DB

H (

m)

CA (sq m)

R2 = 0.68MPE= 0.16 m

for single trees (all species, n = 172). The regression curves are the log-transformed

Table 3Derived regression parameters (95% confidence intervals) for single trees (n = 86). TheMPE values are derived from a separate validation dataset (n = 86).

Equation R2 F-stat

p MPE(m)

ln(DBH) = �2.10229 + 0.61742 ln(Ht) 0.31 37.4 <0.0001 0.16ln(DBH) = �2.40568 + 0.42616 ln(CA) 0.68 181.6 <0.0001 0.16ln(DBH) = �2.64742 + 0.15142

ln(Ht) + 0.38002 ln(CA)0.59 60.1 <0.0001 0.14

0

100

200

300

400

500

600

700

800

5 15 25 35 45

CA

(sq

m)

Ht (m)

Apple Box

Red Gum

Stringy Bark

White Gum

Yellow Box

R2 = 0.30

Fig. 4. Scatter plot of crown projection area (CA) against tree height (Ht) for singletrees (n = 172, 5 species). The solid regression line (and R2) is based on a linearregression between the parameters.

Table 4Summary statistics for tree clusters (all species); n is the total number of trees usedfor model development and validation.

Tree characteristics n Min Max Mean SD

DBHav (m) 52 0.28 1.10 0.56 0.17Htav (m) 9.12 26.50 18.23 4.10CAav (m2) 18.66 443.45 111.75 78.30Number of stems 2 27 5.71 4.20Stem density (/ha) 37.91 535.91 150.71 105.11

Table 5Derived regression parameters (95% confidence intervals) for tree clusters (n = 26).The MPE values are derived from a separate validation dataset (n = 26).

Equation R2 F-stat p MPE(m)

ln(DBHav) = �1.85617 + 0.42221 ln(Htav) 0.14 3.8 0.07 0.10ln(DBHav) = �2.13471 + 0.335344 ln(CAav) 0.67 46.2 <0.0001 0.08ln(DBHav) = �2.5813 + 0.18246

ln(Htav) + 0.31712 ln(CAav)0.69 24.7 <0.0001 0.07

ln(DBHsum) = �2.17648 + 0.523622ln(CAsum)

0.34 11.9 <0.01 0.66

0

1

2

3

4

5

6

7

8

9

10

0 200 400 600 800 1000 1200

DB

Hsu

m(m

)CAsum (sq m)

R2 = 0.34

MPE= 0.66 m

Fig. 6. Scatter plot of DBHsum versus the cluster crown projection area (CAsum) fortree clusters (n = 52). The regression curve is the log-transformed model (Table 5).

N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132 129

projection area and tree height. The scatter plot of crown projec-tion area versus tree height for all species (Fig. 4) illustrates this,with tree height explaining 30% of the variance in crown projectionarea.

It can be concluded at this point that either Ht or CA could beused to infer DBH for single trees, although based on the level ofvariance explained in the DBH by CA, this latter parameter is likelyto provide better precision in predicting DBH on a tree by tree basis.

3.2. Tree clusters

Table 4 lists the descriptive statistics for the tree clusters. Forthe tree clusters, scatter plots of average DBH (DBHav) versus

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.0 5.0 10.0 15.0 20.0 25.0 30.0

DB

Hav

(m)

Htav (m)

R2 = 0.14MPE= 0.10 m

Fig. 5. Scatter plots of DBHav versus (a) average tree height (Htav) and (b) crown projectiomodels for each individual parameter (Table 5).

average tree height (Htav) and average crown projection area (CAav)are given in Fig. 5, along with the curves based on the derivedregression models. Regression statistics are summarised in Table 5.

An assessment of the datasets showed that, again DBHav wasnot normally distributed (Shapiro–Wilk W = 0.953, p = 0.03147).Once log-transformed, the parameter CAav explained a significantlylarger proportion of the variance observed in the average DBH perstem in each cluster as compared to the average tree height withinthe cluster (R2 = 0.67 compared to 0.14). Unlike the single trees,Htav and CAav were not strongly correlated (R2 = 0.04, data notshown), which suggests that a combination of both parametersin a multi-linear regression model (log-transformed inputs) maybe desirable, even though this yields only modest gains in the levelof variance explained in DBHav (R2 = 0.69 compared to 0.67) andprecision in predicting values of DBHav (0.07 m compared to0.08 m). The CAav parameter alone was found to yield a MPE of0.08 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 withinthe clusters (DBHsum) and the total crown projected area (CAsum)of the clusters are also listed in Table 5 and a scatter plot of the

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 50 100 150 200 250 300

DB

Hav

(m)

CAav (sq m)

R2 = 0.67MPE= 0.08 m

n area (CAav) for tree clusters (n = 52). The regression curves are the log-transformed

stem density = 6007.1 x (CAsum)-0.652

R² = 0.35

0

200

400

600

800

0 200 400 600 800 1000 1200

Stem

den

sity

(/h

a)

CAsum (sq m)

single tree envelope

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 thefitted power curve to the cluster data.

Table 6Derived regression parameters (95% confidence intervals) for a combined individualtree–tree cluster model. The parameters DBH� and CA� incorporate data for bothsingle trees (DBH, CA) and the average values of each tree within the measuredclusters (DBHav, CAav). The parameters DBH+ and CA+ incorporate data for both singletrees (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 + 0.426425ln(CA�)

0.71 265.0 <0.0001 0.13(17%)

ln(DBH+) = �3.73606 + 0.70211ln(CA+)

0.59 157.0 <0.0001 0.43(31%)

130 N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132

of DBHsum versus CAsum is given in Fig. 6. Again a log transforma-tion provided the best regression model, although the total crownprojected 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 from1.06 to 8.91 m (�63% and 7.4% error, respectively).

The fact that the net crown projected area for a cluster is onlyweakly related to the sum of DBH values within that cluster isevidence of competition effects (Bella, 1971; White 1981). Fig. 6exhibits increasing scatter at higher CAsum. In the clusters investi-gated in this work, a higher CAsum is correlated with an increase in

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

DB

H*

pred

icte

d (m

)

DBH* actual (m)

single trees

tree clusters

R2 = 0.71MPE= 0.13 m

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Fig. 8. Scatter plots of (a) measured versus predicted DBH⁄ (comprising DBH of single trethe regression model, and (b) measured versus predicted DBH+ (comprising both DBH ofCAsum data in the regression model (n = 112). Solid lines indicate 1:1 equivalence betwe

the number of stems (R = 0.55). At the same time we observe thestem density (stems/ha) – CAsum relationship (Fig. 7) to take theform:

stem density ¼ 6007:1� ðCAsumÞ�0:652 ð3Þ

The larger tree clusters in our farmscape are made up of fewer,older trees, consistent with the self-thinning rule discussed byWhite (1981). Superimposed on Fig. 7 is the curve generated bythe single tree data; the so-called ‘single tree envelope’. Here thestem density (/ha) for the single trees was calculated using theratio 10,000 m2/CA, effectively assuming the inter-stem distanceis limited to the unperturbed canopy envelope of the individualtrees themselves; namely zero influence overlap (Bella, 1971). Itis evident that in our scattered tree clusters, the self-thinningmechanism is resulting in trees expressing similar canopy dimen-sions as isolated trees. With these older trees there is expected tobe an increase in variability in canopy extent due to effects ofweathering, pests and disease (for example Landsberg andOhmart, 1989; Elliott et al., 1993; Neumann, 1993; Köstner et al.,2002) and this may also explain the increasing variability observedat higher values of CAsum. Summing the DBH in Fig. 6 effectivelyaccumulates the effects of individual tree competition, and theresulting departure from the behaviour of single, isolated trees.The act of taking average (per stem) crown projected area andaverage DBH in a cluster most likely reduces this accumulatingerror.

3.3. Combining both single trees and tree cluster datasets

A comparison of the single tree and tree cluster regression mod-els based on crown projected area alone (DBH = f(CA); Tables 3 and5) suggests the parameters derived for single trees, and thosederived on a per-tree basis from tree clusters (stem numbersranging from 2 to 27), are similar. A comparison of the derivedregression equations for the individual trees and tree clustersshows the interaction terms to be non-significant (p = 0.79). Thisimplies that the slopes and the intercepts of the two regressionmodels do not differ significantly; statistically the two equationsgenerate similar estimates of DBH.

The log-transformed regression models derived using thecombined datasets are given in Table 6 and scatter plots of themodel-predicted versus actual DBH are given in Fig. 8. Here themodels are based on the log-transformed DBH (DBH�, DBH+) andthe log-transformed crown projection area (CA�, CA+) values. Thesuperscript ‘�’ denotes the fact that each parameter involves datafor both single trees (DBH, CA), and the average values of each tree

(b)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3 3.5 4

DB

H+

pred

icte

d (m

)

DBH+ actual (m)

single trees

tree clusters

R2 = 0.59MPE= 0.43 m

es and DBHav of tree clusters) using both single tree CA and tree cluster CAav data insingle trees and DBHsum of tree clusters) using both single tree CA and tree clusteren predicted and actual values.

N.K. Verma et al. / Forest Ecology and Management 326 (2014) 125–132 131

within the measured clusters (DBHav, CAav). The superscipt ‘+’denotes the fact that each parameter involves data for both singletrees (DBH, CA), and the net cluster projected canopy area and sumof DBH of each tree within the measured clusters (DBHsum, CAsum).Both models were created using a random selection of half the sin-gle and cluster data for calibration (n = 112) and the remainingdata withheld for subsequent validation (n = 112).

The data for the single tree and tree cluster datasets are shownas separate symbols. The effect of the error in the DBHsum versusCAsum, discussed earlier is again evident in Fig. 8(b), and it is clearthat a single model for both individual trees and tree clustersbreaks down when the DBHsum values are included in DBH+. Inthe DBH� – CA� model that incorporates both DBHav and CAav val-ues of clusters with the data for single trees, the model performswell; it is noteworthy to observe the tree cluster data to be distrib-uted amongst the single tree data points. The MPE in DBH for thetree-averaged data is 0.13 m. While encouraging, this equates toa mean relative prediction error approximately 17% of the DBHvalues encountered in the sampling. An investigation of the rela-tive prediction error on a sample by sample basis did not showany systematic trend towards increasing prediction error withincreasing DBH except for values exceeding 1.2 m.

4. Conclusions

Simple regression models involving crown projection area ofEucalyptus trees (six species), both isolated and in clusters of upto 27 stems ranging from 38 to 536 stems per ha), explained67% and 68%, respectively of the variance in stem DBH. A singlemodel involving both single trees and the tree clusters to predictaverage stem DBH had similar explanatory power (R2 = 0.71) andyielded a mean prediction error in average DBH per stem of±13 cm. We conclude that it is sufficient to use crown projectionarea to infer DBH for these species (and these stem densities andstem numbers). While the results appear encouraging, weacknowledge that the landscape investigated here was only662 ha, and while encapsulating considerable variation in soils,elevation and aspect (for example as reported in Garraway andLamb, 2011), it would be expected that the robustness and preci-sion of the model could potentially be enhanced by includingother landscape parameters. This is the subject of further work.Nevertheless, the use of larger scale data sources such as airborneor satellite imagery to infer DBH from derived values of crownprojection area for both single Eucalypt trees and clusters up to27 stems (between 22 and 536 stems per ha) appears feasibleand worthy of further investigation. Admittedly, the single modeldeveloped here does require knowledge of the number of stemswithin a given tree cluster. However the increasing use of othersensing technologies like LIDAR to both infer total crown projec-tion area and delineate and count the number of stems within acanopy (for example Popescu et al., 2004) potentially offers themeans to achieve this.

Acknowledgments

This work was partially funded by the CRC for Spatial Informa-tion (CRCSI), established and supported under the AustralianGovernment Cooperative Research Centres Programme. One ofthe authors (NKV) wishes to acknowledge the receipt of aPostgraduate ‘Top-up’ Scholarship from the CRCSI. We would liketo thank Ashley Saint and Derek Schneider (UNE-PARG) for theirassistance in conducting the field work and Drs Gregory Falzon(UNE-C4D), Robin Dobos (NSW DPI) and Jackie Reid (UNE) for theirhelpful comments on the statistical analysis.

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