Characterizing forest structural types and shelterwood dynamics from Lorenz-based indicators predicted by airborne laser scanning

ARTICLE

Characterizing forest structural types and shelterwooddynamics from Lorenz-based indicators predicted by airbornelaser scanningRubén Valbuena, Petteri Packalen, Lauri Mehtätalo, Antonio García-Abril, and Matti Maltamo

Abstract: In this study, Lorenz curve descriptors of tree diameter inequality were used to characterize the dynamics of forestdevelopment in a shelterwood-managed Pinus sylvestris (L.) dominated area. The purpose was to stratify the forest area into foreststructural types (FST) from airborne laser scanning (ALS)-based wall-to-wall predictions of the chosen indicators: Gini coefficient(GC) and Lorenz asymmetry (LA). A clear boundary at GC = 0.5 was found, which separated even-sized (below) and uneven-sized(above) areas. Furthermore, a need for including LA in the characterization of the uneven-sized areas was detected, to distinguishbimodal from reverse J-shaped stands. Beta regression was used for the ALS predictions, yielding RMSEs of 19.67% for GC and11.01% for LA. Based on our results, we concluded that forest disturbance decreases GC, whereas seed regeneration increasesGC and, therefore, gap dynamics are characterized by shifts between either side of the GC = 0.5 threshold. In even-sized stands,GC decreases toward maturity owing to self-thinning occurring at the stem exclusion stage. In uneven-sized stands, the skewnessof the Lorenz curve indicates understory development, as ingrowth decreases LA. The possible applications of the resulting FSTmap are discussed; for instance, in identifying areas needing silvicultural treatments or evaluating forest recovery from distur-bances.

Résumé : Dans cette étude, des courbes de Lorenz ont été utilisées comme descripteurs d’inégalité de la distribution diamétraledes arbres pour caractériser la dynamique du développement d’une zone de forêt dominée par Pinus sylvestris (L.) et soumise a unrégime de coupe progressive. Le but était de stratifier la superficie forestière en types structuraux a partir de prédictions encontinu des indicateurs choisis sur la base d’un balayage laser aéroporté (BLA). Ces indicateurs étaient le coefficient de Gini (GC)et l’asymétrie de Lorenz (LA). Une ligne de démarcation claire de GC égale a 0,5 permettait de séparer la strate régulière (GCinférieur a 0,5) de la strate irrégulière (GC supérieur a 0,5). En outre, il était nécessaire d’inclure LA dans la caractérisation de lastrate irrégulière pour distinguer le peuplement a distribution diamétrale bimodale du peuplement a distribution en J inversé.La régression bêta a été utilisée pour produire des modèles de prédiction de GC et LA basés sur le BLA avec une erreur quadratiquemoyenne de 19,67 % pour GC et 11,01 % pour LA. Les résultats obtenus ont permis de conclure que : GC diminue avec laperturbation de la forêt alors qu’il augmente avec la régénération par graine et la dynamique des trouées est caractérisée par lesécarts autour d’un seuil de GC égal a 0,5. Dans les peuplements réguliers, GC diminue avec la maturité en raison de l’auto-éclaircie qui se produit au stade d’exclusion des tiges. En peuplements irréguliers, l’asymétrie de la courbe de Lorenz est unindice de développement du sous-étage car l’augmentation des recrues fait diminuer LA. La discussion porte sur les applicationspossibles de la cartographie des types structuraux ainsi définis, par exemple pour identifier les peuplements nécessitant destraitements sylvicoles ou pour évaluer la récupération de la forêt a la suite de perturbations. [Traduit par la Rédaction]

IntroductionForest dynamics and development are affected by the structural

complexity of forests and the shape of their diameter distribution(Muller-Landau et al. 2006). For this reason, indicators describingforest structure can be a valuable support tool in decision makingand forest management planning. In practice, to identify struc-tural changes over a forest area, its stratification into categorizedforest structural types (FST) must be obtained from concise indi-cators of forest. They can be used as a basis for delineating man-agement units (Mustonen et al. 2008), enabling the application ofcustomized growth, and yield models according to the regenera-tion rates that are appropriate for each of them (Lei et al. 2009;De-Miguel et al. 2012). This way, the assessment of harvestingpotentials can benefit from a precise map of forest stand struc-ture, which may also assist the evaluation of environmental, rec-

reational, and other multifunctional aspects of forests (Pukkalaet al. 2011). However, there is a lack of consensus on the adequateindicators for evaluating the structural complexity of forests, anda need for harmonizing methods for large-scale comparison(Cienciala and Korhonen 2011).

Recent research by Valbuena et al. (2012) pointed out a numberof motivations for using Lorenz curves to describe forest struc-ture, and suggested their derived indicators as a good alternativefor FST identification. The Lorenz curve Mr(xr) is a representationof the inequality among individual trees within a forest commu-nity with respect to a certain variable: basal area, volume, orbiomass (Table 1 includes a concise list of the variable definitionsused throughout this article). The concept was adapted to forestryby De-Camino (1976), who computed them over a grid of cellscovering a forest map, to evaluate changes in stand homogeneity.

Received 16 April 2013. Accepted 14 September 2013.

R. Valbuena. European Forest Institute HQ, Torikatu 34, 80100 Joensuu, Finland.P. Packalen and M. Maltamo. University of Eastern Finland, Faculty of Forest Sciences, PO Box 111, Joensuu, Finland.L. Mehtätalo. University of Eastern Finland, School of Computing, PO Box 111, Joensuu, Finland.A. García-Abril. Technical University of Madrid, School of Forestry, Ciudad Universitaria s/n, 28040 Madrid, Spain.Corresponding author: Rubén Valbuena (e-mail: [email protected]).

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Can. J. For. Res. 43: 1063–1074 (2013) dx.doi.org/10.1139/cjfr-2013-0147 Published at www.nrcresearchpress.com/cjfr on 16 September 2013.

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Weiner (1985) used cumulated biomass so that its applicability wasexpanded to the whole plant ecosystem, and stated the importanceof the Gini coefficient (GC) as simple descriptor of inequality in plantsizes. The GC is an indicator of basal area / volume / biomass concen-tration, i.e., dispersion relative to average, as differences are consid-ered among tree pairs and not as residuals from their mean (Hosking1990). For this reason, Knox et al. (1989) proved that GC describeddiameter distributions than moment statistics such as variance andskewness. Being a statistic of dispersion normalized by quadraticmean diameter (QMD), GC allows for comparing populations at dif-ferent degrees of maturity or developing stages, or the same popula-tion growing over time (Weiner and Solbrig 1984). Thanks to thisproperty, GC has been observed to assort FSTs in a logical order ofhomogeneity that is better than the other alternatives (Lexerød andEid 2006; Duduman 2011).

As it was noted that the asymmetry of a Lorenz curve may yieldthe same GC value for differing plant populations, a Lorenz asym-metry coefficient (S) was developed by Damgaard and Weiner(2000) as a measure of the position of the Lorenz curve’s inflexionpoint. To date, the use of S in forest inventory has been marginal,and only Metsaranta and Lieffers (2008) applied it for the study oftemporal changes in forest development and tree competition.Valbuena et al. (2012) suggested the incorporation of S to expandthe number of different FSTs described by the means of Lorenzcurves, especially among uneven-sized types. S is a joint expres-sion of two structural properties closely related to one another:the proportions of stem density (xQMD) and basal area (MQMD)stocked above QMD (Valbuena et al. 2013). Gove (2004) used thelatter as a descriptor of the shape of diameter distribution and toassist parameter recovery of unknown distributions. This ap-

proach simplified the task of modelling diameter distributions incombination with their basal-area-weighted counterpart, there-fore uniting management methods for even- and uneven-sizedforests (Gove and Patil 1998).

Airborne laser scanning (ALS) is a remote sensing techniquewith a great potential for the structural characterization of vege-tation (Maltamo et al. 2005). ALS allows surveying of broad forestareas, therefore providing automated methodologies for inter-stand comparison, multitemporal change assessment, and quan-tification of the influence of economic uses and silviculturaltreatments (Wulder et al. 2008). A widely used area-based methodconsists of computing ALS metrics from the height distributionsobserved both at the plot area and throughout a grid of simi-larly sized cells covering the whole area surveyed (Næsset 2002;McGaughey 2012). Bollandsås et al. (2008) used this method toevaluate the success of natural regeneration from ALS surveys.Kellner and Asner (2009) observed that a variety of forest distur-bance causes and regimes can be explained under generalizedpatterns found in ALS data sets. Forest structure characteristicsderived from ALS data sets have been used, e.g., to study foresthydrology (Jaskierniak et al. 2011) or improve carbon estimations(Mascaro et al. 2011). For these reasons, forest managers foreseethat the possibilities of ALS for large-scale mapping of forest struc-ture indicators and FSTs will be essential for integrated ecological,economical, and social management of forested environments(Wulder et al. 2008).

This article presents an application of ALS remote sensing tomapping FSTs through the estimation of Lorenz-curve-based indi-cators of forest structure. The description of forest structure wasfirst considered only from the field data, and different stages

Table 1. List of acronyms and symbols.

Symbol (units) Scale Definition

GeneralFST Forest structural typesALS Airborne laser scanningForest variablesj Tree Tree within a plot (j = 1, …, n)n Plot No. of trees in a plotDBHj (cm) Tree Diameter at breast heightgj (m2) Tree Tree basal areag (m2) Plot Total basal area of a plotQMD (cm) Plot Quadratic mean diameterLorenz orderingMr(xr) Plot Lorenz curver Tree Within-plot rank of a tree ordered by decreasing DBHj (r = 1, …, n)gr

# Tree Ranked basal area of a tree within plotMr Tree Cumulated proportion of basal area for tree rxr Tree Cumulated proportion of rank (i.e., stem density) for tree rGC Plot Gini coefficientS, LA Plot Lorenz asymmetry (S) and modified Lorenz asymmetry (LA)M(xQMD) Plot Proportion of basal area accumulated by the trees with DBH above QMDxQMD Plot Proportion of stem density accumulated by the trees with DBH above QMDi DBH increase for sequence of theoretical uniform distributionALS predictorsMax Maximum ALS return altitudeSD Standard deviation of ALS return altitudesP25 25th percentile (1st quartile) of return ALS return altitudesCover Percentage of ALS return altitudes >1 mL.skew Ratio between the third and second L moments of ALS return altitudesStatisticalm No. of plotsxT� Linear combination of predictor variables (see eqs. (3) and (4))� Mean of beta distribution, i.e. expectancy of predicted response� Precision parameter of beta distributionbias Cross-validated difference (eq. (5))RMSE Root mean square error (eq. (6))

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within the rotation cycle of a shelterwood-managed forest werecharacterized by means of Lorenz curves. The relevant descriptorsof these Lorenz curves were identified as indicators of forest struc-ture: GC and Lorenz asymmetry (LA). The objective was to providethe foundations for using the ALS-based predictions of these indi-cators to stratify the forest area into each FST: even-sized, reverseJ, bimodal, and irregular.

Material and methods

Study area and succession of structural types inshelterwood management

The study was carried out in a 384 ha area of the Scots pine(Pinus sylvestris L.) forest of Valsaín, Spain (lat., 40°53=–41°15=N;long., 3°59=–4°18=W), situated in the Sierra de Guadarrama moun-tain range (1.3–1.5 km a.s.l.). Forest management at this study areais based on group shelterwood felling, generating large spatialvariability in forest structure. Selective cuttings are successivelyperformed in patches, with the intention of gradually openinggaps in the canopy to allow natural regeneration (Cañellas et al.2000). Gradual thinnings from above are committed to understoryreinitiation and ingrowth. Parent trees are felled once understoryestablishment has been achieved. As a result of a long regenera-tion phase implemented over small groups, many FSTs can coexistwithin a single forest compartment (Valbuena et al. 2012). Obtain-ing spatially continuous estimations of stand complexity through-out the forest area is, therefore, especially important in this studysite to assess the success of regeneration groups, plan the locationof the next selective cuttings, or evaluate the need for thinning.

An example of representative sample plots have been selected inFig. 1 to illustrate the variety of forest structures found and theirsuccession along the rotation period. Stem exclusion yields even-sized structures (a) characterized by a normal DBH distribution(even-sized FST). The forest ages up to its even-aged mature stage (b),after which gaps begin to be opened, and, hence, seed regenerationstarts at the canopy gaps (c). Gradual thinnings from above and nat-ural forest disturbance (e.g., windthrow) lead to a reverse J-shapedhistogram (d) (FST reverse J). Cleaning is usually carried out to facili-tate the growth of the most competitive stems at the subdominantcohort so that stem frequencies present a descending histogramwith a small peak in the dominant canopy (e). Once a new cohort isestablished under the dominant canopy, a bimodal distribution canbe found with more basal area in the lower layer (f) (FST bimodal),after which the parent trees are removed at the end of the rotationperiod, leading back to (a). In addition to the forest structural typessummarized in Fig. 1, more irregular DBH distributions (FST irregu-lar) can also be found if a systematic sample, either a forest plot or anALS cell, happens to be located straddling at the edge of regenerationpatches (Valbuena et al. 2012).

The Lorenz curve and related indicators of forest structureField survey consisted of 37 systematically located circular plots

with a 20 m radius. Table 2 summarizes plot-level characteristicsof the resulting field sample. The reader may refer to Valbuenaet al. (2013) for further details on plot establishment. For thoseinterested in replicating the method presented, it may be note-worthy to emphasize the importance of including the full DBHrange present in the forest, as Lorenz-curve-related indicators wouldpresent different values if the input data are truncated by any type ofmerchantability limits or the like. Hence, accounting for the pres-ence of understory and seedlings is a methodological requirement.

The starting point for computing empirical Lorenz curves wasto rank all individual trees within a plot (j = 1, …, n) in order ofdecreasing DBH (DBHj, cm) or tree basal area (gj, m2). The vector ofbasal areas g = {gj} consequently became a ranked vector g# � �gr

#�,where r = 1, …, n and g1

# � g2# � … � gn

#. To obtain relative propor-tions, the contribution of each individual tree to the total stemdensity and basal area are expressed in relation to the total num-

ber of trees (n) and basal area of the plot (g, m2), respectively. Bymajorization (Solomon 1979), relative cumulated proportions ofbasal area (Mr � �j�1

r gj#/g) accounted for by each tree are repre-

sented against relative cumulated proportions of stem density(xr � �j�1

r nj#/n � r/n). Therefore, Lorenz curves Mr(xr) contain infor-

mation about both of the histograms used to describe forest struc-ture in Fig. 1, and, hence, Lorenz ordering was considered wellsuited to discriminate among them (Valbuena et al. 2012).

Although we ranked the trees in decreasing order in the presentstudy, obtaining concave Lorenz curves, it is also common to com-pute convex Lorenz curves from increasing ranks, nonethelessdelivering equal results. Lorenz curves are usually compared withthe line of absolute equality obtained when all trees are identicalin size, whether big or small. Should all trees be equal (g1 = g2 = … =gn), the relative increases in basal area and stem density converge(gj

#/g � nj#/n for every j). This line is, therefore, the 1:1 diagonal of

the Lorenz plot representing the minimal inequality among treesizes. The area between a Lorenz curve and the line of absolute equal-ity is the GC, which, therefore, increases for larger inequalitiesamong tree sizes (Weiner 1985). As GC is QMD-independent, it refersto the inequality of tree sizes independently of the amount of basalarea stocked in the forest. GC equals half of the relative mean basalarea differences of all trees within plot. In this study, the bias-corrected sample estimate of mean difference proposed by Glasser(1962) was used to compute GC. The theoretical values of GC have arange of [0,1], whether or not its extremes may actually be ecologi-cally plausible. GC = 0 is the value for any forest plot with all treesequal (line of absolute equality), irrespective of their QMD. The otherextreme of GC = 1 would be a maximally bimodal distribution pre-senting the highest theoretical dispersion.

Furthermore, Valbuena et al. (2012) also considered the impor-tance of comparing Lorenz curves against a theoretical uniformdistribution in tree diameters. This other line of perfect unifor-mity was obtained by following the same Lorenz ordering proce-dure for a simulated sequence of trees with a steady increase (i) intheir diameters over the sample range observed for the wholefield data set (DBHi, DBH2i, …, max DBH). For any range of treesizes considered, the line of perfect uniformity has an asymptotic(n ¡ ∞; i ¡ 0) theoretical value of GC = 0.5.

The shape of the Lorenz curve is described by S, as proposed byDamgaard and Weiner (2000). This index is a measure of the dis-tance between the inflexion point of the Lorenz curve and the axisof symmetry. Being the point where the Lorenz curve is parallel tothe line of absolute equality, its slope equals 1

(1) 1 ��j�1

r�1gj

#/g � �j�1

rgj

#/g

(r � 1)/n � r/n�

gj/g

1/n�

�DBHj2/g

1/n

where � = � × 104/4.Solving eq. (1) for DBHj gives

DBHj � ��g

n� QMD

Therefore, the inflexion point of the Lorenz curve, M(xQMD),coincides with the QMD in its forestry applications. Describingthe asymmetry of the Lorenz curve, therefore, involves the com-putation of the proportions of basal area, M(xQMD,) and stem den-sity, xQMD, accounted for by the trees larger than the QMD. For aLorenz curve to be symmetric, this relation should be balanced sothat its QMD is situated at the axis of symmetry, which orthogo-nally crosses the line of absolute equality at its middle point (seeFig. 1 in Damgaard and Weiner (2000)). Examples of symmetric

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Lorenz curves are M(0.1) = 0.9, i.e., trees above QMD account for90% of the basal area but only 10% of the stem density, or M(0.4) =0.6, i.e., trees above QMD account for 60% of the basal area and40% of the stem density.

The computation of S was based on linear interpolation ofthe empirical diameter distribution (Valbuena et al. 2013). S is thesum of two proportions, as S = M(xQMD) + xQMD; therefore, its valuesmay have a mathematical range of [0,2], with a centre value at S = 1 fora symmetrical Lorenz curve. For this reason, we proposed a modifiedLA coefficient as

Fig. 1. Histograms of diameter distributions (white bars) and proportions of basal area at each 2 cm size class (gray bars). The succession offorest structural types as a consequence of management based on group shelterwood cutting is illustrated with plot-level field samples, with adescription of the dynamics and treatments involved at each step. Vertical broken lines denote quadratic mean diameters (QMDs).

0 4 8 16 24 32 40 48 56 64diameter at breast height (cm)

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Stem Density Prop. (%)

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Basal Area Prop. (%)

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Basal Area Prop. (%)

0 4 8 16 24 32 40 48 56 64diameter at breast height (cm)

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Stem Density Prop. (%)

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Basal Area Prop. (%)

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20.

25

Stem Density Prop. (%)

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20.

25

Basal Area Prop. (%)

Basal Area

Stem Density

0 4 8 16 24 32 40 48 56 64diameter at breast height (cm)

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Stem Density Prop. (%)

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Basal Area Prop. (%)

0 4 8 16 24 32 40 48 56 64diameter at breast height (cm)

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(c)

SEEDCUTTINGS EVEN−SIZED

MATURE

(b)

PREPARATORYCUTTINGS EVEN−SIZED

YOUNG

(d)

CLEANINGUNEVEN−SIZEDREVERSE J

(e)

INGROWTHUNEVEN−SIZEDBIMODAL

SEED REGENERATION

(f)

PARENT OVERSTORY

REMOVAL

(a)

Table 2. Summary characteristics of the field reference data.

Variable Mean (SD) Range

Stem density (stems·ha−1) 732.33 (559.19) 167.11–1917.82Basal area (m2·ha−1) 41.76 (11.24) 21.08–57.74Quadratic mean diameter (cm) 33.14 (12.32) 14.51–48.30Standing volume (m2·ha−1) 390.70 (258.64) 22.95–823.80Gini coefficient [0,1] 0.43 (0.25) 0.15–0.87Lorenz asymmetry [0,1] 0.56 (0.07) 0.43–0.75

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LA � S/2 � [M(xQMD) � xQMD]/2

The reason for this modification was to allow both GC and LA tobe predicted from a similar generalized beta regression model(described later on in this section), where the response consists offractional values between 0 and 1. Moreover, it also had concep-tual sense, as LA was the average the two properties considered inS: the proportions of basal area M(xQMD) and stem density xQMD

stocked above QMD. The outcome was two Lorenz indicators (GCand LA) of similar properties, as LA also ranged as [0,1], with acentre value at LA = 0.5, corresponding to a symmetrical Lorenzcurve.

ALS data and predictor variablesThe ALS data were acquired with a Leica ALS50-II sensor from

Geosystems (Switzerland) on 10 September 2006. The pulse repe-tition rate was set to 55 kHz, and ground speed was 72 m·s−1. Theflight was performed at a height of about 1500 m above terrainlevel, rendering a 665 m swath width with a 40% side lap betweenlines. For the given sensor and flight parameters, the averagepulse density was 1.15 pulses·m−2 and the footprint diameter wasabout 0.5 m at nadir. Further information on survey configura-tion, point cloud positioning, and digital terrain model genera-tion is detailed in Valbuena et al. (2011). With the assistance ofFUSION software (version 3.1; McGaughey 2012), the same ALSpredictors (Table 1) were generated at plot level for model trainingand over a grid of cells covering the whole ALS surveyed area forwall-to-wall prediction of the response variables (Næsset 2002). Athreshold of 1 m was applied, with the intention of masking outthose ALS returns backscattered from the ground without elimi-nating the influence of the understory ingrowth. A prior explor-atory multivariate analysis based on partial least squares andmultimodel inference was devoted to predictor selection (Valbuenaet al. 2013). For the given predictive models, we computed canopycover (Cover) as the percentage of all returns with an altitudegreater than 1 m above ground (McGaughey 2012). The metricCover provides an idea of the density of each forest area, whichinfluences the relation between other metrics and the forest re-sponse and, therefore, it was included in all the models computed.From those same returns, we obtained the maximum (Max), 25thpercentile (P25), and standard deviation (SD) of all above-groundaltitudes within each plot or cell (Magnussen and Boudewyn1998). These were selected for predicting GC, as Valbuena et al.(2013) found that GC is best described by combinations of low andhigh percentiles (Max/P25), along with a descriptor of returnheight dispersion (SD). We also considered the ratio between thethird and second L moment of ALS return altitudes (L.skew),which is a robust expectation of the distribution’s skewness(Hosking 1990). This predictor was chosen for its relation to LA, asValbuena et al. (2013) also suggested the choice of a third-momentstatistic for this purpose.

Beta regression and accuracy assessmentThe suggested responses, y = GC and y = LA, were related to the

ALS metrics at plot level. Beta regression (Ferrari and Cribari-Neto2004) was chosen for modeling this relation, as it is speciallyintended for response variables in the range [0,1], such as concen-tration indices (e.g., GC) and proportions (e.g., LA). Beta regressionis a special type of generalized linear model, which allows regres-sion for distributions other than normal in the response estimatey. In the case of beta regression, the initial assumption is a betadistribution of mean � and variance �.

(2) y � Beta(�, �)

where � � �0,1�.

We used a logit link function that is asymptotic in the range[0,1], subsequently assuring the expectancy for the prediction ofthe response variables, � � GC and � � LA, to lay within theirtheoretical limits.

(3) log �1 � � xT�

The vector of model coefficients being �, and x the set of ALSpredictors that apply for each response variable, as detailed in theprevious section. For

y � GC ⇒ xT� � 0 � 1Max � 2SD � 3P25 � 4Cover

and for

y � LA ⇒ xT� � 0 � 1L.skew � 2Cover

Equation (3) yields the mean function as the inverse of thelogistic and, therefore, the expectation for estimated response is

� �exp(xT�)

1 � exp(xT�)

As a result, the density function of the response estimate can beexpressed as a function of the model coefficients.

(4) y � Beta� exp(xT�)

1 � exp(xT�), ��

which were, therefore, estimated by maximum likelihood for fit-ting of the beta distribution to the plot-level sample (Ferrari andCribari-Neto 2004). The implementation of the method was car-ried out using the betareg package (Cribari-Neto and Zeileis, 2010)in the R environment (version 2.15; R Development Core Team2011). The pseudo R-squared (Rp

2) coefficient of determination, stan-dardized Pearson residuals, and half-normal plots were utilizedfor model evaluation and diagnosis (Ferrari and Cribari-Neto2004). Rp

2 provided an idea of the amount of explained variancefrom the squared correlation coefficient computed between thelink function in eq. (3) and the adjusted xT�, and z tests were usedto assess the significance of model estimates. The overall betaregression model was evaluated with the lrtest function (Zeileisand Hothorn 2002) as a likelihood ratio contrasting the beta den-sity distribution in eq. (4) against the null hypothesis, i.e., thegoodness of fit without predictors as in eq. (2), whose significancewas tested by a �2 distribution.

A jackknifing cross-validation approach was followed for accu-racy assessment so that, after removing one case k out of the totalm number of sample plots, the remaining were used for maxi-mum likelihood estimation of model coefficients (�k) by eq. (4).The resulting coefficient estimates �k were thereafter used forpredicting the removed case (yk). The discrepancy between theobserved (yi) and predicted (yk) values could, therefore, be evalu-ated by their mean difference (bias) and also by their root meansquare error (RMSE). Relative figures (bias% and RMSE%) were alsoobtained by dividing the sample mean values.

(5) bias � �k�1

m(yk yk)/m

(6) RMSE � ��k�1

m(yk yk)2/m

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Results

Describing forest dynamics with Lorenz curvesFigure 2 shows the Lorenz curves obtained by the same plots

used in Fig. 1. For the empirical plots measured in this forest site

under shelterwood management, we found Lorenz curves situ-ated on even-sized areas (Figs. 2a–2c) to always lie below the line ofperfect uniformity (gray broken line), whereas uneven-sized ones(Figs. 2d–2f) were situated above it. Moreover, transitions betweeneven and uneven-sized structures were identified when a skewed

Fig. 2. Lorenz curves obtained by the same sample plots depicted at Fig. 1, illustrating the same cycle dynamics. Concave Lorenz curvesrepresent the cumulated relative proportions of basal area (Mr) and stem density (xr) for the trees in the plot ranked in descending order. Ateach Lorenz curve, a crossed dot denotes the position of the mean quadratic diameter M(xQMD). Gray lines represent the diagonal line ofabsolute equality (solid line), the line of perfect uniformity (broken line), and the axis of symmetry (dotted line).

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SEEDCUTTINGS EVEN−SIZED

MATURE

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PREPARATORYCUTTINGS EVEN−SIZED

YOUNG

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CLEANINGUNEVEN−SIZEDREVERSE J

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INGROWTHUNEVEN−SIZEDBIMODAL

SEED REGENERATION

(f)

PARENT OVERSTORY

REMOVAL

(a)

line−of−absolute−equalityline−of−perfect−uniformityaxis of symmetry

Lorenz curvequadratic mean diameter

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Lorenz curve intersected this line. For the concave Lorenz curvespresented, they were skewed to the right when seed regenerationwas starting (Fig. 2c), whereas they skewed towards the left as thesubdominant cohort increased in basal area (Fig. 2f). Characteriz-ing the symmetry of the Lorenz curve was, therefore, importantfor assessing differences among such situations.

The scatterplot in Fig. 3 shows the capacity of the Lorenz curvedescriptors, GC and LA, to discriminate among the different foreststructural types present. On the x axis, GC clearly distributed theplots on either side of the line of perfect uniformity with GC > 0.5for uneven-sized plots and GC ≤ 0.5 for even-sized plots. Further-more, it was observed that two differing FSTs — reverse J andbimodal — were both located at the right side of the plot, and,therefore, an additional parameter was mandatory for disaggre-gating them. On the y axis of the scatterplot, LA noticeablychanged according to the relative basal area relation betweenthe dominant canopy and the understory. The development ofuneven-sized patches at gaps opened by forest disturbances ateven-sized stands was, therefore, indicated by LA ≥ 0.5, later be-coming LA < 0.5 once ingrowth succeeded. Thus, an intuitiveborder between reverse J and bimodal plots could be stated as LA =0.5, which corresponds to a completely symmetric Lorenz curvewith an inflexion point, M(xQMD), that coincides with the axis ofsymmetry (dotted lines in Fig. 2). No practical application of LAwas observed for even-aged stands (on the left side of this scatter-plot). The reason for this is that Lorenz curves with small GC havethe most trees of diameters close to QMD (Figs. 1a and 1b) and,therefore, they are less affected by asymmetry. Figure 4 summa-rizes the distribution of the FSTs that were observed from a bivari-ate (GC, LA) characterization of the field plots and serves as anabstract of the processes found.

To further study the correspondence between GC and foreststand maturity and development, we observed its relation withQMD. A Spearman’s rank correlation test obtained an r = −0.86(p < 0.001), indicating that an increase in the average DBH of treeswithin a plot led to lower inequality among them. It was further-more detected that plots situated in the drip line between a ma-

ture even stand and a more uneven grove may contain trees fromboth areas. This type of situation would increase the DBH variancefound in an even-sized plot, toward its maximum obtained by atheoretical uniform distribution (upper limit of grey-shaded areasin Fig. 2). Therefore, these plots were considered to tend towardvalues of GC closer to 0.5 than it would be expected from theirlarger QMDs and degrees of maturity. This suggested that valuesclose to GC = 0.5 were to be found in the boundary between even-and uneven-sized areas. For this reason, the presence of an addi-tional FST denominated as irregular was also considered underthe criterion 0.4 < GC ≤ 0.5. In view of these overall results, adecision was made to finally characterize FSTs based upon Lorenzindicators under a simple classification rule expressed in pseudo-code in Table 3. In Fig. 5, we illustrate how the criteria used inTable 3 is reflected with the Lorenz plot and, hence, serves as abasis for FST classification according to the position of M(xQMD).

ALS prediction and FST mappingTable 4 summarizes the results obtained by the beta regression

models. Cross-validation results are also shown, and a more de-tailed evaluation of the predictions can also be made by viewingthe scatterplots in Fig. 6. Rp

2 values showed that the amount ofexplained variance was 89% for GC, whereas it was just 23% for LA(Table 4). Significance tests were positive in all cases, and thecross-validation demonstrated that the beta regression obtainedunbiased estimates with reasonable accuracy. Table 5 reports adetailed leave-one-out contingency table for the FST classification,comparing how applying the decision rules in Table 3 by usingthe ALS-predicted indicators differs from using the field plots.Figure 6 and Table 5 show that in only in few cases did the esti-mation error lead to a FST misclassification (Cohen’s kappa coef-ficient, � = 0.62), according to the thresholds (broken lines) statedin Table 3. Thus, the ALS remote sensing method presented hereseems to be well suited for the purpose of FST classification basedon predictions of GC and LA.

The resulting models were directly used to estimate GC and LAover the whole study area. Thereafter, these predictions were di-rectly used to classify the forest area into FSTs, according toTable 3. First, GC was used to discriminate even- from uneven-sized areas. The method yielded a 0.054 probability of overesti-mating uneven-sized areas, as signified by the inaccuracies of theestimates that led to infringement onto their threshold values atGC ≤ 0.5 (Fig. 6a). Hence, this method for determining the com-plexity of forest structure was slightly biased towards uneven-sized FSTs, in light of our results. Moreover, LA was used toevaluate recruitment in the understory. Thus, the areas classifiedas uneven-sized were disaggregated into reverse J (LA ≥ 0.5) andbimodal (LA < 0.5). Figure 6b shows that an overestimation in LAled to classifying 8.11% of the cases as reverse J while being bi-modal according to their original values (Fig. 3). Thus, the methodpresented here tends to slightly underestimate the success of un-derstory establishment. The final outcome is illustrated in Fig. 7,which shows the results obtained when the prediction maps wereused for classifying the study area into FSTs according to Table 3.As was expected, the resulting map suggested the presence of anedge effect, as the additional so-called irregular FST (in light blue)was predominantly located at boundary areas between even- anduneven-sized groups. Furthermore, using beta regression assuredthat the GC and LA predictions would necessarily fall within the[0,1] range, which was very convenient for the actual applicationof FST classification.

DiscussionIn this study, computing Lorenz curves from the field plots was

a reliable method for describing each of them at the diverse stageswithin the shelterwood rotation period. Results in Figs. 2 and 3illustrate why an index of DBH concentration like GC can be a

Fig. 3. Scatterplot of Lorenz-based indicators GC versus LA. Thepositions of the plots used in Figs. 1 and 2 are shown, illustratingthe changes in the GC–LA plane along the rotation cycle. Thevertical dotted line at GC = 0.5 represents the asymptotic value of auniform distribution. The horizontal broken line at LA = 0.5 is thevalue obtained by a symmetric Lorenz curve for which M(xQMD)would coincide with the axis of symmetry (dotted lines in Fig. 2).

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relevant indicator of forest complexity and structure. This is con-sistent with previous studies (Knox et al. 1989; Lexerød and Eid2006). Moreover, Valbuena et al. (2012) suggested that Lorenzcurves from tree DBH ought to be compared with a theoreticaluniform distribution for which GC = 0.5 asymptotically, in addi-tion to the more traditional diagonal line of absolute equalitywhere GC = 0. Our results supported this hypothesis, as seen in theplots presented in Fig. 2, that Lorenz curves of even-sized areaswere situated below this line of perfect uniformity, whereasuneven-sized areas were above it. This tendency for GC < 0.5 ineven-sized areas and GC > 0.5 in uneven-sized areas also was de-tected by Duduman (2011) in mixed conifer–broad-leaved forests.Therefore, the position of the Lorenz curve’s inflexion point,M(xQMD), can be used as a basis for FST discrimination as well(Fig. 5).

Figure 4 summarizes the dynamics that Lorenz curves can ex-plain in this ecosystem, in the same sense as Weiner (1985) intro-duced them for describing mortality and fecundity in relation tocompetition. Forest disturbance, i.e., events characterized by ahigher probability of senescence for more mature individuals,

induces a decrease in DBH inequality and, hence, GC is lower. Thistendency was clearly observed when removing the overstory(Fig. 2a) after the subdominant cohort was established (Fig. 2f). Onthe other hand, once a canopy gap was opened, the developmentof natural regeneration was identified by an increase in GC(Figs. 2b–2d). This increase was caused by new seedlings sproutingup, as they account for an increment in stem density but not inbasal area. For this reason, the Lorenz curve in Fig. 2c was right-skewed toward LA > 0.5 because of the presence of the seedlings.Once further sprouting was inhibited by the shadow cast from thesubdominant cohort, the understory experiences a significant in-crease in basal area but not in stem density. Thus, the recruitmentof understory ingrowth was indicated by changes in Lorenz asym-metry toward LA < 0.5. Therefore, the application of using LA toassess the success of seed regeneration and identify the areasready for removal of the parent overstory was patent from theresults observed in Fig. 3. However, the authors wish to cast adoubt on the criterion used in this article for discriminating be-tween reverse-J and bimodal FSTs in Table 3, as we consider thatfurther research should be conducted in studying which value ofLA is reached when recruitment succeeds from a silviculturalpoint of view.

It is not only forest disturbance that induces a decrease in DBHinequality, as results also demonstrated a general tendency for GCto decrease with increasing QMD and stand maturity. The ecosys-tem under study reaches a stem exclusion stage when the canopycloses because of the dominance of shade-intolerant species. Treemortality is then mainly explained by self-thinning processesand competition for light and space. In the absence of distur-bances, these conditions drive forest development towardseven-sized structures in which stem density is directly related toQMD (Reineke 1933). It is, therefore, also possible to use GC for

Fig. 4. Schematic diagram representing the general patterns observed as a summary of the results obtained in Fig. 2. Background colourscoincide with those of Figs. 5 and 7, denoting the same forest structural types (FSTs). Please see the Web version of this article for colour.

Table 3. Decision rule in pseudocodefor classifying into forest structural types(FST) according to indicators based on theLorenz curve: Gini coefficient (GC) andLorenz asymmetry (LA).

If GC ≤ 0.5 thenif GC ≤ 0.4 then FST = “even-sized”else FST = “irregular”

Elseif LA ≥ 0.5 then FST = “reverse J”else FST = “bimodal”

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evaluating stand maturity and relative density within even-sizedareas. From this point of view, GC could also serve to analyze themain drivers of tree-size relations within an ecosystem: distur-bance or competition (Muller-Landau et al. 2006). While competi-tion keeps GC under the 0.5 limit, as larger trees limit resourcesand grow at the expense of the smaller ones, forest disturbanceand understory regeneration are characterized by shifts betweeneither side of the GC = 0.5 threshold. Further research could ob-serve results in other types of ecosystems, which may be similar toor different from those observed in, for example, tropical or tem-perate mixed forests.

The Lorenz curve’s axes, xr and Mr, represent relative propor-tions of stem density and basal area (Valbuena et al. 2012). It is,therefore, apparent that for describing forest structure and distin-guishing FSTs, studying the share of stocked density and basalarea above and below QMD is of key importance (Gove 2004). Theframework presented in this article shows that stem density – basalarea relations can be studied in the Lorenz curve as a commonground for even- and uneven-sized forests. Thus, this methodol-ogy could provide new insights for the study of the relative den-sity of fully stocked stands, extending Reineke’s (1933) relations touneven-sized forests.

Furthermore, the scale used in a study always has an effect onthe variables considered in area-based methods for ALS remotesensing (Mascaro et al. 2011). For the presented methodology, itwas found to influence the values of the GC obtained, in terms ofboth the size of the measured field plot and also the dimensions ofthe grid cells where the ALS metrics are computed. Expanding

plot size intrinsically increases the possibilities for finding a treeof a differing size, especially in areas of high spatial variability. Intheory, it can also be affirmed that GC ¡ 0 when decreasing thescale up to a single tree, although in practice it is unlikely thatsuch a small forest plot would be measured. Hence, there is ascale-induced bias towards determining a forest area as uneven-sized, as GC becomes overestimated at coarser spatial scales. Thisis the reason why values close to GC = 0.5 were found on the dripline between differing areas (Fig. 7) and an extra so-called irregu-lar FST was introduced (Table 3). Duduman (2011) also observedthat irregular and bimodal structures can show GC values lessthan 0.5, therefore justifying the consideration of this additionalFST. Applying these estimations over smaller areas may be pre-ferred in forest sites presenting a variety of FSTs and wouldprobably reduce the overestimation of the uneven-sized areasobserved.

In an ALS assessment of forest structure, Jaskierniak et al. (2011)also pointed out the significance of characterizing both over- andunder-story, and Valbuena et al. (2013) indicated the opposingeffects shown by LA components. The methodology presented forstratifying the forest area into FST may have a number of addi-tional applications in forest management, besides identifying ar-eas affected by forest disturbance, assessing the balance betweenoverstory and understory, or evaluating the need for thinning.More accurate forecasts of forest growth and development may beachieved if separate models are applied for each different FST(De-Miguel et al. 2012). Alternatively, GC may be directly includedin growth models, presumably obtaining higher accuracy (Lei

Fig. 5. Classification of forest structural types (FSTs) according to the position of the quadratic mean diameter (QMD) of the Lorenz curve. Theshelterwood cycle in Fig. 2 is hereby summarized. The colours coincide with those of Figs. 4 and 7. Please see the Web version of this articlefor colour.

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Table 4. Model estimates, significance tests, and cross-validation accuracy assessment.

Response Model estimates

0 Max SD P25 Cover �

Gini coefficient (GC)−3.79*** −0.19*** 0.83*** 0.13*** 0.05*** 38.10***

Likelihood ratio Cross-validation

Rp2 LogLik �2 p RMSE RMSE% Bias bias%

0.89 46.60 84.36 <0.001*** 0.084 19.67 −3.22×10−4 −0.076

Response Model estimates

�0 L.skew Cover �

Lorenz asymmetry (LA)1.48** −0.56* −0.02* 69.64***

Likelihood ratio Cross-validation

Rp2 LogLik �2 p RMSE RMSE% Bias bias%

0.23 52.27 9.71 0.008** 0.062 11.01 1.27×10−3 0.227

Note: Rp2, pseudo R2; LogLik, log likelihood; levels of significance: NS, not significant (p > 0.05); *, p < 0.05; **, p < 0.01; ***, p < 0.001. Overall bias and root mean square

errors (RMSE) in absolute (0,1) and relative (percentage of sample mean) units.

Table 5. Leave-one-out contingency table for forest structural type (FST) classification from airbornelaser scanning (ALS).

Predicted (ALS)

Observed (Field) Even-sized Irregular Reverse J Bimodal TotalProducer’saccuracy (%)

Even-sized 18 2 0 0 20 90.0Irregular 3 1 2 0 6 17.0Reverse J 0 0 6 0 6 100.0Bimodal 0 0 3 2 5 40.0Total 19 3 11 2 37User’s accuracy (%) 89.7 33.3 54.5 100.0 Overall accuracy (%) 73.0

Fig. 6. Predicted versus observed plots. The thresholds of GC = 0.5 and LA = 0.5 have been marked by broken lines to illustrate the risk ofmisclassification of forest structural types according to Table 3. The red diagonal is the 1:1 correspondence between the values observed in thefield data and the ALS-predicted values. GC, Gini coefficient; LA, Lorenz asymmetry. Please see the Web version of this article for colour.

0.0 0.2 0.4 0.6 0.8 1.0

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et al. 2009). ALS estimations of LA components can be used toanalyze the balance between two cohorts and evaluate the successof regeneration groups (Valbuena et al. 2013). The application ofthis method in unmanaged forests of differing biogeographic re-gions can, in general, be used to expand the potential of ALS in theunderstanding of ecosystem dynamics with reduced field surveyeffort (Kellner and Asner 2009).

ConclusionThis article describes the rationale for a novel bivariate descrip-

tion of forest structure based on Lorenz curve indicators: Ginicoefficient and Lorenz asymmetry. The analysis of the effect offorest development and silvicultural treatments on the Lorenzcurves revealed their reliability for explaining forest dynamicsand classifying into simple FSTs. Based on our results, we con-cluded that forest disturbance decreases GC, whereas seed re-generation increases GC, and, therefore, gap dynamics arecharacterized by shifts between either side of the GC = 0.5 thresh-old. In even-sized stands, GC decreases toward maturity becauseof the self-thinning occurring at the stem exclusion stage. Inuneven-sized stands, the skewness of the Lorenz curve indicatesunderstory development as ingrowth decreases LA. Based onthese fundamentals, a methodology using ALS remote sensingwas demonstrated to reliably stratify the forest area into FSTs.Therefore, the processes described in the shelterwood cycle can bemonitored by ALS prediction of Lorenz curve indicators. However,a scale-dependent GC overestimation bias was detected as an in-trinsic drawback of using an area-based method. Nonetheless,there are motivations to suggest further research focusing onforest-type invariant relations between the chosen response andALS metrics.

AcknowledgementsRubén Valbuena’s work is funded by a Metsähallitus (Finnish

Forestry Commission) Grant and the Foundation for EuropeanForest Research (FEFR). This study was also partly funded by the

strategic funding of the University of Eastern Finland. Manythanks to Francisco Mauro and Elena Peral (Technical Universityof Madrid) for their assistance with FUSION and to all members ofthe Research Group for Sustainable Forest Management who col-laborated in the field campaign. We appreciate the efforts ofanonymous reviewers who helped in improving the quality of thisarticle. Niina Valbuena and Joanne Fitzgerald (European ForestInstitute) also assisted in language editing and revision.

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Fig. 7. Final map forest structural types (FSTs) characterized fromairborne laser scanning (ALS) predictions of the Gini coefficient (GC)and Lorenz asymmetry (LA) according to Table 2. Colours coincidewith those of Figs. 4 and 5. Please see the Web version of this articlefor colour.

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