Forest Ecology and Management 259 (2010) 1993–2005
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Forest Ecology and Management
journa l homepage: www.e lsev ier .com/ locate / foreco
redicting mature cork biomass with t years of growth from one measurementaken at any other age
oana Amaral Paulo ∗, Margarida Toménstituto Superior de Agronomia, Tapada da Ajuda, 1349-017 Lisbon, Portugal
r t i c l e i n f o
rticle history:eceived 2 October 2009eceived in revised form 8 February 2010ccepted 9 February 2010
eywords:ork biomassuercus suberork backork age
a b s t r a c t
Cork is a natural product that is extracted from the outer bark of the cork oak tree. According to Portugueselegislation, the interval between two consecutive cork extractions on the same tree must be equal orgreater than 9 years. Although the majority of the cork oak stands are debarked at the end of this period,this rotation may not be the optimum in many cases. The existing models for cork weight prediction canonly be used for trees debarked at a 9 years or, in one model, at a 10-years rotation period, since the dataused for its development was limited to these growth periods. The development of a method that allowsfor the prediction of the mature cork biomass with t years of growth, based in one measurement taken atany other age, was the main objective of this work. The method is based on the knowledge that the densityof the cork tissue is constant between the inner and outer cork rings, being significantly different fromdensity of the cork back. It can be implemented using two different equations that were developed duringthis work: a model to estimate cork biomass with 9 years of age and a model to estimate the cork backweight proportion at 9 years of age. For the first model, four different alternative models were developed,considering different levels of information collected during forest inventory. The model to estimate thecork back weight proportion leads to the biomass of cork tissue. Cork biomass at t years is obtained bydecreasing or increasing the biomass of cork tissue according to the difference in cork thickness between
t and 9 years of growth. The proposed method was evaluated by comparing the observed and estimatedvalues of cork biomass from an independent data set with corks with 9, 10 and 11 years of age. Theresults showed similar precision for corks with 9, 10 or 11 years of age. As expected the precision of thepredictions increases when the model to estimate cork biomass with 9 years of age uses more information.The presented method should be an important tool for cork oak stand management, for the predictionof the evolution of carbon stocks in cork oak stands, and will allow analyzing the impact in cork biomass
or ex
production of decreasing
. Introduction
Cork oak is distributed along the western Mediterranean areand in North Africa, mainly in pure cork oak forests and corkak based agro-forestry systems. According to the last Portugueseational forest inventory from 2005/2006, in Portugal the total areaovered with cork oak forest is of 780 × 103 ha (Tomé et al., 2007).his represented an increase since the previous forest inventory,hen the total area was of 713 × 103 ha (DGF, 2001).
Approximately 52.5% of the world’s total cork production is orig-nated in Portugal (157.0 × 103 Mg) leading to a high economic andocial importance of the species in the country: 30% of the totalxports of the forestry sector and 2.3% of the total national exports
(APCOR, 2007). This species is also a key element in the multifunc-tional agro-forestry systems that usually combine the productionof several other non-wood products, and is recognized as a crucialelement for the biodiversity, water retention and soil conservation(Aronson et al., 2009).
Because of the importance of the cork oak to the local ecosys-tems where it prevails as the dominant species, managementoptions are regulated by national legislation in countries along itsdistribution zone. One main aspect that is regulated is the intervalbetween two consecutive cork extractions on the same tree, which,according to Portuguese legislation, must be equal or greater than9 years.
The first cork that the tree produces by the activity of the first
periderm is called the virgin cork. Virgin cork is characterized bydeep fractures and cracks that extend irregularly, but mostly lon-gitudinally, due to the rapid radial growth of wood in young trees.
The removal of the cork layer exposes the phellogen, causingthis layer to consequently die. As a response, a new periderm (the
ig. 1. Structure of cork with 9 years of growth. A—wood; B—phellogen; C—corkelly; D1—inner cork rings; D2—outer cork rings; E—cork back. (Adapted fromatividade, 1950.)
raumatic periderm) is formed in order to protect the living tissueshat were exposed to the environment. The unprotected tissueshat dry out and die during this process fracture easily due to theadial growth of the new periderm that follows. These layers formhe external part of the new cork layer, and are called the cork backPereira, 2007). The thickness of the cork back is highly variable,rom 2 mm to more than 4 mm, and depends mostly on the depthrom which the traumatic periderm is regenerated after the corkxtraction (Fortes et al., 2004).
Internally to the cork back, seasonal cork rings develop until theext cork extraction takes place. The innermost cork layer is calledhe cork belly (Pereira, 2007). Fig. 1 shows the structure of cork,isplaying the cork back, the cork rings and the cork belly.
The cork produced by this second periderm is called the sec-nd cork. It also displays deep fractures and is therefore still nothe most desired for the production of cork stoppers. When theecond cork is removed, the process of traumatic periderm for-ation is repeated and a new layer of cork is produced. The cork
btained from the third and subsequent cork extractions is namedhe mature or the reproduction cork.
Barbato (2004) reported that the differences in the density ofhe cork tissue between the inner and the outer cork rings are notignificant, but that there are significant differences in relation tohe cork back. The cork back has about three times the density of theork: 150–225 kg m−3 in cork layers and 495–608 kg m−3 in corkack (Pereira, 2007).
The water content in the recently extracted cork varies betweenndividuals and even within the cork extracted from the same tree.or this reason, the cork humidity is of great importance to the mar-eting and sale of cork and when assessment of the cork productions necessary for landowners or manufacturers. It is common prac-ice to commercialize cork after it has already been air-dried, sincehe water content variability is much lower at this stage, althoughhe water content still depends on the environmental conditions,uch as temperature, humidity and storage conditions.
The first models for mature cork weight prediction did not con-ider the variation in the weight of freshly extracted cork, and usedhe fresh cork weight as the dependent variable (e.g., Ferreira etl., 1986; Montero et al., 1996). Costa and Oliveira (1998) presenthe first reference in the literature that employed oven-dried corkeight (cork biomass) as the dependent variable. More recently,ázquez and Pereira (2005) also used the oven-dried cork weight,
hile Ribeiro and Tomé (2002) used the air-dried cork weight after
wo weeks of field drying.The earliest models, however, are simple or multiple linear mod-
ls (e.g., Costa and Oliveira, 1998; Guerreiro, 1951; Natividade,950) that use the cork weight as the dependent variable. Het-
anagement 259 (2010) 1993–2005
eroscedasticity is usually present in these models, with largervariability in the model error corresponding to increasing values oftree size. In order to overcome this problem, some authors used thelogarithm of cork weight as the dependent variable, with untrans-formed independent variables (Vázquez and Pereira, 2005), whilesome authors have log-transformed both the dependent and inde-pendent variables (Ferreira et al., 1986; Ribeiro and Tomé, 2002).Some authors have also employed weighted non-linear models(Fonseca and Parresol, 2001; Ribeiro and Tomé, 2002).
The independent variables used in the existing models are thetree variables that are related not only to the tree size and shape,but also to the intensity of cork extraction; where these, to a certaindegree, give an indication of the management decisions related tocork harvesting. In Ribeiro and Tomé (2002), the cork thicknessafter boiling was also tested as an independent variable. It provedto be significant and is presented in two of the final selected modelsin this study. The thickness of the cork is important as it allows forthe comparison of differently estimated cork weight values fromtwo trees with the same size and the same extraction intensity, butwith different cork thickness due to genetic variation and/or thecork age and/or micro-site variation.
With the exception of Vázquez and Pereira (2005), all ofthe models were fitted without considering a nested structureand the resulting spatial autocorrelation. In most cases, the treedata refer to trees measured inside of the plots, and, in somecases, they are grouped by regions. This analytical characteristicmay violate the basic assumption of error independence, sincemultiple observations from the same sampling unit may be cor-related. With the use of a mixed-model approach, the variabilityamong the plots can be modeled with the introduction of randomparameters, which are estimated simultaneously with the fixed-effects.
Vázquez and Pereira (2008) present an extensive literaturereview for cork weight models, as well as the definitions of themain points to consider when new models are being developed.No emphasis, however, is given to the problem of estimating corkbiomass values for different cork ages.
The majority of cork oak stands are debarked with a 9 or 10-yearrotation interval, but the landowners decide to extend the inter-val between two consecutive cork extractions in some cases. Thisdecision is subjective, taking into account one or more of the follow-ing issues: climatic conditions that characterize the cork extractionperiod, the health condition of the stand, the current cork price con-ditions or the expected increase in income due to the increase incork thickness. In some regions characterized by high cork growthpotential, landowners claim that it would be reasonable to decreasethe interval between cork extractions.
With the exception of Vázquez and Pereira (2005), all corkweight models estimate cork weight with 9 years of growth. Someauthors do not even specify the cork age from the data used inthe modeling process (e.g., Costa and Oliveira, 1998; Ferreira et al.,1986). Vázquez and Pereira (2005) developed a model to estimatethe cork weight with 9 or 10 years of growth, testing the significanceof cork age (stripping rotation) with the incorporation of a dummyvariable in the model. This dummy variable was significant in fiveof the nine selected models, but due to the proven strong multi-collinearity, it was only present in three of the final models. Theregional representation of the data set used in this study is limited,being collected in 12 plots from 5 regions, from which 3 had corkaged for 9 years and 9 had cork aged for 10 years.
The development of a single model that allows for the predic-
tion of the mature cork weight for different cork ages, using dummyvariables or the cork age as regressors, requires a data set with corkweight for a wide range of ages. Due to legal restrictions, however,it is impossible to extract and to determine the cork weight for corkwith ages less than 9 years of growth. On the other hand, data con-
J.A. Paulo, M. Tomé / Forest Ecology and Management 259 (2010) 1993–2005 1995
Table 1Variables measured or registered in each tree.
Type of variable Variable Unit Description
Tree size variables du cm diameter under bark at breast height (1.30 m)h m total tree heighths m stem heightnbrd1 – number of debarked first-order branches
Variables related to management options hdv m vertical debarked height (measured to the highest debarked part of the stem or branches)
k thickk thickture c
ct
mra2tcbttb
abtumtotws
2
2y
ctodt
gbi
1
23
Cork variables ctbb mm corctab mm corwcm kg ma
erning cork weight with more than 11 years of growth is difficulto obtain for model fitting and validation.
In addition to the lack of data for the development of such aodel, the introduction of the dummy variables or cork age as
egressors is also a limited solution due to the high variabilitymong trees that is found for cork thickness (Almeida and Tomé,008; Ferreira et al., 2000; Sánchez-González et al., 2007a,b) dueo the effects of the tree genetics and location (micro-site). Whenorks of different trees and origins are being considered the corkiomass does not always increase with cork age. Unless the corkhickness at a certain base age is used as regressor, it is difficulto include the effect of cork age explicitly in a model for corkiomass.
The main objective of this work was the development ofmethod that allows for the prediction of the mature cork
iomass with t years of growth, based on the measurementsaken at any other age. The method can also be operationallysed for cork oak stand management, and as part of a growthodel for cork oak stands. When incorporated in such models,
his method will allow for analyzing the impact of decreasingr extending the interval between two consecutive cork extrac-ions. It will also allow for the prediction of cork biomass growth,hich is essential for the prediction of the evolution of carbon
tocks.
. Data and methods
.1. Developing a methodology to predict the cork biomass with tears of growth from one measurement taken at any other age
This methodology was based on the analysis of the variability ofork density from the back to the belly. Barbato (2004) found thathe differences in the density of the cork tissue between inner anduter cork rings are not significant, but that there are significantifferences between the cork rings (including the cork belly) andhe cork back.
The method for the estimation of cork biomass with t years ofrowth (wcmt), based on cork biomass and cork thickness (afteroiling) with 9 years of growth (wcm9 and ctab9), can be presented
n four different steps:
. Estimate the tree cork biomass for a cork with 9 years of age(wcm9).
. Estimate the cork back weight proportion at 9 years of age (cbp9).
. Estimate the biomass of cork tissue for a cork at 9 years of agefree from the cork back (wcm9 b):
wcm9 b = wcm9
(1 − cbp9
100
)= wcm9 − wcm9
cbp9
100︸ ︷︷ ︸biomass of cork back
ness at breast height (1.30 m) before boilingness at breast height (1.30 m) after boiling
ork biomass
4. Estimate the cork biomass for t years of growth (wcmt):
wcmt = wcm9 bctabt
ctab9︸ ︷︷ ︸biomass of cork tissue
+ wcm9cbp9
100︸ ︷︷ ︸biomass of cork back
,
where ctabt is the cork thickness after boiling with t years ofgrowth and ctab9 is the cork thickness after boiling with 9 yearsof growth.
The use of cork thickness after boiling along the different stepsof the presented method, instead of the respective value beforeboiling, was due to the fact that this variable best represents the realcork thickness from each tree, since it is measured after the internaltensions, caused by the cellular corrugation during cork growth,have been relieved during the water boiling of cork (Pereira, 2007).This effect is particularly important in the radial direction wherecork thickness is measured.
The implementation of the proposed methodology implies thedevelopment of two submodels:
• A model to predict the cork biomass for cork with 9 years of age.• A model to predict the cork back proportion.
If the data are collected when the cork is not 9 years old (corkinventory may be carried out some years before the cork extractionwhen the cork has less than 9 years of age), cork thickness at 9 yearscan be estimated using the Almeida and Tomé (2008) differenceequation model, using the cork thickness measured at any othercork age. Since the Almeida and Tomé (2008) model refers to corkthickness after boiling, it was again reinforced the convenience ofthe method being developed based in cork thickness after boilinginstead of cork thickness before boiling.
An alternative approach when no cork thickness data is availablefor any cork age is to use the average cork thickness for the region(obtained through measurement or predicted with a model such asthe one proposed by Sánchez-González et al. (2007a)).
2.2. Data for tree cork biomass
Two sources of data were used in the modeling of the tree corkbiomass. The first data set refers to the data collected in 131 plots,for a study on the forest inventory design and was obtained insix stands representative of the most important regions for corkproduction. The second data set refers to a sample designed forcharacterization of cork in Portugal and was obtained from 22 per-manent plots, covering all of the cork oak regions in Portugal. On
both data sets plots were circular and had an area of 2827.43 m2
(radius equal to 30 m). Fig. 2 shows the number of plots along thedistribution area of cork oak in Portugal. Tables 1 and 2 respectivelypresent the variables measured on each tree and information aboutthe stand characteristics in the data set.
1996 J.A. Paulo, M. Tomé / Forest Ecology and M
Ft
spicbd
pi
s
•
TS
Ntw
ig. 2. Cork oak distribution in Portugal (pure and dominant stands) and plot loca-ion. Numbers indicate the number of plots in each location.
The trees were debarked and measured during the growingeason, which is the appropriate period for removing the cork. Areliminary analysis of the data set was carried out in order to
dentify trees with extreme values of cork production due to treeharacteristics that affect production (e.g., wounds along the stem,roken branches) or incomplete measurements. These trees wereiscarded from further analysis.
The final data set included 763 trees, 327 from the inventorylots and 436 from the permanent plots. All of the cork produced
n the plots was mature cork aged between 9 and 11 years.Four different debarking heights have been used in previous
tudies:
vertical debarked height, measured vertically from the floor tothe highest point in the tree where cork was removed;
able 2ummary statistics of the stands characteristics for fitting and validation data sets.
is the number of trees per hectare; Gu is the stand basal area (m2) and dug ishe mean quadratic diameter (cm). The two last variables consider measurementsithout/under cork.
anagement 259 (2010) 1993–2005
• total debarked height, considering the stem height plus the sumof the debarked length in each debarked branch;
• maximum debarked height, considering the stem height plus thedebarked length from the branch with longest debarked length;
• average debarked height, considering the stem height plus theaverage of the debarked length in each debarked branch.
In large trees the measurement of the debarked length in thebranches is very difficult to obtain, implying the need to climbthe trees in order to measure the debarked length in the branches.Also, this measurement is not very precise due to the difficulty todefine the beginning of each branch. In the present study it wasdecided to use vertical debarked height, together with the numberof debarked first-order branches, assuming that it provides nearlythe same information about cork production on the tree.
The total extracted cork from each tree was weighted imme-diately after cork extraction, together with a sample taken atbreast height (20 cm × 20 cm) that was used to determine the corkhumidity in the laboratory, and consequently the mature corkbiomass. A second sample, also taken at breast height and withthe same dimensions, was used for cork thickness measurements,both before and after cork boiling (ctbb and ctab).
All of the permanent plots with cork extracted at ages otherthan 9 years were considered for validation purposes. Concerningthe inventory plots (all having cork with 9 years of growth), thefirst three trees measured on each plot were also selected for thevalidation set. The fitting data set consisted of data from all of thepermanent plots with cork extracted at 9 years of age, and all of thetrees from the inventory plots, except the three first trees in eachplot.
Table 3 shows the characterization of the two data sets. Fig. 3shows the relationship between mature cork biomass (wcm) anddiameter at breast height under cork (du) for the fitting and vali-dation data sets.
Notice that variable nbrd1, which refers to the number of first-order branches already debarked, assumes a value of 0 when thetree was never debarked (virgin trees to which this method is notapplied), 1 when the tree was only debarked in the stem, and 2 ormore when there are first-order branches already debarked.
2.3. Data for cork back weight
In order to develop the model to predict the cork back propor-tion cork samples had to be collected, since it was not feasible toseparate and quantify cork back from the rest of the cork alongthe entire the cork extracted in one tree. The cork samples wereextracted at breast height, since it has been recognized that theaverage characteristics from the cork extracted in one tree are rea-sonably assessed by the cork extracted in this level of the tree stem(Corkassess, 2001).
The selection of cork samples to determine the cork back weightwas designed so that the number of samples was as balanced aspossible along the cork quality and the cork thickness classes.
Cork quality is evaluated using the visual observation of a corkplank or a cork sample on the transverse and radial sections, aswell as on the innermost cork layer surface (cork belly). The corkwas classified into seven classes commonly used for industrial pur-poses, ranging from the 1st (best) to the 6th (worst) and includinga refuse class. The characteristics that determine the quality clas-sification are porosity, given by the amount of lenticular channels,and the presence of defects. Due to the subjectivity of the classifi-
cation process, which is carried out by an operator, it is common touse aggregated quality classes. In this study, the sample selectionincludes the 1st to the 3rd representing good quality cork, the 4thand the 5th for medium quality cork and the 6th and refuse for poorquality cork.
J.A. Paulo, M. Tomé / Forest Ecology and Management 259 (2010) 1993–2005 1997
Table 3Summary statistics for fitting and validation data sets.
Data set Cork age (years) Variable Mean Standard deviation Minimum Maximum
ig. 3. Relationship between mature cork biomass and diameter under bark at breith 9 -×-,10 -♦- and 11 -+- years of growth).
To determine the cork back weight, the samples (previouslyoiled) were dried until stabilization of the weight. After the deter-ination of the dry weight, the cork thickness was measured and
he cork back was removed with a sharp knife. Separate weightsere determined for the cork and the cork back and the percentage
f cork back on each cork sample was computed.
Table 4 shows the mean values of the percentage of cork back
nd the number of observations (in parentheses), obtained from aotal of 184 samples that were randomly selected from the set ofork samples collected on the 22 permanent plots corresponding tohe second data set described in Section 2.2. The values for the per-
able 4ean and number of observations (inside the parenthesis) of the percentage of cork back
ight: a) fitting data set (cork with 9 years of growth); b) validation data set (cork
centage of the cork back in the data set vary from 11% to 48%, withan average value of 28%. The cork samples ranged between 9 and16 years of age. Fig. 4 shows the decreasing relationship betweenthe cork back percentage and the cork thickness after boiling.
2.4. Model for cork biomass prediction in cork with 9 years of age
Since the relationship between tree size, management variables,cork thickness and cork biomass is allometric, this type of modelwas considered for modeling cork biomass with 9 years of age.
weight on each combination of thickness and quality class.
ig. 4. Relationship between cork back weight percentage and cork thickness afteroiling.
Four groups of models were considered, each group correspond-ng to a different level of forest inventory information:
(I) Models considering only the diameter at breast height.(II) Models considering other tree size variables, namely the num-
ber of debarked first-order branches.(III) Models considering the tree size and the management options
variables.IV) Models considering the tree size, the management options and
the cork variables.
The group I models represent very simple models, allowing forhe estimation of the cork biomass growth when only the diam-ter at breast height under the bark was measured. The groupI models tested the significance of the additional tree size vari-bles, namely the number of debarked first-order branches and theotal tree height. Group III differs from the previous groups by thentroduction of the vertical debarked height, a variable related to
anagement options. This variable is important to compare treeork production variation when different debarking intensities areeing considered as management scenarios. Finally, group IV allowsor the introduction of cork information, when the cork inventoryas conducted simultaneously with the forest inventory. Estimatesade with these last models include the tree genetic and the local
ite (micro-site) variation, since two trees of the same size that arequally managed will result in different cork biomass values due tohese two variables.
Other than the variables presented in Table 2, the mean standalue of the cork thickness before and after boiling were tested asegressors.
As a starting point for the definition of the allometric mod-ls, these were fitted in non-linear form using the SAS MODELrocedure (SAS Institute Inc., 2004), considering the different com-inations of variables for each group and, as mentioned before,esting some of the regressors for their effect on the allometriconstant associated to other variables.
The selected models were then fitted with a mixed-modelpproach considering the random effect of the stand, using theestricted maximum likelihood (REML) estimation method avail-ble in the SAS NLINMIX macro (Littell et al., 2006).
The general model structure for the mixed non-linear modelsas:
cm9ij = (ˇ0 + s0j)k
˘p=1
[X(ˇp+spj)
pij
]+ eij,
anagement 259 (2010) 1993–2005
where wcm9ij is cork biomass for cork at 9 years of age of tree i instand j, ˇp (p = 0, . . ., k) are the coefficients associated to the inde-pendent parameters and the allometric constant, spj is the randomeffect of stand j, Xpij is the pth covariate at tree level and eij is thetree random effect or pure error.
In models with more than one variable, different random effectswere considered. The procedure started by adding a random effectassociated with each fixed parameter. If the random effect wassignificant (by computing the confidence limits for the covarianceparameter estimates), the full model was retained. If two or morefull models were obtained through this process, a more complexmodel was then fitted considering both random effects associatedwith the two fixed parameters. This more complex model was thencompared with the previous model and the procedure was repeateduntil the final model was obtained.
When comparing two nested models (one being a special case ofanother differing only in their random effects), the AIC values wereused since the Restricted Maximum Likelihood Method invalidatesthe use of the likelihood ratio test (Pinheiro and Bates, 2000).
If random effects proved to be significant in two or more param-eters from the final model (evaluated by the asymptotic t-test),the covariance structure between them was tested for significanceassuming the variance covariance matrix of model parameters wasunstructured.
The normality of the model errors was assessed by analyzingthe QQ-plot of the studentized residuals. Plots of the studentizedresiduals against the predicted values were used to detect the het-eroscedasticity, since it is common to find that the variance of errorsis dependent on the size of the tree (larger trees usually corre-spond to larger variance). When this characteristic was detected,weighted regression was used, testing the significance of the twofollowing variance functions (Fang and Bailey, 2001):
- the power function
f (�ij, ˛) = �˛ij ,
- the exponential function
f (�ij, ˛) = exp(˛�ij),
where �ij is the mean function defined as �ij = E[yij|�j], yij is thecork biomass for cork with 9 years of age of tree i in stand j and�j is the vector parameter for plot j.
Both fixed and random effects parameters were tested for sig-nificance by the computation of the asymptotic t-tests, for a levelof significance of 0.05 (alpha = 5%).
2.5. Model for cork back weight proportion
As mentioned in Section 2.1 the cork back proportion was mod-eled as a function of cork thickness after boiling, so that all themethodology was based in the same variable.
The influence of the cork quality and the cork age was alsostudied, by testing the significance of these variables in the modelparameters (evaluated by the asymptotic t-test).
Since the dependent variable was a proportion, the selectedmodel had to be limited to values from 0 to 1. The cork backweight proportion equal to 1 represents the initial moment
of cork growth immediately after cork extraction, when theonly tissues covering the tree are the ones that correspondto cork back. The cork back weight proportion model musttend to 0 when the cork thickness tends to very large val-ues, representing the decrease of the cork back proportion
J.A. Paulo, M. Tomé / Forest Ecology and Management 259 (2010) 1993–2005 1999
Table 5Models tested for cork back weight proportion modeling.
bp is cork back proportion and ctabt is cork thickness at breast height (1.30 m) afte
hen the cork is not extracted and continues to growndefinitely.
After a review of the models that present these character-stics, two different types of models were considered: convexyperbolic functions (Ratkowsky, 1990) and switch-off functionsSchabenberger and Pierce, 2002). Switch-off functions are math-matical functions that take values between 0 and 1, and have theole of determining how the transaction between the two extremesakes place. After testing several convex hyperbolic functions pro-osed by Ratkowsky (1990), it was determined that only functionsith more than one parameter fitted the collected data. A total
f eight models were considered: four convex hyperbolic func-ions with two parameters and four switch-off models (Table 5).he two-parameter Weibull leads to the S1 model, which allows aigmoidal transition between the maximum and minimum valuesefined for the switch-off function (1 and 0 respectively). The modelresents an inflection point at ctab = a((b − 1)/b)b. S2 is anotherodel that was also applied in forestry research (e.g., Gregoire and
chabenberger, 1996). The Gompertz growth model originated the3 model, which is sigmoidal with an inflection point at ctab = b, butt is not symmetric about the inflection point. S4 (e.g., Valentinend Gregoire, 2001) is a much more flexible model that allowsor a test of whether the fitted model takes a hyperbolic or sig-
oidal shape, since for b > 1 it is sigmoidal with an inflection pointt ctab = a((b − 1)/(b + 1))1/b, and for b < 1 is hyperbolic.
The final model (fitted to cork back proportion values varyingrom 0 to 1) was selected comparing the AIC value of each modelPinheiro and Bates, 2000).
.6. Validation
Validation had the objective of analyzing and characterizing therror of the several cork biomass models, when used jointly withhe model for cork back weight percentage, to predict cork biomassor corks of different ages.
Using the validation data set, the following validation statisticsere computed, for both the bias and the precision assessment.
Model efficiency or the proportion of variation explained by theodel (ef):
f = 1 −∑n
i=1r2i∑n
i=1(yi − y)2.
Mean value of the residuals (Mr) to evaluate bias:
r =∑n
i=1ri
n.
Mean of the absolute value of the residuals (M|r|) to evaluaterecision:
∑n ∣∣ri
∣∣
|r| = i=1
n,
here ri is the residual for observation i, yi is cork biomass valueor observation i, y is the mean value of the cork biomass from allf the observations and n is the number of observations.
Percentiles 95 and 5 of the distribution of the residuals werealso computed in order to help assessing model precision.
Since the final objective was to develop a method that is ableto predict cork biomass for corks with different ages, the valida-tion statistics were also computed separately according to cork age.Since the validation data set only included cork data with 9, 10 and11 years of age, it was only possible to assess if there was a signifi-cant tendency on the method accuracy when cork age increases.
Validation was also carried out for each model separately usingdiameter classes (5 cm range) and cork thickness classes. Corkthickness classes were defined according to the limits presentedin Table 3.
Since the cork biomass increases with tree size, namely withdiameter, it is expected to find larger values of the residuals in largertrees. Nevertheless, these larger values of residuals do not neces-sarily correspond to lower precision of the model, since they mayrepresent a small percentage of the total cork biomass produced bya tree with large dimensions. Therefore, when evaluating precision,the mean of the absolute value of the residuals was computed inpercentage:
M∣∣rj
∣∣ = 100
∑ni=1
∣∣ri
∣∣/nj
cj,
where ri is the residual for observation i, nj is the number of obser-vations in class j and cj is the mean value of cork weight from classj.
Since this statistic was computed as a percentage for the diame-ter classes, this criterion was also maintained for the other variablesalong which precision was investigated along classes.
When computing these statistics using diameter class, it wasfound that in some extreme classes the number of observations wasvery low. For instance, the number of trees with diameter under thecork larger than 70 cm was only 5. For this reason, only one classwas considered for observations with the diameter under cork over70 cm. This guaranteed that all of the classes had a minimum of 5observations. Cork thickness classes were computed according tovalues defined by the cork industry, which were already presentedin Table 4 (Section 2.3).
For validation purposes, it was assumed that no data existedfor the calibration of the cork biomass model for cork at 9 years ofage. This assumption was based on the fact that, in practice, it isnot expected to have data on cork weight for model calibration asthe developed method will be used for two main purposes: 1) as amodule of the cork oak stand growth models; 2) to predict the corkweight before extraction when the cork is younger or older than 9years.
Thus, the expected value 0 was used for all random parameters.The model predictions represent the pattern of typical responsesand define the mean behavior of the cork biomass for cork that is 9
years old.
If cork biomass data will be available, the prediction of randomparameters can be carried out using one of the expressions pre-sented in the literature (e.g., Dorado et al., 2006; Meng and Huang,2009; Trincado and Burkhart, 2006).
2000 J.A. Paulo, M. Tomé / Forest Ecology and Management 259 (2010) 1993–2005
Table 6Selected models for cork biomass prediction for 9 years old corks and respective fitting statistics.
Model Number of parameters Expression AIC
Fixed Random
I 2 2 a du(b+�) 3384.6II 3 2 a du(b+�) (nbrd1)c 3362.7III 4 2 a du(b+�) hdv[c+d ln(nbrd
1)] 3233.7
IV 5 2 a du(b+�) hdv[c+d ln(nbrd1
)] [ln(ctab9)]e 3182.8
F ss witt
3
3
t
aabcmvah
nsns
ovte
s
TP
ig. 5. Plot of the studentized residuals against the predicted values of cork biomahe power variance function in the model.
. Results
.1. Model for cork biomass prediction in cork with 9 years of age
Table 6 displays the selected models, the number of parametershat they include and the AIC statistic used for model selection.
As expected, the AIC value decreased from model I to IV, withn increase in the amount information used by the models. It waslso evident that this increase of information was more relevantetween model II and III than between the other models. This indi-ated that the tree size and the management option variables wereore important to explain the cork biomass values than the cork
ariables. The cork variables are more dependent on tree geneticsnd stand characteristics, such as soil and climate, which seem toave less impact in terms of the biomass extracted.
It is important to notice that total height and stem height wereever significant in any of the fitted models. From all of the treeize variables available, only the diameter at breast height and theumber of debarked first-order branches were incorporated in theelected models.
When considering the cork variables (model IV), the logarithmf cork thickness after boiling proved to be the most significant
ariable. Cork thickness before boiling and the stand mean corkhickness before and after boiling was not significant in any model,ven before the introduction of random effects.
Plots of the studentized residuals against the predicted valueshowed that heteroscedasticity was present in all models, even
able 7arameter estimates and variance components for each of the selected models.
Model Estimated value for fixed parameters Estimated varian
h 9 years, estimated with the model IV, before (a) and after (b) the introduction of
after the introduction of random effects. In all cases, the exponentialvariance function resulted in difficult convergence for the modelsand led to higher values for AIC. The power variance function pre-sented good results in all of the models, with an evident decrease ofthe AIC values when compared to the non-weighted models (val-ues not shown). The new plots of the studentized residuals againstthe predicted values showed that the heteroscedasticity was over-come. Fig. 5 displays this plot considering the predicted values frommodel IV, before and after the introduction of the power variancefunction in the model for heterocedasticity correction.
Normal probability plots for the studentized residuals were alsoconstructed (data not shown), showing that the residuals had adistribution close to normality, in accordance with the assumptionsof the non-linear mixed-effects models theory.
Table 7 presents the estimates of the fixed parameters and theestimated variance of the random effects for each of the resultingmodels.
3.2. Model for cork back weight proportion
From the three cork characteristics considered as regressorsonly the cork thickness (after boiling) presented significant rela-
tionship to cork back proportion, and this variable was used for theconstruction of the model. This relationship is evident in Fig. 4 (seeSection 2.3), where a monotonically decreasing tendency is clear.
Variability among the plots proved to be significant in all of themodels, except for H2. Comparing the results obtained with the
ce of random effects Power variance function Estimated residual variance
J.A. Paulo, M. Tomé / Forest Ecology and Management 259 (2010) 1993–2005 2001
Ff
hSfi
nmamtretCase
t
c
e0
3
ftpfi
Table 8Validation statistics. Mr , mean value of the residuals (kg); M|r| , mean of absolutevalue of the residuals (kg); P5, percentile 5 of residuals (kg); P95, percentile 95 ofresiduals (kg); ef, model efficiency.
ig. 6. Plot of the studentized residuals against the predicted values from the modelor cork back weight proportion.
yperbolic and the switch-off model (values not shown), model2 and the hyperbolic models presented the poorest results whentted to the cork back proportion data.
Model S3 presents an inflection point for ctab = 17.7, which isear the minimum value presented in the data set (ctab = 16.1),eaning that the form of the function does not vary significantly
long the ct values in the data set. When fitting S4 model, the esti-ate for parameter b equals 0.7894. Since this value was smaller
han 1, it was concluded that the function was hyperbolic. Theseesults allowed us to conclude that although the switch-off mod-ls performed best, the cork back percentage as a function of corkhickness was best explained by functions with hyperbolic form.omparing the AIC values of S1 (AIC = −503.8), S3 (AIC = −501.5)nd S4 (AIC = −501.3) models, the first was selected. The plot of thetudentized residuals against the model predicted values (Fig. 6)vidence the inexistence of heteroscedasticity in the model.
The final expression obtained for the cork back weight propor-ion model can be presented as:
bp = exp[−[ctabt/(19.4629 + �)]0.4744]
The estimated variance component of the plot of the randomffect (�) was 5.5803 and the estimated residual variance was.0034.
.3. Validation
Four different systems – each one stemming from one of the dif-erent models for biomass prediction of cork with 9 years of age andhe cork back weight proportion model – were compared. Table 8resents the values of the computed validation statistics for thenal systems of equations.
ig. 8. (a) Bias of the selected models with cork age–mean value of the residuals. (b) Precin percentage).
Fig. 7. Model efficiency along cork age classes for the different models.
When observing the evolution of the validation statistics fromthe model I to model IV, it was evident that model efficiencyincreased when additional information was incorporated in themodel for 9 years old cork biomass estimate. This increase wasmore significant between model II and model III, correspondingto the incorporation of vertical debarked height in addition to thetree size variables. The incorporation of cork variables resulted inthe increase of model efficiency but did not have a large effectin model bias and precision. The diameter at breast height undercork and vertical debarked height proved to be the most importantvariables when cork weight modeling is concerned.
Fig. 7 shows the evolution of the model efficiency along thedifferent cork ages available in the validation data set. All mod-els behave is similar for the three considered cork ages, but it wasevident that the model efficiency increased with model complex-ity. Models I and II performed worse for corks at 11 years of age,but this tendency was not visible for the models III or IV.
When comparing the four different systems through the meanvalue of the residuals (Fig. 8) it was observed that the bias was
generally small, varying from −2.0 to 2.5 kg, representing around6% and 8% of the average cork biomass from the validation data set.However, it was also noticed that models I and II showed a slighttendency to underestimate cork biomass with increasing cork age.No clear tendency was present in models III and IV.
ision of the selected models with cork age–mean of absolute value of the residuals
2002 J.A. Paulo, M. Tomé / Forest Ecology and Management 259 (2010) 1993–2005
Fig. 9. (a) Bias of the selected models with diameter under cork–mean value of the residuals. (b) Precision of the selected models with diameter under cork–mean of absolutevalue of the residuals (in percentage).
F uals.t
fmavfc
owmyb
aafsetc
Fittitp
w
ig. 10. (a) Bias of the selected models with cork thickness–mean value of the residhe residuals (in percentage).
Large differences were found in terms of precision when theour models where compared (Fig. 8). As expected, more complex
odels are much more precise. Model I, which uses only diametert breast height as regressor variable, presents a mean of absolutealue of the residuals around 25%, and this value decreases to 15%or model IV (which included tree size, management options andork variables).
When looking at the evolution of the mean of the absolute valuef the residuals of all four selected models along cork age (Fig. 8), itas concluded that the precision was not very different when theethod was applied for estimating cork biomass with 9, 10 or 11
ears. This showed that the method was able to estimate the corkiomass irrespective of cork age.
Fig. 9 presents the evolution of the mean value of the residu-ls and the mean of absolute value of the residuals (in percentage)long diameter classes. Model I and II tended to be slightly biasedor diameter classes over 50 cm since they only include tree dimen-ion variables and, in these large trees, the intensity of the corkxtraction is quite variable. Again, large differences were found inerms of precision since it clearly increases together with modelomplexity.
Similar plots for the cork thickness classes are presented inig. 10. When observing bias along the cork thickness classest was clear that only the model IV was unbiased for all corkhickness classes. All of the other three models tended to overes-imate the cork biomass for the thinnest corks and under estimate
t for extra large ones. The incorporation of the variable corkhickness after boiling, guaranteed unbiased and more preciseredictions.
When the method was applied with the model IV, the resultsere again more precise for all of the cork thickness classes. Differ-
(b) Precision of the selected models with cork thickness–mean of absolute value of
ences in the model III are evident for the extra thin and the extralarge classes, but they are nearly inexistent for medium cork thick-ness values. Models I and II present higher values for the mean ofthe absolute value of the residuals (in percentage) that vary from17% to 30%.
4. Discussion
The method developed here will allow quantification of theevolution of the existing cork biomass in stands between twocork extractions and further study on the impact of decreasingor extending the cork extraction period on tree biomass produc-tion. Questions are frequently raised by landowners, managers andresearchers about the economic outcomes of different extractionperiods, but these questions were impossible to answer until nowdue to regulations regarding cork harvest. The models of Vázquezand Pereira (2005) include a dummy variable to account for corkbiomass at 10 years, but the models depend on variables that arenot measured in operational forest inventories, even those devel-oped for management purposes (e.g., total debarked height, meancrown radius, cork density). In addition, these models do not allowfor the prediction of cork biomass growth in the first years imme-diately after debarking, or when cork is extracted with more than10 years of growth.
The first three models developed in the present study dependonly on variables usually measured in forest inventories (diameter
at breast height and height measurements), which represent thetwo main sources of variation when the cork biomass is estimated:the tree size and the management options. Additionally, modelsalso include the number of first-order branches already debarked,assuming a minimum value of 1 for trees only debarked in the stem.
and M
Tf
imtlVlaipmae
ioatsobscitwsddrIp2midiooya
(ooPbfiPsoldtfc(bos
faa
J.A. Paulo, M. Tomé / Forest Ecology
his variable does not represent additional complexity or time inorest inventories since it is obtained by simple tree observation.
In accordance with the results obtained by other authors, thentroduction of a variable related to debarking intensity into the
odels increased model efficiency and improved validation statis-ics. Nevertheless, variables like total, maximum or mean strippingength (e.g., Fonseca and Parresol, 2001; Ribeiro and Tomé, 2002;ázquez and Pereira, 2005) imply the measurement of debarked
ength on the branches, very difficult to achieve with high precisionnd being a cost demanding operation, since it requires climb-ng the trees. This procedure is therefore mainly used for researchurposes and in very particular cases of forest inventory, and theodels that use these variables are difficult to use in practice. The
dvantage of using vertical debarked height is the fact that it isasily measured with a hypsometer.
One of the selected models includes a variable directly measuredn the cork (model IV), allowing for the prediction of different valuesf cork biomass for different trees that present the same dimensionnd the same exploration intensity, that are originated by inter-ree and micro-site variation. Four cork variables were tested forignificance in this group of models, previously to the introductionf random effects: tree cork thickness, measured before and afteroiling, and the mean value of cork thickness of the stand, also mea-ured before and after boiling. The mean value of cork thicknessomputed for each stand (before and after boiling) was not signif-cant when added to the model as a potential regressor, contrarilyo the tree cork thickness after boiling. Differences between standsere only significant when random parameters associated to the
tand were added to the model, implying that in order to estimateifferent cork biomass values for two different trees, with the sameimension and with the same management variables, but occur-ing in two different stands, the random parameters from modelV had to be estimated using one of the calibration methods pro-osed in the literature (e.g., Dorado et al., 2006; Meng and Huang,009; Trincado and Burkhart, 2006). Since this implies the deter-ination of cork biomass in some trees during the forest inventory,
t is not to be expected that model calibration will be accomplisheduring the practical use of the model. For this reason, in order to
ncrease the precision of the predictions given by the applicationf the developed method, it is important to use model IV insteadf one of the simpler models when determining cork biomass at 9ears of age, since it is the only model that can capture the inter-treend micro-site variability.
When looking at the existing literature, only Ribeiro and Tomé2002) and Vázquez and Pereira (2005) tested the significancef these types of variables, although the first authors variablenly considered the cork thickness after boiling, and Vázquez andereira (2005) considered both the cork thickness before and afteroiling. Cork thickness after boiling proved to be significant in thenal selected models, a result different from that of Vázquez andereira (2005) that found just cork thickness before boiling to beignificant, but consistent with Ribeiro and Tomé (2002). Since onef the effects of the cork water boiling is the expansion and stabi-ization of the dimension of the cork tissue, particularly in the radialirection where cork thickness is measured, it is thought that corkhickness measured after boiling best represents the cork thicknessrom the tree, since it is measured when the internal tensions thataused the cellular corrugation during cork growth were relievedPereira, 2007). The significance of the variable cork thickness afteroiling in the model for cork biomass at 9 years of age, insteadf cork thickness before boiling, reinforced the model consistency,
ince all the equations that it uses refer to this variable.
The method developed in this study was also based on a modelor cork back weight proportion determination. Although there ishigh variability in cork back thickness between trees (Fortes et
l., 2004), an evident monotonically decreasing tendency between
anagement 259 (2010) 1993–2005 2003
cork back proportion and cork thickness was confirmed in thepresent research during the development of the model for corkback weight proportion. The model refers to samples obtained atbreast height, since it is recognized that the average characteris-tics from the cork extracted in one tree are reasonably assessed bythe cork extracted at that height (Corkassess, 2001). This impliesthat the proposed method is based in the assumption that the corkthickness and the cork back proportion at breast height representthe average from the tree. Natividade (1950) observed a single treefrom 0.10 to 3.10 m of stem height, and noticed a slight decreasein cork thickness, but smaller to 1 cm. The same author also indi-cates that an opposite situation may occur, when local defects arepresent in the stem or in the branches, implying that in these cir-cumstances the cork may be thicker in higher parts of the tree.Montero and Vallejo (1992) looked at the cork thickness variationalong tree height of 100 trees with different diameter sizes andstripping heights. The authors concluded that cork thickness mea-sured at a certain height does not depend neither on diameter sizenot on the stripping height. More recently Taco et al. (2003) studiedthe same variable on 12 cork oak trees (to a maximum debarkingheight of 2.6 m), and found that the within tree variation of corkthickness was not uniform for all trees. Despite the trend of corkthickness variation (decreasing, constant or increasing), the valuesof cork thickness in the same tree did not vary more than 1 cm. Thesame author also concludes that a cork sample taken at 1.3 m ofheight may be used to characterize the tree average in relation tototal thickness.
These results indicate that although the proportion of cork backmay slightly vary along the stem, these differences may not be sig-nificant within one tree and may be opposite for different treesexisting in the same stand. A good estimation of the average value isthen thought to be obtained using the variables measured at 1.3 m,in accordance to the method assumption.
Validation of the method proposed in this paper was in partlimited to the fact that data concerning cork biomass refers to treeswith a minimum cork age of 9 years and a maximum cork age of11 years of growth. This limitation is due to legal and economicconstraints that make it almost impossible to sample trees withcork extraction periods out of this interval.
For cork ages increasing from 9 to 11 years the method provedto be unbiased despite the model used for predict biomass of 9years old cork. The assumption on which the methodology is based– no significant differences in the density of different cork layerswhen cork back is not considered – is confirmed by the good resultsobtained for cork ages 10 and 11. This suggests that the method willalso perform well for other cork ages. Note that the prediction ofthe cork biomass for corks with an age different from 9 years (eitheryounger or older) is not made by an extrapolation of an empiricalmodel, but based on the above-mentioned physical characteristicof the cork tissue.
As an example, Fig. 11 shows the results of applying the pro-posed methodology to predict cork biomass from 1 to 14 years ofage, for two pairs of hypothetical trees. Each pair of trees presentsthe same diameter at breast height, the same exploration intensity(equal number of debarked branches and vertical debarking height)but different value of cork thickness. The following characteristicswere considered for each pair of trees:
• A (dash lines): diameter at breast height of 35 cm, vertical
debarked height of 1.5 m and number of debarked branches equalto 1 (tree only debarked in the stem).
• B (solid lines): diameter at breast height of 65 cm, verticaldebarked height of 4.5 m and number of debarked branches equalto 3.
2004 J.A. Paulo, M. Tomé / Forest Ecology and M
Fo
(twc
pe
••
AdaoDnsabemmwid
ttbtceevdmwtts
5
fo
ig. 11. Estimated values of cork biomass from 1 to 14 years of age, for the two pairf hypothetical trees.
Cork thickness after boiling at 9 years of age for trees A1 and B1graphic lines identified with “o”) was assumed as 25 mm (2nd corkhickness class) and for trees A2 and B2 (graphic lines identifiedith “x”) this variable was assumed as 50 mm (5th cork thickness
lass).The predictions in Fig. 11 seem reasonable from a biological
oint. Note that this application implied the use of two other mod-ls:
to predict tree diameter growth (Tomé et al., 2006).to predict cork growth (Almeida and Tomé, 2008).
Models I and II show some bias for diameter classes over 50 cm.n explanation for this is the fact that tree shape and the treeebarking surface are more complex and variable between treess they grow. When the tree is debarked for the first times (twor three first debarkings) the cork is only extracted from the stem.uring the life of a tree, debarking surface increases as well as theumber of debarked branches that depend on management deci-ions and tree shape. As a result, trees with the same dimensionre more variable in terms of tree shape, and consequently in corkiomass extracted, as their dimension increases. This result againxpresses the importance of using a model that includes manage-ent option variables such as vertical debarked height. The use ofore complex models is more relevant as the percentage of treesith larger diameters increases in the stand. According to the val-
dation results, models I and II should only be used in stands thato not include trees with diameters larger than 45 cm.
Model IV was not only the most precise, but also the only onehat was unbiased for all cork thickness classes. Since the otherhree models do not consider the influence of cork variables in corkiomass prediction, their estimates are similar irrespective of corkhickness and show a trend in bias for increasing cork thicknesslasses. These three models overestimate the cork biomass for thextra thin class and underestimate this variable for the large andxtra large classes. Medium cork thickness classes present the meanalue of the residuals near to zero, showing that for these interme-iate classes the estimates resulting from models I, II and III areore precise than for extreme cork thickness classes. However,hen applying these models at the stand level, it is expected that
he estimates for cork biomass per hectare will be unbiased due tohe large variability in cork thickness that is usually present in theame stand.
. Conclusions
This paper presents a methodology to estimate the cork biomassor corks of different ages. The methodology is very useful for corkak management as it allows:
anagement 259 (2010) 1993–2005
• for accessing the values of existing cork biomass in the stand atany moment and for any given period in time;
• for the studying of the impact of decreasing or extending theinterval between consecutive debarking on the tree biomass pro-duction and on the economic outcomes;
• the choice of the most appropriate model for cork biomass predic-tion, depending on the level of information collected in the forestinventory since the different models were developed consideringdifferent measurement intensities;
• the use of a model for cork biomass prediction that includes verti-cal debarking height as a management option variable, althoughthis does not imply significant additional costs in traditionalinventories;
• the use of a model for cork biomass prediction that includesinformation on cork thickness, collected in cork sampling, takinginto consideration the genetic and local (micro-site) variabilitybetween trees; and
• for modeling the variability among stands by the introductionof random parameters in the model for cork biomass predictionin cork aged 9 years and in the model for the cork back weightproportion.
Acknowledgements
Financial support was provided by project CarbWoodCork(POCI/AGR/57279/2004 and PPCDT/AGR/57279/2004) fromFundacão para a Ciência e Tecnologia (Portugal). This paper ispart of the PhD project of the first author, which is funded by ascholarship (SFRH/BD/23855/2005) granted by Fundacão para aCiência e Tecnologia (Portugal). We also thank Dr. Augusta Costafor help on cork back data collection, and the Coruche forestowners association (APFC).
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