A Variable-Form Taper Function for Pinus Oocarpa ... - Ingenta Connect

NOTES Forest Science, Vol. 36, No. 1, pp. 186-191.

Copyright1990by the Societyof AmericanForesters

A Variable-FormTaper Function for Pinus oocarpa Schiede

in Central Honduras

D. N. Perez, H. E. Burkhart, and C. T. Stiff ABST•ACT. A variable-formtaper equationwas developedfor Pinus oocarpaSchiede in centralHonduras.The modelpredictstree profile as a functionof total height,diameter at breastheight, and relative heightwith a continuousfunctionusinga changing exponentto compensatefor the form of differenttree sections.The proposedfive parametermodelpredictsunderbarkdiameterswith a standarderror of 1.4 cm. The point wherethe tapercurvechangesfrom neiloidto paraboloidform, the inflectionpoint, was assumedto occurat 25% of total height.For the data used,no changesin the predictive ability of the modelwere observedwith differentlocationsof the inflectionpoint. FOR. Sex. 36(1):186-191. ADDITIONAL KEY WORDS. Stem profile, volume.

TAPERFUNCTIONSare very useful when trees are utilized for a variety of products. Sincethe early nineteenthcentury,forestresearchershave searchedfor methodsto expresstree form and taper in terms of easily measuredtree characteristics.With improvedanalytictools, studiesof tree taper changedfrom graphicalto numerical methods(Gray 1956).Duringthe last two decades,a broadarray of numericalmethods have been applied, rangingfrom simpletaper equations(e.g., Kozak et al. 1969, Ormerod 1971, Amidon 1984, Reed and Byrne 1985) to more complex segmented models (e.g., Max and Burkhart 1976, Demaerschalkand Kozak 1977, Cao et al. 1980)which describethe taper of differenttree sections.The resultanttaper equations are usedwidely for estimatingmultiple-productvolumes,heightsto specified diameters,and diametersto given commercialheights. Variation in tree form makes it difficult to formulate general rules readily applicable to a singlespecies,or even to all the stemsin a singlestand(Larson 1963). Recently, Newnham (1988) and Kozak (1988) introduced a "variable-form" taper function which describestree taper with a continuousfunction using a changing exponentto compensatefor the form changesof differenttree sections.The objective of this study was to develop an accuratetaper equation, for trees of Pinus oocarpaSchiedein centralHonduras,that relies on easily observedvariables. DATA

Tree taperdata were collectedas part of a studyaimedat developinga site quality classification systemin the uplandpineforestsof centralHonduras(Stiff et al. 1987). The studyarea, centeredwithin 100km of Siguatepeque,includedthe provincesof Comayagua,La Paz, FranciscoMorazan, and Intibuca (Figure 1). One hundred

D. N. Perezis Professor,EscuelaNacionalde CienciasForestales,CorporacionHondurena de DesarrolloForestal,ApartadoPostal#2, Siguatepeque, Honduras,C.A.; H. E. Burthartis ThomasM. Brooks Professorof Forest Biometrics,Departmentof Forestry, Virginia Polytechnic Institute and State University, Blacksburg,VA 24061; and C. T. Stiff is Assistant Professor,Departmentof ForestResources, Collegeof Forestry,WildlifeandRangeSciences, University of Idaho, Moscow, ID 83843. The data used in this study were collectedunder ProjectAID/SCI 2E-05 supportedby the United StatesAgencyfor InternationalDevelopment Office of Scienceand Technologyand the Corporation Hondurenade Desarrollo Forestal (COHDEFOR). The authorswish to thank forestry techniciansfrom Escuela National de Ciencias Forestales(ESNACIFOR), and T. V. Dechert and F. D. Johnson,University of Idaho, for field data collection.Manuscriptreceived January 17, 1989. 186/FOREST

SCIENCE

ninety-fiveplotswere locatedin naturalstandsofPinus oocarparepresentinga wide rangeof sites,aspects,slopes,and elevations.The plots were clustersof five points arrangedin a circular or linear pattern. Within each stand,overstoryand understory vegetation,soil type, slopeposition, and slopeshapewere nearly uniform. Selected standswere reasonablywell-stockedand relatively free of recent logging,Qre, or insectinfestation.Plot selectionwas not constrainedby speciescomposition,stand density, or age structure.However, plots were located only in standspossessing suitablesiteindextreesofPinus oocarpa.At eachplot, data were collecteddescribing the physicalenvironment,understoryvegetation,and standcharacteristics. Two dominantor codominanttrees and one intermediateor suppressedtree, with diameteroutsidebark at breastheight(dbh at 1.3 m) equal to or greater than 10 cm, were felled and measuredfor total height and crown length on each plot. Dbh was measuredwith a diametertape to the nearestmm before the sampletree was felled. Sectionswere taken at stumplevel (0.3 m), 0.8, and 1.3 m. Above 1.3 m, trees were sectionedinto 10 equal-lengthsegments.The averageoutsideand inside bark diameters of the 12 resultingsectionswere calculatedas the geometricmean of two measurementsrecordedto the nearestmm usinga ruler held at right anglesacrossthe geometriccenter. Double bark thickness(dbt) for each sectionwas calculatedas the differencebetweenthe averageoutsideand insidebark diameter. Dbt was subtracted from dbh to obtain values for diameter inside bark at breast height (d). The edited data containedheight/diameterobservationsfrom 578 destructively sampledtrees. The sampledtrees averaged27.1 cm dbh, 19.1 m total height, and 35 years age at breast height. The range of measurementswere 10.0 to 59.0 cm, 8.2 to 34.4 m, and 11 to 174years, respectively.Sectioningresultedin 6936 stemobservations for fitting and testingtaper models.

The data were randomlydividedinto fitting and validationsubsets.Seventypercentof the sample,or 405trees, was usedfor fitting taper models,and the remaining 173 trees were usedfor model testing. MODEL

SELECTION

The variable-formtaper model proposedby Kozak (1988) has the following form:

d = boDbhl'lb•l'hx c

(1)

FIGu• 1. Study •ea in Honduras(shaded),with dotted•ne denoQngapproximate•ts of •tedor pine forestsw•ch •e dominatedby Pinus oocarpa,but includePinus tecunumanii

•d Pinusm•iminoi at •gher elevations,•d Pinuscaribaeain interiord• v•eys. MARCH 1990/187

where

x = - x/i), C -- b3z '• + b41n(z + 0.001)+ bs•z + b6e z + b7(Dbh/H), d = diameterinsidebark at heighth (cm), h = partial height above ground(m), Dbh = diameter outsidebark at breast height (cm), H = total tree height (m), z = h/H,

I = location of the inflection point,

bi = coefficientsestimatedusinglogarithmicregression. Kozak's (1988) model was linearized usinglogarithmictransformation,

ln(d) = ln(bo)+ b•ln(Dbh)+ ln(b2)Dbh + b31n(X)z 2 + bnln(X)ln(z + 0.001)

+ b51n(X)•/•z + b61n(X)e z + b71n(X)(Dbh/H)

andthenfitted with ordinaryleastsquaresprocedures.This functionhasthe property that diameter(dl) equals0 at the top of the tree. In addition,di is equal to the estimateddiameterat the inflectionpoint and the functionchangesdirectionwhen h,,/H = I. AlthoughKozak's modellooksvery complex,it is basicallya simplepower function, y__- mx c

where rn is the diameterat the inflectionpoint expressedas a functionof dbh. The inflectionpoint, where the taper curve changesfrom neiloid to paraboloidform, is assumedto occur at 25% of total height(Demaerschalkand Kozak 1977). As stated by Grosenbaugh (1966)andFurslund(1982),the powerfunctioncan expresstaperby modifyingthe value of the exponentc to accountfor the shapesof the lower, middle, and upper sectionsof the stem. Rather than usinigdiscretevaluesfor the different sections,the Kozak model(1) assumesthat the value of C varies continuouslyalong the stem.

Sevenindependentvariablesin Kozak's modelindicatethat perhapsstronglinear dependenciesexist amongthem. When this conditionis present, the coefficients from ordinary least squaresproceduresmay not be precisely estimated, thus producingareas in the regressorspacewhere predictioncould be poor (Myers 1986). One meansto combatmulticollinearityis to eliminatevariablesto the point where the quality of fit is not severelycompromised.With this in mind, a more parsimonious version of Kozak's model was fully explored. Before selectingthe "best" model, each was linearizedusinglogarithmictransformationand fitted to the sample data, then carefully scrutinizedusingresidualplots, partial plots, and collinearity diagnostics.The followingcriteria, basedon the fitting data set, were used in the selectionof the "best" reduced model: (1) Mean squareerror (MSE); (2) coefficient

of determination (R2);and(3) predictionsumof squares(PRESS).The bestmodel hadeitherthe highestR• value,or the lowestvaluefor one or moreof the other criteria.

Various variable-formtaper modelswere then evaluated using validation procedureswith the testingdata set to determinethe "best" predictive model. The mean differencebetween predictedand observeddiametersinside bark measuredthe accuracy (or mean bias) of prediction,and the standarddeviationof the differences measuredthe precision of predictions. Models were also compared using the total squarederror (TSE), definedasthe sumof the squaredmeanbiasand varianceof the differences.

RESULTS

AND

DISCUSSION

Alternative variable-formtaper modelswere fitted using logarithmicregressionassumingthat the inflectionpoint occurredat 25% of total height. Settingthe location 188/FOREST SCIENCE

TABLE 1. Estimated coefficientsand their standard errors (in parenthesis)for variable-form taper models1 and 2 basedon 6936 stem observations. Variable-form

model

Estimated • coefficients

Model 1

Model 2

bo

- 0.684788

- 0.432876

b•

(0.043985) 1.138908 (0.019668)

(0.014480) 1.027609 (0.004523)

b2

- 0.004298

--

(0.000759)

b3 b4 b5

0.543205

0.546840

(0.120686)

(0.007340)

- 0.036841

- 0.048592

(0.027232)

(0.003047)

- 0.240226

--

(0.252752)

b6

0.088589

--

(0.137796)

b7

0.150991

0.141629

(0.004706)

(0.004104)

• boandb2for modelI andbofor model2 aregivenin the naturallog scale.

of the inflectionpoint at 15%, 20%, 30%, and 35% of total heighthad little effect on the predictivepropertiesof any model. From the potentialmodelswith comparable fitting quality, preferencewas givento the one with the leastnumberof parameters. The "best" or most parsimoniousmodelthat did not show muchlossin predictive ability as comparedto the "full" modelof Kozak (1988) had the followingform:

d = boDbhb•X c

(2)

where

C = b3z 2 + bnln(z+ 0.001)+ b7(Dbh/H) Model 2 was fitted usingthe following logarithmictransformation,

In(d) = ln(bo)+ b•ln(Dbh)+ b31n(X)z 2 + baln(X)ln(z+ 0.001) + b71n(X)(Dbh/H) The estimated coefficients and associated standard errors for models 1 and 2 are

listed in Table 1. The regressorvariables in model 2 did not show strong linear dependenciesand, as a result, their coefficientestimatesare expectedto be more stable because of less inflated variances. Fit statistics for models 1 and 2 were

basicallyequivalent(Table 2). Table 3 showsthat model 1 is the best predictorbased on the standarddeviation of the differencesand the total squarederror; model 2 is, however, superiorwith regardto mean bias. Figure 2 showsthe meanbias per sectionalongthe stemfor models 1 and 2 based on 2076 independentstem observationsfrom the testing data set. The segmented TABLE 2. Fit statisticsfor variable-form taper models1 and 2 based on logarithmic regression. Model

PRES

MSE

R2

I 2

82.15 82.72

0.016842 0.016977

0.959 0.959

MARCH 1990/189

TABLE 3. Prediction statisticsfor variable-form taper models 1 and 2 based on 2076 stem observations.

Standarddeviation Model

Mean bias

Total squared

of the differences

error

I

-0.0217

1.4075

1.9815

2

- 0.0094

1.4293

2.0429

polynomialregressionmodel of Max and Burkhart (1976) was also fitted to the "fitting" data set, and it is includedin Figure 2 for comparisonpurposes(designated as model 3). In Figure 2, sections1, 2, and 3 refer to diameter measurementsat 0.3, 0.8, and 1.3 m above ground. Sections4 through 12 refer to nine equally spaced observations,eachat (H-1.3)/10 successivepointsup the stem. the mean valueswere computedusingthe 173 trees from the validationdata set. It is evident that both models1 and 2 have nearly the samepredictionalongthe stem and that their predictive ability compareswell with that of the segmentedpolynomial model. The variable-formtaper models1 and 2 havetwo primary weaknesses:(1) numerical integrationmethodsmustbe usedto calculatevolume;and (2) iterative methods must be used to find merchantableheight to a given diameter. In spite of these shortcomings,which are both relatively easily overcomewith modern computing 0.3

Model 2

0.2

0.1

0.0

-0.1

-0.2

odel 3

-0.3

-0.4

-0.5

-I

I

I

I

I

I

I

I

I

I

I

I

I

2

3

4

5

6

7

8

9

I0

II

12

TREE

SECTION

FIGURE2. Mean biasalongthe stemfor variable-formtaper modelsI and 2 and for segmented polynomialmodel(designated model3). Section1 is at stumpheight;section12is nearthe tip of the tree.

190/FOREST SCIENCE

equipment,the variable-formtaper model proved to be a very useful method for predictingdiameterinsidebark alongthe stemfor treesof Pinus oocarpaSchiedein central Honduras.

LITERATURE

CITED

AMIDON, E. L. 1984.A generaltaper functionalform to predictbole volumefor five mixedcontierspeciesin California.For. Sci. 30:166-171. CHo, Q. ¾., H. E. BUgKI•HgT,and T. A. MAx. 1980.Evaluation of two methodsfor cubic volumepredictionof 1oblollypine to any merchantablelimit. For. Sci. 26:71-80. DEMAERSCHALK, J.P., and A. KOZH•. 1977. The whole bole system: A conditioned dual equationsystemfor precisepredictionof tree profiles.Can. J. For. Res. 7:488-497. FU•SLmqD,R. R. 1982.A geometricaltree volumemodelbasedon the locationof the centreof gravity of the bole. Can. J. For. Res. 12:215-221. GRAY,H. R. 1956.The form and taper of forest-treestems.Imp. For. Inst., Oxford Inst. Pap. 32.79 p. GgOSE•qaHUG}t, L. R. 1966. Tree form: Definition, interpolation, extrapolation. For. Chron. 42:444-457.

KOZAK,A. 1988.A variable-exponent taperequation.Can. J. For. Res. 18:1363-1368. KozH•, A., D. D. Mmqgo, andJ. H. G. SMIT}L 1969.Taper functionsand their applicationin forestinventory.For. Chron. 45:278-283. LHgso•q,P. R. 1963.Stemform developmentof foresttrees. For. Sci. Monogr. 5.41 p. MAX, T. A., and H. E. BURKHART.1976. Segmentedpolynomialregressionappliedto taper equations.For. Sci. 22:283-289. MYEgS,R. H. 1986.Classicaland modernregressionwith applications.DuxburyPress.Boston, MA. 359 p. NEWlqHAM,R. M. 1988.A variable-formtaper function.PetawawaNat. For. Inst., Can. For. Serv. Info. Rep. PI-X-83.33 p. OgMEgOD,D. W. 1973.A simplebole model. For. Chron. 49:136-138. REED, D. D., and J. C. BYRNE. 1985. A simple, variable form volume estimationsystem. For. Chron.

61:87-90.

STIFF,C. T., D. N. PEREZ,and F. D. JOHNSON. 1987.Classificationof the uplandpine forests of central Hondurasfor site quality and productivity.P. 128-133 in H. G. Lurid and M. Caballero-Deloya (eds.),Land andresourceevaluationfor nationalplanningin the tropics: Proc. internat. conf. & workshop. USDA For. Serv. Gem Tech. Rep. WO-39.

MARCH 1990/ 191