Three equations for predicting tree height as a function of diameter (outside
bark) at breast height are presented for six species found in coastal regions of
the Pacific Northwest. Foresters can use these “height–diameter” equations
to avoid the time-consuming task of measuring heights of all individual trees
in an inventory, a stand exam, or a timber cruise. Equation coefficients were
estimated with weighted nonlinear regression techniques. Because the relationship
between a tree’s height and diameter depends on the tree’s competitive
position within the stand, alternative equations, including the average
height and average diameter of the 40 largest-diameter trees/ac, are also
presented. These equations are used in the Stand Management Cooperative
version of ORGANON

Hann, David W.

Hanus, Mark L.

Marshall, David D.

Undetermined

ScholarsArchive@OSU

Forest Research LaboratoryOregon State UniversityCollege ofForestryResearch Contribution 25 August 1999HEIGHT–DIAMETER EQUATIONS FOR SIXSPECIES IN THE COASTAL REGIONS OF THEPACIFIC NORTHWESTbyMark L HanusDavid D MarshallDavid W HannRecycledPaperThe Forest Research Laboratory of Oregon State University was established bythe Oregon Legislature to conduct research leading to expanded forest yield,increased use of forest products, and accelerated economic development ofthe State. Its scientists conduct this research in laboratories and forestsadministered by the University and cooperating agencies and industriesthroughout Oregon. Research results are made available to potential usersthrough the University’s educational programs and through Laboratory pub-lications such as this, which are directed as appropriate to forest landownersand managers, manufacturers and users of forest products, leaders of govern-ment and industry, the scientific community, and the general public.The AuthorsMark L Hanus is faculty research assistant, David D Marshall is assistantprofessor, and David D Hann is professor, all in the Department of ForestResources, College of Forestry, Oregon State University, Corvallis.AcknowledgmentsThis study was funded by the Stand Management Cooperative.DisclaimerThe mention of trade names or commercial products in this publication doesnot constitute endorsement or recommendation for use.To Order CopiesCopies of this and other Forest Research Laboratory publications are availablefrom:Forestry Publications OfficeCollege of ForestryOregon State UniversityCorvallis, Oregon 97331-7401Phone: (541) 737-4271FAX: (541) 737-2668email: forspub@cof.orst.eduWeb site: http://www.cof.orst.edu/cof/pub/home/Please indicate author(s), title, and publication number if known.HEIGHT–DIAMETER EQUATIONS FOR SIXSPECIES IN THE COASTAL REGIONS OF THEPACIFIC NORTHWESTbyMark L HanusDavid D MarshallDavid W HannForest Research LaboratoryResearch Contribution 25 August 1999AbstractHanus, ML, DD Marshall, and DW Hann. 1999. Height–Diameter Equationsfor Six Species in the Coastal Regions of the Pacific Northwest. Research Con-tribution 25, Forest Research Laboratory, Oregon State University, Corvallis.Three equations for predicting tree height as a function of diameter (outsidebark) at breast height are presented for six species found in coastal regions ofthe Pacific Northwest.  Foresters can use these “height–diameter” equationsto avoid the time-consuming task of measuring heights of all individual treesin an inventory, a stand exam, or a timber cruise.  Equation coefficients wereestimated with weighted nonlinear regression techniques.  Because the rela-tionship between a tree’s height and diameter depends on the tree’s com-petitive position within the stand, alternative equations, including the aver-age height and average diameter of the 40 largest-diameter trees/ac, are alsopresented.  These equations are used in the Stand Management Cooperativeversion of ORGANON.Table of ContentsIntroduction............................................................................5Data Description .....................................................................6Data Analysis ..........................................................................7Results and Discussion............................................................8Literature Cited ....................................................................115IntroductionThe total height (Ht) of a tree is important for assessing tree volume (Walterset al. 1985; Walters and Hann 1986) and stand productivity through site index(Hann and Scrivani 1987), but accurate measurement of this variable is time-consuming. As a result, foresters often choose to measure only a few trees’heights and estimate the remaining heights with height–diameter equations.Foresters can also use height–diameter equations to indirectly estimate heightgrowth by applying the equations to a sequence of diameters that were eithermeasured directly in a continuous inventory or predicted indirectly by adiameter-growth equation. The diameter-growth prediction approach can bevaluable for modeling growth and yield of trees and stands as is done inORGANON (Hann et al. 1997).A number of studies of height–diameter relationships in northwestern Or-egon, western Washington, and southwestern British Columbia have alreadybeen published. Curtis (1967) investigated several equations for Douglas-firthat included tree diameter outside bark at breast height (DBH) as an explana-tory variable. Larsen and Hann (1987) and Wang and Hann (1988), using avariant of Curtis’s (1967) recommended model, found that an equation whichincluded tree diameter and site index was a better height predictor for 6 of 16species in the mid-Willamette Valley. Krumland and Wensel (1988) includedtop height and quadratic mean diameter in their height–diameter equation.Garman et al. (1995) used a Chapman-Richards model to characterize the ex-pected asymptotic behavior of height at large diameters.Our objective in this study was to develop equations for predicting treeheight as a function of DBH by itself or in conjunction with the average diam-eter and height of the 40 largest-diameter trees/ac (D40 and H40 respectively),for the following six tree species found in the Pacific Northwest’s coastal re-gions:Douglas-fir Pseudotsuga menziesii (Mirb.) FrancoRed alder Alnus rubra Bong.Sitka spruce Picea sitchensis (Bong.) Carr.Western hemlock Tsuga heterophylla (Raf.) Sarg.Western redcedar Thuja plicata Donn ex D. DonWestern white pine Pinus monticola Dougl. ex LawsThe height–diameter equations developed in this study are being used inthe Stand Management Cooperative (SMC) version of the ORGANON standsimulator (Hann et al. 1997). In the SMC model they are used to predict heightsof unmeasured trees and to predict height growth in hardwood species.6Data DescriptionThe data for this analysis came from the SMC modeling database, whichcontains tree measurements from 3387 plots within 378 installations in south-western British Columbia, western Washington, and northwestern Oregon, westof the Cascades. Our data subset came from 3345 plots within 372 installa-tions. These fixed-area plots ranged in size from 0.1 to 1.2 ac (average size0.17 ac), and were located between 42.00° N and 50.63° N and between120.7° W and 127.68° W.For a given tree to be included in our sample, it had to have a measuredDBH, a Ht >4.5 ft, and an undamaged top. By species, Douglas-fir data camefrom 1309 pure-species plots on 187 installations, and the western hemlockdata came from 630 pure-species plots on 83 installations. (Plots with at least80% of stand basal area in a single species are considered pure-species stands;thus, we also obtained data on other species growing in these plots.) TheDouglas-fir site index (SI) (Bruce 1981) averaged 105.7 ft with a range of 56–162 ft; we calculated the individual values by solving Bruce’s (1981) domi-nant height equations for site index and calculating site index from dominantheight and age. Douglas-fir breast height age averaged 27.8 yr and rangedfrom 3 to 108 yr; standing plot basal area averaged 175.4 ft2/ac and rangedfrom 3 to 411 ft2/ac. For western hemlock plots, site index averaged 101.5 ftwith a range of 43-131 ft; breast height age averaged 45.7 yr with a range of8–108 yr; and study plot basal area averaged 225.1 ft2/ac with a range of25.7–411.0 ft2/ac.In the SMC modeling set, tree diameter at breast height (DBH), measuredto the nearest 0.1 in. using a diameter tape, was recorded for all trees. ActualTable 1. Descriptive statistics for the height–diameter modeling data set.Tree-level statisticsDBH (in) Height (ft)Number Standard StandardSpecies of trees Mean deviation Range Mean deviation RangeDouglas-fir 71,887 7.54 4.97 0.1–46.0 53.78 32.29 5.0–192.0Red alder 899 10.0 2.84 2.6–19.4 78.8 10.22 25.6–113.8Sitka spruce 115 13.9 7.44 1.7–28.9 97.17 41.52 14.4–157.2Western hemlock 17,984 7.19 4.24 0.1–30.6 58.48 31.28 4.6–159.1Western redcedar 819 4.6 2.84 0.1–19.8 34.1 18.8 5.0–103.3Western white pine 191 5.6 2.71 1.0–16.8 38.7 16.77 7.5–107.6Plot-level statisticsD40 H40 SINumber Standard Standard   StandardSpecies of plots Mean deviation Range Mean deviation Range Mean  deviation RangeDouglas-fir 1,309 11.65 4.49 1.9–31.8 69.8 26.95 13.6–182.1 105.7 4.49 56.1–162Western hemlock 630 13.6 4.5 3.5–27.0 84.82 29.48 20.3–163.2 101.5 16.12 43–1317tree height measurements were made on a sample of trees from each plot. Tocalculate H40, unmeasured total tree heights were estimated with plot-levelheight–diameter fits (Flewelling and de Jong 1994). Table 1 summarizes theresulting height–diameter modeling data set.Data AnalysisMany of the height–diameter equations presented in the literature use alog-linear model form (Curtis 1967; Wykoff et al. 1982). However, some re-searchers have found that the residuals of these log-linear equations are notnormally distributed (Larsen and Hann 1987), a situation that makes log biascorrections difficult (Flewelling and Pienaar 1981). Therefore, we chose thefollowing nonlinear equation form, recommended by Curtis (1967), to char-acterize the relationship of Ht to DBH for this study:where a0, a1, and a2 are parameters to be estimated by nonlinear regres-sion from the data (Table 1).Krumland and Wensel (1988) proposed that the following model, whichincludes H40 and D40 as variables, explains more of the residual variation forsome species:where b0, b1, and b2 are parameters to be estimated by nonlinear regres-sion from the data (Table 1).This model constrains the height–diameter curve to equal H40 when DBHequals D40.Garman et al. (1995) fitted the following Chapman-Richards model to adata set from western Oregon, believing that this model form best accountedfor the expected asymptotic behavior in the height–diameter relationship:wherec1 = asymptotic heightc2 = steepness parameterc3 = curvature parameter.We fitted Eq. [1] to all species, Eq. [2] to Douglas-fir and western hem-lock, and Eq. [3] to Douglas-fir—in all cases with weighted nonlinear regres-sion techniques (weight = 1.0/DBH) (Larsen and Hann 1987). However, be-cause users are most interested in predicting unweighted heights, we reportthe unweighted coefficients of determination here.Ht H b DBH b D40b b H b b H= + −( ) × ×⎛⎝⎞⎠ ×⎛⎝⎞⎠⎡⎣⎢⎤⎦⎥+ −( )( ) + −( )( )4 5 40 4 5 040 4 5040 4 51 2 1 2. . exp / exp. .Ht c c DBHc= + × − ×( )( )4 5 101 2 3. . expHt a a DBHa= + +( )4 5 0 1 2. exp8Results and DiscussionTables 2 and 3 contain the regression coefficients, mean square errors(MSEs), and unweighted adjusted coefficients of determination (–R2) for equa-tions [1] and [2]. Table 4 contains the regression coefficients, MSEs, and 44–R2 values for Eq. [3]. The fit for red alder was significantly worse than the fitsfor any of the conifers. This poor fit is probably due to the greater difficulty inmeasuring the heights of species with weak apical dominance, and a greatervariability in the height–diameter relationship.Garman et al. (1995) proposed using a Chapman-Richards model form tobetter extrapolate into larger diameter classes. They reported their results byspecies, geographic region, and site class. As the SMC modeling data set didnot include the latter two categories, we compared the Chapman-RichardsTable 2. Regression coefficients and associated unweighted statistics for all species in the modeling data set, fitted toEq. [1].Coefficientsa0 a1 a2Species (std. error) (std. error) (std. error) MSE–R2Douglas-fir 7.262195456 -5.899759104 -0.287207389 16.453 0.6641(0.047807485) (0.038884463) (0.00406176)Red alder 4.41820972 -12.00274935 -2.13835482 21.7689 0.2975(0.01435762) (2.96861022) (0.170420047)Sitka spruce 5.404491308 -6.570862442 -0.819705048 61.77849 0.6199(0.151873318) (0.872852109) (0.125177723)Western hemlock 6.555344622 -5.137174162 -0.3645508 24.421 0.5348(0.07869243) (0.05409665) (0.011147923)Western redcedar 7.232880669 -5.746899904 -0.271564741 16.54749 0.6792(0.529191522) (0.470945659) (0.038874446)Western white pine 7.946192109 -6.278973035 -0.208892429 21.60565 0.5098(2.119025826) (1.975460938) (0.099823967)Table 3. Regression coefficients and associated unweighted statistics for Douglas-fir and western hemlock, fitted to Eq. [2].Coefficientsb0 b1 b2Species (std. error) (std. error) (std. error) MSE–R2Douglas-fir -2.857232223 -0.393885195 -0.000521583 3.037 0.938(0.005046006) (0.003860332) (0.000069927)Western hemlock -2.790360488 -0.235470605 -0.002374673 7.2786 0.8613(0.009938554) (0.007385901) (0.000120054)9model form to the SMC models by refitting to the SMC data set. The resultsof the fit are presented in Table 4.  For Douglas-fir, the adjusted coefficient ofdetermination for the Chapman-Richards model (0.6644) is greater than thatfor the Larsen and Hann equation (0.6641) but less than that for the Krumlandand Wensel equation (0.938). The coefficient for the asymptotic height fromthe fit of the SMC modeling data set to the Chapman-Richards model form is167.68 ft. This value is unrealistic for Douglas-fir (Garman et al. 1995) andprobably results from the young age of the trees in the modeling data set.When we fixed the asymptotic height value to the average of the Garman etal. (1995) reported asymptotic height values for Douglas-fir in northern Or-egon coastal and Cascade regions, the fit declined (–R2 = 0.6596). This showsthat the model form is sensitive to the limited range of heights in the SMCmodeling data set. The Garman et al. (1995) data sets included larger trees,which provided better estimates of the asymptotic height parameter (c1). Theresiduals from the Chapman-Richards fit (Eq. [3]) show a trend against H40(Figure 1). In stands where H40 is large, the Chapman-Richards modelunder-predicts the height, due to the low estimate for the asymptotic height.The Chapman-Richards model form also exhibits high intrinsic curvature whichcan produce biased parameter estimates (Ratkowsky 1983).Including H40 and D40 in the model, as is done in Eq. [2], is another wayto account for the tendency of even-aged pure-species stands to have a tighterheight–diameter relationship than uneven-aged or mixed-species stands. Scalingthe projected height–diameter ratio to the observed H40:D40 point results ina smaller range in predicted heights.Table 4. Regression coefficients and associated unweighted statistics for Douglas-fir, fitted to Eq. [3].Coefficientsc1 c2 c3Species (std. error) (std. error) (std. error) MSE–R2Douglas-fir 167.6806252 -0.0700404 1.3063557 16.438 0.6644(1.4108489) (0.0009659) (0.0057041)70503010-10-30-50-70010 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190H40Y ResidualFigure 1. Residuals from the fit of Eq. [3].10Figure 2. The height–diameter relationships of Eqs. [1], [2], and [3] for Douglas-fir.Figure 2 shows the height–diameter relationships for Douglas-fir predictedby the three models. Equation [2] is applicable to trees in even-aged, pure-species stands. The DBH measurements in these stands do not normally spanthe entire range of the modeling data set, but rather fall within a smaller range,as shown by the three curves in Figure 2. For each of the curves, the slope ofthe predicted Ht values declines as DBH increases; in other words, trees withsmaller-than-average DBH values have a larger Ht:DBH ratio. If a stand is nottruly even-aged—for instance if it includes a few large residual trees from aprevious stand—Eq. [2] may over-predict the heights of the smaller trees. Itcan also be seen that the predicted heights for trees with small DBH values willbe much greater using Eq. [2] than using Eq. [1]. Equations [1] and [3] predictvery similar Ht values for DBHs of less than 20 in. For greater DBHs, predictedHt from Eq. [3] asymptotes to a value of 167 ft, while predicted Ht from Eq. [1]continues to increase, yielding reasonable values of Ht over the full range ofDBH in the modeling data set.In general, we recommend use of Eq. [2] for even-aged stands with at least80% of the stand basal area in Douglas-fir or western hemlock. However, usersshould consider the additional expense or effort associated with collecting theadditional information this model requires.In summary, the equations developed in this study show that tree height isstrongly correlated to diameter for the species examined. Including D40 andH40 improves the precision of predicting tree height. These three height–di-ameter equations provide new and useful information about tree species growingin the coastal regions of the Pacific Northwest. They have been incorporatedinto the SMC version of the ORGANON (Hann et al. 1997) stand simulator.2502001501005000 10 20 30 40 50DBH (in.)Height (ft)H40=73D40=10H40=142D40=25H40=188D40=40Exponential form (Eq. [1])Krumland-Wensel form (Eq. [2])Chapman-Richards form (Eq. [3])11Literature CitedBruce, D. 1981. Consistent height-growth and growth rate estimates forremeasured plots. Forest Science 27: 711–725.Curtis, RO. 1967. Height–diameter and height–diameter–age equations forsecond-growth Douglas-fir. Forest Science 13: 365–375.Flewelling, JW, and RJ de Jong. 1994. Considerations in simultaneous curvefitting for repeated height–diameter measurements. Canadian Journal of For-est Research 24:  1408–1414.Flewelling, JW, and LV Pienaar. 1981. Multiplicative regression with lognormalerrors. Forest Science 27: 281–289.Garman, SL, SA Acker, JL Ohmann, and TA Spies. 1995. Asymptotic Height–diameter Equations for 24 Species in Western Oregon. Research Contribution10,  Forest Research Laboratory, Oregon State University, Corvallis.Hann, DW, and J Scrivani. 1987. Dominant-Height-Growth and Site-Index Equa-tions for Douglas-fir and Ponderosa Pine in Southwest Oregon. Research Bulle-tin 59, Forest Research Laboratory, Oregon State University, Corvallis.Hann, DW, AS Hester, and CL Olsen. 1997. ORGANON User’s Manual Edition6.0.  Department of Forest Resources, Oregon State University, Corvallis.Krumland, BE, and LC Wensel. 1988. A generalized height–diameter equationfor coastal California species. Western Journal of Applied Forestry 3: 113–115.Larsen, DR, and DW Hann. 1987. Height–diameter Equations for Seventeen TreeSpecies in Southwest Oregon. Research Paper 49, Forest Research Laboratory,Oregon State University, Corvallis.Ratkowsky, DA. 1983. Nonlinear Regression Modeling: A Unified Practical Approach.Marcel Dekker, New York NY.Walters, DK, and DW Hann. 1986. Predicting Merchantable Volume in Cubic Feetto a Variable Top and in Scribner Board Feet to a 6-inch Top for Six Major Coni-fers of Southwest Oregon. Research Bulletin 52, Forest Research Laboratory,Oregon State University, Corvallis.Walters, DK, DW Hann, and MA Clyde. 1985. Equations and Tables PredictingGross Total Stem Volume in Cubic Feet for Six Major Conifers of Southwest Or-egon. Research Bulletin 50, Forest Research Laboratory, Oregon State Uni-versity, Corvallis.Wang, CH, and DW Hann. 1988. Height–diameter Equations for Sixteen Tree Speciesin the Central Western Willamette Valley of Oregon. Research Paper 51, ForestResearch Laboratory, Oregon State University, Corvallis.Wykoff, WR, NL Crookston, and AR Stage. 1982. User’s Guide to the Stand Prog-nosis Model. General Technical Report INT-133, USDA Forest Service, Inter-mountain Forest and Range Experiment Station, Ogden UT.Oregon State University is an affirmative action equal opportunity employer.Forestry Publications OfficeOregon State University256 Peavy HallCorvallis, OR 97331-5704Address Service RequestedNon-Profit Org.U.S. PostagePAIDCorvallis, ORPermit No. 200

Height-diameter equations for six species in the coastal regions of the Pacific Northwest

https://ir.library.oregonstate.edu/downloads/3n2043903

Height-diameter equations for six species in the coastal regions of the Pacific Northwest

Abstract

Similar works

Full text

Available Versions

ScholarsArchive@OSU