Modeling Aboveground Biomass Components and Volume-to-Weight

For. Sci. 62(5):463– 473 http://dx.doi.org/10.5849/forsci.15-129 Copyright © 2016 Society of American Foresters

FUNDAMENTAL RESEARCH

biometrics

Modeling Aboveground Biomass Components and Volume-to-Weight Conversion Ratios for Loblolly Pine Trees Dehai Zhao, Michael Kane, Robert Teskey, and Daniel Markewitz With large taper and biomass measurement data sets from destructively sampled loblolly pine trees, we developed weight-to-volume ratio equations and biomass allocation equations based on tree dbh and height. The weight-to-volume ratio equations were fitted simultaneously with stem green weight with bark and stem outside-bark volume equations or stem wood dry weight and stem inside-bark volume equations, using a weighted nonlinear seemingly unrelated regression approach to account for the inherent correlation among the equations and the heteroscedasticity. The biomass component proportions were modeled with Dirichlet, fractional multinomial logit, and log-ratio regression approaches, which guarantee the constraint of all nonnegative proportions summing to 1. Both weight-to-volume ratios and biomass component proportions varied with tree size. The Dirichlet regression was superior to the fractional multinomial logit and log-ratio regression approaches. Keywords: Dirichlet regression, fractional multinomial logit model, log-ratio regression, destructive sampling, fractional biomass composition

L

oblolly pine (Pinus taeda L.) is the most important commercial timber species in the southeastern United States (Smith et al. 1994). Pine plantations in the southern United States have been among the most intensively managed forests in the world. They not only produce traditional industrial wood (Prestemon and Abt 2002) but also contain a high percentage of the sequestered forest carbon in the conterminous United States (Johnsen et al. 2001) and have a great potential for biomass energy. Accurate estimates of tree volume and biomass are crucial for estimating both carbon storage and biomass energy at regional or national scales. Depending on the available inventory data, several methods exist for estimating tree total biomass or component biomass, such as stem wood, stem bark, branch, or foliage. When dbh and total height (HT) measurements are available, total and/or component biomass can be estimated directly from biomass allometric equations (Zhao et al. 2015). When an estimate of stem volume is available, it can be converted to stem biomass by a weight-to-volume ratio or more precisely to stem biomass and then to total or other component biomass by biomass component ratios. Biomass component ratios can also be used to estimate component biomass from

total biomass or to estimate total or other component biomass from an available specific component. Constant values may be used; however, these ratios vary with forest type, growing conditions, stand density, and climate (Intergovernmental Panel on Climate Change 2003). For example, biomass component proportions were shown to vary in relation to growing stock (Boudewyn et al. 2007, Teobaldelli et al. 2009), stand age (Pajtik et al. 2011, Sanquetta et al. 2011), and tree diameter or height (Food and Agriculture Organization of the United Nations (FAO) 2005, Pajtik et al. 2008, Sanquetta et al. 2011). Therefore, it is necessary to develop equations for these ratios, depending on age or size of trees (Teobaldelli et al. 2009, Sanquetta et al. 2011, Longuetaud et al. 2013). In practice, the weight-to-volume ratios might be derived from either previously developed allometric equations or calculated from the estimated weight and volume (FAO 2005). However, the ratios ˆ /Vˆ ) are biased, because the expecestimated in this way (RWˆ/Vˆ ⫽ W tation of the ratio between weight and volume [E(W/V ) ⫽ RW/V ] is not equal to the ratio between the estimated weight and the estimated volume (i.e., RˆW/V ⫽ RWˆ/Vˆ ). Furthermore, using this approach we cannot perform a formal statistical test of whether the ratios vary in relation to tree size.

Manuscript received September 25, 2015; accepted May 10, 2016; published online July 7, 2016. Affiliations: Dehai Zhao ([email protected]), Warnell School of Forestry and Natural Resources, The University of Georgia, Athens, GA. Michael Kane, The University of Georgia. Daniel Markewitz, The University of Georgia. Robert Teskey, The University of Georgia. Acknowledgments: This research was funded by the USDA National Institute of Food and Agriculture (Agreement No. 2011-67009-30065) and the USDA Forest Service (Agreement No. 11-JV-11242305-109). We thank the Plantation Management Research Cooperative (PMRC) technical staff for their hard work in field sampling and data collection and thank the PMRC members for their permitting access to sample sites and stands. We also thank the associate editor and two anonymous referees for insightful comments on earlier drafts. Forest Science • October 2016

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Biomass allocation can be described as the proportion of total biomass allocated to components such as stem wood, stem bark, branches, and foliage. Traditionally, equations for the various components have been developed without any constraint that they remain nonnegative and sum to 1. To address this constraint, alternative approaches such as the Dirichlet regression model (DRM) (Hijazi and Jernigan 2009), the fractional multinomial logit model (FMLM) (Papke and Wooldridge 1996), and the log-ratio regression model (LRM) (Aitchison 1986) should be considered for biomass allocation analysis. These approaches are seldom used in forestry research. Using our combined legacy and new biomass sampling data sets and with new modeling approaches, our objective in this study was to develop a series of ratio equations for loblolly pine trees: (1) ratio between stem green weight and stem outside-bark volume, (2) ratio between stem wood dry weight and stem inside-bark volume, (3) proportions of stem wood and bark in stem biomass, and (4) proportions of stem wood, bark, branch, and foliage components in total tree aboveground biomass. The DRM, FMLM, and LRM approaches were compared for fitting biomass component proportion equations.

Materials and Methods Data Description

Data used in this study are from three data sets. The first data set consisted of measurements on 1,280 individual trees obtained in 1977–1983 from 376 sample plots located in plantations in the Coastal Plain and Piedmont physiographic provinces of North Carolina, South Carolina, Georgia, Florida, and Alabama. In most cases, four sample trees without any obvious stem abnormalities (one below-average dbh tree, one average tree, and two dominant and codominant trees) were felled on each sample plot. Each sample tree was measured for dbh and, after felling, for total height. The felled trees were then cut into 1.52-m bolts (5 ft) starting at ground level up to a top diameter of less than 5.1 cm (2 in.). Each bolt was measured for diameter inside and outside bark at the small end. In addition, the first bolt from the stump (i.e., butt bolt) was measured for middle diameter inside and outside bark. The green weight of each cut bolt was measured in the field. For each tree, total stem outside-bark volume (Vob) was calculated using Bailey’s (1995) overlapping bolts method, and green weights of all cut bolts of that tree were summed to total green weight of stem wood including bark (GWob). The second data set consisted of measurements on 274 trees that were sampled in 2003–2004 from stands located in the Coastal Plain and Piedmont regions of North Carolina, South Carolina, Georgia, Florida, Alabama, and Mississippi. Each sample tree was measured for dbh. After felling, total height was measured and discs of about 3-cm thick were removed from the trees at butt (0.18 m), 1.37 and 3.04 m, and subsequently at 1.52-m intervals up to a top diameter of ⬍5.1 cm. Diameters inside and outside bark were measured on each disc. In the laboratory, each disc was measured for density of green wood with bark and density of dry wood without bark. Vob and total stem inside-bark volume (Vib) for each tree were obtained using Bailey’s (1995) overlapping bolts method. Weights (W: green weight including bark or dry weight of wood without bark) for each stem section were calculated as W ⫽ [2D1A1 ⫹ D1A2 ⫹ D2A1 ⫹ 2D2A2] ⫻L/6, where D1 and D2 are densities as determined for the discs at both ends of the bolt, A1 and A2 are cross-sectional areas for the discs at both ends of the bolt, and L is the length of the 464

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Table 1. Summary statistics for dbh (cm) and HT (m) for the loblolly pine trees used for fitting Rgw/vob and Rdw/vib equations and biomass component proportion equations. Data for Rgw/vob equations Rdw/vib equations Biomass proportion equations

Variable DBH HT DBH HT DBH HT

n

Range

Mean

SD

1,941 3.30–43.18 17.96 6.33 1,941 3.87–31.64 14.82 4.46 747 7.62–43.18 20.36 6.44 747 10.39–31.64 17.81 3.63 480 7.62–43.18 18.60 6.67 480 10.39–31.64 16.88 3.74

Rgw/vob, ratio between stem green weight to stem outside-bark volume; Rdw/vib, ratio between stem-wood dry weight to stem inside-bark volume.

bolt. The weights of the tip of the tree were calculated as W ⫽ D ⫻ V, where D is the density determined for the last disc, and V is the volume of the tip. Thus, dbh, total height, Vob, Vib, GWob, and total dry weight of stem wood excluding bark (DWib) were available for each tree in this data set. The third data set consisted of measurements on 481 trees from destructive biomass sampling in 2010 –2015 and included taper measurements, green weights of cut bolts, green weight of branch with foliage, green weights of discs, subsampled branch with foliage, and dry weights of disk wood, bark, branch, and foliage. For more detailed information about this biomass sampling and field and laboratory measurement protocols, refer to Zhao et al. (2012, 2015). For each sampled tree, Vob and Vib were also obtained using Bailey’s (1995) overlapping bolts method. In addition, GWob, DWib, dry weight of bark (DWbk), dry weight of branches (DWbh), and dry weight of foliage (DWfl ) were calculated for each tree using the method of Zhao et al. (2015). The ratio between stem green weight and stem outside-bark volume (Rgw/vob) was calculated as Rgw/vob ⫽ GWob/Vob for trees in these three data sets, and the ratio between stem wood dry weight and stem inside-bark volume (Rdw/vib) was calculated as Rdw/vib ⫽ DWib/Vib for each tree in the second and third data sets. For trees in the third data set, stem wood proportion (Pstemwood) and stem bark proportion (Pstembark) in stem biomass were calculated as Pstemwood ⫽ DWib/DWstem and Pstembark ⫽ DWbk/DWstem, respectively, where DWstem ⫽ DWib ⫹ DWbk. Proportions of component biomass in tree aboveground biomass were calculated as Pwood ⫽ DWib/DWt, Pbark ⫽ DWbk/DWt, Pbranch ⫽ DWbh/DWt, and Pfoliage ⫽ DWfl/DWt for stem wood, stem bark, branch, and foliage components, respectively, where DWt is total tree aboveground biomass. Summary statistics for dbh and HT of the trees used for developing the equations for weight and volume ratios and biomass component proportions are shown in Table 1. Total stem outside-bark volume, stem green weight with bark, the ratio between stem green weight and stem outside-bark volume and their relationships with tree dbh and HT are shown in Figure 1. Total stem inside-bark volume, stem wood dry weight, the ratio between stem wood dry weight and stem inside-bark volume related to tree dbh and HT are shown in Figure 2.

Model Specification and Estimation Ratios of Stem Weight to Volume

Visual inspection of the data of stem green weight with bark, stem outside-bark volume, stem wood dry weight, stem inside-bark volume, and their ratios Rgw/vob and Rdw/vib (Figures 1 and 2) indicated that they might be modeled as power functions of tree dimension variables such as dbh (DBH) and total height (HT) as

Figure 1. Relationships between total stem outside-bark volume (Vob), stem green weight including bark (GWob), the GWob/Vob ratio of the trees, and tree dbh and total height.

再

Vob ⫽ a 0 DBH a 1HT a 2 ⫹ ␧ vob GWob ⫽ b 0 DBH b 1HT b 2 ⫹ ␧ gw R gw/vob ⫽ c 0 DBH c 1HT c 2 ⫹ ␧ gw/vob

Fractional Biomass Components

(1)

and

再

Vib ⫽ a 0 DBH a 1HT a 2 ⫹ ␧ vib DWib ⫽ b 0 DBH b 1HT b 2 ⫹ ␧ dw R dw/vib ⫽ c 0 DBH c 1HT c 2 ⫹ ␧ dw/vib

(2)

To estimate the parameters of ratio equations, the systems of stem volume, stem green or dry weight, and ratio equations were fitted by weighted nonlinear seemingly unrelated regression (NSUR), using the SAS/ETS MODEL procedure (SAS Institute, Inc. 2011a) and following the four-step fitting approach of Zhao et al. (2015).1 This approach takes into account the contemporaneous correlations among different equations and the heteroscedasticity. Fitting the equation systems leads to stem outside-bark volume, stem inside-bark volume, total stem green weight, and stem insidebark dry weight equations that have a high practical value.

DRM Let {m1, …, mk} represent the k individual component biomass, total biomass mtotal ⫽ ⌺ki⫽1mi, and then the component proportions yi ⫽ mi/mtotal. That is, the vector of component proportions is y ⫽ (y1, …, yk)⬘ with constraints y 僆(0, 1) and ⌺ki⫽1yi ⫽ 1. First, we assume that the fractional components y follow the Dirichlet distribution, a generalization of the beta distribution:

写 k

1 f 共y兩␣兲 ⫽ y ␣i⫺1, (␣i ⬎ 0) B(␣)i⫽1 i

(3)

The normalizing constant is the multinomial beta function and is expressed in terms of the gamma function: B共 ␣兲 ⫽

k ⌫共␣i 兲 ⌸i⫽1

冉冘冊

⌫

k i⫽1

␣i

⫽

k ⌫共␣i 兲 ⌸i⫽1 , ⌫共␣0 兲

冘

␣)


465

␣ ⫽ 共␣1 ,. . ., ␣k 兲 and (␣0 ⫽

k i⫽1 i

Figure 2. Relationships between stem inside-bark volume (Vib), stem wood dry weight (DWib), the DWib/ Vib ratio of the trees, and tree dbh and total height, based on the second and third data sets.

Then, the expected values, the variances, and the covariances of the component proportions are E共 y i 兲 ⫽

␣i ␣i 共␣0 ⫺ ␣i 兲 , , VAR(yi ) ⫽ 2 ␣0 ␣0 共␣0 ⫹ 1兲 ⫺ ␣i ␣j COV(yi , yj ) ⫽ 2 共i ⫽ j兲 ␣0 共␣0 ⫹ 1兲

respectively. The full log-likelihood of the Dirichlet distribution is defined in Equation 4: l i 共y兩␣兲 ⫽ log⌫

冉冘冊冘 k i⫽1

␣i ⫺

log⌫共␣i 兲 ⫹

k i⫽1

冘

k i⫽1

共␣i ⫺ 1兲log共yi 兲 (4)

The log link functions for the shape parameter of each component can be related to tree dimension variables as log(␣i ) ⫽ fi 共Xi ␤i 兲共i ⫽ 1,. . ., k兲 466


(5)

So we have the DRM with ␣ parameters depending on the covariates Xi. The maximum likelihood estimates of ␤ parameters are obtained with the likelihood function, and then ␣ˆ i ⫽ exp {fi(Xi, ␤ˆ i)}, so ˆ兴 ⫽ E 关 y i兩 ␣

␣ˆ i ␣ˆ 0

(6)

After trying several model forms, the following log link functions were used for biomass component proportions: log(␣i ) ⫽ ␤0i ⫹ ␤1i log(DBH) ⫹ ␤2i log(HT ) 共i ⫽ 1,. . ., k兲

(7)

The DRMs for biomass component proportions were fitted using R Package DirichletReg (Maier 2014). FMLM Setting the first component (i.e., stem wood proportion y1) as the baseline, the odds of other components to the baseline is defined as yj yk y2 ,. . ., ,. . ., , 共 j ⫽ 2,. . ., k兲 y1 y1 y1

冉

冊

The logit model with the predictors X are

log

冉冊

yj ⫽ X␤j , 共j ⫽ 2, . . ., k) y1

(8)

The model 8 has k ⫺ 1 equations, with separate parameters for each. When k ⫽ 2, the model simplifies to ordinary logistic regression. After simultaneous fitting of these equations, the models to predict proportions of total biomass in each component are

Table 2. Parameter estimates and their asymptotic SE and p values for the equations of Vob, GWob, Rgw/vob, Vib, DWib, and Rdw/vib based on tree dbh (cm) and HT (m). Equation

Variable

Vob DBH HT GWob

y1 ⫽

DBH HT

1

1⫹

冘

k j⫽2

e

ˆ X␤ j

Rgw/vob DBH HT

ˆ X␤ 2

y2 ⫽

e

1⫹

冘

Vib

ˆ k X␤ j j⫽2

DBH HT

e

DWib

...

DBH HT

ˆ

yk ⫽

eX␤k

1⫹

冘

k j⫽2

Rdw/vib ˆ X␤ j

e

Once again, we assume that the log odds equations are functions of DBH and HT, as follows

冉冊

yj ⫽ ␤0j ⫹ ␤1j log(DBH) ⫹ ␤2j log(HT ), 共 j ⫽ 2,. . ., k兲 log y1 (10) The FMLMs for biomass component proportions were fitted using an SAS program based on the SAS/STAT LOGISTIC procedure (SAS Institute, Inc. 2011b) or can be fitted with the Stata FMLOGIT module (Buis 2009). LRM Unlike the multinomial logit model 8 in which all equations were fitted simultaneously, in log-ratio analysis, the ratios of component proportions to baseline component proportion (i.e., odds) were transformed by the logarithm. Then the k ⫺ 1 logarithmic ratios were separately fitted to the predictors with the ordinary least squares:

冉冊

yj ⫽ yj* ⫽ X␤j ⫹ ␧j , (j ⫽ 2, . . ., k) log y1

(11)

Although having a fitting method different from that of the FMLM model, the LRM results in the same model form 9 used to predict proportions of total biomass in each component based on tree sizes, such as DBH and HT. log

冉冊

yj ⫽ yj* ⫽ ␤0j ⫹ ␤1j log(DBH ) ⫹ ␤2j log(HT ) y1 ⫹ ␧j , 共 j ⫽ 2,. . ., k兲

HT

(9)

Parameter

Asymptotic estimate

Asymptotic SE

a0 a1 a2 b0 b1 b2 c0 c1 c2 a0 a1 a2 b0 b1 b2 c0 c2

4.5901E–5 1.8837 1.0379 0.0258 1.9054 1.1762 547.6490 0.0267 0.1426 2.0872E–5 1.9397 1.1890 0.0059 1.9465 1.3389 281.5783 0.1569

6.3562E–7 0.0072 0.0084 0.0005 0.0103 0.0119 8.7425 0.0075 0.0091 6.7756E–7 0.0116 0.0173 0.0003 0.0117 0.0209 13.5661 0.0167

All p ⱕ 0.0001. Vob, stem outside-bark volume; GWob, total stem green weight with bark; Rgw/vob, ratio between GWob and Vob; Vib, stem inside-bark volume; DWib, stem-wood dry weight; Rdw/vib, ratio between DWib and Vib.

Table 3. Weight function and fit statistics for the equations of Vob, GWob, Rgw/vob, Vib, DWib, and Rdw/vib based on tree dbh (cm) and HT (m). Equation Vob GWob Rgw/vob Vib DWib Rdw/vib

3.629

DBH DBH 3.566HT 1.012 HT ⫺1.194 DBH 3.375HT 13.332 DBH 3.973HT 2.018 HT ⫺0.824

The DRM, FMLM, and LRM approaches were used to model proportions of stem biomass found in stem wood and bark, and proportions of total aboveground biomass allocated to stem wood, stem bark, branch, and foliage for loblolly pine trees. To briefly show how the weight-to-volume ratios and biomass component proportions vary with relation to tree size, a Mitscherlich equation HT ⫽ a ⫺ b exp(⫺cDBH) was fitted for the HT and DBH relationship using all trees in the three data sets. This equation

MABE

RMSE

R2

0.001 0.690 0.705 ⫺0.000 ⫺0.171 0.012

0.014 15.994 50.967 0.019 9.640 33.487

0.028 29.427 66.821 0.032 15.602 41.156

0.985 0.981 0.299 0.983 0.982 0.105

Vob, stem outside-bark volume; GWob, total stem green weight with bark; Rgw/vob, ratio between GWob and Vob; Vib, stem inside-bark volume; DWib, stem-wood dry weight; Rdw/vib, ratio between DWib and Vib.

was used to convert the resulting ratio and proportion equations into ones depending on tree dbh only. Model Assessment and Evaluation

For each modeling approach, four criteria: mean residual (E), mean of the absolute value of residuals (MABE), root mean square error (RMSE), and the coefficient of determination (R2) were calculated for each original response variable of interest and were used to evaluate the modeling approaches. Mathematical expressions of these criteria are E⫽

(12)

E

Weight function

冘共Y ⫺ Yˆ 兲

MABE ⫽

RMSE ⫽

(13)

n

冘兩Y ⫺ Yˆ 兩

(14)

n

冑冘

R2 ⫽ 1 ⫺

(Y ⫺ Yˆ )2 n

冘共Y ⫺ Yˆ 兲冘共Y ⫺ Y៮ 兲

(15)

2 2


(16) 467

Table 4. Parameter estimates, SEs, and p values for the DRM, FMLM, and LRM fitted to fractional components in stem biomass with tree dbh (cm) and HT (m). Model DRM Wood: log(␣1) Bark: log(␣2) FMLM Bark versus wood LRM log(Pbark/Pwood)

Parameter

Estimate

SE

p

Intercept Log(DBH) Intercept Log(HT)

5.8216 0.0647 6.3554 ⫺0.8998

0.1095 0.0308 0.1500 0.0483

⬍0.0001 0.0357 ⬍0.0001 ⬍0.0001

Intercept Log(DBH) Log(HT)

0.5243 ⫺0.0683 ⫺0.8927

2.0558 0.6738 1.0180

0.7987 0.9193 0.3806

Intercept Log(DBH) Log(HT)

0.4415 ⫺0.0602 ⫺0.8748

0.0965 0.0321 0.0487

⬍0.0001 0.0619 ⬍0.0001

In this two-component case, the DRM becomes a beta regression and the FMLM becomes a fractional logistic regression.

Figure 3. Ratio between stem green weight and stem outsidebark volume (Rgw/vob, F) and ratio between stem wood dry weight and stem inside-bark volume (Rdw/vib, Œ) changed with tree dbh. The dotted line represents 958 kg/m3 of Rgw/vob, and the dashed line represents 470 kg/m3 of Rdw/vib reported by Miles and Smith (2009).

Table 5. Fit statistics for the DRM, FMLM, and LRM fitted to fractional components in stem biomass with tree dbh and HT. Model

E

MABE

RMSE

R2

DRM FMLM LRM

0.0000 ⫺0.0000 ⫺0.0009

0.0112 0.0112 0.0112

0.0142 0.0142 0.0143

0.6319 0.6318 0.6282

In this two-component case, the DRM becomes a beta regression and the FMLM becomes a fractional logistic regression.

evaluated using the goodness-of-fit statistics described above. Our models were compared with the findings of other published work.

Results Stem Volume, Weights, and Ratios

Figure 4. Stem wood and bark proportions in stem biomass changed with tree size (dbh). The solid line is the predicted stem wood proportion, and the dashed line is the predicted bark proportions with the DRM. The dotted line represents bark proportion 0.104 reported by Miles and Smith (2009).

where Y, Yˆ, and Y៮ are the observed, predicted, and average values of the response variable, respectively, and n is the total number of observations. In this study, the models were fitted to the entire data set. Kozak and Kozak (2003) found that cross-validation techniques such as data-splitting do not provide any addition information on model performance compared with the statistics obtained for models fit to the entire data set. Our leave-one-out validation experiences for modeling volume and weight equations also supported their conclusion (Zhao et al. 2007, 2015). The models fit in this study were 468


The two systems (Equations 1 and 2) were fitted using the NSUR, and different expressions of residual weights were used for each of the components in Equations 1 and 2. dbh and HT were highly significant in all volume and weight equations (Table 2). The positive coefficients of dbh and HT imply that dbh and HT were positively related to the volume and weight. The fit statistics indicated that the volume and weight equations predicted tree stem outside- and inside-bark volumes, total green weight of stem with bark, and stem wood dry weight well (Table 3). The ratio between total green weight of stem with bark and stem outside-bark volume significantly depended on tree dbh and HT, whereas the ratio of stem wood dry weight to stem inside-bark volume significantly depended on tree HT only (Table 2). The fitted Mitscherlich equation for tree DBH and HT was HT ⫽ 40.08 ⫺ 38.13 exp(⫺0.0235DBH). Combining this equation with the weight-volume ratio equations shows the trend of the ratios varying with tree dbh (Figure 3). Fractional Components in Tree Stem Biomass

For fractional components in stem biomass (k ⫽ 2: stem wood and stem bark proportions) (Figure 4), the DRM is simplified to a beta regression, the FMLM becomes fractional logistic regression, and the LRM needs only one equation to fit. With these three modeling approaches, stem wood and bark proportions in stem biomass were fitted to tree dbh and HT. The parameter estimates are shown in Table 4, and fit statistics are shown in Table 5. When proportion equations associated with dbh and HT were fitted using the DRM, dbh significantly related to the first parameter

Figure 5. Comparisons of stem wood and bark proportions in stem biomass with the predictions for the DRM (Equation 17), FMLM (Equation 18), and LRM (Equation 19) based on dbh and HT.

(␣1) and HT related to the second parameter (␣2). Parameters of dbh and HT were significant in the log-ratio regression. When fitted using the FMLM, however, all the parameters were not significant at a significance level of 0.1 (Table 4). The predicted stem wood and bark proportions with the DRM based on dbh and HT are shown in Figure 4. The proportional models had almost the same performance, no matter which modeling approach was used (Table 5; Figure 5). The proportion of stem wood in stem biomass can be estimated by either of the following equations (17–19): DRM: Pstemwood ⫽

337.5116DBH 0.0647 337.5116DBH 0.0647 ⫹ 575.5925HT ⫺0.8998 (17)

FMLM: Pstemwood ⫽

1 1⫹1.6893DBH ⫺0.0683 HT ⫺0.8927 (18)

LRM: Pstemwood ⫽

1 1⫹1.5550DBH ⫺0.0602 HT ⫺0.8748

(19)

Then, the proportion of stem bark in stem biomass is estimated by Pstembark ⫽ 1 ⫺ Pstemwood. Fractional Components in Total Tree Aboveground Biomass

Fractional composition of tree aboveground biomass plotted against tree dbh (Figure 6) shows that the proportions of stem wood, stem bark, branch, and foliage changed with increasing tree dbh. The proportions were fitted to tree dbh and HT, using the DRM, FMLM, and LRM, respectively. In the DRM, all coefficients of DBH and HT in both ␣1 and ␣2 were significant, the coefficient of HT was significant in ␣3, and no coefficient of dbh or HT was significant in ␣4 (Table 6). In the LRM, the regression coefficients associated with dbh and HT in all the log-ratios were significant at a significance level of 0.1. For the FMLM, dbh and HT were significant only in the equation of the ratio between branch and stem wood proportions. Forest Science • October 2016

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Figure 6. Relationships between proportions of stem wood, stem bark, branch, and foliage components in tree total aboveground biomass and tree size (dbh). The solid, dashed, dotted, and dot-dash lines are stem wood, bark, branch, and foliage component proportions predicted with the DRM. Table 6. Parameter estimates, SEs, and p values for the DRM, FMLM, and LRM fitted to fractional components in tree aboveground biomass with tree dbh (cm) and HT (m). Model DRM Stem-wood: log(␣1) Stem-bark: log(␣2) Branch: log(␣3) Foliage: log(␣4) FMLM Bark/wood Branch/wood Foliage/wood LRM log(Pbark/Pwood) log(Pbranch/Pwood) log(Pfoliage/Pwood)

Parameter

Estimate

SE

p

Intercept log(DBH) log(HT) Intercept log(DBH) log(HT) Intercept log(HT) Intercept

0.4715 ⫺1.2765 2.9255 0.9391 ⫺1.3144 2.0281 0.5190 0.9694 2.0669

0.2146 0.0381 0.0869 0.2606 0.0595 0.1120 0.2338 0.0821 0.0386

0.0280 ⬍0.0001 ⬍0.0001 0.0003 ⬍0.0001 ⬍0.0001 0.0264 ⬍0.0001 ⬍0.0001

Intercept log(DBH) log(HT) Intercept log(DBH) log(HT) Intercept log(DBH) log(HT)

0.5454 ⫺0.0666 ⫺0.9017 ⫺0.0635 1.3096 ⫺1.9468 1.7378 1.1702 ⫺2.8705

2.2671 0.7469 1.1264 1.8161 0.6106 0.9307 3.1811 1.0417 1.5936

0.8099 0.9289 0.4234 0.9721 0.0320 0.0365 0.5849 0.2613 0.0717

Intercept log(DBH) log(HT) log(DBH) log(HT) Intercept log(DBH) log(HT)

0.4415 ⫺0.0602 ⫺0.8748 1.2873 ⫺1.9561 1.5761 1.2886 ⫺2.9425

0.0965 0.0321 0.0488 0.0580 0.0596 0.2135 0.0711 0.1078

⬍0.0001 0.0619 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001 ⬍0.0001

Table 7. Fit statistics for the DRM, FMLM, and LRM fitted to fractional components in total tree aboveground biomass with tree dbh (cm) and HT (m). Model

Component

E

MABE

RMSE

R2

DRM

Wood Bark Branch Foliage Wood Bark Branch Foliage Wood Bark Branch Foliage

0.0002 ⫺0.0013 0.0015 ⫺0.0004 ⫺0.0000 ⫺0.0000 0.0000 ⫺0.0000 ⫺0.0048 0.0004 0.0027 0.0017

0.0266 0.0103 0.0246 0.0102 0.0265 0.0101 0.0248 0.0101 0.0265 0.0101 0.0246 0.0101

0.0350 0.0131 0.0332 0.0142 0.0349 0.0130 0.0330 0.0141 0.0353 0.0130 0.0332 0.0143

0.6662 0.5410 0.5135 0.5163 0.6667 0.5482 0.5183 0.5236 0.6601 0.5456 0.5122 0.5107

FMLM

LRM

Pstemwood ⫽ 2.5577DBH ⫺1.3144 HT 2.0281 /␣0 Pbranch ⫽ 1.6803HT 0.9694 /␣0 Pfoliage ⫽ 7.9003/␣0

(20)

FMLM: c ⫽ 1 ⫹ 1.7253DBH ⫺0.0666HT ⫺0.9017 ⫹ 0.9385DBH1.3096HT ⫺1.9468 ⫹ 5.6848DBH1.102HT ⫺2.8705 Pstemwood ⫽ 1/c

The models with dbh and HT fitted well for each fractional component (Table 7; Figure 7), no matter which modeling approach was used. Biomass allocation in total tree aboveground biomass can be estimated by the dbh and HT models fitted using either DRM, FMLM, or LRM:

Pstembark ⫽ 1.7253DBH⫺0.0666 HT ⫺0.9017 /c Pbranch ⫽ 0.9385DBH1.3096 HT ⫺1.9468 /c Pfoliage ⫽ 5.6848DBH1.102 HT ⫺2.8705 /c

LRM: c ⫽ 1⫹1.5550DBH ⫺0.0602HT ⫺0.8748 ⫹ DBH 1.2873HT ⫺1.9561

DRM:

␣0 ⫽ 7.9003 ⫹ 1.6024DBH ⫺1.2765 HT 2.9255 ⫹ 2.5577DBH ⫺1.3144 HT2.0281 ⫹ 1.6803HT 0.9694 Pstemwood ⫽ 1.6024DBH ⫺1.2765 HT 2.9255 /␣0 470

(21)


⫹ 4.8361DBH1.2886HT ⫺2.9425 Pstemwood ⫽ 1/c Pstembark ⫽ 1.5550DBH ⫺0.0602 HT ⫺0.8748 /c

Figure 7. Comparisons of fractional components in tree total aboveground biomass with the predictions for the DRM (Equation 20), FMLM (Equation 21), and LRM (Equation 22) based on dbh and HT.

Pbranch ⫽ DBH1.2873 HT ⫺1.9561 /c Pfoliage ⫽ 4.8361DBH 1.2886 HT ⫺2.9425 /c

(22)

Discussion Although one main objective in this study was to develop volume-to-weight conversion ratio equations for loblolly pine trees, individual tree stem volume and weight equations were developed along with the ratio equations. Because these weight or volume equations were developed based on a large data set with a wide range of dbh and HT across the geographic range of loblolly pine, they can be used to estimate total stem outside- and inside-bark volumes, total green weight of stem with bark, and total stem wood dry weight of loblolly pine trees across the region. The weight-to-volume ratio equations were fitted simultaneously with the stem volume and weight equations using NSUR. With the NSUR approach, the inherent correlations among tree volume, weight, and their ratio were taken into account, resulting in more efficient estimates of the system of equations. The heteroscedasticity in model residuals existed not only in tree volume and weight regressions but also in ratio equations. To achieve minimum variance estimates and reliable prediction intervals, the heteroscedasticity was addressed by modeling the variances as a power function of dbh and/or HT for each equation (Table 3).

Our results show that weight-to-volume ratios, either the ratio between stem green weight and stem outside-bark volume or the ratio of stem wood dry weight to stem inside-bark volume, are not reasonably constant. Both ratios increase with tree size. As tree dbh increased from 4 to 44 cm, the ratio between stem green weight and stem outside-bark volume increased from 722.2 to 966.9 kg/m3 (Figure 3). For loblolly pine trees of ⬍40 cm dbh, the ratio of stem green weight to stem outside-bark volume is smaller than 958 kg/m3 as reported by Miles and Smith (2009). Stem wood density results here were generally consistent with the average value of 470 kg/m3 reported by Miles and Smith (2009); however, rather than seeing a constant pattern of wood density with tree size, a gradual increase in wood density with increasing tree dbh was observed here. The average wood density increased from 366.6 to 471.0 kg/m3 as tree dbh increased from 4 to 44 cm. Biomass allocation data are compositional data that are nonnegative proportions of disjoint biomass components adding to 1. This unit-sum constraint severely complicates analysis, and the usual multivariate statistical methods are rarely used for compositional data (Hijazi and Jernigan 2009). In this study, we used the DRM, FMLM, and LRM approaches to model biomass component proportions. These approaches guarantee that proportions of all components sum to 1. Forest Science • October 2016

471

The DRM approach assumes that the proportion variable Y follows the Dirichlet distribution, and the LRM approach assumes that the proportion Y follows the additive logistic normal distribution. Different from the DRM and LRM with specific distributions, the FMLM is a quasi-likelihood method that does not assume any distributions but directly models the conditional mean of the response as an appropriate function, i.e., E(Y円X) ⫽ G(Xⴕ␤). The traditional multinomial logit model (MLM) is used to predict the probability of the different possible outcomes of categorically distributed dependent variables. When Boudewyn et al. (2007) modeled proportions of total tree biomass, they implemented some nonstandard data processing steps to create “proportions” for using a MLM, rather than directly fitting the actual biomass proportions as a set of dependent variables. The FMLM used in our study, however, directly modeled biomass proportions that each ranges between 0 and 1 and always add up to 1 for each observation. In terms of goodness-of-fit statistics, our results show that these three approaches have almost the same performances (Tables 5 and 7). We may assign different sets of predictors to the Dirichlet distribution parameters (␣) in the DRM approach or to different log ratio equations in the LRM approach. Based on the significance of coefficients, the equations can be modified accordingly with the DRM and LRM (Tables 4 and 6). In the FMLM, however, the log odds equations need to keep the same functional form. The predictor coefficient might be significant in one log odds equation but not in others (Table 6). Therefore, if one is interested in testing the effects of variables or treatments on the composition, e.g., tree size on biomass allocation, the choice of DRM or LRM is supported by this analysis, whereas that of FMLM is not. Our biomass component proportion models clearly showed that biomass allocation in stem biomass or in total aboveground biomass changed with tree size. The biomass component proportions can be estimated from tree size: dbh and HT. On a green volume basis, Miles and Smith (2009) reported average woody density of 470 kg/m3, average bark density of 330 kg/m3, and average bark volume as 16.6% of wood volume for loblolly pine trees. According to these percentages, after a simple calculation, we found that loblolly pine bark mass is about 10.4% of total stem dry weight. Our results showed that bark proportion in stem biomass decreased from 16.1 to 6.6% as tree dbh increased from 6 to 44 cm (Figure 4). The finding of Miles and Smith (2009) agree with ours for trees between about 16 and 18 cm, but not for smaller or bigger trees.

Conclusion We developed stem weight-to-volume ratios and proportional biomass component equations for loblolly pine trees across a wide portion of the species’ geographic range. The equation of the ratio between stem green weight and stem outside-bark volume was fitted simultaneously with equations for stem outside-bark volume and total green weight of bark with bark, and the equation of the ratio between stem wood dry weight and stem inside-bark volume was fitted simultaneously with equations for stem inside-bark volume and stem wood dry weight in the same manner, using the NSUR approach to account for the inherent correlation among the equations and heteroscedasticity. The resulting equations for stem outside-bark and inside-bark volumes, green weight of stem with bark, and stem wood dry weight, as intermediate equations, should be useful in regional prediction applications because of the large data set used in this study. The weight-to-volume ratios vary with tree size and can be estimated by tree dbh and HT. Proportional biomass 472


components can be modeled with tree dbh and HT. The proportional biomass composition equations were fitted with the DRM, FMLM, and LRM approaches. These approaches guarantee the constraint that all component proportions are nonnegative and sum to 1. The Dirichlet regression approach can be used to model biomass fractional composition and perform statistical tests about the effects on biomass allocation. Endnote 1. The SAS programs are available from the corresponding author.

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