Evaluating and improving of tree stump volume prediction models in the eastern United States. Ethan Jefferson Barker

Transcription

1 Evaluating and improving of tree stump volume prediction models in the eastern United States Ethan Jefferson Barker Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science In Forest Resources and Environmental Conservation Philip J. Radtke Harold E. Burkhart John W. Coulston May 1 st, 2017 Blacksburg, Virginia Keywords: regression; non-linear least squares; carbon sequestration; biomass

2 Evaluating and improvement of tree stump volume prediction models in the eastern United States Ethan Jefferson Barker ABSTRACT Forests are considered among the best carbon stocks on the planet. After forest harvest, the residual tree stumps persist on the site for years after harvest continuing to store carbon. A bigger concern is that the component ratio method requires a way to get stump volume to obtain total tree aboveground biomass. Therefore, the stump volumes contribute to the National Carbon Inventory. Agencies and organizations that are concerned with carbon accounting would benefit from an improved method for predicting tree stump volume. In this work, many model forms are evaluated for their accuracy in predicting stump volume. Stump profile and stump volume predictions were among the types of estimates done here for both outside and inside bark measurements. Fitting previously used models to a larger data set allows for improved regression coefficients and potentially more flexible and accurate models. The data set was compiled from a large selection of legacy data as well as some newly collected field measurements. Analysis was conducted for thirty of the most numerous tree species in the eastern United States as well as provide an improved method for inside and outside bark stump volume estimation.

3 Evaluating and improvement of tree stump volume prediction models in the eastern United States Ethan Jefferson Barker GENERAL AUDIENCE ABSTRACT Forests are considered among the best carbon stocks on the planet, and estimates of total tree aboveground biomass are needed to maintain the National Carbon Inventory. Tree stump volumes contribute to total tree aboveground biomass estimates. Agencies and organizations that are concerned with carbon accounting would benefit from an improved method for predicting tree stump volume. In this work, existing mathematical equations used to estimate tree stump volume are evaluated. A larger and more inclusive data set was utilized to improve the current equations, and to gather more insight in to which equations are best for different tree species and different areas of the eastern United States.

4 iv Table of Contents List of figures... v List of tables... vi Introduction... 1 Materials and Methods... 4 Description of Study Area... 4 Field Data... 7 Model Forms... 7 Raile Alterations Model Testing and Evaluation Stump profiles Stump volumes Results Taper Analysis Diameter outside bark Diameter inside bark Stump form class Volume Analysis Volume outside bark Volume inside bark Discussion Literature cited... 41

5 v List of figures Figure 1 Geographic distribution of field measurements Figure 2 Residual plot for Raile model dob predictions for eastern white pine Figure 3 Residual plot for the Raile model dib predictions for eastern white pine Figure 4 Fit line using predicted values from the Clark model (solid) and the species-specific constant produced from the Raile model (dashed) laid over a scatterplot of observed stump form classes for loblolly pine Figure 5 Residuals plot for the Raile 3 model ob volume predictions for shortleaf pine Figure 6 Residuals plot for the Clark ob 2 model ob volume predictions for shortleaf pine Figure 7 Residuals plot for the Wensel dib 2 model [12] ib volume predictions for eastern white pine

6 vi List of tables Table 1 Description of data including count of tree, minimum, mean, and maximum dbh of each species Table 2 AIC, RMSE, and Percent bias results for dob models averaged over all species for each model form Table 3 AIC results for dob prediction. Best column corresponds to which equation number had smallest value Table 4 RMSE results for dob prediction. Best column corresponds to which equation number had smallest value Table 5 Percent bias results for dob prediction. Best column corresponds to which equation number had smallest value Table 6 AIC, RMSE, and Percent Error results for dib prediction averaged over all species for each model form Table 7 AIC results for dib prediction. Best column corresponds to which equation number had smallest value Table 8 RMSE results for dib prediction. Best column corresponds to which equation number had smallest value Table 9 Percent bias results for dib prediction. Best column corresponds to which equation number had smallest value Table 10 RMSE and percent bias for outside bark volume predictions averaged over all species for each model form Table 11 RMSE results for ob volume prediction. Best column corresponds to which equation number had smallest value Table 12 Percent bias results for ob volume prediction. Best column corresponds to which equation number had smallest value Table 13 RMSE and percent bias for inside bark volume predictions averaged over all species for each model form Table 14 RMSE results for ib volume prediction. Best column corresponds to which equation number had smallest value Table 15 Percent bias results for ib volume prediction. Best column corresponds to which equation number had smallest value Table 16 Regression coefficients for the Raile models [1] and [2] Table 17 Regression coefficients for the Raile 3 models [15] and [16] Table 18 Regression coefficients for the dib Wensel and Olson model [11] Table 19 Regression Coefficients for both the Wensel models dob 2 [11] and dib 2 [12] Table 20 Stump dob regression coefficients reported by Raile (1982) Table 21 Stump dib regression coefficients reported by Wensel and Olson (1995) Table 22 Description of data used by Wensel and Olson to fit stump models

7 1 Introduction Interest in forest biomass production and carbon sequestration has led to an increased need for models capable of predicting standing tree volume and biomass contents (Weiskittel, et al. 2015). Past research has focused largely on predicting the merchantable stem contents of trees or sections of standing tree stems from the top of the stump to an upper stem diameter such as four or six inches. Another common goal has been to predict total stem volume or biomass (Wensel and Olson 1995). Stem volume equations have existed for many years, but they were often designed for predicting only whole stem or merchantable stem volumes (Biging 1984,Parresol, et al. 1987), typically not stump volumes. Stem taper equations, which predict diameters or squared diameters at any given height, came in to use because of their overall flexibility in predicting stem volume (Li, et al. 2012). Taper functions can be formulated so their mathematical integrals give accurate predictions either of whole stem volumes or for predicting volumes of individual sections of interest. Although some taper functions can predict stump volumes, they typically have not been designed with this use in mind. Taper functions tend to be most accurate for the part of the stem above the stump and below a merchantable top. In many applications, this is adequate because it corresponds to the part of the stem most widely used for commercial purpose. Tree stumps have proven to be difficult to model accurately because of the presence of swelling, flares, or buttressing at heights close to ground level (Biging 1984,Parresol, et al. 1987). Some investigators have created models that use stump measurements to predict diameter at breast height for investigating timber theft (Pond and Froese 2014,Westfall 2010). Although these models are useful, they do not predict the stump contents; thus, another solution is necessary if stump volume is of interest.

8 2 Numerous functional forms have been used to develop models of stem diameter-height profiles, i.e. taper models. Second order polynomial equations or greater have received attention in the past, because of their accuracy and flexibility in predicting upper stem diameters. Some work has shown that models of this type are unable to accurately describe the stumps in trees having significant buttressing (Biging 1984). Kozak (2004) developed a taper equation that is considered to be accurate for predicting total tree volumes as well as stem diameters, even though it exhibits significant bias in the stump and tip sections of the tree (Li, et al. 2012). Bruce and others (1968) used equations having polynomial powers as high as 40 to describe the stem profile of red alder (Alnus rubra Bong.) with success, but their use is challenging due to the large number of coefficients and lengthy equation forms involved. Segmented taper models reported by Max and Burkhart (1976) and Clark (1991) accurately predict stem profile and volume. Equations that split the stem in to segments that are modeled separately but constrained to join smoothly. These models are flexible and accurate, although selecting the number and location of join points can pose analytical challenges. Ideally, any segments involved must produce the same diameters at their join points as well as identical first derivatives at the join points (Burkhart and Tomé 2012). Measurements used to develop and fit taper equations are collected either inside bark (ib) or outside bark (ob). When predicting volumes either approach may be suitable, depending on the goal of the work being done. For example, when wood volume is desired ib predictions are appropriate, because ob measurements will over predict the wood volume by including the volume of bark (Stangle, et al. 2016). On the other hand, ob predictions would be appropriate for applications where wood and bark volumes are needed. For practical purposes, it is preferable to have equations for predicting either ib or ob volumes. One method is to have separate equations

9 3 available for either diameter measurements such as the work of Raile (1982). A second option is to combine an ib diameter equation with a bark thickness equation, as long as the bark thickness equation has the ability to predict at any point along the stem (Li and Weiskittel 2011). Models for accurately predicting stump volume or biomass should address current limitations and needs, especially in carbon monitoring applications. Organizations and agencies like the United Nations, U.S. Environmental Protection Agency, and U.S. Forest Service that are concerned with global or national carbon inventories should benefit from improved models to predict biomass in tree stumps. Groups like these often use existing tree-level equations for estimating above-stump biomass and carbon, and different equations for quantifying belowground carbon. Often stump carbon contents are determined from broad, regional factors that remain largely untested (Raile 1982,Weiskittel, et al. 2015). Stump volume equations would allow for improved accuracy in predicting stump contents in either standing trees or residual stumps following forest harvesting. Either approach should improve global and national estimates of carbon stored in forest ecosystems. The objective here was to develop and test stump taper models for both inside and outside bark diameters, designed for the purpose of accurate stump volume prediction in standing trees. This project defines a tree stump as the part of a tree between ground level and any specified height up to breast height, 4.5 feet above the ground. As a part of the overall objective three specific goals were pursued, (1) the evaluation and fitting of multiple model forms for predicting stump diameter profiles; (2) use of stump taper models in integral form to predict stump volumes; (3) and characterizing the accuracy of stump volume predictions obtained using stump taper equations.

10 4 Materials and Methods Description of Study Area The study area for development of stump models included the states east of the Great Plains including Minnesota, Iowa, Missouri, Oklahoma, and East Texas (Figure 1). This region was selected because of the identified need for improved stump models and availability of suitable sources of information for model development. Data sources included a large collection of legacy data and some new field data. Utilization of both data types allows development and testing of models for thirty different species over the entire study area. The available data were comprised of field measurements of stump diameters at 61,800 locations in the study region. Legacy Data Figure 1 Geographic distribution of field measurements. Legacy data compiled by Radtke, et al. (2015) was the primary source of data for models developed here. Legacy data were compiled from electronic files and paper documents from the U.S. Forest Service, private companies, universities, and other institutions with suitable stump

11 5 measurement data for use in individual stump volumes. The data mainly consisted of dimensional measurements from ground-line to breast height measured on standing and felled trees. The majority of measurements were made after felling trees by direct measure of either ib or ob diameters, or both, at multiple heights up to breast height. The data downloaded for this study, the database included 220 individual data sets with suitable taper measurements from 151,000 trees (Radtke, et al. 2015). Variables needed for the work pursued here included tree species, geographic location information, total tree height, diameter at breast height (dbh), height of measurement, and diameters ob and ib at the specified height. Trees to be used in model development were limited to those having at least one diameter measurement below breast height. This requirement ensured that every tree had at least two measurement points including dbh (Table 1). Species with fewer than 1000 measured trees were excluded in order to ensure that enough data were available for model fitting and to include species that comprise roughly three-fourths of current growing stock in the eastern United States. Not all trees included both ib and ob measurements but many included both ib and ob observations at one or more measurement points.

12 6 Table 1 Description of data including count of tree, minimum, mean, and maximum dbh of each species. Common name Genus Species Count of trees Minimum DBH Mean DBH Maximum DBH FIA species code USDA plant code balsam fir Abies balsamea 1, ABBA black oak Quercus velutina 1, QUVE black spruce Picea mariana 1, PIMA blackgum Nyssa sylvatica 1, NYSY chestnut oak Quercus prinus 2, QUPR2 eastern white pine Pinus strobus 2, PIST hickory spp. Carya spp. 2, CARYA jack pine Pinus banksiana 3, PIBA2 laurel oak Quercus laurifolia 1, QULA3 loblolly pine Pinus taeda 33, PITA longleaf pine Pinus palustris 6, PIPA2 northern red oak Quercus rubra 2, QURU paper birch Betula papyrifera 1, BEPA pond pine Pinus serotina 1, PISE post oak Quercus stellata 1, QUST quaking aspen Populus tremuloides 2, POTR5 red maple Acer rubrum 3, ACRU red pine Pinus resinosa 2, PIRE scarlet oak Quercus coccinea 1, QUCO2 shortleaf pine Pinus echinata 6, PIEC2 slash pine Pinus elliottii 15, PIEL southern red oak Quercus falcata 1, QUFA sugar maple Acer saccharum 1, ACSA3 swamp tupelo Nyssa biflora 1, NYBI sweetgum Liquidambar styraciflua 6, LIST2 Virginia pine Pinus virginiana 3, PIVI2 water oak Quercus nigra 2, QUNI white oak Quercus alba 5, QUAL white spruce Picea glauca 1, PIGL yellow-poplar Liriodendron tulipifera 4, LITU

13 7 Field Data New field measurements were also collected to supplement the legacy data and provide additional information on stump volume and biomass. The goal of collecting the additional data was to fill gaps in legacy data sets to achieve as wide as possible diameter ranges for the species of interest. Outside bark diameters at breast height (dbh, 4.5 ft), 3.5 ft, 2.5 ft, 1.5 ft, and 0.5 ft above ground line were recorded before felling trees. Following felling the total tree height was measured and a disk was cut from the stem at breast height. Double-bark thickness was determined by measuring ib diameters at the stump, the height where the tree was felled, and on the disk taken back to the lab. Bark thickness was then determined by subtraction. The bark thicknesses are then subtracted from the remaining outside bark diameter measurements to estimate the rest of the inside bark diameter measurements in stumps. In the lab, disks were weighed and measured for ib and ob diameters and disk thickness to calculate disk volumes. Model Forms Stump volume equations were developed based on existing taper and volume functions. Some modifications were made to achieve desired properties of ib or ob diameter predictions, and the prediction of either diameter or diameter squared. Some of the initial models of interest included equations by Raile (1982), Max and Burkhart (1976), Clark (1991), Ormerod (1973), Wensel and Olson (1995) and a few simple mathematical functions chosen for comparison. The Raile function for predicting ob diameters [1] is a one-parameter model that predicts ob diameter (d, inches) at any given height above ground (h, feet) up to 4.5 feet, with the regression coefficient (β) and a tree level predictor dbh (D, inches). The equation form ensures that when h is equal to breast height then d is equal to D. This is a desirable trait for these models, because it ensures that the model goes through dbh.

14 8 d = D + βd 4.5 h h+1 [1] The Raile function for predicting inside bark diameters [2] is similar to the outside bark equation, but with a second parameter related to bark thickness. For this equation, the intercept term (β 1 ) can be interpreted as the breast-height ratio of diameter ib to diameter ob, which is a common metric for bark thickness (Martin 1981). d = β 1 D + β 2 D 4.5 h h+1 [2] The Clark et al. (1991) taper function, hereafter Clark, is a widely known and accepted segmented taper function. This particular segmented function has three join points, the lowest at breast height. Here only the equation [3] for the lowest segment was evaluated. The Clark model requires total tree height, which is unusual for a stump model. Total tree height is represented by H with regression coefficients c, e, and r. The remaining variables hold the same definitions as in equations [1] and [2]. This model was fit twice, once in diameter form [3] and again in diametersquared form [4]. Both Clark forms were fit to both ib and ob. d = D 2 {1 + (c + e (1 h D3) H )r ( H )r } 1 ( H )r 1 2 [3] d 2 = D 2 {1 + (c + e (1 h D3) H )r ( H )r 1 ( H )r } [4] Ormerod (1973) developed a one-parameter model [5] that can be made linear by a logarithmic transformation. As with some of the other models tested the Ormerod equation ensures d =D where h = 4.5 ft. A regression parameter (α) was added to equation [5] to create

15 9 equation [6] for fitting ib values. Like β1 in equation [2], in equation [6] quantifies the bark thickness ratio. d = D ( H h H 4.5 )β [5] d = αd ( H h H 4.5 )β [6] Max and Burkhart (1976) developed a model with three stem segments described by polynomials. The geometric form assumed for the stump segment of the Max and Burkhart model is a neiloid [8], where β0, β1, and β2 are regression coefficients and hd is the height of the first join point. Here, hd was assigned to coincide with breast height, i.e. hd = 4.5 feet. This model does not go through dbh, which means in cases where h is set to breast height, d does not equal D. This makes sense for ib diameters, similar to equation [2] with β0 in equation [7] representing the squared bark thickness ratio for dib measured at breast height. This model will be analyzed for ib and ob measurements. d 2 = β h h D β 1 + β 2 h 2 d h2 [7] d An alternative geometric formulation was derived from a parabola, with the basic parabolic function being a two-parameter model altered in a way to force the equation through dbh when h is set to breast height [8]. d i 2 D 2 = 1 + β 1(x 1) + β 2 (x 2 1) [8] where x = h 4.5

16 10 When predicting ground line diameters this model will produce a stump form class that is species dependent and depends only on the parameters β1 and β2. The final model examined was developed by Wensel and Olson (1995), hereafter Wensel. The model form for ob diameters [9] and ib diameters [10] are below. d = De β 1(4.5 h) [9] d = (1 β 0 Z)De β 1(4.5 h) [10] This model was also fit in two forms, the original form that outputs diameter at height h, and a squared form that outputs diameter squared for a given h. This model will also be evaluated for ob [11] and ib [12] diameters. d 2 = (De β 1(4.5 h) ) 2 [11] d 2 = ((1 β 0 Z)De β 1(4.5 h) ) 2 [12] Raile Alterations To investigate possible improvements to the Raile models, several alterations were proposed. An alteration of the Raile outside bark model, denoted Raile 2 [13] here, was formulated with both sides of equation [1] squared. The purpose of squaring both sides is to avoid bias in volume estimation when using the model in its integral form. All of these altered equations will give d 2 = D 2 when h = 4.5 ft. This is ideal for ob models but not for ib models. When fitting these models with ib diameters the 1 in each equation can be replaced with an intercept term (β0), equation [14] is the Raile 2 model form for ib values. d 2 = D 2 (1 + β 4.5 h h+1 )2 [13]

17 11 d 2 = D 2 (β 0 + β h h+1 )2 [14] A further alteration denoted as Raile 3 was made to determine whether the single parameter in [13] was suitable for use in a quadratic form [15]. d 2 = D 2 (1 + β 1 x + β 2 x 2 ) [15] where x = 4.5 h h + 1 d 2 = D 2 (β 0 + β 1 x + β 2 x 2 ) [16] Model [15] was derived by expanding the factors in [13] to get (1 + 2βx + β 2 x 2 ) then replacing the coefficients on x and x 2 with two regression parameters. Equation [16] is the Raile 3 form used for ib values. The final Raile alteration, Raile 4 [17] is simply the original Raile model that has been squared. This formulation was done to understand if models that are squared outperform standard models in predicting stump volume. Equation [18] is the Raile 4 form used for ib values. d 2 = D 2 [1 + β ( 4.5 h h+1 )]2 [17] d 2 = D 2 [β 0 + β 1 ( 4.5 h h+1 )]2 [18]

18 12 Model Testing and Evaluation Stump profiles The individual models were estimated using nonlinear least squares, which is a common regression technique. Taper functional forms were evaluated first using root mean squared error (RMSE), Aikaike information criteria (AIC), and percent bias. RMSE can be defined as the square root of the average of squared errors. AIC can be thought of as a method of model selection that is a balance between goodness of fit and model complexity. RMSE and AIC were calculated for each model-species combination, separately for diameter and diameter-squared dependent variables because each species data set was fitted to alternative models. This approach facilitated direct comparison of RMSE and AIC as well as percent bias. Percent bias was calculated as observed minus predicted, divided by the observed value, and then multiplied by one hundred. The values were then averaged overall for each model-species combination. In addition to the summary statistics for each model and species, residual plots were made and investigated for all fitted model- species combinations. Stump volumes To evaluate stump volume predictions, Smailian s formula was used to calculate observed volumes for each tree stump in the data set. Vol (h i, h i + 1) = (d i 2 + d 2 i+1 ) 2 (h i+1 h i ) Where di is the ith diameter measurement (inches) and hi is the height (feet) at which di was measured with hi < hi+1. The number of height-diameter pairs measured on a stump was defined as n. Since no measurements above breast height were used, breast height measurements were denoted as dn and hn = 4.5 feet. In many trees, the lowest measurement point was observed

19 13 at h1 > 0, meaning no ground-line diameter was observed. For consistency in calculations, if h1 > 0, the volume below hi was calculated as a cylinder and denoted as Vol (0, h1). Vol (0, h 1 ) = d i 2 h i If d1 was measured at ground-line, i.e. hi = 0, then Vol (0, hi) was set to zero. The observed volume of any stump was obtained by summing the calculated volumes from Smailian s formula and a cylinder based at ground-line volume if necessary. n 1 Vol (0, h n ) = Vol (i, i + 1) i=0 To predict stump volumes fitted taper functions were integrated between the lowest taper measurement and 4.5 feet. In trees where h1 > 0, i.e. the lowest measurement point was above ground-line, cylinder volume was assumed for the stump below hi. Thus, predicted stump volume was calculated using equation [19]. Vol (0, h n ) 2 h = d 1 h i n d 2 dh [19] h 1 Volume residuals were calculated as observed minus predicted volumes for each individual tree. Vol residual = Vol (0, h n ) Vol (0, h n )

20 14 Results Taper Analysis Diameter outside bark Model fit statistics showed the Raile model and the Clark models performed best in terms of AIC and RMSE values averaged overall (Table 2). These two models both performed well in terms of percent bias with the Clark model [3] showing slightly less bias overall. The Wensel model fitted to diameter outside bark (dob) was third lowest in all statistics. The Ormerod model was the poorest performing model for all three averaged summary statistics. Table 2 AIC, RMSE, and Percent bias results for dob models averaged over all species for each model form. Model AIC RMSE Per. Bias Raile [1] Clark [3] Wensel [9] Ormerod [5] Raile 2 [13] Raile 3 [15] Raile 4 [17] Clark [4] Wensel [11] Neiloid [7] Parabolic [8] Because fit statistics averaged over many species did not allow for evaluation of speciesspecific models, detailed fit statistics were compiled for each species-model combination (Tables 3 5). When examining the best column, the number for each species corresponds to the equation number. AIC ranks for dob were largely consistent with the overall averages showing the Raile model [1] was best for twenty species and the Clark model [3] best for the remaining ten (Table 3). The same pattern was evident from dob model RMSE statistics (Table 4). The

21 15 Raile [1] and Clark [3] models demonstrated smallest biases for twenty-two of the thirty species and the Wensel [9] model has the lowest bias for the remaining eight. The Ormerod model [5] was never designated as having the lowest bias, AIC, or RMSE for any species for dob prediction. For diameter-squared models, the Clark model [4] and modified Raile models [13], [15], and [17] generally had the lowest average AIC and RMSE values. The Raile 3 model [15] performed best in all three summary statistics, although several other models gave results nearly equally good (Table 2). The Raile 3 model [15] and the parabolic model [8] both showed average bias of less than one percent and were thus the best performers when averaged across all species. Species-specific AIC and RMSE results were mostly consistent with Table 2 results, showing overall accuracy was best for the Raile 3 [15] and Clark models [4] in all but two species, paper birch (Betula papyrifera) and quaking aspen (Populus tremuloides). Percent bias results (Table 5) were notably less consistent between species-specific results and the overall averages for dob 2 models. Every dob 2 model examined showed the smallest bias for at least two species. Five models had the smallest bias for at least four species, the Raile 2 [13], Raile 4 [17], Clark [4], Neiloid [7], and parabolic [8]. The Clark model [4] dob 2 bias was lowest for eight species but also highest in seven species showing the tendency to over predict dob 2.

22 16 FIA code Table 3 AIC results for dob prediction. Best column corresponds to which equation number had smallest value. Raile [1] Clark [3] Wensel [9] Ormerod [5] Raile 2 [13] Raile 3 [15] Raile 4 [17] Clark [4] Wensel [11] Neiloid [7] Parabolic [8] Best

23 17 Table 4 RMSE results for dob prediction. Best column corresponds to which equation number had smallest value. FIA code Raile [1] Clark [3] Wensel [9] Ormerod [5] Raile 2 [13] Raile 3 [15] Raile 4 [17] Clark [4] Wensel [11] Neiloid [7] Parabolic [8] Best

24 18 Table 5 Percent bias results for dob prediction. Best column corresponds to which equation number had smallest value. FIA code Raile [1] Clark [3] Wensel [9] Ormerod [5] Raile 2 [13] Raile 3 [15] Raile 4 [17] Clark [4] Wensel [11] Neiloid [7] Parabolic [8] Best

25 19 Residuals plots for all species-model combinations were created and evaluated to check model fit. The residual pattern indicated a slight over prediction in the smallest diameters and relatively stable variance over the range of 10 inches to 20 inches dob (Figure 2). Generally, the same patterns held for all species when using the Raile model [1]. The main features of dob residuals were slight curvature in diameters less than about 5 inches and an increasing trend in residuals with increasing diameters. Figure 2 Residual plot for Raile model dob predictions for eastern white pine. Diameter inside bark For diameter inside bark (dib) models, The Raile model [2] has the lowest AIC and RMSE values when averaged across all species (Table 6). The Wensel model [10] has the lowest percent bias for the dib models averaged across all species. The Raile model [2] had the lowest AIC and RMSE in 28 of 30 species and the Clark [3] and Wensel [10] models each were lowest for one species, swamp tupelo (Nyssa biflora) and laurel oak (Quercus laurifolia) (Table 7). In Table 8, species-specific model biases were mainly consistent with the overall averages (Table 6, Table 9). Besides the Wensel model [10], which has the smallest biases for 19 species, the Raile

26 20 model [2] was smallest for 10, and the Clark model [3] had the smallest bias for another species, laurel oak. Species-specific fit statistics showed that the Raile 3 [16] fit the dib 2 data best for 27 of 30 species. The only exceptions being balsam fir (Abies balsamea) and paper birch, for which the Raile 2 model [14] performed slightly better, and black spruce (Picea mariana) for which the Neiloid model [7] had slightly lower AIC and RMSE values (Table 7, Table 8). In terms of bias the Wensel model [12] results were lowest for 20 of 30 species while four other models showed lowest biases, although just slightly lower, for either two or three species. Similar to the dob 2 results, the Clark model [4] showed notably larger biases for a number of species compared to the other dib 2 models tested. In several species, the Clark model [4] biases exceeded the closest competing models by as much as 4 to 10% or more. Table 6 AIC, RMSE, and Percent Error results for dib prediction averaged over all species for each model form. Model AIC RMSE Per. Bias Raile [2] Clark [3] Wensel [10] Ormerod [6] Raile 2 [14] Raile 3 [16] Raile 4 [18] Clark [4] Wensel [12] Neiloid [7]

27 21 FIA code Table 7 AIC results for dib prediction. Best column corresponds to which equation number had smallest value. Raile [2] Clark [3] Wensel [10] Ormerod [6] Raile 2 [14] Raile 3 [16] Raile 4 [18] Clark [4] Wensel [12] Neiloid [7] Best

28 22 Table 8 RMSE results for dib prediction. Best column corresponds to which equation number had smallest value. FIA code Raile [2] Clark [3] Wensel [10] Ormerod [6] Raile 2 [14] Raile 3 [16] Raile 4 [18] Clark [4] Wensel [12] Neiloid [7] Best

29 23 Table 9 Percent bias results for dib prediction. Best column corresponds to which equation number had smallest value. FIA code Raile [2] Clark [3] Wensel [10] Ormerod [6] Raile 2 [14] Raile 3 [16] Raile 4 [18] Clark [4] Wensel [12] Neiloid [7] Best

30 24 Residuals plots for dib predictions were made for all species model combinations. The Raile model [2] demonstrated a similar pattern to the plots for dob predictions (Figure 2). The same curvature for small trees was still evident as well as the increasing residual variance with increasing tree size (Figure 3). In addition, the location of the outliers were of concern. In the middle of the diameter range, the outliers have high positive residuals meaning they are under predicting by up to eight inches. At the end of the diameter range, the outliers are over predicting by almost six inches. Although, the body of the data is centered around zero and does not show a pattern like the outliers. Figure 3 Residual plot for the Raile model dib predictions for eastern white pine. Stump form class Attention was given to the relationship each model exhibited between ground-line diameter (gld) and dbh. Models such as the Raile model [1] and its variations reduce to a constant ratio between gld and dbh. More specifically when predicting the ground line diameter

31 25 by setting height above ground to zero these models will simplify to an expression that multiplies dbh by a species-specific constant. Conversely, models such as the Clark or Ormerod simplify to expressions where the gld to dbh ratio, referred to here after as the stump form class, varies with tree dbh or size. The Clark model [3] will give larger stump form classes for Figure 4 Fit line using predicted values from the Clark model (solid) and the species-specific constant produced from the Raile model (dashed) laid over a scatterplot of observed stump form classes for loblolly pine. smaller trees, especially those smaller than five inches dbh. This pattern was generally supported by examining how observed stump form class varied. The curve was calculated by predicting ground line diameters using the Clark model for loblolly pine (Pinus taeda) divided by dbh (Figure 4). Volume Analysis Volume outside bark When averaged across all 30 species the Clark dob [3] and Clark dob 2 [4] and Raile 3 [15] models gave similar results (Table 10). Except for its somewhat larger RMSE, the Ormerod model [5] performed well, giving the smallest volume bias of any models unlike the fit statistics for dob versus dob 2 models. RMSE and bias for volume predictions could be compared directly across all model forms with this being the case it was noted that two dob 2 models, Raile 3 [15]

32 26 and Wensel [11] were more accurate in 23 of 30 species than the most accurate dob volume model (Table 11, Table 12). Differences in volume prediction RMSE were often similar for two or more models, meaning tradeoffs between some models would be minor in terms of volume prediction. Table 10 RMSE and percent bias for outside bark volume predictions averaged over all species for each model form. Model RMSE Per. Bias Raile [1] Clark [3] Wensel [19] Ormerod [5] Raile 2 [11] Raile 3 [13] Raile 4 [15] Clark [4] Wensel [11] Neiloid [7] Parabolic [8]

33 27 Table 11 RMSE results for ob volume prediction. Best column corresponds to which equation number had smallest value.. FIA code Raile [1] Clark [3] Wensel [9] Ormerod [5] Raile 2 [13] Raile 3 [15] Raile 4 [17] Clark [4] Wensel [11] Neiloid [7] Parabolic [8] Best

34 28 Table 12 Percent bias results for ob volume prediction. Best column corresponds to which equation number had smallest value. FIA code Raile [1] Clark [3] Wensel [9] Ormerod [5] Raile 2 [13] Raile 3 [15] Raile 4 [17] Clark [4] Wensel [11] Neiloid [7] Parabolic [8] Best

35 29 The models were further evaluated through residuals plots that were made for all speciesmodel combinations. The Raile 3 [15] demonstrated increasing variance that we have seen in all models, but also the residuals were not centered on zero (Figure 5). Most of the residuals were larger than zero, which translates to the model under predicting ob volume. The Clark ob 2 [4] demonstrated a similar pattern of the residuals being larger than zero for most of the data, but for the larger trees, greater than 20 inches, the residuals tended to be less than zero which translates to over prediction (Figure 6). This pattern was present in most species for the Clark model [4] when predicting shortleaf pine (Pinus echinata). Figure 5 Residuals plot for the Raile 3 model ob volume predictions for shortleaf pine.

36 30 Figure 6 Residuals plot for the Clark ob 2 model ob volume predictions for shortleaf pine. Volume inside bark When averaged across all 30 species the Wensel dib [10], Wensel dib 2 [12], and the Raile 3 [16] performed similarly in both RMSE and bias (Table 13). The Wensel dib 2 [12] had slightly lower RMSE, but the Wensel dib [10] had less bias. Volume prediction RMSE values were often similar for two or more models, meaning tradeoffs between some models would be minor in terms of volume prediction (Table 14). For species-specific RMSE values the Wensel dib 2 [12] had the smallest value for 8 species, the Wensel dib [10] and the Raile 4 [18] both were smallest 7 species each (Table 14). Bias was more complicated; no individual model had the smallest bias for more than 5 species (Table 15). The models that had the smallest bias for 5 species each were the Raile 3 [16], Ormerod [6], and Wensel dib [10]. The Clark dib [3], Clark dib 2 [4], and Wensel dib 2 [12] had the smallest bias for 3 species each. As previous stated, the Clark dib 2 [4] model had the smallest bias for a few species, but in most cases had much larger bias as compared to the other models.

37 31 Table 13 RMSE and percent bias for inside bark volume predictions averaged over all species for each model form. Model RMSE Per. Bias Raile [2] Clark [3] Wensel [10] Ormerod [6] Raile 2 [14] Raile 3 [16] Raile 4 [18] Clark [4] Wensel [12] Neiloid [8]

38 32 Table 14 RMSE results for ib volume prediction. Best column corresponds to which equation number had smallest value. FIA code Raile [2] Clark [3] Wensel [10] Ormerod [6] Raile 2 [14] Raile 3 [16] Raile 4 [18] Clark [4] Wensel [12] Neiloid [7] Best

39 33 Table 15 Percent bias results for ib volume prediction. Best column corresponds to which equation number had smallest value. FIA code Raile [2] Clark [3] Wensel [10] Ormerod [6] Raile 2 [14] Raile 3 [16] Raile 4 [18] Clark [4] Wensel [12] Neiloid [7] Best

40 34 Further insights for ib volume predictions were provided by residual plots created for all species-model combinations. Wensel dib 2 [12] resulted in the lowest RMSE for 8 species, and performed well in terms of bias (Table 14, Table 15). The residual plot for Wensel dib 2 [12] had the increasing variance with increasing tree size seen with all the models (Figure 7). The model also tended to under predict models similarly to other models, especially for trees between 5 inches and 15 inches in diameter. Outliers also existed in the same diameter range that were under predicted approximately 2 cubic feet or more. Unlike some other models the Wensel dib 2 [12] does not exhibit curvature for smaller trees (Figure 7). Figure 7 Residuals plot for the Wensel dib 2 model [12] ib volume predictions for eastern white pine.

41 35 Table 16 Regression coefficients for the Raile models [1] and [2]. dob [1] dib [2] Common name β β 1 β 2 balsam fir white spruce black spruce jack pine shortleaf pine slash pine longleaf pine red pine pond pine eastern white pine loblolly pine Virginia pine red maple sugar maple paper birch hickory spp sweetgum yellow-poplar blackgum swamp tupelo quaking aspen white oak scarlet oak southern red oak laurel oak water oak chestnut oak northern red oak post oak black oak

42 36 Table 17 Regression coefficients for the Raile 3 models [15] and [16]. dob dib Common name β 1 β 2 β 0 β 1 β 2 balsam fir black oak black spruce blackgum chestnut oak eastern white pine hickory spp jack pine laurel oak loblolly pine longleaf pine northern red oak paper birch pond pine post oak quaking aspen red maple red pine scarlet oak shortleaf pine slash pine southern red oak sugar maple swamp tupelo sweetgum Virginia pine water oak white oak white spruce yellow-poplar

43 37 Table 18 Regression coefficients for the dib Wensel and Olson model [11] Wensel and Olson Common name b 0 b 1 balsam fir white spruce black spruce jack pine shortleaf pine slash pine longleaf pine red pine pond pine eastern white pine loblolly pine Virginia pine red maple sugar maple paper birch hickory spp sweetgum yellow-poplar blackgum swamp tupelo quaking aspen white oak scarlet oak southern red oak laurel oak water oak chestnut oak northern red oak post oak black oak

44 38 Table 19 Regression Coefficients for both the Wensel models dob 2 [11] and dib 2 [12]. dob dib Common name β 1 β 0 β 1 balsam fir black oak black spruce blackgum chestnut oak eastern white pine hickory spp jack pine laurel oak loblolly pine longleaf pine northern red oak paper birch pond pine post oak quaking aspen red maple red pine scarlet oak shortleaf pine slash pine southern red oak sugar maple swamp tupelo sweetgum Virginia pine water oak white oak white spruce yellow-poplar

45 39 Discussion Raile (1982) successfully modeled stump volumes for northern tree species. Overlap exists between the species Raile modeled and the species modeled here. Species shared between both studies are eastern white pine (Pinus strobus), jack pine (Pinus banksiana), white spruce (Picea glauca), black spruce, balsam fir, paper birch, and quaking aspen. Raile also modeled oaks (Quercus spp.) and maples (Acer spp.), but they were analyzed by genus as opposed to by species (Raile 1982). Comparison of regressions coefficients to the coefficients Raile reported demonstrate similar coefficients produced by both authors (Table 16, Table 20). Error from each study were reported differently. Although error for the Raile models [1, 2] was reported as being low by both authors. Additionally, the sample size used for work performed here has a minimum of 1,000 individuals per species (Table 1). Four species in the work of Raile (1982) has a sample size over 1,000 individuals per species. The diameter ranges for each species are also similar. Table 20 Stump dob regression coefficients reported by Raile (1982).