Evaluating and improving of tree stump volume prediction models … · 2020-01-21 · The objective...

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 1st, 2017

Blacksburg, Virginia

Keywords: regression; non-linear least squares; carbon sequestration; biomass

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.

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.

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 ...................................................................................................................................... 10

Model Testing and Evaluation ................................................................................................................ 12

Stump profiles ..................................................................................................................................... 12

Stump volumes ................................................................................................................................... 12

Results ......................................................................................................................................................... 14

Taper Analysis ......................................................................................................................................... 14

Diameter outside bark ........................................................................................................................ 14

Diameter inside bark ........................................................................................................................... 19

Stump form class ..................................................................................................................................... 24

Volume Analysis ...................................................................................................................................... 25

Volume outside bark ........................................................................................................................... 25

Volume inside bark ............................................................................................................................. 30

Discussion.................................................................................................................................................... 39

Literature cited............................................................................................................................................ 41

v

List of figures

Figure 1 Geographic distribution of field measurements. ............................................................................ 4

Figure 2 Residual plot for Raile model dob predictions for eastern white pine. ........................................ 19

Figure 3 Residual plot for the Raile model dib predictions for eastern white pine. ................................... 24

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. ................................................................................................................................................ 25

Figure 5 Residuals plot for the Raile 3 model ob volume predictions for shortleaf pine. .......................... 29

Figure 6 Residuals plot for the Clark ob2 model ob volume predictions for shortleaf pine. ...................... 30

Figure 7 Residuals plot for the Wensel dib2 model [12] ib volume predictions for eastern white pine. ... 34

vi

List of tables

Table 1 Description of data including count of tree, minimum, mean, and maximum dbh of each species.

...................................................................................................................................................................... 6

Table 2 AIC, RMSE, and Percent bias results for dob models averaged over all species for each model

form............................................................................................................................................................. 14

Table 3 AIC results for dob prediction. Best column corresponds to which equation number had smallest

value. ........................................................................................................................................................... 16

Table 4 RMSE results for dob prediction. Best column corresponds to which equation number had

smallest value. ............................................................................................................................................ 17

Table 5 Percent bias results for dob prediction. Best column corresponds to which equation number had

smallest value. ............................................................................................................................................ 18

Table 6 AIC, RMSE, and Percent Error results for dib prediction averaged over all species for each model

form............................................................................................................................................................. 20

Table 7 AIC results for dib prediction. Best column corresponds to which equation number had smallest

value. ........................................................................................................................................................... 21

Table 8 RMSE results for dib prediction. Best column corresponds to which equation number had

smallest value. ............................................................................................................................................ 22

Table 9 Percent bias results for dib prediction. Best column corresponds to which equation number had

smallest value. ............................................................................................................................................ 23

Table 10 RMSE and percent bias for outside bark volume predictions averaged over all species for each

model form. ................................................................................................................................................ 26

Table 11 RMSE results for ob volume prediction. Best column corresponds to which equation number

had smallest value.. .................................................................................................................................... 27

Table 12 Percent bias results for ob volume prediction. Best column corresponds to which equation

number had smallest value. ........................................................................................................................ 28

Table 13 RMSE and percent bias for inside bark volume predictions averaged over all species for each

model form. ................................................................................................................................................ 31

Table 14 RMSE results for ib volume prediction. Best column corresponds to which equation number

had smallest value. ..................................................................................................................................... 32

Table 15 Percent bias results for ib volume prediction. Best column corresponds to which equation

number had smallest value. ........................................................................................................................ 33

Table 16 Regression coefficients for the Raile models [1] and [2]. ............................................................ 35

Table 17 Regression coefficients for the Raile 3 models [15] and [16]. ..................................................... 36

Table 18 Regression coefficients for the dib Wensel and Olson model [11] .............................................. 37

Table 19 Regression Coefficients for both the Wensel models dob2 [11] and dib2 [12]. ........................... 38

Table 20 Stump dob regression coefficients reported by Raile (1982). ..................................................... 39

Table 21 Stump dib regression coefficients reported by Wensel and Olson (1995). ................................. 40

Table 22 Description of data used by Wensel and Olson to fit stump models. .......................................... 40

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.

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

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.

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.

Figure 1 Geographic distribution of field measurements.

Legacy Data

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

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.

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 1,315 1.5 7.7 19.9 12 ABBA black oak Quercus velutina 1,701 2.0 7.7 20.9 837 QUVE black spruce Picea mariana 1,098 1.0 8.5 30.5 95 PIMA blackgum Nyssa sylvatica 1,177 3.3 10.4 20.5 693 NYSY chestnut oak Quercus prinus 2,384 1.0 10.5 33.0 832 QUPR2 eastern white pine Pinus strobus 2,533 1.0 8.7 22.6 129 PIST hickory spp. Carya spp. 2,974 0.7 13.1 41.9 400 CARYA jack pine Pinus banksiana 3,165 0.8 10.7 26.4 105 PIBA2 laurel oak Quercus laurifolia 1,216 1.0 12.2 45.0 820 QULA3 loblolly pine Pinus taeda 33,826 1.0 9.6 41.8 131 PITA longleaf pine Pinus palustris 6,375 1.7 8.5 21.3 121 PIPA2 northern red oak Quercus rubra 2,333 1.0 13.0 35.2 833 QURU paper birch Betula papyrifera 1,269 1.0 10.2 32.7 375 BEPA pond pine Pinus serotina 1,257 0.7 10.2 38.3 128 PISE post oak Quercus stellata 1,087 1.1 10.3 27.0 835 QUST quaking aspen Populus tremuloides 2,023 1.0 11.8 31.1 746 POTR5 red maple Acer rubrum 3,789 0.7 8.6 22.4 316 ACRU red pine Pinus resinosa 2,517 1.0 12.4 36.7 125 PIRE scarlet oak Quercus coccinea 1,557 1.0 10.1 44.9 806 QUCO2 shortleaf pine Pinus echinata 6,311 2.1 10.6 25.8 110 PIEC2 slash pine Pinus elliottii 15,194 1.0 9.9 25.1 111 PIEL southern red oak Quercus falcata 1,894 1.0 10.2 38.7 812 QUFA sugar maple Acer saccharum 1,442 1.0 11.3 33.8 318 ACSA3 swamp tupelo Nyssa biflora 1,740 1.0 12.1 44.4 694 NYBI sweetgum Liquidambar styraciflua 6,159 1.0 9.8 28.2 611 LIST2 Virginia pine Pinus virginiana 3,159 0.7 12.1 37.5 132 PIVI2 water oak Quercus nigra 2,247 1.0 13.8 44.3 827 QUNI white oak Quercus alba 5,194 1.0 11.8 48.9 802 QUAL white spruce Picea glauca 1,206 0.1 10.4 26.1 94 PIGL yellow-poplar Liriodendron tulipifera 4,715 1.0 10.3 32.7 621 LITU

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.

8

𝑑 = 𝐷 + 𝛽𝐷4.5−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 (𝛽1) 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 = 𝛽1𝐷 + 𝛽2𝐷4.5−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 diameter-

squared form [4]. Both Clark forms were fit to both ib and ob.

𝑑 = 𝐷2 {1 + (𝑐 +𝑒

𝐷3) ∗ (

1−ℎ

𝐻)

𝑟−(

1−4.5

𝐻)𝑟

1−(1−4.5

𝐻)𝑟

}

1

2

[3]

𝑑2 = 𝐷2 {1 + (𝑐 +𝑒

𝐷3) ∗ (

1−ℎ

𝐻)

𝑟−(

1−4.5

𝐻)𝑟

1−(1−4.5

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

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.

𝑑2

𝐷2 = 𝛽0 + 𝛽1ℎ

ℎ𝑑+ 𝛽2

ℎ2

ℎ𝑑2 [7]

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].

𝑑𝑖2

𝐷2= 1 + 𝛽1(𝑥 − 1) + 𝛽2(𝑥2 − 1) [8]

where

𝑥 = ℎ

4.5

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 = 𝐷𝑒𝛽1(4.5−ℎ) [9]

𝑑 = (1 − 𝛽0𝑍)𝐷𝑒𝛽1(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.

𝑑2 = (𝐷𝑒𝛽1(4.5−ℎ))2 [11]

𝑑2 = ((1 − 𝛽0𝑍)𝐷𝑒𝛽1(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 d2 = D2 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.

𝑑2 = 𝐷2(1 + 𝛽4.5−ℎ

ℎ+1)2 [13]

11

𝑑2 = 𝐷2(𝛽0 + 𝛽14.5−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].

𝑑2 = 𝐷2(1 + 𝛽1𝑥 + 𝛽2𝑥2) [15]

where

𝑥 = 4.5 − ℎ

ℎ + 1

𝑑2 = 𝐷2(𝛽0 + 𝛽1𝑥 + 𝛽2𝑥2) [16]

Model [15] was derived by expanding the factors in [13] to get (1 + 2𝛽𝑥 + 𝛽2𝑥2) then

replacing the coefficients on x and x2 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.

𝑑2 = 𝐷2 [1 + 𝛽 (4.5−ℎ

ℎ+1)]

2

[17]

𝑑2 = 𝐷2 [𝛽0 + 𝛽1 (4.5−ℎ

ℎ+1)]

2

[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 (ℎ𝑖 , ℎ𝑖 + 1) = .005454 ×(𝑑𝑖

2 + 𝑑𝑖+12 )

2 × (ℎ𝑖+1 − ℎ𝑖)

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

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, ℎ1) = 0.005454 × 𝑑𝑖2 × ℎ𝑖

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.

𝑉𝑜𝑙 (0, ℎ𝑛) = ∑ 𝑉𝑜𝑙 (𝑖, 𝑖 + 1)𝑛−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].

𝑉𝑜�̂�(0, ℎ𝑛) = .005454 × 𝑑12 × ℎ𝑖 + 0.005454 × ∫ 𝑑2̂ℎ𝑛

ℎ1 𝑑ℎ [19]

Volume residuals were calculated as observed minus predicted volumes for each

individual tree.

𝑉𝑜𝑙𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = 𝑉𝑜𝑙 (0, ℎ𝑛) − 𝑉𝑜�̂�(0, ℎ𝑛)

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] 25350.5 1.05 0.70

Clark [3] 25699.9 1.05 0.64

Wensel [9] 27813.4 1.16 -1.05

Ormerod [5] 32151.3 1.44 -3.53

Raile 2 [13] 94226.4 46.39 1.28

Raile 3 [15] 93695.2 44.73 -0.70

Raile 4 [17] 94226.4 46.39 1.28

Clark [4] 94247.4 45.46 -2.13

Wensel [11] 96392.1 50.08 -3.30

Neiloid [7] 94447.6 46.01 -1.10

Parabolic [8] 94459.3 46.03 -0.69

Because fit statistics averaged over many species did not allow for evaluation of species-

specific 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

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 dob2 models. Every dob2 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] dob2 bias was lowest for eight

species but also highest in seven species showing the tendency to over predict dob2.

16

Table 3 AIC 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

12 5681 6010 6409 7109 28663 28632 28663 28882 29140 28768 28767 1 94 5936 5972 6084 7692 29579 29547 29579 29623 29655 29559 29557 1 95 2585 2467 2787 4071 21082 20926 21082 20795 21091 20922 20920 3

105 8669 9266 9320 14488 52466 52105 52466 52877 52954 52150 52148 1 110 37819 40585 45619 58447 176172 174851 176172 177951 184035 178236 178303 1 111 85057 84956 91737 112567 318565 316446 318565 318148 324395 318666 318695 3 121 46233 47489 53415 60573 183887 183267 183887 184198 189709 185002 185036 1 125 7510 7050 7430 8694 44683 44087 44683 43947 44159 44033 44031 3 128 12144 12504 13527 14940 44513 44439 44513 44762 45766 44778 44781 1 129 16805 16993 18164 21211 71686 71122 71686 71699 72694 71621 71622 1 131 141895 146082 159650 188974 544584 541852 544584 546424 563040 547810 547919 1 132 14832 16622 19956 23107 74638 74551 74638 75895 79982 76588 76617 1 316 32626 33138 34953 36554 107714 107607 107714 107783 109343 108013 108018 1 318 9844 9913 10605 11063 33809 33784 33809 33793 34422 33960 33963 1 375 4295 4408 4676 5217 19004 19005 19004 19083 19353 19100 19100 1 400 26465 26977 29312 32488 91934 91700 91934 92032 94291 92394 92404 1 611 40190 39361 42551 50430 136103 134498 136103 135122 138148 135259 135273 3 621 38522 39484 41803 43340 126847 126792 126847 127446 129237 127674 127680 1 693 10334 10507 10707 11788 33588 33454 33588 33693 33917 33605 33606 1 694 26431 24639 24778 30125 68938 67460 68938 67416 67531 67476 67477 3 746 9104 9191 10008 11740 37281 37283 37281 37364 38084 37344 37344 1 802 47987 48279 51415 55290 153817 152969 153817 153333 156588 153815 153830 1 806 17995 18300 19894 21541 57055 57032 57055 57207 58847 57512 57517 1 812 13862 13836 15146 17093 48683 48077 48683 48143 49328 48306 48311 3 820 12590 12168 12668 14891 38598 38241 38598 38288 38507 38292 38292 3 827 14363 14238 15482 17914 47951 47454 47951 47671 48795 47730 47735 3 832 22745 22663 24567 26603 78799 78380 78799 78304 79838 78604 78606 3 833 22487 22011 23290 25333 70053 69454 70053 69412 70334 69557 69558 3 835 9084 9234 10166 11437 30903 30795 30903 30877 31852 31100 31105 1 837 16422 16653 18285 19817 55198 55046 55198 55255 56728 55554 55564 1

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

12 0.59 0.62 0.66 0.73 21.28 21.17 21.28 22.01 22.92 21.62 21.63 1 94 0.60 0.60 0.61 0.79 22.55 22.44 22.55 22.70 22.82 22.48 22.47 1 95 0.38 0.37 0.39 0.49 9.19 8.94 9.19 8.74 9.20 8.93 8.93 3

105 0.45 0.47 0.47 0.69 10.61 10.34 10.61 10.93 10.99 10.37 10.37 1 110 0.62 0.67 0.75 1.04 19.59 18.96 19.59 20.48 23.84 20.63 20.66 1 111 0.87 0.87 0.96 1.32 29.13 28.22 29.13 28.95 31.80 29.18 29.19 3 121 0.79 0.82 0.95 1.14 27.10 26.67 27.10 27.31 31.46 27.88 27.91 1 125 0.49 0.47 0.49 0.55 16.01 15.14 16.01 14.94 15.24 15.06 15.06 3 128 0.92 0.96 1.07 1.25 32.52 32.26 32.52 33.42 37.33 33.48 33.49 1 129 0.77 0.78 0.85 1.05 34.04 32.74 34.04 34.06 36.49 33.88 33.88 1 131 0.87 0.90 1.02 1.33 33.02 32.21 33.02 33.57 39.00 33.99 34.03 1 132 0.55 0.60 0.72 0.86 14.57 14.50 14.57 15.61 19.53 16.21 16.24 1 316 1.30 1.34 1.47 1.59 62.38 62.03 62.38 62.59 67.84 63.34 63.36 1 318 1.12 1.13 1.26 1.36 47.07 46.89 47.07 46.94 51.80 48.18 48.21 1 375 0.64 0.65 0.69 0.78 17.51 17.51 17.51 17.81 18.94 17.88 17.88 1 400 1.16 1.20 1.38 1.66 56.47 55.69 56.47 56.79 64.94 58.02 58.06 1 611 1.17 1.14 1.29 1.75 50.74 47.64 50.74 48.82 54.98 49.08 49.11 3 621 1.35 1.41 1.56 1.67 68.97 68.80 68.97 70.83 76.72 71.55 71.57 1 693 1.22 1.25 1.29 1.53 46.00 45.04 46.00 46.75 48.43 46.11 46.12 1 694 2.60 2.22 2.24 3.63 119.00 104.18 119.00 103.76 104.86 104.33 104.34 3 746 0.72 0.72 0.80 0.98 20.64 20.65 20.64 20.84 22.72 20.79 20.80 1 802 1.44 1.46 1.64 1.89 73.82 71.52 73.82 72.50 81.83 73.81 73.85 1 806 1.51 1.55 1.83 2.16 79.67 79.47 79.67 80.89 95.57 83.44 83.49 1 812 1.18 1.18 1.37 1.71 63.33 59.08 63.33 59.52 68.17 60.64 60.68 3 820 1.78 1.66 1.80 2.56 109.68 103.63 109.68 104.38 108.10 104.46 104.47 3 827 1.20 1.19 1.36 1.79 51.40 48.62 51.40 49.80 56.48 50.13 50.17 3 832 1.15 1.14 1.30 1.49 53.10 51.59 53.10 51.32 57.01 52.39 52.40 3 833 1.63 1.57 1.75 2.08 92.53 87.94 92.53 87.62 94.77 88.71 88.72 3 835 1.13 1.16 1.35 1.68 45.53 44.69 45.53 45.32 53.47 47.06 47.11 1 837 1.27 1.30 1.54 1.79 64.27 63.28 64.27 64.63 75.03 66.61 66.68 1

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

12 -1.0 -0.2 -1.3 -1.6 -3.7 -3.3 -3.7 -0.9 -4.2 -3.2 -2.9 3 94 -0.5 -0.2 -0.8 -1.6 -1.8 -1.9 -1.8 -1.2 -2.2 -1.8 -1.8 3 95 -0.5 -0.1 -0.6 -1.1 -2.1 -2.0 -2.1 -0.4 -1.9 -1.8 -1.8 3

105 0.0 -0.3 -0.5 0.9 -0.3 -0.9 -0.3 -1.8 -1.3 -0.9 -0.9 1 110 1.2 1.1 0.0 1.5 2.5 1.7 2.5 1.1 1.0 1.7 2.1 9 111 0.8 0.3 -1.7 -3.2 0.5 -2.2 0.5 0.9 -6.0 -3.2 -2.7 3 121 0.5 0.3 -1.0 -0.3 0.9 -0.3 0.9 -1.2 -2.7 -0.6 -0.2 8 125 0.2 -0.1 0.1 -0.4 0.2 0.3 0.2 -1.2 0.4 0.4 0.4 9 128 0.8 0.4 -1.3 -2.7 1.4 0.1 1.4 -0.1 -4.0 -0.6 -0.2 4 129 0.3 0.5 -0.4 -2.8 1.3 0.1 1.3 -3.1 -0.5 -0.2 -0.1 8 131 0.6 0.7 -1.0 -3.0 0.9 -0.5 0.9 -0.1 -2.7 -1.1 -0.6 4 132 1.2 1.0 -0.3 -1.3 3.0 2.4 3.0 0.7 -0.3 1.7 2.1 11 316 0.4 1.1 -1.3 -5.7 0.8 -0.7 0.8 -4.9 -4.4 -1.6 -1.1 1 318 -0.3 0.3 -0.9 -2.4 -1.0 -1.5 -1.0 -5.2 -2.8 -1.4 -0.9 1 375 -0.4 0.0 -0.7 -1.0 -1.6 -1.5 -1.6 -1.1 -2.2 -1.6 -1.2 3 400 1.3 1.7 -0.8 -5.7 2.4 0.5 2.4 -0.8 -3.5 0.0 0.5 7 611 0.4 0.7 -2.1 -7.4 -0.1 -3.6 -0.1 -2.9 -6.4 -4.3 -3.9 13 621 0.1 0.5 -1.5 -4.4 -0.4 -1.1 -0.4 -2.8 -4.7 -2.0 -1.6 1 693 2.0 1.7 0.3 -5.7 5.4 3.5 5.4 -6.8 1.4 3.1 3.5 9 694 4.3 0.0 -1.2 -2.9 9.6 -1.3 9.6 -14.2 -4.3 -2.9 -2.3 3 746 -0.9 -0.2 -1.5 -1.0 -3.0 -3.0 -3.0 -1.0 -4.5 -2.8 -2.6 3 802 2.0 1.4 -0.8 -4.3 5.4 2.2 5.4 -3.3 -2.5 1.8 2.4 9 806 0.1 0.9 -2.4 -4.4 -1.2 -2.3 -1.2 2.0 -8.6 -2.7 -1.9 1 812 2.3 1.8 -0.5 -4.1 5.9 1.8 5.9 -1.9 -2.7 1.3 1.8 9 820 0.3 0.4 -3.9 -17.3 0.3 -7.0 0.3 -3.2 -11.8 -8.1 -7.5 1 827 0.7 1.4 -1.8 -10.1 1.5 -2.3 1.5 -1.7 -6.1 -3.0 -2.4 1 832 1.2 0.6 -0.7 -2.3 3.3 1.2 3.3 -4.1 -1.2 1.0 1.3 3 833 0.7 0.1 -1.7 -2.9 1.4 -2.4 1.4 -3.9 -5.3 -2.3 -2.0 3 835 2.5 2.3 -0.1 -4.7 5.7 3.7 5.7 -0.7 -1.0 3.2 3.9 9 837 0.8 1.2 -1.2 -4.1 1.2 -0.5 1.2 -0.3 -4.0 -0.9 -0.2 8

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

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 dib2 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 dob2

results, the Clark model [4] showed notably larger biases for a number of species compared to

the other dib2 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] 23902.5 1.03 -1.30

Clark [3] 27800.2 1.18 -3.77

Wensel [10] 25721.1 1.13 -1.10

Ormerod [6] 29867.5 1.39 -3.77

Raile 2 [14] 86353.2 40.97 -4.95

Raile 3 [16] 86137.7 40.31 -4.55

Raile 4 [18] 86353.2 40.97 -4.95

Clark [4] 89035.3 44.05 -10.92

Wensel [12] 88007.6 43.75 -3.75

Neiloid [7] 86838.8 41.43 -4.57

21

Table 7 AIC 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

12 3632 4501 4217 4703 18789 18790 18789 19483 19261 18910 2 94 4740 5694 4861 5955 22966 22943 22966 23661 23036 22971 2 95 2650 3805 2706 3665 18032 17869 18032 18890 17899 17857 2

105 7649 10469 8409 10242 42910 42891 42910 44924 43554 43149 2 110 39658 55953 44937 55029 168623 168281 168623 183820 174799 171280 2 111 87111 101513 92027 111617 308732 308109 308732 316060 313406 310323 2 121 45777 58030 52222 59589 177837 177777 177837 186611 183518 180024 2 125 8544 13670 8608 9715 44434 44160 44434 49696 44375 44191 2 128 11948 14683 13052 15056 42879 42845 42879 44598 43842 43139 2 129 14974 19688 15619 17367 60917 60515 60917 65047 61204 60718 2 131 139868 175009 151402 184249 519158 517849 519158 542840 531625 523015 2 132 14804 18948 18917 22755 71664 71662 71664 74754 76007 73544 2 316 28025 28711 29657 31493 92368 92331 92368 92579 93442 92647 2 318 4681 4814 4825 5060 15052 15012 15052 15097 15118 15031 2 375 3641 4096 3939 4244 15156 15158 15156 15541 15410 15281 2 400 25271 27598 27562 31258 87060 86971 87060 88034 88879 87642 2 611 40149 40602 41937 49588 132452 131658 132452 132503 134055 132357 2 621 35836 38568 38158 40318 115675 115656 115675 117193 117392 116378 2 693 8799 9441 8933 9722 27596 27525 27596 28087 27726 27598 2 694 24858 24106 23900 27415 66180 65518 66180 65585 65523 65530 10 746 3479 3799 3716 4080 13251 13247 13251 13493 13418 13311 2 802 43668 45584 46321 51174 141524 141057 141524 142129 143474 141761 2 806 16802 17548 18347 20223 53144 53143 53144 53443 54473 53649 2 812 13220 14462 14387 16636 46776 46527 46776 47047 47425 46757 2 820 12214 12088 12383 14302 37842 37694 37842 37758 37868 37748 3 827 13879 14277 14873 17291 46600 46375 46600 46773 47370 46678 2 832 20018 22643 21364 23908 68604 68442 68604 70024 69408 68647 2 833 18024 18528 18865 20786 56922 56747 56922 57024 57365 56896 2 835 8332 9155 9231 10520 28211 28176 28211 28583 28949 28462 2 837 14824 16021 16257 18064 49242 49203 49242 49778 50409 49672 2

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

12 0.52 0.63 0.59 0.66 13.12 13.12 13.12 15.21 14.51 13.46 2 94 0.61 0.74 0.63 0.77 21.53 21.43 21.53 24.65 21.82 21.54 2 95 0.41 0.52 0.42 0.51 9.37 9.07 9.37 11.15 9.12 9.05 2

105 0.47 0.60 0.50 0.59 9.87 9.85 9.87 11.75 10.43 10.07 2 110 0.66 1.00 0.76 0.98 17.66 17.50 17.66 25.99 20.66 18.89 2 111 0.90 1.11 0.97 1.30 25.15 24.91 25.15 28.08 26.98 25.75 2 121 0.78 1.07 0.92 1.12 23.21 23.17 23.21 29.07 26.85 24.55 2 125 0.54 0.88 0.55 0.61 16.15 15.74 16.15 26.56 16.06 15.78 2 128 0.90 1.22 1.02 1.27 27.16 27.06 27.16 32.82 30.20 27.95 2 129 0.79 1.14 0.83 0.95 29.36 28.44 29.36 40.64 30.03 28.90 2 131 0.87 1.20 0.97 1.31 28.15 27.82 28.15 34.97 31.56 29.17 2 132 0.55 0.70 0.70 0.86 13.31 13.31 13.31 15.82 16.97 14.78 2 316 1.28 1.33 1.41 1.57 58.31 58.18 58.31 59.05 62.15 59.28 2 318 1.25 1.31 1.32 1.43 48.22 47.54 48.22 48.99 49.37 47.85 2 375 0.66 0.75 0.72 0.78 16.20 16.21 16.20 18.02 17.39 16.77 2 400 1.13 1.30 1.30 1.63 49.04 48.77 49.04 52.04 54.79 50.81 2 611 1.19 1.21 1.27 1.72 45.73 44.32 45.73 45.82 48.73 45.56 2 621 1.34 1.53 1.50 1.66 61.06 61.00 61.06 65.65 66.28 63.14 2 693 1.27 1.43 1.30 1.51 43.87 43.28 43.87 48.12 44.95 43.87 2 694 2.28 2.13 2.09 2.87 95.18 89.65 95.18 90.19 89.70 89.75 10 746 0.78 0.87 0.84 0.95 20.81 20.77 20.81 22.57 22.01 21.22 2 802 1.35 1.45 1.50 1.81 63.22 62.06 63.22 64.74 68.25 63.81 2 806 1.46 1.58 1.72 2.10 71.13 71.11 71.13 73.43 81.99 75.07 2 812 1.13 1.30 1.29 1.68 56.11 54.50 56.11 57.90 60.51 55.98 2 820 1.68 1.64 1.72 2.33 97.28 95.00 97.28 95.98 97.69 95.83 3 827 1.15 1.20 1.28 1.68 45.24 44.10 45.24 46.12 49.32 45.63 2 832 1.11 1.36 1.23 1.50 45.17 44.62 45.17 50.33 48.03 45.32 2 833 1.52 1.60 1.66 2.01 80.18 78.75 80.18 81.01 83.89 79.96 2 835 1.10 1.28 1.30 1.64 40.99 40.72 40.99 43.85 46.88 42.90 2 837 1.26 1.43 1.47 1.80 57.44 57.18 57.44 60.96 65.39 60.24 2

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

12 -1.0 -3.4 -1.0 -3.4 -3.6 -3.6 -3.6 -8.9 -3.6 -3.5 10 94 -1.1 -3.4 -0.9 -3.4 -3.2 -3.0 -3.2 -8.3 -2.6 -2.8 10 95 -1.4 -4.1 -1.0 -4.1 -5.3 -4.4 -5.3 -8.6 -3.7 -4.1 10

105 -1.3 -4.5 -1.3 -4.5 -5.0 -5.0 -5.0 -9.9 -4.8 -5.2 2 110 -1.7 -6.2 -1.3 -6.2 -5.9 -5.6 -5.9 -14.1 -4.1 -5.1 10 111 -2.9 -7.3 -2.7 -7.3 -11.7 -11.3 -11.7 -16.2 -10.1 -11.4 10 121 -1.2 -5.6 -1.1 -5.6 -4.2 -4.1 -4.2 -13.2 -3.6 -4.1 10 125 -0.1 -4.1 0.3 -4.1 -0.3 0.2 -0.3 -9.0 0.9 0.4 2 128 -1.6 -7.0 -1.7 -7.0 -5.8 -5.6 -5.8 -15.9 -5.5 -5.8 2 129 -0.1 -4.3 0.3 -4.3 0.0 0.3 0.0 -11.7 1.0 0.3 14 131 -2.6 -7.6 -2.2 -7.6 -9.2 -8.7 -9.2 -17.6 -7.0 -8.8 10 132 0.1 -2.1 0.2 -2.1 0.2 0.2 0.2 -5.7 0.6 0.0 7 316 -0.5 -1.1 -0.6 -1.1 -2.6 -2.6 -2.6 -9.2 -2.7 -3.2 2 318 -1.0 -3.2 -1.1 -3.2 -3.5 -3.3 -3.5 -14.8 -3.6 -3.4 2 375 -0.7 -2.8 -0.5 -2.8 -2.9 -2.9 -2.9 -7.1 -1.5 -2.6 10 400 -1.3 -3.7 -1.1 -3.7 -5.2 -4.9 -5.2 -10.2 -4.0 -5.0 10 611 -3.4 -3.6 -3.0 -3.6 -11.6 -11.1 -11.6 -9.1 -9.6 -11.3 10 621 -1.5 -3.9 -1.5 -3.9 -6.6 -6.6 -6.6 -12.2 -6.3 -6.5 10 693 0.2 -2.5 0.6 -2.5 0.8 1.2 0.8 -13.2 2.1 1.4 2 694 -2.3 -3.0 -1.5 -3.0 -9.3 -6.1 -9.3 -19.1 -4.6 -5.6 10 746 -1.3 -3.2 -1.3 -3.2 -5.2 -5.1 -5.2 -7.7 -4.8 -5.2 10 802 -0.3 -1.7 -0.1 -1.7 -1.5 -0.9 -1.5 -9.5 0.2 -0.8 10 806 -1.5 -2.9 -1.4 -2.9 -5.9 -5.8 -5.9 -5.6 -3.9 -6.0 10 812 -0.3 -2.6 -0.1 -2.6 -4.5 -3.9 -4.5 -9.4 -3.1 -3.9 10 820 -4.3 -3.0 -4.1 -3.0 -13.9 -13.1 -13.9 -8.1 -12.2 -13.4 3 827 -2.3 -2.2 -1.8 -2.2 -7.4 -6.7 -7.4 -7.8 -5.2 -7.1 10 832 -0.9 -4.9 -0.9 -4.9 -3.4 -3.2 -3.4 -14.8 -3.1 -3.3 2 833 -1.5 -3.2 -1.5 -3.2 -5.6 -5.3 -5.6 -10.8 -4.9 -5.6 10 835 0.3 -2.6 0.6 -2.6 -0.7 -0.3 -0.7 -10.1 1.2 -0.2 7 837 -1.4 -3.8 -1.1 -3.8 -5.6 -5.3 -5.6 -9.6 -4.0 -5.4 10

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

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

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 dob2 [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 dob2 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 dob2 models, Raile 3 [15]

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.

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] 0.64 4.94

Clark [3] 0.59 2.63

Wensel [19] 0.83 9.34

Ormerod [5] 0.70 1.87

Raile 2 [11] 0.65 5.25

Raile 3 [13] 0.59 2.67

Raile 4 [15] 0.65 5.25

Clark [4] 0.60 3.56

Wensel [11] 0.60 -2.38

Neiloid [7] 0.60 3.72

Parabolic [8] 0.61 3.94

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

12 0.23 0.23 0.28 0.25 0.22 0.23 0.22 0.23 0.22 0.24 0.24 11 94 0.36 0.32 0.45 0.38 0.35 0.33 0.35 0.32 0.30 0.34 0.34 11 95 0.19 0.16 0.24 0.18 0.18 0.16 0.18 0.16 0.14 0.17 0.17 11

105 0.19 0.17 0.23 0.24 0.19 0.18 0.19 0.17 0.17 0.19 0.19 11 110 0.27 0.27 0.38 0.33 0.28 0.25 0.28 0.28 0.30 0.26 0.26 15 111 0.44 0.38 0.58 0.49 0.43 0.38 0.43 0.39 0.38 0.38 0.39 15 121 0.34 0.33 0.45 0.39 0.34 0.32 0.34 0.34 0.37 0.33 0.34 15 125 0.29 0.25 0.35 0.26 0.28 0.25 0.28 0.25 0.24 0.25 0.25 11 128 0.44 0.43 0.56 0.44 0.44 0.41 0.44 0.44 0.43 0.43 0.43 15 129 0.46 0.45 0.55 0.46 0.47 0.43 0.47 0.45 0.43 0.44 0.45 11 131 0.46 0.43 0.62 0.51 0.46 0.42 0.46 0.45 0.45 0.43 0.44 15 132 0.20 0.20 0.29 0.23 0.21 0.19 0.21 0.22 0.23 0.20 0.21 15 316 0.79 0.77 0.94 0.83 0.80 0.77 0.80 0.78 0.80 0.79 0.79 3 318 0.61 0.61 0.73 0.66 0.62 0.60 0.62 0.61 0.61 0.64 0.64 15 375 0.21 0.20 0.25 0.22 0.21 0.21 0.21 0.21 0.20 0.21 0.21 11 400 0.73 0.67 0.97 0.82 0.73 0.68 0.73 0.70 0.71 0.71 0.72 3 611 0.71 0.64 0.97 0.79 0.72 0.62 0.72 0.66 0.65 0.64 0.64 15 621 0.87 0.86 1.07 0.87 0.87 0.85 0.87 0.88 0.88 0.86 0.87 15 693 0.64 0.65 0.77 0.67 0.67 0.63 0.67 0.66 0.66 0.63 0.63 15 694 1.99 1.66 2.51 2.51 2.11 1.67 2.11 1.66 1.66 1.67 1.67 3 746 0.30 0.28 0.37 0.35 0.30 0.30 0.30 0.28 0.28 0.32 0.32 4 802 0.92 0.88 1.30 0.96 0.98 0.87 0.98 0.91 0.93 0.91 0.92 15 806 0.97 0.92 1.31 1.09 0.95 0.92 0.95 0.96 0.96 1.00 1.01 15 812 0.85 0.73 1.28 0.98 0.92 0.73 0.92 0.75 0.76 0.76 0.77 3 820 1.81 1.63 2.19 1.97 1.83 1.60 1.83 1.62 1.60 1.61 1.61 11 827 0.73 0.69 1.01 0.80 0.75 0.66 0.75 0.70 0.69 0.68 0.69 15 832 0.67 0.62 0.83 0.75 0.70 0.64 0.70 0.63 0.70 0.65 0.66 3 833 1.13 1.02 1.42 1.26 1.15 1.02 1.15 1.02 1.04 1.06 1.06 15 835 0.53 0.50 0.76 0.60 0.56 0.51 0.56 0.53 0.57 0.53 0.54 3 837 0.80 0.76 1.10 0.86 0.80 0.74 0.80 0.79 0.78 0.79 0.81 15

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

12 2.0 1.9 5.1 0.3 1.1 2.1 1.1 2.0 -1.2 3.2 3.3 5 94 3.9 1.9 6.5 2.2 3.5 2.5 3.5 1.7 0.3 2.8 2.8 11 95 4.4 3.5 6.8 1.8 3.7 2.5 3.7 3.6 0.6 3.2 3.2 11

105 4.4 0.7 6.9 4.7 4.5 2.9 4.5 0.7 0.2 4.0 4.0 11 110 3.6 1.6 6.7 3.6 3.9 2.5 3.9 3.5 -2.7 2.9 3.1 3 111 6.7 2.7 12.4 1.8 6.4 2.5 6.4 4.1 -3.9 3.1 3.3 5 121 3.2 1.9 7.0 0.3 3.3 1.6 3.3 2.4 -4.1 2.6 2.8 5 125 3.5 1.8 5.4 1.0 3.4 2.1 3.4 1.5 0.3 1.7 1.7 11 128 4.8 2.7 9.3 -1.3 5.0 3.3 5.0 4.1 -3.8 4.4 4.7 5 129 3.9 2.5 6.0 1.7 4.4 2.9 4.4 3.3 -0.7 3.3 3.4 11 131 4.8 3.1 9.0 0.4 4.9 2.8 4.9 4.7 -3.3 3.7 4.0 5 132 3.8 2.5 7.7 -1.0 4.3 3.5 4.3 4.5 -3.8 4.3 4.6 5 316 4.3 3.4 8.6 -0.6 4.7 2.8 4.7 4.6 -3.4 4.3 4.5 5 318 2.9 3.7 6.0 -0.2 3.0 2.3 3.0 3.7 -1.6 3.8 4.0 5 375 2.5 2.1 5.2 0.7 2.2 2.4 2.2 2.6 -0.7 2.7 2.8 5 400 4.9 3.1 9.7 1.2 5.1 2.8 5.1 4.8 -3.4 4.5 4.8 5 611 6.0 2.7 11.5 3.0 6.4 2.2 6.4 4.1 -3.0 3.3 3.6 15 621 3.7 2.6 7.9 -1.5 3.7 2.7 3.7 3.4 -3.8 3.4 3.7 5 693 4.8 1.4 8.6 3.4 6.0 3.7 6.0 4.0 -2.3 4.2 4.4 3 694 15.8 3.5 24.8 19.1 18.2 5.2 18.2 5.1 0.9 4.3 4.6 11 746 2.5 1.6 5.8 1.7 2.0 1.9 2.0 1.5 -1.1 3.5 3.6 11 802 6.1 3.2 12.2 -0.1 7.6 3.9 7.6 5.1 -3.6 5.8 6.1 5 806 5.3 4.3 11.6 0.8 4.8 3.5 4.8 6.2 -4.3 6.7 7.1 5 812 5.7 2.7 11.5 1.1 7.0 2.2 7.0 3.5 -3.5 3.7 4.0 5 820 10.3 3.5 18.2 7.1 10.8 1.6 10.8 3.2 -2.9 2.3 2.7 15 827 5.7 3.0 11.7 2.6 6.2 1.8 6.2 3.6 -3.6 3.5 3.9 15 832 4.1 2.2 7.9 -0.5 5.0 2.2 5.0 2.9 -3.4 3.3 3.5 5 833 5.3 2.5 9.9 0.6 5.8 1.5 5.8 2.5 -2.7 3.3 3.5 5 835 5.0 3.2 10.8 1.9 6.1 3.6 6.1 5.3 -3.8 5.2 5.6 5 837 4.7 3.6 9.7 0.4 4.8 2.7 4.8 4.7 -3.1 4.7 5.0 5

29

The models were further evaluated through residuals plots that were made for all species-

model 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 ob2 [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.

30

Figure 6 Residuals plot for the Clark ob2 model ob volume predictions for shortleaf pine.

Volume inside bark

When averaged across all 30 species the Wensel dib [10], Wensel dib2 [12], and the Raile

3 [16] performed similarly in both RMSE and bias (Table 13). The Wensel dib2 [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 dib2 [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 dib2 [4], and

Wensel dib2 [12] had the smallest bias for 3 species each. As previous stated, the Clark dib2 [4]

model had the smallest bias for a few species, but in most cases had much larger bias as

compared to the other models.

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] 0.64 4.70

Clark [3] 0.55 -2.86

Wensel [10] 0.53 -0.39

Ormerod [6] 0.64 2.21

Raile 2 [14] 0.54 1.68

Raile 3 [16] 0.53 1.27

Raile 4 [18] 0.55 -0.79

Clark [4] 0.55 -2.87

Wensel [12] 0.52 -0.84

Neiloid [8] 0.55 2.45

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

12 0.17 0.14 0.14 0.16 0.16 0.15 0.16 0.14 0.14 0.16 12 94 0.39 0.30 0.29 0.37 0.33 0.31 0.38 0.30 0.29 0.31 12 95 0.19 0.12 0.14 0.18 0.16 0.13 0.18 0.12 0.13 0.13 4

105 0.14 0.13 0.13 0.14 0.14 0.13 0.13 0.13 0.13 0.13 18 110 0.26 0.38 0.25 0.30 0.25 0.25 0.31 0.38 0.25 0.26 10 111 0.40 0.37 0.36 0.40 0.36 0.36 0.36 0.38 0.35 0.37 12 121 0.32 0.41 0.31 0.35 0.31 0.31 0.38 0.41 0.31 0.32 10 125 0.24 0.30 0.23 0.24 0.26 0.25 0.24 0.29 0.24 0.24 10 128 0.36 0.44 0.35 0.38 0.36 0.36 0.45 0.44 0.35 0.38 10 129 0.39 0.51 0.40 0.38 0.41 0.41 0.44 0.48 0.40 0.41 6 131 0.40 0.46 0.38 0.42 0.38 0.38 0.44 0.46 0.38 0.39 12 132 0.21 0.21 0.19 0.21 0.19 0.19 0.19 0.20 0.18 0.20 12 316 0.86 0.72 0.73 0.85 0.73 0.73 0.72 0.72 0.72 0.74 18 318 0.56 0.46 0.48 0.56 0.49 0.48 0.48 0.47 0.48 0.49 3 375 0.14 0.15 0.14 0.15 0.15 0.14 0.15 0.15 0.14 0.14 10 400 0.73 0.62 0.62 0.74 0.63 0.63 0.61 0.62 0.62 0.65 18 611 0.81 0.58 0.59 0.77 0.59 0.59 0.60 0.58 0.58 0.60 8 621 0.81 0.78 0.75 0.80 0.76 0.76 0.75 0.78 0.75 0.78 10 693 0.64 0.72 0.64 0.64 0.63 0.63 0.65 0.68 0.63 0.63 16 694 1.93 1.42 1.42 1.86 1.43 1.42 1.56 1.41 1.42 1.43 4 746 0.28 0.26 0.26 0.28 0.27 0.27 0.26 0.26 0.26 0.27 3 802 1.01 0.78 0.75 0.97 0.77 0.76 0.75 0.77 0.75 0.80 18 806 1.08 0.81 0.83 1.06 0.84 0.84 0.78 0.81 0.81 0.90 18 812 1.05 0.73 0.77 1.05 0.77 0.75 0.71 0.73 0.74 0.78 18 820 1.94 1.47 1.47 1.85 1.47 1.46 1.52 1.47 1.46 1.47 12 827 0.84 0.62 0.61 0.80 0.62 0.61 0.61 0.62 0.60 0.63 12 832 0.58 0.64 0.57 0.62 0.57 0.57 0.64 0.64 0.57 0.58 10 833 1.15 0.93 0.94 1.14 0.95 0.95 0.95 0.93 0.94 0.98 3 835 0.56 0.49 0.47 0.56 0.48 0.48 0.47 0.48 0.47 0.50 12 837 0.83 0.67 0.68 0.83 0.68 0.68 0.64 0.67 0.67 0.71 18

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

12 2.8 -1.2 0.5 1.7 2.2 1.8 2.7 -1.8 -0.7 2.7 10 94 5.1 -0.2 0.6 3.8 3.3 2.1 5.0 -0.2 0.1 2.2 12 95 4.8 -0.2 0.8 3.3 2.8 0.7 4.2 -0.9 -0.5 0.0 7

105 1.5 -2.5 0.2 0.1 1.7 0.6 0.3 -2.7 -1.2 0.2 6 110 -1.5 -7.9 -1.1 -3.9 0.4 0.4 -6.0 -7.9 -1.8 1.1 16 111 2.6 -7.2 -0.9 -1.7 -0.3 -0.5 -9.1 -8.0 -3.5 0.7 14 121 -0.9 -8.1 -1.7 -4.0 0.9 0.9 -7.5 -8.3 -2.1 2.3 16 125 0.7 -3.0 0.6 -0.4 2.9 2.0 0.9 -2.9 0.9 1.8 6 128 -0.6 -9.1 -1.0 -4.1 1.6 1.5 -9.9 -9.2 -1.8 3.4 2 129 1.0 -3.5 0.0 -0.4 3.1 2.6 0.1 -2.8 0.7 2.9 10 131 -0.1 -8.1 -1.4 -3.7 0.1 -0.1 -8.1 -8.4 -2.9 1.1 16 132 2.9 -3.0 -1.4 0.2 2.5 2.5 -3.0 -2.2 -0.6 4.0 6 316 7.7 0.2 -1.0 5.1 2.0 1.8 0.7 0.8 -1.2 3.4 3 318 3.9 -1.1 -1.0 2.0 1.1 0.5 1.0 -2.0 -2.4 1.9 16 375 1.6 -2.0 0.0 0.6 1.7 0.9 1.6 -1.7 -0.1 0.8 10 400 4.5 -3.1 -0.6 1.7 1.3 1.2 -2.7 -3.1 -1.2 3.3 10 611 8.4 -0.8 -0.3 5.6 0.1 0.0 0.7 -0.9 -1.5 1.4 16 621 3.0 -3.3 -1.4 0.5 1.2 1.1 -3.8 -3.4 -2.2 2.5 6 693 2.3 -4.5 -0.9 0.0 2.7 2.5 -1.4 -2.9 0.6 3.2 6 694 20.5 0.8 2.4 17.2 4.3 3.4 11.4 1.8 2.8 3.6 3 746 2.3 -1.8 0.5 0.8 2.2 1.5 1.0 -2.0 -0.5 1.4 12 802 9.3 -0.8 -0.5 6.6 2.7 2.6 2.1 0.0 0.8 4.8 4 806 8.5 -0.5 -0.6 5.2 2.4 2.4 -0.4 0.0 0.2 5.9 4 812 6.5 -2.5 -0.3 3.7 0.0 -0.2 -1.7 -3.6 -2.0 1.9 14 820 16.4 1.0 0.4 12.5 1.0 0.0 4.3 0.0 -1.5 1.3 4 827 9.3 -0.2 -0.4 6.4 0.9 0.9 2.0 -0.3 -0.6 3.0 3 832 1.0 -6.2 -1.1 -1.6 1.0 0.7 -5.9 -6.4 -1.6 2.1 16 833 6.9 -1.5 -0.6 4.5 1.2 1.0 0.4 -2.0 -1.0 2.7 18 835 5.5 -2.7 -0.7 2.4 2.3 2.3 -1.5 -2.3 0.3 4.6 12 837 5.0 -2.6 -0.6 2.4 1.2 1.2 -1.5 -2.6 -1.0 3.5 10

34

Further insights for ib volume predictions were provided by residual plots created for all

species-model combinations. Wensel dib2 [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 dib2 [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 dib2

[12] does not exhibit curvature for smaller trees (Figure 7).

Figure 7 Residuals plot for the Wensel dib2 model [12] ib volume predictions for eastern white pine.

35

Table 16 Regression coefficients for the Raile models [1] and [2].

dob [1] dib [2]

Common name β β1 β2

balsam fir 0.11029 0.943875 0.101997

white spruce 0.112857 0.956736 0.106345

black spruce 0.108573 0.949428 0.099936

jack pine 0.089224 0.931992 0.064234

shortleaf pine 0.091752 0.893057 0.076584

slash pine 0.121696 0.872957 0.103595

longleaf pine 0.089969 0.88806 0.079845

red pine 0.090883 0.927579 0.072773

pond pine 0.094938 0.868636 0.083414

eastern white pine 0.09441 0.92199 0.076527

loblolly pine 0.10156 0.879249 0.087655

Virginia pine 0.093352 0.922137 0.083559

red maple 0.105426 0.956208 0.096301

sugar maple 0.105315 0.946492 0.088844

paper birch 0.113373 0.939862 0.09967

hickory spp. 0.121059 0.914233 0.107215

sweetgum 0.126276 0.960186 0.108816

yellow-poplar 0.099085 0.915248 0.089062

blackgum 0.114948 0.920452 0.093084

swamp tupelo 0.211616 1.024012 0.159324

quaking aspen 0.093452 0.927142 0.079218

white oak 0.146188 0.948049 0.127065

scarlet oak 0.145329 0.924244 0.132532

southern red oak 0.136522 0.922957 0.119591

laurel oak 0.150009 0.990593 0.126636

water oak 0.14333 0.960012 0.126367

chestnut oak 0.093294 0.900212 0.081587

northern red oak 0.127573 0.943491 0.10968

post oak 0.137864 0.912175 0.122915

black oak 0.134127 0.916411 0.11851

36

Table 17 Regression coefficients for the Raile 3 models [15] and [16].

dob dib

Common name β1 β2 β0 β1 β2

balsam fir 0.1800 0.0478 0.8978 0.2120 -0.0076

black oak 0.2997 -0.0221 0.9143 0.2955 -0.0353

black spruce 0.3122 -0.0288 0.9000 0.3657 -0.0723

blackgum 0.2486 -0.0217 0.8818 0.1906 -0.0220

chestnut oak 0.2276 -0.0066 0.8011 0.1680 -0.0024

eastern white pine 0.3628 -0.0148 0.7577 0.2675 -0.0067

hickory spp. 0.2282 -0.0058 0.7924 0.1604 0.0017

jack pine 0.2695 -0.0373 0.8490 0.2215 -0.0347

laurel oak 0.2363 -0.0035 0.7501 0.1863 -0.0020

loblolly pine 0.2424 -0.0125 0.8371 0.1935 -0.0135

longleaf pine 0.2666 -0.0073 0.7752 0.2107 -0.0057

northern red oak 0.2031 0.0020 0.8508 0.1564 0.0052

paper birch 0.2611 -0.0038 0.9080 0.2313 -0.0030

pond pine 0.2399 0.0000 0.8793 0.2728 -0.0191

post oak 0.2245 0.0188 0.8768 0.2817 -0.0442

quaking aspen 0.3118 -0.0049 0.8258 0.2574 -0.0023

red maple 0.3861 -0.0211 0.8883 0.3451 -0.0191

red pine 0.2289 0.0016 0.8447 0.1870 0.0022

scarlet oak 0.2929 -0.0131 0.8111 0.2475 -0.0143

shortleaf pine 0.9096 -0.0922 0.8815 0.7469 -0.0747

slash pine 0.1968 0.0096 0.8611 0.2183 -0.0154

southern red oak 0.3980 -0.0143 0.8550 0.3543 -0.0142

sugar maple 0.3412 0.0111 0.8490 0.2769 0.0149

swamp tupelo 0.4125 -0.0240 0.8295 0.3508 -0.0178

sweetgum 0.5721 -0.0458 0.9086 0.4953 -0.0374

Virginia pine 0.4307 -0.0204 0.8813 0.3770 -0.0162

water oak 0.2581 -0.0142 0.8017 0.2097 -0.0106

white oak 0.4015 -0.0267 0.8566 0.3305 -0.0180

white spruce 0.3431 -0.0052 0.8147 0.2825 -0.0011

yellow-poplar 0.3409 -0.0030 0.8334 0.2743 0.0014

37

Table 18 Regression coefficients for the dib Wensel and Olson model [11]

Wensel and Olson

Common name b0 b1

balsam fir 0.0564 0.0424

white spruce 0.0473 0.0512

black spruce 0.0549 0.0506

jack pine 0.0717 0.0383

shortleaf pine 0.1436 0.0628

slash pine 0.1944 0.0962

longleaf pine 0.1530 0.0686

red pine 0.0796 0.0404

pond pine 0.1826 0.0784

eastern white pine 0.0947 0.0508

loblolly pine 0.1683 0.0774

Virginia pine 0.1230 0.0705

red maple 0.0988 0.0791

sugar maple 0.0833 0.0631

paper birch 0.0688 0.0482

hickory spp. 0.1466 0.0885

sweetgum 0.1132 0.0905

yellow-poplar 0.1290 0.0735

blackgum 0.1329 0.0765

swamp tupelo 0.1132 0.1347

quaking aspen 0.0863 0.0493

white oak 0.1324 0.1035

scarlet oak 0.1492 0.1061

southern red oak 0.1527 0.0999

laurel oak 0.1146 0.1118

water oak 0.1193 0.1005

chestnut oak 0.1509 0.0736

northern red oak 0.1232 0.0902

post oak 0.1588 0.1002

black oak 0.1447 0.0929

38

Table 19 Regression Coefficients for both the Wensel models dob2 [11] and dib2 [12].

dob dib

Common name β1 β0 β1

balsam fir 0.0525 0.0551 0.0462

black oak 0.0524 0.0459 0.0523

black spruce 0.0526 0.0517 0.0537

blackgum 0.0463 0.0644 0.0393

chestnut oak 0.0515 0.1352 0.0600

eastern white pine 0.0805 0.2037 0.1063

hickory spp. 0.0565 0.1564 0.0713

jack pine 0.0419 0.0805 0.0396

laurel oak 0.0633 0.1937 0.0857

loblolly pine 0.0494 0.0960 0.0490

longleaf pine 0.0637 0.1699 0.0815

northern red oak 0.0569 0.1298 0.0720

paper birch 0.0654 0.1112 0.0852

pond pine 0.0538 0.0865 0.0690

post oak 0.0521 0.0699 0.0492

quaking aspen 0.0734 0.1641 0.0988

red maple 0.0771 0.1223 0.0975

red pine 0.0620 0.1349 0.0783

scarlet oak 0.0616 0.1360 0.0741

shortleaf pine 0.1242 0.1183 0.1359

slash pine 0.0476 0.0848 0.0519

southern red oak 0.0848 0.1570 0.1123

sugar maple 0.0894 0.2016 0.1306

swamp tupelo 0.0822 0.1639 0.1093

sweetgum 0.0998 0.1270 0.1210

Virginia pine 0.0856 0.1404 0.1111

water oak 0.0568 0.1481 0.0734

white oak 0.0757 0.1332 0.0962

white spruce 0.0789 0.1825 0.1099

yellow-poplar 0.0764 0.1618 0.1028

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).

40

Although there is no species overlap and the study areas are different, comparison to

Wensel and Olson (1995) provided further information on the quality of work done here.

Regression coefficients reported by Wensel and Olson (1995) are similar to pine (Pinus spp.), fir

(Abies spp.), and spruce (Picea spp.) coefficients reported here (Table 18,Table 21). The sample

size used by both authors is similar, although the sample size used here is slightly larger. The

upper limit for the diameter range used by Wensel and Olson is much larger than the work done

here (Table 1, Table 22). Although that is to be expected, because their work included much

larger tree species from the western United States.

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.

41

Here two separate equations were used to predict dib and dob values, but others such as

Nunes et al. (2010) utilized a system of equations which allows dib, merchantable volume, and

dob, total tree volume, values to be predicted using a single equation. The system combined a dib

equation with a bark thickness equation. This method allows the use of a single equation to

model either dib or dob values, diameter or volume, dependent on what information is desired.

Nunes et al. (2010) were able to achieve accurate predictions. This is an example of what could

be another option to perform the type of work done here.

Based on the work here, for accurate stump volume predictions, ob or ib, it is suggested

the use of either the Raile 3 model [15, 16] or Wensel models [11, 12]. These models resulted in

similar fit statistics and the lowest biases of all the models. If the purpose of work is specific to a

particular species, it is suggested to identify the best model for that species in the tables with

species-specific statistics. Although, the suggested models performed well in almost all cases

with minor tradeoffs in accuracy and bias. Regression coefficients for the Raile 3 model dob [15]

and dib [16] are available in Table 17. Regression coefficients for the Wensel models [11, 12]

are in Table 19.

42

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