Developing Inventory 

Projection Models 

Using Empirical Net 

Forest Growth and 

Growing-Stock Density 

Relationships Across 

U.S. Regions and 

Species Groups

Prakash Nepal

Peter J. Ince

Kenneth E. Skog

Sun J. Chang

United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Paper

FPL–RP–668
July 2012

Nepal, Prakash; Ince, Peter J.; Skog, Kenneth E.; Chang, Sun J. 2012. 

Developing inventory projection models using empirical net forest growth 

and growing-stock density relationships across U.S. regions and species 

groups. Research Paper FPL-RP-668. Madison, WI: U.S. Department of 

Agriculture, Forest Service, Forest Products Laboratory. 20 p.

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Abstract

This paper describes a set of empirical net forest growth 

models based on forest growing-stock density relationships 

for three U.S. regions (North, South, and West) and two 

species groups (softwoods and hardwoods) at the regional 

aggregate level. The growth models accurately predict 

historical U.S. timber inventory trends when we incorporate 

historical timber harvests. The models also project future 

timber inventory trends when linked to a model of regional 

timber harvest, forest product markets, and trade, specifically the U.S. Forest Products Module (USFPM) within 

the Global Forest Products Model (GFPM). The market 

model also takes into account the timber supply and market 

impacts of projected trends in U.S. timber inventory, and results show the sensitivity of U.S. regional timber inventory 

projections to alternative timber market scenarios. Given 

the parsimonious nature of the model and its simplicity, the 

developed net forest growth models can be very useful in 

providing projections of growing-stock inventory trends 

across U.S. regions and species groups for alternative 

U.S. and global timber market scenarios.

Keywords: Forest Inventory and Analysis (FIA); Forest 

and Rangeland Resource Planning Act (RPA); growingstock density; growing-stock inventory projection; net 

forest growth; nonlinear regression; prediction error

Acknowledgments

We acknowledge the support of the USDA Forest Service 

for funding this project, and especially Dr. Linda Heath of 

the Forest Service who helped to plan and arrange support 

for the project. We also acknowledge the guidance provided 

by Dr. Joseph Buongiorno of UW-Madison in helping 

design this study. The authors thank three reviewers whose 

comments helped improve this paper.

Contents

Introduction..........................................................................1

Theoretical Framework........................................................3

Methods................................................................................3

 Data ..................................................................................3

 Model...............................................................................6

 Model Validation and Application ...................................7

Results and Discussion ......................................................10

 Net Forest Growth and Growing-Stock Density 

 Relationship ...................................................................10

 Model Comparison and Application ..............................12

Conclusions........................................................................16

Literature Cited ..................................................................19

 
Developing Inventory Projection Models 

Using Empirical Net Forest Growth and 

Growing-Stock Density Relationships 

Across U.S. Regions and Species Groups

Prakash Nepal, Post-Doctoral Research Associate

Louisiana State University Agricultural Center

Baton Rouge, Louisiana

Peter J. Ince, Research Forester

Kenneth E. Skog, Supervisory Research Forester

Forest Products Laboratory, Madison, Wisconsin

Sun J. Chang, Professor

Louisiana State University Agricultural Center

Baton Rouge, Louisiana

Introduction

Forest sector models have been widely used in projecting 

forest product market trends, timber harvest levels, and 

overall forest resource conditions and trends. Such models 

differ in their geographic scope (e.g., regional, national, 

global), economic scope (e.g., partial market equilibrium, 

general equilibrium), and temporal scope (e.g., static, dynamic). One important aspect of such models is the link 

between timber supply and forest products demand (Turner 

and others 2006). Among variables that govern long-run 

shifts of timber supply, available timber growing-stock inventory is often considered to be the most important based 

on the logic that supply of timber is proportional to physical 

volume of growing stock available in the forest, which is 

influenced over time by forest growth and harvest levels. 

Therefore, a reliable projection of growing-stock inventory is a critical aspect of all kinds of forest sector models. 

Growing-stock inventory is projected by adding growth and 

subtracting harvest quantity to the initial growing stock. 

Harvest quantity is an element related to the timber market 

and is generally determined by modeling equilibrium timber 

market demand and supply, whereas forest growth rate is a 

biological concept and generally modeled as some function 

of biological or silvicultural measures related to forest resources (e.g., growing-stock density).

The USDA Forest Service, as mandated by the Forest and 

Rangeland Renewable Resources Planning Act of 1974 

(RPA), is responsible for assessing nationwide timber demand, supply, and conditions of forest resources. For several 

decades, forest sector models have played a key role in producing long-range RPA projections of U.S. forest resources, 

conditions, and trends (Adams and Haynes 2007). The 

USDA Forest Service has used a combination of forest 

sector models, including the Timber Assessment Market 

Model (TAMM) (Adams and Haynes 1996); North American Pulp and Paper Model (NAPAP) (Zhang and others 

1996; Ince 1994); and Aggregate Timberland Assessment 

System (ATLAS) (Mills and Kincaid 1992). Recently, the 

USDA Forest Service designed a new forest sector modeling 

framework for the 2010 RPA assessment. This new modeling system consists of a group of submodels, collectively 

known as the U.S. Forest Assessment System (USFAS). As 

a part of this system of models, the U.S. Forest Products 

Module (USFPM) (Ince and others 2011) was developed at 

the USDA Forest Products Laboratory (FPL). USFPM is a 

dynamic partial market equilibrium model of the U.S. forest 

product sector that operates within the Global Forest Products Model (GFPM) (Buongiorno and others 2003). The 

USFPM/GFPM produces long-range projections of U.S. 

forest products markets, forest product trade, and regional 

U.S. timber markets (Ince and others 2011; Buongiorno 

and others 2003).

Some of these forest sector models project growing-stock 

inventory based on very complex growth models that require more effort to gather and analyze input data (e.g., the 

ATLAS model, or the forest dynamics module of USFAS), 

whereas others have only limited capability to project U.S. 

forest inventory (e.g., USFPM, GFPM). Growth in ATLAS 

is modeled as a function of relative density change as defined by the ratio of the existing inventory volume per acre 

and the corresponding base-yield table volume per acre 

(Mills and Kincaid 1992). The requirement of the base-yield 

tables and specification of relative density change parameters makes it complex to use because such parameters and 

base-yield tables are not available for every species in the 

United States. For RPA purposes, ATLAS has now been 

replaced by a more complex set of probabilistic forest 
Research Paper FPL–RP–668

2

transition models developed at the USDA Southern Research Station to simulate the dynamics of U.S. forest 

growth and changes in forest inventory for tens of thousands of forest sample plots within each region. Such plot 

transition models have been developed for the U.S. North 

and South regions, and the southern regional plot transition 

model was used also in the recently completed Southern 

Forest Futures Project (Wear and others 2011). Such a plotlevel forest transition model runs stochastic simulations of 

harvest and management responses for all forest plots, but 

one complex aspect of such a stochastic simulation model 

is that it may typically produce different results each time 

the model is run, requiring many model runs to derive a 

representative sample of results (Wear and others 2011). 

Another example of this approach to modeling forest growth 

presented by Abt and others (2000) used Forest Inventory 

and Analysis (FIA) plot-level data to estimate forest growth 

per acre by species groups, physiographic regions, and management types in the U.S. South. Growth was modeled as a 

function of state, owner, stand age, and interaction between 

owner and stand age. One problem with using such a model 

is that plot age data are not always available. 

At the opposite end of the complexity spectrum, GFPM 

estimates change in forest growth using a single nonlinear functional relationship between net forest growth and 

growing-stock density applied to all countries (Turner and 

others 2006). This functional relationship was derived from 

cross-sectional analysis of data on aggregate forest growth 

and forest growing-stock density of 180 countries, and the 

GFPM used the same elasticity of growth rate with respect 

to growing-stock density across all countries (as estimated 

in Turner and others 2006). Note that although the GFPM 

employs the same elasticity of growth rate with growingstock density across all countries, the base year growth 

rate and growing-stock density in a given country may be 

different and thus may change differently over time as determined by the elasticity of growth rate with growing-stock 

density. For example, a typical projection of the growth rate 

of growing-stock inventory in Russia remains quite different from that in Germany because of different initial growth 

rates and growing-stock densities in those countries.

Nevertheless, this global approach is more generalized and 

may not precisely account for U.S. regional or species group 

differences in the relationship between forest growth and 

stocking density (if the same elasticity relationship that is 

applied uniformly to all countries and all timber species is 

also applied to U.S. subregions and species groups). Such 

an approach to modeling forest growth is of course less 

complex, based on global data that are readily available and 

may be appropriate for analysis of global forest trends in 

aggregate. However, in our earlier attempts to predict U.S. 

regional growth and timber inventory, we found that the 

global elasticity relationship employed in the GFPM did 

not accurately represent the dynamics of observed forest 

growth and growing-stock density relationships among 

species groups and regions in the United States. In this 

sense, the current versions of the GFPM and USFPM have 

limited capability to project U.S. regional growing-stock inventory with acceptable accuracy (and for RPA applications, 

the much more complex stochastic plot transition model of 

the USFAS was employed to produce U.S. forest inventory 

projections, but the model simulations were only indirectly 

related to the USFPM/GFPM timber market and timber harvest projections).

Therefore, we needed tools to precisely project U.S. timber 

inventory trends or changes in forest carbon stocks in direct 

relation to USFPM/GFPM timber market scenarios and had 

a compelling need to develop basic forest growth models 

that are simple to apply, yet credible enough to reliably 

project growing-stock inventory across U.S. regions and 

species groups in relation to U.S. and global timber market 

scenarios. Accordingly, this study had two purposes. The 

first purpose was to develop empirical net forest growth and 

forest growing-stock density relationships for three U.S. 

regions (North, South, and West) and two species groups 

(softwoods and hardwoods) that would allow USFPM to 

accurately predict U.S. timber inventory trends when including projected aggregate timber harvest, forest product 

markets, and trade at the U.S. regional and national level. 

Note that the USFPM/GFPM forest products market model 

projects production, consumption, and trade of timber and 

forest products markets at the national level in all other 

countries (but production and timber supply at a regional aggregate level for the U.S. only). Therefore, one of the goals 

of this study was to develop regional timber growth models 

that can make accurate projections of timber inventory by 

U.S. regions and species groups so that these projections can 

be consistently applied in conjunction with a broader-scale 

market projection model such as USFPM/GFPM.

While detailed stand-level growth models can be developed 

as functions of several variables related to forest ecology 

such as relative density (e.g., stand density index), stand 

age, and site quality and they might provide more precise 

projections of timber inventory at the stand level, such 

detailed models are generally not necessary in the context 

of more generalized national or global level timber market 

models. This is because market parameters such as timber 

harvest, stumpage price, lumber demand, and supply are 

generally not analyzed at a stand level but only at regional 

and/or national aggregate levels.

Our secondary purpose in this study was to develop longterm inventory projections that would be suitable for broadly modeling forest carbon and timber market implications of 

different carbon offset and timber market scenarios at U.S. 

regional and national levels (an objective we plan to pursue 

in subsequent research). This paper focuses primarily on 

methods used to estimate our regional timber growth and 

inventory models in relation to growing-stock density and 

also the linkage of U.S. timber growth and inventory models 

to USFPM/GFPM.
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

3

Theoretical Framework

Modeling aggregate timber growth as a function of growing-stock density is an approach that was first applied decades ago. This approach to modeling timber growth was 

based on a notion that the rate of timber growth is reduced 

as the level of growing stock increases (Binkley and Dykstra 1987; Mills and Kincaid 1992; Turner and others 2006). 

For example, in the context of global forest products market 

modeling, Binkley and Dykstra (1987) modeled aggregate 

timber growth as a function of growing-stock density for 

coniferous and nonconiferous species in eastern and western 

United States, Canada, and selected countries in Asia, Africa, Australia, Europe, and Latin America. Similarly, Mills 

and Kincaid (1992) applied this approach to model timber 

inventory in the United States, whereas the same approach 

was applied by Turner and others (2006) to model global 

timber growth.

In addition, literature related to silviculture and forest ecology has indicated also that individual tree growth (per unit 

volume) and overall forest stand growth (per unit area) can 

both be regarded as functions of stand density, although the 

two relationships differ (Long and others 2004; Jack and 

Long 1996; Smith 1986; Long 1985). These studies present conceptual relationships that describe a theory of tree 

growth, stand growth, and the level of growing stock. The 

theory includes differences in growth response between individual trees and a forest stand. When relative stand density 

is low, individual tree growth rate is maximized because 

individual tree leaf area exposed to direct sunlight is at its 

maximum, but stand growth is below its potential because 

stand leaf area is below its upper limit (Long and others 

2004; Jack and Long 1996). At high relative densities, the 

opposite is true; stand growth approaches its potential 

because of stand level leaf area reaching its upper limit, but 

individual tree growth decreases because of crowding and 

reduced leaf area (Long and others 2004; Jack and Long 

1996). These relationships can be further clarified in that 

gross stand growth (growth including mortality per unit of 

land area) generally continues to increase over the entire 

range of relative stocking densities, but net stand growth 

(growth net of mortality) decreases with further increase in 

relative density because of increasing tree mortality. Consequently, net growth rate of forest inventory (net annual 

growth as a percentage of live growing-stock volume) 

generally declines with increasing stocking density. 

Figure 1 represents implicit relationships between growth in 

growing-stock volume (m3/ha/y) and growing-stock density 

(m3/ha) and between net annual growth rate as a percentage 

of inventory (growth rate/y) and growing-stock density 

(m3/ha).

In this paper, we use regional forest growth data that are 

based on the U.S. timber inventory concept of “net annual 

growth” as defined in Smith and others (2009), specifically 

“The average annual net increase in volume of trees during 

the period between inventories. Components include the 

increment in net volume of trees at the beginning of the specific year surviving to its end, plus the net volume of trees 

reaching the minimum size class during the year, minus the 

volume of trees that died during the year and minus the net 

volume of trees that became cull trees during the year.” This 

measure of growth is thus net of tree mortality, and is therefore akin to the concept of net stand growth (growth net of 

mortality, but not net of harvest).

The U.S. regional net annual growth rate models developed 

in this study were based on the theoretical relationship 

between net stand growth rate (measured by timber “net 

annual growth” as a percentage of inventory volume) and 

growing-stock density. This theoretical framework was 

chosen because the county-level forest net annual growth 

and growing-stock density data used in this study showed a 

similar empirical relationship as one would expect between 

the net annual growth rate and growing-stock density (i.e., 

a declining relationship, similar to the implicit relationship 

shown in Fig. 1). Specifically, for a given species group and 

region, the counties with higher growing-stock density had 

generally lower growth rates per unit of inventory and vice 

versa. In addition, based on the empirical data (Table 1), we 

found unique growth/density relationships for softwoods 

and hardwoods and for each region. For example, softwoods 

in the U.S. South are observed to have the most productive 

growth rate (annual growth rate as percentage of inventory) 

among all regions and species groups (Table 1).

Methods

Data

This study used cross-sectional data of net annual forest 

growth and growing-stock density by county obtained from 

the FIA database of the USDA Forest Service (Miles 2011). 

County level cross-sectional data were used because crosssectional data aggregated at U.S. state level (50 observations) or time series data aggregated at U.S. regional level 

(9 observations, 1953, 1963, 1970, 1977, 1987, 1992, 1997, 

0.000

0.005

0.010

0.015

0.020

0.025

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 10 20 30 40

Gro

wth rate (%/y)

Gro

wth (m3

/ha/y)

Growing stock density (m3/ha)

Growth (m3/ha/y)

Growth rate (%/y)

Figure 1—Implicit relationships between growth in growing 

stock volume (m3/ha/y) and growing stock density (m3/ha) 

(left axis) and between net annual growth rate as percentage of inventory (growth rate/y) and growing stock density 

(m3/ha) (right axis).
Research Paper FPL–RP–668

4

2002, 2007) did not show expected relationships between 

net annual forest growth and growing-stock density, probably due to relatively small sample size. The growth equations derived from cross-sectional analysis of county-level 

data represent regional aggregate level relationships, based 

on an aggregate region-wide data (the counties within each 

region). Parenthetically, the aggregate growth and growingstock density relationship estimated at the regional level 

do not necessarily apply to any individual county, but will 

reflect the range of growth and growing-stock density data 

observed at the county level.

The EVALIDator tool (USDA 2011) was used to retrieve net 

annual growth and growing-stock density data according to 

two species groups (softwoods and hardwoods) in various 

counties in three U.S. regions (Fig. 2). Figure 3 shows the 

empirical relationships between net growth in growing stock 

(%/y) and growing-stock density (m3/ha) by U.S. region and 

species group. The cross-sectional county-level net annual 

growth data were not available for five states in the Western 

region (Alaska, California, Hawaii, Oregon, and Washington). Thus, the empirical relationship analyzed in this study 

for the U.S. West does not include those five states, but we 

made adjustments to our model for the West so that it would 

accurately predict growth for the entire region (including the 

Pacific Coast states). For the projections in this study, we 

calibrated the estimated growth parameters for the West so 

that the model would project timber growth and inventory 

for the entire West quite similar to the region-wide net 

annual growth as reported for 2006–2007 in the Forest Resources report from the Forest Service (Smith and others 

2009). This data calibration was a final step in our model 

estimation procedure and was necessary because our origiTable 1—Descriptive statistics of the variables used in the model 

Variable N Mean SD Minimum Maximum 

North, softwoods 

Net forest growth 

(as annual percentage of inventory), Yit

660 3.97 4.92 –9.41 28.43 

Growing-stock density, Xit (m3

/ha/y) 660 17.31 18.98 0.04 92.73 

North, hardwoods 

Net forest growth 

(as annual percentage of inventory), Yit

988 3.01 2.43 –5.89 27.11 

Growing-stock density, Xit (m3

/ha/y) 988 98.31 43.03 0.78 305.32 

South, softwoods 

Net forest growth 

(as annual percentage of inventory), Yit

990 5.91 3.97 –5.72 24.72 

Growing-stock density, Xit (m3/ha/y) 990 38.93 26.07 0.05 213.93 

Proportion of artificially regenerated area 990 0.18 0.17 0.00 0.74 

South, hardwoods 

Net forest growth 

(as annual percentage of inventory), Yit

1,013 3.10 1.76 –5.13 15.57 

Growing-stock density, Xit (m3

/ha/y) 1,013 70.88 41.02 0.10 215.20 

West, softwoodsa 

Net forest growth 

(as annual percentage of inventory), Yit

273 2.50 4.94 –9.95 27.10 

Growing-stock density, Xit (m3/ha/y) 273 88.34 60.10 0.074 298.37 

West, hardwoodsa 

Net forest growth 

(as annual percentage of inventory), Yit

369 3.76 5.78 –9.55 28.82 

Growing-stock density, Xit (m3

/ha/y) 369 36.86 51.60 0.02 373.60 

a

The descriptive statistics do not include five Pacific Coast states (Alaska, California, Hawaii, Oregon, 

and Washington) because county-level growth data were not available for those five states. 

Figure 2—The three U.S. subregions of the USFPM, North, 

South, and West, for which the study developed net forest 

growth and growing stock density models according to 

softwoods and hardwoods species group. County data were 

used to develop original estimates of growth parameters 

for each region (Table 4), although county data for the West 

excluded the five Pacific Coast states (Alaska, California, 

Hawaii, Oregon, and Washington), so the growth model 

for the West was adjusted so that predicted growth for the 

West matched actual region-wide growth in 2006–2007.
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

5

–10

–5

0

5

10

15

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25

0 50 100 150 200 250

Net gro

wth of gro

wing stock (%/y)

Observed

Predicted

–10

–5

0

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0 50 100 150 200 250

Net gro

wth of gro

wing stock (%/y)

Observed

Predicted

–10

–5

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0 50 100 150 200 250

Net gro

wth of gro

wing stock (%/y)

Observed

Predicted

–10

–5

0

5

10

15

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25

0 50 100 150 200 250

Net gro

wth of gro

wing stock (%/y)

Observed

Predicted

0

5

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15

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25

0 50 100 150 200 250

Net gro

wth of gro

wing stock (%/y)

Growing stock density (m3/ha)

Observed

Predicted

0

5

10

15

20

25

0 50 100 150 200 250

Net gro

wth of gro

wing stock (%/y)

Observed

Predicted

–10

–5

–10

–5

Growing stock density (m3

/ha)

Growing stock density (m3

/ha)

Growing stock density (m3

/ha)

Growing stock density (m3

/ha)

Growing stock density (m3

/ha)

(a) (b)

(c) (d)

(e) (f)

Figure 3—Empirical relationship between net growth in growing stock (%/y) and growing stock density (m3/ha) for (a) softwoods in the U.S. North, (b) hardwoods in the U.S. North, (c) softwoods in the U.S. South, (d) hardwoods in the U.S. South, 

(e) softwoods in the U.S. West, and (f) hardwoods in the U.S. West.
Research Paper FPL–RP–668

6

nal estimates of growth parameters for the West were based 

on county-level data that did not include Pacific Coast 

states. We know those states have generally much lower 

average timber growth rates than the region as a whole (as 

a percentage of timber inventory volume) partly because of 

relatively high timber inventory volumes within the Pacific 

Coast states.

Table 1 presents the descriptive statistics of the county-level 

data used in the study (excluding the Pacific Coast states). 

The number of observations (number of counties with timber data) ranged from as low as 273 for softwoods in the 

West to as high as 1,013 for hardwoods in the South. The 

softwood forest inventory in the U.S. South was characterized by the largest average net annual growth as percentage 

of inventory (5.91%/y), followed by softwoods in the North 

(3.97%/y). Similarly, the largest average growth in hardwood forest inventory was observed in the West (3.76%/y), 

followed by the South (3.10%/y). The lowest softwood 

inventory growth rate was observed in the West (2.50%/y), 

and the lowest hardwood inventory growth rate was observed in the North (3.01%/y). The lowest average net 

growth rate was also accompanied by the largest growingstock density. For example, hardwood forest inventory in 

the North and softwood forest inventory in the West had the 

largest average growing-stock density (98.31 and 88.34 m3/

ha, respectively) (Table 1).

According to Smith and others (2009), net annual growth of 

softwoods in the entire U.S. West in 2006 was 0.173 billion 

m3, whereas net annual growth of hardwoods in the entire 

U.S. West was 0.022 billion m3. Also, total volume of 

softwood growing-stock in the entire U.S. West in 

2007 was 10.049 billion m3, whereas total volume of 

hardwood growing-stock in the entire U.S. West was 

1.152 billion m3 (Smith and others 2009). These data include all Pacific Coast states, including Alaska and Hawaii, 

as well as Oregon, Washington, and California. On the basis 

of these data, net annual growth rates as percentages of inventories in the entire West were 1.72% for softwoods and 

1.91% for hardwoods. These growth rates are clearly lower 

than the Table 1 growth rates for the West obtained as mean 

values from county data that excluded the Pacific Coast 

states, which explains why we must make adjustments to 

growth parameters that we estimate for the West.

Softwood and hardwood growth and stocking density data 

used in this study pertain to total timberland area in all ownership categories across the three U.S. regions. Therefore, 

the growth models developed in this study are regional models of softwoods and hardwoods that include all ownership 

categories. Furthermore, stocking densities for both species 

groups (hardwood and softwood) were computed on the basis of total timberland area, so growing-stock densities 

are naturally the highest for species that are regionally 

dominant on the landscape, specifically hardwoods in 

the East (North and South) and softwoods in the West 

(Table 1). In addition, softwood timber in the South has 

by far the highest mean growth rate as a percentage of 

inventory, largely because of widespread plantations and 

productive management of Southern Pines with relatively 

short timber rotations. Mean growth rate for softwood is 

much lower in the West (at 1.72% of growing stock) because a large share of forest land is in public ownership with 

relatively passive timber management practices and long 

timber rotations that contribute to high stocking density and 

slower growth rates for softwood in the West.

Model

The study used nonlinear least squares regression models to 

estimate empirical relationships between net annual growth 

and growing-stock density across U.S. regions for the two 

species groups, hardwoods and softwoods. The regression 

model was represented by the following functional form 

(Turner and others 2006):

(1)

where Gi is net annual growth as percentage of growingstock inventory (%/y) in county i, Si

 is growing-stock density (m3/ha) (growing-stock inventory divided by timberland 

area) in county i; Pi is proportion of area in plantations in 

county i; α and γ are parameters to be estimated; and ui is 

an error term. Such a functional form is consistent with 

theoretical relationships described in the literature between 

net forest growth rate and growing-stock density (Turner 

and others 2006; Oliver and Larson 1996; Smith and others 

1996). For a given species group and region, the counties 

with lower stocking density generally have higher growth 

rates per unit of inventory and vice versa. Thus, γ should be 

negative. In addition, counties that have more timberland 

areas under plantations will generally have higher percentage net forest growth. Thus, α1 should be positive. The FIA 

reporting of data does not include timberland area under 

plantations. However, it reports stand origin indicating 

whether the stand was naturally or artificially regenerated 

(trees were planted). Although artificial regeneration does 

not necessarily mean that a stand is an intensively managed 

tree plantation, it can be used as a proxy indicator for the 

higher productivity of plantations because artificially regenerated stands certainly include managed plantations, and 

artificial regeneration is also likely to enhance forest growth.

We used the proportion of timberland area that is artificially 

regenerated as a proxy for Pi in Equation (1). Such a proxy 

variable was applied for the softwoods in the South only 

because the proportion of the timber resource that is artificially regenerated and intensively managed in the South is 

relatively high compared with other regions. Although there 

may be some degree of uncertainty about the FIA stand origin data because of a perceived difficulty in determining the 

origin of older stands on survey plots, the data indicate that 

planted timberland area is 15% of the timberland area in the 

Pacific Coast region. In the West as a whole (both Rocky 

Mountain and Pacific Coast) planted timberland area is only 

8% of total timberland area, and in the North the planted 

Gi = (α + α1Pi)(Si)γ + ui
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

7

models in projecting inventory. The reliability of such 

models was assessed by comparing historically observed 

growing-stock inventory with predicted inventory using 

the developed growth models and historical harvest data 

(Figs. 4, 5). The predictive accuracy of the model was 

evaluated based on the estimated root mean square error 

as follows (Greene 2003):

(2)

where, n0 is the number of periods being predicted, yi is 

the individual observation for a particular year, and i yˆ is 

the predicted value for a particular year. The statistical understanding of the prediction error (PE) over the predicted 

periods was estimated by taking percentage of root mean 

squared error against the mean of the observed historical 

values over the predicted period as follows:

(3)

For this comparison, we used nine different historical 

years (1953, 1963, 1970, 1977, 1987, 1992, 1997, 2002, 

and 2007) for which U.S. inventory, growth, harvest, and 

area data were available as reported in various USDA Forest 

Service Forest Resources reports. The historically observed 

growing-stock inventory was projected by subtracting the 

harvest quantity and adding the growth predicted by the 

developed models (using regional historical growing-stock 

density) with two separate approaches. The first approach 

was to use historical inventory and growing-stock density 

data at each time period separately. In other words, we used 

1953 actual inventories, harvest quantities, and growingstock densities to project 1963 inventory, and we used 1963 

actual inventories, harvest quantities, and stocking densities 

to project 1970 inventory, and so forth. Another approach 

was to predict timber inventory for all historical years using 

only the initial inventory in 1953 and the historical harvest 

from 1953 to 2007. Results of these two inventory projection approaches are shown in Figures 4 and 5 in comparison 

with actual historical inventory data.

The developed growth models were applied also in making 

a long-range growing-stock inventory projection for three 

U.S. regions under three alternative 2010 RPA scenarios and 

using timber harvest projections from USFPM/GFPM (the 

model used to produce timber harvest projections for the 

2010 RPA). The RPA scenarios were based on various assumptions about global economic growth, global wood energy consumption, population changes, and climate changes 

and were expected to have different effects on future U.S. 

forest resources conditions and trends (Ince and others 

2011). These scenarios were termed A1B, A2, and B2 in line 

with similar scenarios developed by the Intergovernmental 

Panel on Climate Change (IPCC) (Nakicenovic and Swart 

2000). The A1B scenario represented the highest global 

economic growth and wood energy consumption coupled 

timberland area is about 4% of total timberland area (Smith 

and others 2009). By contrast, about 22% of the total timberland area is artificially regenerated in the South (Miles 

2011; Smith and others 2009).

Furthermore, most artificially regenerated stands in the 

South are planted pines, and much of the pine forest area in 

the South consists of managed pine plantations (Wear and 

others 2011). By contrast, managed plantations are a much 

smaller share of the timber resource in the West, even for 

species that are most commonly planted, such as Douglasfir. Planted stands account for only 13% of the growingstock inventory of Douglas-fir timber in the West, whereas 

planted stands account for 46% of the inventory of longleaf–

slash pine and 43% of the loblolly–shortleaf pine inventory 

in the South (Smith and others 2009). Therefore, we did 

not include Pi as an additional variable in our inventory 

growth models for the North and West or for hardwoods in 

the South, because in those cases the proportion of the total 

resource that is artificially regenerated and intensively managed region-wide was thought to be relatively small compared with southern softwoods. However, if county-level 

data were to become available for the Pacific Coast region, 

we would consider extending our analysis by including Pi as 

a variable for the softwood inventory growth model in the 

West because the Pacific Coast has the next largest area of 

planted timberland area after the South.

The model parameters were estimated by nonlinear least 

squares method (Marquardt’s method, SAS 1999). Because 

the functional form used in this study represents a constant 

elasticity functional form, the estimates of γ provide a direct 

measure of elasticity. It tells us the percentage change in net 

growth for a 1% change in stocking density. The plausibility 

of each model was evaluated based on the conformity of 

the estimated parameters with expected signs, the statistical 

significance of the overall model, and individual parameters. 

The validity of estimated parameters for meaningful inferences was determined using Hougaard’s measure of skewness 

(Ratkowsky1990; Hougaard 1985). According to Ratkowsky (1990), when the parameter estimates from nonlinear 

regression models are “close-to-linear,” then the leastsquares estimators of the parameters are close to being 

unbiased, are normally distributed, and have minimum variance, thus allowing their standard errors and confidence intervals to be safely used for inferences. Hougaard’s measure 

of skewness (g1i) can be used to assess the extent to which 

a parameter exhibits close-to-linear behavior (Ratkowsky1990; Hougaard 1985). The value of |g1i

| < 0.1 indicates 

that parameter estimates are very close-to-linear in behavior, 

and 0.1 < |g1i| < 0.25 indicates that parameter estimates are 

reasonably close-to-linear (Ratkowsky1990). 

Model Validation and Application

One goal of the growth models developed in this study is 

to provide a reliable basis for timber inventory projection 

across U.S. regions and species groups. Therefore, it was 

important to evaluate and understand the reliability of these 

RMSE = √ 1

n0 Σ

i

( yi – ŷi

 )2

PE = RMSE 

Σ

i

yi

×100 1

n
Research Paper FPL–RP–668

8

Figure 4—Comparison of observed historical growing stock inventory and predicted inventory (using 

predictions for each time period separately) for 

(a) softwoods in the U.S. North, (b) hardwoods in 

the U.S. North, (c) softwoods in the U.S. South, 

(d) hardwoods in the U.S. South, (e) softwoods in 

the U.S. West, (f) hardwoods in the U.S. West, 

(g) softwoods, U.S., (h) hardwoods, U.S., and 

(i) softwood and hardwoods combined, United 

States.

0.00

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(e) (f)
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

9

0.00

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(g) (h)

(i) Figure 5—Comparison of observed historical growing stock inventory and predicted inventory, (using 

only 1953 inventory and 1953–2007 harvest data) for 

(a) softwoods in the U.S. North, (b) hardwoods in the 

U.S. North, (c) softwoods in the U.S. South, (d) hardwoods in the U.S. South, (e) softwoods in the U.S. 

West, (f) hardwoods in the U.S. West, (g) softwoods, 

U.S., (h) hardwoods, U.S., and (i) softwood and hardwoods combined, U.S.
Research Paper FPL–RP–668

10

with slowing population growth, whereas the A2 and B2 

scenarios were associated with considerably lower economic growth and lower wood energy consumption. The A2 

scenario assumed higher global population growth coupled 

with lower economic growth compared with other scenarios, 

and the B2 scenario assumed lowest population growth with 

mid-level global economic growth. Table 2 summarizes 

various assumptions of the selected scenarios.

We made timber inventory projections for these scenarios 

by adding to the base year (2006) inventory the net forest 

growth (predicted by the empirical growth models developed in this study) and subtracting annual harvest quantities 

by regions based on USFPM projected timber harvest levels 

(from 2006 to 2060). In addition, the regional U.S. timber 

supply equations in USFPM were shifted over time by our 

projected regional timber inventories (using an inventory 

elasticity of 1.0 for the supply equations). The Gauss–Seidel 

technique of iterative solutions (Jeffreys and Jeffreys 2000) 

was used to derive convergent equilibrium solutions for 

each scenario by linking our spreadsheet model of regional 

timber growth and inventory to USFPM/GFPM. The spreadsheet model used USFPM projections of timber harvest by 

region, and in turn the spreadsheet model provided projections of growing-stock inventory to USFPM, where they 

were used to shift regional timber supply curves. The models were run iteratively until a reasonably stable convergent 

equilibrium was obtained (usually less than 12 iterations). In 

a separate analysis, we compared inventory projections with 

projections reported in other recent literature related to U.S. 

forest resources.

Results and Discussion

Net Forest Growth and Growing-Stock Density 

Relationship

Table 3 presents estimates of model parameters and related 

statistics. All regression models were statistically significant 

at 1% or better significance level (F ≤ 0.0001). The models 

also fulfilled an a priori expectation on the signs of the estimated parameters, a negative sign on the growing-stock 

density coefficient (γ) and a positive sign on the coefficient 

associated with fraction of area under artificial regeneration in the U.S. South (α1). Individual parameter estimates 

were strongly significant at 1% or better significance level 

(t ≤ 0.0001) for all regions and species. The coefficient of 

determination (R2), interpreted as the proportion of variation in dependent variable explained by variation in independent variable, has meaning only in case of least squares 

regression applied to a linear equation with a constant term 

(Greene 2003). This usual interpretation of R2 does not apply in the case of nonlinear regression without a constant 

term, as used in this study. Therefore, we did not use R2

as a basis of evaluating our models. However, an analogous measure of R2 called Pseudo-R2 (Gujarati 2003) was 

calculated based on residual and total corrected sums of 

squares and is reported in Table 3. The absolute value of 

Hougaard’s measure of skewness for all parameter estimates 

ranged from less than 0.01 to 0.25 for all regions and species groups, indicating that the parameter estimates were 

close-to-linear and that their standard errors and confidence 

intervals could be safely used for inferences.

Figure 3 presents observed and predicted relationships between net annual growth rates and growing-stock inventory 

for the three U.S. regions and two species groups. Results 

indicated a higher percentage net growth rate for lower 

growing-stock densities and a lower percentage net growth 

rate for higher stocking densities in all regions and species. 

However, the decline in net growth rate with increasing 

growing-stock density differed among regions and species 

groups. For example, the net growth rate declined strongly 

with higher growing-stock density for hardwoods in the 

North (γ = –0.47). A gamma value of –0.47 indicated that 

Table 2—Summary of the three alternative RPA scenariosa

Summary A1B A2 B2 

General

description 

Globalization,

economic convergence 

Heterogenic 

regionalism, less trade

Localized solutions, 

slow change 

Social development 

themes 

Economic growth, 

new technologies, 

capacity 

Self-reliance,

preservation 

of local identities 

Sustainable

development,

diversified technology 

Global real GDP 

growth (2010–2060) 

High (6.2×) Medium (3.2×) Medium (3.5×) 

Global population 

growth (2010–2060) 

Medium (1.3×) High (1.7×) Medium (1.4×) 

U.S. GDP 

growth (2006–2060) 

Medium (3.3×) Low (2.6×) Low (2.2×) 

U.S. population 

growth (2006–2060) 

Medium (1.5×) High (1.7×) Medium (1.3×) 

Global expansion 

of primary biomass energy 

production (2000–2060) 

High (5.9×) Medium (3.1×) Medium (3.2×) 

a

Ince and others (2011). 
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

11

doubling of growing-stock density would decrease the net 

growth rate by almost half (–47%). A similar response to 

increase in stocking density was observed for softwoods 

in the West (γ = –0.41). In contrast, softwoods in the North 

and hardwoods in the West had relatively smaller declines 

in net growth rates in response to higher stocking density 

(γ = –0.28 and –0.20, respectively). The observed response 

of net growth to stocking density was markedly lower in 

the South for both softwoods and hardwoods, relative to the 

North and West. With a doubling of stocking density, the 

hardwood forest in the South showed a decline of only 16% 

in net annual growth, and similarly only 15% for softwoods. 

Furthermore, southern softwood stands showed significantly 

different rates of growth for naturally and artificially regenerated stands. For a given stocking density, net annual 

growth as a percentage of inventory was significantly higher 

(five times) in artificially regenerated softwood stands relative to naturally regenerated softwood stands in the South. 

Our result that growth was five times higher in artificially 

regenerated stands for softwoods in the South is consistent 

with estimates provided in other studies. For example, Ryan 

and others (2010) reported that planted pines with improved 

seedlings, competition control, and fertilization grow four 

times faster than naturally regenerated second-growth pine 

stands in the U.S. South.

Overall, our results showed a relatively smaller decline in 

percentage net growth for an increase in growing-stock 

density for all U.S. regions and species groups relative to 

estimates reported by Turner and others (2006), who investigated a similar relationship at the global level using a singular nonlinear functional relationship. Their results indicated 

that doubling of growing-stock density would decrease net 

forest growth by 81%. Results from this study did not indicate as strong a decline in U.S. forest stock growth with 

increasing density as reported globally by Turner and others 

(2006). The differences in results between this study and 

those of Turner and others (2006) can be explained by the 

Table 3—Parameter estimates of net forest growth (%/y) and 

growing stock density (m3

/ha/y) relationships by three U.S. regions 

and softwood and hardwood species groups 

Variable Estimatea

Approximately 95%

confidence limits Hougaard’s

measure of 

skewness (g1i)

b Lower Upper 

North, softwoods 

α 6.1323 (0.2314)*** 5.6779 6.5867 –0.000003† 

γ –0.2809 (0.0191)*** –0.3200 –0.2400 –0.0195† 

F value 351.1100*** — — — 

Pseudo-R2 0.2100 — — — 

North, hardwoods 

α 23.2566 (1.4047)*** 20.5001 26.0132 0.0306† 

γ –0.4648 (0.0152)*** –0.4986 –0.4390 0.1088‡ 

F value 1119.4900*** — — — 

Pseudo-R2 0.2300 — — — 

South, softwoodsc

 

α 5.6224 (0.3274)*** 4.9800 6.2649 0.0021† 

α1 22.8506 (2.8764)*** 17.2060 28.4952 0.2523‡ 

γ –0.1453 (0.0220)*** –0.1885 –0.1021 0.1039‡ 

F value 1028.2600*** — — — 

Pseudo-R2 0.1600 — — — 

South, hardwoods 

α 5.3815 (0.3932)*** 5.0599 6.6031 0.0480† 

γ –0.1584(0.0171)*** –0.1920 –0.1248 0.1284‡ 

F value 1659.4100*** — — — 

Pseudo-R2 0.0500 — — — 

West, softwoods 

α 9.3639 (0.5017)*** 8.3763 10.3516 –0.00092† 

γ –0.4097 (0.0265)*** –0.4619 –0.3574 –0.1600‡ 

F value 175.1500*** — — — 

Pseudo-R2 0.4500 — — — 

West, hardwoods 

α 5.7107 (0.4185)*** 4.8876 6.5337 –0.0157† 

γ –0.1992 (0.0309)*** –0.2600 –0.1384 0.0133† 

F value 101.6400*** — — — 

Pseudo-R2 0.1000 — — — a

Numbers in parentheses indicate approximate standard error; ***, statistically significant 

at 1% significance level. 

b

†, parameters are very close-to-linear; ‡, parameters are reasonably close-to-linear. c

The model includes proportion of artificial regenerated area as additional variable. 
Research Paper FPL–RP–668

12

geographic coverage of the analyses and by the fact that we 

used different data sets. The single functional relationship 

for the world developed by Turner and others (2006) was 

derived from cross-sectional analysis of FAO global data 

on the net growth rate and forest stocking density among 

180 countries. 

This study was designed to account for regional and species 

group differences by developing separate empirical relationships based on cross-sectional analysis of FIA county-level 

data for each region, as opposed to global data by Turner 

and others (2006). Therefore, it was logical to expect differences between our estimated regional growth models and 

the generalized global formula developed by Turner and 

others (2006). Also, preliminary research conducted at FPL 

indicated that the global relationship employed in the GFPM 

did not accurately fit forest growth and growing-stock 

density data for any of the U.S. regions or species groups. 

However, employing such relationships specific to U.S. regions and species groups as developed in this study showed 

a much improved fit, indicating that the general theory of a 

growth-to-growing-stock density relationship is valid, and 

growth models developed in this study more accurately represented growth dynamics for the U.S. regions and species 

groups.

Model Comparison and Application

Figures 4 and 5 compare historically observed and predicted 

growing-stock inventory for U.S. regions and species groups 

using two approaches to predict inventory (predictions made 

at each time period separately versus using only 1953 inventory and 1953–2007 harvest data). The comparison shows 

that the region- and species-specific net annual growth and 

growing-stock density models developed in this study were 

successful in predicting growing-stock inventory trends and 

inventory levels very close to historically observed inventory when predictions were made at each timber period separately. The overall differences between predicted and actual 

historical observations, as measured by the percentage of 

root mean square error over the mean of the historically 

observed inventories (Eq. (2)), ranged from 3% to 6% for 

hardwoods and 4% to 7% for softwoods for individual regions (using each time period prediction approach). The prediction showed a 38% increase in total U.S. growing-stock 

inventory during 1963–2007, compared with a 41% increase 

in actual historical trend during the same period. However, 

when predictions were made using only 1953 inventory and 

1953–2007 harvest data, the overall average prediction error for individual region and species group increased and 

ranged from 12% for hardwoods in the North to as much 

35% for hardwoods in the South (Fig. 5). The predictions 

for the U.S. aggregate inventory (Figs. 5g–i), however, remained very close to the actual historical observations.

To further validate our regional inventory projections across 

species groups, we compared them with corresponding 

inventories projected for the 2005 RPA base scenario. The 

2005 RPA base scenario was developed based on assumptions about the overall future U.S. economy and biological 

and market conditions affecting U.S. timber supply on both 

public and private lands (Haynes and others 2007). The 

2005 RPA base scenario projected rising trends in U.S. 

softwoods, hardwoods, and total inventories, which we 

reconciled with trends in our projections. Specifically, we 

applied the same harvest levels and timberland area assumptions employed in the 2005 RPA assessment to our 

spreadsheet model and compared the resulting inventory 

projections. Figures 6a–d show that our original estimates 

of growth model parameters (Table 3) yield inventory projections for the North and for softwood in the South quite 

close to 2005 RPA projections when using the same projected timber harvest and land area assumptions. However, 

our original estimates of growth model parameters yield 

large over-projections of timber inventories in the West and 

hardwood timber inventory in the South. The inaccuracy 

of our original estimates of growth parameters for the West 

can be attributed as explained earlier to our lack of county 

data for the Pacific Coast states, which have generally much 

lower net annual growth as a percentage of growing-stock 

inventory than the West as a whole (Smith and others 2009). 

Therefore, we made calibrating adjustments to our growth 

parameters for the West (Table 4), and as a result our timber 

inventory projections for the West come much closer to the 

2005 RPA projections, as shown in Figures 7a–d. We also 

made some additional minor adjustments to growth parameters for other regions to bring our inventory projections 

more closely in line with the 2005 RPA projections, particularly for hardwood inventory in the South.

The most likely reason for our over-projection of hardwood 

inventory in the South relative to the 2005 RPA was that our 

model did not take into account large-scale future conversions of hardwood forest types to planted pines, as projected 

by Haynes and others (2007). Using ATLAS to model forest 

growth, Haynes and others (2007) projected a big decline 

in the area of upland hardwoods in the South because of 

conversions to planted pine, resulting in less region-wide 

hardwood growth and smaller hardwood inventories. In ATLAS, when hardwood area was converted or removed from 

the timberland base in the 2005 RPA projections, area was 

removed across a range of age classes and not just older, 

slower growing stands (John Mills, personal communication, USDA Forest Service, Pacific Northwest Research 

Station, October 18, 2011). Our broad region-wide growth 

model (based on cross-sectional county-level data of just 

one year) cannot capture effects of such forest cover changes or management conversions occurring at different time 

periods except by making exogenous adjustments to growth 

parameters. Without such adjustments, our model produces 

over-projections of hardwood inventories in the South.

In general, our region-wide growth model parameters can 

be further adjusted to reflect other assumptions about future 

forest resource conditions and trends: for example, timberland area changes from forest type conversion, increased 

forest management intensity, increased plantation area, and 
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

13

so forth. As shown in Table 4, we actually made modest 

adjustments to the alpha (α) or gamma (γ) parameters (or 

both) for every region and species group to produce the inventory projections shown in Figures 7a–d, which matched 

quite closely to the 2005 RPA inventory projections. The 

necessary adjustments were very small for the North and 

also for softwood in the South, but they were larger for the 

West and for hardwood in the South. In almost all cases 

(except for hardwoods in the West) the adjusted parameter 

values were still within the 95% confidence limits of our 

original estimates (Table 4).

Figure 8 presents historical and projected U.S. growingstock inventories for three alternative 2010 RPA scenarios 

for the period between 1953 and 2060, based on iterative 

convergent solutions between the USFPM/GFPM model 

and the U.S. regional forest growth and inventory models 

that we developed in this study (without calibration to 2005 

RPA base scenarios). Specifically, the projected regional 

growing-stock inventories took into account our originally 

0

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(a) (b)

(c) (d)

(e) (f)

Figure 6—Projected growing stock inventories (billion cubic meters), 2002–2050, by U.S. regions and species 

group in 2005 RPA base scenario by Haynes and others (2007) and our inventory projections for (a) softwoods, 

North, (b) hardwoods, North, (c) softwoods, South, (d) hardwoods, South, (e) softwoods, West, and (f) hardwoods, 

West. Blue bars represent projections of the 2005 RPA base scenario by Haynes and others (2007); red bars represent projections using the original estimates of growth model parameters (Table 3) developed in this study along 

with harvest quantities and timberland areas as projected in the 2005 RPA base scenario.
Research Paper FPL–RP–668

14

estimated forest growth and stocking density relationships 

(Table 3) and also the projected regional timber harvest 

levels from USFPM, and in turn the USFPM regional timber supply equations were shifted by the projected regional 

inventories for hardwoods and softwoods. Consistent with 

the assumed economic growth, energy consumption level, 

and population growth, the projections (Fig. 8) indicated 

higher inventory levels over the entire projection period 

for both hardwoods and softwoods in A2 and B2 scenarios, 

which featured modest expansion of timber harvest and 

represented modest growth in the economy, population, 

and wood energy consumption. In contrast, forest inventory 

levels were lower and declining in the A1B scenario, which 

represented a scenario with the largest economic growth and 

wood energy consumption and highest timber harvest levels. 

Relative to 2010, softwoods, hardwoods, and total inventory increased by 56%, 76%, and 65%, respectively, by 

2060 in the B2 scenario. Similarly, increases in softwoods, 

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Figure 7—Projected growing stock inventories (billion cubic meters), 2002–2050, by U.S. regions and species 

group in 2005 RPA base scenario by Haynes and others (2007) and our inventory projections after adjusting estimated growth parameters (see Table 4) for (a) softwoods, North, (b) hardwoods, North, (c) softwoods, South, (d) 

hardwoods, South, (e) softwoods, West, and (f) hardwoods, West. Blue bars represent projections of the 2005 RPA 

base scenario by Haynes and others (2007); red bars represent projections using our adjusted growth model parameters along with harvest quantities and timberland areas as projected in the 2005 RPA base scenario.
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

15

0

5

10

15

20

25

1953 1970 1987 1997 2007 2010 2020 2030 2040 2050 2060

Billion cubic meters

Historical A1B A2 B2

0

10

20

30

40

50

1953 1970 1987 1997 2007 2010 2020 2030 2040 2050 2060

Billion cubic meters

Historical A1B A2 B2

0

5

10

15

20

25

1953 1970 1987 1997 2007 2010 2020 2030 2040 2050 2060

Billion cubic meters

Historical A1B A2 B2

(c)

(b)

(a)

Figure 8—Historical and projected U.S. growing stock inventories (billion cubic 

meters), 1952–2060, by three alternative 2010 RPA scenarios, (a) softwoods, (b) hardwoods, and (c) total, based on USFPM harvest projections for 2010 RPA scenarios 

linked to our regional growth and inventory models without any adjustments except 

for the West (Table 3).
Research Paper FPL–RP–668

16

hardwoods, and total inventory for the A2 scenario were 

45%, 53%, and 48%, respectively. Projected total inventories in these two scenarios were fairly close to the historical 

trend in total inventory from 1953 to 2007, during which 

total inventory increased by 51% (Smith and others 2009). 

Consistent with higher global economic growth and high 

wood energy consumption inducing by far the largest projected timber removals in the A1B scenario, U.S. softwood, 

hardwood, and total inventory levels increased only by 30%, 

11%, and 22%, respectively, in the A1B scenario by 2060 

relative to 2010.

Finally, we used the adjusted parameters of growth equations (Table 4) that matched the timber inventory projections reported in 2005 RPA in conjunction with the USFPM/

GFPM model to produce projections of U.S. growing-stock 

inventories for the three alternative 2010 RPA scenarios. 

The results are shown in Figure 9 along with historical 

trends in growing-stock inventories (billion cubic meters) 

1952–1960. Our adjusted growth model parameters yield 

generally lower projected inventories for the 2010 RPA 

scenarios than the projections obtained without such adjustments (Fig. 8). We also compared the actual historical 

inventories and the predicted inventories using the adjusted 

parameters of growth equations. The comparison of the actual historical inventories with those inventories predicted 

from the adjusted growth equations (Fig. 10) showed larger 

deviations than those inventories from unadjusted growth 

equations (Fig. 4 or 5) for both individual regions and the 

U.S. aggregate. Because of better performance in predicting 

historical inventory, we plan to apply our unadjusted growth 

equations in future studies.

Conclusions

This study presents a parsimonious model of regional net 

forest growth that is linked to a regional and global forest 

product market model to project future U.S. timber inventory by region and species group (hardwood and softwood). 

For each region and species group, the model requires data 

for only one independent variable and growing-stock density but allows for the inclusion of plantation area as an 

additional variable (for softwoods in the South). Data and 

models support theoretical relationships between net forest 

growth and growing-stock density. By estimating separate 

regional growth models according to species group, this 

study was successful in taking into account distinguishing 

relationships between net forest growth and stocking density across U.S. regions and species groups. The developed 

growth models can be easily linked via a spreadsheet model 

to USFPM/GFPM or other forest sector market models.

Results showed a different response in net forest growth to 

changes in growing-stock density across U.S. regions and 

species groups. Inventory projections for the 2010 RPA scenarios showed contrasting dynamics between the A1B scenario and other scenarios because of large differences 

Table 4—Original and adjusted parameters required to 

reconcile projected inventory in this study with that of 2005 

RPA base scenarioa

Variable

Original 

estimate 

Approximately 95%

confidence limits Adjusted

parameter value 

Lower Upper (if any) 

North, softwoods 

α 6.1323 5.6779 6.5867 — 

γ –0.2809 –0.3200 –0.2400 –0.2950 

North, hardwoods 

α 23.2566 20.5001 26.0132 — 

γ –0.4648 –0.4986 –0.4390 –0.4880 

South, softwoods 

α 5.6224 4.9800 6.2649 — 

α1 22.8506 17.2060 28.4952 — 

γ –0.1453 –0.1885 –0.1021 –0.1470 

South, hardwoods 

α 5.3815 5.0599 6.6031 5.1000 (2010–2050) 

γ –0.1584 –0.1920 –0.1248 –0.1900 (2002–2010) 

–0.1850 (2010–2050) 

West, softwoods 

α 9.3639 8.3763 10.3516 8.3763 

γ –0.4097 –0.4619 –0.3574 –0.4619 

West, hardwoods 

α 5.7107 4.8876 6.5337 4.8876 

γ –0.1992 –0.2600 –0.1384 –0.3200b

a

Haynes and others 2007. 

b

The adjustment required for the ߛ parameter is outside the 95% confidence 

interval.
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

17

0

5

10

15

20

1953 1970 1987 1997 2007 2010 2020 2030 2040 2050 2060

Billion cubic meters

Historical A1B A2 B2

0

10

20

30

40

1953 1970 1987 1997 2007 2010 2020 2030 2040 2050 2060

Billion cubic meters

Historical A1B A2 B2

0

5

10

15

20

1953 1970 1987 1997 2007 2010 2020 2030 2040 2050 2060

Billion cubic meters

Historical A1B A2 B2

(a)

(b)

(c)

Figure 9—Historical and projected U.S. growing stock inventories (billion cubic meters), 1952–1960, by three alternative 2010 RPA scenarios for (a) softwoods, (b) hardwoods, and (c) total, based on our growth models adjusted to match 2005 RPA base 

scenario inventory projection (Table 4).
Research Paper FPL–RP–668

18

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

0.00

1.00

2.00

3.00

4.00

5.00

6.00

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

0.00

2.00

4.00

6.00

8.00

10.00

12.00

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

0.00

1.00

2.00

3.00

4.00

5.00

6.00

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

(a) (b)

(c) (d)

(e) (f)

Figure 10—Comparison of observed historical growing stock inventory and predicted inventory (using 

only 1953 inventory and 1953–2007 harvest data) after adjusting estimated growth parameters to match 

2005 RPA baseline projection (see Table 4) for 

(a) softwoods in the U.S. North, (b) hardwoods in 

the U.S. North, (c) softwoods in the U.S. South, 

(d) hardwoods in the U.S. South, (e) softwoods in 

the U.S. West, (f) hardwoods in the U.S. West, 

(g) softwoods, U.S., (h) hardwoods, U.S., and 

(i) softwood and hardwoods combined, U.S.

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

0.00

2.00

4.00

6.00

8.00

10.00

12.00

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

0.00

2.00

4.00

6.00

8.00

10.00

12.00

1963 1970 1977 1987 1992 1997 2002 2007

Billion cubic meters

Historical

Predicted

(g) (h)

(i)
Developing Inventory Projection Models Using Empirical Net Forest Growth and Growing-Stock Density Relationships

19

in projected timber harvest. The RPA scenario resulting 

in highest removals (A1B) was accompanied by lowest 

projected inventory levels, and the scenario resulting in 

lowest removals (B2) resulted in highest projected inventory levels. The A1B scenario, which represented the highest economic growth coupled with high demand for wood 

energy, induced largest removals, leading to a decline in 

softwood, hardwood, and total U.S. growing-stock inventory (Fig. 8). In contrast, with modest economic growth and 

modest removals, inventories were projected to increase 

over the entire projection period in the B2 scenario.

Prior to the development of our growth models, USFPM/

GFPM could not be used to accurately simulate regional 

impacts of climate change policies on forest carbon storage. 

However, analysis of climate change policies is important 

for predicting trends in future forest resource conditions and 

trends. This study is one step toward enabling the USFPM/

GFPM to provide such analysis. Earlier unpublished work 

at the Forest Products Laboratory had shown a strong linear 

relationship between growing-stock inventory and forest 

carbon stocks. This suggests that the developed inventory 

growth models can equip USFPM with forest carbon stock 

accounting capability by projecting growing-stock inventory. Such a capability in forest carbon accounting will allow USFPM/GFPM to simulate the impact of national and 

global climate change policies (e.g., forest carbon offset 

policies) on U.S. timber markets and trade and conversely 

to analyze how market and trade scenarios affect forest 

carbon stocks. By enhancing USFPM/GFPM’s capability to 

model changes in the U.S. forest sector carbon inventory in 

response to changing global timber markets, wood energy 

demands, forest growth, and forest carbon offset credits, this 

study can contribute to a more meaningful analysis of U.S. 

and global forest carbon offset strategies. Using USFPM/

GFPM and our growth models, we plan to evaluate how 

forest sector carbon offset credits will influence U.S. forest 

carbon sequestration and forest products markets, focusing 

initially on the strategy to pay forest landowners for reducing or avoiding timber harvests.

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