The Enhanced Forest 

Inventory and Analysis 

Program—National 

Sampling Design and 

Estimation Procedures

William A. Bechtold and 

Paul L. Patterson, Editors

United States

Department of 

Agriculture

Forest Service

Southern 

Research Station

General Technical 

Report SRS–80
William A. Bechtold, Research Forester, USDA Forest Service, Southern 

Research Station, Asheville, NC 28802; and Paul L. Patterson, Mathematical 

Statistician, USDA Forest Service, Rocky Mountain Research Station, Fort 

Collins, CO 80526-1891.

The Authors:

June 2005

Southern Research Station

P.O. Box 2680

Asheville, NC 28802

Cover: The cover graphic illustrates FIAʼs sampling frame superimposed 

onto a landscape. The patchwork of green and brown under the hexagonal 

grid represents forest and nonforest areas. Each hexagon contains one 

ground plot symbolized by the cluster of four points located within each 

hexagon. The ground plots are not drawn to scale.

Cover artwork: Greg C. Liknes.
i

The Enhanced Forest Inventory 

and Analysis Program—

National Sampling Design and 

Estimation Procedures

William A. Bechtold and Paul L. Patterson, Editors
ii
iii

Table of Contents

Executive Summary ............................................................................ vii

1 The Enhanced Forest Inventory and Analysis Program ............ 1

 Ronald E. McRoberts

1.1 Purpose and Scope of the Documentation ................................. 1

1.2 Overview of the Enhanced Forest Inventory and

 Analysis Program ...................................................................... 2

 1.2.1 Historical Perspective ........................................................... 2

 1.2.2 Forest Health Monitoring ..................................................... 3

 1.2.3 Annual Inventories ............................................................... 3

 1.2.4 National Consistency ............................................................ 5

1.3 Overview of the Documentation ................................................ 8

 1.3.1 Sampling Frame ................................................................... 8

 1.3.2 Plot Design ........................................................................... 8

 1.3.3 Sample-Based Estimators ..................................................... 9

 1.3.4 Combining Data for Multiple Panels .................................... 9

 1.3.5 Notation and Glossary .......................................................... 9

 1.3.6 Web-Based Supplementary Documentation ......................... 9

1.4 Literature Cited ........................................................................ 10

2 The Forest Inventory and Analysis Sampling Frame ............... 11

 Gregory A. Reams, William D. Smith,

 Mark H. Hansen, William A. Bechtold,

 Francis A. Roesch, and Gretchen G. Moisen

2.1 Overview of Forest Inventory and Analysis Sampling 

 Design ...................................................................................... 11

 2.1.1 Forest Inventory and Analysis Populations ........................ 11

 2.1.2 Phase 1 ................................................................................ 12

 2.1.3 Phase 2 ................................................................................ 15

 2.1.4 Phase 3 ................................................................................ 16

2.2 Development of the Phase 2/Phase 3 Sampling Frame ........... 18

 2.2.1 Hexagonal Sampling Frame ............................................... 18

 2.2.2 Division of the Sampling Frame into Panels ...................... 20

 2.2.3 Populating the Sampling Frame ......................................... 22

 2.2.4 Deviations from the Five-Panel Annual System ................. 22

 2.2.5 Theoretical Basis for the FIA Sampling Frame ................. 23

2.3 Literature Cited ........................................................................ 26
iv

3 The Forest Inventory and Analysis Plot Design ........................ 27

 William A. Bechtold and Charles T. Scott

3.1 Overview of the FIA Plot Design ............................................ 28

 3.1.1 Motivation Behind the FIA Plot Design ............................. 30

 3.1.2 Field Protocols for Mapping Plots ..................................... 31

 3.1.3 Differences in Mapping Forest and Nonforest Plots .......... 32

3.2 Condition Classification Based on Direct Field 

 Observation .............................................................................. 33

 3.2.1 Discrete Variables That Trigger Recognition of a

 Unique Condition Class ..................................................... 33

 3.2.2 Ancillary Condition-Class Variables .................................. 33

3.3 Computed Attributes ................................................................ 34

 3.3.1 Computed Tree-Level Attributes ........................................ 34

 3.3.2 Condition Classification Based on Computed Attributes .... 35

 3.3.3 Calculations of Area in Each Condition Class ................... 38

3.4 Special Cases ........................................................................... 39

 3.4.1 Population Boundaries ....................................................... 39

 3.4.2 Slivers ................................................................................. 39

 3.4.3 Nonsampled Plots and Plot Replacement ........................... 41

3.5 Literature Cited ........................................................................ 42

4 Sample-Based Estimators Used by the Forest Inventory 

 and Analysis National Information Management System ....... 43

 Charles T. Scott, William A. Bechtold, Gregory A. Reams,

 William D. Smith, James A. Westfall, Mark H. Hansen,

 and Gretchen G. Moisen

4.1 Panels and Estimation .............................................................. 43

4.2 Phase 1 ..................................................................................... 44

 4.2.1 Satellite Classification and Known Strata Weights ............ 44

 4.2.2 Aerial Photography and Estimated Strata Weights ............ 45

 4.2.3 Combining Small Strata and Populations ........................... 46

4.3 Phase 2 ..................................................................................... 46

 4.3.1 Estimation of Area by Stratum by Domain 

 (Row and Column) Attributes ............................................ 47

 4.3.2 Estimation of Population Totals and Their Variances ......... 50

 4.3.3 Estimation of Other Attributes ........................................... 52

 4.3.4 Estimation of Ratios ........................................................... 55

 4.3.5 Computation of Sampling Errors ....................................... 59

 4.3.6 Components of Change ...................................................... 59

4.4 Expansion Factors ................................................................... 65
v

4.5 Phase 3 ..................................................................................... 66

 4.5.1 Estimation Procedures Used for FHM (Phase 3) Data ....... 66

 4.5.2 Combining with Phase 1 .................................................... 66

 4.5.3 Linking with Phase 2 .......................................................... 66

4.6 Literature Cited ........................................................................ 67

5 Combining Panels for Forest Inventory and Analysis 

 Estimation ..................................................................................... 69

 Paul L. Patterson and Gregory A. Reams

5.1 Single Panels vs. Multiple Panels ............................................ 69

5.2 Combining Panels .................................................................... 71

 5.2.1 The Moving Average Method ............................................. 71

 5.2.2 The Temporally Indifferent Method ................................... 73

5.3 Literature Cited ........................................................................ 73

6 Notation for Equations ................................................................ 75

6.1 Indices ...................................................................................... 75

6.2 Double Sampling ..................................................................... 76

6.3 Area Related ............................................................................ 76

6.4 Tree and Plot Related .............................................................. 77

7 Glossary of Terms ........................................................................ 79

8 Web-Based Supplementary Documentation ............................. 85
vi

Abstract

The Forest Inventory and Analysis (FIA) Program of the U.S. Department 

of Agriculture Forest Service is in the process of moving from a system of 

quasi-independent, regional, periodic inventories to an enhanced program 

featuring greater national consistency, annual measurement of a proportion 

of plots in each State, new reporting requirements, and integration with the 

ground sampling component of the Forest Health Monitoring Program. This 

documentation presents an overview of the conceptual changes, explains 

the three phases of FIA’s sampling design, describes the sampling frame 

and plot configuration, presents the estimators that form the basis of FIA’s 

National Information Management System (NIMS), and shows how annual 

data are combined for analysis. It also references a number of Web-based 

supplementary documents that provide greater detail about some of the more

obscure aspects of the sampling and estimation system, as well as examples 

of calculations for most of the common estimators produced by FIA.

Keywords: Annual inventory, FIA, forest health monitoring, forest 

inventory, plot design, sampling frame. 

Acknowledgments

The authors extend special thanks to the nine subject-matter specialists who 

reviewed this manuscript. Their valuable feedback helped us construct a 

solid foundation for this important national program and improved the usefulness of the associated documentation for the wide range of customers and 

stakeholders served by Forest Inventory and Analysis. We are particularly 

grateful to Alan R. Ek, University of Minnesota, Department of Forest 

Resources, who acted as associate editor and coordinated a double-blind 

review process.
vii

Executive Summary

The Forest Inventory and Analysis (FIA) Program of the U.S. Department 

of Agriculture Forest Service has begun a move from quasi-independent, 

regional, periodic inventories to the national Enhanced FIA Program, 

which features greater national consistency, a complete and systematic 

annual sample of each State, new reporting requirements, and integration 

with the ground sampling component of the Forest Health Monitoring 

Program. The transition to Enhanced FIA has both resulted from and caused 

conceptual changes in approaches to conducting forest inventories, which, 

in turn, have required corresponding technical changes. This document 

provides background on the conceptual changes and articulates technical 

aspects of the Enhanced FIA Program as a means of ensuring a common 

understanding and practice among the regional programs. It facilitates 

further development of the national core program, provides a defensible 

statistical basis for the program’s sampling and estimation components, and 

fosters greater credibility with users and stakeholders. Although this version 

of the documentation primarily is intended for an internal FIA audience, 

additional documentation of more general interest will be developed as the 

program matures.

The Enhanced FIA Program is conducted in three phases with objectives as 

follows: 

1. In Phase 1, FIA personnel stratify land areas to increase the precision of 

 estimates. 

2. In Phase 2, FIA field crews obtain observations and measurements of the 

 traditional FIA suite of variables. 

3. In Phase 3, FIA field crews obtain observations and measurements of 

 additional variables related to the health of forest ecosystems. 

These three phases provide the basic framework for the program’s structure.

The documentation emphasizes current, nationally consistent FIA practices 

where they exist. Examples include the glossary of terms, the plot configuration, sampling design and protocols, condition classes and mapped plots,

computed variables, and estimation procedures. The documentation also 

notes aspects of the FIA Program in which regional options are not only 

practiced but are deemed necessary (e.g., the regional approaches to stratification). Although it is beyond the scope of this documentation to provide 

detail on all regional differences, such differences and options have been 

noted where essential to the topics discussed. 
viii

After completion of the first draft of this manuscript, an external editor coordinated technical reviews by subject-matter and statistical specialists from 

the university, government, and forest industry sectors. The editor solicited 

nine reviewers who participated in a double-blind review process (i.e., the 

identities of the authors and reviewers were not known to each other). After 

several rounds of reviews and revisions, the editor gave written approval to 

release this documentation as a peer-reviewed publication.

Finally, documentation of the Enhanced FIA Program is an ongoing, dynamic

process. Additional documentation of other aspects of Enhanced FIA is 

currently planned. This includes such topics as Phase 3 estimation, modelbased updating procedures, data quality assurance and quality control, longterm program direction, and more detail regarding regional differences. 

Regional options and enhancements to the prescribed national procedures 

discussed herein are currently documented at the discretion of the regional 

programs. Although an important focus of the Enhanced FIA Program is 

national consistency, it is through regional enhancements that innovations 

and new technologies will be introduced. As these demonstrate their value, 

stabilize, and become generally accepted, many will be incorporated into 

Enhanced FIA, and additional documentation will ensue.
1

1 The Enhanced Forest Inventory 

and Analysis Program

Ronald E. McRoberts1

1.1 Purpose and Scope of the Documentation

The Agricultural Research, Extension, and Education Reform Act of 1998 

(Public Law 105–185), also known as the 1998 Farm Bill, prescribed 

conceptual changes in approaches to forest inventories conducted by the 

Forest Inventory and Analysis (FIA) Program of the U.S. Department 

of Agriculture (USDA) Forest Service. Realization of these conceptual 

changes required development and implementation of technical changes, 

some of which are substantial. The underlying purposes for documenting 

the conceptual and technical features of the resulting Enhanced FIA 

Program are fourfold: 

1. To ensure a common understanding and practice among the regional FIA 

 programs

2. To facilitate further development of the national core program, including 

 FIA’s National Information Management System (NIMS) 

3. To provide a defensible statistical basis for the sampling and estimation 

 components of the program

4. To promote credibility with users and stakeholders 

The primary intended audience for this version of the documentation is the 

national FIA Program itself. Nevertheless, our users and stakeholders will 

also find it useful for understanding FIA methods and, as the Enhanced FIA 

Program matures and as internal issues are resolved, documentation specifically intended for external use will be published. To the extent possible at 

the present time, this documentation addresses the full range of conceptual 

issues, technical details, and statistical techniques for sample-based 

estimation.

1

 Ronald E. McRoberts, Mathematical Statistician, USDA Forest Service, North Central 

Research Station, St. Paul, MN 55108.
2

1.2 Overview of the Enhanced Forest Inventory and 

 Analysis Program

1.2.1 Historical Perspective

Forest inventories conducted under the auspices of the national FIA Program

historically have been commodity oriented, with emphasis on estimating the 

area and volume of the Nation’s timber supply. These statewide inventories 

typically have been conducted on productive timberland2 but not on other 

forest lands, not on reserved forest lands, and depending on the region, 

not on National Forest System lands. The design and implementation of FIA

inventories have been the responsibility of the five regional FIA Programs 

that administer them; and plot configurations, sampling designs, measurement protocols, analytical techniques, and reporting standards frequently 

have been tailored to regional requirements. Those inventories were characterized as periodic surveys because field crews were concentrated in one

or two States until the measurement of all plots was completed. States were 

selected for inventories on a rotating basis with time intervals between inventories for the same State ranging from 6 to 18 years (Gillespie 1999). The plot

measurement component of periodic inventories for any given State required

from 1 to 4 years, depending on the magnitude of the resource, the number 

of plots to be measured, and the number of field crews. The analysis component required an additional 2 to 5 years, during which time FIA staff 

analyzed the data and then compiled, published, and distributed reports. 

The timeliness, quality, and usefulness of estimates obtained from periodic 

inventories came under scrutiny in the 1980s and 1990s. Estimates obtained 

from these inventories were degraded by the effects of conducting them 

over multiple years, while the bias and uncertainty of estimates increased 

over time due to factors such as change in land use, tree growth, tree 

mortality, and removals between inventories. The periodic nature of these 

inventories made consistent interstate estimation difficult, even within 

regions, while interregion estimation was even more difficult due to varying 

plot configurations, sampling designs, and measurement protocols. These 

factors, when compounded by the lack of measurements on all forested 

lands, caused national compilations to depend on a variety of ad hoc 

techniques. Finally, the environmental and forest ecosystem health interests 

of groups challenging the commodity focus of FIA inventories were difficult 

to address using only traditional FIA measurements. 

2

 The first use of a glossary term in each chapter is in bold face.
3

FIA clients recognized the deficiencies inherent in these regional, periodic 

inventories. They registered their dissatisfaction and proposed solutions. 

Concerns related to cycle length led to proposals to increase the sampling 

intensity, reduce cycle lengths, and conduct midcycle updates. Clients also 

advocated consistency across regions and measurement of all vegetation on 

all forested lands. Various solutions were proposed to resolve these issues, 

but most were expensive to implement and represented only a piecemeal 

approach to dealing with problems inherent in the periodic inventories.

1.2.2 Forest Health Monitoring

In response to user concerns regarding the health of forest ecosystems, the

Forest Health Monitoring (FHM) Program was established in 1990. It was

an independent, cooperative effort among multiple State and Federal 

Agencies focused on assessing and monitoring the health and sustainability 

of the Nation’s forests using nationally standardized inventory procedures. 

The FHM Program consists of four primary activities. In the first—detection 

monitoring—field crews measure selected biotic and abiotic features of 

forests called “indicators” during a baseline period. The same features are 

remeasured at regular intervals to identify changes associated with natural 

forest succession and ecosystem disturbances. In the second activity—

evaluation monitoring—teams of ecologists, entomologists, hydrologists, 

pathologists, silviculturists, and others conduct intensive field sampling and

provide combined interpretations when the causes of detected changes are

unknown. In the third activity—intensive site monitoring—long-term 

research is conducted on watershed-sized sites that have diverse forest 

types and biomes typical of those found in the United States. In the fourth 

activity—research on monitoring techniques—researchers focus on developing and refining indicator measurements to improve the efficiency and

reliability of data collection and analysis. Together, these four FHM activities permit predictions of where and how future ecosystems might change 

under various environmental and management conditions. 

1.2.3 Annual Inventories

The impetus for the transition from regional, periodic inventories to 

nationally consistent, annual inventories came from two pilot studies in the

1990s and the reports of two Blue Ribbon Panels. In 1990, the North Central

Research Station began a pilot study with the objective of producing annual,

statewide inventory estimates that were no more costly and no less precise

than those obtained from periodic inventories in the year of their completion.

The 1992 report of the first Blue Ribbon Panel (BRP-I) recommended a 

nationally consistent approach to the collection, analysis, and reporting of

forest inventory data (American Forest Council 1992). The first step toward
4

this goal was a directive issued in 1995 by the USDA Forest Service, Deputy

Chief for Research instructing FIA to adopt the FHM plot configuration as a 

national replacement for the various regional plot configurations.3 In the

mid-1990s, the Southern Research Station began a second pilot study that 

featured annual inventories augmented with State support and based on a

5-year measurement cycle. The cumulative effect of the North Central and

Southern Research Stations’ pilot studies, the report of BRP-I, and the report 

of the 1998 Blue Ribbon Panel (American Forest and Paper Association 

1998), which affirmed the recommendations of BRP-I, was passage of the 

1998 Farm Bill [Agricultural Research, Extension, and Education Reform 

Act of 1998 (Public Law 105–185)]. 

The 1998 Farm Bill directed the Secretary of Agriculture to produce a strategic

plan for forest inventory with several features—an annual forest inventory

program; State reports every 5 years; a set of prescribed core variables, 

standards, and definitions; and integration of the ground sampling components

of the FIA and FHM Programs. This legislation, together with the coalescing

of the two pilot studies and national cooperation in standardizing inventories,

resulted in an annual forest inventory program, designated Enhanced FIA, 

with identifiably new features: 

1. A nationally consistent plot with four fixed-radius subplots

2. A systematic national sampling design for all lands

3. A complete, systematic, annual sample of each State

4. Reporting of data or data summaries within 6 months of completion of 

 designated proportions of plot measurements

5. Provision for several estimators to combine data from multiple panels, 

 some of which incorporate updating techniques

6. State inventory reports every 5 years 

7. Integration of the FIA field component and the ground sampling 

 component of the FHM detection monitoring activity 

Implementation of the last feature was facilitated by the 1995 USDA Forest 

Service directive instructing FIA to adopt the FHM plot configuration, and 

takes advantage of efficiencies gained by consolidating the field components 

of these two forest inventories. 

3 Sesco, Jerry A. 1995. Letter dated March 30 to Station Directors and WO Research Staff 

Directors. On file with: FIA National Program Leader, USDA Forest Service, Forest Inventory 

and Analysis, Rosslyn Plaza, 1620 North Kent Street, Arlington, VA 22209.
5

1.2.4 National Consistency

Concern among FIA clients regarding the lack of program consistency was

comparable to their concerns about the lack of timely FIA estimates and the

need to monitor forest ecosystem health and sustainability. As a means of

emphasizing national consistency in the Enhanced FIA Program, its technical

aspects are described using the Ends-Ways-Means strategic planning model.

Ends are the criteria that must be satisfied for the program to be characterized

as nationally consistent; Ways are the procedures and protocols that lead to 

achieving the Ends; and Means are the resources that are committed to the 

effort. The following discussion focuses on selecting a set of Ends that lead 

to national consistency, and identifying technical Ways to achieve them. 

Some Ends require prescribed Ways to achieve them, while others require 

flexibility in the selection of Ways. The guiding principle is that Ways are to 

be facilitating, not inhibiting, and that they are to be prescribed only where

necessary to achieve the Ends. Thus, the Ends leading to national consistency

are achieved, while creativity and innovation to further enhance the FIA 

Program are encouraged.

The Enhanced FIA Program is described in terms of six Ends:

End 1: a standard set of variables with nationally consistent meanings and 

 measurements

End 2: field inventories of all forested lands

End 3: consistent estimation:

 (a) the ability to obtain estimates for areas larger than the single 

 county level for which FIA usually reports estimates

 (b) the ability to obtain data and estimates for user-defined 

 applications and areas of interest

End 4: national precision guidelines

End 5: consistent reporting and data distribution

End 6: credibility with users and stakeholders

These Ends describe the major foci of the Enhanced FIA Program and 

provide direction for methodological research.

To ensure that the 6 Ends are achieved, 10 Ways have been prescribed. To 

achieve End 1, Ways 1 and 2 have been prescribed:

Way 1: a national set of prescribed core variables with a national field 

 manual that prescribes measurement procedures and protocols for 

 each variable
6

Way 2: a national plot configuration

Core variables are identified in the national field manual and are measured 

on each plot in every State by field crews from each of the five regional FIA 

programs. Additional enhanced prescribed core variables and nonprescribed optional variables may be included, but no prescribed core variable may be deleted. Agreement on measurement protocols and procedures 

has required compromise among representatives of the five regional FIA 

programs, but that has led to a broad national consensus. 

To achieve End 2, Way 3 has been prescribed:

Way 3: a national sampling design

The national sampling design emerged as a result of its development and 

implementation in one regional FIA program and its subsequent acceptance 

and implementation in two others. Both the national sampling design and 

the national plot configuration represent broad consensus, and both are 

described in greater detail in chapters 2 and 3. 

Achieving the two components of End 3 is facilitated by the three Ways 

already described. However, to fully achieve End 3a, a fourth Way also has 

been prescribed:

Way 4: estimation using standardized formulae for sample-based estimators

End 3b was deemed necessary to accommodate the large number of nonFIA researchers, both within and outside the USDA Forest Service, who 

seek to use FIA data for their own applications and areas of interest. To 

achieve End 3b, two additional Ways have been prescribed:

Way 5: a national FIA database with core standards and user-friendly 

 public access

Way 6: a national information management system

Ways 5 and 6 are necessary to make user friendly FIA data available to the 

general public while also yielding estimates that are consistent with those 

presented by the FIA Program. In addition, Ways 5 and 6 greatly facilitate 

Way 4, in particular, and estimation within the FIA Program, in general.

Compliance with the national FIA Program’s precision guidelines (U.S. 

Department of Agriculture Forest Service 1970) associated with End 4 

requires flexibility in prescribing Ways. The current guidelines primarily 

relate to estimates of forest area and inventory volume, and are formulated 

quantitatively as:
7

where 

 the estimate of the attribute of interest

 the estimate of the variance of 

S = a scaling factor—one million acres for area estimates and one billion 

cubic feet for volume estimates

PREC = the target precision per S units, which is 0.03 for area estimates 

or 0.05 for volume estimates in the Eastern United States and 0.10 in the 

Western United States

Guidelines for the precision of estimates of other attributes may be 

considered on a case-by-case basis as the need arises. 

Budgetary constraints and natural variability among plots prohibit sample 

sizes sufficient to satisfy precision guidelines, unless the estimation process 

is enhanced using ancillary data. With FIA’s traditional sample-based estimation, enhancement has been achieved via stratified estimation using 

remotely sensed data as the basis for stratification. However, regional 

differences in species diversity, topography, forest management practices, 

and other factors may require qualitatively different approaches to stratification to achieve End 4. Thus, no Ways regarding stratification are prescribed 

other than that the stratifications should be statistically defensible and 

feasible for incorporation into NIMS. However, assuming that the historical 

levels of sampling variability and the benefits of stratified estimation would 

continue, the national sampling intensity in terms of number of plots per 

unit area was selected so that, on average across much of the Nation, 

compliance with the precision guidelines would be achieved. 

End 5 reflects FIA’s response to users and stakeholders who desire consistency and temporal compatibility in cross-State and cross-region estimates 

of prescribed core variables. 

Two Ways are prescribed to ensure that End 5 is achieved:

Way 7: a nationally consistent set of tables of estimates of prescribed core 

 variables

Way 8: publication of statewide tables of estimates of prescribed core 

 variables at 5-year intervals
8

Finally, achieving End 6, credibility with users, will require that the FIA 

Program not only develop and implement a nationally consistent program, 

but that technical details of the program be transparent and subject to stakeholder and public scrutiny. Thus, two additional Ways have been prescribed:

Way 9: the technical aspects of the FIA Program, including procedures, 

 protocols, and techniques, are documented

Way 10: the technical documentation is peer-reviewed and published for 

 general access

1.3 Overview of the Documentation

The documentation is divided into chapters, of which the following sections 

are brief overviews. 

1.3.1 Sampling Frame

Chapter 2 describes in detail the Enhanced FIA Program’s three phases in 

the context of sample-based estimation. Phase 1 is designed to produce 

stratifications of land area in the population of interest to reduce variance in

the estimates. It entails the use of ancillary data, including remotely sensed 

imagery in the form of aerial photography and/or satellite imagery, to stratify

the land area in a population of interest and to assign plots to strata. In

Phase 2, field crews visit permanent ground plots and measure the traditional suite of FIA variables. The Phase 2 sample is based on a national 

array of approximately 6,000-acre hexagons containing one permanent 

ground plot each. It is designated the Federal base sample. In Phase 3, field

crews measure additional variables related to the health of forest ecosystems.

The Phase 3 sample comprises a 1/16th subset of the Phase 2 plots, resulting 

in a sampling intensity of one plot per approximately 96,000 acres. Because 

Phase 3 plots (previously denoted FHM plots) are also Phase 2 plots, they 

include all measurements made on Phase 2 plots, plus measurement of the

FHM biotic and abiotic features associated with forest and ecosystem health.

The chapter 2 documentation describes the three phases in greater detail, as 

well as the genesis of the national sampling frame and its theoretical basis.

1.3.2 Plot Design

Chapter 3 describes the history, rationale, and configuration of Phase 2 and 

Phase 3 ground plots, as well as explanations of protocols used for mapping 

(or partitioning) plots by condition class. The basic FIA area and tree 

attributes are described, including those measured in the field and those 

computed from field measurements. Special situations that influence data 

collection and processing also are addressed.
9

1.3.3 Sample-Based Estimators

Chapter 4 focuses on the sample-based approach to estimation and describes 

the estimators used for area and forest attribute totals under assumptions for 

simple random estimation, stratified estimation, and double sampling for 

stratification. Variance estimators are provided so that sampling errors may 

be calculated for each cell in every output table. Estimators are derived to 

accommodate sample plots that contain multiple land uses and/or straddle 

population boundaries. Ratio-of-means estimators are described so that 

estimates may be computed on per-unit-area, per-tree, and per-stand bases. 

Methods are also described for estimating the components of change

between measurements such as growth, mortality, and removals.

1.3.4 Combining Data for Multiple Panels

Chapter 5 provides a brief overview of considerations when selecting techniques to obtain multiple panel estimates of forest attributes for the required 

5-year reports. Several strategies are noted including moving average estimation, a temporally indifferent method that ignores the multiyear nature of

the multiple panel data, and model-based updating. Special attention is given

to the moving average and the weighted moving average methods. As of the 

date of this publication, no national default estimator had been selected.

1.3.5 Notation and Glossary

Chapter 6 provides a reference for the consistent mathematical and statistical notation used throughout this documentation, and chapter 7 provides a 

comprehensive glossary of terms and expressions.

1.3.6 Web-Based Supplementary Documentation

The supplementary documents referenced in this manuscript are posted on the

Web site http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm.

The purpose of these documents is to provide details about the algorithms, 

equations, and other specifics of the national FIA Program that are too 

technical for the chapter discussions. These documents are posted on the 

Internet because they are dynamic. They are currently in various stages of 

completion, and periodically will be updated and revised to accommodate 

changes in protocol and demand for technical detail. More information may 

be added to this Web site in the future as unresolved and new issues are 

presented and addressed.
10

1.4 Literature Cited

American Forest Council. 1992. Report of the Blue Ribbon Panel on Forest Inventory and 

Analysis. Arlington, VA: American Forest Council. 14 p.

American Forest and Paper Association. 1998. Second report of the Blue Ribbon Panel on 

Forest Inventory and Analysis. Arlington, VA: American Forest and Paper Association. 17 p.

Gillespie, A.J.R. 1999. Rationale for a national annual forest inventory program. Journal of 

Forestry. 97: 16-20.

U.S. Department of Agriculture Forest Service. 1970. Operational procedures. First report of 

the Blue Ribbon Panel on forest inventory and analysis. Forest Service Handb. 4809.11. 

Washington, DC: 11: 1-1. Chapter 10.
11

2 The Forest Inventory and Analysis 

Sampling Frame

Gregory A. Reams, William D. Smith,

Mark H. Hansen, William A. Bechtold,

Francis A. Roesch, and Gretchen G. Moisen1

2.1 Overview of Forest Inventory and Analysis Sampling Design

2.1.1 Forest Inventory and Analysis Populations

For purposes of sampling and estimation, Forest Inventory and Analysis 

(FIA) subdivides the total land area of the United States into mutually 

exclusive populations2

 and subpopulations. Populations are usually defined 

by county boundaries or by public ownerships that may or may not cross 

county boundaries (e.g., national forests). In cases where the sample size 

for individual counties is insufficient, groups of counties may be combined 

into a super-county to form a single population with adequate sample size. 

Based on user request, counties occasionally are split into subpopulations 

to accommodate enumerated (known) acreages supplied by public agencies 

(e.g., National Forest System and The Bureau of Land Management). This is

done to ensure that FIA totals match the county-level acreages reported by

the requesting agencies. Each FIA population and subpopulation has a known

number of plots and a known area of land, obtained from the U.S. Census 

Bureau, from which population estimates are derived. Each is sampled and

processed as a separate entity, so estimates of grand totals and their variances for groups of populations and subpopulations are additive. For 

example, State-level estimates are obtained by totaling the estimates from 

all populations and subpopulations bounded by the State.

Note that FIA estimation is based on land area which excludes census water

(4.5 acres in size and at least 200 feet wide). Census water is thus subtracted 

1 Gregory A. Reams, FIA National Program Manager, USDA Forest Service, 1601 North Kent

Street, Arlington, VA 22209; William D. Smith, Research Quantitative Ecologist, USDA Forest

Service, Southern Research Station, Research Triangle Park, NC 27709; Mark H. Hansen, 

Research Forester, USDA Forest Service, North Central Research Station, St. Paul, MN 55108;

William A. Bechtold, Research Forester, USDA Forest Service, Southern Research Station, 

Asheville, NC 28802; Francis A. Roesch, Mathematical Statistician, USDA Forest Service, 

Southern Research Station, Asheville, NC 28802; and Gretchen G. Moisen, Mathematical 

Statistician, USDA Forest Service, Rocky Mountain Research Station, Ogden, UT 84401.

2

 First use of a glossary term in each chapter is in bold face.
12

from the total area of land and water at the beginning of the estimation 

process. We anticipate that estimation eventually will be based on total area, 

including census water, when precise digitized census water boundaries 

become available from the U.S. Census Bureau. The capacity to tabulate 

the area of census water from digitized data will improve FIA’s ability to 

generate forest statistics for user-defined polygons.

FIA engages in three types of sampling—Phase 1, Phase 2, and Phase 3.

All three types are performed for each population or subpopulation of 

interest. The sample points associated with each phase are subsets of the 

previous phase, but from the descriptions that follow it should be clear that 

this is not intended to be an application of classical three-phase sampling.

2.1.2 Phase 1

Phase 1 is designed to reduce variance through stratification. Although the 

details have differed among FIA units, all have used double sampling for 

stratification since aerial photography became widespread in the 1950s. 

For a given population of interest, a supplemental grid of Phase 1 sample 

points (i.e., photo points or satellite pixels) is superimposed over the Phase 

2 sample points, such that Phase 2 can be viewed as a subset of Phase 1.

All sample points, both Phase 1 and 2, are then assigned to strata based on 

their classification from remote-sensing imagery. 

The remote sensing medium selected to accomplish Phase 1 is left to the 

discretion of the FIA regions, but satellite imagery is replacing aerial 

photography as discussed in section 2.1.2.1. The number of photo points 

or pixels classified and the frequency of Phase 1 sampling are regional 

decisions. The number of strata, if any, and the definitions of these strata 

also are left to the discretion of the regions, but most recognize a minimum 

of two—forest and nonforest. Factors influencing the details of Phase 

1 stratification include the homogeneity of the population; the timing, 

availability, and cost of remote imagery; and the availability of personnel 

available to perform the work. Nationally prescribed core methodology 

related to Phase 1 is limited to:

• If available, acquisition of new imagery at least once for each new cycle 

 of panel measurements (e.g., every 5 years for States on a 5-year panel 

 system)

• Application of the double sampling for stratification estimation techniques

 described in chapter 4 

The only difference in estimation techniques associated with the details of 

Phase 1 sampling is whether the strata weights are treated as estimated or 
13

known. With wall-to-wall satellite classification, strata weights are known. 

When photo points or satellite pixels are sampled, strata weights are treated 

as estimated. Even when wall-to-wall imagery is available, pixels are often 

sampled to ease the computational burdens of working with high-resolution 

imagery and multiple data layers. FIA units thus can choose between two 

approaches to stratified sampling: 

1. The double sampling for stratification approach used when strata weights 

 are estimated (sec. 4.2.2)

2. The stratified estimation approach used when the weights are known 

 (sec. 4.2.1)

The choice is largely based on local efficiencies, but most units are moving 

toward the latter as satellite imagery replaces aerial photography. 

2.1.2.1 Aerial Photography vs. Satellite Imagery

Since the 1950s and prior to satellite imagery, FIA used aerial photography 

to assign plots to strata at Phase 1 and in some cases to estimate forest area

(Bickford 1952). The intensity of photo plots has varied over time and among

FIA regions, ranging from one photo point per 230 acres in the South and

Northeast, to one point per 248 acres in the Rocky Mountain region (assuming

that each photo point has a radius of 50 feet). Photo points usually were 

established by overlaying a systematic grid on 1:40,000 black-and-white 

aerial photos, although other scales (e.g., 1:20,000) and media (e.g., color 

infrared) also have been used. Decisions regarding scale and media have been

based on availability, timing, price, and coverage. A good historical overview of FIA Phase 1 sampling is provided by Frayer and Furnival (1999). 

All FIA units have begun replacing photo-point classifications with satellitebased (pixel) classifications of land use. The primary source of FIA Phase 

1 satellite imagery is the Landsat Thematic Mapper (TM) series. The TM 

sensor has a repeat cycle of 16 days and a swath width of 115 miles. This 

multispectral sensor has 6 nonthermal bands—three in the visible, one in the 

near infrared, and two in the midinfrared, all with 100-foot resolution. TM is 

the remote sensing platform of choice due to:

• Historic and planned continuity of wall-to-wall land cover classifications 

• Moderate spatial and spectral resolution of the sensor

• A scale of resolution appropriate for matching ground-truth units to 

 pixels for the computation of standard error estimates 

Because TM satellite imagery has been used more often and with more 

success for forest assessments than any other satellite sensor, there is a 
14

vast body of literature on classification algorithms using various analytical 

approaches including unsupervised, supervised, and various hybrid classification approaches. From this work it is known that land cover classification

accuracies > 80 percent are difficult to achieve with satellite imagery, which 

is notably less than the 95-percent accuracies attained by experienced FIA 

photo interpreters. This difference in accuracy should not be disregarded 

because the gap is even wider when classification is attempted beyond forest 

and nonforest cover types. It also means that aerial photography will remain 

a useful tool even after the transition to satellite imagery is complete. 

Although photo classification is demonstrably more accurate than satellite 

classification on a point-by-point basis, satellite classification has several 

distinct advantages when compared to aerial photos (Wayman and others 

2001, Wynne and others 1999):

• Satellite classification accuracy is expected to improve as classification 

 algorithms and ancillary ground-truth data improve.

• The gain in precision from 80 percent accuracy with wall-to-wall satellite 

 coverage offsets the 95 percent accuracy attained from a comparatively 

 small sample of photo points. 

• Satellite-derived thematic maps usually are generated from objective and 

 consistent processes (although some human interpretation is needed to 

 label classified cover types and other land features). 

• Satellite imagery provides an opportunity for more frequent updates.

• Spectral change detection is relatively easy and particularly useful when 

 analyzing change associated with timber removals, as well as catastrophic 

 disturbances.

• Spatially explicit enumerations of the entire landscape (i.e., maps) can be 

 automated.

The FIA Program has national precision standards of 3 percent per million 

acres of timberland and 5 percent per billion cubic feet of growing-stock 

volume in the Eastern United States. Recent Phase 1 applications using TMbased classifications for the National Land Cover Data (NLCD) indicate that

FIA can come very close to meeting the precision standards (Hansen 2001). 

With a forest/nonforest stratification based on the most recent NLCD, 

the FIA North Central Research Station region produced sampling errors 

ranging from 2.83 to 3.71 percent per million acres of timberlands for four 

States (Indiana, Iowa, Minnesota, and Missouri). For these same States, 

sampling errors ranged from 6.03 to 6.73 percent per billion cubic feet of 

growing-stock volume.
15

Improved Phase 1 techniques offer an efficient opportunity to meet or 

exceed the stated precision goals, and the FIA Program plans to continue 

investigating alternative methods for improved stratification. TM image 

classification can be improved by auxiliary information from other sources 

(see Web page http://www.fs.fed.us/ne/rsb). Potentially useful auxiliary 

information currently under study includes the Gap Analysis Program, the 

Moderate Resolution Imaging Spectroradiometer, the Natural Resource 

Information System, topographic and ecological data layers, and highresolution low-altitude photography and satellite images. Use of highresolution imagery, with either visual or digital interpretation, may increase 

FIA’s ability to classify highly fragmented landscapes more precisely, but it 

may not be cost effective for more general applications. One alternative to 

stratification, which has been used with moderate success in Alaska, is use 

of regression methods to correlate plot data with individual pixel values. 

This allows pixels to be summed to provide estimates for the area of interest 

and is actively being investigated for small-area applications. However, 

because it can become quite cumbersome operationally, this technique is not 

ready for general application at the State level (Scott 1986).

2.1.3 Phase 2

Phase 2 relates to FIA’s network of permanent ground plots, which has 

a spatial sampling intensity of approximately one plot per 6,000 acres. 

Field crews install, monument, and measure ground plots if any portion 

of a plot contains a forest land use. Detailed field remeasurements of 

forest plots are repeated at regular intervals as long as the plots remain in 

forest (note that protocols for handling plots that cannot be sampled due 

to access restrictions are discussed in section 3.4.3). Nonforest plots are 

assigned a nonforest use code (nonforest land, census water, or noncensus 

water) and checked at each scheduled inventory for potential reversion 

to forest. Forest plots are installed if reversion occurs. Note that neither 

LANDSAT imagery nor stratification is used in the decision to visit a 

ground plot. Field crews physically visit all ground plots that have any 

chance of being forested. However, to avoid unnecessary costs in extensive 

areas of nonforest or in inner cities, some FIA regions use recent aerial 

photography to identify and assign land uses to plots that obviously have 

no chance of being forested. Phase 2 plots are assigned to strata based on 

their classification at Phase 1, which may or may not be consistent with the 

land use assigned by field crews at Phase 2. Discrepancies can result from 

misclassification or from changes since the imagery was obtained, and are 

factored into the estimation process described in chapter 4. 

FIA’s ongoing remeasurement process is designed to accommodate changes 

in protocols and plot design over time. This is accomplished by remeasuring 
16

the previous plot installation at each new inventory. For example, as FIA

moves from horizontal point samples to fixed-area mapped plots, the program is being careful to preserve change estimates that span the transition 

period. To complete this calibration over time, the horizontal point samples 

are remeasured for change estimates when new mapped plots are installed. 

The mapped plots will be remeasured at future visits. 

To be classified as forest, an area must be at least 10 percent stocked with 

tree species, at least 1 acre in size, and at least 120 feet wide. Stocking

protocols are further discussed in section 3.3.2.2.1, as well as the supplementary document “National Algorithms for Stocking Class, Stand Size 

Class, and Forest Type” at http://srsfia2.fs.fed.us/publicweb/statistics_band/

stat_documents.htm. Previously forested land that is not stocked, and which 

has not been developed to another land use, is still considered forest (e.g., 

clearcuts). Land that meets the minimum requirements for forest, but is 

developed for a nonforest land use, is considered nonforest (e.g., city parks 

or campgrounds). 

Discussions are underway within the FIA Program about presenting both 

use and cover estimates of land area. Researchers involved with remote 

sensing have been exploring the development of landcover estimates based 

on percent tree-crown cover, but this work has yet to be used operationally. 

The most significant impediment to estimating attributes of interest by 

cover class is the cost of increasing the scope of FIA such that field crews 

are required to measure trees and detailed area attributes on land that is 

simply classified as nonforest under current protocols.

2.1.4 Phase 3

Phase 3 plots include all of the features of Phase 2, plus additional measurements such as tree-crown assessment, soil sampling, lichen communities, 

understory vegetation structure, ozone bioindicators, and down woody 

material. Every 16th Phase 2 plot is also a Phase 3 plot, so Phase 3 sample 

intensity is approximately one plot per 96,000 acres. All Phase 3 plots are 

combined with Phase 2 plots for Phase 2-based estimations of attributes 

common to both plot types (i.e., double sampling for stratification applies). 

Attributes unique to Phase 3 are estimated directly from the Phase 3 subset. 

Use of Phase 1 stratification and Phase 2 samples to enhance the estimation 

of attributes unique to Phase 3 is currently being studied. Because Phase 

3 is a subsample of Phase 2, the use of double sampling with regression is 

being considered for estimating some Phase 3 attributes. Detailed estimation 

procedures for attributes specific to Phase 3 will be provided in future 

documentation.
17

A summary of the general attributes associated with Phases 1, 2, and 3 is 

provided in table 2.1.

Table 2.1—Summary of general attributes associated with FIA Phase 1, 

Phase 2, and Phase 3 sampling

Attribute Phase 1 Phase 2 Phase 3

Sample type Photo point or Ground plot, subset Ground plot, subset 

satellite pixel of Phase 1 of Phase 2

Sample Point or pixel Cluster of four Same as Phase 2a

configuration 1/300-acre microplots, four 1/24-

acre subplots, and 

optional four 1/4-

acre macroplots

Purpose Stratificationb

 of the Samples FIA tradi- Samples FIA tradilandscape for the tional attributes of tional attributes of

purpose of variance interest, primarily interest,c plus addireduction related to tree tional attributes

species of all sizes associated with

forest health

Tesselation Supplemental Systematic national Systematic national 

method regional grid super- hexagonal cell grid hexagonal cell grid

imposed over the (subset of Phase 2

population of grid)

interestd

Base-grid At the discretion One plot per every One plot per every

intensity of each FIA unit 6,000-acre hexa- 1/16 6,000-acre hexagonal cell gonal cell (i.e., one 

per 96,000 acres)

FIA = Forest Inventory and Analysis.

a

 Note that additional sample designs associated with forest health indicators (to be described 

in a future document) are superimposed over the Phase 2 sample configuration on Phase 3 

plots. 

b

 Most FIA units recognize a minimum of two strata—forest and nonforest. Census water is 

currently subtracted from the total area prior to any stratification or estimation.

c

 Phase 3 plots also double as Phase 2 plots for estimation of attributes associated with Phase 2. 

Phase 3 plots are unique only when used to estimate attributes unique to Phase 3.

d

 Regional Phase 1 grids are systematic grids of varying density (up to wall-to-wall) that are not 

necessarily linked to the national hexagonal grid. The only prescribed requirement is that Phase 

2 plot centers must be a subset of the Phase 1 points.
18

2.2 Development of the Phase 2/Phase 3 Sampling Frame

With passage of the 1998 Farm Bill [The Agricultural Research, Extension, 

and Education Reform Act of 1998 (Public Law 105–185)], Congress directed

major changes in the way FIA conducts inventories. This legislation prescribes an annual inventory where a proportion of plots in each State must 

be measured every year. The switch from a variety of regional periodic 

surveys to a nationally standardized annual inventory required FIA to implement a new sampling frame. The 1998 law also precipitated the integration 

of Forest Health Monitoring (FHM) with FIA. When the two programs 

merged, FHM had already implemented a national sampling frame with a 

plot network that was systematically divided into panels measured on an 

annual basis. A national sampling grid was viewed as a more convenient 

and consistent method for tessellating the landscape and populating the 

sample frame than the county-by-county approach previously used by most 

FIA units; especially because county boundaries occasionally change, and 

counties may be divided into different subpopulations at different times. 

When the two programs integrated it was decided to build on the existing 

FHM sampling frame, where the FHM panels were redefined as subpanels 

of the larger FIA plot network.

2.2.1 Hexagonal Sampling Frame

The U.S. Environmental Protection Agency’s Environmental Monitoring 

and Assessment Program originally developed the sampling frame used by 

FHM (Overton and others 1990, White and others 1992). This framework is 

actually based on a triangular grid, but the cells surrounding each point on 

a triangular grid form a hexagonal shape, so the sampling frame can also be 

viewed as a network of hexagonal cells. The hexagonal frame was projected 

onto the landscape by centering a large base hexagon over the continental 

United States (fig. 2.1). Similar hexagons were then extended from the base

hexagon to tessellate the planet. The result is described as a truncated icosahedron (White and others 1992) made up of 20 hexagon faces and 12 pentagon faces, which give the framework a “soccer ball” appearance (fig. 2.2).

To achieve the desired sample intensity for FHM, the base hexagon was then

subdivided into approximately 28,000 smaller hexagons with centers about 

17 miles apart. To avoid alignment with property boundaries that follow the

public land survey system, the hexagon configuration was randomly offset

from cardinal directions. To accommodate the sampling intensity and 

frequency desired by FIA, the hexagonal sampling frame was further 

modified as described in the next section 2.2.2.
19

Figure 2.2—Truncated icosahedron made up of 20 hexagon faces 

and 12 pentagon faces.

Figure 2.1—Base hexagon positioned over the conterminous United 

States.
20

2.2.2 Division of the Sampling Frame into Panels

The original FHM sampling frame conveniently accommodated 3-, 4-, 7-, 

9-, and 11-panel rotations, and multiples of these. In other words, the 

centers of the hexagons in a given panel formed a triangular pattern of 

equidistant points for these panel rotations. Figure 2.3 shows the triangular 

pattern of the four-panel system originally used for FHM. FIA requires a 

five-panel rotation to accommodate the measurement frequency mandated 

by the 1998 Farm Bill (20 percent per year). Unfortunately, the five-panel

system does not conform to an equidistant triangular configuration. 

To satisfy the desired sampling frequency for FIA, the program used a 

parallelogram-shaped pattern of hexagon centers to assign hexagons to 

panels (fig. 2.4). Although hexagons within a given panel are no longer 

2

4

1

3

4

2

1

3

4

2

4

1

4

1

3

2

3

1

1

4

4

2

3

2

3

2

3

4

3

5

4

5

5

1

2 4

3

2 4

1

2

4

3 5

1

1

2 4

3

2 4

3

5 2 4

5

1

5

5

Figure 2.3—Hexagon panel assignments illustrating the 

triangular pattern of a four-panel rotation.

Figure 2.4—Hexagon panel assignments illustrating the 

parallelogram pattern of a five-panel rotation.
21

equidistant, the parallelogram configuration provides the most uniform 

spatial arrangement possible for five-panel rotations, and multiples thereof. 

To satisfy the desired sampling intensity for FIA, FHM hexagons (approximately 160,000 acres) were subdivided in 27 smaller hexagons, resulting in 

hexagons of 5,937 acres. Figure 2.5 [from Brand and others (2000)] shows 

the spatial arrangement of the FIA hexagons relative to the original FHM 

hexagons. Figure 2.6 from Brand and others (2000) details the systematic 

coverage resulting from the panel assignment process. Again, note the 

parallelogram pattern that results from connecting the hexagons in any 

given panel.

FHM hexagon

FIA hexagon

3.27 mi

2 4 1 5 2 4 1 3 5

3 5 2 4 1 3 5 2 4

2 4 1 3 5 2 4 1 3 5

3 5 2 4 1 3 5 2 4

2 4 1 3 5 2 4 1 3 5

3 5 2 4 1 3 5 2 4

2 4 1 3 5 2 4 1 3 5

3 5 2 4 1 3 5 2 4

2 4 1 3 5 2 4 1 3 5

3 5 2 4 1 3 5 2 4

3

Figure 2.5—The FIA hexagon lattice (each black dot 

is at the center of an FHM hexagon).

Figure 2.6—Assignment of hexagons to one of five panels 

(shown by number).
22

2.2.3 Populating the Sampling Frame

Once the FIA hexagon frame was established, and hexagons were assigned 

to panels, one field plot was allotted to each hexagon as follows:

1. If the FIA hexagon contained an FHM plot, the existing FHM plot was 

 selected.

2. If not, then an existing FIA plot was selected.

3. If there were multiple FIA plots in the hexagon, the one closest to 

 hexagon center was selected and the others were abandoned.

4. If there were no FHM or FIA plots in the hexagon, a new sample 

 location was selected based on a random azimuth and distance from 

 hexagon center.

Because FHM plots originally were measured on a four-panel annual system,

some additional constraints were necessary when reassigning those plots to 

FIA panels. The following constraints resulted in minor perturbations of the 

parallelogram pattern, which were accepted so that historic measurement 

sequences and cohorts would remain unchanged:

1. No existing FHM plots were dropped.

2. FHM panels retained their historic measurement sequence, so colocated 

 Phase 2 and Phase 3 plots kept their preexisting FHM panel number (this 

 constraint was relaxed in States that had intensified FHM sampling 

 frames). 

3. The subset of Phase 3 plots was increased by 20 percent to accommodate 

 a fifth panel (to preserve the same annual sampling intensity established 

 under the four-panel FHM system).

Additional technical details related to panel assignments and population 

of the sampling frame are available in the supplementary document “The 

Hexagon/Panel System for Selecting FIA Plots Under an Annual Inventory” 

at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm.

2.2.4 Deviations from the Five-Panel Annual System

Panels and their associated plots, are scheduled for measurement based on 

their panel assignment. Panels are measured in sequence, one at a time. 

After all five panels have been completed the process is repeated. Ideally, 

exactly one panel per year would be completed in each State. However, the 

realities of budgetary constraints and logistical problems (e.g., forest fires) 

prevent some States from being inventoried at the prescribed rate of one 
23

panel per year. This situation can lead to “panel creep,” where the length of 

time to complete an inventory panel exceeds 1 year. This situation is most 

common in States that do not have additional resources to move from the 

federally financed 7-year cycle length to a 5-year cycle length. 

The concept of subpaneling the five-panel system is an alternative that will

be implemented if the measurement cycle becomes too protracted. A number

of subpaneling schemes could be developed to yield timelier inventory 

results and still retain uniform spatial coverage. For example, the FIA 

Western Pacific Northwest and Rocky Mountain regions are now funded to

collect data on a 10-year cycle. To accommodate the funding disparity, those

two regions are using the five-panel design where each panel has been divided

into two subpanels (each with complete spatial coverage); one subpanel is

scheduled for measurement each year. This is analogous to a 10-panel system.

2.2.5 Theoretical Basis for the FIA Sampling Frame 

It is clear from the previous discussion that the current FIA sampling frame

was forged from a variety of preexisting regional FIA and national FHM 

sampling frames. The goal of this approach was to maintain linkage with 

historical data to the extent possible (to preserve temporal consistency and

continuity for trend estimation), and to smooth the transition from the numerous variations of periodic systems. This approach relates to established 

sampling theory in a number of ways. In this section we give one general 

description of the joint distribution resulting from the marriage of various 

periodic designs with the common annual design. 

Sample plots are linked to a systematic triangular grid with time-interpenetrating panels. In a triangular grid, the cells surrounding each grid point 

are hexagonal and the grid is systematically divided into panels. Assuming 

one panel per year is measured for T consecutive years, then every T years 

the panel measurement sequence begins again. If panel 1 were measured 

in 1998, it also would be measured in 1998+T, 1998+2T, and so on. Panel 

2 would be measured in 1999, 1999+T, and 1999+2T. Of the numerous 

methods that might have been used to choose existing sample-point locations

for retention in the new design, the preferred option was to assign existing 

plots to the nearest triangular grid point (i.e., hexagon center). Extra plots 

in each grid cell (hexagon) were subsequently deleted, and new plots were 

randomly added to empty grid cells. Although the methodology does not 

produce a regular grid of sample points at a fine scale (i.e., grid-point intersections); it does at a coarse scale (i.e., grid-point cells). This feature has the 

advantage of masking the exact location of ground plots, which is required 

by law. 
24

Assume that the sample points from the entire collection of periodic inventories constitute a random sample from the infinite set of points contained 

within the boundaries of the United States. Panel assignments are made to

hexagons in a systematic fashion. Although panel assignments are not random

with respect to the triangular grid, they are random with respect to the underlying area-based population, due to the random establishment of the grid

combined with a scale-dependent assumption of randomly arranged population elements. The entire sampling frame is a three-dimensional cube—two 

dimensions incorporate the land area and the third represents time. 

Assume that random tessellation of the land area into identical, mutually 

exclusive hexagons (H) defines two samples:

S1 = A selection from the previous randomly chosen plot locations, where 

each chosen point is assigned to the hexagon from within which it was 

selected. The individual element of S1 for each hexagon j is denoted s1j.

S2 = A sample of random points resulting from a random tessellation of the 

land area into identical, mutually exclusive hexagons. A random point is 

chosen for the sample from the infinite set of points within each hexagon. 

The individual element of S2 for each hexagon j is denoted s2j.

Let:

Then a single sample point is chosen for each hexagon j such that

sj = I

j

s1j + (1 – I

j

)s2j, j = 1,..., N .

I

j randomly selects an element from 1 of 2 random samples. We also assume 

that H randomly assigns one of the T panels to each sample element.

Adding the dimension of time to the two dimensions that constitute the land

area of the United States produces a population which is a three-dimensional

cube. The primary sampling unit (PSU) is a series of line segments, linear 

in time. That is, when the time dimension is collapsed down onto the area 

dimensions, each series of line segments collectively appear as a single 

point on the area. When the area dimensions are collapsed down to the time 

dimension, each line segment within a series is of an approximate length 

of 1 day. Individual segments occur every T + 1 years within each series. 

 iff a previous sample point was selected from within hexagon j

 otherwise
25

Within a sufficiently small segment of time, all points within the land area 

dimensions of the volume common to each area segment created by the 

overlapping inclusion areas of all possible subsets of trees occurring on the

land area [in the sense of Roesch and others (1993)] could be viewed as a

temporally specific sampling unit. However, because these segments change 

over time, the PSU appears as a point in the temporally specific land area 

dimensions of the cube. That is, if we slice the population into, say, annual 

volumes, such that land area constitutes the base and time constitutes the

height, and then view the annual subpopulation from the top, we’ll observe

a set of 1/T points on the land area base. Each point exists within a temporally

specific inclusion area for a specific subset of trees. The temporal slices 

actually could be of any height; however, the thinner the slice, the smaller 

the sample per land area of interest. The wider the slice, the fuzzier the segment boundaries, as the subsets of trees change. For most of FIA’s purposes, 

annual slices will constitute the minimum height that forms a reasonable 

compromise between temporal specificity and land-area generality.

The plot measurements provide support to the point (line) from which they

were chosen. The plot measurements for an individual sample point (sample 

line) are multiplied by the inverse of the land area (land area/temporal volume)

upon which they were based, resulting in a value per acre for each sample

point (a value per acre per temporal unit for each sample line). The collection of sample points per area of interest (sample lines per area/temporal 

volume) contributes to the estimates for that area (area/time volume). The 

sampling units have known inclusion probabilities, which are used in the 

estimation equations.

This discussion supports the detailed estimation procedures described in 

chapter 4, which assumes that the FIA systematic sample for Phase 2 and 3

can be treated as a simple random sample. The systematic coverage provided by the hexagonal grid eliminates the clumping of samples and loss of 

precision that would occur with a purely random assignment of plots. The 

use of the hexagonal grid also increases the chances of sampling infrequent 

forest types. Given that plot locations are randomly assigned within hexagons,

the chance of the sample network coinciding with a systematic land feature

or spatially periodic phenomenon is greatly reduced. Research on the periodicity concern indicates that the hypothetical has little chance of occurring 

(Milne 1959). Cochran (1977) provides the following justification for the 

use of simple random-sample-based estimates for systematic samples: 

Consider all N! finite populations which are formed by the N! permutations and any set of numbers y1

, y2,..., yN. Then, on the average over 

these finite populations, E(Vsy)=Vran. Note that Vran is the same for all 

permutations.
26

Madow and Madow (1944) state that if the order of the items in a specific 

population can be regarded as drawn at random from the N! permutations, 

systematic sampling is (on average) equivalent to simple random sampling.

2.3 Literature Cited

Bickford, C.A. 1952. The sampling design used in the forest survey of the Northeast. Journal of 

Forestry. 50(4): 290-293.

Brand, G.J.; Nelson, M.D.; Wendt, D.G.; Nimerfro, K.K. 2000. The hexagon/panel system for 

selecting FIA plots under an annual forest inventory. In: McRoberts, R.E.; Reams, G.A.; Van 

Deusen, P.C., eds. Proceedings of the first annual forest inventory and analysis symposium. 

Gen. Tech. Rep. NC–213. St. Paul, MN: U.S. Department of Agriculture, Forest Service, 

North Central Research Station: 8-13.

Cochran, W.G. 1977. Sampling techniques. 3rd ed. New York: John Wiley. 428 p.

Frayer, W.E.; Furnival, G. M. 1999. Forest survey sampling designs: a history. Journal of 

Forestry. 97(12): 4-10.

Hansen, M.H. 2001. Remote sensing precision requirements for FIA estimation. In: Reams, 

G.A.; McRoberts, R.E.; Van Deusen, P.C., eds. Proceedings of the second annual forest 

inventory and analysis symposium. Gen. Tech. Rep. SRS–47. Asheville, NC: U.S. 

Department of Agriculture Forest Service, Southern Research Station: 43-51.

Madow, W.G.; Madow, L.H. 1944. On the theory of systematic sampling. Annals of 

Mathematical Statistics. 15: 1-24.

Milne, A. 1959. The centric systematic area sample treated as a random sample. Biometrics. 

15: 270-297. 

Overton, W.S.; White, D.; Stevens, D.L. 1990. Design report for EMAP (environmental 

monitoring and assessment program). EPA/600/3–91/053. Washington, DC: U.S. 

Environmental Protection Agency, Office of Research and Development. 43 p.

Roesch, F.A., Jr.; Green, E.J.; Scott, C.T. 1993. An alternative view of forest sampling. Survey 

Methodology. 19(2): 199-204.

Scott, C.T. 1986. An evaluation of sampling with partial replacement. In: Oderwald, R.G.; 

Burkhart, H.E.; Burk, T.E., eds. Use of auxiliary information in natural resource inventories. 

SAF 86–01. Blacksburg, VA: Society of American Foresters: 74-79.

Wayman, J.P.; Wynne, R.H.; Scrivani, J.A.; Reams, G.A. 2001. Landsat TM-bases forest area 

estimation using iterative guided spectral class rejection. Photogrammetric Engineering and 

Remote Sensing. 67(10): 1155-1166.

White, D.; Kimerling, A.J.; Overton, W.S. 1992. Cartographic and geometric components 

of a global sampling design for environmental monitoring. Cartography and Geographic 

Information Systems. 19: 5-22.

Wynne, R.H.; Oderwald, R.G.; Reams, G.A.; Scrivani, J.A. 1999. Optical remote sensing for 

forest area estimation. Journal of Forestry. 98(5): 31-36.
27

3 The Forest Inventory and 

Analysis Plot Design

William A. Bechtold and Charles T. Scott1

This chapter describes the prescribed core plot2 design currently used by

Forest Inventory and Analysis (FIA) for Phase 2 and Phase 3 ground 

sampling. FIA ground plots relate to the sampling frame discussed in the 

previous chapter as follows:

• One plot has been randomly located within each 6,000-acre hexagon. 

• Each plot has been assigned to one of five panels as described in section 

 2.2.2.

• Each plot has been designated Phase 2 or Phase 3 based on the rules 

 outlined in section 2.2.3. 

• The center point of each plot constitutes the primary sampling unit 

 (PSU) described in section 2.2.5.

• The area and vegetation data gathered on each plot serve to support and 

 quantify the information associated with each PSU. 

The plot design characteristics, field protocols, and calculations discussed in 

this chapter are intended to provide additional background to the estimation 

procedures outlined in chapter 4; some explanation of the most important 

derived values produced by FIA; and discussion of sampling and estimation 

issues associated with the plot design. More detail is provided in the referenced supplementary documentation, and a complete description of all field 

measurements can be found in the FIA Phase 2 field guide3 available on the 

Web site http://fia.fs.fed.us/library.htm#manuals.

All of the measurements described herein likewise apply to Phase 3, because

Phase 3 plots are a subset of Phase 2. Additional detailed measurements 

associated with Phase 3 forest health “indicators” (e.g., tree crowns, soils, 

1

 William A. Bechtold, Research Forester, USDA Forest Service, Southern Research Station, 

Asheville, NC 28802; and Charles T. Scott, FIA Program Manager, USDA Forest Service, 

Northeastern Research Station, Newtown Square, PA 19073. 

2

 First use of a glossary term in each chapter is in bold face.

3

 U.S. Department of Agriculture Forest Service. 2004. Forest inventory and analysis national 

core field guide: field data collection procedures for phase 2 plots, Version 2.0. [Not paged]. 

Vol. 1. Intern. Rep. On file with: USDA Forest Service, Forest Inventory and Analysis, Rosslyn 

Plaza, 1620 North Kent Street, Arlington, VA 22209. 
28

lichens, downed woody material, and understory vegetation) are described 

in the Phase 3 field guides, also posted on the above Web site. Some of these 

indicators have specialized plot designs superimposed over the basic Phase 

2 plot design. Specialized designs unique to Phase 3 indicators are beyond 

the scope of this manuscript but are covered in detail in the field guides. 

Separate manuscripts that document specialized Phase 3 plot designs and 

associated estimation procedures have been written and are currently in 

review.

The prescribed core plot design originated with the Forest Health Monitoring

(FHM) Program in 1990 and was selected as the national standard for FIA 

in 1995. Shortly thereafter, FIA began converting its various regional plot 

designs to the national standard. Most FIA units had been using 5- or 10-

point clusters of prism points arranged in a variety of patterns. While all FIA

units change to the national plot design, previously installed plot configurations are being remeasured to provide estimates of change (growth, removals,

and mortality). As earlier designs are remeasured to estimate change, the 

new design is simultaneously installed to yield current inventory estimates 

and to provide the basis for change estimation upon future remeasurement.

3.1 Overview of the FIA Plot Design

Phase 2 and Phase 3 ground plots are clusters of four points arranged such

that point 1 is central, with points 2 through 4 located 120 feet from point 1 

at azimuths of 0, 120, and 240 degrees (fig. 3.1). Each point in the cluster is 

surrounded by a 24-foot fixed-radius subplot where trees 5.0 inches diameter at breast height (d.b.h.) and larger are measured. All four subplots 

total approximately 1/6 acre. Each subplot contains a 6.8-foot fixed-radius 

microplot where saplings (1.0 to 4.9 inches d.b.h.) and seedlings are measured. All four microplots total approximately 1/75 acre. Microplots are 

offset from subplot centers (12.0 feet at an azimuth of 90 degrees) to minimize trampling. Each subplot is surrounded by a prescribed optional

58.9-feet fixed-radius macroplot, which can be useful for sampling rare

occurrences such as large trees (e.g., 40.0 inches d.b.h. and greater) or 

mortality. Macroplots encompass subplots, as well as the additional area

from 24.0 to 58.9 feet beyond the subplot circumference. All four macroplots

total approximately 1 acre. When used together, microplots, subplots, and 

macroplots constitute a tri-areal plot design for sampling trees in three 

different tree-diameter ranges. In regions where the optional macroplots are 

not used, the plot design is bi-areal. 

For attributes such as large trees that are always measured within subplots, 

whether or not macroplots are utilized, it is sometimes useful to describe 
29

the trees between 24.0 and 58.9 feet on tri-areal plots as being located in an 

annular ring. For example, this description avoids the need to redefine the 

range of diameters sampled on the subplot if the diameter threshold for

sampling large trees is changed. Theoretically, distinctions between macroplot trees in the annular and inner portions of the macroplot do not have any 

implications for the estimation procedures described in chapter 4.

In addition to the trees measured on FIA plots, data are also gathered about 

the area or setting in which the trees are located. Area classifications are 

particularly useful for partitioning the forest into meaningful categories 

(i.e., domains) for analysis. Some of these area attributes are measured

(e.g., percent slope), some are assigned by definition (e.g., ownership group), 

and some are computed from tree data (e.g., percent stocking).

To enable division of the forest into various domains of interest for analytical purposes, it is important that the tree data recorded on these plots are 

properly associated with the area classifications. To accomplish this, plots 

are mapped by condition class. Field crews assign an arbitrary number to

the first condition class encountered on a plot. This number is then defined 

by a series of predetermined discrete variables attached to it (i.e., land use,

Macroplot:

58.9 feet radius 

Subplot:

24.0 feet radius

Azimuth 1 - 2 = 360 ̊

Azimuth 1 - 3 = 120 ̊

Azimuth 1 - 4 = 240 ̊

Microplot:

6.8 feet radius center

is 12 feet horizontal

@ 90 ̊ azimuth from

the subplot center

Distance between

subplot centers is

120 feet horizontal

2

1

4 3

Figure 3.1—FIA plot design.
30

forest type, stand size, regeneration status, tree density, stand origin, ownership group, and disturbance history). Additional conditions are identified if 

there is a distinct change in any of the condition-class variables on the plot. 

Further details are provided in section 3.2.

Sometimes a plot straddles two or more distinct condition classes. Boundaries of condition classes can bisect the subplots, or may occur between 

subplots. When they bisect a subplot, condition boundaries are mapped 

using two or three azimuths as described in section 3.1.2. Similarly, microplots and macroplots, if used, also are mapped. So for each ground plot, the 

microplot, subplot, and macroplot area in each condition class are known, 

as is the location and condition class of every tree tallied. Because FIA 

primarily is concerned with the classification and monitoring of forest land, 

no tree data are recorded for nonforest land uses.

At first glance, an unwieldy number of condition-class permutations seems 

likely at the regional scale, especially because condition classes from the

same dataset must be processed in different combinations from one inventory

summary table to the next depending on the domain of interest. However, 

most plots have only one or two condition classes and data summarizations 

are easily managed with indicator functions as described in chapter 4.

3.1.1 Motivation Behind the FIA Plot Design

FIA has historically used cluster plots, primarily because they reduce 

between-plot variance and, therefore, the total number of plots necessary to 

achieve a given accuracy standard (Scott 1993). The 4-point cluster was

chosen because experience with FHM and FIA pilot studies showed that on

average, crews can complete one 4-point cluster plot per day. It is conceivable that the number of points comprising the national standard may be 

revised in the future if the field workload changes such that a different 

number of cluster points is more efficient.

The mapped-plot feature of the design arose from the need to correctly 

match tree data with area classifications when plots straddle multiple conditions. Before the advent of mapped plots, some FIA units moved plots into 

a single, uniform condition. This generated a bias by altering the selection 

probabilities of trees, especially those near condition edges (Williams and 

others 1996). Other FIA units addressed the problem by prohibiting the 

movement of plots, but then blended area data from distinctly different 

conditions. Although unbiased for area and volume totals, this procedure 

resulted in domain misclassifications. For example, a plot that straddled 

a pure oak forest type and a pure pine forest type might be classified as a 

mixed oak/pine forest type.
31

In 1991, FIA project leaders and inventory specialists met with a panel of 

university and forest industry biometricians to discuss the problem and 

explore a variety of potential alternatives. A committee was subsequently 

appointed by the FIA project leaders to review the alternatives and recommend

a solution (Hahn and others 1995). The tri-areal, fixed-radius mapped design 

was ultimately selected because it solved both the bias and classification 

problems, it had the flexibility needed to satisfy a growing FIA customer 

base, and it permitted greater use of the data for such nontraditional 

purposes as forest health monitoring.

The tri-areal design is a departure from the polyareal plot sampling (pps) 

approach (Husch and others 1982) originally implemented by FIA in the 

early 1960s. The pps design is more efficient for sampling timber and estimating volume, but fixed-radius plots add versatility by preserving information about the spatial relationships among trees. Fixed-sized plots also are 

more compatible with mapped-plot designs, because the area that must be 

mapped is constant. In addition, pps sampling in conjunction with mapped 

plots often leads to a situation where the full range of tree sizes is not sampled in all conditions (Scott and Bechtold 1995). This has negative consequences that are difficult to correct when area attributes such as forest type 

and stand size are computed from the tree data. This problem rarely occurs 

with the tri-areal design and is much easier to manage when it does occur. 

3.1.2 Field Protocols for Mapping Plots

Field crews specify and define (if not previously defined) the condition class 

at each subplot center, as described in section 3.2. If a subplot straddles two 

or more conditions, they then specify the condition class that contrasts with 

the condition at subplot center. Standing at subplot center and facing the 

contrasting condition, they record the two azimuths where the conditionclass boundary crosses the subplot perimeter. A third azimuth (with a 

distance) is permissible if the boundary contains a sharp curve or a corner 

(fig. 3.2). All trees tallied are then assigned to the condition class in which 

they occur. Horizontal distance and azimuth to the center of each tree are 

recorded for remeasurement purposes and to establish spatial relationships 

among the sampled trees. Microplots (and macroplots, if used) are mapped 

in a similar fashion. It is not necessary to match boundaries at the edge 

of each plot type, so microplots, subplots, and macroplots all are mapped 

independently. Microplot, subplot, and macroplot areas in each condition 

class are computed from field measurements when the data are processed 

(sec. 3.3.3). 

Field crews are trained to recognize and map only those boundaries that are 

distinct and obvious. A variety of logic checks are programmed into field 
32

data recorders to check boundary data for errors and to verify that the tree

location and condition observations are consistent with condition-class and

boundary observations (Scott and Bechtold 1995). Most condition-class 

boundaries do not require mapping because they occur between subplots, or 

are indistinct. The frequency of mapped subplots depends on the homogeneity of the landscape, but is typically < 5 percent of the total number of 

subplots in a region. There is some concern over the level of detail to which 

plots are mapped, as well as the repeatability of boundary recognition and 

placement. We expect that mapping protocols will be evaluated as part of 

the FIA quality control program and adjusted as necessary.

3.1.3 Differences in Mapping Forest and Nonforest Plots

Any plot that intersects with a forest land use is designated as a forest plot. 

Otherwise, the plot is classified as nonforest, census water, noncensus 

water, or (if inaccessible) nonsampled. In order to reduce the field costs 

associated with nonforest plots, mapping is initiated only in the presence of 

accessible forest. For those plots that contain no accessible forest, only the 

condition status (e.g., nonforest, census water) at the center of subplot 1 is 

recorded, and that condition is assigned to the entire plot.

Similarly, forest plots may include individual subplots or macroplots where 

no accessible forest is present. There, only the condition status at subplot 

center is recorded and that condition is assigned to the entire subplot. Thus, 

when two or more condition classes occur within a subplot or macroplot, 

boundaries between them are mapped only when one or both conditions are

classified as accessible forest. Boundaries between adjacent nonforest, census

water, and nonsampled conditions are mapped only on subplots containing 

some accessible forest.

Left azimuth

Corner azimuth 

Right azimuth

Figure 3.2—Using azimuths to reference a boundary to 

a subplot center.
33

3.2 Condition Classification Based on Direct Field Observation

Some condition-class variables trigger the mandatory identification of a 

distinct condition class in the field; others are ancillary, recorded only after a 

new condition class is recognized. Both mandatory and ancillary conditionclass variables typically are used to specify the domains of interest (e.g., a 

specific forest type and physiographic class) for which population estimates 

are generated for some attribute of interest (e.g., acres or volume). 

3.2.1 Discrete Variables That Trigger Recognition of a 

 Unique Condition Class 

There are seven discrete condition-class variables that require recognition 

of a unique condition in the field: condition status (land use), reserve status, 

owner group, regeneration status, tree density, forest type, and stand size. If 

one of these variables changes during plot measurement, a new condition is

defined and mapped if necessary. All are subjective field calls, some of 

which have guidelines and/or subsampling protocols to assist crews make a 

determination.

3.2.2 Ancillary Condition-Class Variables 

Ancillary condition-class variables are recorded in the field whenever unique

conditions are defined, but these variables do not trigger the recognition of 

new condition classes. These ancillary variables, obtained for all forested 

conditions include detailed owner class, private owner industrial status, 

artificial regeneration species, stand age, disturbance history, treatment 

history, and physiographic class.

3.2.2.1 Site Index Equations and Site Productivity Classes

In addition to ancillary condition-class variables, one or more site trees are 

measured in each unique condition class if qualified trees are available. If 

there is no reason to suspect a difference in site quality among condition 

classes, the same site tree(s) may be used for multiple conditions on a plot.

Site trees are used in determining site quality (i.e., the capacity of forest 

land to grow trees). A site index or site productivity class is thus associated

with each forested condition class. 

Site index is the average total height that the dominant and codominant trees 

in fully stocked, even-aged stands will obtain at key ages (Husch and others 

1982). Site productivity class, also known as site class or yield capacity, is

the maximum mean annual increment in cubic feet per acre that can be 

expected in fully stocked, natural, even-aged stands. Using regionally specific

equations, site index is computed as a function of the stand age and the 
34

average height of dominant or codominant trees as determined from the 

species, d.b.h., and total height of qualifying site trees on or near the plot. 

The following selection criteria are preferred for site trees: acceptable 

species, free of damage, dominant or codominant crown position, between 

15 and 120 years old, and at least 5.0 inches d.b.h. Resulting site-index 

values are then applied to site-productivity equations, or look-up tables, 

to determine the maximum mean annual increment for a given condition 

class. The site index and site class equation references used by the various 

FIA units are Brickel (1970), Clendenen (1977), and Edminster and others 

(1985) in the Interior West; Carmean and others (1989) in the North Central 

region;4

 Scott and Voorhis (1986) in the Northeast; Hanson and others 

(2002) in the Pacific Northwest; and Vissage and Greer in the South.5

3.3 Computed Attributes

In contrast to attributes that are observed and classified directly in the field

such as the condition-class variables in the previous sections; others are computed. Some attributes are computed at the tree level; some are computed at 

the condition-class or plot level. Computed attributes can be the measures 

upon which population estimates are based (e.g., acres, numbers of individuals, and volume) or they might be used to specify domains. An example 

of the latter would be placement of continuous variables into discrete classes

(e.g., volume-per-acre class) in order to estimate the area in each class. Some

attributes are computed in addition to being observed directly (i.e., forest 

type and stand size), so these have both field-assigned and computed values. 

3.3.1 Computed Tree-Level Attributes

Tree-level attributes are variables associated with the individual trees tallied 

on FIA ground plots. Expressions of tree volume and weight are among the

most basic statistics reported by FIA. The functions used to compute these 

values are typically statistical models developed or calibrated by State or

region. The most commonly reported volume and weight statistics are 

described in the supplementary document “FIA Volume Calculations” at 

http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm.

4

 For site-index to site-class conversions used by the North Central FIA unit see the 

supplementary document “Site Productivity Assignment for the North Central FIA unit” 

at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm. [Date accessed: 

December 9, 2004].

5

 Vissage, John S.; Greer, Travis R., Jr. Site class and site index – two estimates of site quality 

for the Southern Research Station Forest Inventory and Analysis unit. 12 p. Internal document. 

On file with: J.S. Vissage, USDA Forest Service, North Central Forest Experiment Station, 

1992 Folwell Ave., St. Paul, MN 55105.
35

3.3.2 Condition Classification Based on Computed Attributes 

Computed condition-class attributes further describe condition classes and 

the domains they represent. These attributes usually are derived from treelevel (vegetation) data. They can be discrete (e.g., forest type) or continuous 

(e.g., percent stocking).

3.3.2.1 Condition-Level Per-Acre Ratios

For modeling, or for summarizations of area data by discrete per-acre classes,

tree-level statistics (e.g., volume or basal area) can be used to compute 

per-acre ratios for individual plots, or per-acre ratios for specific condition 

classes within a plot. Condition-level ratios are computed by summing the 

tree-level attribute of interest (e.g., basal area) for all trees in the condition 

class and then dividing by the area of the plot in that condition:

 

(3.1)

where 

 the attribute of interest associated with tree t on microplot, 

subplot, or macroplot j covering condition k on plot i

 area used to observe the attribute of interest (microplot, subplot, 

or macroplot j) covering condition k on plot i

When combining subplot and microplot values, condition-level ratios are:

 

(3.2)

where 

= the attribute of interest associated with tree t on microplot j

covering condition k on plot i

 area of condition k on microplot j on plot i
36

3.3.2.2 Condition-Level Attributes Based on Stocking

Stocking class, stand-size class, and forest type are important conditionlevel attributes of interest that are calculated from the stocking contributions 

of individual trees. Although forest type and stand-size class are assigned 

in the field, these attributes are also computed from the tree tally when the 

field data are processed. The primary purpose of the field assignments is 

to delineate the unique condition classes encountered on each plot, and to 

supply an alternative value in case the area sampled is too small to derive a 

calculated value (i.e., calculated values for forest type and stand size are not 

produced for conditions that occupy < 25 percent of a plot). 

3.3.2.2.1 Algorithm to Assign Stocking Values to 

 Individual Trees

FIA uses a complicated algorithm to assign stocking percentages to individual trees, the details of which are provided in the supplementary document 

“National Algorithms for Stocking Class, Stand-Size Class, and Forest Type”

at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm. 

A simplified explanation follows to summarize the concept. The algorithm 

is based on “A-line” values, which are described by Gingrich (1967) as 

the number of trees per acre where average maximum stocking occurs in 

undisturbed stands. A-line values are negatively correlated with mean stand 

diameter, so they vary by diameter class as well as by species. The formula 

used to assign percent-stocking values to individual tally trees is:

 

(3.3)

where 

= the percent stocking assigned to tree t on (microplot, subplot, or 

macroplot j) covering condition k on plot i

Amax = the A-line value (trees per acre) associated with the species and 

diameter class of tree t

Once is assigned, division by adjusts the sample tree to a per-acre 

basis as shown in equation 3.1 when is substituted for . Equation 

3.1 then yields the total percent stocking for condition class k on plot i. For 

reporting purposes, condition-level stocking percentages commonly are 

grouped into the classes listed below:
37

Stocking class Class stocking range

 percent

Nonstocked 0 – < 10

Poorly stocked 10 – < 35

Moderately stocked 35 – < 60

Fully stocked 60 – < 100

Overstocked > 100

3.3.2.2.2 Stand-Size Class of Each Condition Class 

 Based on Stocking Algorithm

Tree-level stocking values also are used to categorize each condition by 

stand-size class. Each tree is first assigned to one of the following size 

classes based on its d.b.h.:

Stand-size class Stand-size class d.b.h. range

Seedling-sapling d.b.h. < 5.0 inches

Poletimber 5.0 inches < d.b.h. < 9.0 inches for softwoods

5.0 inches < d.b.h. < 11.0 inches for hardwoods

Sawtimber 9.0 inches < d.b.h. for softwoods

11.0 inches < d.b.h. for hardwoods

For a given condition class, stocking values for each tree (i.e., / ) are 

then summed across all trees in each stand-size class, and for all stand-size 

classes combined. Stand-size class is then assigned on the basis of which of 

the following stocking requirements is satisfied first:

Stand-size class Stocking requirement

Nonstocked Total stocking across all size classes < 10 percent

Seedling-sapling Seedling-sapling stocking > 50 percent of total stocking

Poletimber Poletimber stocking > sawtimber stocking

Sawtimber Poletimber stocking < sawtimber stocking
38

3.3.2.2.3 Forest Type of Each Condition Class Based on 

 Stocking Algorithm

Tree-level stocking values also are used to categorize each condition by 

forest type. The forest type assignment algorithm is quite complicated 

and still undergoing evaluation. Details are provided in the supplementary 

document “National Algorithms for Stocking Class, Stand-Size Class, and

Forest Type” at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_

documents.htm.

3.3.3 Calculations of Area in Each Condition Class

3.3.3.1 Mathematical Functions

Geometric and trigonometric functions can be used to calculate the area 

within each condition class as specified by Scott and Bechtold (1995).

3.3.3.2 Computer Simulation

Mathematical functions are useful for calculating areas at any one time, or

when performing data recorder logic checks; but they can be unwieldy for 

some applications—particularly the calculation of area change matrices

between inventories. Change matrices are necessary to quantify land-use

and condition-class change and to enable the partitioning of growth, removals,

and mortality by condition-class attributes at either the initial or terminal 

inventory of a measurement cycle. 

Change matrices are produced by overlaying a computer-generated map of

each subplot at time t (the previous inventory) with a similar map of the 

same subplot at time t+1 (the current inventory) (fig. 3.3). The area of the 

2 1

Time t

3

Time t + 1 

Intersection of

Time t and t + 1

2 - 3 1 - 3

2 2 - 2 1 - 2

Figure 3.3—Condition-class change matrix between two points in time.
39

intersection of all combinations of initial and terminal condition classes is

then calculated by using a computer to count dots on an electronically generated grid superimposed onto the intersected maps (Bechtold and others 

2003). For each cell of the intersection matrix, an observation is created 

that includes area percent, as well as all of the condition-class variables at 

both time t and time t+1. Subplot-level change data are then combined to 

produce plot-level matrices. Similar matrices are produced for microplots 

and macroplots. Note that it is not necessary for field crews to retain specific 

condition-class numbers over time. Numbers assigned to conditions remain 

arbitrary and are defined by the series of condition-class variables attached 

to them.

3.4 Special Cases

3.4.1 Population Boundaries

Plots (and portions of plots) are assigned to the population in which they are

located with indicator functions. Plots (and portions of plots) in the population of interest are then pooled to compute population estimates as described 

in chapter 4.

FIA sampling protocols recognize four types of population boundaries: 

(1) County, (2) National, (3) Federal agency (e.g., National Forest System, 

Bureau of Land Management), and (4) Census water.

All except county boundaries are currently mapped in the field. County 

boundaries are often not observable in the field, so whole plots are assigned 

to the county in which the center of subplot 1 is located. The inability to 

recognize population boundaries is not considered a problem because this 

implies that forest conditions are the same on both sides of the boundary, 

so no bias is introduced by sampling area outside the population of interest 

(i.e., the numerators and denominators of ratio-of-mean (ROM) estimators 

are incremented proportionately). 

3.4.2 Slivers

Slivers are defined as conditions that occupy less than one full subplot and

not encountered on any other subplots of a given plot. Slivers have the potential to create data processing or analysis issues, but in most cases the difficulties are minor. Alternative designs that do not require mapping create 

different or more serious problems such as classification anomalies and tree

selection probabilities that result in biased estimators (Hahn and others 1995). 
40

3.4.2.1 Slivers and Continuous Variables

One concern regarding slivers is that extremely small values in the denominators of per-acre ratios might unreasonably inflate estimates of per-acre values.

This is not a problem because the ROM estimators prescribed in chapter 4 

avoid the use of plot-level ratios. Slivers are pooled with similar conditions 

from other plots in the calculation of population means and totals. Although 

users of FIA data should be aware that a sliver may yield an unrealistic 

mean per-acre ratio for a rare domain in a small population, that possibility 

is usually of little consequence for standard FIA estimations. A domain 

that rarely would not be isolated in any standard output and, for reporting 

purposes, would be pooled with other domains.

Modeling is the only application where slivers and potentially inflated peracre values are isolated and used as individual observations. The modeler 

has a variety of options to deal with this problem, such as accepting the 

increased variance, pooling slivers with other conditions on the same plot, 

pooling slivers with similar conditions from different plots, or deleting 

slivers from the analysis. 

3.4.2.2 Slivers and Classification Variables Based on Tree Data

Slivers have the potential to inflate per-acre continuous variables that are 

computed for individual condition classes and then grouped into discrete 

classes for presentation in summary tables (e.g., area by volume-per-acre 

class). Such inflated values are rare and never stand alone. They are simply 

grouped into the highest class presented in the summary table. The most 

serious consequence is increased within-class variance caused by estimates 

from plots of different sizes.

Slivers can pose a slightly different problem for computed classification 

variables that are not per-acre estimates (e.g., forest type). When the tree 

tally on a given plot falls below a certain threshold, sufficient data may 

not be available to make an accurate classification. In such cases it is 

necessary to accept the computed classification at face value, revert to a 

subjective field classification, or engage in auxiliary sampling to obtain 

enough field data to compute the classification. The amount of field data 

required for reliable area classifications depends on the spatial scale of 

the vegetation upon which the classification is based (Williams and others 

2001). FIA is still evaluating the minimal areas required for classifications 

most commonly computed from tree data, forest type and stand size, so 

subjective field calls are available in addition to computed values for these. 

Preliminary analyses suggest that computed values for forest type and stand 

size are unreliable for conditions that occupy areas smaller than one full 

subplot.
41

A related problem involves attributes, usually indices, for which it is important to base the classification on plots of equal size (e.g., species diversity 

index). This is not a common application of FIA data and there is no prescribed method for handling this situation, but an analyst has the option to

adjust the index (e.g., species/area curves), delete partial plots from the analysis, or pool data from other conditions on the plot. It is noteworthy that pooling vegetation data is not an option for plots that contain nonforest land uses,

because FIA protocols ignore vegetation on land uses defined as nonforest. 

3.4.3 Nonsampled Plots and Plot Replacement

For various reasons, some plots (or portions of plots) within a given population cannot be sampled at the time they are scheduled for measurement. Such

plots are classified as “nonsampled”, a “nonsampled reason” is assigned, 

and no additional data are recorded. The magnitude of the problem has not

been fully evaluated, but nonsampled plots have the potential to be a significant factor in populations with relatively few forested plots (e.g., Plains 

States). FIA currently assumes these plots are randomly distributed, so nonsampled plots assume the strata means for all estimated values as discussed

in chapter 4, which ensures that estimates are produced for the entire population—not just the accessible portion. This approach presumes that conditions

on nonsampled plots are missed at random, which may not be valid under 

some circumstances. More precise methods of assigning attributes to nonsampled plots are being considered, and will be implemented if they yield 

better results. 

Access refusal by landowners is the most common reason for nonsampled 

plots, but plots occasionally are inaccessible due to hazardous situations 

encountered by field crews. To avoid altering the sampling network such 

that it becomes nonrepresentative of the population, inaccessible plots 

usually are not replaced. However, inaccessible plots may be replaced where 

nonreplacement causes inadequate sample size, or where there is evidence 

that replacement results in estimators that are less biased.

Less often, missing data or corrupted plot files are discovered after the field 

measurements for a panel are completed. When it is impractical to correct 

the situation, such plots also are classified as nonsampled. They are then 

resampled at their next scheduled measurement.

Field crews occasionally fail to relocate previously established plots. Upon

verification that a plot is truly lost, a replacement is installed at the approximate location of the original, and the lost plot is retired from the panel. 
42

3.5 Literature Cited

Bechtold, W.A.; Heravi, N.E.; Kinkenon, M.E. 2003. A simulation algorithm to approximate 

the area of mapped forest inventory plots. Gen. Tech. Rep. SRS–67. Asheville, NC: U.S. 

Department of Agriculture Forest Service, Southern Research Station. 8 p.

Brickell, J.E. 1970. Equations and computer subroutines for estimating site quality of eight 

Rocky Mountain species. Res. Pap. INT–75. Ogden, UT: U.S. Department of Agriculture 

Forest Service, Rocky Mountain Research Station. 22 p. 

Carmean, W.H.; Hahn, J.T.; Jacobs, R.D. 1989. Site index curves for forest species in the 

Eastern United States. Gen. Tech. Rep. NC–128. St. Paul, MN: U.S. Department of 

Agriculture Forest Service, North Central Forest Experiment Station. 142 p. 

Clendenen, G.W. 1977. Base-age conversion and site index equations for Engelmann spruce 

stands in the central and southern Rocky Mountains. Res. Note INT–223. Ogden, UT: U.S. 

Department of Agriculture Forest Service, Rocky Mountain Research Station. 6 p.

Edminster, C.B.; Mowrer, T.H.; Shepperd, W.D. 1985. Site index curves for aspen in the central 

Rocky Mountains. Res. Note RM–453. Ogden, UT: U.S. Department of Agriculture Forest 

Service, Rocky Mountain Research Station. 4 p.

Gingrich, S.F. 1967. Measuring and evaluating stocking and stand density in upland hardwood 

forests in the Central States. Forest Science. 1: 38-53. Vol. 13.

Hahn, J.T.; MacLean, C.D.; Arner, S.L.; Bechtold, W.A. 1995. Procedures to handle inventory 

cluster plots that straddle two or more conditions. Forest Science Monograph. 31: 12-25.

Hanson, E.J.; Azuma, D.L.; Hiserote, B.A. 2002. Site index equations and mean annual 

increment equations for Pacific Northwest Research Station Forest Inventory and Analysis 

inventories. Res. Note PNW–RN–533. Portland, OR: U.S. Department of Agriculture Forest 

Service, Pacific Northwest Research Station. 24 p.

Husch, B.; Miller, C.I.; Beers, T.W. 1982. Forest Mensuration. 3rd ed. New York: John Wiley. 

402 p.

Scott, C.T.; Bechtold, W.A.1995. Techniques and computations for mapping plot clusters that 

straddle stand boundaries. Forest Science Monograph. 31: 46-61.

Scott, C.T. 1993. Optimal design of a plot cluster for monitoring. In: Rennolls, K.; Gertner, 

G., eds. The optimal design of forest experiments and forest surveys: Proceedings, IUFRO 

S.4.11 conference. London, UK: University of Greenwich. 233-242 p.

Scott, C.T.; Voorhis, N.G. 1986. Northeastern forest survey site index equations and site 

productivity classes. Northern Journal of Applied Forestry. 3: 144-148.

Williams, M.S.; Schreuder, H.T. Reich, R.M. 1996. The extent of bias caused by substituting 

points in forest survey: A simulation study. Res. Pap. RM–327. Fort Collins, CO: U.S.

Department of Agriculture Forest Service, Rocky Mountain Forest and Range Experiment 

Station. 7 p.

Williams, M.S.; Schreuder, H.T.; Czaplewski, R.L. 2001. Accuracy and efficiency of area 

classifications based on tree tally. Canadian Journal of Forest Research. 31: 556-560.
43

4 Sample-Based Estimators 

Used by the Forest Inventory and 

Analysis National Information 

Management System

Charles T. Scott, William A. Bechtold, 

Gregory A. Reams, William D. Smith, 

James A. Westfall, Mark H. Hansen, and 

Gretchen G. Moisen1

4.1 Panels and Estimation

This chapter outlines prescribed core2 procedures for deriving population

estimates from attributes measured in conjunction with the Phase 1 and 

Phase 2 samples. These estimation procedures also apply to those Phase 

3 attributes in common with Phase 2. Given the sampling frame and plot

design described in the previous two chapters, many estimation approaches 

can be applied. In fact, one goal of the overall design is to maximize flexibility, so Forest Inventory and Analysis (FIA) data can be used to address a 

variety of analytical needs. 

Much of the flexibility inherent in the Enhanced FIA design is derived 

from the way panels are combined for analysis. This topic is addressed in 

chapter 5. For estimations involving a single panel or a periodic survey, 

the approach to estimation would proceed directly as outlined in chapter 4. 

When the estimation is for some combination of annual panels, then the

estimation procedures discussed herein may require modification depending 

on the method used to combine the panels. Related modifications are 

discussed in conjunction with the two specific methods for combining 

panels presented in chapter 5 (secs. 5.2.1 and 5.2.2).

1

 Charles T. Scott, FIA Program Manager, USDA Forest Service, Northeastern Research 

Station, Newtown Square, PA 19073; William A. Bechtold, Research Forester, USDA 

Forest Service, Southern Research Station, Asheville, NC 28802; Gregory A. Reams, FIA 

National Program Manager, USDA Forest Service, 1601 North Kent Street, Arlington, VA 

22209; William D. Smith, Research Quantitative Ecologist, USDA Forest Service, Southern 

Research Station, Research Triangle Park, NC 27709; James A. Westfall, Research Forester, 

USDA Forest Service, Northeastern Research Station, Newtown Square, PA 19073; Mark H. 

Hansen, Research Forester, USDA Forest Service, North Central Research Station, St. Paul, 

MN 55108; and Gretchen G. Moisen, Mathematical Statistician, USDA Forest Service, Rocky 

Mountain Research Station, Ogden, UT 84401.

2

 First use of a glossary term in each chapter is in bold face.
44

4.2 Phase 1

Stratification is a statistical tool used to reduce the variance of attributes 

of interest by partitioning the population into homogenous strata, such as 

forest and nonforest. It may also involve partitioning out a highly variable 

but small portion of the population. 

To use stratified sampling methods, the strata sizes (weights) must be either 

known or estimated. As discussed in chapter 2, strata weights commonly 

are estimated by classifying points on aerial photographs or pixels on 

satellite imagery. Full enumeration of an entire landscape (population) is 

possible with wall-to-wall satellite imagery, resulting in strata weights that 

can be treated as known values in the variance formulations. Whether the 

strata weights are known or estimated, strata classifications (e.g., forest 

and nonforest) from remote sensing are never perfectly accurate. Although 

this increases the variance of the resulting population estimates, it does 

not introduce any bias due to the interaction between Phases 1 and 2 in the 

estimation process.

4.2.1 Satellite Classification and Known Strata Weights

Using wall-to-wall satellite imagery and computer-aided classification, 

the population can be divided into strata of known size—typically forest, 

noncensus water, nonforest, or combinations of these. In this case the 

classified imagery divides the total area of the population (AT) into pixels 

of equal and known size (e.g., 30 m square with Landsat TM), and the 

classification assigns one of H strata to each of these pixels. Here, the 

stratum weight, Wh (h = 1,...,H), typically is determined as the number 

of pixels classified as stratum h divided by the total number pixels in the 

population of interest. The weights are known quantities that are fixed by 

the classification system and selected strata. As used here, “pixel” indicates 

either an individual pixel, or a non-overlapping block of multiple pixels. 

Each stratum, h, then contains nh ground plots, each selected with known 

probability, where the Phase 2 attributes of interest are observed. Note that 

the strata sample sizes, nh, are random because ground-plot locations are 

chosen prior to stratification.

Satellite classification systems separate the reflectance values from each 

pixel into a set of H distinct values (i.e., ranges). Such systems can range 

from very complicated functions dependant on a number of different 

reflectance band values and ancillary data layers, to very simple step 

functions of a single reflectance band. A variety of automated classification 

schemes are available, but these rarely match FIA stratification requirements. These must usually be modified or additional classifications must be 

performed to produce strata that are relevant to FIA estimation needs. 
45

When using classified satellite imagery for stratification, one must know 

the location of each plot and each pixel, so that plots and pixels can be linked

in the estimation process. FIA assigns each plot to one and only one stratum 

using the pixel corresponding to plot center. Typically this is done using 

a global positioning system (GPS) instrument for ground plots and georeference information associated with the imagery for pixels. If GPS field 

data are not available, plot locations can be digitized manually from aerial 

photos or maps. 

Most of the estimation presented here is based on the assumption of equal 

probability sampling, where all elements of the population have the same 

probability of being sampled by a ground plot. This assumption can be 

violated if ground-plot information is used to help classify satellite imagery. 

Inevitably, ground-plot data will be used in the development of some classification algorithms. Breidt and Opsomer (2004) show that for algorithms 

based on general linear models, this approach provides valid results and 

variances. Other types of classification algorithms (e.g., nonparametric) 

have not been similarly evaluated. 

It is also important to remember the importance of the pixel-plot link when 

changing or considering a change in the stratification algorithm. Two similar 

algorithms may divide a population into very similar strata that have similar 

or identical names. The application of a different classification system will 

not only change the Wh (even when they have identical names), but also the 

strata assignments of ground plots.

4.2.2 Aerial Photography and Estimated Strata Weights

In some cases, especially when using aerial photos, it is not practical or cost 

effective to divide imagery into strata of known sizes. Strata sizes then may 

be estimated using photo-plot sampling with manual interpretation, and 

applying double sampling for stratification as opposed to the stratified 

estimation that occurs when strata weights are known. When double 

sampling for stratification is used in conjunction with satellite imagery or 

other spatial layers, only a sample of pixels is classified, as opposed to the 

wall-to-wall estimation described in the previous section. This relieves some 

of the computational burden of more complex computer-aided classification 

methods. The loss of precision when such methods are applied to a subset 

of pixels is minimal when large Phase 1 sample sizes are used (Moisen and 

Edwards 1999).

Under double sampling for stratification, a set of sample points (i.e., Phase 1

plots) is placed on aerial photo or satellite coverage of the population. Phase 

1 plots are then assigned to strata by photo interpretation or computer-aided 
46

classification algorithm. Phase 1 plots are usually placed on a grid, typically 

every 200 to 300 acres. The Phase 2 and Phase 3 plots usually are a subset 

of the Phase 1 photo plots. Any Phase 2 and Phase 3 plots that are not must 

also be classified, but their values are not used in developing the estimates 

of strata sizes. Here: 

= total number of Phase 1 plots (pixels) sampled in the population

= number of Phase 1 plots (pixels) classified as belonging to stratum h 

(h = 1,...,H, H = total number of strata)

4.2.3 Combining Small Strata and Populations

Because sample sizes are random with respect to strata, some may contain 

too few plots to compute a reliable mean and variance. Experience indicates

that a minimum of four Phase 2 plots per stratum are required. If less than

four, then similar strata must be collapsed (combined) until the minimum is

attained. Because stratification schemes may vary regionally to accommodate resource differences, there are no prescribed rules by which strata are 

combined.

The minimum-sample-size rule also affects some small populations or populations with little forest (e.g., some counties in the Eastern or Great Plains 

regions of the United States). Populations with fewer than 12 forested Phase 2

plots should thus be combined with adjoining counties (i.e., populations) 

forming what is termed as a “super-county.” The use of super-counties will 

be especially important when using only a subset of panels.

4.3 Phase 2

This chapter builds on chapter 3 by specifying the population estimators 

used for calculation of area and other attributes of interest associated with 

mapped plots. Generally, these attributes are summed for each plot (after 

adjustment to correct for any plots that overlap the population boundary), 

and then averaged across plots in the stratum. The strata means are then 

combined using the Phase 1 strata weights to form a population mean. This

mean is then expanded to a total using the total land area in the population. 

This approach to estimation is first described for area attributes, and then 

other attributes, because the formulas are somewhat different. Examples of

the estimation procedures discussed below are provided in the supplementary

documents “Examples of FIA Point-in-Time Estimation Procedures for 

Several Common Cases” and “Examples of FIA Change-Component 

Estimation Procedures for Several Common Cases” at http://srsfia2.fs.fed.

us/publicweb/statistics_band/stat_documents.htm.
47

4.3.1 Estimation of Area by Stratum by Domain 

 (Row and Column) Attributes

For a given attribute of interest (e.g., forest area), estimates of population 

totals and domain totals (e.g., area by forest type and stand size class) are 

produced similarly. The only difference is that different restrictions (filters) 

are placed on each cell of an output table depending on which domain (row 

or column) the cell is in. Mapped condition classes (e.g., forest type) are 

often used to specify domains, but other attributes also can be used (e.g., 

species).

Row and column attributes must be discrete categorical variables. Continuous attributes can be converted into discrete categories by dividing them into 

classes (e.g., diameter classes). The attribute of interest is “summed” into a 

cell only when it satisfies the row and column requirements (i.e., when it is

in the domain of interest). Each plot and tree has an associated indicator 

function, (or ), which is 1 if the attribute is in the domain d of

interest, or 0 if not. For example, when estimating the area in northern hardwood (row) sapling stands (column), the indicator function is 1 when the 

specified forest type and stand-size requirement are satisfied; otherwise it is 0.

This method is described in Cochran (1977) for estimating domain (cell) 

means. 

4.3.1.1 For Each Table Cell, Compute the Attribute of Interest 

 for Each Plot

Each plot is assigned to only one stratum based on the Phase 1 stratification 

of the plot center. For area estimation, the attribute of interest is the 

proportion of the plot in the domain of interest: 

 

(4.1)

where

 proportion of plot i in the domain of interest d, for plots assigned 

to stratum h, adjusted for stratum h plots that overlap the population 

boundary

 mapped area (acres) of subplot (macroplot) j covering condition

k on plot i assigned to stratum h (The area is computed using the largest 

area mapped, which is the subplot except in the Pacific Northwest (PNW) 

where the macroplot is used.)
48

 zero-one domain indicator function, which is 1 if condition k

on subplot (macroplot) j of plot i assigned to stratum h belongs to the 

domain of interest d

 the number of conditions that exist on subplot (macroplot) j of plot 

i assigned to stratum h

am = total area of the largest-sized plot on which area attributes are 

mapped (i.e., four times the subplot or macroplot area)

mh = mean proportion of stratum h mapped plot areas falling within the 

population (see equation 4.2)

Table margins (row and column totals) are treated like any other cell. 

Additivity is a property of the table construction, not the estimators. Most 

tables will be additive (i.e., the cells in the table body will add to totals in 

the margins such as area by forest type and stand size). However, a table of 

the area containing combinations of species and diameter classes will not 

be additive, because each cell represents the number of acres on which the 

particular species and diameter class combination occurs.

4.3.1.2 Adjustment for Partial Plots Outside the Population

Equation 4.1 essentially treats all areas sampled on stratum h plots equally—

every square foot is expanded equally whether it is part of a partial plot 

or not. For a given population, dividing by mh in equation 4.1 adjusts the 

plot observations to account for any portions of stratum h plots falling 

outside the population. Note that 0 for condition classes outside the 

population in equation 4.1, because conditions outside the population are 

never conditions of interest. Reasons for condition classes being outside the 

population include: 

• Partial plots that straddle an international boundary (i.e., Canada or 

 Mexico)

• Partial plots that straddle a mapped ownership population boundary 

 (e.g., national forest)

• Whole or partial plots within the population that are nonsampled (e.g., 

 denied access or hazardous conditions)

• Whole or partial plots in census water (unless census water is estimated, 

 i.e., included in AT) (Note that FIA currently subtracts census water from 

AT, but anticipates that census water will be estimated when precise 

 digitized census water boundaries become available from the U.S. Census 

 Bureau. If and when census water is estimated from the FIA sample, a 

 Phase 1 stratum will likely be created for it.) 
49

If all plots are entirely within the population boundaries, then mh would 

be 1. Otherwise, the average mapped-plot area actually sampled within the 

population is divided by the standard plot area, am. This approach was taken 

as a way of handling the potential bias introduced by ignoring portions of 

plots straddling population boundaries. Essentially, this creates a buffer 

around the population to ensure that plots which are only partially inside the 

population of interest are included in the estimation. More area is sampled, 

but more plots are also taken into the sample. Multiplying the totals by the 

larger area is on average equivalent to the adjustment made by using mh:

 

(4.2)

where

nh = number of ground plots with Phase 1 assignments to stratum h (For 

initial area tables, this includes all plots sampled with any portion of a 

subplot (macroplot) in the population. For subsequent tables, any plots 

that are entirely nonsampled are excluded.)

 zero-one in-sample indicator function, which is 1 if condition 

k on subplot (macroplot) j of plot i assigned to stratum h is within the 

boundaries of the population (Nonsampled areas are included in initial 

area tables in order to estimate their areas, but are zero otherwise. 

Missing values are also treated as zero values.)

Clearly, there are cases where mh will not be 1, meaning that mh is a constant for a given stratum, but varies between strata and populations. The 

variation due to nonsampled plots and plots extending beyond population 

boundaries is expected to be small enough to be ignored. One particular 

area of concern is the checkerboard ownership pattern in the West where 

National Forest System (NFS) boundaries may be treated as population 

boundaries. FIA is evaluating the frequency of plots that straddle NFS 

boundaries and may switch to the ratio-of-means estimators described by 

Zarnoch and Bechtold (2000), if necessary. 

4.3.1.3 Compute Strata Means and Variances

Plot values are averaged within each stratum. In the case of simple random 

sampling, this is the final estimate because simple random sampling is just 

stratified sampling with a single stratum.

The stratum mean is the sum of the plot observations, Phid, divided by the 

number of plots in the stratum, nh:
50

 

(4.3)

with variance estimator:

 

(4.4)

Note that is the mean of field-based observations in domain d within 

each Phase 1 stratum. This means that some plots classified as forest on the 

ground may have been assigned to a nonforest stratum and vice versa. These 

are not viewed as “misclassifications”, but as strata with less than ideal 

homogeneity. The estimators remain unbiased. 

4.3.2 Estimation of Population Totals and Their Variances

Generally, individual counties are the populations of interest (i.e., the basic

building blocks for estimation). Counties may be divided into subpopulations that are processed independently. This is the case when a portion of a 

county has an intensified Phase 2 sampling grid, has enumerated acreages, 

or has a measurement cycle that differs from the rest of the county. These 

scenarios are not uncommon when sampling land owned by public agencies 

[i.e., NFS, Bureau of Land Management (BLM), and National Park Service 

(NPS)]. Because populations and subpopulations are mutually exclusive, 

estimated totals are additive. Likewise, because different populations and 

subpopulations are independent, the associated variance estimates are also 

additive. Thus, totals from groups of counties can be combined to formulate 

State and regional totals; or segments of NFS land, by county, can be combined to yield totals for a specific national forest. County areas provided 

by the U.S. Census Bureau, which are used in the estimation of population 

totals, are updated at least every 10 years. NFS and BLM provide similar 

area totals for their lands, and totals by county if they have intensified or 

otherwise altered the sampling effort on their lands. 

As first noted in equation 4.2, nonsampled plots present an estimation 

problem that requires more attention than plots that are otherwise out of the 

population. Because field crews cannot reliably ascertain the actual land 

use of such plots, initial area tables in FIA reports will report the proportion 

of total area designated as nonsampled. However, in all subsequent tables, 

these plots (or the nonsampled portions of them) will be processed as if they 

hd
51

were out of the population. This approach either reduces the sample size 

(nh), or the mean proportion of stratum h observed plot areas falling within 

the population, or both. The result is to adjust upward the number of acres 

represented by the accessed plots, or portions of them, in each stratum. Thus 

the area that could not be accessed is redistributed based on the accessible 

plots within each stratum. This essentially replaces nonsampled plots with 

the strata means and increases the strata variances due to the reduction in 

sample size. This approach has the advantage of simplicity, but has the 

potential to incur bias if the nonsampled plots are not representative of the

rest of the population. Other methods of accounting for nonsampled areas

are under investigation, including remote sensing both for direct measurement of a subset of attributes and for use in identifying similar plots for 

imputation purposes. 

In the simple random sampling case, there is only one stratum, so totals 

are estimated by multiplying the population mean by the total area in the 

population. FIA rarely uses simple random sampling, but it has been used 

for Forest Health Monitoring (FHM) and is useful in responding to special 

requests where stratification data may be lacking. Processing protocols for 

the FIA plot design under a simple random sampling approach using ratios 

of means are discussed in detail by Zarnoch and Bechtold (2000).

To estimate the total area in each domain when the population has been 

stratified, the strata means are averaged using the strata weights and then 

multiplied by the total land area in the population. The estimated total is 

given for the stratified estimation and double sampling for stratification 

cases, respectively, as: 

 

(4.5)

and

where

AT = total area in the population in acres

 estimated proportion of the population in the domain of interest d

Wh = weight for stratum h (i.e., the proportion of the population area, AT, 

that is in stratum h)
52

An approximation of the variance of the total area in the stratified estimation (and the simple random sampling) case, where strata sizes are known, 

is adapted from equation 5A.42 in Cochran’s (1977) finite sampling framework. An estimate of this population variance (finite population correction 

factors ignored) is given by:

 

(4.6)

The first term is for stratification, assuming proportional allocation, and the 

second term reflects the fact that the sample sizes are not fixed by strata in 

advance. 

Double sampling for stratification applies when the strata weights are estimated. The variance of total area in this case is adapted from equation 12.32 

in Cochran (1977), again ignoring finite population correction factors. The 

second term accounts for the fact that strata sizes are estimated:

 

(4.7)

As noted above, totals and their variances from different populations are

additive. Thus the variance of a total across multiple populations or subpopulations is the sum of their variances. Examples of how to apply equations 

4.1 through 4.7 are given in spreadsheet form in the supplementary document “Examples of FIA Point-in-Time Estimation Procedures for Several 

Common Cases” at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_

documents.htm (see sections “plot summary”, “problem number 1”, and 

“problem number 2”).

4.3.3 Estimation of Other Attributes

Population totals for attributes other than area usually are calculated by 

summing attributes to the plot level and then averaging at the stratum level.

Indicator functions are used to identify the attribute of interest (e.g., total 

volume of white oak) in the domain of interest (e.g., oak-pine stands). The

attribute of interest is summed for each plot and then divided by the observed

plot area and the mean proportion of stratum h observed plot areas falling 

within the population, yielding an estimate of the attribute of interest on a 

per-unit-area basis:
53

 

(4.8)

where

 attribute of interest for tree t on macroplot, subplot, or microplot j

of plot i assigned to stratum h

 zero-one domain indicator function, which is 1 if tree t on subplot

j of plot i assigned to stratum h belongs to the domain of interest d

ao = total area normally used to observe the attribute of interest on a plot 

(i.e., four times the microplot, subplot, or macroplot area)

oh = mean proportion of stratum h observed-plot areas falling within the 

population (see equation 4.9)

In equation 4.8, dividing by oh adjusts the plot observations to account for 

any portions of stratum h plots falling outside the population:

 

(4.9)

where

 area normally used to observe the attribute of interest (microplot, 

subplot, or macroplot j) covering condition k on plot i assigned to stratum h

In equation 4.8, dividing by the observed plot area and by the proportion of 

plots outside the population allows attributes such as number of trees across 

all diameter classes to be summed across plot types while accounting for any

differences in the proportion of the various plot types that are outside the population. For example, if yhid is the number of trees 1.0-inch diameter at breast

height (d.b.h.) and larger sampled on microplots and subplots, then equation

4.8 should include a term for each plot type. This adjustment is the main 

difference between the estimators described herein and the ratio-of-means 

estimators outlined by Zarnoch and Bechtold (2000), where population 

totals are computed on the basis of each plot size separately, then summed

for all plot sizes. Because totals for the latter alternative are not independent,

the variance is complicated by the need to include a covariance term among

plot sizes. We opted for a simpler approach, where all estimates are combined

at the plot level and then treated as a single attribute of interest when calculating population totals. Using this method, the variance is much easier to
54

compute, and the resulting estimate (equation 4.10) is equivalent to the 

separate approach. When combining subplot and microplot values, the 

attribute of interest is computed as:

 

(4.10)

where

 attribute of interest for tree t on microplot j of plot i in stratum h

 zero-one domain indicator function, which is 1 if tree t on microplot j of plot i assigned to stratum h belongs to the domain of interest d

a9

o

 = total microplot area

 = mean proportion of stratum h microplot areas falling within the 

population (equation 4.9)

The plot attributes from either equation 4.8 or equation 4.10 are then 

summed across all plots within each stratum and divided by the total 

number of plots in the stratum to yield the stratum mean of the yhid:

 

(4.11)

with estimated variance:

 

(4.12)

As was the case for area, the strata means are averaged using the strata 

weights, then multiplied by the total land area. The estimated total is given 

for the stratified estimation and double sampling for stratification cases, 

respectively, as:

 

(4.13)

oh
55

or

where

population mean of the attribute of interest in the domain of interest d

As with the area estimate when strata weights are known, an approximation 

to the variance of the attribute total in the stratified estimation (and the 

simple random sampling) case was developed in the same way as for 

equation 4.6:

 

(4.14)

The estimated variance of the attribute total in the double sampling for 

stratification case is again adapted from equation 12.32 in Cochran (1977):

 

(4.15)

As noted in section 4.3.2, totals and their variances from different populations are additive for a given domain. Thus the variance of a total across 

multiple populations or subpopulations is the sum of their variances. Although

the additivity property is also true for means, sums of means generally are

not useful. To obtain a mean over populations, the totals should be added 

and divided by the total area of the populations. The result is a mean weighted

by the population areas. An example of how to apply equations 4.8 through

4.15 is given in problem number 3 of the supplementary document “Examples of FIA Point-in-Time Estimation Procedures for Several Common Cases”

at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm.

4.3.4 Estimation of Ratios

Often, interest is not in the totals but in the attribute of interest expressed on 

a per-acre, per-condition (stand), or per-tree basis. An approach that is also 

compatible with the aforementioned estimates of population totals is the 

ratio-of-means estimator, wherein the numerator is the estimated attribute 

total and the denominator depends on the ratio to be estimated. The three 

cases can all be estimated using one of the following general formulas. 

The first is for stratified estimation where strata weights are known and the 

second for double sampling where strata weights are estimated:
56

or (4.16)

The strata means for the denominator, , are computed in the same 

manner as the numerator. Note that the numerator and denominator have 

different domains of interest, with d being a subset of d9 (e.g., 12-inch 

oaks in oak-hickory stands). If the denominator is an area attribute, then 

xhid9 is estimated using equation 4.1 and the area attribute replaces Phid. If 

the denominator is a tree or other attribute, then xhid9 is estimated using 

equation 4.8 and the attribute replaces yhid. The choice of the individual plot 

observations, xhid9, is described in sections 4.3.4.1 through 4.3.4.3 for three 

common situations. 

The variance estimator from equation 5.6.10 in Särndal and others (1992) is:

 

(4.17)

In the stratified estimation case, the right-hand-side variances are computed 

using equation 4.6 for area attributes and equation 4.14 for other attributes. 

The covariance is estimated as:

(4.18)

where

 

(4.19)
57

In the double sampling for stratification case, the variance is approximated 

by assuming the covariance can be computed similarly to the area attributes 

(equation 4.7) and other attributes (equation 4.15) 

(4.20)

Equations 4.16 and 4.17 can be used for all three estimation of ratio cases—

to express values on a per-acre, per-tree, or per-condition basis.

4.3.4.1 Estimation on a Per-Acre Basis 

Often interest is in expressing the attribute of interest on a per-acre basis. 

This can be estimated by dividing in equation 4.13 by the total surface 

area in the population, AT. Because the area is known, the variance of the 

ratio is simply the variance of the total (equation 4.14) divided by the square 

of the total area, AT. 

However, interest is more commonly in the attribute total expressed on a perforested-acre basis—a ratio estimate. The denominator of equation 4.16 can 

be the estimate of total forest area computed using equation 4.5, or the 

area of another domain such as the area in a specific forest type. When the 

denominator is derived from equation 4.5, the xhid9 is equal to Phid in equation 

4.1 which is then used to compute the X values in equations 4.16 through 

4.20.

Indicator functions are used to specify the domains and attributes of interest

that define yhi and xhi. These may or may not change for various cells in the 

tables, depending on the ratio of interest. For example, the value of xhid9 can

be the same for all values in a table, such as those used in computing .

This is useful for estimating additive tables, such as stand tables with numbers

of trees per acre by species and diameter class. Alternatively, the value of 

xhid9 might change for each cell, such as the area by forest type and stand size,

so that the attribute average can be estimated for each combination (domain).

An example of how to apply equations 4.16 through 4.20 for estimation on a 

per-acre basis is given in problem number 4 of the supplementary document 

“Examples of FIA Point-in-Time Estimation Procedures for Several Common

Cases” at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.

htm.
58

4.3.4.2 Estimation on a Per-Condition (Per-Stand) Basis 

A few attributes are only observed at the condition (or stand) level, such as 

stand age. To compute average stand age requires a slight modification of 

the approach to the yhid attribute. The approach is to weight the condition 

attribute, yhik, by the area in the condition of interest on the plot: 

 

(4.21)

The denominator reflects the proportion of the area sampled in the condition 

of interest:

 

(4.22)

The result is the average of the attribute of interest weighted by the area in

which it occurs. These values of yhid and xhid9 are then used in equations 4.16

through 4.20. An example of how to apply equations 4.16 through 4.22 for

estimation on a per-stand basis is given in problem number 5 of the supplementary document “Examples of FIA Point-in-Time Estimation Procedures 

for Several Common Cases” at http://srsfia2.fs.fed.us/publicweb/statistics_

band/stat_documents.htm.

4.3.4.3 Estimation on a Per-Tree Basis 

For some applications, the attribute of interest is an individual tree attribute, 

such as average tree height or conks per tree, where the ratio of interest is

expressed on a per-tree basis. The ratio estimator then becomes the estimate 

of the population total for the attribute of interest, , divided by the total

number of trees, , in the population. Thus, yhid is the sum of tree attributes

observed on plot i in stratum h in the domain of interest; and xhid9 is the 

number of trees observed on plot i in stratum h in the domain of interest. 

Those values are then used in equations 4.16 through 4.20. An example of 

how to apply equations 4.16 through 4.20 for estimation on a per-tree basis 

is given in problem number 6 of the supplementary document “Examples of 

FIA Point-in-Time Estimation Procedures for Several Common Cases” at 

http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm.
59

4.3.5 Computation of Sampling Errors

Sampling errors are used by FIA to reflect the accuracy of the estimates. 

Expressed on a percentage basis in order to enable comparisons between the 

precision of different estimates, sampling errors are computed by dividing 

the estimate into the square root of its variance:

 

(4.23)

Assuming normality of the distribution of estimates, the percent sampling 

error can be used to compute an approximate 67 percent confidence interval. 

If the sampling error is doubled, then an approximate 95 percent confidence 

interval can be formed. A better approximation is achieved by using the 

appropriate Student’s t-values.

4.3.6 Components of Change

FIA inventories are designed to measure net change over time, as well as the

individual components of change that constitute net change (e.g., growth, 

removals, mortality). Change estimates are computed for two sequential measurements of each inventory panel. Upon remeasurement, a new initial 

inventory is established for remeasurement at the next scheduled inventory. 

As such, computation of change components is not intended to span more

than one inventory cycle. Rather, the change estimation process is repeated 

cycle by cycle. This simplifies field protocols and ensures that change estimation is based on short and relatively constant time intervals (e.g., 5 years).

Change estimates for individual panels are combined across multiple panels 

in the same manner as panels are combined to obtain current inventory 

parameters such as total standing volume.

FIA recognizes the following components of change as prescribed core 

variables; they usually are expressed in terms of growing-stock or all-live 

volume, where t is the initial inventory of a measurement cycle, and t+1 is 

the terminal inventory:

GS = survivor growth—the growth on trees tallied at time t that survive until 

time t+1.

I = ingrowth—the volume of trees at the time that they grow across the 

minimum d.b.h. threshold between time t and time t+1. The estimate is 

based on the size of trees at the d.b.h. threshold which is 1.0 inch for alllive trees and 5.0 inches for growing-stock trees. This term also includes 

trees that subsequently die (i.e., ingrowth mortality), are cut (i.e., ingrowth 
60

cut), or diverted to nonforest (i.e., ingrowth diversion); as well as trees that 

achieve the minimum threshold after an area reverts to a forest-land use 

(i.e., reversion ingrowth).

GI = growth on ingrowth—the growth on trees between the time they grow 

across the minimum d.b.h. threshold and time t+1.

R = reversion—the volume of trees on land that reverts from a nonforest 

land use to a forest land use (or, for some analyses, land that reverts from 

any source to timberland) between time t and time t+1. The estimate is 

based on tree size at the midpoint of the measurement interval. Tree size at 

the midpoint is modeled from tree size at time t+1.

GR = reversion growth—the growth of reversion trees from the midpoint of 

the measurement interval to time t+1. Tree size at the midpoint is modeled 

from tree size at time t+1. This term also includes the subsequent growth on 

ingrowth trees that achieve the minimum diameter threshold after reversion.

M = mortality—the volume of trees that die from natural causes between 

time t and time t+1. The estimate is based on tree size at the midpoint of the 

measurement interval (includes mortality growth). Tree size at the midpoint 

is modeled from tree size at time t.

GM = mortality growth—the growth of trees that died from natural causes 

between time t and the midpoint of the measurement interval. Tree size at 

the midpoint is modeled from tree size at time t. This term also includes the 

subsequent growth on ingrowth trees that achieve the minimum diameter 

threshold prior to mortality.

C = cut—the volume of trees cut between time t and time t+1. The estimate 

is based on tree size at the midpoint of the measurement interval (includes 

cut growth). Tree size at the midpoint is modeled from tree size at time t. 

Trees felled or killed in conjunction with a harvest or silvicultural operation 

(whether they are utilized or not) are included, but trees on land diverted 

from forest to nonforest (diversions) are excluded.

GC = cut growth—the growth of cut trees between time t and the midpoint 

of the measurement interval. Tree size at the midpoint is modeled from tree 

size at time t. This term also includes the subsequent growth on ingrowth 

trees that achieve the minimum diameter threshold prior to being cut.

D = diversion—the volume of trees on land diverted from forest to nonforest 

(or, for some analyses, this may also include land diverted to reserved 

forest land and other forest land), whether utilized or not, between time 

t and time t+1. The estimate is based on tree size at the midpoint of the 

measurement interval (includes diversion growth). Tree size at the midpoint 

is modeled from tree size at time t.
61

GD = diversion growth—the growth of diversion trees from time t to the 

midpoint of the measurement interval. Tree size at the midpoint is modeled 

from tree size at time t. This term also includes the subsequent growth 

on ingrowth trees that achieve the minimum diameter threshold prior to 

diversion.

FIA recognizes the following components of change as prescribed optional 

variables:

CI = cull increment—the net reduction in growing-stock volume due to 

reclassification of growing stock trees to cull trees between two surveys. 

Cull increment is the volume of trees that were growing stock at time t, 

but cull at time t+1. The estimate is based on tree size at the midpoint of 

the measurement interval (includes cull increment growth). Tree size at the 

midpoint can be modeled from tree size at time t, time t+1, or both. 

GCI = cull increment growth—the growth to the midpoint of the measurement

interval between time t and t+1 of trees that were growing stock at time t, 

but cull trees at time t+1. Tree size at the midpoint can be modeled from tree 

size at time t, time t+1, or both.

CD = cull decrement—the net gain in growing-stock volume due to reclassification of cull trees to growing stock trees between two surveys. Cull decrement is the volume of trees that were cull at time t, but growing stock at time

t+1. The estimate is based on tree size at the midpoint of the measurement 

interval. Tree size at the midpoint can be modeled from tree at time t, time 

t+1, or both.

GCD = cull decrement growth—the growth from the midpoint of the measurement interval to time t+1 on trees that were cull at time t, but growing stock 

at time t+1. Tree size at the midpoint can be modeled from tree size at time 

t, time t+1, or both.

Except for R, D, GD, and GR, all components listed above are computed from

plot areas where land use is defined as forest at both time t and time t+1. 

Note that it is not possible to measure the terminal d.b.h. of all trees that 

were cut, died, or diverted to a nonforest land use. To minimize potential 

bias associated with the growth of these trees, estimates of GM, GC, GD, D,

R, and GR are modeled on the basis of the measurement-interval midpoint. 

The midpoint is calculated as ∆t / 2, where ∆t is the time in years (rounded 

to the nearest 10th) between measurements for an individual plot. Models 

to predict midpoint tree sizes are developed regionally and may include 

a variety of factors, including terms to account for slowed growth on 

mortality trees.
62

The use of midpoint tree sizes creates special situations where careful attention is needed to account for all change components. Particularly notable are

ingrowth trees that assume a status other than survivor. For instance, between

time t and time t+1, a tree may cross the 5.0-inches d.b.h. threshold (ingrowth),

grow to 5.4-inches d.b.h., and then die. Under those circumstances, there would

be three components of change: (1) Ingrowth (I), the volume at 5.0-inches 

d.b.h.; (2) Mortality growth (GM), the volume growth from 5.0-inches d.b.h.

to 5.4-inches d.b.h.; and (3) Mortality (M), the volume at 5.4-inches d.b.h. 

Similar circumstances occur where ingrowth trees are associated with cutting, reversions, and diversions. Such situations implicitly require that midpoint tree sizes be modeled for all trees 1.0-inch d.b.h. and larger in order 

to check for trees that may have crossed the tree diameter threshold before 

removal or death.

For reporting growth and change, the individual components are usually 

combined as follows, and expressed either in terms of growing-stock or alllive volume:

Gross ingrowth = I + R 

Accretion = GS + GI + GR + GM + GC + GD

Gross growth = gross ingrowth + accretion

Mortality = M 

Removals = C + D 

Net growth = gross growth – mortality 

Net change = net growth – removals 

The above terms for accretion and net growth are modified as follows for 

FIA regions that elect to produce additional output containing optional 

expressions of cull increment and decrement. Note that these optional terms 

are relevant only when components of change are expressed in terms of 

growing-stock volume: 

Accretion = GS + GI + GR + GM + GC + GD + GCI + GCD

Net growth = gross growth – mortality + CD – CI

In addition to volume, all change components may also be expressed in terms

of basal area or weight. More commonly, some (I, R, M, C, D, CI, and CD) 

are occasionally expressed as numbers of trees.

A variety of estimators have been proposed for the various components of 

change (Beers and Miller 1964, Gregoire 1993, Roesch and others 1989, 

Van Deusen and others 1986). When only one fixed-size plot is involved, 
63

estimation of change components is very straightforward and most estimators

are equivalent. This is the case for most FIA reporting purposes because 

volumes generally are reported in terms of growing stock, which is based 

on trees 5.0-inches d.b.h. and larger that are only recorded on the subplot 

(except PNW which also uses macroplots). 

For regions using the macroplots, or for expressions of growth involving 

microplot trees (e.g., growth of trees 1.0-inch d.b.h. and larger), change 

estimation is complicated by trees that grow from one plot size to the next. 

This requires techniques designed for variable-radius plots. Historically, 

FIA units have used one of two methods for calculating components of 

change for variable-radius plots—Beers and Miller (1964) or Van Deusen 

and others (1986). Because the Van Deusen estimator is more appropriate 

for prism sampling and FIA has moved away from prism sampling, we have 

decided to use the simpler Beers-Miller approach.

The Beers-Miller estimator weights all survivor growth (GS

) on the basis of 

plot size at time t:

 (4.24)

where

s2 = tree size at time t+1 weighted on the basis of plot size at time t

s1 = tree size at time t weighted on the basis of plot size at time t

Ingrowth, I, is defined as those trees on the microplot that grew across the 

1.0-inch threshold:

 (4.25)

where

sdbh=1.0 = the size of an ingrowth tree at the 1.0-inch d.b.h. threshold and 

growth on ingrowth as

 (4.26)

Both s2 and the size of the tree at 1.0-inch d.b.h. (sdbh=1.0) are weighted on the 

microplot basis for trees on the microplot that were < 1.0-inch d.b.h. at time 

t but greater than 1.0 inch at time t+1. Note that the Beers-Miller estimator 

ignores trees that grow onto the subplot from outside the microplot.

When estimating change, individual trees are placed into the appropriate 

change-component category(s). Tree attributes associated with the changecomponent of interest are then summarized to the plot level using equations 
64

4.8 and 4.10, annualized by dividing plot-level periodic values by the number

of years between the initial and terminal measurements of each plot, and 

summarized to the population level as specified in equations 4.11 and 4.13.

The components of change are converted to average annual values as 

follows, where ∆t

 is the time in years (rounded to the nearest 10th) between 

measurements for an individual plot:

Annual gross ingrowth = (I + R) / ∆t

Annual accretion = (GS + GI + GR + GM + GC + GD ) / ∆t

Annual gross growth = annual gross ingrowth + annual accretion

Annual mortality = M / ∆t

Annual removals = (C + D) / ∆t

Annual net growth = annual gross growth – annual mortality 

Annual net change = annual net growth – annual removals 

Annualized values for accretion and net growth for regions that include the 

optional expressions of cull increment and decrement would be: 

Annual accretion = (GS + GI + GR + GM + GC + GD+ GCI + GCD ) / ∆t

Annual net growth = annual gross growth – annual mortality + (CD – CI) / ∆t

Observe that some of the change components pertain to trees on conditions 

that remained in forest for an entire inventory cycle; some are based on trees

and areas that become forest between the initial and terminal inventory of 

a cycle (reversions); and some pertain to trees on conditions removed from 

the forest land base (diversions). Improved estimates of change can be 

obtained by stratification on the basis of both initial and terminal land use: 

(1) Forest to forest, (2) Forest to nonforest, (3) Nonforest to forest, and (4) 

Nonforest to nonforest. This might also be expanded to include the finer 

subsets of forest recognized by FIA such as timberland, reserved forest land, 

and other forest land. 

This requires classification of the same Phase 1 points at both time t and 

time t+1—either photo interpretation of the same photo plots if photography 

was used or classification of the same pixels if satellite imagery is used. 

Note that the same collapsed strata should be used for both current and 

change estimates if the results are to be compatible. 

Once the strata weights are assigned, estimation by domains within strata 

follows the same procedure outlined for current inventory parameters. For
65

standard reporting purposes, domains identified or partitioned by forest 

attributes (e.g., owner group) usually are defined on the basis of initial classifications for forest-nonforest and forest-forest parameters; and domains for

nonforest-forest inventory attributes are defined on the basis of terminal 

classifications. Examples of computational procedures for change estimation

are provided in the supplementary document “Examples of FIA ChangeComponent Estimation Procedures for Several Common Cases” at http://

srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm.

4.4 Expansion Factors

When periodic inventories and flat files were the FIA standards, it was convenient to calculate a small set of expansion factors by which individual 

plot-level or tree-level observations could be converted to their populationlevel equivalents. This allowed population totals to be obtained via summation, which greatly simplified the estimation process. Expansion factors were

popular with external FIA clients, many of whom used this concept to build 

their own processing systems. The tradeoff for such simplicity is that the 

use of expansion factors precludes the ability to calculate variances. At best,

the variances of estimators derived from expanded values can only be approximated, and these approximations are known to be poor (Alegria and Scott 

1991).

Expansion factors are less practical with panelized inventory systems, which

are designed to increase analytical flexibility by allowing panels to be combined in a variety of ways. Each different panel combination produces a 

unique set of expansion factors, rendering expansion factors associated with 

panel systems less stable than those produced by periodic systems.

The use of expansion factors is discouraged because they prohibit accurate 

variance estimation and they no longer have the advantage of simplicity. 

However, there is still a demand for them, and it may take a while to convert 

processing systems to the estimation procedures specified in this chapter. 

Therefore, FIA will continue to offer expansion factors until a demand is 

no longer apparent. The derivation of expansion factors is described in the 

supplementary document “Computation of FIA Plot Expansion Factors” at 

http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_documents.htm.
66

4.5 Phase 3

4.5.1 Estimation Procedures Used for FHM (Phase 3) Data

The FHM Program did not utilize any Phase 1 stratification prior to merging 

with FIA. In order to avoid reporting regional and State totals that conflicted 

with those reported by FIA, FHM statistics usually were presented as population-level means based on either simple random sampling or Generalized 

Leased Squares (GLS). Zarnoch and Bechtold (2000) developed one 

such approach based on simple random sampling and ratio-of-means 

estimators. Smith and Conkling (2004) used GLS estimation procedures, 

where population-level current mean values and annual change estimates 

are obtained from linear models for repeated measurements (Gregoire and 

others 1995, Urquhart and others 1993, Van Deusen 1996). The estimate of 

current value is:

 (4.29)

where

yt = predicted value of the attribute at year t

bt-1 = the mean value of attribute at year t-1

b1 = the annual change in y from Year t-1 to Year t

Both bt-1 and b1 are computed by estimated generalized least squares using 

Proc MIXED® (SAS Institute 1999). 

4.5.2 Combining with Phase 1

Because the FIA and FHM inventory systems have merged, it is now possible to combine Phase 1 and Phase 3 data using stratified random estimation 

and double sampling for stratification. We are studying use of Phase 1 stratification to enhance the estimation of attributes unique to Phase 3 and plan to

document estimation procedures for attributes specific to Phase 3 in subsequent manuscripts.

4.5.3 Linking with Phase 2

For inventory parameters common to both Phase 2 and Phase 3, usually 

there is no advantage in generating population estimates from the smaller 

Phase 3 subset, because the estimates will not match and the Phase 3 variances will be larger due to a reduced n. Phase 3 plots, therefore, should be

combined with Phase 2 plots when estimating inventory parameters common

to both phases. 
67

For those inventory parameters that are unique to Phase 3, it is possible to 

model relationships between unique Phase 3 parameters and parameters 

common to both Phase 2 and Phase 3. Such models can then be used to 

extrapolate estimates of Phase 3 parameters onto the more intensive Phase 2 

sampling network. 

4.6 Literature Cited

Alegria, J., Scott, C.T. 1991. Generalized variance function applications in forestry. Res. Note 

NE–345. Radnor, PA: U.S. Department of Agriculture Forest Service, Northeastern Forest 

Experiment Station. 4 p.

Beers, T.W.; Miller, C.I. 1964. Point sampling: research results, theory, and application. Res. 

Bull. 786. Lafayette, IN: Purdue University. Agricultural Experiment Station. 56 p. 

Breidt, F.J.; Opsomer, J.D. 2004. Endogenous and post-stratification in surveys: classifying 

with a sample-fitted model. Preprint Series #02-23. Ames, IA: Department of Statistics, Iowa 

State University. 25 p. 

Cochran, W.G. 1977. Sampling techniques. New York: John Wiley. 428 p.

Gregoire, T.G. 1993. Estimation of forest growth from successive surveys. Forest Ecology and 

Management. 56: 267-278.

Gregoire, T.G.; Schabenberger, O.; Barret, J.P. 1995. Linear modeling of irregularly spaced, 

unbalanced, longitudinal data from permanent-plot measurements. Canadian Journal of 

Forest Research. 25: 137-156.

Moisen, G.G.; Edwards T.C., Jr. 1999. Use of generalized linear models and digital data in a 

forest inventory of northern Utah. Journal of Agricultural, Biological, and Environmental 

Statistics. 4: 372-390.

Roesch, F.A.; Green, E.J.; Scott, C.T. 1989. New compatible estimators for survivor growth 

and ingrowth from remeasured horizontal point samples. Forest Science. 35: 281-293.

Särndal, C.E.; Swensson, B.; Wretman, J. 1992. Model Assisted Survey Sampling. New York: 

Springer-Verlag. 694 p.

SAS Institute. 1999. SAS/STAT® user’s guide. Version 8. Cary NC: SAS Institute Inc. 2552 p. 

Vol. 2.

Smith, W.D.; Conkling, B.L. 2004. Analyzing forest health data. Gen. Tech. Rep. SRS-077. 

Asheville, NC: U.S. Department of Agriculture Forest Service, Southern Research Station. 

33 p.

Urquhart, N.S.; Overton, W.S.; Birkes, D.S. 1993. Comparing sampling designs for monitoring 

ecological status and trends: impact of temporal patterns. In: Statistics for the Environment. 

London: John Wiley: 71-85.

Van Deusen, P.C. 1996. Incorporating predictions into an annual forest inventory. Canadian 

Journal of Forest Research. 26(10): 1709-1713.

Van Deusen, P.C; Dell, T.R.; Thomas, C.E. 1986. Volume growth estimation from permanent 

horizontal points. Forest Science. 32: 415-422.

Zarnoch, S.J.; Bechtold, W.A. 2000. Estimating mapped-plot attributes with ratio of means. 

Canadian Journal of Forest Research. 30: 688-697.
68
69

5 Combining Panels for Forest 

Inventory and Analysis Estimation

Paul L. Patterson and Gregory A. Reams1

5.1 Single Panels vs. Multiple Panels

The term panel2 denotes a set of samples where the same elements are measured on two or more occasions. Historically, Forest Inventory and Anaysis 

(FIA) has used a single panel to conduct periodic surveys. Annual panels, 

however, allow greater flexibility because they can be combined in a variety 

of ways. Note that FIA assumes complete spatial coverage for each panel 

across the population of interest. When estimating inventory attributes for 

a single panel, the estimation approach proceeds as outlined in chapter 4. 

When estimating inventory attributes for combined panels, however, such 

procedures may require modification, depending on how the panels are 

combined. Related modifications are discussed in conjunction with the two 

specific methods presented in sections 5.2.1 and 5.2.2.

Dividing a single large periodic survey into a series of smaller surveys by 

measuring panels, one at a time, has several noteworthy advantages:

1. Individual panels can yield information about variations that occur with-

 in a measurement cycle; they can estimate year to year as well as long- 

 term cycles and trends. This greatly improves our ability to understand 

 the causes and timing of changes in the resource, as opposed to assuming 

 linear trends. 

2. Successive measurement of panels can provide quicker feedback to facili-

 tate decisions that depend on knowledge of fluctuations in the survey 

 attributes. If necessary, field protocols can be modified at the next sche-

 duled panel, rather than waiting for a full inventory cycle to be completed.

3. Panels are highly responsive to widespread catastrophic events. The 

 impact of a catastrophic event that occurs in a single year (e.g., fire or 

 hurricane) can be gauged immediately. In the past, alternative methods 

 such as interim periodic surveys were used to deal with catastrophic 

 events (Sheffield and Thompson 1992).

4. Panels provide a natural, temporal link to other annual ancillary data.

1

 Paul L. Patterson, Mathematical Statistician, USDA Forest Service, Rocky Mountain 

Research Station, Fort Collins, CO 80526-1891; and Gregory A. Reams, FIA National 

Program Manager, USDA Forest Service, 1601 North Kent Street, Arlington, VA 22209.

2

 First use of a glossary term in each chapter is in bold face.
70

Although annual inventories are considered superior to periodic inventories 

for FIA applications, conversion comes at a price. Some advantages of 

periodic surveys include:

1. Travel cost is minimized. Annual inventories require field crews to travel 

 across the entire population each time a panel is measured. 

2. Change estimates apply to just two points in time (although field measure-

 ments often take 2 to 3 years to complete). With multiple panels, change 

 estimates are staggered over two inventory cycles, rather than just one.

3. The length of time required to measure individual panels can be incon-

 sistent and difficult to manage. Budget constraints, regional issues, and 

 logistical problems all influence the time needed to complete individual 

 panels and sets of panels. As a result, the time required to complete a 

 panel (or subpanel) typically does not equal exactly 1 year. From State 

 to State, the time needed to finish one complete set of inventory panels 

 can range from 3 to 10 years due to panel acceleration or panel creep as 

 discussed in chapter 2.

4. The sample size is nearly always sufficient. A single panel has sampling 

 errors that are times larger than when using all P panels. However, 

 as FIA moves to the annual system this will cause a short-term problem 

 for analysts who must report results from only one or two panels of data. 

 The problem also may make some of the more sophisticated methods for 

 combining panels inappropriate for small samples. For example, recent 

 data from 220 counties in Indiana, Missouri, and Illinois, where two 

 panels have been completed, showed an average of only two forested 

plots per county per panel. 

Additional discussion of the advantages and disadvantages of multiple or 

single panels is provided by Köhl and Scott (2000).

FIA uses panels to measure both current inventory and change. Change can 

be estimated in a multitude of ways. One method uses the net difference 

between two sequential, but different, panels. Assuming this approach 

involves independent samples, the variance of the difference is the sum of

the variances, roughly . Measuring different panels over time yields

estimates of net change, but only remeasured panels can provide information

about specific components of change behind the net change. The latter are

particularly useful for researching the dynamics of causation and associated

relationships. There, change is directly observed, so the variance is reduced

by the correlation, R, between measurements, roughly (1 – R2)s2/n. Alternatively, this can be expressed as the reduction in the variance of the difference

due to the covariance between occasions: 
71

 . The remeasured panels 

approach is generally preferred for its robustness, efficiency, and ability to 

isolate individual components of change.

5.2 Combining Panels

There is currently no prescribed core procedure for combining panels. Due 

to different spatial, temporal, and forest characteristics within and among 

regions, it is not clear if any single technique will work for all. Whatever 

estimation strategy is used to estimate current conditions, variance reduction 

usually can be attained by combing current data with earlier data from 

previous panels. Several estimation strategies have been devised to take 

advantage of previous data (Czaplewski 1995, Reams and Van Deusen 1999);

those now being investigated by FIA include:

1. The moving average (MA)

2. The temporally indifferent (TI) method

3. Modeling [updating plots, mixed estimators (Van Deusen 2002), Kalman 

 filters (Brockwell and Davis 1996), and various time series models 

 (Johnson and others 2003)] 

The first two are relatively straightforward, highly compatible with the estimators presented in chapter 4, and discussed in further detail in sections 5.2.1

and 5.2.2. The third technique, modeling, has so many possible variations 

that potential approaches are beyond the scope of this chapter. 

5.2.1 The Moving Average Method

Let P denote the number of panels to be combined for analysis. Let denote

the true quantity for panel , where ; and let denote the estimate of obtained using the appropriate technique from chapter 4. Note 

that each panel is treated as an independent estimate, which permits:

1. The weighting of individual panels 

2. Phase 1 stratification instruments to differ among panels (i.e., different 

 maps may be used to stratify different panels)

Using the above notation, the MA estimator is given by:

 

 (5.1)
72

where 

is a set of constant positive weights that sum to 1 across all

combined panels (Roesch and Reams 1999) 

The variance formula for is:

 (5.2)

where 

 for each panel is calculated as specified in chapter 4 

The MA estimator is appealing because it is simple and the use of previous 

panels can lead to a substantial reduction in the variance over the variance 

of an estimate based on an individual panel (Gillespie 1999).

Roesch and Reams (1999) suggest that equal weighting of all panels 

( , for ), or heavier weighting of more recent panels would

be appropriate in equation 5.1. Johnson and others (2003) have shown, with 

simulation based on FIA data, that in most situations the moving average 

with equal weights has the smallest mean squared error.

Because the MA is a weighted sum of estimates across all panels of interest, 

it can be viewed as an estimate of the attribute of interest at some time 

between the first and last years of the time period from which the panels 

were drawn. The specific point in time depends on the weights used, as well 

as the direction and magnitude of change that have influenced that attribute 

from panel to panel. Also, moving averages and related techniques result 

in estimators that dampen trends by obscuring annual fluctuations, and in 

that sense do not measure the current status of a finite population, but rather 

a temporal average of that population. Such estimators will make changes 

appear smaller than they are, and the use of older panels potentially creates 

a lag bias when estimating current conditions. However, in the absence of

some widespread catastrophic event, the smoothing and lag effects of moving

averages usually will be inconsequential and more than offset by the reduction in variance acquired from using the maximum number of available panels

(Johnson and others 2003). Still, there is some concern that potential lag bias

may mask time trends (Roesch and Reams 1999). FIA is now researching 

whether lag bias significantly influences trends associated with the attributes 

of interest occurring on forest lands. Obviously, in the presence of a widespread catastrophic event, lag bias cannot be ignored. The best way to adjust 

methods for such situations is also an area of ongoing research.
73

Finally, the MA approach does not require separate Phase 1 stratification 

for each panel. Thus, the weighting feature of the MA estimator may still be 

used when applying the same Phase 1 stratification to any or all panels. 

5.2.2 The Temporally Indifferent Method

The temporally indifferent (TI) method differs from the MA method in that 

all panels of interest are pooled into the equivalent of one large periodic 

inventory, and the same Phase 1 stratification is applied across all panels. 

Although this approach lacks some of the flexibility offered by alternative 

methods of combining panels, it does have advantages over periodic inventories because individual panels can be used to produce spatially unbiased 

estimates before the results of a complete periodic inventory are available. 

Note that in the simple random sampling case, the TI method is equivalent 

to the MA with weights proportional to the number of plots in each panel. 

The TI method is simpler than the MA approach in that estimation proceeds 

directly as specified in chapter 4, without the added complication of weighting (i.e., equations 5.1 and 5.2 are not used). In addition to simplicity, use of

the TI method may be advantageous when sample sizes per panel are small.

For example, when the MA approach is used in conjunction with stratification, the variance estimates for individual panels may be inflated by small 

sample sizes within each stratum. This could offset the variance reduction 

obtained through the MA estimator’s weighted sum. When this is the case, 

the larger sample sizes per stratum attained with the TI method could reduce 

the variance considerably more than the MA alternative. 

Finally, potential smoothing of temporal trends and lag bias associated with 

catastrophic disturbances present the same problems described for the MA 

estimator, with the added disadvantage that no weights are used to adjust for 

these effects. If weighting is necessary to overcome lag bias or to adjust for 

catastrophic events, the MA method is preferred. 

5.3 Literature Cited

Brockwell, P.J.; Davis, R.A. 1996. Introduction to time series and forecasting. New York: 

Springer Verlag. 420 p.

Czaplewski, R.L. 1995. Continuous adaptive monitoring of status and trends in ecosystem 

conditions. Proceedings of the 1995 Society of American Foresters conference. Bethesda, 

MD: Society of American Foresters. 482 p.

Gillespie, A.J.R. 1999. Rationale for a national annual inventory program. Journal of Forestry. 

97: 16-20.

Johnson, D.S.; Williams, M.S.; Czaplewski, R.L. 2003. Comparison of estimators for rolling 

samples using forest inventory and analysis data. Forest Science. 49(1): 50-63.
74

Reams, G.A.; Van Deusen, P.C. 1999. The southern annual forest inventory system. Journal of 

Agricultural Biological and Environmental Statistics. 4: 346-360.

Köhl, M.; Scott, C.T. 2000. A comparison of periodic versus permanent surveys. In: Burk, 

T.E.; Hansen, M.H., eds. Integrated tools for natural resources inventories in the 21st century: 

Proceedings of the IUFRO Conference. GTR–NC–212. St. Paul, MN: U.S. Department of 

Agriculture Forest Service, North Central Research Station: 94-103 p.

Roesch, F.A.; Reams, G.A. 1999. Analytical alternatives for an annual inventory system. 

Journal of Forestry. 97: 33-37.

Sheffield, R.M.; Thompson, M.T. 1992. Hurricane Hugo: effects on South Carolina’s forest 

resource. Res. Pap. SE–284. Asheville, NC: U.S. Department of Agriculture Forest Service, 

Southeastern Forest Experiment Station. 51 p.

Van Deusen, P.C. 2002. Comparison of some annual forest inventory estimators. Canadian 

Journal of Forest Research. 32: 1992-1995.
75

6 Notation for Equations 

6.1 Indices

H = number of strata

P = number of complete panels being combined for estimation

T = number of years necessary to complete P panels

N = number of sampling units in the population

n = number of plots

h = stratum index, 

 weight for stratum h (i.e., proportion of the population that is in 

stratum h)

 number of ground plots with Phase 1 assignments to stratum h

(For initial area tables, includes all plots sampled with any portion of a 

subplot (macroplot) in the population. For subsequent tables, any plots that 

are entirely nonsampled are excluded.)

d = domain of interest index, typically referring to a cell in a table, such as a 

combination of stand size class (column) and species (row)

i = plot index, 

j = subplot index, 

t = tree index, 

 number of condition classes on subplot (macroplot) j of plot i

assigned to stratum h

k = condition-class index, k=1,...,Khij

p = panel index, 

 weight assigned to panel p
76

6.2 Double Sampling

 total number of Phase 1 plots (pixels) sampled in a population

 number of Phase 1 plots (pixels) classified as belonging to stratum h, 

6.3 Area Related

 total acres in a population

 estimated acres within a domain of interest d

 zero-one domain indicator function, which is 1 if condition k on 

subplot (macroplot) j of plot i assigned to stratum h belongs to the domain 

of interest d

 zero-one in-sample indicator function, which is 1 if condition k on 

subplot (macroplot) j of plot i assigned to stratum h is within the boundaries 

of the population (Nonsampled areas are included in initial area tables in 

order to estimate their areas, but are zero otherwise. Missing values are also 

treated as zero values.)

 total area of the largest size plot on which area attributes are mapped 

(i.e., four times the subplot or macroplot area)

 mapped area (acres) of subplot (macroplot) j covering condition k

on plot i assigned to stratum h (The area is computed using the largest area 

mapped, which is the subplot except in the Pacific Northwest (PNW) where 

the macroplot is used.)

mh = mean proportion of stratum h mapped plot areas falling within the 

population

 total area normally used to observe the attribute of interest on a plot 

(i.e., four times the microplot, subplot, or macroplot area)

 area normally used to observe the attribute of interest (microplot, 

subplot or macroplot j) covering condition k on plot i assigned to stratum h

oh = mean proportion of stratum h observed-plot areas falling within the 

population
77

 total microplot area

 = mean proportion of stratum h microplot plot areas falling within the 

population

 proportion of plot i in the domain of interest d, for plots assigned to 

stratum h, adjusted for stratum h plots that overlap the population boundary

 mean of the plot proportions in the domain of interest d assigned to 

stratum h

 estimated proportion of the population in the domain of interest d

6.4 Tree and Plot Related

 attribute of interest for tree t on macroplot, subplot, or microplot j of 

plot i assigned to stratum h

 zero-one domain indicator function, which is 1 if tree t on subplot j

of plot i assigned to stratum h belongs to the domain of interest d

 attribute of interest for tree t on microplot j of plot i assigned to 

stratum h

 zero-one domain indicator function, which is 1 if tree t on microplot 

j of plot i assigned to stratum h belongs to the domain of interest d

 the estimate of the attribute of interest for plot i in stratum h in 

domain of interest d

 stratum h mean of the plot estimates of the attribute of interest in the 

domain of interest d

 mean of the attribute of interest in domain of interest d

 estimated total for the attribute of interest in domain of interest d

 ratio of means estimator (per acre, per condition, or per tree) in 

domains of interest d and d9

 the estimate for the attribute of interest for panel p

= the moving average estimator for the attribute of interest for P panels

oh

hd

hd
78
79

7 Glossary of Terms 

area change matrix: the area of the intersection of all combinations of 

initial and terminal condition classes between two points in time, compiled 

for a microplot, subplot, macroplot, or plot.

attribute: a discrete or continuous variable, usually associated with the 

classification or measurement of area or vegetation.

bi-areal plot: a plot design that incorporates two different plot sizes at each 

sample location for the purpose of measuring trees in two different treediameter ranges.

bioindicator: the use of a biological entity’s condition, frequency, and 

abundance as an indicator of ecosystem quality.

boundary (condition class): the border between two distinctly different 

condition classes. 

boundary (population): the border of a population or subpopulation.

census water: areas of permanent water > 4.5 acres or > 200 feet wide.

classified satellite imagery: a map (satellite image) that defines and 

displays the spatial arrangement of each classified stratum on a pixel basis.

components of change: the different subdivisions of the changes that can

occur to a tree between measurements, such as growth, mortality, and 

removals. 

condition class (or condition): the combination of discrete attributes that 

describe the area associated with a plot. These attributes include condition 

status (land use), forest type, stand origin, stand size, owner group, reserve 

status, and stand density, as well as other ancillary and computed attributes.

contrasting condition: The condition class that differs from the condition 

class located at the subplot center (for boundaries on the subplot or macroplot) or at the microplot center (for boundaries on the microplot), i.e., the 

condition class present on the other side of a boundary. 
80

cycle: one sequential and complete set of panels.

cycle length: the period of time required to measure a complete set of 

panels (synonymous with measurement cycle).

diameter at breast height (d.b.h.): the diameter of a tree stem, located at 

4.5 feet above the ground (breast height) on the uphill side of a tree. The 

point of diameter measurement may vary on abnormally formed trees.

diameter at root collar (d.r.c.): the diameter of a shrub-like “woodland” 

tree species, measured outside bark at the ground line or stem root collar.

domain: a class (or combination of classes) for which a population estimate 

is made for some attribute of interest. Domains are typically the row and 

column margins of tabular output tables (e.g., saw timber stands on publicly 

owned timberland).

double sampling for stratification: a sampling method whereby a large 

sample of plots are stratified in Phase 1, then a subsample are measured for 

all attributes in Phase 2. When the strata are homogeneous with respect to 

the attribute, then the estimators are more accurate versus simple random 

sampling.

down woody material: dead pieces of wood > 3.0 inches in diameter. 

Down woody material includes downed, dead tree and shrub boles, large 

limbs, and other woody pieces that are severed from their original source of 

growth or are leaning more than 45 degrees from vertical.

enhanced prescribed core variable: all FIA units produce a value for these

variables and there is a prescribed national protocol for measuring or calculating these variables. However, a given FIA unit is collecting data in greater

detail than national protocol requires, and the detailed data can be aggregated

to core specifications. Examples: fifth micro-plot in NE, additional disturbance codes beyond the prescribed codes.

forest (or forest land): land that is at least 10 percent stocked by forest trees

of any size, or land formerly having such tree cover, and not currently developed for a nonforest use. The minimum area for classification as forest land 

is one acre. Roadside, stream-side, and shelterbelt strips of timber must be 

at least 120 feet wide to qualify as forest land. Unimproved roads and trails, 

streams and other bodies of water, or natural clearings in forested areas are 

classified as forest, if less than 120 feet in width or one acre in size. Grazed 

woodlands, reverting fields, and pastures that are not actively maintained are 
81

included if the above qualifications are satisfied. Forest land includes three 

subcategories: timberland, reserved forest land, and other forest land. 

hybrid classification: a combination of supervised and unsupervised 

classification (e.g., guided classification).

indicator function: a variable with a value of 0 or 1 that is used to specify 

attributes of interest (e.g., white-oak growing-stock volume), and domains 

of interest (e.g., northern hardwood forest types) in the estimation process.

macroplot: a circular area with a fixed horizontal radius of 58.9 feet (1/4 

acre). Macroplot centers are co-located with subplot centers. Macroplots are 

used in the optional tri-areal design, primarily for sampling relatively rare 

events.

mapped plot: a plot that has been partitioned into unique and distinct condition classes by establishing the boundaries between them.

microplot: a circular area with a fixed horizontal radius of 6.8 feet (1/300 

acre), primarily used to sample trees less than 5.0 inches at d.b.h./d.r.c.

moving average: a weighted average of the estimates for distinct panels.

noncensus water: bodies of water from 1 to 4.5 acres in size and water 

courses from 30 feet to 200 feet in width.

nonforest: areas defined as nonforest land, census water, or noncensus 

water.

nonforest land: land that does not support, or has never supported, forests, 

and lands formerly forested where use for timber management is precluded 

by development for other uses. Includes areas used for crops, improved 

pasture, residential areas, city parks, improved roads of any width and 

adjoining rights-of-way, power line clearings of any width, and noncensus 

water. If intermingled in forest areas, unimproved roads and nonforest strips 

must be more than 120 feet wide, and clearings, etc., more than 1 acre in 

size, to qualify as nonforest land.

nonprescribed core variable: all FIA regions must produce a value for the 

variable; but there is no prescribed protocol for measuring or calculating the 

variable. Examples: tree volume, site index.
82

nonprescribed optional variable: a value is produced for the variable at the

discretion of the FIA regions, and there is no prescribed national protocol 

for measuring or calculating the variable.

other forest land: forest land other than timberland and reserved forest 

land. It includes available and reserved low-productivity forest land, which 

is incapable of producing 20 cubic feet of growing stock per acre annually 

under natural conditions because of adverse site conditions such as sterile 

soil, dry climate, poor drainage, high elevation, steepness, or rockiness.

panel: a sample in which the same elements are measured on two or more

occasions. FIA divides plots into five panels that can be used to independently

sample the population. 

periodic survey: a noncontinuous inventory system. A survey strategy 

whereby a set of inventory panels is measured simultaneously over a short 

time frame, often 1 to 3 years in the case of FIA, and there is a time lag, 

often many years, before the panels are remeasured.

phase 1: FIA activities related to remote-sensing, the primary purpose of 

which is to obtain strata weights for population estimates.

phase 2: FIA activities conducted on the network of ground plots. The 

primary purpose is to obtain field data that enable classification and summarization of area, tree, and other attributes associated with forest land uses.

phase 3: a subset of Phase 2 plots where additional attributes related to 

forest health are measured.

pixel: picture elements—the elements of a digitized picture. The resolution 

of a picture is dependent on the size and number of elements of which it 

consists.

plot: a cluster of 4 points arranged such that point 1 is central, with points 

2, 3, and 4 located 120 feet from point 1 at azimuths of 360, 120, and 240 

degrees, respectively. Each point includes a microplot, a subplot, and an 

optional macroplot.

population: a basic building block of land area for which the number of 

plots and the land area being sampled are known. Typically, this is the 

county, but some counties may be grouped into super-counties due to small 

numbers of forested plots or to mask a large landowner.
83

prescribed core variable: all FIA regions produce a value for these variables and there is a prescribed national protocol for measuring or calculating 

these variables. Examples: d.b.h., azimuth, distance, species.

prescribed optional variable: a value is produced for these variables at 

the discretion of the FIA regions; but, when measured, the protocol must 

conform to prescribed national standards. Examples: magnetic declination, 

subplot condition list, sapling damage, percent rough cull.

ratio of means: an estimator which is computed as the ratio of the means 

of two random variates (attributes), such as the volume per acre of forested 

land.

reserved forest land: land permanently reserved from wood products utilization through statute or administrative designation. 

sampling unit: the sampling unit is the basic unit of selection and observation. All FIA units use the center point of the 4-point cluster of subplots as 

the primary sampling unit.

simple random sample: a method of selecting n units out of the N such that 

every one of the samples has an equal chance of being chosen.

site index: the average total height that dominant and codominant trees in 

fully-stocked, even-aged stands will obtain at key ages, usually 25 or 50 

years.

site productivity class (or site class): the maximum mean annual increment in cubic feet per acre that can be expected in fully-stocked, natural 

even-aged stands.

sliver: a condition class that occupies less than 25 percent of a plot (less 

than one full subplot and not encountered on other subplots).

stocking: at the tree level, stocking is the density value assigned to a sampled

tree, usually in terms of numbers of trees or basal area per acre, expressed 

as a percent of the total tree density required to fully utilize the growth 

potential of the land. At the stand level, stocking refers to the sum of the 

stocking values of all trees sampled.

strata: nonoverlapping subdivisions of the population such that each primary

sampling unit is assigned to one and only one subdivision (or stratum). The 

relative sizes of these strata are used to compute strata weights. 
84

stratification: a statistical tool used to reduce the variance of the attributes 

of interest by partitioning the population into homogenous strata. It may also

 involve partitioning a highly variable but small portion of the population.

stratified estimation: estimation of population attributes using the total 

area of the population, strata means, and known strata weights. Strata means 

and weights are obtained from the stratification of the population either 

before or after the selection of sampling units.

subplot: a circular area with a fixed horizontal radius of 24.0 feet (1/24 acre),

primarily used to sample trees at least 5.0 inches at d.b.h./d.r.c.

subpopulation: a subdivision of a population for which the area sampled 

is known and controlled for, such as the area within a county in national 

forest ownership. Sub-populations are not necessarily a subset of one single 

population. 

super-county: a group of counties that have been combined to form a single 

population. Counties are combined into super-counties when the sample 

size for individual counties is too small. 

supervised classification: training sites with known properties are used to 

extract spectral statistics from an image data set by interactively identifying 

sites in the image. These statistics are used to establish starting values for 

cluster means, and a clustering algorithm is used to classify the image.

systematic sample: a method of selecting n units out of the N such that 

restrictions are placed on the samples chance of being chosen.

timberland: Forest land that is producing or capable of producing in excess 

of 20 cubic feet per acre per year of wood at culmination of mean annual 

increment (MAI). Timberland excludes reserved forest lands.

tri-areal plot: a plot design that incorporates three different plot sizes at 

each sample location for the purpose of measuring trees in three different 

tree-diameter ranges.

unsupervised classification: radiance values of an image data set are 

used in a statistical clustering algorithm. The clusters are labeled after the 

classification.
85

8 Web-Based Supplementary

Documentation 

The supplementary documents referenced in this manuscript are posted on 

the Web site at http://srsfia2.fs.fed.us/publicweb/statistics_band/stat_

documents.htm.

The purpose of these documents is to provide details about the algorithms, 

equations, and other specifics of the FIA National Program that are too 

technical for the chapter discussions. These documents are posted on the 

Internet because they are dynamic. Currently they are in various stages of 

completion; periodically they will be updated and revised to accommodate 

changes in protocol and demand for technical detail. More information may 

be added to this Web site in the future as unresolved and new issues are 

presented and addressed. Supplementary documents available at the time of 

publication include:

• “The Hexagon/Panel System for Selecting FIA Plots Under an Annual 

 Inventory”

• “Site Productivity Assignment for the North Central FIA Unit”

• “FIA Volume Calculations”

• “National Algorithms for Stocking Class, Stand-Size Class, and Forest 

 Type”

• “Examples of FIA Point-in-Time Estimation Procedures for Several 

 Common Cases”

• “Examples of FIA Change-Component Estimation Procedures for Several 

 Common Cases”

• “Computation of FIA Plot Expansion Factors”
86
Bechtold, William A.; Patterson, Paul L., Editors. 2005. The enhanced Forest 

Inventory and Analysis program—national sampling design and estimation 

procedures. Gen. Tech. Rep. SRS-80. Asheville, NC: U.S. Department of 

Agriculture, Forest Service, Southern Research Station. 85 p.

The Forest Inventory and Analysis (FIA) Program of the U.S. Department of 

Agriculture Forest Service is in the process of moving from a system of quasiindependent, regional, periodic inventories to an enhanced program featuring 

greater national consistency, annual measurement of a proportion of plots in each 

State, new reporting requirements, and integration with the ground sampling 

component of the Forest Health Monitoring Program. This documentation 

presents an overview of the conceptual changes, explains the three phases of 

FIA’s sampling design, describes the sampling frame and plot configuration, 

presents the estimators that form the basis of FIA’s National Information 

Management System (NIMS), and shows how annual data are combined for 

analysis. It also references a number of Web-based supplementary documents 

that provide greater detail about some of the more obscure aspects of the 

sampling and estimation system, as well as examples of calculations for most of 

the common estimators produced by FIA.

Keywords: Annual inventory, FIA, forest health monitoring, forest inventory, 

plot design, sampling frame.
The Forest Service, United States Department of 

Agriculture (USDA), is dedicated to the principle 

of multiple use management of the Nation’s 

forest resources for sustained yields of wood, water, 

forage, wildlife, and recreation. Through forestry research, 

cooperation with the States and private forest owners, 

and management of the National Forests and National 

Grasslands, it strives—as directed by Congress—to provide 

increasingly greater service to a growing Nation.

The USDA prohibits discrimination in all its programs and 

activities on the basis of race, color, national origin, sex, 

religion, age, disability, political beliefs, sexual orientation, or 

marital or family status. (Not all prohibited bases apply to all 

programs.) Persons with disabilities who require alternative 

means for communication of program information (Braille, 

large print, audiotape, etc.) should contact USDA’s TARGET 

Center at (202) 720-2600 (voice and TDD).

To file a complaint of discrimination, write USDA, Director, 

Office of Civil Rights, Room 326-W, Whitten Building, 1400 

Independence Avenue, SW, Washington, D.C. 20250-9410 

or call (202) 720-5964 (voice and TDD). USDA is an equal 

opportunity provider and employer.