United States

Department

of Agriculture

Forest Service

Rocky Mountain

Research Station

General Technical

Report RMRS-GTR-107

September 2003

Statistical Strategy for Inventorying and

Monitoring the Ecosystem Resources of

the Mexican States of Jalisco and

Colima at Multiple Scales and

Resolution Levels

H. T. Schreuder

M. S. Williams

C. Aguirre-Bravo

P. L. Patterson
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Abstract

Schreuder, H. T.; Williams, M. S.; Aguirre-Bravo, C.; Patterson, P. L. 2003. Statistical strategy for

inventorying and monitoring the ecosystem resources of the Mexican States of Jalisco and Colima

at multiple scales and resolution levels. Gen. Tech. Rep. RMRS-GTR-107. Ogden, UT: U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station. 15 p.

The sampling strategy is presented for the initial phase of the natural resources pilot project in the Mexican

States of Jalisco and Colima. The sampling design used is ground-based cluster sampling with poststratification based on Landsat Thematic Mapper imagery. The data collected will serve as a basis for

additional data collection, mapping, and spatial modeling products to be described in detail in later

documents. Estimation described in this document will be primarily useful for strategic planning at the state

and national levels. Because it is a pilot study, the approach for the actual statewide and national inventories

is likely to be modified as results are obtained and studied.

Keywords: spatial information, natural resources survey, strategic and management survey pilot study,

pixel-based system

The Authors ____________________________________________________

H. T. Schreuder is a Mathematical Statistician, U.S. Department of Agriculture, Forest Service, Rocky Mountain

Research Station, Fort Collins, CO.

M. S. Williams is a Mathematical Statistician, U.S. Department of Agriculture, Forest Service, Rocky Mountain

Research Station, Fort Collins, CO.

C. Aguirre-Bravo is Research Coordinator for the Americas, U.S. Department of Agriculture, Rocky Mountain

Research Station, Fort Collins, CO.

P. L. Patterson is a Mathematical Statistician, U.S. Department of Agriculture, Forest Service, Rocky Mountain

Research Station, Fort Collins, CO.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 1

Introduction ____________________

In Mexico, individual agencies address the extent,

status, and trends of selected natural resources in

response to agency missions. No agency or group of

agencies examines the interactions and interdependence among multiple natural resource components

from an integrated ecosystem perspective. Available

information often is at scales that have limited value

for state planning and policymaking. While useful for

strategic national level planning, this information has

limited use for supporting regional and local decisionmaking resource applications. Similarly, there is no

integrated program to periodically assess multiple

natural resources at regional scales, local scales, and

multiple resolution levels. Accordingly, the ecosystem

resource monitoring initiative, of which this document

is a part, will enable comprehensive assessments of

Jalisco’s and Colima’s natural resources for the management of their sustainability.

This ecosystem resource monitoring initiative addresses critical issues that stakeholders in the States

of Jalisco, Colima, and elsewhere in Mexico are facing

to ensure the environmental sustainability for present

and future generations. Land management agencies

charged with this responsibility have come to the

realization that current data and information are

insufficient to confront successfully the ecological and

economic challenges of ecosystem resource sustainability. In light of the above, the purpose of this

technical document is to describe the statistical approach for inventorying and monitoring the ecosystem

resources of the Mexican States of Jalisco and Colima,

at multiple scales and resolution levels. Only the

design and estimation theory for large-scale national

and statewide (strategic) surveys are presented here.

Design-based inference is used to ensure a minimum

number of assumptions.

Putting this strategy into action will enable us to

develop quality assured data, knowledge, and assessment syntheses of ecosystem resources condition and

Statistical Strategy for Inventorying and

Monitoring the Ecosystem Resources of

the Mexican States of Jalisco and Colima

at Multiple Scales and Resolution Levels

H. T. Schreuder

M. S. Williams

C. Aguirre- Bravo

P. L. Patterson

trends. The information will be scientifically credible

and technically defensible, and have the greatest social and economic utility for the purpose of planning

and decisionmaking for sustainability.

Overview of the Sampling

Design ________________________

Unlike many national level inventories, such as the

U.S. Department of Agriculture’s Forest Service Forest Inventory and Analysis program (FIA), the goal of

the Jalisco-Colima pilot project (JCPP) is to design an

integrated inventory system that provides both traditional assessments of population totals and means as

well as spatially realistic maps that describe the

location and distribution of various attributes of the

population. In other words, an inventory is designed

that meets strategic, management, and local needs.

The products produced by the inventory include reliable estimates of means and totals for important

ecosystem characteristics and maps describing the

spatial and temporal properties of the ecosystem.

In designing an integrated multiresource inventory

and monitoring system to evaluate the condition and

change of variables and indicators for sustainable

ecosystem resource management (forest, rangeland,

agriculture, wildlife, water, soils, biodiversity, and so

forth) one needs some baseline data for comparison.

Because one is generally dealing with complex systems, it is not wise to focus on only one or two variables

for ecological monitoring purposes. Also, analyzing

these variables independently of one another may lead

to incorrect conclusions because of their interdependencies. One approach is to model the spatial relationship between key indicator variables. In ecosystem

resource management, for example, this information

can then be used to identify forest habitats that are

either conducive or a deterrent to the presence of

ecologically important plant and/or animal species.

Techniques commonly used in describing spatial relationships between two or more variables include
2 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003

regression analysis and a variety of geo-statistical

procedures that take into consideration the spatial

dependency (Cliff and Ord 1981). The proposed ecosystem resource monitoring system will rely on information collected at different spatial scales of resolution

and sampling intensities (fig. 1) to provide detailed

information at the local level for ecosystem resource

planning and management purposes.

The JCPP is an integrated inventory and monitoring program that is designed to address numerous

objectives at different spatial scales. The components

of the inventory are broken down into five components, which are referred to as “levels,” each addressing different spatial scale and information needs. This

document provides a brief overview of all five levels of

the inventory and a detailed description of levels 1

and 2. Other documents will address levels 3, 4, and 5

in more detail.

Spatially Continuous Monitoring (Level 1)

Landsat Thematic Mapper (TM) data will be used to

provide a complete and uniform census of individual

Environmental Accounting Units (EAU) across jurisdictional domains (private lands, federal and state

lands, ejidos, communities,municipalities, regions, and

so forth). An EAU is a watershed used to assess

ecological and human activity assessment as well as

for resource decisionmaking and planning applications. This approach will provide data or measurements collected as a series of contiguous and simultaneous measures across land tenure units. It will also

provide the capability of monitoring EAU’s for changes

in spectral and spatial characteristics that can be

applied over a range of spatial and temporal scales

appropriate for addressing specific ecosystem resource

issues.

SPATIALLY CONTINUOUS LEVEL MONITORING

(REMOTE SENSING) – LEVEL 1

STATE LEVEL SUBSAMPLE MONITORING – LEVEL 3

NATIONAL LEVEL SUBSAMPLE MONITORING – LEVEL 2

REGIONAL LEVEL SUBSAMPLE

MONITORING – LEVEL 4

LAND-OWNER LEVEL

SUBSAMPLE MONITORING

LEVEL 5

SITE

SPECIFIC

LEVEL

MONITORING

LEVEL 6

INCREASING NUMBER OF SITES

INCREASING SPATIAL REPRESENTATION

INCREASING SITE CHARACTERIZATION

SCALE INTEGRATION

INSTITUTIONAL PROGRAMS INTEGRATION

SPATIAL AND TEMPORAL INTEGRATION

DATA INTEGRATION

INFORMATION INTEGRATION

Figure 1—Conceptual model for integration of monitoring design and institutional processes.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 3

Remote sensing data have been used primarily to

develop static maps of vegetation cover or land use to

guide management decisions in planning applications.

However, a current component missing in ecosystem

resource monitoring is its use for change detection and

spatial modeling of ecosystem properties. When compared to traditional plot data, remotely sensed data

can be quite helpful in detecting changes in the land

base. Many critical indicator variables such as site

productivity, presence or absence of threatened or

endangered species, and biological diversity are simply not being measured with sufficient spatial coverage and frequency to allow evaluation of current and

future trends. In our approach, Landsat TM data are

treated as an auxiliary variable while the field data

are treated as the primary variable of interest in

describing specific properties of the region at any

desired spatial scale (1 to 30 m). Appropriate use of the

approach can greatly increase the value of outputs

from monitoring programs (Metzger 1997).

The decision to use Landsat TM data over other

remote sensors (for example, SPOT [10 m resolution],

LEWIS [5 to 300 m resolution], Space Imaging [1 to

4 m resolution], Earthwatch [3 to 15 m resolution],

Earthwatch Quickbird [1 to 4 m resolution], OrbView

2 [1,100 m resolution], OrbView-3 [1 to 8 m resolution],

SPOT 4 Vegetation Sensor [1,000 m resolution], and

MODIS [250 to 1,000 m resolution]) is based on the

following reasons. The cost of acquiring data from

these high resolution sensors is significantly higher

than what it currently costs to acquire Landsat TM

data. Users of these sensors have shown that in terms

of modeling large-scale spatial variability (Metzger

1997), Landsat TM data are superior to currently

available SPOT data, even though the latter has a

finer resolution, but is also more expensive. Also the

resolution associated with some of these new sensors

are not the same across different spectral bands making it difficult to integrate with field data with the

same level of precision. For this approach to work, it is

important to have a wide range of spectral data at a

consistent resolution. An additional benefit is that the

spatial modeler involved in this work has considerable

experience in working with TM. Therefore, Landsat

TM data will be used as the remote sensing sensor of

choice until some of these other sensors have been

thoroughly evaluated.

In addition to the Landsat data, GIS grids of elevation, slope, and aspect will be developed from digital

elevation models. Grid coverages for each topographic

variable will be resampled (Resample function, nearest neighbor, Grid Module [ARC/INFO®, ESRI 1995])

to provide a 30 m spatial resolution so that an average

elevation, slope, and aspect can be assigned to each

pixel.

An independent ground sampling effort has been

undertaken to collect data that will be used to construct a classified map of the cover types of Mexico.

These ground data comprise approximately 2,600 purposively chosen points. At each point, the Global

Positioning System (GPS) coordinates and the cover

type are recorded. These data will be used to determine the spectral signature of the TM data and landscape characteristics associated with each cover type.

This will allow us to generate a preliminary land cover

map. Details on the techniques used to produce the

map will appear in a later document.

Design-Based Inference for Inventory and

Monitoring (Level 2)

The development of the sampling and plot designs is

complicated by the variety of indicators to be assessed,

the need to assess the ecosystem resources at a range

of scales, the need to monitor the indicators over time,

and the need to do so efficiently. To meet national level

objectives for ecosystem monitoring and large-scale

assessments, a traditional grid-based sampling design is used. The plot locations are treated as a simple

random sample of the landscape, and post-stratification is employed as a variance reduction technique.

This will be referred to as the Tier 1 inventory. This

proportion of the inventory is funded by the national

government.

The next level of the survey is designed to meet state

level inventory objectives. This is referred to as the

Tier 2 inventory. The only feature that distinguishes

the Tier 1 and 2 inventories is the increased sampling

intensity in Tier 2.

Model-Based Inference for Monitoring and

Inventory (Levels 3–4)

For the remaining objectives involving estimation at

local scales (levels 3–4), the plot data from the Tier 1

and 2 inventories will be enhanced with additional

ground plots to provide information needed to develop

geostatistical models. These models will be used to

describe the location and distribution of various resource attributes and to estimate key attributes at all

locations within the sampled population.

To ensure that the spatial variability in the study

area is captured, additional sample plots will be located to enhance the spatial models for use in local

management applications. The location of these plots

can be based on a prestratification scheme that locates

plots in areas with the highest errors in the spatial

models or where the spectrally derived strata may not

capture all of the unique spatial features on the

landscape. The primary sampling unit (psu) is a 90 m x

90 m square.
4 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003

Site Specific Monitoring (Level 5)

Because of the biological importance of certain

areas in terms of threatened and endangered plant

and animal species, there is a great need to initiate

species and/or site specific research and monitoring.

Very specific information will be collected at this

stage through specific projects if funding is available.

In addition to this information, data should also be

collected that is compatible with key indicator variables collected at other monitoring levels. All data

should be georeferenced to allow the integration with

information collected at the different levels using

spatially explicit models. Further development of

this level of the inventory is not possible without

knowing more specific details for each specific site

and the characteristics of the population to be studied (for example, a study of animal species would

require very different methods than would a study of

endangered mosses and lichens).

Analytical Issues

The analysis of the level 1 and 2 inventory data is

straightforward once the data have been edited and

stored in an electronic file. Validation data sets will be

made available using techniques described in Williams and Patterson (2002). An analysis program for

level 1 and 2 data incorporating the estimators discussed later in this paper is now being developed in the

computer language R (Thaka and Gentleman 1996).

Analytical techniques for levels 3 and 4 of the inventory will be more complicated, but software and training will be made available. Analyses for level 5 of the

inventory will be developed as needed.

Study Area _____________________

The Pilot Study Area consists of the Mexican Southwestern States of Jalisco and Colima with a continental area of approximately 9 million hectares (22.5

million acres). Though Jalisco is larger in area (90

percent), the state of Colima (10 percent) plays a very

distinctive role in the economy of the whole region and

diversifies the Pilot Study Area considerably. Four

major ecological regions provide the natural resources

and environmental conditions that make this region

one of the most prosperous ones of Mexico (fig. 2).

The eco-regions are: the Transversal Neo-Volcanic

System, the Southern Sierra Madre, the Southern

and Western Pacific Coastal Plain and Hills and

Canyons, and the Mexican High Plateau. Linked to

these ecological regions, there are several important

hydrological regions (HR) that drain to the Pacific

Ocean (HR12 Lerma-Santiago, HR13 Huicicila, HR14

Ameca, HR15 Costa de Jalisco, HR16 ArmeriaCoahuayana, HR18 Balsas, and HR37 El Salado). One

of the watersheds, the Lerma-Santiago Hydrological

Region, is connected to Lake Chapala, the most important source of water for the City of Guadalajara.

Precipitation ranges from roughly 300 mm/year in

some locations to more than 1,200 mm/year in the

higher elevations, with the principal precipitation

coming in summer monsoons. Spatially, the ecological

systems of this region cut across the boundaries of

other Mexican States. For example, several major

watersheds drain through the tropical and subtropical

forests of the State of Colima. Mostly in the State of

Jalisco, water from surface and underground sources

is heavily used for agriculture and industrial activities, though a significant portion goes to meet the

domestic needs of approximately 10 million people.

While on average Colima is wet, the lack of water in

the State of Jalisco threatens the sustainability of

urban and rural ecological and economic systems.

Most of the land (85 percent) in the State of Jalisco is

privately owned. Small private landowners are the

main driving force of economic development in agriculture, forestry, and rangeland economic activities.

In contrast to Colima, for example, a small portion of

Jalisco’s land is owned by ejidos (10 percent), communities (3 percent), and the government (2 percent).

Recently, as a result of trade liberalization brought

about by NAFTA policies, new industries have been

established in these two States and natural resource

utilization has increased due to high population

growth rates.

The region’s biophysical heterogeneity blends itself

to bring about unique habitat conditions for a large

diversity of plant and animal species. Within its boundaries, there are a significant number of species of

mammals and birds, many of which are severely

Figure 2—Geographic location of pilot study area in Mexico.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 5

threatened by human activities. Some of the plant and

animal species are endemic to specific locations within

the ecological regions that comprise the Pilot Study

Area. Extensive areas of pine-oak forest are home to

“specialty” birds such as the thick-billed parrot, the

Mexican-spotted owl, and woodpeckers. It is thought

that habitat loss is the single most important element

affecting bird populations in this ecosystem complex.

Not much is known about how (in other words, what,

when, where, why) plant and animal species are being

impacted by human activities. Water and other biological resources are an integral part of these ecological regions, whose products transcend geopolitical

domains and jurisdictions.

The Design and Estimation for the

Tier 1 and 2 Inventories __________

There is no “perfect” solution for the sample design

and estimators for a large-scale inventory. Limitations associated with remote sensing, GPS, and the

amount of time required by the field crews for data

collection influence the design of the survey (Williams

and Eriksson 2002; Overton and Stehman 1996). The

sampling strategy chosen for the JCPP is an efficient

and flexible strategy.

An efficient sampling design for collecting information over large geographical areas is a systematic grid

with equal spacing and a random start. It has the

advantage of spreading the sample units uniformly

throughout the population. However, the systematic

grid design tends to be the least efficient for spatial

modeling because the equal spacing between plots

provides less information for modeling purposes than

a design with uneven spacing. In order to avoid periodicity in the resource and to permit plots to occur at

random distances for spatial modeling, the plots will

be located randomly within hexagonal cells formed

by a triangular grid. This grid for the Tier 1 inventory

will uniformly cover all of Mexico with a spacing of

30 km x 30 km. To facilitate across border assessments, the grid for Mexico will be similar to the one

developed for the FIA inventory in the U.S.A. In

addition to the 30 km x 30 km grid, five more triangular grids of increasing intensities are available to

support sampling decision-making processes (25 km x

25 km, 20 km x 20 km, 15 km x 15 km, 10 km x 10 km,

and 5 km x 5 km). These grids can be used to intensify

the Tier 1 sample for meeting state level inventory

needs. This state level intensification is the Tier 2

component of the inventory. For the pilot study, the

Tier 2 inventory uses a spacing of 5 km x 5 km with the

Tier 1 ground plots being a subsample of the Tier 2

plots.

The field crews will locate all of the ground plots at

the UTM coordinates at the center of the TM pixel

given to them—accurate location of the points is important both for spatial modeling as well as future

relocation of these permanent plots. The grid density

will be set first to meet state needs. It will then be

intensified to meet other multiple scale needs, based

on available funding. The intensity can also be altered

in each of the main hydrological regions or municipalities. Plot locations will be kept secret. The opportunity

to intensify also exists for local areas within land

tenure units, EAUs, or administrative units, as funding is made available.

The other major advantage of using Landsat TM

data is that the pixels can be used to construct an area

frame comprised of equal area sample units. The

advantage of this area frame approach (Eriksson 1995;

Husch and others 1982) is that it is a true equal

probability sample regardless of the type of stratification used, with the inclusion probabilities of the sample

units (for example, trees, coarse woody debris, and

other vegetation) along the boundaries of different

subpopulations not needing adjustment. This is especially advantageous when the sample data will be

poststratified, because every division (stratification)

of the population creates new subpopulations with

new boundaries. These new boundaries would require

adjustments to the inclusion probabilities of all population elements (trees) along the new boundaries, and

it is unlikely the data needed for these could or would

be collected (see Williams and Eriksson 2002 for a

discussion of this topic).

For estimation, the plots will be poststratified using

the classified level 1 Landsat TM data. At a minimum,

the entire population and each EAU will be stratified

into areas that are predominantly forest or nonforest.

The plots will also be assigned to these strata, and

then the appropriate results for poststratified random

sampling will be applied (Cochran 1977; Schreuder

and others 1993). The estimators are given below. For

large areas, the sample size in each stratum will be

approximately proportional to the size of each stratum.

Description of the Ground Sample Unit

The primary sampling unit (psu) measures 90 m x

90 m (fig. 3) and is constructed with a 3 x 3 block of

Landsat TM pixels. The psu will be subsampled by

five 10 m x 10 m secondary sampling units (ssus).

Each primary sampling unit will be centered on the

coordinates assigned to it and will be laid out in a

north-south, east-west manner. PSU locations will

be verified using a GPS with an estimated accuracy

of within 3 m.

In order to maintain an equal probability sample

design and because the locational and registration

errors in remote sensing and GIS technologies prevent

an accurate subdivision of a pixel, a psu is considered

either totally within a population or totally out of a
6 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003

population (or stratum). Thus the population boundaries are redefined to follow the area frame population

boundaries created by the edges of the psus. A psu is

“in” when the center of the psu falls inside the population or stratum boundary. This creates a jagged stairstepped edge along the population boundaries, so a

small portion of the actual landmass of an area segment will lie within the boundary of another area

segment. The size of the difference should be minimal,

generally less than 0.05 percent for the watersheds in

Jalisco and Colima. For example, a minuscule quantity of the landmass of Colima may be included within

the area frame for the Jalisco land mass.

Because these will be permanent plots, the psu

center will be monumented on the ground. A sample of

5 of the 81 ssu’s will be selected for measurement,

using a circular plot of 5 m radius. This plot will be

referred to as the 5 m plot. One 5 m plot will be located

at the psu center. The other four 5 m plots will be

located at fixed angle directions and constant distance

with respect to the psu center (fig. 3).

Several kinds of subplots will be located within each

of the 5 m plots (fig. 4), and different measurements

will be made on each plot type. All large trees (≥ 12.5 cm

d.b.h.) will be measured on each of the 5 m plots.

Observed attributes will be specified in the field sampling and indicator measurement manuals. Saplings

(2.5 cm ≤ d.b.h. < 12.5 cm) will be measured on a

circular plot (3 m radius) colocated at the center of

each tree subplot. The term “3 m plot” will be used to

denote this plot. Within each of the 5 m plots will be

three square plots, each measuring 1 m x 1 m. These

will be referred to as 1 m quadrats. The first 1 m

quadrat will be located at the center of the 5 m plot.

The remaining two are located 6 m from the center plot

on a diagonal of the 5 m plot (fig. 4). Seedlings (height

≥ 30 cm and d.b.h. < 2.5 cm) will be sampled on three

30m 30m 30m

90m

PSU SSU

10 METERS

10 METERS

SECONDARY SAMPLING UNIT

5 m RADIUS

Figure 3—Plot layout for primary (psu) and secondary (ssu) sampling units.

10 METERS

10 METERS

SECONDARY SAMPLING UNIT

1 m2 cover subplot

10 m Diagonal Fuels

Line

5m radius tree plot

φ

3 m radius

Sapling subplot

Random angle

Figure 4—Layout of secondary sampling unit.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 7

1 m quadrats. In addition to the number of seedlings,

the percent cover of herbaceous plants, shrubs, and

tree species < 30 cm tall will be recorded.

To estimate fuel loadings, a 10 m transect with a

random orientation will be established. This will be

referred to as the 10 m transect. Line intersect techniques will be used to estimate fuel loadings of large

woody material (sound and rotten) ≥ 7.5 cm in diameter. All large woody material intersecting the 10 m

transect will be counted and cross-sectional areas

measured by genus. Medium size woody materials

(2.5 to 7.4 cm in diameter) that are intersecting the

first 5 m of the transect will be counted. In addition,

the first, center, and last 1 m of the diagonal transect

will be used to count fine woody materials (0.01–2.4 cm

in diameter). In each case, the mean height of fuels in

each sampled diameter class, as well as the slope of the

diagonal transect, will be measured and reported.

Soil attributes will be observed on each 5 m radius

plot. Any destructive soil samples will be collected on

the west side (270 degree Azimuth) of the primary

sampling unit at a distance of 5 m of the plot boundary line.

Most of the indicator variables are compatible with

those used by the USDA Forest Service and Canadian

ecosystem resource monitoring programs. Other indicator variables can be integrated into this pilot study

as resources become available and the need dictates to

ensure comparability and interoperability of indicators with participating government agencies from the

United States of America and Canada.

Selection of the Ground Plot Locations

and the Assignment of Measurement

Periods

The system used to derive the Jalisco/Colima Pilot

Project (JCPP) ground plot locations was developed to

facilitate cross-border analyses between the United

States and Mexico. The Environmental Monitoring

and Assessment Program (EMAP) of the Environmental Protection Agency (EPA) originally developed it.

EMAP developed a worldwide sampling grid to evaluate status, change, and trends in the condition of

ecological resources. White and others (1992) used a

truncated icosahedron as the basis for the grid of

the Forest Health Monitoring Program of the U.S.

Forest Service. The truncated icosahedron was made

up of 20 hexagon faces and 12 pentagon faces that give

it a “soccer ball” appearance (fig. 5).

The plot locations were arranged on a triangular

systematic grid with a random start, with plot locations falling at the intersection of a triangular grid

with approximately a 27 km point spacing. This grid

allows for a 4-year time span between remeasurement

of each ground plot on an interpenetrating rotating

panel design. Thus, the entire grid and each individual

panel is evenly distributed within the area, with a

sampling intensity of approximately one ground plot

for every 6,800 hectares (167,000 acres) for the FHM

program.

The JCPP will adopt the same sampling intensity as

FIA with one ground plot for every 2,400 hectares and

maintain across-border consistency by meshing the

hexagon grid covering Mexico with the existing grid

covering the U.S.A. However, with the increased focus

on spatial modeling, the placement of ground plots

within each hexagon will not be on the systematic grid.

Instead, the plot location within each hexagon will be

completely random. This is done by generating a

random (x, y) coordinate that falls within each hexagon. The justification for this practice is that the

estimation of correlograms and variograms is difficult

when ground plot locations are equally spaced, especially when the scale of spatial patterns is expected to

be much finer than the spacing of the grid. For the

purpose of estimating population means and totals,

this sample is still treated as a simple random sample

of the land base, as is common with most environmental surveys. The justification for this practice is summarized by Ripley (1981: 19–21).

The purpose of the hexagon grid is twofold. In the

first place it is used to establish a pseudo-systematic

sample (with a random start) of ground plots across

the JCPP study area. Secondly, it is used to allocate

ground plots to each yearly panel so that a good spatial

representation of all conditions is represented in each

panel. The hexagon grid has no other use in the

estimation process. The resulting sample can be best

described as a hybrid of centric- and nonalignedsystematic sampling (Ripley 1981: 19–21).

The assignment of each plot to a panel is loosely

based on the original design used by Forest Health

Figure 5—Hexagon tessellation of the Earth.
8 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003

Monitoring (FHM) and the current FIA design. The

EMAP grid was developed for interpenetrating

rotating panels of 3, 4, 7, 9, and 11 years or any

multiple. The FIA design is designed for sampling

periods of 5, 7, or 10 years. The States of Jalisco and

Colima are tentatively planning a 5-year period of

remeasurement.

Estimators for the TIER 1 and 2

Inventories

The estimators for the Tier 1 and 2 inventories are

simple. The purpose of this section is to present the

estimators in detail. This allows the readers to progress

only as far into the notation as necessary. Further

justification for this approach and simulation results

can be found in Williams and Patterson (2003).

One topic that will not be discussed in detail is the

method used to assign stratum values to each 90 m x

90 m psu. Instead, it is assumed that an effective

stratification algorithm will be developed for various

regions that will combine the individual pixel attributes, cover map information, and other auxiliary

information contained within the area frame. The

algorithm may reflect regional forest and landscape

characteristics and may depend on the type of analysis

being performed. However, at this point in time there

is too little known about available information to

define a single technique for the study area.

Slightly different approaches are used to derive the

estimators of various cover types versus the estimators associated with vegetation. Thus, these topics will

be treated separately.

Estimators for total area of land by cover

type or condition classes

To estimate the total area of forest or other condition

class, the five 5 m plots of the psu are viewed as the

second stage of a two-stage cluster sample. Multistage

cluster plot sampling results are used to divide each

psu into M = 81 secondary sampling units, from which

a sample of size m = 5 is chosen (fig. 3).

While the estimators derived from the smaller plot

sizes (for example, the 3 m and 1 m plots) can be

combined with the one derived from the 5 m plots, the

nested structure of these plots will produce highly

correlated estimates. This correlation between the

estimators of forest area is such that essentially no

reduction in variance exists when estimating forest

areas on anything other than the largest plot used

(Williams and Patterson 2003). Thus, the area of

forest or condition class is estimated using only the

information from the largest plot size implemented.

Regardless of the plot size or the method of assignment to a stratum for each psu, the goal is to estimate

the proportion of specific condition classes in each

stratum. To accomplish this, then n ground plots are

treated as a poststratified sample with a random

sample size of nh in each stratum. For each ground

plot, the proportion of the psu covered by forest is

either estimated by a subsample of ssu’s or by measuring the proportion of the psu covering the condition

class of interest. Thus, given that the assumptions are

reasonable, the estimator of the area of forest is a poststratified, two-stage cluster sample (Cochran 1977),

given by

ˆ

A Ap A ˆ , N

N

p

mn

F F

h

h

H

hij

F

j h

m

i

nh

= =















= = =  ∑ ∑∑1 1 1

[1]

where phij

F

is the proportion of forest in the j

th 5 m

subplot of the i

th psu within the hth stratum.

Since the variance of ˆ

AF is equal to A Var pF

2 ( ˆ ), it

suffices to determineVar pF ( ˆ ). The derivation of the

variance combines results from the variance of a

simple random poststratified sample and the variance of a stratified two-stage sample (Cochran

(1977), §5A.9 and §10.9). Conditioning on the nh’s,

Var p Var E p n E Var p n F Fh Fh ( ˆ ) [[ = + ˆ ]] [ [ ˆ ]].Since for

fixed stratum sample sizes, pˆF is an unbiased estimator of pF , the first term is zero. For fixed nh,

Var p N

N nN

S m M

mn

S F

h

h

H

h h

h

h

h ( ˆ ) / = 



 



 − 



 



 + − 













 =

∑

1

2

1

2

2

11 1 2

= 



 



 − −





 



 































−

=

∑ 1 11

1

2

1

2

2

2

2 1

2

n

N

N

S

m M

S N

N

S

h h

H

h

h h

h Figure 6—Numbering of each hexagon for h [2]

the panel assignment.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 9

(Cochran 1977). The next task is to find the expectation of 1/nh. Assuming the sample size is sufficiently

large to preclude nh = 0 then to order n

–2,

E

n nN N

N N

h h nN N

h

h

1 1 1

2 2











= + −

( /)

/

( /) .

Using results derived in Cochran (1977) it can be

shown that an unbiased estimator for the above approximation of Var pFˆ [ ]is given by

var p n

N

N

n

N s nNm M s NN n s F

h

H

h h h h h [ ] ˆ = − , 



 



 + − 



 



 









+ − 



 



 =

∑ 1 1 1 1 1 1

1

1

2

2

2

2 1

2

[3]

where s1h

2 and s2h

2 are the between cluster and within

cluster sample variances for stratum h(Cochran 1977).

Due to the large number of psu’s (N) in comparison to

sample size (n), the second term within the square

brackets is small relative to the first term for both the

Tier 1 and 2 inventories.

Vegetation or tree estimators

In this section, the estimator that utilizes the tree

information from the entire ground plot is given.

Unlike the estimator for forest area, where the assumption is made that the psu is divided into secondary sampling units, a different approach is taken. This

is because the tree and other vegetation attributes

are gathered from transect samples and across the 1 m,

3 m, and 5 m plots, which would require different M

values and the calculation of the correlations between

the estimators derived from each of the plot sizes.

Thus, the proposed estimators of tree and other vegetation attributes within each psu are expressed as a

triareal sampling design comprised of three different

size subplots (Husch and others 1982). The estimator

for the total within each psu is expressed using a

Horvitz-Thompson estimator. Two assumptions are

necessary. The first is that systematic nature of the

three subplots within each psu is equivalent to having

the points randomly located within the psu. In the

literature, this assumption has been justified by assuming that the location of trees in the forest has a

random pattern within the psu (de Vries 1986). When

this assumption is not realistic, the variance estimator tends to overestimate the variance. The second

assumption is that none of the points falls close enough

to the boundary of the psu so no adjustments to the

inclusion probabilities of trees are required. This allows a constant inclusion probability to be assigned to

all trees tallied on each of the three types of subplots.

This assumption is necessary because it is impractical

to obtain the actual probabilities of selection for border

trees.

The n ground plots are treated as a post-stratified

sample of psu’s, with a random sample size nh in each

stratum. Within each psu, a tree-centered circle, or

1 m quadrat in the case of seedlings (Husch and others

1982) is placed around each tree. The size of the circle

is determined by the diameter at breast height limits,

which in turn determine the inclusion probabilities.

Then a tree is included in a sample drawn by a single

point if the point lies within the tree-centered circle.

Let a

1m, a3m, a5m denote the area of the 1 m quadrat-

, 3 m-, and 5 m radius plots respectively and let apsu =

8,100m

2 denote the area of the psu. The probability

that a tree is included in the sample drawn by a single

point within the psu depends on the size of the tree and

is defined as:

a

1m/aPSU

π = a

3m/aPSU,

a

5m/aPSU

where a

1m is the area of the 1 m quadrat, a3mis the

area of the 3 m radius plot, and a

5m is the area of the

5 m radius plot.

The estimator given here is for the 3 m and 5 m

radius plots, where m = 5 plots are sampled within

each psu. The estimator for the 1 m quadrats is defined

analogously with the exception being the change in

sample size to m = 15. Estimation for the psu is based

on the selection of 5 points within the psu and measurements of the subplots of 3 m and 5 m radius

respectively centered at these points. Let Thij be the

number of trees tallied at the j

th point of psu i in

stratum h. Then the estimator of the total of the tree

attributes within the psu for the j

th point of psu i, in

stratum h, is given by

ˆ

Y y

hij

hijt

t hijt

Thij

=

=

∑

1 π , with yhijt and πhijt respectively the

value of interest and the probability of selecting tree t

at point j, psu i, in stratum h,

with the estimator for the total of the psu derived from

the m = 5 points being

 ̃ ˆ Y . m hi hij Y

j

m

=

=

∑1

1

An unbiased estimator of Var Yhi

 ̃ [ ]is given by

var Y m m Y Y hi hij hi

j

m

 ̃ ( ( )) ˆ  ̃ . [ ] = − − ( ) −

=

1 ∑

1

1

2

This estimator is derived by summing across the psus

and strata. Then the estimator is

Y N ˆ  ̃

n

h Y

h

H

h

hi

i

nh

=

= =

∑ ∑

1 1

1 . [4]
10 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003

An approximation of the variance of the estimator is

given by

Var Y N N

N n

N

h N n

H

h h [ ˆ] ≈ + − 



 



 



 



 =

∑ 2

1

2

1 1 1

[5]

VY N N

NnN

N

N n

hi S h hh [  ̃ ] , + − 



 



+ − 



 



 









 2

2

1 1 2 1 1

where V Yhi hi Var Y N i

N

h

h [ ˆ ] [  ̃ = ] / ∑ =1

S YY N h hi i

N

h h

2 h

1

2 = − −1 = ∑ ⋅ and ( ) / ( ).

An unbiased estimate of the above approximation

of Var Y[ ˆ] is given by

var Y N N

N n

N

h N n

H

h h [ ˆ] = + − 



 



 



 



 =

∑ 2

1

2

1 1 1

[6]

var Y N N

NnN

N

N n

s hi

h h

h [  ̃ ]+ −  ̃ , 



 



+ − 



 



 









 2

2

1 1 2 1 1

where var Y var Y n hi i

n

hi h

h [ ˆ ] [  ̃ = ] / ∑ =1

and  ̃  ̃  ̃ s YY n / . h hi h i

n

h

2 h

1

2

= − 1 ( )⋅ ( ) − ∑ =

The above estimation assumes that there are no

plots or subplots that cannot be measured. A nuisance

that arises with almost any type of field sampling with

plots is that parts of the psu or subplots will fall

outside the population of interest or are inaccessible to

sampling either because of difficulty of terrain or the

necessity to trespass on private land. Not all landowners will allow the crews on or across their land even to

measure other lands. How to treat inaccessible plots or

subplots is described in the appendix. The following

estimator deals with the situation of missing plots or

subplots. Poststratification, as considered above, is

not shown in the below formulation since it is a

straightforward extension of what is given by simply

applying the estimator shown to each stratum.

The most appropriate estimator to be used in estimating population totals with missing subplots is:

ˆ ˆ / ˆ Y A AY A s ijk ijk

k

n

j

n

i

n

ijk

k

n

j

n

i

s ss ss n s

= 





























= = = = = =  ∑ ∑ ∑ ∑ ∑∑1 1 1 1 1 1

[7]

and

ˆ ˆ / ˆ Y A AY A ij ij

j

n

i

n

ij

j

n

i

s s n

= 





























= = = =  ∑ ∑ ∑ ∑1 1 1 1

[8]

where ˆ

Aij and ˆYij are the estimated sampled area and

value of interest respectively in subplot j of plot i, n is

the number of plots in the sample, ns is the number of

subplots in the sample for plot i where ˆ

Ys is the estimator for subplot size s and subplot ss, and mi is the

number of subplots in the sample for plot i (Max and

others 1996). We recommend the use of bootstrap

variance estimators here as discussed in Schreuder

and others (1993).

Line intersect sampling estimators

The line intersect sample (LIS) is drawn using m =

5 randomly oriented lines with the centers covering

each of the 5 m plots. The estimator employed is the

LIS estimator as described in Kaiser (1983) and

Gregoire (1998), with the key difference being that the

sample locations are a systematic subsample of the

psu rather than five random locations. There should

be little if any detectable bias in the resulting estimators, but the variance estimator should overestimate

the true sample variance. The LIS estimator for the

amount of woody debris on the psu estimated from the

m = 5 and L = 10 m lines with random orientation is

given by

 ̃ ˆ

cos

Y , mY

m

a

L

g

hi hij

j

m psu

j

m hijk

k hijk

Pj

= =

= ==

∑ ∑∑ 1 1

1 11 2

π

δ [9]

where ghijk is the cross-sectional area of the log measured perpendicular to the long axis of the log (m

3

) for

the kth piece of debris on the j

thtransect, Pj is the

number of pieces of debris intersected by the j

th line,

and δhijk is the slope of the log. In relatively flat terrain,

δhijk, may be set to zero. However, a bias in excess of 10

percent can occur whenever the slope exceeds 25

percent. The derivation and further details for this

estimator can be found in Kaiser (1983 example 2c).

The variance estimator given in equation [6] applies.

Other estimators

While area and vegetation totals are the primary

forest attributes that are estimated, other estimators

are also of interest. The first class is mean per tree

estimators (for example, average number of conks or

disease agents per trees). The form of this estimator is

ˆ ˆ/ ˆ Y YN = , where Nˆ is the estimator of the number of

trees (eq. [4] with yhijt = 1 ). Another class of estimators

is used to assess change. These have the form

∆ ˆˆ ˆ YY Y = 2 1 − , where ˆ

Y2 and ˆY1 are the estimators at

times 2 and 1, respectively.

Variance estimators for the mean per tree and difference estimators are
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 11

var Y

N

var Y Y

N

var N Y

N

( ˆ) cov Y N ˆ [ ( ˆ) ˆ

ˆ ( ˆ) ˆˆ ( ˆ, ˆ ≈ + − )], 1 2

2

2 [10]

and

var Y var Y var Y cov Y Y ( ˆ) ( ˆ ) ( ˆ ) ( ˆ , ˆ ∆ = + − ). 2 1 21 2 [11]

Alternative estimators

At this point in time there are a number of alternative estimators that may offer advantages over the

estimators given above.

In various circumstances we may have complete

knowledge on a covariate associated with the variable

of interest for which we know all the values in the

population or we can get those with relative ease.

Usually this information is combined with the information on the variable of interest measured on a

subsample of the units in the population. Denoting by

y = variable of interest and x = covariate, numerous

estimators are possible. We focus only on the generalized regression and the ratio-of- means estimator, the

others are generally not desirable.

A very general, efficient estimator is the generalized

regression estimator developed by C. E. Sarndal:

ˆ

Y ny aN bX x // / gr i

i

n

i gr i

i

n

gr i i

i

n

= + − 



 



+ − 



 



 = ==

∑ ∑∑

1 11

ππ π 1

= +

= =

∑ ∑ y e ˆ / i

i

N

i

i

n

i

1 1

π [12]

where

y a bxe y y ˆ , ˆ , i gr gr i i i i =+ = −

a y vb x v v gr i i i gr i i i

i

n

i

n

i i

i

n

= − 









 = = =

∑ ∑ /( ) /( ) π ππ ∑ / /( )

1 1 1

1

b v xy v y v x v gr i i i i ii i i i i i i

i

n

i

n

i

n

i

n

= − 







 = = = =  ∑1 ∑ ∑ ∑

1 1 1 1

/( ) /( ) /( ) /( ) π π ππ

/ /( ) /( 1 ) /( ) 2

1

2

1 1

ππ π ii i i i i i i

i

n

i

n

i

n

v x v xv −





























 = = =  ∑ ∑ ∑

with variance:

VY e e gr i

i j

N

j ij i i j j ( ˆ ) ( / ( )( / / ) = − −

≠

1 2∑ 2 ππ π π π [13]

with 2 possible variance estimators given in

Schreuder and others (1993) but we recommend

using a bootstrap variance estimator here. This is

the estimator to be used when possible.

Another estimator that is not a special case of

the above is called the generalized ratio of means

estimator:

ˆ // / ( ˆ / ˆ Y y x XY X X) rm i i i i

i

n

i

n

= HT HT





 



 =

= =

∑ π π ∑

1 1

[14]

with approximate variance

V Y( rm HT HT HT HT V Y RCov Y X R V X ˆ ) ( ˆ ) ( ˆ , ˆ ) ( ˆ = −2 + ). 2

[15]

There is a good discussion in Schreuder and others

(1993) a variance estimation but, we recommend using a bootstrap variance estimator. Both the generalized regression and the ratio-of-means estimator are

asymptotically unbiased.

There will be opportunities to use ˆ

Ygr in the pilot

study. With the heavy emphasis on remote sensing

platforms we will examine what we can do with that in

relation to the ground sampling to obtain improved

estimates for some parameters.

Variance estimators can also be derived that utilize

resampling techniques, with the most common being

bootstrap and jackknife estimators. Bootstrap

samples are generated by repeating the sampling

design with replacement at the same sample sizes as

the actual sample design and then generating estimates for each bootstrap sample. The variance between those estimates is the variance estimate for

the estimator of interest. Usually 2000 bootstrap

samples are generated. This is applicable for all

estimators above.

Strategy for the Remeasurement

of Plots ________________________

During the first year, all plots will be measured.

Following 2 years of reporting and analysis, a 5-year

remeasurement cycle involving the remeasurement of

a systematic and representative subsample will begin.

The remeasurement cycle is not decided at this time

but will be at most 5 years.

Landsat TM imagery of the study area will be obtained just prior to the periodic reports, such as every

five years. The only criterion is that the image be

cloud free to ensure no loss of information in modeling the spatial relationship between the field data and

the Landsat imagery. This imagery must then be

classified into forest and nonforest at a minimum.

During the second measurement cycle, it may also be

advantageous to add change strata (forest to nonforest and nonforest to forest).

All permanent plots will be remeasured at least

once during a 5-year cycle. The plots will be divided
12 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003

into five or fewer systematically ordered panels, each

representative of the whole study area. One panel

will be measured each year.

For site-specific monitoring plots, the remeasurement

cycle will depend on the specific objectives of the

research study, or survey, as well as the funding

available. Variables and measurement protocols will

be compatible across plot types and monitoring levels.

When the same population is repeatedly sampled,

the sampler is in a unique situation to make realistic

cost and variance estimates and to use this information to apply techniques that lead to optimum efficiency of sampling (Cochran 1977). The current practice of remeasuring forest inventory plots on a yearly

basis is to divide the sample plot into K groups (panels)

and remeasure one group each year for T years. This

allows us to estimate change on a T-year cycle, but the

disadvantage of this approach is that the remeasured

sample plots provide limited information on year-toyear changes. As an alternative, one could randomly

sample, with replacement, P percent of the sample

plots each of the T years. At the end of the T years

some plots will have been remeasured zero times, one

time, two times, three times, and so forth, at a 1 year

interval, 2 year interval, and so forth. Constraints

could be added to the sampling protocol to ensure that

all sample plots are remeasured at least once during

the T-year cycle. This would provide the information

necessary to estimate current values as well as yearly

changes.

To this point, the focus has been on the estimation of

forest area and vegetation attributes for a single

panel. In an annualized survey, the individual estimators for each panel are combined into a single overall

estimator. Individual panel estimates will, however,

have less precision due simply to the smaller sample

size measured each year. With this in mind, it has been

often suggested that the estimator of the current

condition should take advantage of data collected in

the previous years (panels).

There have been several different estimation strategies proposed that would take advantage of previous

data. One estimator of the current population statistic, ˆ

Y , that appears often in the literature is the

moving average (MA) given by

ˆ ˆ , Y wY MA t t T h ,

h

T

= − + tT h

=

∑ − +

1

[16]

where ˆ

YtT h − + , h = 1,...,T, is an unbiased estimator of

YtT h − + and wtT h − + is a set of constants which sum to 1.

The constant, T, is equal to the number of panels in

the survey, or equivalently, the number of years until

the entire inventory is completed. If 20 percent of

the plots selected are sampled each year then T = 5.

An advantage of the MA estimator is that it will have

a substantially lower annual variation than any one

of the yearly estimators. This fact may appeal to

consumers of FIA data, who may distrust estimates

that vary too much from year to year (Gillespie 1999).

The major problem with the MA estimator is that it

is design biased as an estimator of the current year’s

population parameter. Particularly major changes in

the population from one year to the next could cause

serious bias. The rationale for the moving average

estimator is that by using past data you are “borrowing strength” in terms of sample size, from previous

years. While this creates a “lag” bias when estimating

current conditions, it is more than compensated for by

a reduction in variance. This unequal trade between

variance and bias leads to a lower mean squared error

for the moving average estimator.

Another problem is selection of the weights. Roesch

and Reams (1999) suggest that equal weights could be

used for each year but note that equal weighting might

mask time trends. However, the equal weighting estimator can be thought of as an unbiased estimator for

the population parameter at some time approximately

in the middle of the rotation cycle, so if the time trends

are long enough in duration, the equal weighting

estimator will react to the trends. Eriksson (unpublished manuscript) has shown that if the population

parameter is relatively constant over time, the equal

weighting estimator is approximately optimal in terms

of a squared error criterion. In order to track time

trends more accurately, more weight could be placed

on more recent estimates, but this will increase the

variance of the estimator. Johnson and others (2002)

also tested numerous estimators using a data set

comprised of FIA plot data and found that the MA

estimator has the smallest mean squared error in

almost every case studied. Thus, in the absence of a

catastrophic event or other large-scale change, the lag

bias is expected to be small in comparison to the

variance, and the MA estimator with equal weights

can be considered a nearly unbiased estimator of

current forest conditions. If a catastrophic event such

as a large fire or a hurricane occurs, then the use of a

different estimator will have to be considered. Johnson

and others (2002) present several estimators that can

be used in such a situation.

The panel system used by the FIA program produces

independent yearly estimates from each panel, hence

the variance of ˆ

YMA t, is

VY w VY MAj t T h h

T

tT h [ ˆ ] [ ˆ = ]. − + = ∑ − +

2

1 [17]
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 13

Summary and

Recommendations ______________

The statistical strategy presented in this document

significantly benefits from the experience on national

forest surveys in Canada and the United States. In

light of how rapid technology has been evolving, this

strategy incorporates new design elements so that

remote sensing and land measurement protocol processes can be unified for purposes of making reliable

statistical inferences. Consistency of data collection

protocols at national and state levels is essential for

this strategy to work properly and generate estimates

of known confidence. Its implementation on the Pilot

Study Area will produce valuable information for

improving design elements and data analysis approaches. How this strategy can be integrated with

spatial estimation strategies constitutes one of the

questions to be addressed through the implementation of this project.

References _____________________

Cliff, A.D. and Ord, J.K. 1981. Spatial processes: models and

applications. London: Pion. 266 p.

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

Wiley & Sons. 428 p.

De Vries, P. 1986. Sampling theory for forest inventory. New York:

John Wiley & Sons. 399 p.

Efron, B and Tibshirani, R.J. 1993. An introduction to the bootstrap.

New York: Chapman and Hall. 436 p.

Eriksson, M. 1995. Design-based approaches to horizontal point

sampling. Forest Science 41: 890-907.

Gillespie, A.J.R. 1999. Rationale for a national annual inventory

USDA Forest Service North Central Forest Experiment Station

Research Bulletin NC-165.

Gregoire, T.G. and Scott, C.T. 1990. Sampling at the stand boundary: A comparison of the statistical performance among eight

methods. In: Proc. XIX World Forestry Congress IUFRO; 1990

August 5-11; Montreal, Canada. Publ. No. FWS-3-90. Blacksburg,

VA: Virginia Polytechnic Institute and University: 78-85.

Husch, B., C., Miller, C.I. and Beers, T.W. 1982. Forest mensuration. 3rd ed. New York: John Wiley & Sons. 402 p.

Johnson, D. S., Williams, M.S. and Czaplewski, R.L. 2003. A test of

some estimators for rolling samples using Forest Inventory and

Analysis data. Forest Science 49: 50-63.

Kaiser, L. 1983. Unbiased estimation in line-intersect sampling.

Biometrics 39: 965-976.

LePage, R. and Billard, L., eds. 1992. Exploring the limits of

bootstrap. (Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics Section). New York: John

Wiley & Sons. 426 p.

Max, T.A., Schreuder, H.T., Hazard, J.W., Oswald, D.D., Teply, J.

and Alegria, J. 1996. The Pacific Northwest Region vegetation

and monitoring system. Research Paper RP-493. Portland, OR:

USDA Forest Service, Pacific Northwest Research Station. 22 p.

Metzger, K. 1997. Modeling forest stand structure to a ten meter

resolution using Landsat TM data. Fort Collins, CO: Colorado

State University. 123 p. Thesis.

Overton, W.S. and Stehman, S.V. 1996. Desirable design characteristics for long-term monitoring of ecological variables. Environmental Ecology and Statistics 3: 349-361.

Schreuder, H.T., Gregoire, T.G. and Wood, G.B. 1993. Sampling

methods for multiresource forest inventory. New York: John

Wiley & Sons. 446 p.

Thaka, R. and Gentleman, R. 1996. R: a language for data analysis

and graphics. Journal of Computational and Graphical Statistics

5: 299-314.

Williams, M.S. and Eriksson, M. 2002. Comparing the two paradigms for fixed-area sampling in large-scale inventories. Forest

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estimation techniques for established surveys using remotely

sensed and ground data. Forest Science 49: 110-118.
14 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003

Appendix ______________________

Definition of Symbols

A = Total area in population in ha

N = number of sample units in the population

n = number of sample units in the sample

ns = number of subsample units in each sample unit

Y YN i

i

N

=

=

∑

1

/ = average per sample unit in the population for the variable of interest

Y = total for the variable of interest in the population, Y Y NY i

i

N

= =

=

∑

1

Population variance for y =σ y i

i

N

YY N 2

1

2 = −

=

∑( )/

πi = probability of selecting sample unit I

πij= joint probability of selecting sample units I and j

Nh = number of possible sample units in stratum h

nh = number of sample units in stratum h

 phij

F = the proportion of forest in the j

th subplot of the ith psu within the hth stratum

 pˆhij

F = estimated proportion of forest in the jth subplot of the i

th psu within the hth stratum

 ˆ

AF = estimated area in forest

 pˆF = estimated proportion of the total area in forest

 ˆ

Aij = estimated sample area in subplot j of plot I

 ˆ

Yij = estimated sampled value of interest in subplot j of plot i

 n = the number of plots in the sample

ns = number of subplots in the sample for plot I

ˆ

Ys = estimator for subplot size s and subplot ss

mi = number of subplots in the sample for plot i (Max and others 1996)
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-107. 2003 15

Treatment of Missing Subplots

The approach used in the pilot study for Jalisco and Colima closely resembles what is done in Region 6 of the

U.S. Forest Service. The approach used there to treat missing subplots is presented here and will be used in the

pilot study.

The basic rule for measuring plots that either intersect the boundary of the population or are partially

inaccessible due to, say hazardous conditions, require adherence to the following basic principles:

(1) Decisions concerning whether or not to measure a subplot or what part of a subplot to measure are made

independently on each subplot of a psu, regardless of subplot number, in other words, the psu center, subplot

number 1, is no different than any other subplot center;

(2) If a subplot center is inaccessible, then no measurements are taken on this subplot, since the subplot center

cannot be established and referenced by standard field methods. In cases where part of the subplot is accessible

or within the population boundary, even though the subplot center is inaccessible or outside the population, then

this results in a part of the population that is selected to be included in the sample but is not measured. Hence

this is a source of potential bias in the estimation process.

(3) If the subplot center is accessible and within the population boundary, then all regular measurements are taken

on that part of the plot that is accessible. The subplot must be mapped, with relevant measurements, in sufficient

detail so that the amount of area of the subplot that was actually measured can be calculated accurately. This area

provides the information essential for computing the proper area weight for the partially measured subplot.

(4) For purposes of making decisions about measurement, a psu can be thought of as consisting of distinct and

separate, although contiguous, subplots. Each subplot has nested area-based measurement units as described in

the text.

Inaccessible PSU’s

In this section we address the situation where some, or possibly all, of a psu is inaccessible. Inaccessibility is

usually caused by hazardous conditions, for example the existence of cliffs, that prevents safe access to the area

covered by the plot.

1. Subplots where subplot center is inaccessible.

a.- Entire area of subplot is inaccessible. Follow the basic rule, part (2). The entire subplot is not

measured. This lack of measuring a part of the selected sample area is a source of potential bias in

estimation. If the subplot center that is inaccessible is subplot 1, the psu center, then the location of the

psu must be monumented with respect to a subplot whose center is accessible. This is a deviation from

standard field procedure in which the location of the psu is referenced with respect to the psu center

coincident with the center of subplot 1.

b.- Some area of subplot is inaccessible. Follow the basic rule, part B (2). The entire subplot is not

measured because the center of the subplot is inaccessible and cannot be establish using usual field

procedures. This lack of measuring a part of the selected sample area is, again, a source of potential bias

in estimation. If the subplot center that is inaccessible is subplot 1, the psu center, then the location of

the psu must be monumented with respect to a subplot whose center is accessible. This is a deviation from

standard field procedure in which the location of the psu is referenced.

2. Subplots where subplot center is accessible.

The basic rule is followed, in this case, without exception. Follow the basic rule, part B (2). The part of

this subplot within the population boundary is really part of the population. To access and measure this

part of the subplot requires establishing the subplot center that actually is located on the adjacent

property. To avoid gaining access to this adjacent property, we forgo measuring the part of the subplot

on NFS land. There is a potential bias associated with not measuring the part of this subplot that is within

the population boundary. If the subplot center that is outside the population boundary is subplot 1, the

psu center, then the location of the psu must be monumented with respect to a subplot whose center is

within the population boundary. This is a deviation from standard field procedure in which the location

of the psu is referenced with respect to the psu center, coincident with the center of subplot 1.

3. Subplots where center is within the population.

The basic rule is followed, in this case, without exception.

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