United States

Department

of Agriculture

Forest Service

Rocky Mountain

Research Station

Research Paper

RMRS-RP-92

July 2012

Paul L. Patterson

Photo-Based Estimators for the 

Nevada Photo-Based Inventory
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Abstract

The U.S. Department of Agriculture, Forest Service, Forest Inventory and Analysis Program 

conducted the Nevada Photo-Based Inventory Pilot in an effort to improve precision in estimates 

of forest parameters, reduce field data collection costs on margin lands that are covered by slow 

growing woodland species, and address the potential of strategic-level inventory on lands not traditionally sampled. One part of the project involved the use of large-scale aerial photography instead 

of traditional field plot visits to produce three types of estimates: (1) area by a variety of forest and 

nonforest types; (2) percent cover of object types in the landscape; and (3) percent cover of object 

types within forest or nonforest type. In the context of the infinite sampling paradigm and using the 

support region construct, we construct the estimators used in the project, derive the variance of the 

estimator, and give an unbiased estimate of the variance. All estimates are constructed under the assumptions of a stratified sample of an infinite population with an independent simple random sample 

of each strata and a simple random sample of points in the support region around each sample point 

in the stratified sample.

Keywords: infinite sampling, support region, inclusion field, photo-based inventory, large-scale aerial 

photography (LSP)

Authors

Paul L. Patterson is a Statistician for the U.S. Forest 

Service, Rocky Mountain Research Station, Interior-West 

Forest Inventory and Monitoring Program, Ogden, UT.

Patterson, Paul L. 2012. Photo-based estimators for the Nevada photo-based inventory. 

Res. Pap. RMRS-RP-92. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, 

Rocky Mountain Research Station. 14 p.

Acknowledgments

The author thanks all the reviewers for their comments 

that greatly improved the quality of the manuscript, and 

Tracey Frescino for programming the estimators.
USDA Forest Service Res. Pap. RMRS-RP-92. 2012 1

Introduction ______________________________________________________

The U.S. Department of Agriculture, Forest Service, Forest Inventory and Analysis 

(FIA) Program is a national program that conducts an annual forest inventory on a 

permanent grid of plots across the United States (Bechtold and Patterson 2005). In an 

effort to improve precision in estimates of forest parameters, reduce field data collection 

costs on margin lands that are covered by slow growing woodland species, and address 

the potential of strategic-level inventory on lands not traditionally sampled by FIA, the 

Interior-West region of FIA (IW-FIA) conducted the Nevada Photo-Based Inventory 

Pilot (NPIP) project (Frescino and others 2009). One part of the project involved the 

use of large-scale aerial photography instead of traditional field plot visits to produce 

three types of estimates: (1) area by a variety of forest and nonforest types; (2) percent 

cover of object types in the landscape (for example, tree cover, cover of specific trees 

species or species grouping, bare ground, and human structures); and (3) percent cover 

of object types within forest or nonforest type (for example, percent cover of pinyon 

within pinyon juniper woodland group and percent cover of bare ground within the 

pinyon juniper woodland group).

The IW-FIA sample is organized into 10 panels, with 1 panel measured each year. 

Each panel has the same geographic distribution properties as the entire sample, only 

at one-tenth of the sampling intensity, so each panel can be used to construct estimates 

under the same methods that pertain to the overall FIA design. For the NPIP project, 

the State of Nevada was pre-stratified using a pixel-based, 250-m resolution map of 

predicted timberland, woodland, and nonforested areas (Frescino and others 2009). 

Within the forested stratum (timberland and woodland classes), all IW-FIA plot locations were selected, and within the nonforested stratum, one panel was selected. For 

this subset of the IW- FIA plot locations, high-resolution photographs were obtained of 

an area containing the FIA plot location and a 250-m radius photo-plot was installed on 

the photograph with the photo-plot center collocated with FIA plot center. For a sample 

of points within the photo-plot, a photo interpreter assessed two properties for each 

point: first, the characteristics of the vegetation cover and second, the object the point 

fell on. The photo-based estimates were constructed using these data.

Historically, FIA derives the properties of estimators based on the finite sampling 

paradigm, in other words, under the assumption of a finite population. This approach 

has the theoretical difficulties of specifying what the population unit is and whether area 

is subdivided into distinct, non-overlapping population units. A different approach is to 

construct an estimator and derive its properties using the infinite sampling paradigm 

where the probability sample is a set of points from a continuous population (Cordy 

1993). From an infinite sampling perspective, the NPIP sample is a set of clusters of 

sample points (the clusters are the photo-interpreted points within each photo-plot) 

with the clusters centered on the FIA plot centers. From this perspective, determining 

Photo-Based Estimators for the 

Nevada Photo-Based Inventory

Paul L. Patterson
2 USDA Forest Service Res. Pap. RMRS-RP-92. 2012

the probability of selection for a population element is difficult, and this difficulty is 

compounded for photo-plots that straddle a stratum boundary. Instead, one can consider the photo-plot as a support region for a measurement assigned to the FIA plot 

center, where a support region is the region over which the measurement is calculated 

(for example, the proportion of bare ground on a photo-plot is the measurement that is 

assigned to the photo-plot center and is calculated on the support region of the photoplot). Then the sample is the FIA plot centers and the photo-interpreted points within 

each photo-plot are a separate point sample of the support region. An advantage of using 

a support region is that the independent samples of each stratum are maintained; the 

disadvantage is the value assigned to the FIA plot center is an average of the attribute 

of interest over the support region. Stevens and Urquhart (2000) introduced conditions 

on the support region so that using the average of attribute of interest over the support 

region produces the same results as using the value of the attribute of interest at the FIA 

plot center. Cordy (1993) introduced an extension of the Horvitz-Thompson estimator 

to sampling from an infinite universe. 

The purpose of this paper is to: (1) use Steven and Urquhart’s results on support 

regions along with Cordy’s (1993) definitions and results to construct an unbiased estimate of the total and unbiased estimated variance of the estimated total; (2) construct an 

unbiased estimate of the covariance between two estimates; and (3) apply these results 

to construct estimates and estimated variances for the NPIP study. All estimates are 

constructed under the assumptions of a stratified sample of an infinite population with 

an independent simple random sample of each strata and a simple random sample of 

points in the support region around each sample point in the stratified sample.

The remainder of the paper is two sections: the first contains the construction of the 

estimator, the derivation of the covariance between two estimates, and a derivation of 

an unbiased estimator of the covariance; and the second section applies these results to 

the NPIP study, with a numeric example provided. 

Estimator and Estimated Variance____________________________________

This section is organized into three sub-sections. The first presents the necessary 

background material of sampling from infinite population and support regions. After 

presenting Cordy’s (1993) extend of the Horvitz-Thompson estimator in the second subsection, we construct the unbiased estimator. In the third sub-section, the covariance of 

two estimates is derived, and an estimator of covariance is presented and shown to be 

an unbiased estimator of covariance. Restricting to a single estimate yields a formula 

for an unbiased estimator of the variance of the estimator.

Sampling From Infinite Populations and Support Regions

In infinite sampling, the probability sample is a set of points from a continuous 

universe. Let R be the region that will be sampled; the population characteristics of 

interest are Iy = ∫R y(s)ds, where y(s) is the value of the attribute of interest at point s. 

For example, if we are interested in percent cover of bare ground over R, let y(s) be 

equal to 1 if s falls on bare ground, and 0 if s does not fall on bare ground. Then, the 

percent bare ground is Iy /||R|| * 100, where ||R|| is the area of R. However, we are interested in an estimate of Iy , in instances where an attribute is not measured at a point 

but is averaged over a neighborhood of support of the point (the neighborhood will also 

be referred to as a plot). Let ȳD(s) denote the average value of the attribute over the 
USDA Forest Service Res. Pap. RMRS-RP-92. 2012 3

neighborhood of support and D(s) denote the plot. The quantity being estimated 

is ∫R ȳD(s)ds instead of ∫R y(s)ds. Stevens and Urquhart (2000) derive a sufficient 

condition for ∫R ȳD(s)ds to equal ∫R y(s)ds. The condition is stated in terms of area 

of D(s) and the area of the inclusion field D–1(t), where the inclusion field is the set of 

all point s with t ∈ D(s). The sufficient condition is that the area of plot and the area of 

inclusion field are equal for all points, in other words,

[1]

If the plot is a circle, then the condition is satisfied for all “internal” points in the 

region, R, in other words, a point for which the distance to the nearest edge of R is 

greater than the radius of the circle. For points near the edge, part of the plot falls outside 

of R. Near the edge, the plot needs to be deformed. Stevens and Urquhart (2000) give 

examples of plot designs that are circles for interior plots and deformed for points near 

the edge so that equation [1] holds. An example is shown in Figure 1 (from Stevens 

and Urquhart 2000). We will assume that a plot design has been adopted that satisfies 

equation [1]. 

Construction of Two-Step Estimator

We wish to construct a strategy for estimating ∫R ȳD(s)ds, where ȳD(s) is the 

average of y over the support region s centered at D; where a strategy is a combination of a sampling design and an estimator, in other words, an equation (Sarndal 1992). 

The sample design for the proposed estimator is a two-step process: the first step is a 

stratified sample of the points in R with sample values ȳD(·). Then, a sample of the 

support region around each point in the stratified sample is drawn to estimate ȳD. This 

is not two-stage sampling with the support regions being the clusters; the support regions 

Figure 1—Adapted from Stevens and Urquhart’s (2000) figure 12. The part of the plot indicated by the vertical lines is outside the boundary of the region. This portion of the plot is repositioned inside the region—the sections with horizontal lines—so that the condition is preserved.
4 USDA Forest Service Res. Pap. RMRS-RP-92. 2012

can overlap, which violates the assumption of disjoint clusters. The process of sampling the 

support region is mentioned in Stevens and Urquhart (2000). The equation for the estimator is constructed using Cordy’s (1993) extension of the Horwitz-Thompson estimator. The 

specifics of the construction follow.

Cordy (1993) extended the Horwitz-Thompson estimator to continuous domains. We 

summarize his development and concurrently develop notation. For the region R, we assume z is a Lebesgue integrable function on R and there is a probabilistic sample design 

defined on R. If S is a sample drawn based on a sample design, then an estimator of Ιz = 

∫R z(s)ds is defined by

where π(s) is the inclusion density function; the inclusion density function is the infinite sampling version of first order inclusion probability in finite sampling (Sarndal 1992). 

Cordy (1993) showed that Îz(S) is an unbiased estimator of Ιz.

At times, the region will be explicitly included in the notation. For the population 

characteristic, the region will be denoted using a subscript, in other words, ΙR,y; for the 

estimator, the region is designated by ÎR ,y(S) = ∑s∈S (R) y (s)/πR(s), where πR is the inclusion 

density function for the sample design on R, S(R) denotes a sample of the region R, and to 

simplify the notation, the sample S inside the parenthesis refers to the sample of the region 

in subscript, that is, S(R).

From this point on, we focus on estimating R y D(s )ds , ∫ where y D(s )= 1

D  D(s ) y(t )dt ∫ . 

We use stratified sampling and the associated estimator. That is, if the region R has 

been partitioned into H subregions, R1, R2, ..., RH, the linearity of the integral implies 

If an independent sample S(Rh) is drawn from each Rh, h = 1, H, the stratified estimator of

ΙR,ȳD

 is given by

where, to simplify notation, the sample S inside the parentheses refers to the sample of the 

region in the subscript, that is, S(R) or S(Rh) in this equation.

When using stratified sampling, the deformation of the plots that is required so that 

equation [1] holds is relative to the boundary of the entire region and not relative to the 

boundary of each stratum. For example, if there are two strata, R1 and R2, and s ∈ R1, then 

Ds can contain points in R2. This may appear to violate the constraint in stratified sampling 

that each population element is in one, and only one, stratum. For infinite domains, the 

population element is the center point, not the plot; the information on the plot is collapsed 

to the value that is assigned to the center point of the plot, and the center point of the plot 

is the population element.

We restrict our attention to ΙR,ȳD

 and the estimator

[2]

where πh is the plot-level inclusion density function on Rh.
USDA Forest Service Res. Pap. RMRS-RP-92. 2012 5

Instead of calculating y D(s )= 1

D  D(s ) y(t )dt ∫ , we estimate 

by using an infinite domain Horvitz-Thompson estimator based on a simple random 

sample of D(s). So

[3]

where s represents a sample of D(s) and πD is the inclusion density function on D(s). 

Substituting equation [3] into equation [2], the final form of the estimator is

[4]

where, by the convention adopted earlier, the symbol S in the middle expression represents either a sample of subscript Rh or a sample of the subscript D(s).

The variance of the infinite domain Horvitz-Thompson estimator contains an extension of the finite sampling second order inclusion probability. In the next result, which 

follows directly from example 1 in Cordy (1993), we evaluate inclusion density function 

and the pairwise inclusion density function for the stratified sampling design on the region 

R and simple random sample design on support region D(s).

Lemma 1:

(a) For stratified sampling with simple random sample of size nh within each of the 

strata, (Rh), the inclusion density function and the pairwise inclusion density function is given by 

(b) Using simple random sampling for the i

th support region in the hth stratum with a 

sample size of mhi, the inclusion density function and the pairwise inclusion density 

function is given by

Denote the sample elements of Rh by shi, i = 1, ..., nh, and denote the sample elements 

of the simple random sample of D(shi) by shij, j = 1, ..., mhi, . Using this notation and 

lemma 1, equations [2] though [4] can be rewritten as

[5]

where S indicates a sample of Rh.
6 USDA Forest Service Res. Pap. RMRS-RP-92. 2012

[6]

where S indicates a sample of D(shi). Combining, we get

[7]

where, to simplify the notation, we have dropped the (S) in the two most left-hand symbols. It is clear from the construction process that ˆ

ˆ

I Rh,yD

 is an unbiased estimator of I Rh,yD.

Variance and Estimated Variance of the Two-Step Estimator

Cordy (1993) derived the variance of Îy (S) and gave an unbiased estimator of 

Var (Îy(S)); the following proposition extends this to the covariance between estimates 

of two functions. The proof is a straightforward extension of the proof in Theorem 2 

in Cordy (1993).

Proposition 1: Suppose the functions y and z are bounded, π(s) >0 for each s ∈ R, and 

∫R(1/π(s))ds < ∞.

(a) Then Cov (Îy(S), Îz(S)) exists and is given by

(b) If, in addition, π(s,s′) > 0 for all s,s′ ∈ R, then

is an unbiased estimator of Cov (Îy(S), Îz(S)).

The following equation is used in the proof of the next proposition and is a straightforward calculation using Lemma 1. For stratified sampling with simple random sample 

of size nh within each of the strata, (Rh),

[8]

We extend the use of the subscripts h and D to indicate conditioning with respect to 

sample design within the stratum and within the plot, respectively. 

Proposition 2: Let y and z be bounded functions on the region Rh, and let D(s) be a plot 

design that satisfies equation [1] on a region R containing Rh. If a simple random sample 

of size nh, , is drawn from Rh, and for each plot D(shi) a simple random sample 

is drawn, then for the estimator defined in equations [4] and [7]
USDA Forest Service Res. Pap. RMRS-RP-92. 2012 7

(a) the

(b) and

is an unbiased estimator of Cov ˆ

ˆ

I Rh,yD, ˆ

ˆ

I Rh,ZD

⎛

⎝

⎞

⎠

.

Proof: (a) We use the standard result

[9]

We start by calculating the EhCovD term on the right-hand side of equation [9].

where the second equality follows since the photo-plots are sampled independently. The last expression is the product of ||Rh||/nh and since 

 is a function defined on Rh and ˆ

I Rh,y(S )= Rh 

nh s∈S (Rh) ∑ y(s ).

Taking the expectation with respect to the sample design on Rh yields

[10]

We now turn our attention to calculating Covh ED(⋅),ED ⎡ (⋅) ⎣ ⎤

⎦ of equation [9]. Note that
8 USDA Forest Service Res. Pap. RMRS-RP-92. 2012

Thus 

[11]

Adding equations [10] and [11] completes the proof of part (a).

(b) To calculate the expectation, we use E = EhED. Using the identity ∑iai

∑jbj = 

∑iaibi + ∑i∑j,j≠iaibj, we first express Cˆ ˆ

ˆ

I Rh,yD, ˆ

ˆ

I Rh,zD

⎛

⎝

⎞

⎠

 in the equivalent form

[12]

The expectation of the first term of expression [12] is

where ED was distributed across the sum and we used the identity CovD(X,Y) = 

ED(XY) – ED(X)ED(Y). The first term in this expression is the product of ||Rh||/nh

and ˆ

IRh,

CovD ˆID(s ),yD,

ˆ ( ID(s ),zD), so we can rewrite the above expression in the equivalent form

Since Î is an unbiased estimator, this expression is equal to

[13]

Before continuing with this calculation, we calculate the expectation of the second term 

of expression [12].
USDA Forest Service Res. Pap. RMRS-RP-92. 2012 9

Distributing the expectation, ED, across the sum in combination with the independent 

sampling on the plots implies the previous expression is equal to

Since Î is an unbiased estimator, the previous expression is equal to

[14]

By part (b) of Proposition 1 in conjunction with equation [10], the addition of the second 

term of expression [13] and expression [14] is equal to Eh Cˆ

h ˆ

I Rh,ID(s ),yD, ˆI Rh,ID(s ),zD ( ) ⎡

⎣ ⎤⎦. By 

Proposition 1, Ĉh is an unbiased estimator of Covh, hence the addition of expressions 

[13] and [14] is equal to

and the proof is complete.

In Proposition 2, the case of y and z being the same function gives the equation for 

the variance and estimated variance of ˆ

ˆ

I Rh,yD

. For those familiar with two-stage sampling 

for finite populations, there is a similarity between the variance and estimated variance 

in Proposition 1 and the equations for variance and estimated variance for two-stage 

sampling for finite populations (Cochran 1977: Theorems 10.1 and 10.2). However, there 

is a striking dissimilarity. If we view the plots as clusters, then as with finite sampling, 

the the variance contains a between cluster component, Var

h ˆ

I Rh,ID(s ),yD

⎡

⎣ ⎤⎦, and a within 

cluster component, IRh,VarD ˆ

ID(s ),yD ( )

; what is dissimilar is the estimated variance in Proposition 2 does not contain a within cluster component. In both this setting and the twostage sampling for finite populations, the between cluster component of the estimated 

variance contains information about the within cluster component of the variance. The 

difference is, in this setting, the expectation of the between cluster component of estimated variance contains both an unbiased estimate of the between cluster component of 

variance and the within cluster component of the variance. In two-stage sampling, the 

expectation of estimate for the between cluster component of the variance contains an 

unbiased estimate of the between cluster component of the variance and a biased estimate of the within cluster component of the variance and hence needs to be “adjusted” 

with a within cluster component to produce an unbiased estimate of the variance. This 

remark is from an intuitive point of view; formally, the plots are not clusters since they 
10 USDA Forest Service Res. Pap. RMRS-RP-92. 2012

can straddle strata boundaries and a primary sampling unit (cluster) can be in one, and 

only one, stratum. In the construction of ˆ

ˆ

I Rh ,yD, the information on the plot is collapsed 

to the primary sampling unit, which is the center of the plot; the sample of the plot is 

not a subsample but a separate sample from the sample of the region.

The following restatement of Proposition 2 will be useful in the next section.

Application to Nevada Photo-Based Inventory _________________________

A complete description of the sample design for the NPIP is contained in Frescino 

and others (2009). We present the details that are germane to the development of the 

estimators and their estimated variances.

The FIA sample design has been described as a quasi-systematic sample in that plot 

locations were randomly selected within hexagons, which are a systematic tiling of the 

United States (Bechtold and Patterson 2005). For the NPIP project, the State of Nevada 

was pre-stratified into predicted timberland/woodland forest and nonforested areas 

(Frescino and others 2009). Within the forested stratum, all IW-FIA plot locations were 

selected, and within the nonforested stratum, one panel was selected. For the purpose 

of estimation, FIA treats the FIA quasi-systematic sample as a simple random sample 

(Bechtold and Patterson (2005: p. 25). We treat the NPIP sample as a pre-stratified 

sample, with a simple random sample of each stratum.

For each 250-m radius photo-plot that was co-located at the FIA plot location, a systematic grid of 49 points was located within each photo-plot. Cochran (1977) showed 

that systematic sampling has a smaller variance than simple random sampling when units 

within systematic samples have greater variability than the population as a whole. We 

assume the systematic grid that was located within each photo-plot does not coincide 

with any systematic land feature within the photo-plot; hence, treating the systematic 

sample as a simple random sample produces a conservative estimate of the variance.

The other assumption that was used in the development of the estimators in the "Estimator and Estimated Variance" section is the plot design satisfied equation [1]. All of 

the photo-plots were sufficient distance from the estimation unit boundaries that they 

did not cross the estimation unit boundaries; thus, they satisfy equation [1].

For the photo-plot portion of NPIP, three classes of population characteristics are 

of interest: (1) area of land classified as being in a condition, for example, proportion 

of land in pinyon-juniper forest type, or proportion of land that is privately owned and 

in pinyon-juniper forest type; (2) the percent cover of the land by an object type, for 

example, percent cover of land by bare ground, or the percent cover of land by pinyon 

trees; and (3) the percentage of a condition that has an attribute, for example, percentage 
USDA Forest Service Res. Pap. RMRS-RP-92. 2012 11

of bare ground in the pinyon-juniper forest type, or the percentage of pinyon-juniper 

forest type covered by juniper trees. Estimates of these population characteristics use 

the following three functions:

All the population characteristics can be stated in terms of proportions; for example, 

the area of land classified as being in condition c, IR,yc

, equal to R Pc , where Pc is the 

proportion of area classified as being in condition c. The rest of this paper is devoted 

to estimating the proportion of land by characteristic or type (classes 1 and 2) and the 

proportion of land within a condition that has an attribute (class 3). 

Proportion of Condition or Cover

When estimating the proportion of a region R that is classified as being in condition 

c or the proportion of R that is covered by object o, the function yc is used for conditions, while the function yo is used for objects. The equations are stated in terms of the 

proportion of land classified as being in condition c. To obtain the proportion of land 

covered by object o, simply substitute yo for yc in the equations. 

Under the assumption that the photo-plot design satisfies equation [1], then the 

proportion of land in region R classified as being in condition c is Pc = 1

R 

I R ,ycD,which 

we can estimate using

[15]

The functional notation used in "Estimator and Estimated Variance" section was useful for the derivation and proofs, but the notation is not standard to FIA. We introduce 

notation that is more familiar to FIA users. 

Let shij be the jth point of the ith photo-plot in stratum h. If we let

[16]

then from equations [6] and [7],

[17]

Combining equations [15] and [17] yields

[18]
12 USDA Forest Service Res. Pap. RMRS-RP-92. 2012

An unbiased estimated variance of Pˆ

c follows from first noting that since the samples 

of the strata are independent, equation [17] implies Vˆ(Pˆ

c ) is equal to R 

−2

h=1

H

∑ Vˆ ˆ

ˆ

IRhycD

⎛

⎝

⎞

⎠

,

which, combining with Proposition 2.A and ˆ

ID,yD= pch, implies

[19]

with the usual calculational form

[20]

Percent Cover Within Condition

We denote the proportion of the condition c covered by attribute o byPo⊂c . Since 

spatial distribution of condition c is unknown, Po⊂c is the ratio of the proportion that is 

both classified as condition c and covered by object o to the proportion that is classified 

as condition c, that is Po⊂c = Po|c/Pc . An estimate of Po⊂c is given by

[21]

where 

[22]

An approximate variance is calculated using Taylor linearization (see Särndal and 

others 1992: p. 172-175 and bottom p. 177 through p. 178).

An estimate of the approximate variance is obtained by estimating all of the unknown 

quantities, that is,

[23]

where, by Proposition 2.A and equation [22]

[24]

Which, for computational proposes, can be expressed as

[25]
USDA Forest Service Res. Pap. RMRS-RP-92. 2012 13

Example To illustrate the use of these estimators, a couple of results, using the NPIP data are 

presented. To estimate the percentage of Nevada covered by pinyon-juniper forest type, 

use equation [18] (multiplied by 100) with yc(s) equal to 1 if the point is in the pinyonjuniper forest type and 0 otherwise. The estimate is 13.5 percent with a standard error 

of 0.55, with the standard error calculated by multiplying the square root of equation 

[20] by 100. The estimates of percent of land cover by forest type are standard for FIA; 

what is not standard is coverage by object type. For instance, the percentage of land 

covered by the various sage species (denoted sage complex) is 12.3, with a standard 

error of 0.49 (see Frescino and others [2009] for the definition of sage complex). The 

estimate was calculated using equations [18] and [20] with the function yo(s) equal to 

1 if the point s falls on a member of the sage complex and 0 otherwise. Similarly, the 

percentage of land covered by other shrub types is 19.7, with a standard error of 0.5, 

and the percentage of land covered by soil or rock is 45.1, with a standard error of 0.66. 

The above percentages are for the entire State of Nevada; what can also be estimated 

is the percentage of the pinyon-juniper forest type that is covered by sage complex, 

other shrubs, and soil or rock. These are estimated using equation [21] and the standard 

error estimated using the square root of equation [23]. First, estimate the percentage 

of land covered by sage complex within the pinyon-juniper forest type; this estimate is 

calculated using equation [15] and the function yo|c(s), which is 1 if the point s falls on 

a sage complex and is in pinyon-juniper forest type and is 0 otherwise. The percentage 

is 1.1; the ratio of this estimate and the estimate of pinyon-juniper forest type cover 

yields the estimate of 8.2 percent of pinyon-juniper forest type is covered by the sage 

complex, with a standard error of 0.61. Similarly, the percentage of the pinyon-juniper 

forest type covered by other shrubs is 7.4, with a standard error of 0.43; while the percentage of pinyon-juniper forest type covered by soil or rock is 44.9, with a standard 

error of 0.95. As would be expected, the percentage of soil or rock cover within forest 

types that typically have a high density of tree cover is significantly less; for example, 

8.5 percent of aspen forest type is covered by soil or rock, with a standard error of 2.

Conclusion_______________________________________________________

An unbiased estimator was derived for the total of a population attribute from an 

infinite population. The assumptions are: (1) a stratified sample of an infinite population 

with an independent simple random sample of each stratum; and (2) a simple random 

sample of points in the support region around each sample point in the stratified 

sample. Readers who are mostly interested in calculating estimates should reference 

equation [7]. The forms of an unbiased estimated variance and an unbiased estimated 

covariance for use in calculations are presented in Proposition 2.A. 

These estimators were applied to the situation of using high-resolution photographs 

to estimate for a region the (1) area of land classified as being in a condition; (2) the 

percent cover of the land by an object type; and (3) the percentage of a condition that 

is covered by an object type. For items 1 and 2, the form estimator used in calculations 

is given in equation [18] and the estimated variance is presented in equation [20]. For 

item 3, a ratio estimator is used (equation [21]), and an estimated variance is presented 

in equations [23] and [24].
14 USDA Forest Service Res. Pap. RMRS-RP-92. 2012

References _______________________________________________________

Bechtold, W.A.; Patterson, P.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.

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

Cordy, C.B. 1993. An extension of the Horvitz-Thompson theorem to point sampling from a continuous 

universe. Statistics & Probability Letters. 18: 353-362.

Frescino, Tracey S.; Moisen, G.G.; Megown, K.A.; Nelson, V.J.; Freeman, E.A.; Patterson, P.L.; Finco, M.; 

Brewer, K; Menlove, J. 2009. Nevada photo-based inventory pilot (NPIP) photo sampling procedures. 

Gen. Tech. Rep. RMRS-GTR-222. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, 

Rocky Mountain Research Station. 30 p.

Särndal, C.; Swensson, B.; Wretman, J. 1992. Model assisted survey sampling. Springer-Verlag, New 

York. 694 p.

Stevens, D.L.; Urquhart, S. 2000. Response designs and support regions in sampling continuous domains. 

Enviornmetrics. 11: 13-41.



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