NOTE / NOTE

Forest inventory estimation with mapped plots

Paul C. Van Deusen

Abstract: Procedures are developed for estimating means and variances with a mapped-plot design. The focus is onfixed-area plots, and simulations are used to validate the proposed estimators. The mapped-plot estimators for meansand variances are compared with simple random sampling estimators that utilize only full plots. As expected, themapped-plot estimates have smaller mean squared errors than the simple random sampling estimates. The theory forfixed-area plots is easy to apply, although additional work is required to map plots in the field. Corresponding theoryfor variable plots is developed but not tested with simulations. The difficulty of applying these methods to variableplots is greater, but not prohibitive.

Résumé : Des procédures ont été développées pour estimer les moyennes et les variances à l’aide du dispositif de pla-cettes cartographiées. Les auteurs se sont concentrés sur des placettes à superficie fixe en validant par simulation lesestimateurs proposés. Les estimateurs des moyennes et des variances basés sur les placettes cartographiées sont compa-rés aux estimateurs basés sur un échantillonnage aléatoire simple qui utilisent seulement des placettes complètes. Telqu’anticipé, les estimations basées sur les placettes cartographiées ont des erreurs moyennes quadratiques plus petitesque les estimations basées sur un échantillonnage aléatoire simple. La théorie des placettes à superficie fixe est facile àappliquer, bien qu’un effort additionnel soit requis pour cartographier les placettes sur le terrain. Une théorie corres-pondante pour les placettes à superficie variable est développée, mais elle n’a pas été testée par simulation. L’applica-tion des estimateurs proposés aux placettes à superficie variable est plus difficile, mais non insurmontable.

[Traduit par la Rédaction] Van Deusen 497

Introduction

The Forest Inventory and Analysis (FIA) program of theUSDA Forest Service is currently using a mapped-plot de-sign. Field plots that sample more than one condition aremapped so that the plot measurements can be allocated toparticular conditions. Conditions are defined by characteris-tics such as land use, tree species, tree size, and tree density.In the past, some FIA regions rotated plots so that they onlycontained single conditions. While this made intuitive sense,it led to a small bias in the estimates (Birdsey 1995) and isno longer an accepted practice.

Hahn et al. (1995) recommended the mapped-plot designas an alternative to rotating straddler plots. Details on how tomap condition boundaries for circular plots are availableelsewhere (Scott and Bechtold 1995). Comparison of tradi-tional methods to estimate growth components (Williamsand Schreuder 1995) indicated that these procedures canwork with mapped plots.

The purpose of this paper is to look at the mapped-plotdesign in relation to a very simple forest inventory model.This approach makes it easy to develop some useful estima-

tors for means and variances. It then becomes apparent thatthese estimators can be applied to the somewhat more com-plicated situation that arises with FIA use of mapped plots.

A simple inventory model

Consider a simple forest inventory model that consists oftwo conditions, say C and B.

Interest is primarily in condition C, which is surroundedby another condition denoted as B (Fig. 1). Condition C isdepicted as a circular island (Fig. 1) surrounded by conditionB, but in practice condition C could have any spatial ar-rangement. Sampling is either systematic or simple randomand is based on fixed-area circular plots with diameter d.There is a perimeter band of width d that overlaps the outeredge of area C. The perimeter band represents the zonewhere a plot center could be located such that the plot con-tains both conditions.

The following notation is used:ai is the proportion of the area of plot i that is within con-

dition C.A is the area of condition C.~A is the area in the perimeter band that is outside of con-

dition C plus the area of C.r is the ratio of the area of C (A) to the area of C plus the

outer perimeter band (~A).

yi is a variable that can be measured on each randomly lo-cated plot that completely or partially overlaps condi-tion C. For plots that don’t overlap C, yi = 0.

Can. J. For. Res. 34: 493–497 (2004) doi: 10.1139/X03-209 © 2004 NRC Canada

493

Received 9 May 2003. Accepted 2 September 2003.Published on the NRC Research Press Web site athttp://cjfr.nrc.ca on 1 March 2004.

P.C. Van Deusen. National Council for Air and StreamImprovement, 600 Suffolk Street, Fifth Floor, Lowell, MA01854, U.S.A. (e-mail: PVanDeusen@ncasi.org).

µ is the per unit area mean of variable y for condition C,e.g., cubic metres per hectare pine volume.

~µ is the per unit area mean of variable y in condition C in-clusive of the outer perimeter area. The outer perimeterarea does not contain variable y by definition, and µ =~/µ r.

n is the number of plots that completely or partially over-lap condition C.

The sampling design used here involves randomly locat-ing plot centers within the model forest area (Fig. 1) and re-cording the value of variable y in the plot. Note that yi = 0and ai = 0 when plot i does not overlap condition C. It fol-lows immediately that a nearly unbiased estimator for the ra-tio r is

[1] � /r a nii

n

==∑

1

Equation 1 is not exactly unbiased because the samplesize is random (Singh and Narain 1989). Likewise, an esti-mator of ~µ is

[2] ~� /µ ==∑ y nii

n

1

Combining eqs. 1 and 2 produced an estimator for themean value of y within condition C:

[3] �µ =∑

∑

y

a

ii=

n

ii=

n1

1

A model-based approach is used to rederive eq. 3 in thenext section.

Model-based derivation of estimatorsThe derivations for eqs. 1, 2, and 3 are based on the as-

sumption that there is a variable yi that can be measured oneach plot and adjusted to a per-acre or per-hectare estimate.On average, the value of y diminishes in proportion to theplot area. The relationship is captured by the followingequation:

[4] yi = aiµ + ei

where ei is a random error with mean 0, and E(eiej) = aiσ2

for i = j and 0 otherwise.It is reasonable to assume that Var(ei) = aiσ2 because vari-

ance must be zero when ai = 0 and must increase as ai in-creases. There is little to be gained by the alternativeassumption that variance increases with ai

2, because both aiand ai

2 range between 0 and 1.Now rewrite eq. 4 in matrix notation as

[5] Y = Aµ + e

where Y is a vector of plot measurements, A is a vector ofthe corresponding ai values, and e is a vector of error terms.The variance–covariance matrix of the error vector isE(ee′) = ∑ = σ2 diag(A). The least squares estimate (Maddala1977) of µ is

[6] �µ = [ ] 1′ ′−A A A YΣ Σ−1 −1

Equation 6 reduces to eq. 3, which means that eq. 3 is amodel-unbiased estimator for the mean, even though it is notdesign unbiased.

It follows from eq. 6 that the variance of �µ is

[7] Var( ) = [ ] 1�µ σA A′ =−

=∑

Σ−12

1

aii

n

An estimate of σ2 is required to use eq. 7, and the follow-ing estimator is proposed:

[8] �

�

σ2

2

1

11

1

=−

=

=

∑

∑

e

a

ii

n

i

n

where

� �e y ai i i= − µ

Equations 3, 7, and 8 all involve the ∑ ai. A further intu-itive justification for these equations is that ∑ =a ni whenall plots are completely in the condition of interest. There-fore, these equations reduce to the usual equations for sim-ple random sampling when all plots are fully in thecondition of interest.

A finite population correction factor (FPC) has been ig-nored (Cochran 1977) because forest sampling usually in-volves such large areas. When a FPC is deemed appropriate,it should be multiplied by Var( �)µ (eq. 7). The FPC takes theform of (N – n)/N, where N is the total number of potential

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494 Can. J. For. Res. Vol. 34, 2004

Fig. 1. An area of condition C surrounded by condition B. Con-dition C is bounded by the broken line that is contained in aperimeter band equal to the diameter of the fixed-area circularplots. Plots with centers that fall within the band will containsome of both conditions. All other plots contain only one condi-tion. One plot that is fully in condition C is shown along with aplot that overlaps the boundary.

sample plots in the forest, and n is the actual sample size.Both N and n are difficult to determine with mapped plots,but it is usually clear that the ratio of n/N is very small.

Horizontal point samplesHorizontal point samples (HPS) require a more complex

adjustment than fixed-area plots when multiple conditionclasses are straddled. HPS plot size is determined by tree di-ameter. Therefore, the plot area adjustment involves integra-tion over the range of possible diameters. The plot radius fordiameter x and basal area factor F is

[9] Rx

F=

2

where F is the prism factor (m2/ha or ft2/acre). It’s conve-nient to think of a HPS as consisting of a series of concen-tric circles, with each circle representing the plot size for adifferent diameter (Fig. 2). A single adjustment, ai, is re-quired for a fixed-area plot, but it is clear (Fig. 2) that an in-tegrated adjustment is required for HPS.

The following formula gives a value that is proportional tothe expected plot-level value of some tree-level variable y:

[10] E y y x x f x x( ) ( ) ( )∝∞

∫ 2

0

d

where y(x) is the function of interest, f x( ) is the underlyingdiameter distribution, and x2 accounts for the fact that treesare being selected proportionally to diameter squared. A nor-malizing constant (Van Deusen 1986) could be included tomake this into an exact equality, but it is not needed in thiscase. Now consider a continuous function, ai(x), that givesthe proportion of plot i within the desired condition, C, as afunction of diameter. When all “tree circles” are fully withinC, then ai(x) = 1. For many plots, ai(x) = 1 for smaller diam-eters (Fig. 2) and ai(x) < 1 for larger tree circles that overlapan adjoining condition.

An adjustment factor for plot i that is analogous to thefixed-plot adjustment factor can be derived as follows:

[11] a

a x y x x f x x

i

i

=

∞

∫ ( ) ( ) ( )2

0

d

E(y)

The normalizing factors cancel each other, which resultsin an equality for this equation. The resulting area adjust-ment factors, ai, could be used in exactly the same way asthe adjustment factors for fixed plots. This involves assump-tions about knowing ai(x), y(x), and f x( ). A volume functionmight be used for y(x), and one could fit a Weibull distribu-tion to the survey data to obtain f x( ) by adjusting for thesize-biased sampling distribution. It needs to be stressed thatf x( ) is not the size-biased distribution (Gove 2000) that re-sults from HPS, but the underlying distribution that is ob-tained from simple random sampling. The continuousadjustment function, ai(x), must be developed for each plotusing methods from Scott and Bechtold (1995).

A discrete alternative to the continuous formulation ineq. 11 can be derived. Start by creating diameter classes in-dexed from k = 1 to K, where class K contains the largest

trees in the forest. It is necessary to have an estimate of therelative number of trees in each diameter class, n(xk). Thevalues for n(xk) might come from an assumption that the di-ameter distribution has an inverse J shape. Likewise, a func-tion y(xk) provides an estimate of the tree-level variable ofinterest for each diameter class. The discreet formula for themean of y is

[12] y y x x n xk

K

k k k∝=∑

1

2( ) ( )

The discrete area adjustment equation is ai(xk), whichgives the proportion of the tree circle associated with diame-ter class xk that is in the condition class The discrete versionof the integrated plot adjustment factor is

[13] a

a x y x x n x

y

ik

K

k k k k

i = =∑

1

2( ) ( ) ( )

Simulation

Simulated data are used to test the proposed mapped-plotestimators of mean and variance for a fixed-plot design. They values that are to represent plot measurements are drawnfrom a normal distribution with a mean of 1000 and a stan-dard deviation of 300. There are 1000 replications with 100samples each. The ratio of the condition area to thecondition-with-perimeter area, r, is varied from 0.75 to 1.0.

The following procedure is used to generate the pairs (yi,ai), which represent plot values and overlap areas. In a firststage, a y value is determined to be totally in the condition ifu < r1, where u is a uniform (0, 1) random variable. Other-wise, the y value overlaps the perimeter area. The proportionof overlap is assigned in a second stage by drawing anotheruniform (0, 1) random variable. The first stage will, on aver-age, result in r1n full plots where ai = 1. The second stage

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Van Deusen 495

Fig. 2. An area of condition C surrounded by condition B. Con-dition C is bounded by the broken line. A horizontal point sam-ple is depicted by a center point with three concentric circlesrepresenting plot sizes for three diameter classes. The smallestcircle is completely in condition C, and the larger circles arepartially in condition B.

will, on average, assign ai = 0.5 to the remaining (1 – r1)nplots. This results in a simulated ratio with an expectedvalue of E(r) = (r1 + 1)/2. In the simulation, r1 ranges from0.5 to 1 in increments of 0.01, so that E(r) ranges from 0.75to 1.

The mapped-plot estimators are used to compute estimatesfor each of the 1000 replications. Simple random sampling(SRS) estimates for means and variances are also computedfor simulated plots where ai = 1. The SRS alternative is a vi-able method for handling mapped plots, but it amounts tothrowing away some of the data. The comparison betweenthe mapped-plot estimates and the SRS estimates indicateshow much is lost by using SRS.

Both estimation schemes result in virtually unbiased esti-mates of the mean. In fact, there is no practical differencebetween the mean estimates based on SRS and the mappedestimates. The maximum relative difference among themeans for 1000 replications at each level of r (Fig. 3) isabout 0.0015. It is also clear (Fig. 3) that means from SRSand mapped estimates converge as r approaches 1.0, as theyshould. Variance estimates (Fig. 4) indicate that the mappedestimator is more efficient than SRS, since it uses the datamore completely. As expected, the variances converge as rapproaches 1.0.

The variance of the SRS means and the mapped plotmeans can be estimated independently from the 1000 simu-lated replications for each level of r. The relative differencebetween simulation and SRS variances is shown (Fig. 5),along with the relative difference between simulation andmapped-plot variances. The graph (Fig. 5) indicates that themapped-plot variance estimator performs well and follows atrend similar to that of the SRS estimator. This confirms theutility of the mapped-plot variance estimator given in eq. 7.

Discussion and conclusions

Selection probabilities have not been mentioned, but itshould be clear that mapped plots have no effect on them.For the model forest used here, selection probabilities can beignored in the sense that all of the area in condition C has anequal chance of being selected. For example, a tree near theborder of C has the same selection probability as a tree inthe center of condition C. However, the total number of treesin a border plot is likely to be reduced in proportion to thearea of the plot that falls outside of condition C. This is truefor fixed plots and HPS, but the relationships for HPS aremore complicated because of variable selection probabilities.

Selection probabilities are altered in the special situationwhere a condition boundary corresponds to the boundary ofthe sampling population, but this would occur even withoutplot mapping. In this case, techniques for adjusting selectionprobabilities that were developed for boundary overlap be-come relevant (Barrett 1964; Beers 1966; Gregoire 1982).

The current FIA design uses a rigid configuration of fourfixed-area subplots. In practice, this configuration can betreated as a single fixed-area plot, and the developments forhandling multiple condition classes given here are applica-ble. FIA computes the proportion of each subplot that fallsinto a condition and provides corresponding plot-level pro-portions. These proportions can be used in the formulasgiven above for fixed-area plots.

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496 Can. J. For. Res. Vol. 34, 2004

Fig. 3. Relative difference between means using mapped-plot es-timators and means using only full plots and simple randomsampling estimators. Results are based on 1000 simulated plotvalues at each level of condition overlap. Condition ratio is theproportion of the sample plots that overlap the desired condition.

Fig. 4. Comparison of variances using mapped-plot estimatorsand variances using only full plots and simple random samplingestimators. Results are based on 1000 simulated plot values ateach level of condition overlap. Condition ratio is the proportionof the sample plots that overlap the desired condition.

Fig. 5. Relative variance differences between mapped-plot esti-mators and simulation variances and between simple randomsampling estimators that utilize only full plots and simulationvariances. Results are based on 1000 simulated plot values ateach level of condition overlap. Condition ratio is the proportionof the sample plots that overlap the desired condition.

References

Barrett, J.P. 1964. Correction for edge-effect bias in point sam-pling. For. Sci. 10: 52–55.

Beers, T.W. 1966. The direct correction for boundary-line slopoverin horizontal point sampling. Purdue University Agricultural Ex-periment Station, West Lafayette, Ind. Res. Prog. Rep. 224.

Birdsey, R.A. 1995. A brief history of the “Straddler Plot” debates.For. Sci. Monogr. 31: 7–11.

Cochran, W.G. 1977. Sampling techniques. 3rd ed. John Wiley &Sons, New York.

Gove, J.H. 2000. Some observations on fitting assumed diameterdistributions to horizontal point sampling data. Can. J. For. Res.30: 521–533.

Gregoire, T.W. 1982. The unbiasedness of the Mirage CorrectionProcedure for boundary overlap. For. Sci. 28: 504–508.

Hahn, J.T., MacClean, C.D., Arner, S.L., and Bechtold, W.A. Pro-cedures to handle inventory cluster plots that straddle two ormore conditions. For. Sci. Monogr. 31: 12–25.

Maddala, G.S. 1977. Econometrics. McGraw-Hill Book Company,New York.

Scott, C.T., and Bechtold, W.A. 1995. Techniques and computa-tions for mapping plot clusters that straddle stand boundaries.For. Sci. Monogr. 31: 46–61.

Singh, R., and Narain, P. 1989. Method of estimation from sampleswith random sample sizes. J. Stat. Plan. Inference, 23: 217–225.

Van Deusen, P.C. 1986. Fitting assumed distributions to horizontalpoint sample diameters. For. Sci. 32: 146–148.

Williams, M.S., and Schreuder, H.T. 1995. Documentation andevaluation of growth and other estimators for the fully mappeddesign used by FIA: a simulation study. For. Sci. Monogr. 31:26–45.

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