A Sampling Method for Inventorying Noble Fir Stands
to Determine Merchantable Bough Weight
by
Roger C. Chapman
Professor
Department of Natural Resource Sciences
Washington State University
PO Box 646410
Pullman, WA 99164-6410
Keith A. Blatner
Professor and Chair
Department of Natural Resource Sciences
Washington State University
PO Box 646410
Pullman, WA 99164-6410
William E. Schlosser
Vice President & CEO
Pacific Rim Taiga, Inc.
PO Box 187
Pullman WA 99163
and
Roger D. Fight
Principal Economist
PNW Research Station
PO Box 3890
Portland, OR
2
Abstract
Two-stage cluster sampling is proposed as a sampling design to estimate the merchantable yields of noble fir (Abies procera Rehder) boughs in the Westside cascades
of Washington and Oregon. This sampling design uses the noble fir bough weight
model developed by Blatner et al. (In Review). In addition to describing the sampling
design, this paper describes calculations for estimating the stand yield of merchantable
boughs and the associated standard error. Estimators are also presented to determine
the number of plots to sample and number of trees per plot to measure for a given total
budget, plot and tree measurement costs, or desired variance of total bough weight.
3
Introduction
Noble fir (Abies procera Rehder) is an important tree species in the Cascade
Mountain range of western Washington and Oregon. In its native range, noble fir occupies high elevation sites, above 1,000 m, characterized by steep slopes, and where the
summer to winter variation in environmental conditions is especially pronounced (Franklin 1983). From an economic standpoint it is less well known for its value as a timber
species than it is as a Christmas tree species and supplier of boughs for use in the
Christmas ornamentals industry (Schlosser et al. 1991, Schlosser et al. 1995). Since
the mid 20th century, noble fir has been cultivated in regions of western Washington and
Oregon at lower elevations (Murray et al. 1991), to areas north of its natural range in
Canada (Xie and Ying 1994), and even in northern European plantations (Bang 1979).
Land management agencies such as the USDA Forest Service, the Washington State
Department of Natural Resources, and the Oregon Department of Lands have included
commercial Christmas tree and bough harvests from noble fir as a component of their
land management schemes.
Noble fir boughs are harvested annually in October and November for use in
making wreaths, table decorations, and other ornaments for the holiday season.
Schlosser et al. (1991) determined that nearly 10,000 tons of noble fir boughs with a
raw material value of $6.7 million were harvested in 1989 in the northwestern US and
southern British Columbia. Many working in the industry believe this volume has increased over the past decade, although no other estimates of the annual volume or value of boughs harvested have been published.
4
While estimation procedures for determining merchantable log volumes present
in a forest have been used for decades, methods for estimating the weight of boughs
that can be harvested from a noble fir stand are largely lacking. Blatner et al. (In Review) developed an estimation model capable of predicting individual bough weight and
cumulative bough weight per tree. Blatner et al.’s prediction model must be combined
with a sampling method to generate stand level data. This manuscript describes a
sampling method that can be used in conjunction with the noble fir bough estimation
model to estimate the weight of harvestable noble fir boughs present in a given stand.
We also provide a spreadsheet template developed for Microsoft™ Excel® which allows
users to enter data collected from a noble fir stand, compute aggregated tree bound estimates for individual sample trees, and compute stand bough weight statistics when a
two-stage cluster sampling design is used.
Sampling Design
Although a variety of sampling methods will provide estimates of total bough yield
our choice of sampling design was based on (1) minimizing field cost, and (2) the ability
to provide statistically valid estimates and associated standard errors. There is no claim
that the design described in this manuscript is optimum in the sense of providing a minimum variance (shortest confidence interval). This research is designed to provide the
first step in developing estimators of bough yields that are non-subjective, quantifiable
with an associated measure of reliability. In a sense, these preliminary efforts mimic the
efforts of the foresters who developed the first volume equations and subsequently developed stand sampling procedures.
5
Many challenges differentiate bough cruising from traditional timber cruising.
Since noble fir bough plantations and forests are commonly younger forest stands and
noble fir has the propensity to retain a dense layer of lower branches, the prospect of
implementing a prism (angle gauge) sampling was considered to be impractical. The
traditional fixed area plot in which all “merchantable” trees within the plot boundary are
measured was considered impractical, particularly in larger area plots, because of the
time needed to measure all of the “merchantable” trees.
A two-stage cluster sampling design1 (Scheaffer et al. 1996), an adaptation of the
traditional fixed area plot design where all the trees in the plots are measured, is recommended (Shiver and Borders 1996, Scheaffer et al. 1996). In this design, the primary sampling unit is the collection of “merchantable” trees in a randomly selected fixed
area plot2. In each plot the number of “merchantable” noble fir trees are recorded. A
random sample of merchantable noble fir trees in each plot is then selected for careful
measurement. Each sample tree requires measurement of selected tree characteristics, aggregated whorl characteristics and competition data from an environmental habitat circle (radius, 20.4 feet) centered on the sample tree. The environmental habitat circle is associated with estimation of tree bough weight and is not a function of the sampling design being used.
Cunia (1986) describes a “two-stage random sampling of clusters” design that
could be used to estimate tree biomass. Cunia’s design is similar to that described here
in that a regression equation is used to estimate individual tree yields. It is unlike the
1
A two-stage cluster sampling design is also widely known in the statistical literature as a two-stage design. In this study plots (clusters of trees) and trees within a cluster were randomly selected.
2 Selection of primary units (plots) with probability proportional to number of merchantable trees is impractical because a user generally has no knowledge of the number of merchantable trees on each of the plots.
6
design described in this paper in that its primary units are stands and its secondary
units are plots or points within the sampled stands in which all trees are measured. If
our design were extended geographically using Cunia’s design the resulting design
would be classified as a three-stage cluster design.
It is essential that only one plot size be used in each stand to be sampled because the number of primary units N, number of plots in a stand used in the statistical
analysis is computed using the relationship, Area in acres/plot area in acres. Problems
with plots intersecting borders can be handled using the Mirage Method (Shiver and
Borders1995). A population consisting of multiple stands can be estimated using either
the methodology of stratified random sampling with two-stage clusters or the methodology of a three-stage cluster design in which the first stage is the stand and the second
and third stages are plots and trees in the stands.
Although it is possible to measure an unequal number of sample trees on each
plot, the authors for reasons of simplicity and ease of field measurement suggest that
the same number of sample trees be measured on each plots. If at all possible at least
two trees per plot should be carefully measured in order to quantify the variability within
the trees of the plot. Plots containing zero or less than two merchantable trees are valid
observations and must not be ignored in the estimation of the total weight of merchantable boughs in the stand. This design not only provides an estimate of the total weight,
it also provides an estimate of the associated standard error.
If all of the merchantable trees in each plot are measured, the two-stage design
becomes the traditional fixed area plot sampling design. The selection of a limited
number of trees per plot (two-stage cluster sampling) rather than the measurement of all
7
trees in the population of interest reduces the cost of sampling. This is an important
consideration when estimating a product that has a relatively low unit value or a high
cost of estimation.
The notation and mathematical structure associated with the two-stage cluster
design of a single stand is shown below.
A
=
unit area in acres
P
=
plot size in acres
Yij
=
bough weight of the jth tree on the ith plot
Mi =
total number of merchantable trees in the ith plot (total tree count)
mi =
number of trees measured on the ith plot (second-stage of sampling)
n
=
total number of plots measured
M
=
n
Mi
, average number of merchantable trees per plot
i
N
=
(A/P), number of plots of size P in the unit,
T̂p,i  Mi  yi , estimated weight of boughs in the ith plot
(1)
1 mi
 yi,j , average accumulated bough weight per tree on
mi j1
the ith plot (second stage)
n
1

 Tp,i , estimated average plot bough weight per plot
n i1

yi
T̂p
Tˆ Tot

(2)
(3)
N  Tˆp , estimated total bough weight in the unit  A * 1 Tˆ 
P p


(4)
The variance of the estimate total bough weight in a unit is given by:
s2T
Tot
 N2    N  
   sB2     s2w 
 n    n  
(5)
The components of the estimated variance of the estimated total
Nn 2
 sBT
 N 
sB2  
(6)
8
s
2
BT

 Tˆ  Tˆ
p,i
p
 

n 1
i 1 


2

 , variation between plots



M m 
s2
s2w   wi Mi2  i i  , variance of aggregated average within plot variances
i1 mi
 Mi 
s
2
wi

y
i,j
 yi 
mi  1
(7)
(8)
2
, variation between trees in the ith plot
(9)
Estimating Sample Size
The sample size number of plots, n and number of trees per plot to measure, m0
that minimizes the variance for a fixed total cost when the same number of trees (mo),
are measured in each plot and the cost of a sampling project is given by:
C = Co + (n  C1 )+ (n  mo  C2 )
(10)
Where:
Co
is the fixed cost of associated with planning the cruise, obtaining necessary permits etc,
C1
is the cost of establishing the fixed area plot, which includes planning costs attributable to the number of plots, travel cost to unit apportioned to each plot and
travel between plots within a unit, the cost of determining and recording plot
characteristics,
C2
is the cost of measuring a merchantable tree in a fixed area plot.
The number of trees to measure on each plot m0 is given by the expression:
mo 
V2  C1
V1  C2
(11)
9
The number of plots to measure, n for a pre-assigned total cost is given by the expression:
n
C  C0    V1 C1 
 V1  C1    V2  C2 
(12)
where
2




T̂
M
1


Tot
i
V1  N2  

y

  i N  M  
n  1  i  M


 

 1 
 Mi 2  mi  s2wi  
-        1  
 ,
 n 
 M   Mi  mi  
a measure of variation among Stage I plots
(13)
V2  N Mi2s2wi , aggregate within plot variance
(14)
i
The estimation of n for a specified variance V is shown below.
The theoretical basis for determination of the number of plots to measure, n and
number of merchantable trees to measure on each plot (mo) are found in Som (1973).
The number of plots to measure, n in a given stand is a function not only of the ratio of
the variation among and within plots, but also of the relative plot and tree costs, the cost
per plot (C1), the total project cost (C), and the fixed cost of the cruise (C0) (Figure 1).
The calculations for determining the number of trees, m0 per plot to measure
(Equation 11) can be simplified by expressing the cost per tree as a simple proportion of
the cost per plot (C2/C1) and the ratio of the variance between plots to the aggregated
variance between trees within plots. These ratios allow users to estimate the number of
plots to measure n and number of trees to measure on each plot and mo using only
general information about the relationships between the respective costs and between
the respective variances. The formula for the number of trees to measure on each plot
(mo) becomes:
10
mo 
 V2 
 
 V1 
 C2 
 C1 
 
(15)
Similarly, equation 16 can be used to estimate the number of plots needed if the
ratio of variances, the total budget, the cost per plot, and the total variable cost is
known. The first component of the equation represents a set of site-specific costs, while
the second more general relationship depends on the interaction between the estimated
variance and cost ratios.
n

 C  Co  
C1
1
 
 C2 
  V2   
 C1 
1  
 
V1
 
 




 
 
(16)
is to round up. If n and mo are both rounded up the resulting cost will obviously exceed
the desired total cost C.
The previous paragraphs have been focused on determining the number of plots
to measure and the number of trees within each plot to measure when the total cost of
the inventory, C is fixed. Often a sampler wishes to determine the number of plots and
trees to measure when the size of the variance of estimate is specified. In many texts
this is specified in terms of the 95% half confidence. Because mo, the number of trees
to measure on each plot or number of fixed area plots, n based on equations 15 and 16
are rarely whole numbers the user has the option of rounding mo or n either up or down.
The conservative approach in terms of obtaining a smaller variance interval is to round
up.
11
When total inventory cost is expressed as shown in Equation 10, the estimator of mo is
that shown in Equation 15. However the estimator of n, the number of plots is given by
n
V1c1  V2c 2
(17)
c
V 1
V1
where
V
v1
v2

n n  m0
(18)
Application
The variance between trees on each plot (s2wi), and the variation between fixed
area plots (s2B), can be estimated prior to sampling by measuring preliminary fixed area
plots and trees in the area to be inventoried and using the resulting estimates in the calculations for the number of plots to be sampled. Plot costs (C1) can be approximated
using prior field experience associated with determining density with fixed area plots in
similar noble fir stands.
In order to reduce the computational effort required for the calculation of the estimated bough weights and standard errors, we developed a Microsoft™ Excel ® spreadsheet template. The template is composed of four linked spreadsheets: 1) an instructions sheet, 2) a tree input sheet, 3) a sheet of calculations and plot level statistics, and
4) a report sheet that summarizes important cruise data. The spreadsheet template is
available (free) on the Internet at http://www.fs.fed.us/pnw/data/soft.htm.
The method for selecting which trees to measure on each fixed area plot must be
such that the selection is without personal prejudice so that all noble fir with merchantable boughs have the same chance of being sampled. Suggestions for unbiased selec-
12
tion of sample trees can be found in Bell et al. (2000). The choice of plot size (area of
primary sampling units) associated with the estimate of the number of merchantable
noble fir per acre is a function of stand density, cost, and the desired precision. No theoretical problems arise because the habitat information circle associated with each
measured tree overlaps another habitat information circle or extends beyond the
boundary of a primary unit (Figure 2).
Example of Sample Size Calculations
The subsequent discussion illustrates the computations associated with calculating the number of fixed area plots to sample n and the number of trees to measure per
plot m0, on the assumption that the same number of trees, m0 will be measured on each
fixed area plot. In order to estimate the number of plots to measure for a particular project and the number of trees to measure in each plot, we need the budget for the inventory (C), the fixed cost of conducting the cruise (Co), and an average plot cost (C1). In
addition, we need an estimate of the ratio of variances (V2/V1) and the ratio of costs
(C2/C1, e.g., the cost per tree relative to the cost per plot) as indicated in equation 16. In
this example, we will assume the following conditions:
C
=
$1,000 (budget for conducting the cruise)
C0
=
$150 (fixed cost of conducting the cruise)
C1
=
$50 (estimated Stage I plot cost)
C2
=
$37.50 (estimated Stage II plot cost of measuring each sample tree)
V2/V1 =
3.1 (estimated)
C2/C1 =
0.75 (calculated from above)
An estimate of the number of trees per plot to measure, mo can be obtained by
using equation 15 and approximate values of the respective ratios.
13
 V2 
 
V
3.10
mo   1  
 2.03, trees per plot
C
2
0.75
 
 C1 
 
(19)
As the ratio of costs (C2/C1) decreases, or as the ratio of variances (V2/V1) increases, the number of trees per plot (m0) to measure increases (Figure 1). The number of plots to measure (n) and the number of trees to measure per plot (m o) are very
sensitive to the ratio of variances (V2/V1). When there are few merchantable trees per
plot, V2 tends to be very small, because the proportion of the trees not measured in
each plot is small and the resulting variance associated with the estimated population
total approaches the variance associated with simple cluster sampling.
Based on this calculation we would want to measure at least two trees on each
plot. Based on equation (16) the number of plots n is 6.72, which we would normally
round up to 7.
n 
 C  Co 
 
 C2 
  V2   
 C1 
1  
 
V1
 
 


  C
1



 

1,000  150 
1  


 850 
= 
 = 6.72
 126.2 
3.1 .75   50


(20)
In this example, most users would probably measure 7 fixed area plots with 2 measured
trees per plots. The results in a total cost of $1025 for the inventory, which exceeds the
amount allotted by only $25 – a trivial amount. However, if we were more conservative
and measured three trees on each of the seven plots the difference between the expected total cost and the estimated total cost is +$287.50 or a 28% cost overrun. Thus,
there is no assurance that the conservative approach of simply rounding the number of
14
plots and/or trees to measure to next highest whole integer will result in a trivial increase
in cost.
Calculation of Estimated Weights and Associated Statistics -- An Example
Tables 1 and 2 illustrate the calculations for a noble fir parcel of 30 acres located
on the west side of the Cascades of Washington State. This example provides a
demonstration of the mechanics of calculating estimated total stand merchantable
weight of boughs and the associated standard error. We have used an inventory plot
size of 1,307.4 ft2 (3% of an acre). Seven plots were visited, corresponding to about
one plot for every four acres. On each plot we counted the number of merchantable
noble fir trees and randomly selected and estimated the weight of three trees on each
plot containing merchantable boughs. Estimating the bough weight on sample trees requires that we count the number of trees including the sample tree within the habitat information circle of the sample tree and record selected tree and whorl data (Blatner et
al. In Review). Estimates of the total weight and associated statistics for the example
shown are presented in Tables 1 and 2.
In this example, the fixed plot area of 3% of an acre coincides with the area of
environmental habit circle associated with the measured trees. The fixed plot area can
be of almost any size that usually contains at least two merchantable sample trees. Table 1 presents the individual sample tree bough weights for three sample trees per fixed
area plot and calculations for average weight per tree, variance, and the total number of
trees on each plot. Using this data, we are able to complete the calculations shown in
Table 2 and determine that the estimate of the total weight of merchantable noble fir
boughs on the unit is 207,643 pounds with a standard error of 35,384 pounds.
15
Conclusions
The arrangements under which noble fir bough material is made available to harvesters include bidding a lump sum and bidding a price per pound with payment based
on the weight harvested. It is especially important with a lump sum sale to have a reasonable estimate of the volume that is available for harvest. Poor estimates may lead to
bankrupt purchasers, cautious bidding, or prices being paid that are unfair to the buyer
or the seller. The method of estimating the volume of boughs available for harvest from
an area presented here provides a means to replace subjective estimates of harvestable noble fir boughs with unbiased sample-based estimates. Although some judgments
related to boundaries, accessibility, and adherence to cutting guidelines are still required; an unbiased sample of harvestable material should provide final estimates that
are more precise.
In this example the estimates of sample size are those that minimizes variance
subject to a cost constraint. Under this approach, increasing or deceasing the amount
of funds allocated to the inventory will change the number of plots to measure and the
resulting standard errors. As in any random sample, it is up the to the resource manager to evaluate the resulting inventory data and assess whether or not, he/she feels that
the resulting information has a sufficient level of reliability. Estimates of cruise variances (V1 and V2) can be based on preliminary estimates and past cruise data. The
costs used in the calculations may or may not represent costs on a particular bough
sale.
16
Literature Cited
Bang, C. 1979. Grontudbyttet ved forskellige klippemetoder og - intensiteter I Nobilis
(Translated title: Various lopping methods and intensities for the production of
decorative greenery from Abies procera). Forstlige Forsogskommission. Kobenhavn, Det Kommission, Denmark 37(1): 1-22. (In Danish)
Bell, J. F., Iles, K., Johnson, G. P., Marshall, D. D. 2000, Inventory and cruising newsletter, a compilation of the first 50 issues. Corvallis: John Bell & Associates.
Blatner, K.A. R.D. Fight, N. Vance, M. Savage, R.C. Chapman. In Review. A model to
estimate noble fir bough weight. Western Journal of Applied Forestry
Cunia ,T, 1986 , On the Error of Biomass Estimates in Forest Inventories, Part 2:The
Error Component from Sample Plots. Faculty of Forestry Miscellaneous Publications Number 9, SUNY College of Environmental Science and Forestry.
Franklin, J.F. 1983. Ecology of noble fir. In: Silvics of North America, Volume I Conifers
US Forest Service, Washington D.C., Agricultural Handbook 654. Pp. 80-87.
Murray, M.D., D. Coble, and R.O. Curtis. 1991. Height growth of young Pacific silver fir
and noble fir established on clearcuts in the Pacific silver fir zone of western
Washington. Canadian Journal of Forest Research, 21(8): 1213-1221.
Scheaffer, R., W. Mendelhall, and L. Ott. 1996. Elementary survey sampling. Belmont,
Cal. Wadsworth Publishing Co.
17
Schreuder, H., T. Greiure, and G. Wood. 1993. Methods for multiresource inventory.
New York: John Wiley.
Schlosser, W. E., K. A. Blatner, R. C. Chapman. 1991. Economic and marketing implications of special forest products harvest in the coastal Pacific Northwest. Western Journal of Applied Forestry 6(3): 67-72.
Schlosser, W. E., K. A. Blatner, E. G. Schuster, M. S. Carroll. 1995. Potential for expansion of the special forest products industry in the Northern Rockies. Western
Journal of Applied Forestry 10(4): 138-143
Shiver, B.D. and B.E. Borders. 1995. Sampling techniques for forest resource Inventory.
New York: John Wiley & Sons.
Som, R.K. 1973. A manual of sampling techniques. Heinemann Educational Books Ltd,
London.
Xie, C.Y. and C.C. Ying. 1994. Adaptedness of noble fir (Abies procera Rehd.) beyond
its northern limits. Forest Science 40(3): 412-428
18
Table 1. Bough weight per tree, number of merchantable trees/plot, average weight per
tree and variance of weight per tree for each plot for seven hypothetical 3% acre (20.4
feet) fixed area plots.
Bough Weight per Tree of
Sample Trees (Pounds)
Plot
1
2
3
4
5
6
7
Tree 1
11.6
43.5
6.0
3.5
5.0
24.0
45.5
Tree 2
5.0
38.5
17.0
3.5
28.0
10.3
9.25
Tree 3
20.1
8.5
22.1
30.5
9.5
42.3
103.0
Average Weight
Variance
per Tree
(Weight/tree) Number of Mer(Pounds)
s2wi
chantable Trees
12.2
57.3
28
30.2
358.3
4
15.0
67.7
7
12.5
243.0
9
14.2
148.6
18
25.5
257.8
10
52.6
2234.9
5
19
Table 2. Estimated total bough weight in pounds and the associated variance components based on the cruise data presented in Table1.
A
N
P
m
=
30
= 1,000
=
0.03
=
3
n
=
7
M
=
n
unit area in acres
number of plots in area (A  P)
plot size in acres
number of trees measured on each plot (second stage of sampling)
total number of plots measured
Mi
11.57(average number of merchantable trees per plot)
i
T̂p
Tˆ
s
2
BT
1 n
 Tp,i
n i1
 N  Tˆp

207.64lbs. (average bough weight per plot)
207,642.86 lbs. (estimated total bough weight in the unit)
 [Tˆp,i  Tˆp ]2
 

n 1
i1 




8,780.15 (variation between plots)

 N  n  2   1,000  7 
sBT   
 8,780.15  = 8,718.69 (variance of the total)


 N 
  1,000 

sB2  
2
wi
s

s2w

s2Tˆ
y
i,j
 yi 
2
mi  1
(variance between trees on the ith plot (i=1, …, n))
2  M  mi 
Mi  i
 = 4,569.6(variance of aggregated average within plot variances)
i1 mi
 Mi 
 N2    N    1,0002 
  1,000 

   sB2     s2w   
4569.6   1,252,054,912
 8718.69   


 n    n    7 
  7 
s2wi
(overall variance of the total)
SE(T) =35,384.4
20
Figure 1: The number of trees to measure within a fixed area plot in two-stage sampling
mo-number of trees to
measure in a plot
for a range of cost, C2/C1and variance [V2/V1] ratios .
5
V2/V1=0.5
V2/V1=1.0
V2/V1=1.5
V2/V1=2.0
V2/V1=2.5
V2/V1=3.0
V2/V1=3.5
4
3
2
1
0
0
0.5
C2/C1
1
1.5
21
Figure 2. Example of a fixed area plot with three measured trees selected each identified with an "x" and their habitat environment circles defined.