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Experiment Station

Research Note Lodgepole Pine

PNW-417

ae Marlin E. Plank and James M. Cahill

EXPERIMENT STATION
mNOr
STATION LIBRARY COPY |
Abstract Ta A sample of 351 ponderosa pine (Pinus ponderosa Dougl. ex

Laws.) and 509 lodgepole pine (Pinus contorta Dougl. ex

Loud.) logs were used to evaluate the performance of three
commonly used formulas for estimating cubic volume. Smalian’s
formula, Bruce's formula, and Huber's formula were tested to
determine which would provide the best estimate of cubic volume
when it was applied to tree-length logs. Smalian's formula
overestimated the volume by 19 percent, Bruce's formula
underestimated by 16 percent, and Huber's formula
underestimated by 2 percent. Huber's formula provided the
closest estimate and is recommended. Accuracy and bias tests
are shown.

Keywords: Cubic volume, log volume, Smalian's formula, Huber’s
formula, Bruce's formula, scaling.

Introduction Coniferous trees have a central woody stem comprised of many
geometric shapes (fig. 1). The geometric form of logs cut from
these trees can vary depending on their position in the tree.
Butt logs, for instance, approximate the shape of a concave
paraboloid and logs cut from the middle of the tree, usually a
convex paraboloid; logs cut from the top are either cones or
paraboloids.

The geometric shape of a log dictates the formula to be used to
estimate cubic volumes. Smalian's and Huber's formulas are
often used to estimate the volume of midstem and upper-stem
logs; both formulas assume a parabolic shape. Smalian's
formula can also be used on butt logs after the large-end
diameter has been reduced to account for basal flare. An
equation developed by Bruce (1982) considers the neiloid shape
of butt logs and estimates volume with no adjustment to the

MARLIN E. PLANK is a research forest products technologist and
JAMES M. CAHILL is a research forester at Forestry Sciences
Laboratory, P.O. Box 3890, Portland, Oregon 97208.

x U.S. GOVERNMENT PRINTING OFFICE: 1984—593-311

Cone or paraboloid

Middle logs > Convex paraboloid

Butt log Concave paraboloid

Stump

Figure 1--Geometric shapes in a
coniferous tree stem.

measurement of the large end. Bruce's equation has proved
accurate and unbiased2/ for coast Douglas-fir (Pseudotsuga
menziesii (Mirb.) Franco var. menziesii) logs, and it is
being evaluated by the forest products industry and the USDA
Forest Service in the Western United States.

Selecting an accurate and unbiased formula to estimate the cubic
volume of small diameter “tree-length" logs is difficult. Tree-
length logs usually extend from the stump into the live crown
and can have the concave shape of the butt, the convex shape of
the midstem, and the conic shape of the top within a single long
log. The purpose of this paper is to compare the accuracy and
bias of volume estimates made by Bruce's, Smalian'’s, and Huber's
formulas on small diameter, tree-length ponderosa pine (Pinus
ponderosa Dougl. ex Laws.) and lodgepole pine (P. contorta
Dougl. ex Loud.) logs. None of these equations were designed
specifically for tree-length logs, but because they are all in
common use, knowledge of their accuracy will be helpful for
scalers and mensurationists.

1/max, T. As (Cahill, ie. Mand
Snellgrove, T. A. Validation of butt
log estimator for Douglas-fir. Sub-
mitted to Forest Science in May 1984.

Methods
Data Base

Log Measurements

The data used in this analysis were Scribner log scale
measurements recorded on 860 tree-length logs. The sample
included 351 ponderosa pine logs from Colorado, Arizona, and
South Dakota, and 509 lodgepole pine logs from Wyoming and
Oregon. The logs were from several product recovery studies.
Generally, trees selected for product recovery studies include
the range of stem quality that exists within a geographic area.
We think these samples represent a good range of stem forms for
small diameter ponderosa and lodgepole pine trees. The
following tabulation shows the range of diameters and lengths
for the tree-length logs in the sample:

Number of Range in diameter Range
logs Large Small in length
(Inches) (Feet)
Ponderosa pine 352 7-20 5- 9 20-50
Lodgepole pine 509 6-18 4-10 20-50

Lengths and diameters of the tree-length logs were measured and
recorded by USDA Forest Service scalers in the mill yard. After
the logs were bucked for milling, the dimensions of the short
logs were also recorded. Length of the short logs varied from

4 to 20 feet, depending on whether the log was processed into
veneer (4 feet), studs (8 feet), or random-length dimension
lumber (8 to 20 feet). All measurements were taken according to
the USDA Forest Service Log Scaling Handbook (1973) rules.

Midpoint diameters were not directly measured on the logs in
the data base. Estimates of the midpoint were made from the
short log scale measurements as in the following example. For
a 42-foot tree-length log bucked into three short logs, the
dimensions of the short logs are as follows:

Butt log: 16 x 12 inches x 16 feet
Middle log: 12x 9 inches x 16 feet
Top log: 9 x 6 inches x 10 feet

The midpoint of the tree-length log (21 feet) occurred in the
second short log. Diameters at each end of that log were 12 and
9 inches and the log length was 16 feet; the average taper was
(12 - 9)/16 = 0.187 inch per foot. Because the midpoint of the
tree-length log was 5 feet from the large end of the second
short log, the midpoint was estimated to be 12-(5 x 0.187) =
11.1 inches. Midpoint diameters were rounded to the nearest
inch.

Computation of Actual For our purposes, actual volume of a tree-length log is the sum

Log Volume of the cubic volumes of short logs bucked from the log. The
majority of tree—length logs were cut into two or more segments
and were processed into lumber. The volume of the butt segment
was computed by Bruce's (1982) formula, and the volume of all
other segments was computed by Smalian'’s formula. For the
example shown above, the dimensions and volumes of the short

logs are:

Butt log: (16 x 12 inches x 16 feet) volume = 15.0 cubic feet
Middle log: (12 x 9 inches x 16 feet) volume = 9.8 cubic feet
Top log: (9 x 6 inches x 10 feet) volume = 3.2 cubic feet

Actual volume of the tree-length log (V,) is computed by
summing the volumes of the short logs: Vg, = 15.0 + 9.8
+ 3.2 = 28.0 cubic feet.

Validation of Actual To test whether a reasonable estimate of actual volume can be

Volume made by adding the volumes of short logs, we used 84 lodgepole
pine logs from our data base; measurements had been taken at
4-foot intervals. If the sum of the volume for the 4-foot
segments approximated the volume estimated for the entire short
log, then our method of calculating short-log volume, and thus
the tree-length volume, was accurate. We did not have similar
measurements on the ponderosa pine and had to assume that the
results of the lodgepole pine validation would apply to the
ponderosa pine. The following procedure was used to make this
test:

Each of the 84 tree-length logs was divided into a 16-foot butt
log and an upper log of variable length. The actual volume for
the shorter logs was computed by summing the volume of the
4-foot segments. The volume of each segment was computed by
assuming that the segment was a frustum of a cone; that is,
with the formula:

V= ee (D Z + D : + D_ D_)L;
Se oe s L Sil
3
where:
V = volume of the 4-foot segment (cubic feet);
L = length;
D, = small end diameter inside bark of the segments
(inches);
Dy = large end diameter inside bark of the segments
(inches) ;
0.005454 = conversion constant.

Computation of
Estimated Volume

The volumes summed from the 4-foot segments were validated by
comparing them with the estimates of the short-log volume
computed by applying Bruce's (1982) formula on the butt 16-foot
short-log segment and Smalian's formula on the top segment.

For the 84 test logs, we found that Bruce's equation
underestimated the volume of the butt segments by an average of
5.3 percent, and Smalian's formula overestimated the volume of
the top segments by an average of 0.4 percent. For the entire
tree-length log, the estimated volume, obtained by summing the
volumes of the short logs, underestimated the actual volume by
3 percent. We considered this an acceptable level of accuracy.

Estimated cubic-foot volumes of the tree-length logs were
computed by Bruce's (1982), Huber's, and Smalian's formulas.
The formulas are shown below:

0.005454(0.25 D a +0775 D. dL

Bruces's: Volume (cubic feet) L

Huber’s: Volume (cubic feet) 0.005454(D_*)L;

Smalian's: Volume (cubic feet) ro

0.0027274(D 2 + D
s L

where:

L is log length;

Ds is small end diameter;

Dm is midpoint diameter;

Dy is large end diameter;

and 0.005454 and 0.0027274 are conversion constants.

By use of the dimensions of the same tree-length log, the
cubic-foot volumes estimated by the three formulas are:

0.005454(0.75(6-) + 0.25(167))42.0

20.8 Ft?

Bruce's estimate: Yb

Huber's estimate: V. 0.005454(11°) 42.0 Ahad ft>.

3

Smalian's estimate: ve 0.0027274(6- + 167)42.0 = 33.4 ft

Computation of Bias
and Accuracy

Results

The deviations between actual volume (V,) and the three
estimates of actual volume--Vp, Vp, and V,—--were computed

for each log. The mean deviation was used to estimate the
average bias for the ponderosa and lodgepole pine samples. The
square root of the mean squared deviation was used as an
estimate of accuracy. The computational formulas for bias and
accuracy are shown below:

Bias = ) (Vg - Ve)/N;

ut
Accuracy = (1 (vg - Ve)2/N)) 2

?

where:

V,g is the actual volume of a tree-length log;

Ve is the volume estimated by either Bruce's, Smalian's, or
Huber’s formula; and

N is the number of logs in the sample.

Volume estimates made by Huber's formula had the least amount
of bias and were the most accurate (table 1). This was
consistent for the ponderosa and lodgepole pine data. On a
percentage basis, Huber's formula underestimated the actual
volume for both species combined by 2 percent, whereas Bruce's
(1982) underestimated it by 16 percent and Smalian’s
overestimated it by 19 percent.

Table 1--Bias and accuracy of Huber’s, Bruce’s, and Smalian’s
formulas for predicting the cubic volume of ponderosa and lodgepole
pine tree—length logs

Bias Accuracy

Species
Huber Bruce Smalian Huber Bruce Smalian
(1982) (1982)
Cubic feet

Ponderosa
pine 0.39 3.20 —-5.31 1.88 5.44 eral
Lodgepole
pine .38 3.17 —2.96 Loz 4.39 4.32

Discussion

Literature Cited

We recommend using Huber's formula for estimating cubic-foot
volume for ponderosa and lodgepole pine tree-length logs. For
our data, volume estimates made by Huber's formula were less
biased and more accurate than estimates made by either Bruce's
(1982) or Smalian's formula. Caution is necessary when these
results are extrapolated beyond the range of diameters and
lengths included in our samples.

Measuring only the midpoint diameter, as required by Huber'’s
formula, has advantages and disadvantages. The obvious
advantage is that only one diameter measurement is required on
each log. This may represent a reduction in scaling costs.
The disadvantages are: Scalers must have access to the middle
of the log for caliper measurements, and estimates of bark
thickness at the midpoint must be made. These are serious
limitations if logs are scaled on trucks but would not be a
problem when logs are rolled out in a yard. Finally, computing
volumes by Huber's formula is easy and can be done directly in
the field. Scalers need only log length and the midpoint
diameter to obtain volume estimates by use of tables (USDA
Forest Service 1978) of half cylinder cubic volumes.

Bruce, David. Butt log volume estimators. Forest Science. 28(3):
489-503; 1982.

U.S. Department of Agriculture, Forest Service. National Forest
log scaling handbook. FSH 2409.11. Amend. 4. Washington, DC;
£973.

U.S. Department of Agriculture, Forest Service. Cubic scaling
handbook. (A review draft.) Washington, DC; 1978.

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Agriculture is dedicated to the principle of multiple
use management of the Nation’s forest resources
for sustained yields of wood, water, forage, wildlife,
and recreation. Through forestry research,
cooperation with the States and private forest
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Origin.

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Experiment Station

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PO. Box 3890

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se