Forest Mensuration. Brack and Wood

Primary measurements to be considered are:

• length;
• diameter;
• volume;
• weight.

Log length

Length may be measured directly by placing a copy of the basic unit of length against the log, e.g. a graduated tape measure made of linen, steel or plastic; or it may be measured indirectly by scale stick, revolving wheel, photogrammetrically, infra-red scanning, etc.

Care must be taken to ensure that:

• instruments are not biased;
• measurements are taken along the correct plane;
• correct precision of measurement is adopted.

Log diameter

• Direct measurements can be made at the ends of logs with rulers and at intermediate points with calipers (two readings at right angles should be made unless a large number of logs is involved).
• Indirect measurements can be made with a diameter tape (graduated in PI units), Biltmore stick, photogrammetrically, infra-red scanner, etc.

Diameter tapes for log measurement are usually made of steel. They should be wound tightly around the circumference of the log in a plane at right angles to the long axis of the log.

Sectional area (recorded in m^2) is usually needed to derive log volume and is derived from calculations on the measurements of diameter. Errors in assuming that the log cross-section is circular can be judged from the table below.

Table 1 Cross-sectional area (SA) of elliptical cross sections estimated in various ways. Figures are based on two values of each of maximum (d1) and minimum (d2) diameter.

		d1=66 cm; d2=60 cm		d1=26 cm; d2=20 cm
Parameter	Formula	SA (m^2)	% Error (%)	SA (m^2)	Error (%)
Girth,	SA = C^2/(4PI)	0.3121	+ 0.34	0.0414	+ 2.70
Arithmetic mean	SA =PI/16 (d1+d2)^2	0.3117	+ 0.23	0.0415	+ 1.72
Geometric mean	SA =PI/4 (d1d2)	0.3110	+ 0	0.0408	+ 0
Quadratic mean	SA =PI/8 (d1^2+d2^2)	0.3124	+ 0.45	0.0423	+ 3.68
d1 alone	SA =PI/4 d1^2	0.3421	+10.00	0.0531	+30.15
d2 alone	SA =PI/4 d2^2	0.2817	- 9.09	0.0314	-23.04

Even an ellipse is not necessarily the true shape of the cross-section of a log; rather, it may be described as a "closed convex region". The geometric mean of at least two calipered diameters will lead to minimising the error in this assumption.

Log volume

Excurrent trees: Stems of excurrent trees approach in general outline the forms of a limited number of solids of revolution, i.e. neiloid, conoid, paraboloid or cylinder; and most logs are regarded as a frustum of one or other of these solids. All result from the revolution about a vertical axis of a curve of general formula (Grosenbaugh, l966):

y^2  = kx^b  

where y =  the radius of cross section
      x =  the distance from the apex

      b determines the way the solid tapers, i.e. the shape of the solid
and   k determines the rate of taper within the specified shape.

Shape	b value	Formula
Cylinder	0	y^2 = kx^o, i.e. y = k
3 degree paraboloid	2/3	y^2 = kx^2/3 or y^3 = kx, i.e. y = kx^1/3
2 degree (quadratic) paraboloid	1	y^2 = kx, i.e. y = kx^1/2
Conoid	2	y^2 = kx2, i.e. y = kx
Neiloid	3	y^2 = kx3, i.e. y = kx^3/2

Deliquescent trees: Stems of deliquescent trees approximate frusta of the above solids. In mature trees, the frusta are generally parabolic. Seldom are they conic or neiloidic.

Portions of a tree stem may assume several geometric forms:

Top - cone (paraboloid on rare occasions). Main bole - frustum of paraboloid Butt - frustum of neiloid (rarely a cylinder)

Swell in the butt section of a tree (buttswell) varies with species, tree age and environment. Thus, log shape depends on five factors:

• Species
• Tree type - excurrent or deliquescent
• Position in tree
• Environment (incl. silvicultural treatment)
• Age

Returning now to the question of the volume of a log. This is its cubical content measured in m^3 and is a function not only of diameter and length, but also of shape and taper.

The generalised formula for the volume of a full solid figure is:

v = [1/(b+1)] PI/4 d0^2 l

where v = volume
      d = diameter at base
      l = length from base to tip
and   b = a constant which varies with shape, viz.:
          0 for a cylinder
          2/3 for a paraboloid (third degree)
          1 for a paraboloid (second degree)
          2 for a conoid
          3 for a neiloid

Huber's formula assumes that the average sectional area of a log is found at its mid-point. Unfortunately, this is not always so.

Huber's formula is intermediate in accuracy between Newton's and Smalian's formulae when shape is not a 2 paraboloid, but it has limitations in use because -

• bark removal, bark measurement or an empirical bark deduction is required to obtain mid-dub;
• the mid-points of stacked logs are mostly inaccessible.

Despite this, Huber's formula is the common basis of log tables giving log volume for various lengths and centre diameters or girths. Ellis and Duff (l973) in New Zealand found Huber's formula to be more robust than other formulae when long intervals between measurements are used. They point out that even with an interval of l0 m or more, underestimation by Huber's formula was only about 5% of the total volume. Given the low intensity of measurement with Huber's formula, the value of the information obtained from the small end and irregular butt end, as required by Newton's and Smalian's formulae, is poor compared to that from the mid-point.

Thus, when the volume of a butt or long log has to be measured, it is preferable to use Huber's formula. Measurement by caliper along two axes at right angles to each other at the mid-length position may be the easiest to apply to logs on the ground.

However, recent research indicates clearly that the centroid method of sampling is more accurate than all three of these standard formulae (Wood and Wiant 1990, Wiant et al. 1992, Patterson et al. (1992)).

When large numbers of logs are to be measured, one diameter reading by caliper at the mid-length of each log may be sufficient. If a tape is used to estimate the diameter, ensure that it is not twisted and it is in contact with the surface of the log around its entire circumference.

Smalian's formula, requiring measurements at both ends of a log, is the easiest to apply (which explains why it has the widest acceptance world-wide for log scaling) but it is also the least accurate of the three formulae if log shape is not a 2 degree paraboloid. This applies particularly to butt logs with flared ends. Errors in the volume estimate increase rapidly as intensity of measurement decreases exceeding 8% when the interval between measurements is greater than 5 m, e.g.

Effect of section length on the error of the volume estimate using various formulae.

Formula	Material	Section Length (m)	Error (%)	Reference
Huber	Log	1.2 4.9	- 3.5 -3.6 to -5.1	1
Huber	Tree	1.2 3.0 6.1	- 0.4 -2.0 -2.9	2
Smalian	Log	1.2 4.9	- 4.6 +8.7 tp +10	1
Smalian	Tree	0.9 1.2 1.6 3.0 5.0 6.1	Unbiased +1.5 Unbiased +6.1 8+ +14.7	5 2 5 2 4 2
Newton	Tree	1.2 6.1	+ 0.2 +3.0	2 2

References:
1. Young, Robbins and Wilson l967. IUFRO Section 25: 546-62.
2. Ellis and Duff, l973. N.Z. For. Serv., FRI For. Mensuration Rept. 50. 
3. Carron and McIntyre, l959. Aust. For. 23(1): 50-60.
4. Goulding, l979. N.Z. J. For Sci. 9(1): 89-99.
5. Dargavel and Ditchburne, l97l. Aust. For. 35(3): 191-8.

Note that Newton's and Huber's formulae are applicable only when the interval between measurements is uniform. Where trees have been measured with irregular section lengths, as in the dendrometry of standing trees, Smalian's formula alone (of the standard formulae) can be used.

Goulding (l979) showed that integrating a cubic spline curve fitted through the data points can give an estimate of volume with only 60% of the error of Smalian's formula. Hence, the spline curve function is particularly useful for processing dendrometer measurements of standing trees.

The extent of error in estimating log volume using each of the formulae is governed by actual log shape and the representativeness of the points of measurement. Too little attention is given to the latter in log measurement, e.g. the end- and mid-diameters are mostly taken as fixed points even though the points may coincide with nodal swellings.

Note also that Smalian's formula involves calculating the mean of the end sectional areas, not the sectional area of the mean of the end diameters. The two means are not equal. The error can be important if the end diameters differ appreciably, e.g. for a log of end diameters 25 and 40 cm respectively, the error in sectional area (and hence volume) is 5%.

When log shape is not a second degree paraboloid, the errors given by both Huber's and Smalian's formulae are proportional to log length and the square of the difference between the two diameters, i.e. the longer the log and the greater the taper, the greater is the error.

Newton's formula and centroid sampling can be used to estimate the volume of any of the forms which a tree stem or portion assumes. These are the most accurate of the formulae for log volume measurement but, being the most expensive to apply, use may be limited to research.

For perfect cylinders and frusta of truly 2 degree paraboloids, all the above formulae yield accurate results.

For conic or neiloidic frusta, Huber's and Smalian's formulae under- and over-estimate volume respectively. With these shapes, theoretical considerations favour the use of Huber's formula because the negative error incurred is less than (actually one-half of) the positive error incurred by using Smalian's formula.

Volumes of logs in Australia, New Zealand and many other countries are often derived from log volume tables, which give underbark volumes from either:

• length and d.u.b. at mid-length, or
• length and small-end diameter (s.e.d.) underbark

The New Zealand Forest Research Institute (J.C. Ellis, 1982: FRI Bull. 20) have developed what they term a 'Universal Formula' for determining volume under bark of round wood (logs) given log length and diameters under bark at both the large and small ends. The formula is 3-dimensional and requires input of length, small-end diameter and log taper. It has been used to compile ten log volume tables, based on average tapers from 0.4 cm/m to 2.2 cm/m, which apply to any species of conifer anywhere in New Zealand (and presumably in Australia and elsewhere).

The "Universal Formula" is:

V = EXP (1.944 157 x ln L + 0.029 931 x Ds + 0.884 711 x ln ((D - Ds) / L) - 0.038 675) x 10^-3

    + 0.000 078 540 Ds^2 x L

where V      = total log volume (m^3)
      EXP    = antilog or e^x  
      L      = log length (m)
      D      = large end diameter under bark (cm)
      Ds     = small end diameter under bark (cm)
     (D-d)/L = log taper (cm/m)

The formula estimates volume to within 16% and 2% at 95% probability for one log and 50 logs respectively.

Sometimes, for sale purposes, an estimate may be needed of the volume of logs in standing timber. Such estimates require compilation from sample tree data of sales volume functions or models. An example of such a model is that developed by the NSW Forestry Commission for east coast hardwoods based on Kendall sample tree data. The form of this model is:

      Vs  =  a  +  b x d  +  c x L  + e x D^2 x L

where Vs  =  gross sale volume (m^3)
      D   =  dbhob (cm)
      L   =  log length (m)
      a, b, c, e denote coefficients

Measurement of log quantity by weight (weight scaling)

New Zealand experience has established that weight scaling is the most efficient method of assessing "log volume", and it is used whenever access to or installation of a weighbridge is practicable (85% of cases). Log volume is then derived from green (wet) weight using conversion factors established by continuous sampling, e.g.:

Volume (m3) = tonnes green weight x 0.94.

Weight scaling is used extensively in North America and Europe, particularly Scandinavia. In Norway, a separate company has been established by law to measure and/or weigh all types of logs throughout the country. Pulpwood is regularly sampled for moisture content. They use a modified chainsaw and sample the chips.

Weight scaling has many attractive advantages when large quantities of material are involved. It is fast, easy and objective. Conversion to volume requires continuous sampling rather than predetermined conversion factors.

Advantages and disadvantages of measuring log quantity by weight

            Advantages                         Disadvantages

Measurement is objective.            Industry is more or less geared to
Eliminates log by log computations   the volumetric system of 
thus minimising opportunities for    measurement.
error.                               A considerable change involving
                                     legal, accounting and sales 
                                     matters must occur.  Change will be
                                     difficult and slow.
Less dangerous.  Log measurers risk  
physical injury from rolling logs.

Provides incentive for better        Strategic distribution of weigh-
loading of trucks and thus incre-    bridges is essential.  Weighing
ases volume handled.                 cannot be enforced if trucks have
Stimulates delivery of green logs    to be diverted from a direct route
free from stain - desirable for      to a mill to cross a weighbridge.
pulping purposes. 
Is quick, requires no special        Not suited for logs where grade is a
handling, and saves time for both    factor determining selling price.
buyer and seller.                    Grade involves both log size and
Inventories are more easily          quality (freedom from knots, bends,
maintained.                          spiral grain, etc.).
Makes possible uncontested spot
payment for delivered logs.
 

Determination of wood weight

1. Weighbridge - need to make allowance for bark, mud and similar detritus
2. Production line - weigh each log individually somewhere along the line
3. Tree weight tables - based on independent variables similar to those used in tree volume tables. Particularly useful for standing trees.

With all methods, conversion factors may be needed to give dry weight equivalent.

Allowance for defect

Defect presents a problem in estimating log quantity. Log grading rules attempt to overcome the problem.

Scale deductions may be made for rots, insect flight passages, ring shakes, checks, splits and crooks but not for sizes and frequencies of sound knots, spiral grain, sapstains, resin pockets, bark inclusions, etc. These latter, however, affect quality of the sawn produce and will be taken into account during grading in the mill.

Allowance for pipe is usually computed assuming the cross section of the pipe is a square of side equal to pipe diameter. A common practice in making the allowance is to measure pipe diameter at each end of the log (to the nearest centimetre above) and mean the two diameters.

Making deductions for log defect requires two steps:

• determine type and extent of defect
• compute the loss.

Grosenbaugh (1952) provides a consistent set of rules (imperial units). These are useful for reference purposes particularly for those wishing to develop similar allowances for logs measured in metric units. These rules apply to:

• defect affecting entire cross-section
• defect affecting a wedge-shaped cross-section
• log sweep
• crook
• interior defect, enclosed by a circle or ellipse.

http://online.anu.edu.au/Forestry/mensuration/LOGS.HTM
Cris.Brack@anu.edu.au
Mon, 14 Apr. 1997