Can’t See the Forest for the Trees (II)

I’m not a logger or a sawyer, so I can’t speak exactly from those perspectives. I have logged and I have sawn logs many times in my life, not enough to be any sort of expert in either matter, but I’ll offer what I can on the subject.

Let’s say you find some logs, like what you see, and you want to buy them. Or, perhaps you have some logs and want to sell them. What if you wanted to sell logs internationally? How is the price to be negotiated?  By number of logs? By length, by width, by volume? By the amount of usable timber or by the total volume of the log including bark? Or without bark?

Saw logs are conventionally sold on a ‘cubic meter’ volumetric basis, and this has been the case since the late 18th century.

You would think that a ‘cubic meter’ – 1m. wide x 1m. deep x 1m. tall  -would itself be a fairly well-defined and universally agreed sort of thing, but it is not so when it comes to logs. A cubic meter here is not the same as a cubic meter there.

It boils down to the one question: how do you measure a log?

What comes into consideration though, before the matter of measuring of volume, is the assumption as to which geometric shape the log conforms as we calculate volume based on a geometric form. The candidates are:

1. Cylinder

 

2. Cone

3. Paraboloid

‘Paraboloid’ itself can be considered a simplification. We might also assume the tree trunk has varied forms according to which section of the trunk is considered:

Those seem to be the choices in front of us. Notice in all four that the base or horizontal plane cutting through the geometric form is a circle. Any horizontal plane cutting through the log would also render a circle. Trouble is, trees are rarely, if ever, perfectly circular in cross-section, and some are a good deal non-circular. We could have added in an illustration of a cone with an elliptical base, for example. Not that any tree has a perfectly elliptical cross section mind you!

No tree trunk is a pure cylinder, however old Sitka Spruce trees do look a lot like gigantic pencils in the forest and the logs therefore come very close to cylinders. Some trees are much more tapered in form, like a cone. Cedar trees come to mind. Some tree trunks taper evenly, others stay fairly cylindrical and then taper radically at the crown portion. Some trees flare strongly at the butt, others not so much.

Between the forms of a cone and a parabola, neither is an absolute ringer for any tree I’ve come across, however the cone is certainly a simpler form upon which to base calculations. Assuming a conical shape for a log was the standard way to do things way back in 1765, as noted by Oettelt in Germany.

Assuming that a log is a section of a cone is a simplistic one, however the fact is that no tree truly conforms to any geometric form, not precisely at least. So we proceed from a simplistic assumption and nothing more, simplicity taken as a virtue over ultimate accuracy. Refining the accuracy of the measurement is seen to be more expense in term of the time involved than is realized as a benefit resulting from the extra effort. Relative accuracy, therefore, is what is sought, and that is simply the maximum accuracy that is profitable and possible to obtain in practice.

Right, given that, we can at least start from the basis of assuming each log is a portion of a cone, and calculate volume from there, yes? The formula for determining the volume of a cone is easy to remember, it seems to me, IF you find the formula for the volume of a cylinder easy to remember. How’s that for a bit of, er, circular reasoning?

The volume of a cylinder is that of the base circle area (πr^2) times the height, h:

A cone’s volume is simply 1/3 the volume of a cylinder, or (πr^2h/3):

An easy way to think about how a cone and a cylinder relate to one another is to imagine a pair of paper drinking cups, one a cone and one a cylinder. Both have the same total height, and the same circular base. If we filled the conical cup with water or salt, say, and poured it out into the cylindrical cup, it would fill that cup 1/3 of the way:

One can see thereby the vastly different estimate of volume one would obtain whether one were to consider a log as a cylinder or a cone.

One niggling point though in regards to cones: when we are looking to buy logs, we’re not typically looking for the pointy end of the stick (i.e., a pointed cone), as that is the end with all the branches. That bit is perhaps best left behind in the forest. We are looking instead at a section of the trunk, which is a truncation of a cone. Another name for a truncated cone is ‘frustum’ cone, a frustum being that portion of the cone laying between the two parallel planes cutting it:

The word frustum comes from Latin and means ‘piece’, or ‘crumb’.

Now, the formula for a frustum of a cone is a little bit trickier than the formula for the volume of a cone. A simple way to do it, if you had both portions of the original cone before truncation, would be to perform two separate calculations, one for the entire uncut cone, and a second for the piece which was lopped off on top. Subtract the volume from that upper piece from the total and you will find the volume for the frustum. That’s an inefficient way to do things, and it is also the case that when looking at a log you only have the frustum to consider. So we need a formula just for that portion:

The volume of a frustum of any cone is equal to one-third of the product of the altitude and the sum of the upper base, the lower base, and the mean proportional between the two bases.

Put into mathematical language, the above sentence becomes:

The uppercase ‘R’ in the formula denotes the radius for the base of the frustum, and the lowercase ‘r’ is the radius of the upper cutting plane. The area of the base is given by π R^2 while the area of the upper plane is π r^2 – you can see those formulas as part of the larger formula above. The piece under the square root symbol in the formula above, “√ π R^2( π r^2)” is the “mean proportional between the two bases”. The ‘mean proportional’ of two different quantities, let’s call them a and b, is a quantity r such that a is to r, as r is to b (a:r = r:b). for instance, the mean proportional between 2 and 18 would be 6 (as 2:6 = 6:18). The Golden mean, 1:1.6180339…, is a particularly famous proportional, known to many readers I’m sure.

Consider then that the formula (a:r = r:b) can be simplified algebraically to r^2 = ab. Solving for r, we get: r = √ab, and that is the core idea of the portion of the formula above we see as √π R^2 (π r^2). Of course, when we see a formula in which we take a square root of an entity that is squared, in effect we cancel out both the square root and the squares, as √6^2 = 6. Therefore, √π R^2 (π r^2) becomes simply πRr. We thereby produce this formula:

This formula can then be cleaned up a bit by extracting π out from the parentheses:

Well, not quite so easy to remember as the formula for the volume of a regular cone, but it does fall short of what one would call a complex formula. The math geeks out there will possibly be yawning at this point.

So, let’s say we have a log, and it measures 300 centimeters long, has a butt end of 100 centimeter diameter (R of 50 cm), and a skinny end of 80 cm diameter (r of 40cm). By the above formula we would write:

V = ⅓ π ((502 wh+ 402 wh + (50 x 40)) x 300

That gives us:

V = ⅓ π (2500 + 1600 + 2000) x 300

Simplifying to:

V = ⅓ π (6100) x 300

Pi (π) equals 3.14159 or thereabouts. To complete the solution, we obtain:

V = ⅓ (3.14159 x 6100) x 300
V = ⅓ (5,749,109.7)
V = 1,916,369.9

The answer, since we reckoned the problem in centimeters, is in cubic centimeters. Logs are sold on a cubic meter basis, and 1 meter is 100 cm. Therefore, 1 cubic meter (1 m³) equals (100 cm/m)³ = 100³ cm³ = 1,000,000 cm³. We therefore divide the above obtained value in cubic centimeters, 1,916,369.9 by 1,000,000, to obtain the volume in cubic meters: 1.916.

Keep in mind that when we performed the above calculation we were working with nice round numbers, however lengths and diameters of actual logs are more often given with nominal or rounded forms, possibly in 2-foot length multiples, for instance, or the calculated answer may also be rounded, the 1.916 above being rounded to 2 cubic meters.

If you measured the log in inches or feet, it is much the same process, however at the end one would need to convert over to cubic meters from cubic feet or inches. One cubic meter equals 35.315 cubic feet. The conversion is simple enough: take the volume in cubic feet and divide by 35.315 to obtain cubic meters. If working from cubic inches, divide by 61024 to obtain cubic meters.

If only things were so simple as plugging data into a formula, but the situation turns out to be very much different in the world of log scaling today. Because taking a simple mathematical calculation of a log to obtain volume does not give us the actual likely amount of convertible timber, does it? Logs vary, and in some species by quite an amount. Between species it becomes more varied yet. Some trees, for instance, have thin bark while others have thick bark. In the US, log volume excludes the bark but in other countries the bark is incorporated in the calculation. Some tree trunks contain enormous amounts of sapwood with little commercial value (Gabon ebony would be a good example). American Black Walnut is a tree which also has a hefty band of sapwood, however the case is different from the ebony as the wood can be steamed so as to homogenize the color of the sapwood with the heartwood. Young trees have a higher proportion of juvenile wood than do old growth trees. Some species are prone to defects of various kinds, or have characteristic areas of rot, also limiting the amount of convertible timber. Some logs are almost exclusively converted into veneer, while others are nearly always sawn for timber. And on it goes. Log scaling is simply the process for estimating the weight or volume of a log while allowing for features that reduce product recovery.

One could come up with different scaling methods to calculate the useable extractable volume of timber from a log on a species by species basis, however it is fortunate that things haven’t gotten quite that fractionated – but it is close. Welcome to the world of log scaling, where complexity, as they say, rules.

In reckoning log volume, some systems average the log end areas, while others average the log end diameters. Some assume different geometric shapes for the log. Here are 7 basic systems, with the actual formulas left out:

1. Smalian (assumes a parabolic log shape, and is the standard log rule (in metric form) in British Columbia and the basis of the Interagency Cubic Foot scaling System)
2. Bruce’s Butt Log (as Smalian’s formula assumes a paraboloid shape, it tends to overestimate volume of butt logs. David Bruce came up with his rule in 1982 to adjust for the butt portion of the tree trunk)
3. Huber (this formula assumes the average cross section area is found at the midpoint of the log, an assumption which is not always true of course)
4. Sorenson (derived from the Huber formula and assumes a taper of 1 inch per 10 feet of log length. The taper assumption is not always correct however)
5. Newton (the most accurate, requiring measurement of both end diameters of the log as well as the midpoint diameter, however this measuring is more time consuming to execute)
6. Subneloid (similar to the Smalian, however configured so as to allow a multiplication by 12 to obtain the board foot measure, becoming the “Brererton Log Foot Scale”)
7. Two-end Conic (assumes the log shape is a cone, and is the basis of the “Northwest Cubic Foot Log scaling Rule”)

There is also the Hoppus rule, derived in Britain. It is the most widespread system internationally; assumes an assumption about processing loss. Sometimes called the “quarter girth formula”, it considers the volume of a log in cubic meters to be:

V = (C/4)2 wh x L/10,000

Where, V is volume, C is log circumference in centimeters and L is the log length in meters. The result obtained by this formula can be compared to the math formula for volume we worked earlier, using the same numbers. We would have to pick a circumference though, so I’ll average the small end (r = 40cm) and the large end (R=50cm) to obtain a radius of 45cm. Radius converts to circumference as 2πr, so a 45cm radius gives a circumference of π90, or 282.743…

Plug that into the formula and let’s see what we get:

V = (282.743 / 4)2 wh x 3/10,000

V =  4996.48722805 x 3/10,000

V = 1.49894616842 cubic meters

Compared to the strict mathematical estimate for cubic volume, the Hoppus rule gives a volume which accounts, on average, for about 78.5% of the cubic volume of the log-  the other 21.5% is lost as edgings, sawdust, and slabbed offcuts. Another name for the cubic volume measure obtained by the Hoppus rule is the Francon Cubic Meter, so as to distinguish it from the solid cubic meter.

As mentioned above, there is the Interagency Cubic Foot System, developed in the US in 1991. This takes the Smalian formula and applies it to any log segment of 20′ or less. If the log is longer than 20′, then it is divided into segments and the taper of each segment is estimated as the difference between diameters at each end. This method aims to reduce the bias which results from the Smalian formula otherwise, which assumes a log to be a paraboloid. As you can see from the 4th picture in the series of geometric forms of logs shown above, the log usually does not conform perfectly to the paraboloid form.

That is an overview of the situation in western countries, however other places have different log rules yet. For instance there are four different log rules used in Japan (these rules are also used in Korea):

1. Japanese Agricultural Standard (JAS): the log diameter is measured at the small end only, and taken in two axes, which are averaged, towards obtaining what is called a ‘scaling diameter’ (D). The scaling diameter (D) receives adjustments which are done using a table. Scaling length (L) is in 20cm intervals. Volume = D2 whL / 10,000 if L is ≤ 6meters. If L is ≥ 6 meters, then the formula becomes V= [D -INT(L) -4) / 2 ]2 wh x L/10,000. “INT(L)” is the length of the log rounded down to the nearest meter, and the expression INT(L) -4) / 2 is a taper adjustment of 1cm per meter of length. This formula views a log as a square cant with a side equal to the scaling diameter.
2. Revised JAS: modifies the formula for logs 6 meters and longer with a factor to adjust the original taper assumption.
3. South Sea Log Scale (SSL): also known as the Brererton. Measure the long and the short axis of each log end and round them down to the nearest 2cm (i.e., 77.1 cm becomes rounded to 76 cm). then average the two rounded measures to the nearest whole centimeter. This gives the ‘D’ (diameter) value. Volume = 0.7854 D2 whL /10,000. This formula is the metric equivalent of the Subneloid formula described in the earlier table.This formula is typically applied to hardwood logs from tropical Asian sources.
4. Hiragoku or Heiseki scale: Uses the same procedure for finding the ‘D’ measure as the SSL method above. Volume = D2 whL /10,000. This method considers the log as being a square cant with each side equal to the recorded small end diameter, and there is no adjustment for taper.

In fact, in Japan the traditional measure of log volume has not been the cubic meter until recently. Much the same as Japanese carpentry, which is done to the measurement system of Shakkan-hō using shaku 尺 (30.3030cm) sun 寸 (3.0303cm) and bu 分 (3.0303mm), logs are reckoned by koku 石, a unit of volume equal to 10 cubic shaku. A koku was originally defined as a quantity of rice, supposedly equal to the amount required to feed one person for a year. The unit for koku, 石, is in fact the same character as is used for stone – if you find this confusing, welcome to the world of Kanji. For internal purposes, I would suspect a lot of log scaling in Japan is done on the basis of koku, however when importing or exporting, the koku value needs to be converted to cubic meters, one koku equals 0.2782 m³ (9.826 cubic ft.).

Indonesia, Malaysia and the Phillipines often use the SSL system, however there are regional variations in how dimeters are taken and recorded which lead to variances.

British Columbia, a Canadian province larger than many countries and a major timber supplier, has its own log rule system, the ‘BC Metric Scale’ which is the metric form of Smalian’s formula. Here it is: V = (r1^2 + r2^2) L 0.0001570976, where r1 and r2 denote the top and bottom radii, rounded to the nearest even number, and L is the length in meters recorded to the nearest 0.2m, with the convention that exact odd numbers are rounded down (i.e., 13.5m becomes 13.4m).

Let’s see how the BC Metric Scale compares to our previous reckonings using a 3m log, 40 cm radius at the skinny end and 50 cm radius at the fat end:

V = (402wh + 502 wh) 3 x  0.0001570976

V =  (4100) 3 x  0.0001570976

V = 1.93230048 cubic meters.

This formula lead to an answer in which the volume of wood was slightly greater than for the geometric form of the frustum of a cone, which is not surprising since the Smalian method presupposes the log is a paraboloid. 

Chile uses the JAS method. Russia uses a system called GOST 2708-75, which has similarities to the JAS and Hiragoku systems, and to the Huber formula. New Zealand has its own systems, one for logs used domestically, and three other systems specially for logs exported to other Countries. For export to Japan they use a system which is an Imperial form (that is, in feet and inches) of the metric Hoppus formula described above. Or they will use the JAS system. For logs exports to China, likely to be the bulk of business these days, the William Klemme system is used, which calculates the volume a if a log were a simple cylinder.

In practice, a given location and a given mill will likely be using one scaling system all the time, so once a log scaler is familiar with the particular scaling system, it is fairly straightforward process. When international trade gets factored in, the picture can be a good deal more complex and buyer beware. Those selling logs may well have incentive to use systems which tend to bump up the apparent net volume of timber converted.

This post has been but an overview of the fairly convoluted world of log scaling. Not wishing to write a book on the topic myself, certain details have been omitted or simplified in the interest of brevity. There is plenty of reading to be found online in regards to log scaling, and I recommend interested readers have a look around.

Next post in this series will be a look at board foot measures for lumber and we’ll find that the picture there is no tidier than the ones for cubic volume measure. Stay tuned, and thanks for visiting. Part III follows.