The alinement chart method of preparing tree volume tables

UNIVERSITY OF CALIFORNIA PUBLICATIONS

IN

AGRICULTURAL SCIENCES

Vol. 4, No. 9, pp. 233-245, 3 figures in text December 20, 1921

THE ALINEMENT CHART METHOD OF
PREPARING TREE VOLUME TABLES1

BY

DONALD BRUCE

The chief use of the alinement chart2 is to express with the simplest
possible system of lines a law the equation of which is known. The
underlying principle is so flexible that almost any formula can be
expressed thereby, although the most striking advantages over a system
of rectangular coordinates do not appear unless three or more variables
are involved. The axes may be parallel or converging, straight or
curved, and graduated either uniformly or with intervals which vary
in accordance with some given law, the form of the graph depending
on the form of the equation. It follows from this very flexibility that
such charts are, in general, unsuited for use with empirical data. The
following pages, however, describe an exception to this general rule in
which one type of alinement chart may be advantageously used in the
preparation of tree volume tables, although the form of the equation
of such a table is yet unknown.

The most suitable type of chart can be determined by working
out an approximate algebraic expression for the volume of a tree in
terms of its diameter and height. This expression is complicated by
the fact that, for almost all American tables, volumes must be com-
puted in board feet as scaled by some log rule, instead of in cubic feet.
The starting point must therefore be the equation of the volume of a

1 Acknowledgment is made to Professor Frank Irwin, of the Department of
Mathematics of this University, for assistance in connection with the analytic
features of this study.

2 A complete discussion of the theory of alinement charts may be found in
such works as: J. Lipka, Graphical and Mechanical Computation; J. B. Peddle,
The Construction of Graphic Charts; and M. d'Ocagne, Traite de Nomographic.
For a discussion of the application of certain simple types to some formulae of
forest mensuration see D. Bruce, "Alinement Charts in Forest Mensuration,"
Journal of Forestry, XVII, 7, 773 (November, 1919).

\

234 University of California Publications in Agricultural Sciences [Vol. 4

log in board feet first formulated by Professor Daniels,3 i.e., v = ad'2
+ bd + c (I). For simplicity, let us apply this formula to a tree of
uniform taper up to its merchantable top limit, that is, one which is a
frustum of a cone.

Let V = volume of a tree in feet b. m.

Let v = volume of a log in feet b. m.

Let D = d. i. b. stump (assumed equivalent to d. b. h.).

Let d = top diameter of a log.

Let H = height in logs of tree.

Let t = top d. i. b. of tree.

It is evident that the taper of the tree = D — t, and that the taper

per log = — z- — Therefore the top diameters of the several logs

. D—t

of the trees are the terms of the following series : t, t + — fj~}

t _j_ 2(^~0 f t + 3(D—t)} to jj terms, and the volumes of the

H H

same are the terms of the following series, each top diameter being

successively substituted in I :
at2+bt+c;

at' +j± (D - t) + jp(D - ty + bt + | (D - t) + c;
at' + *£(D-t)+ J (D -ty + U + 2^(D-t)+ c;
at2 + to* {D _ t) + g {D _ ty + bt + g {D _ t) + c;

at2 + 8J {D _ t) + go (Z) _0, + w + «(jD_0 + c.

. . . . to H terms.

V = the sum of this series to H terms. This may be obtained by the
differential method in which a new series of first differences is derived
by subtracting each term from that which follows it, and this process
is repeated, successively obtaining a series of second differences, third
differences, etc., until all terms become zero and the series vanishes.

-Sec A. L. Daniels, Measurement of Sawlogs, Vermont A.gr. Exp. Sta., Bulletin
L02, L903.

1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 235

m, ,, , . 11 (n — 1) , . (n — 1) (n — 2) , ,
Ine sum then equals na -\ ^r dx -\-n ~ - a\ +

LA LA

.... where n = number of terms, a = the first term of the original
series, d1 = the first term of series of first differences, d2 = the first
term of series of second differences, etc.

Series of first differences is :

^ . (D - t) + jp (D - ty +A (Z) - 0;

2f(D-t)+^(D-ty+±(D-t);
|-< (D - t) +g(Z> - ty +jj (D - t);

~ (D - I) + J (Z) - ty + ~ (D - t); etc.

Series of second differences is :

2a (D - t)\ 2a (D - t)\ 2a (D - t)2
H2 ; H2 ; H2

Series of the third differences is :
o; o; o; etc.
And V = sum of this series = H (at2 + bt + c) + — — t>

^(D - t) +%(D - ty- + ±(d - t)}+H (H ~ ^H -V

\%(D-ty.

; etc.

Expanding and rearranging in terms of D, this becomes :

. f (2afi + Zbt + &c)H . {Zaf- + 3bt) at2 .
+ 6 + 6~~ " + 6/7 '

\

236 University of California Publications in Agricultural Sciences [Vol. 4

Rearranging in terms of H, this may also be written :

V=^{ 2aD2 + (2at + 35) D + 2a/2 + Sbt + 6c } — Y2 j aD2 +

ID

— t(at+b) \ H (D2 — 2tD + t2) (III)

j 6#

Let us now apply this general formula to a specific case, for example,
that of trees scaled by the Scribner log rule to a six-inch top cutting
limit. A close approximation formula for this log rule (for 16-foot
lengths) is:

V = .765d2 — .55d — 21.

We therefore may assume

a =.765
b=— .55

c = — 21

Substituting these values in III, we have :

V = H (.255Z>2 + 1.2550 — 13.47) — (.3825D2 — .275/) —

1
12.12) + — (.1275Z)2 — 1.532) + 4.59) . (IV)

H

Typical equations for the height class curves of a volume table in
graphic form can now be found by substituting in IV given values
of H; for example :

For H = 2, F = .19125Z)2 + 2.02Z) — 12.52 (V)

H=6, V = 1.16875/)2-}-7.55/) — 67.93 (VI)

H = 10, V= 2.18025/)2 + 12.672/) — 122.121 (VII)

Similarly, typical diameter class curves are :

9 04
For D = 10, V = 24.58/7 - 23.38 + n^ (VIII)

ti

D = 20, V = 113.63// - 135.4 + ^° (IX)

73 4-4-
D = 30, V = 253.68// - 323.88 + -^P (X)

94 fi S4
/) = 50, 7 = 686.78// - 930.38 + =^£^ (XI)

ti

1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 237

It will readibly be seen that V, VI, and VII are equations of parabolas,
while VIII and IX and X and XI are hyperbolas. These deductions
agree so well with the actual results obtained in volume tables con-
structed by the conventional method on a similar basis that it seems
probable that the general form of the equation should apply at least
approximately to actual trees as well as to the cone frusta on which
it is based. Furthermore, it has been tentatively established, and with-
out any conflicting evidence coming to light, that, in the case at least
where a fixed top cutting limit is used, frustum form factors are func-
tions of diameter and not of height. If this is true, such equations as
VIII can be corrected to apply accurately to any given species by
multiplying into them the proper form factors, which would merely
change the values of the constants without affecting the form. Finally,
the ease with which the alinement chart devised to apply to cone
frusta works out for actual trees is the best proof of the adequacy of
the equation.

Next, it is necessary to determine this alinement form. Unfortu-
nately a difficulty at once presents itself. The equation appears to be
one of those rare instances which cannot be thus expressed.4 It has
been found by experiment, however, that if two parallel axes be as-
signed to V and H, the former graduated uniformly upward and the
latter uniformly downward, all lines expressing a single value of D
(taken from a table of values of volumes of cone frusta in board feet
or calculated by the above formulae) will intersect nearly (but not
quite) in a common point, and that these common points for a series
of values of D lie almost (but not quite) in a straight line, which if
produced will pass through the zero point on the V axis. Figure 1
illustrates this fact, although a larger scale is needed to bring out the
failure of the lines to intersect perfectly.

The reason for this becomes evident upon analysis. Let the lower
left-hand corner of figure 1 serve also as the origin of a system of
rectangular coordinates with one unit equaling ten of the small squares.
Also let b equal the width of the paper. Then any straight line
used in solving values by the alinement chart can be expressed

4 Only those equations can be expressed by an alinement chart that can be
put in the determinant form

A Or h

f2 g2 Ju = 0
U 9z \

where fi, gi, and h-t are functions of x, (i = 1, 2, 3).

238 University of California Publications in Agricultural Sciences [Vol. 4

f v)

as connecting the points (0, 22 — 2H) and-< b, —^ for by the equation

100

I V 1 X

Y= \~q - 22 + 2H y + 22 - 2#.

r

(XI)

10

.fcf

'S3

Figure 1

Alinement Form giving approximately correct results for volumes
of cone frusta in feet b.m.

O c3
>

Now from equation IV, V = AII -\- B -\ (where A, B, C are func-

II
lions of D) and the equations of two such lines corresponding to any
values oi* //, such as //, and II 2, and having a common value of D, will
then be (from equation XI) :

1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 239

Y =

AH1 + B +

J7

and

100

AH2 + B +

(I

100

- 22 + 2H2

X

- 22 + 2#! I 6

+ 22 - 2#i

X

+ 22 - 2#2

The point of intersection of these two lines can now be found by
solving these two equations simultaneously, and when this is done the
following values of X and Y are obtained :

X

2006

A -

C

Hi Hi

+ 200

C

- =^=. !> + 25 + ^ + ^
ill £li tli tl<i

Y =

A -

C

Hi Hi

+ 200

(XIII)

(XIV)

It will be seen that, for such a range of values for D and H as are
actually encountered, these two equations approximate quite closely to

X =

2006

A + 200

22A + 2ff

A+ 200

(XV)

(XVI)

Since both X and Y in this last pair of equations are independent of
H, this approximation explains the approximate common intersection
of all lines having a given value of D. They can, moreover, be com-
bined to give an equation of the curve upon which all these common
intersections fall, but the result is in a form too complicated to be of
much value, although it can be readily identified as an equation of a
conic section (obviously a very straight portion of a hyperbola). The
curve can be plotted more simply by means of equations XV and XVI,
and will be found to be very nearly a straight line passing through
the zero point on the V axis.

240 University of California Publications in Agricultural Sciences [Vol. 4

It seems as if a regraduation of the H axis might be made to result
in perfect instead of approximate intersections. It will be found that
this can be readily accomplished for any given value of D on the as-
sumption that the H graduating distance from a fixed point = KH

M

-\- L -| where K, L, and M are constants. But unfortunately these

H
three constants prove to be themselves functions of D. In other words,

different sets of graduations would have to be used for each value of
the diameter, which is obviously impracticable.

Each value of H, however, is in practice associated with a rather
narrow range of D values. It is therefore possible to modify the
positions of the H graduations empirically so as to result in a decided
improvement in the intersections, and at the same time a slight re-
adjustment of the D axis can be made with advantage. The best
results appear to be obtained by the following plan :

1. Graduate the V axis as already described.

2. Graduate the H axis as already described but omit all values
under that of 5 logs.

3. Select a few definite values of D well distributed over the de-
sired range, and for each calculate the volumes for three or more values
of H (H = 5 or over). If a table of cone frusta is available, these
volumes may, of course, be taken therefrom.

4. Draw straight lines from each value of H to the corresponding
value of V, and select points which appear to be the averages of the
intersections of lines relating to each common value of D.

5. Draw a curve through the points thus selected. For most work
a sufficiently close approximation will be found to be a straight line
passing through the V origin.

6. Enter on this curve or straight line the D graduations indicated
by the straight lines of step 4.

7. Obtain two or three values for V corresponding to the lower
values of H (under 5 logs) and to the values of D already graduated.
By drawing the appropriate straight lines their intersections with the
II axis will indicate the best average positions of the smaller H gradu-
ations. These should then be entered on that axis.

8. Complete the graduation of the D axis by intersection, using
for each value of D appropriate values of .IT.

The following table, worked out by the above process, may also be
used directly to save time and labor.

1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 241

TABLE I

Graduation of H Axis

logs
with th<

Distance from fixed point on axis

A

Height in
tersection

r

Values to be used

where the heights

are in logs and

tenths of logs

3

Values to be used

where the heights

are in feet

diagonal

axis

1.26

20.16

2

•2.08

33.28

3

3.03

48.48

4

4.01

64.16

5

5

80

6

6

96

7

7

112

8

8

128

9

9

144

10

10

160

11

11

176

In the application of this theory to the making of a volume table
the successive steps may be as follows :

1. Prepare an incomplete alinement graph (fig. 2) such as has
been illustrated in Figure 1, but with the H axis graduated in accord-
ance with the table just given.

2. Draw a diagonal straight line representing the D axis between
the point representing its intersection with the H axis and the zero
point on the V axis.

3. Graduate the D axis as follows: Each tree measurement (a
sufficient number of which are supposed to be at hand) is used to draw
a straight line between the point on the H axis corresponding to the
height of the tree and the point on the V axis corresponding to the
volume of the tree. The intersection of this with the D axis is an
indication of the position of the D graduation corresponding to its
diameter. The indications of a number of trees will naturally be
more or less conflicting and the results must therefore be evened off
by a graduating curve such as indicated in figure 3. In this curve
the distance of each intersection above the base of the graph of figure
2 is plotted over its corresponding diameter. When all the points
are thus plotted a smooth curve is drawn through them, and from this
curve the D axis is finally graduated.

242

University of California Publications in Agricultural Sciences [Vol. 4

4 £

10

'53

7 -—

Figure 2
An alinement volume table.

O °3

Table II gives the basic tree data used in this illustrative instance.
The broken line in figure 2 shows how the first values of this table
are plotted. The left-hand point on figure 3 is plotted from the re-
sulting intersection with the D axis.

It is immaterial whether each tree is thus used to determine a point
on the graduating curve or whether the average volume, height, and
diameter of the height-diameter classes are used. In the latter case,
each point should, of course, be weighted in accordance with the num-
ber of trees which are thus averaged together. Figure 3 was drawn
in accordance with the latter method.

4. The volume table can now be read from the completed alinement
chart, the result being given in Table III.

1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 243

TABLE II

Basic Tree Data, White Fir, Stanislaus National Forest

(Measurements taken by U. S. Forest Service)

o. of trees
in class

Average
D. B. H. inches

Average

merchantable

height in 16-ft.

logs

Average
Vol. ft. B. M.

7

18.1

4.0

280

3

19.5

3.9

330

8

19.9

4.0

340

5

19.8

5.0

500

6

21.9

4.4

540

7

22.2

5.1

640

6

21.8

6.0

670

1

22.8

6.5

810

2

23.9

4.1

510

1

23.4

4.2

490

9

23.7

5.2

640

6

24.1

6.2

970

2

24.6

7.3

1230

4

25.5

5.0

660

9

25.9

5.6

950

12

26.3

6.0

970

9

26.2

7.0

1270

1

25.8

7.6

1510

4

27.7

5.6

1050

10

28.3

6.3

1250

7

28.1

7.5

1450

1

27.6

8.3

1760

1

29.2

4.6

1090

7

29.9

6.0

1300

7

29.8

6.4

1280

12

29.7

7.3

1640

3

30.2

7.9

1720

16 31.6 6.6 1600

TABLE III

Volume Table Eead From Figure 2

Height in 16-ft. logs

. B. H.

r
4

5

6

7

8

18

280

380

20

350

480

22

430

590

740

900

24

510

690

880

1060

26

790

1000

1220

1430

28

850

1130

1370

1600

30

1010

1280

1550

1820

32

1140

1450

1750

2060

244

University of California Publications in Agricultural Sciences [Vol. 4

In most cases it will be found that the graduating curve of figure
3 can be entered on the same sheet as the alinement chart without
confusion and with a saving in time and convenience. In step 3,
moreover, it is not necessary actually to draw the various straight lines,
which are apt to become confusing if many tree measurements are
available. Instead, a straight edge may be laid across the proper values
and its intersections' with the intermediate axis noted.

1300
1200

1100

1000

900

"1 800
I 700
5 600

■■■ " :t'-il •■■:!"' Ij !■--:' •;! i--|:"l I ■■■-,■ ■ I ■; "I ::: i J V-

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
D.B.H. inches

Figure 3

The graduating curve.

There are several advantages in this method of preparing a volume
table. In the first place, the curve drawing is simplified. In place
of the system of curves which have to be harmonized in the usual plan
only a single graduating curve need be drawn, and since this is based
on all of the tree measurements available it is much better denned
and more easily and accurately located. As compared with the frus-
tum form factor method, which also uses a single curve, the use of the
alinement method saves considerable time, since the calculation of the
form factors is avoided; the result is, however, practically identical,
for if the frustum form factors of such a table as Table III be calcu-
lated they will be found to vary with diameter but not materially with
height.

Secondly, exterpolations are easily (perhaps almost too easily)
made, especially in height, and with far more certainty than is possible
by the normal system of curve extension.

Lastly, the resulting alinement chart can be read with great accu-
racy for fractions of inches in diameter and fractions of logs in height.

1921] Bruce: Alinement Chart Method of Preparing Tree Volume Tables 245

This is not true of the conventional method, where graphic interpo-
lations between the harmonized curves are both slow and inaccurate
and where arithmetical interpolations in the final table are exceed-
ingly laborious. For certain problems of forest mensuration this
advantage is highly important, although it is of little weight in
connection with ordinary timber cruising. The method appears su-
perior in accuracy to the ordinary plan, especially where the amount
of data available is limited. Volume tables have been made by both
methods from tree data for three species, including the species used in
illustrating this paper. The results appear as follows:

Aggregate difference between Average deviation between

all trees as actually scaled individual tree volumes as

and as read by table scaled and as read by table

. A . . A .

/■ ^ /■ ~s

Basic data Conventional Alinement Conventional Alinement

145 trees, western larch 1.5% 0.2% 5.8% 5.1%

166 trees, western white pine 2.1% 0.3% 3.8% 3.9%

166 trees, white fir 0.5% 0.2% 7.1% 6.3%

It will be observed that the result by the alinement method is much
superior as a whole for each species, and is better, on the average, in
detail as well.