U.S .. FOREST SERVICE Research Paper PNW- 7 ·
Procedure for· developing a
·SITE INDEX ESTIMATING SYSTEM
from Ste�11 Analysis Data
by
FLOYD A. JOH NSON
and
NORMAN
P.
WORTHINGTON
December 1963 .
\
PACIFIC NORTHWEST
FOREST AND RANGE EXP RIMEN'r STATION.
Philip A. Briegleb, Db;·ector
Portland, Oregon
U.S. DEPARTMENT OF AGRICULTURE
FOREST SERVICE
I N T RODU C T IO N
Site index curves for red alder.!J were developed from stem analysis
data by a procedure which has since been adopted for lodgepole pine in eastern
Oregon and for aspen in Alaska. In these. three cases, the results have been so
encouraging that a detailed description of the procedure seems justified. The
red alder data has been used to illustrate the proced re in the description which
follows.
TH E DA TA
Forty-three red alder trees from 16 locations scattered throughout the
red alder stands of western washington were selected, felled, and cut into'
sections during 195 7 for the site index phase of the red alder yield-table project,
Height abov,e.ground and tree age corresponding to this height were deter­
mined for: each point of cutting on every tree, Measurements recorded for tree
number 26 have be en listed in table 1 to illustrate the n;3.ture of the basic data,
Table 1.--Basic data_ for tree No. 26
Tree age
Tree height
Tree age
2
3
5
8
12
15
18
22
8
16
24
32
40
48.
53
58
25
29
34
37
41
52·
58
68
Tree height
63
68
73
78
83
88
93
96
\
Tree ages at the time of felling varied: 13 trees we.re less than 50 years,
3 trees were over 8 0 years, and the others were somewhere between 50 and 8 0
years,
Each of the 43 trees was dominant at time of cutting ·and was considered
to have been dominant throughout its entire life. Evidence to the. contrary. would
h<we been cause for rejection,
}:_/ Bishop, .Daniel M. , Johnson, Floyd A., and Staebler, George R.
Site curv.es for red alder. U.S. Forest Serv. Pac. NW. Forest & Range Expt.
Sta. Res, Note .1.62, .7 pp , illus.. 1958
•.
.•
1
1
TH E P' RO C EDU R E
Heights of the various cuts were plotted over corresponding ages for
ch of the 43 trees, and adjacent points were connected by straight lines.
.
gure 1 illustrates this step for tree number 26.
TREE HEIGHT
( feet )
100
90
80
'70
60
50
40
30
20
10
0
0
10
.20
30
40
50
60
70
TREE AGE ( years)·
Figure I. -Height over age for tree No. 26
\
Heights from the segmented curves constructed in step 1 ere listed for
ch of the following ages: 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, and 70,
tese heights are shown in table 2 for age 30. All 43 trees were at least 30
ars old when cut. The uppermost segment of the curve for each of the 13
es which were less than 50 years old when cut was extended to inter ept
e 50, Thus, the table for age 50 corresponding to table 2 also contained 43
es. These 13 extrapolations covered only a few years at the most and were
nsidered entirely acceptable. Tables for ages 60 and 70 had fewer than 43
es because extrapolations to ages 60 and 10 were not acceptable for all trees,·
may be
l example of whq.t was considered to be an acceptable extrapolatio
en in fig re 1 as the broken line extension of the last segment to age 70.
-2-
Table 2. -Tree heights at age 30 for 43 trees
Tree
number
Height
at
age 30
Tree
number
Feet
Height
at
age 30
Tree
number
12
13
14
15
16
17
18
19
20
21
22
Height
at
age 30
Tree
number
F eet
Feet
Feet
--
--
81.5
78.5
80.0
68.5
69.5
71.0
67.0
51 0
63.5
59.0
48.5
1
2
3
4
5
6
7
8
9
10
.11
Height
at
age 30
23
24
25
26
27
28
29
30
31
32
33
53 5
50.0
58.0
59.5
60.0
61.0
79.0
88.0
82.0
86.0
89.0
34
35
36
37
38
39
40
41
42
43
94.5
80.0
72.0
69.0
66,0
68.0
63.0
57.5
57.0
47.0
53.0
82.5
80.0
83.0
80.0
74.5 .
73 0
81.5
86.0
70.0
73,0
Average height at age 50 as calculated from all trees at a location was
.rbitrarily. called the site index of that location, These site indexes are listed
n table 3 for each of the 43 trees.
For each of 10 ages (10, 15, 20, 25, 30, 3 5!
:0, 45, 60, and 70), site indexes were.then plotted over corresponding heights.
his step is illust rated in figure 2 for age 3 0, Note th a:t site indexes in table 3
vere p lotted over heights in table 2 to give the scatter diagram in figure 2,
.
I
SITE INDEX
12Kl
0
\
100
80
60
. 40
20
(j
0
10
20.
30
40
so
60
70
80
. 90
100
HEIGHT at AGE 30 (feet)
Figure .2. -Scdtter dio.groni. Site index over height at age 30.
":'3 -
Table 3.--Site indexes for 43 trees
Tree
umber
1
2
3
4
5
6
7
8
9
10
11
Site
index
Tree
number
Site
index
12
13
14
15
16
17
18
19
20
21
22
67.0
67. 0
80. 0
78.0
78. 0
78.0
103.5
103. 5
103.5
103. 5
103.5
100.0
100.0
100.0
88.5
88. 5
88.5
88.5
78. 0
78.0
78.0
- 67. 0
Tree
number
23
24
25
26
27
28
29
30
. -31
32
33
-
Site
index
Tree
number
Site
index
111.0
111.0
100.0
87.5
87. 5
87. 5
67. 5
67. 5
67.5
62. 0
62.0
34
35
36
37
38
39
40
41
42
43
103. 5
103. 5
103,5
91.5
91. 5
91.5
94. 5
98. 0
98. 0
98. 0
A linear least squares line was calculated for the relationship of site
ex to height with a separate ine for each -of the 10 ages.
The line for age
is 'shown in figure z
There was no indication from these plottings that
rthing other than a straight line would have been appropriate, Each line
•.
made subject to the condition that site index was zero when height was
·o. Subsequent steps in the procedure proved to be intractable unless this
3
Ldition was imposed,
In
addition, the· condition was not hard to accept. Foi\
:h of the 10 scatter· diagrams, the calculated line was as obviously accept­
.
e as- it was in figure 2.
The .mathematical expression for each line was:
S
=
bH ---------- - ----------- --- :.. --
where S
=
site index
H
=
height at a giyen a ge
b
=
.
.
.
regress1on coe ff'1c1ent
-4­
=
-·
z;SH
-2
Z:H
( 1)
j
15
The 10 regression coefficients from the 10 analyses (table 4) were
ted over age, and a smooth progression of b values resulted (fig. 3),
Table 4.--Regression coefficients for the relationship of
site index on height
Untransformed
Age (A)
10
15
20
25
30
35
40
45
60
70
Transformed
Regression
coefficient (b)
·.
x
1/A-1/50
=
2.59
L90
1.54
1.36
1. 26
1.16
1.10
1.04
.95
, 90
0,080
.047
,030
.020
,013
,009
,005
,002
-,003
-.006
y
b-1
1.59
,90
.54
.36
.26
.16
,1(.)
,04
-,05
-.10
6
Data in the first two columns of table 4 were transformed as follows to
:l
I
the condition b= L 00 when age = 50 (obviously, b must be 1. 0 0 when age
because site index is equal by definition to height at age 50):
y =
b- 1
X =
1 IA
-
1 I 50
where A= age
transformed values in the last two columns of table 4 were then used to
c for this expression:
y = ex
..
whe-re
-2c = L:yx
L:x
-5-
\
Thus:
y
=
19. 452x
and after reconverting y and x:
b
=
0 . 611.
+
19. 452/A----------- --------- (2)
This curve is shown in figure 3.
REGRESSION COEFFICIENT (b) 2.60
2.40
2.20
2.00
1.80
..
1.60
1.40
\
\
\
\(
"
'
·-
1.20
..
'
1.00
0.80
0
I
I
I
30
20
. 10
I
40
I.
50
I
60
. 70
TREE AGE (years)
Figu e 3. -Regression coefficients for the relationship of site index tci height;plotted over age.
5tep 7
An equation which generates· site index. for any combination of age
d
b.eight resulted when equation 2 was substi_ tuted for b in equation 1.
Thus:
·s =
(0. 611
+
19. 452/A)H---------- -- ----- (3)
D
. AHMS 1
M 0 D IF I CA T I 0 N
An evolution of the above proced/ re is already underway.
In developing
took the height of only the tallest of six
site curv es for lodgepole pine; Dahms
_
trees on a plot at each of several stand ages. Although this left him with only
::me-sixth the amount -of data that would have· been avail<:�-ble under the procedure
iescribed in this report, there is a good chance, at least in his case, that he
;�.vo.ided a serious bias. Except for this one modification, Dahms used the pro­
cedure described above.
\
l
2/
Dp..hms, Walter G. Correction for a possible bias in developing site
lndex curves from sectioned tree data. Jour, Forestry 61: 25-27, illus, 1-963 .
-
6
-
I
·I
D IS CUSS IO N
Equation 3 is not the same as the comparable equation published in 11Site
.rves for Red Alder11]_/ because a few minor errors were discovered in the
lginal work. Since both equations give almost ident1cal results, these errors
re of no practical importance,
When equation 3 is used to develop curves of height over age for each of
veral site indexes, one curve will be proportional to another in the sense that
l ratio of heights for a given age from the two curves will be constant over all
as. This proportionality is caused by the arbitrary restriction on the curves
site index over height (see step 4) which forces them through the zero-zero
ndinates, The only justification for the proportionality is the empirical
ldence (see fig. 2) which suggests that the restriction is reasonable.
Thus, .
S ite index for an individual tree. should stay constant over all·-ages
equation 3 were to be applied to the basic data from a single tree, there should
no evidence of a trend when estimated site index is plotted over age. This was
1e for each of the 43 trees and the results for tree number 26 (see table 1) are
)Wll in fi ure 4.
·
ESTIMATED
SITE INDEX
120
i 10
· -
0
0
100
90
0
=-- -
0
0
-e-o.
0
0
80
0
-
.
.
- -rr-C-
-
0
70. r-
I
60
0
10
I
·20
I
30
I
I
I
40
50
60
\
70
TREE AGE (years)
·
Figure 4. -Estimated
.
ˆ
ite index plotted over age for tree No. 26. Estimated site index was found from these plottings to be very erratic
mage was le.ss than 10 years. Above 10 years, 17. tr·ees gave no evidence
icating a trend of estimated site index on age (tree 26 was one of these 17).
iositive trend was apparent on 15 trees, and a negative trend was apparent on·
remaining 11 trees. Perhaps this can be taken to mean that equation 3 is
:eptable when age is 10 or greater.
]_/ See footnote 1.
... ?-
Differences between estimated and actual site iridex, as ·calculated· from
all measurements on all trees, were plotted over age in figure 5, Here, a nega­
tive bias is evident below 10 years, and a lack of bias is evident when age is 10
or greater, Standard deviations of these differences were found to be 16 site
. index units for age class 1·0 to 20 years, . 8 site index units for age class 20 to
30 years, 5 site index units for age class 30 to 40 years, and 4 site index units
for age class 40 to 50 years.
+60
:t50
+40
+30
+20
+10
0
-10
-20
-30
- 40
.. so
-60
-70
-80
-9 0
10 .
·20.
30
40
60
50
.TREE AGE
' i
fig\lre· 5. -Differences bet)Vee · estimated. and actual s te index plottc:d
70
(years )
ver 'tree age (.43 'trees ) ,
.
The complete set of red alder data has ·been made available in table 5
for the benefit of anyone who might wish to experiment further with procedures
for developing a site· index estimating system,
·
-8-
·
..
Table s ..
oca
Tree
ion
No.
1
1
10
20
30
40
50
60.
70
75
80
85
90
95
A
6
8
9
11
16
17
19
26
29
32
38
43
50
H
2
10
?0
30
40
50
60
70
75
80
85
90
95
100
A
1
4
5
8
10
13
16
22
27
31
36
40
46
50
H
10
20
30
40
50 . 60
70
75
80
85
90
95
100
102
A
9
12
15
18
21
23
24
27
30
36
41
48
51
H
2
4
10
20
40
50
60
65
70
75
so
85
90
A
2
5
6
7
30
3i
11
15
20
23
25
32
34
37
47
50
58
H
10
20
30
40
50
55
60.
64
68
.74
78
so
84
88
94
A
5
6
8
15
17
22
24
25
29
41
46
52
58
H
4
10
2
30
40
50
65
70
75
35
37
2
33
8J)
85
90
..100
105
108
A
2
22
29
34
.3 8
43
.
60 .
H
4
3
70,,,: .75
..8_ o
. 85
A
3
37
51
·n
2
34
A
H
· · · ---·
A,
2
10
- -
3
4
2
5
6
7
3
8
9.
. ·<·-
..
I
• A
11 . - il
12
!4
6
10
12
13
20 '.
30
40
50
55
60
6. .
14
'·15"
18
24
29
31
·
18
6. . n
24
30
35
40
45
50
55
60
65 : 68
1
4
7
11
13
i7
18
20
24
29
33
37
40
3
8
.. 16
24.
32
40
45
50
. 55
60
65
70
74
.. !.'!.
16
18
22
26
32
38
. 42
10
6·
i
3
3-
15
16
8 · --1:2:
8
.'i6
T. .. X
....:s. .! ,.
9
..
22.
..
11..
:32 . 40·
•·
--'r•:
·n.-· .14.
. 4.
32
17
..
40
A
2
4
"8' "10., 16
2
8
16
'24'
32
1
4
8
13
17
•M
1·
4
8
16
24
32
A
1
5
8
14
17
.·H
2
40
R
2
A
1
H
2
A
7
. . ·-·
3 .. ·.6
2
45
50
55
6!J
65
J9
. ..
45
22
27
31.
36
5o
.55
33
6·o
..
...
.
72
79
83
_95
100.
105
62.
64.
76
56
:
70
7l
39 ·
40
4
·-
.
.-
-
.
..
64
43
46
60
65
65
27
31
38
41
42
40 . 45
50
55
60
65
24
28
38·
52
56
60
43
47
· 48
30
64
68
72
76
80
29
1
35
38
40
42
48
27
31
45
50
21
23.
20
44
98
90·
55
21
40
· - --·-
.
·
.•
96
-- ··-
8
24
'32
3
6
. ti··
"8
16
24'
32
36
40
48
52
56
64
68
78
2
li
..,
10
11
16
'21
24
27
a
24
36
40
42
48
· 52
60
64
68
71
79'
3
4
32
37
46
16
34
· 6
11
13
16
18
21
24
26
30
46
2h
32
40
44
48
52
56
60
64
36
39
16
i 34
A · "1"
6
100
5.
H
A
13·
5
Heights and ages for 43 red alder trees
H
2
4
Basic red alder stem analysis data
·
..
2f
2:i
.17 . . 19
56
17
H
A
1
2
3
5
9
11
13
15.
18 . 22
27
36
39
42
44
18
H
3
8
16
24
32
40
44
48
52
60
64
68
72
76
82
88
92
96
100
104
107
10
11
15
16
17
19
1
24
25
32
37
38
51
58
62
52.
56
64
70
80
84
88
92
96
100
104
108
43
100
104
108
112
116
41
44
54
61
67
2
g ..
•
68
73
79
A
2
3
6
7
19'
H
3
12
40
A
2
3
28
8
11
13
14
16
17
19
25
29
30
34
35
38
5q
55
20
H
2
16
24
48
56
64
68
72
76
88
92
96
A
6
10
12
14
17
19
21
22
24
27
·so
84
4
32
40
1
:a
34
36
40 .
H
2
12
24
32
40
48
52
56
60
68
76
80
84
32
88
92
A
1
3
5
7
9
12
15
18
20
22
26
27
H
12
24
32
40
48
52
56
66
72
80
84
88
33
36
2
2
90
A
1
2
3
5
6
10
11
14
17
20
26
27
j4
21
. 22
8
_58
-9-
28
29
96
100
104
. 41
45
53
96
100
104
108
·112
40
46
47
55
65
108
.
58
Table
Loeation
Tree
24
9
25
10
26
27
28
-.-
11
29
.
. 30
. 31
. ..
32
:12
..
13
I
4
12
24
32
4o
48
54
60
66
74
. 80
84
88
92
96
100
104 .
108
112
A
2
3
4
6
7
8
10
1
14
15
17
20
23
26
33
34
40
43
46
52
H
3
8
20
32
48
52
56
60
64
68
72
76
80
84•
88
92
96
100
102
108
112
116
A
2
5
6
8
13
15
17
18
20
21
22
25
30
32
35
40
42
44
46
51
62
68
H
3
8
16
24
32
40
48
60
64
68
72
76
0
84
' 88
92
100
104
108
·20
54_
23
25
28
30
31
32
37
42
46
50
55
60
83
88
93
96
41
52
58
68
98
A
2
7
9
10
18
19
H
8
16
24
32
40
48
53
58
63
68
73
78
A
2
3
5
8
12
15
18
22
25
29
34
37
22
H
2
24
32' . 40
48
53
63
68
73
78
83
88
93
A
1
8
13
16
20
23
27
34
38
43
47
53
65
73.
H
2
8
16
24..
32
40
48
53
58
63
68
73
78
83
88
93.
98
103
107
A
1
2
7
11
12
15
. 18
20
2'1'
24
30
. 34
37
.42
46
64'
69
79
H
2·
8
55
. 16
24
32
40
44
56
60
64
68
72
57
..
·_·
48. . 52
A
1
3
4
5
6
9
13
15
17
20
25
31
41
H
2·
8
16
24
32
44
48
.52
56
60
64
68
. A'
4u-
1
3
5
7
11
12
15
16
18
25
38
42'
·56
H
2
8
-16
24-
32
40
44
"48
52
56
60
64
69
A
1
4
5
10
12
18
20
. 23
26
28
36
42
62
H
1
8
16-
24
32
36
40
52
56
60
69
A ..
1
33
H
1
A
34:
43 red alder trees
H
...
35
14
Heights and ages for
No,
23
8
5. uBasic red alder stem analysis data--Continued
L
44
48
...
·'
9
15
18
20
23
33
36
41
49
59
8
16
24
32
36
40
44
48
52
56
60
63
1
5
7
12
15
16
19
23
25
29
32
37
47
-
R
1
8
17
25
33
41
49
57
65
69 .
75
80
85
88
93.
1
2
3
5
7
10
11
14
. 17
18
23
27
33
36
42
46
H
1
8
17
25
33
41
49
57
65
69
. 75
80
85
89
93
100
97
101
105
106
51
57
60
104
108
1
2
3
4
7
11
13
15
17
19
25
30
33
36
42
46
49
60
H
1
9
17
25
33
41
49
57
65
70
75
80
85
90
93
io6
1.05
112
'A
1
5
6
8
9
'11
14
16
19
21
23
27
32
35
42
45
49
60
37
-H
2
8
16
24
32
40
48
56
64
72
76
80
85
90
94
A
1
3
4
5
6
9
12
15
18
23
25
30
36
42
.
52
H
2
8
16
24
32
40
48
56
60
64
68
72
76
80
84
88
92
A
1
3
11
16
19
20
23
25
28
31
35
39
43
52
39
15
40'
16
41
42
43
-
7.
8.
H
2
8
16
24
32
40
48
56
64
68
72
76
80
84
88
94
A
1
2
5
9
11
13
17
21
24
27
29
33
40
41
46
54
H
2
8
1.6
32
40
48
64
72
76
80
84
88
92
96
102
A
1
2
3
8
10.
13
19
22
26
28
33
37
43
54
63
H
1
9
17
24
37
49
58
66
74
78
82
86
90
·.94
98
A
1
2
5
7
8
13
14
17
21
23
26
30
32
36
42
H
1
9
21
3;l
43
51
59
67
71
75
79
84
.
A
1
5
6
7
10
16
18
27
31
33
38
43
H
1
9
17
25
33
49
57
62
66
70 . 74
82
86
90
92
A
1
3
6
8
11
14
19
20
21
24
27
35
37
41
43
-10-
. ,.
. "
- ..
-
...
A
4 '
'·-
....
A
38
_·. .·
..
36.
-
....
·-:
·-· ·
7
___
116
·-
..
, •
'