Equations and computer subroutines for estimating site quality of eight Rocky Mountain species

Historic, archived document

Do not assume content reflects current
scientific knowledge, policies, or practices.

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RESEARCH PAPER INT-75
1970

EQUATIONS AND COMPUTER
SUBROUTINES FOR ESTIMATING
SITE QUALITY OF EIGHT

ROCKY MOUNTAIN SPECIES

James E. Brickell >
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FOREST SERVicp

Sevens) INTERMOUNTAIN FOREST AND RANGE EXPERIMENT STATION
Ogden, Utah 84401

THE AUTHOR

JAMES E. BRICKELL is an Associate Mensurationist in the
Station's forest survey research work unit located in
Moscow at the Forestry Sciences Laboratory. He re-
ceived his bachelor's degree in forest management from
Washington State University in 1961 and joined the Sta-
tion's staff in 1964 after taking graduate studies at the
University of Washington. He gratefully acknowledges
the assistance and advice given him by Albert R. Stage,
leader of the Station's timber measurement work unit at
the Laboratory in Moscow.

USDA Forest Service
Research Paper INT -75
1970

EQUATIONS AND COMPUTER
SUBROUTINES FOR ESTIMATING
SITE QUALITY OF EIGHT

ROCKY NOUNTAIN SPECIES

James E. Brickell

INTERMOUNTAIN FOREST AND RANGE EXPERIMENT STATION
Forest Service
U.S. Department of Agriculture
Ogden, Utah 84401
Joseph F. Pechanec, Director

CONTENTS

INtEOGUCHLON. cra. “steel ns Gaye” 2 Hee eaten
Site Index and Yield Capability Equations .
Western: White Pine. . css. i- 2.76
Ponderosa Pine
Lodpepole Pine: s:ke: 7 t 'e se. Sh She t eime) feu reyes ace
Western Larch
Engelmann Spruce .
Inland Douglas- fir,
GEANGGE EDs a icc a eae a) sane ety he
Quaking Aspen

Literature Cited ,

INTRODUCTION

Computer programs for such purposes as the compilation of forest inventory data
must contain instructions for operations that were done by hand only a few years ago.
At that time, tables, graphs, and alinement charts were indispensable to the forester.
The information which was contained therein is still indispensable, but today we must
be able to store this information in a computer program in a form readily accessible
for use. Computer use of charts or graphs to describe site or yield functions is
impractical, but information stored in tabular form could be used. However, this
would require either a very extensive table for each species or interpolation within a
small table. The core storage required for a number of extensive tables, in addition
to the rest of the program and the data being processed, might well be more than is
available on smaller computers, while interpolation can lead to inaccuracy.

The existence of an alinement chart or a smooth curve, or tables made therefrom,
implies the existence of a mathematical relationship between the variables involved.
Sometimes the relationship is known, as when a table is made by repeated solution of
an equation. Some charts and tables were derived altogether by graphic methods. When
this is the case, an approximate mathematical relationship can usually be derived.
However, the resulting approximation equation, complex and sophisticated as it may
appear, can never be any better than the hand-drawn curve that it is supposed to
represent.

Data collected 40 years ago represent our only source of information on site
quality and yield capability for some species. Yield capability, as used by Forest
Survey, is defined as mean annual increment of growing stock attainable in fully
stocked natural stands at the age of culmination of mean annual increment. Yields may
be substantially higher with thinning and other intensive management. Yield capability
is measured in cubic feet per acre per year.

Site index, together with associated yield information, provides a means of
assessing yield capability or, in other words, productive capacity of the land.
However, all variation in yield capability from one site to another cannot be explained
by conventional ''site index.'"' Short of direct measurement of stand growth itself, it
is not generally known what Stand parameter(s) could be used to account for variation
not explainable by site index.

Despite its shortcomings, site index does provide a means of rating sites in terms
of volume yield capacity--a more meaningful expression than site index alone.

Another factor may affect yield capability of a stand--the extent to which an area
may be stockable. It may be that available soil is interspersed with areas of rock or
gravel that will not support tree growth. Any yield capability figure expressed in
this report on a per-acre basis will refer to an acre of tree-supporting soil, not to
an acre of land containing some nonstockable area. Where full stocking is not possible,
the yield capability estimate should be adjusted. This might be done by multiplying
the yield capability estimate by the ratio of productive land area to total land area.
The result would be a yield capability estimate for the entire area, including non-
stockable portions.

Growing stock is defined, for Forest Survey purposes, as the live, noncull trees
of commercial species in the stand 5.0 inches d.b.h. and larger. However, the sources
of information from which the material presented in this report was derived used
differing size standards, so the definition of growing stock that was necessarily used
for each species will be given with the equations for that species.

The equations presented in this report can be used to estimate site index and/or
yield capability for each of the following species:

1. Western white pine (Pinus monticola Dougl.)

2. Ponderosa pine (Pinus ponderosa Laws.)

3. Lodgepole pine (Pinus contorta var. latifolia S. Wats.)
- 4, Western larch (Larix occidentalis Nutt.)

5. Engelmann spruce (Picea engelmannii Parry)

6. Inland Douglas-fir (Pseudotsuga menziesii var. glauca
(Mirb.) Franco)

7. Grand fir (Abies. grandis (Dougi.) Lindl.)
8. Quaking aspen (Populus tremuloides Michx.)

For western white, ponderosa, and lodgepole pines, and for western larch and
grand fir, the yield capabilities that are supposed to be equivalent to given levels
of site index are shown in table 1. For Engelmann spruce, inland Douglas-fir, or
quaking aspen, yield capability equations cannot be given because the necessary informa-
tion was not available.

Table 1.--Yield capability according to site index for western white pine,
ponderosa pine, lodgepole pine, western larch, and grand fir

Site index Yield capability
(50-year Western Ponderosa Lodgepole Western Grand
base age) white pine®/ pine® pine!) larch2/ fir
aa eet Cubase féét per acre pereyear -==----=--=- = =
20 49.5 19.4 12.4 -- 56.5
25 581 2452 16.4 -- 65:22
30 66.8 29.6 21.6 21.8 TS509
55 75.4 39\s5 218 2657 82.6
40 84.1 42.1 evomall 55.8 O15
45 SZ 49.6 43.5 42.5 100.0
50 101.4 51.9 550) 52a2 108.7
55 110...1 67.3 63.6 6227 117.4
60 11S 27 Tin S35 LSao 12681
65 127.4 89.9 88.0 85:25 134.8
70 13620 103.4 101.8 97.2 143.5
i 144.7 LS .7 116.8 109.3 15222
80 153.5 136.1 162.7 12 Te, 160.9

Based on all surviving trees in the stand at the age of culmination of cubic
mean annual increment.
“Based on all trees larger than 5.0 inches d.b.h.

SITE INDEX
AND
YIELD CAPABILITY

EQUATIONS

The information in this section was derived either from published material or from
data in the files of the Intermountain Forest and Range Experiment Station, as will be
noted for each particular species. The mathematical details of equation construction
or derivation have been omitted because use of the equations or computer subroutines
does not require a complete description of the means by which they were derived.

Except for quaking aspen, all site indices are referred to at a base age of 50
years. For the user's convenience, computer programs that contain the calculating
procedure are presented whenever the calculations required to obtain site index or
yield capabilities for a species involve more than the simplest equation. The programs
are written in FORTRAN IV. They have been tested on an IBM 360, Model 67 computer and
they can be expected to function on any IBM 360 having the minimum of a FORTRAN IV,
Level E compiler available. The programing might not always be the most efficient for
the IBM 360 because the programs are designed to allow modification to other versions
of FORTRAN. These programs have been written as subroutines because it is expected
that their use will be implemented through a larger program.

The user is cautioned that extrapolation of statistically fitted equations beyond
the range of the basic data may lead to inaccurate predictions. The probability that
this will occur is greater when the equations are more complex. Mistakes in input
data, poor selection of site trees, or the sampling of a site quality not represented
in the data basic to an equation can prevent the iterative process from reaching a
solution. It may well be that this trouble will never be encountered, but it is
something of which the user should be aware.

WESTERN WHITE PINE

Site Index

Site index for even-aged stands of western white pine can be estimated by the

equation:

S = boH [1 - by - exp (bsA)] 3: where

S = site index at a base age of 50 years

H = average height of dominant and codominant trees

A = age of the oldest dominant tree in the even-aged ‘stand
bo= 0.37504453

by= 0.92503

bo= -0.0207959

b3= -2.4881068.

This equation was obtained by fitting an equation of suitable form to the height/age
data in table 24 of Haig's (4) yield tables, then algebraically solving the latter
equation for site index. Written in FORTRAN IV (IBM 360) the equation is:

Yield capability (cubic feet per acre per year)

S = 0.375045 * H * (1.0 -0.92503 * EXP (-0.020796*A) )**(-2.48811).

WESTERN WHITE PINE

0 10 20 30 40 50 6 70 80

Site index (height in feet at 50 years)

Figure 1.--Relationship between site index and yteld capability.

Yield Capability
Yield capability of western white pine can be estimated by the equation:

YC = by + b,S, where

YC = yield capability in cubic feet per acre of mean annual
increment on all trees at the age of culmination of
increment

S = site index at a 50-year base age

bo = 14.849891

b, = 1.7311563.

This equation was derived from table 7 of Haig's (4) yield tables which show average
mean annual increment of all trees in so-called normal stands for various ages and
levels of site index. For western white pine, cubic mean annual increment culminates
at a stand age of about 105 years, regardless of site quality, according to Haig's
table. Consequently, this equation expresses the relationship between site index and
cubic volume mean annual increment at a stand age of 105 years, as shown in figure l.

PONDEROSA PINE
Site Index

The estimating procedure for even-aged stands of ponderosa pine is based on the
curves developed by Lynch (7). The index base age was changed from 100 years to 50
years and the base of sae ty in the equation was changed so as to use natural
rather than common logarithms. The surface of dominant stand height over age and site
index remains unchanged, because the conversion of the curves resulted only in a
rescaling of the independent variables. Lynch's system of site index curves takes into
account the reduction in height growth due to overdense stocking on lower quality
sites. Because one or the other of two equations is to be used, depending on whether
or not stocking and site conditions have reduced height growth, the estimation pro-
cedure has been put into a computer program subroutine, shown in figure 2.

SLBERGLTINE PPSITECA shes th sS150,Sv20
U=1.40.2432*(105./A—-1.)
PI=(H¥10.**(C.O437#(1L0C e/A-1Le))IFE( LSU)
SD=B¥(Oe2918t0 eo CUSGSFH—C eo G46T*H/ A429 6 7T52/4A)
TFC SE=—100 207335351
1 TECPI—-75..9) 25333
22a (SE=— VCO. PC (b= PT}
RZ=EXP(-02383123E-03¥*2Z490.27E3CHE—U6*ZLZ*Z)
GC TO 4
Rl=1.
4 PA=EXP(-1.19222*(10C./A-2-))
GA=Cso 19562354 10Ce/A=2e4
SI5C={(H/(PARRZ) )*¥*#(12/(1e+GA))
RETURN
cND

Figure 2.--Stte tndex
estimating subroutine
for ponderosa pine.

Ww

YNatural or Naperian logarithms are to the base e, where e = 2.71828. Common loga-
rithms are to the base 10. In conventional notation, which will be used in this report,
ln X means the natural logarithm of X, while log X means the common logarithm of X.

The mnemonics used in the subroutine are as follows:

A = total stand age
H = average height of trees in the dominant stand
B = stand basal area per acre

SI50 = site index at a 50-year base age

SD = stand density in percent as expressed by Lynch (7), where
density is relative to the average for a
particular age and site index. That is, when
SD = 100 percent the stand is of average density; when
SD = 50 percent the stand is half average density,
and when SD = 200 percent twice average density is indicated.

Average density of Lynch's sample plots was a little less than normal for most ages and
sites. The extremes were 83.3 percent and 111 percent of what Meyer (8) called normal.

Yield Capability

The plot data used in Lynch's study are the basis for the equation or procedure.
A regression equation was fitted to the relationship between age, site index, stand
density, and net cubic foot volume yield of the plots. When stands are assumed to be
of average density for the sample (SD = 1), the equation reduces to:

In Y = botb,1n S-bjA~2 -b31n S-by +(bs5-bg)S-b7A~1S7!,

or
Y = coS! + exp (coS - boA~2 - byA7487}),
where
Y= net yield-of all trees: inucubice: feet per. acre
S = site index at a 50-year base age
A = stand age

Cg = exp(bo-by) = 13,100.281
Ci = by ml b3 = -0.4930327

0. 26782874E-01 2

Cp = bs - de

bo = 467.59461

oO
N
iT]

1843.6671

then, mean annual increment = Y/A.

2 ; ‘
2 according to FORTRAN notation, E+n following a number means that the number is to

be multiplied by 166) In this instance, the E-01 indicates that the decimal point
should be moved one place to the left (0.026782874).

6

On any particular site, the maximum mean annual increment is reached at the age when
dm.a.i./d A = 0.
Evaluating the indicated derivative and solving the equation for age, we find that
mean annual increment is a maximum when
peesell obs -S) AUN,

A Se eae

so the age at which mean annual increment is maximized can be expressed as a function
of site index. Then,

YC = cy S°! - exp (c9S - byA~2 - bzA™!S71) + An},

where

>
iT]

the age of mean annual increment culmination expressed
as a function of site index, as shown above.

These equations have been put into a computer subroutine, which is shown in figure 3,
and the relationship between site index and yield capability is shown in figure 4.

SUBROUTINE PPYCAP (S,YCAPP)

A=(1843.67/StSQRT(.3399LLEtI/S(S*S)4+2.374076E+4) 9/2.

YCAPP=(13100.281*S5S**(-0.4930327)*EXP(0.2673829E-01 *S—-467.595 /(A
I*A)—-18432.67 J/1A¥S)))/A

RETURN

ENC

Figure 3.--The ponderosa pine yteld capability subroutine.

|
tt

PONDEROSA PINE

Yield capability (cubic feet per acre per year)

Site index (height in feet at 50 years)

Figure 4.--Relationship between site index and yield capability.

LODGEPOLE PINE
Site Index

The procedure for estimating site index is based on the curves developed by
Alexander (1). Like Lynch's curves for ponderosa pine, Alexander's curves for lodge-
pole pine can be used when stand density is heavy enough to have inhibited height
growth of the dominant stand. This is thought to take place at densities where the
crown competition factor is greater than 125. Thus, two different procedures have
been devised for estimating site index; where CCF is 125 or less a straightforward
calculation by means of two equations will suffice.

The equations are:
SI = H + by (1m A - In 100) + by (A* - 1002) + b3 (A? - 1003)
+ by [(H/A) - (H/100)] + bs [(H/A*) - (H/1002)] +

bg [(H/A%*°) - (H/100?-5)],

where
SI = site index at a base age of 100 years by = -0.21973907 X 10.4
H = height of dominant stand b3 = 0.61670435 X 10°”
A = total age of an even-aged stand by = -64.32135
b, = 18.310745 Be. = 9528-25711
be = -34848.289
and
S.. | Cpt -cy Si,
where
S = site index at a 50-year base age
Cg = 1.029546
€q = 0.6297251.

In stands where CCF exceeds 125 site index must be adjusted upward to compensate
for the reduction in height of the dominant stand due to stocking density. Let g(A) be
a function of age:

g(A) = b,[(1/A) - 0.01] + bs[(1/A2) - 0.0001] + be[(1/A2-5) - 0.00001]
and

k = 0.8188667 X 107° (CCF=125):
Then

P = 1-k[g(A) + 1].

Spor an explanation of crown competition factor, see Krajicek, Brinkman, and
Gingrich (6).

8

Within the range of the data on which these equations were based, P will range in
value between zero and unity, so dividing SI, as computed above, by P gives the proper
adjustment to compensate for the reduction in stand height.

These equations were not based on data including ages over 200 years. Therefore we
recommend that ages older than 200 be reduced to 200 years.

SUBROUTINE LPSITE(A,H,CCF,S)
DIMENSTON A(9)
DATA B/18¢ 310745 4006 2197399 TD-02 »00 616 7943 50-059 5409 32135,9578 oe 3711,
1348486 289, 0. 81 8 867020-0 3 910029546 4026297251 /
MEGA 42orL
PPE 2 5 2% 3
2 WRITE( 6,201)
S=0,0
RETURN
CCFT=CCF-125.0
FA=B(1 )*(ALOG(4)—45 6051 702) -B (2) *(A*A—-10000,0)4B8(3) *(A**3-10,90%*6)
GA=-B(4)*(1.C/A-e OLY FBI S) (LoS (A*AV KH 0 IND] N—-B (6) FE (AX*(-2,5)-,00001)
PO=FA+H*(GAt+120)
IF (CCFT) 49495
4 CCET=0 Pe)
5 EK=B(7)*CCFT
PL=EK*(GA4+1.0)-1.%
ST =—PO/P1
S=B(8)+B(9)*ST
RETURN
201 FORMAT (62HCEITHER AGE OR HEIGHT OF A LOOGEPOLE PINE IS NEGATIVE OR
1 ZERO. )
END

Ww

Figure 5.--The stte index estimating subroutine for lodgepole pine.

pee

LODGEPOLE PINE

Yield capability (cubic feet per acre per year)

20 30 40 50 60 70 80

Site index (height in feet at 50 years)
Figure 6.--Relattonshtp between site index and yield capability.
Yield Capability

The equation for lodgepole pine was obtained by projecting yields of hypothetical
stands according to the method published by Myers (9). The projection procedure dif-
fered from that of Myers in that we maintained a different stocking level for each
level of site index. Myers used stocking levels of 80 for poorer sites and 100 for
better sites. By 80 and 100 are meant stocking regimes that peeve the stand to 80 or
100 square feet per acre of basal area at the time when average stand diameter at
breast height (d.b.h.) is 10 inches. Thereafter the stocking level is kept by thinning
at 80 or 100 square feet of basal area per acre, even though average stand d.b.h. will
increase. In making stand projections to derive a yield capability equation, the
levels of stocking used for each site index were a linear function of site index, as
shown below.

Site index at a

50-year base age Stocking level
20 74.0
30 84.5
40 94.8
50 105.0
60 TPS52
70 125% 7,
80 136.0

the d.b.h. of the tree of average basal area.

10

Site index at a 50-year base age was used rather than at 100 years as in Myers'
published example. Stand projection by Myers' method depends on the use of a predicting
equation for future d.b.h., which has site index as one of the independent variables.
Projections were made for hypothetical stands at higher site index levels than were
found on any of the plots that served as the data base for Myers' equation. Neverthe-
less, the results obtained appeared entirely reasonable, so projections of stands at
the higher site index levels were included in the estimates used to derive the yield
capability equation.

For each site index, a curve of net yield before thinning was plotted over age.

It was assumed that the volumes removed in thinnings would be lost to mortality in
natural stands. The age of culmination of net cubic volume mean annual increment was
estimated by drawing a line from the origin of coordinates tangent to the yield curve.
The point of tangency indicated the age of m.a.i. culmination and the net stand yield
at that age. Dividing the yield thus obtained by the culmination age gave the m.a.i.
at the age of culmination, or the yield capability associated with that site index.
Plotting yield capability over site index resulted in the curve shown in figure 6.
This curve is described by the equation:

YC = bo + b,S 7 boS*>.
where
YC = yield capability bo = 6.9091271
S = site index at a 50-year base age b,; = -0.16172109
b> = 0.021683019
WESTERN LARCH
Site Index

The following equation for height of western larch in even-aged stands was
developed by Arthur L. Roe:

log H = log S = b, .(1/A - 1/50),
where
log H = the common logarithm (to the base 10) of dominant
stand height

log S = the common logarithm of site index at a 50-year

base age
A = total stand age
b, = 21.036.

This equation can be easily solved for site index:
log S = log H + by (1/A - 1/50)
S = 0.37956 H: exp (48.4372/A)

The last equation shown above can be used in computer programs to estimate site index
of even-aged stands of western larch.

5 principal Silviculturist, Intermountain Forest and Range Experiment Station.

11

| | |
WESTERN LARCH

T i tae
|

at
|

Yield capability (cubic feet per acre per year)

it) 10 20 30 40 50 60 70 80

Site index (height in feet at 50 years)

Figure 7.--Relationship between site index and yteld capability.

Yield Capability

The yield capability equation for western larch is based on current work being
done by Roe and John H. Wikstrom. For cubic volume yield capability in stems 5.0
inches d.b.h. and larger, the equation is:

YC = bo + Dy S + b./S,

where
YC = yield capability in cubic feet per acre per year
of mean annual increment
S = site index (50-year base age)
bo = =126.05
b, = 2.7974081
by = 1919.3157.

This equation cannot be used when site index is less than 26 feet, because the table
from which it was made gives no information below site index 30.

ee P : 2 :
principal Economist, Intermountain Forest and Range Experiment Station.

12

To derive this equation, the yield curves were plotted for each level of site
index. Then a series of lines was drawn, passing through the origin of coordinates
and tangent to each yield curve. From the point of tangency the following was read:
(a) the age of culmination of mean annual increment, and (b) the yield at that age for
each level of site index. The mean annual increment at its culmination age for each
site index level was obtained by dividing (a) into (b). The relationship between site
index and yield capability was approximated by the least squares fitting of a suitable
curve form to the series of points plotted from site index and yield capability. This
is shown in figure 7.

ENGELNMANN SPRUCE
Site Index

An equation with which site index can be estimated directly from measurements of
tree age and tree height is given by Brickell (2). This equation is:

11

H + >, Dex,
: Lot
t=l1

S =

where
S = site index at a 50-year base age
H = total tree height
A = total tree age

and
b; = 0.10717283X102 X; = (1n A - 1n 50)
bo = 0.46314777X10~2 X>-= [(1019/A5) -- 32]
b3 = 0.74471147 X, = H[(107/A2) - 4]
by = -0.26413763x108 X= Hk see
bs = -0.42819823x10-! X; = H (In A - In 50)?
be = -0.47812062X10~2 Xe-= H*[(10*/A*) = 4]
bz = 0.49254336X10-5 Xz = H2[(1029/A°) - 32]
bg = 0.21975906x10~® Xa = Ho 1(10- 97a) = 32|
by = 5.1675949 tes HRM -eeoe
by9= -0.14349139X10~7 Xi9= H*[ (100/A) - 2]
by ,= -9.481014 Recta eee ae

The standard error of estimate (Syx) for this equation is 0.69 foot of site index units.
An example of how this equation might be programed in FORTRAN IV (IBM 360) is shown in
figure: 8:

13

SUBROUTINE ESSITE(AsH,ST)

DIMENSION X(11),8411)

Q=A

P=H

DATA B/1060 71728 9 Co 46314 7BE=2 4 Oo 74471155-2641 36 769-06 4281 98 EH-1,—-004
L721 20 6F=- 249 Co 4925434F-55 N60 2197591 F—6950157595 9-30 143491 4E-7, -90 4810
214/

ST=S20

Q2=0*9

Q3=Q2*Q

Q94=92*Q2

Q5=92*Q3

RECO5=120/05

Pp2=PxPp

P3=P2*D

P4=P2?xP2

X(1)=ALOG(0)—35 912023
X(2)=1Le0FLO*RETCQ5-32.0

X( 3) =P *( 12 0F4/02-4. 0)
X(4)=P*(SORT(RECQ5)-5,556854E-5)
X(5)=P*X(1)*X(1)

X( 6)=P¥*XK( 3)

X( 7) =P 2*XK( 2)

X(8)=P*X07)

X( 9V=P3*(SORT(RECOS *SOR T(120/Q))-20.127318E-5)
X( 19) =P4*(19C20/0-2.0)

KC 11)=P4( (150/04) / SORT (Q) 20 262742E-8)
DO 1 T=1.11

S1T=ST+8(1)*xX{(T)

SI=SI+H

RETURN

END

Figure 8.--A site index estimating subroutine for Engelmann spruce.

—

If a shorter but less precise equation is desired, the following is recommended:

1=1 a
where
k; = 0.32158242 Z,.= H [(10*/A2) - 4]
ky = -0.98468901X10* 7 ou cor)
k3 = -0.12253415X10°* Z3 = H2[(104/A2) - 4]
k, = 1.0662061 Tie ko Oe a
ke = -0.80894818x10 © Ze =Ht [| (100/A-= 2].

For this equation, Sx meee, fete

These equations are valid for trees between the ages of 20 and 200 years and for
site indices ranging from 10 to 95. A site index estimate should be computed for each
sample tree. The average of the individual tree estimates will be the average site
index for the stand. If trees older than 200 years must be used as site trees, estimate
the site index as if tree age were 200. Above that age, height increases very little.

14

INLAND DOUGLAS-FIR

Site Index

A site index equation, similar to the one given for Engelmann spruce, has been
derived for inland Douglas-fir by Brickell (3). The equation is:

1
S =H+ > b Xo
1=1

where
S = site index at a 50-year base age
H = total height of a sample tree in the dominant stand
A = total age of a sample tree

and
b; = 40.984664 te eso 5
by = 4521.1527 ea A iS Oncee)
b3 = 123059.38 ee Cena
b, = -0.5332868X10 ~§ ky = (At = 50")
bs = 0.37808033xX10-!0 Xe =9(A> = 50°)
be = 216.64152 Xe SoA 50. 7)
bz = -158121.49 X7 = H(A-* - 50-4)
bg = 1894030.8 | Xg = H(A~5 - 50-5)
bg = -0.10230592X10~9 Xg = H(A* - 50*)
by 9= -6.0686119 X19= H2(A~? - 5074)
by 1= -25351.090 Xi ;= H2(A-5 = 5075)
by9= 0.33512858X10-+ X,2= H2(A - 50)
b,3= 0.17024711X10 -2 X,3= H3(A-! - 50-1)
bi y= 398.36720 Xzy= H3(A75 - 5075)
by >= -0.88665409X10-§ epee a0)
big= 0.40019102x10-1" Xig= H3(A* - 50°)
b17= -0.46929245x1078 hes HG 0)

byg= -0.16640659X10-2° Xig=, HCAs? = 50°*>).

The standard error of estimate for this equation is 0.24 of site index. A computer
program in FORTRAN IV (360) to evaluate this equation is shown in figure 9.

SUBROUTINE DFSITECAsH,STI)

DIMENSION B(18),X(18)

Q=A

P=H

DATA 8/40. 98466 94521015 34123059%e 4-5 332868E—-9y 30 TROBO3ZE-11 92160664
1159-15 81210591 89403 le 9-10023059F-10 9-60 06861 29 -2535100949 30 351286F-
259 lo 702471 F-39398 36729 —Be 866541 E-9 940 OOL IL E-15 9 —40692924E-9 9-10 66
340466E-21/

Si=0.0

P2 =P «xp

P3=P2*P

P4=P2x*P2

P5=P4xP

Q12=SORT(QO)

015=0*Q12

Q2=Q*9

04=02*02

X(1)=100/012-C0o 14142132

X( 2) =1LeO/( Q2*01 2)-52655854E-5

X(3)=160/(Q2*0*012)-101 31371 E-6

K(4)=(02-250000)*(Q2+2500. 0)

X(5)=0 *04-3,125E8

X(6) =P *€ 160/015-2598 2842 6E-3)

X(7)=P*(1.0/04-126E-7)

X( 8) =P*( 1. 0/(04%*0)-3.2F-9)

X(9)=9%X(4)

X(10)=P2*( 160/902-0002 0004)

X(11)=P*xX( 8)

X(12)=P2*(90-5020)

X(13)=P3¥(169/0-Ce.02)

X(14)=P*X(11)

X( 15) =P3*( 915-353, 5534)

X(16)=P2*xX (9)

X(17)=P5*X(1)

X(18)=P5*(04*Q12-00 4419416E8)

D9. 1 I=1,18
1 ST=SI+B(T) *X(T)

ST=SI+H

RETURN

END

Figure 9.--A site index estimating subroutine for tnland Douglas-fir.
Although it provides less precise results, a shorter equation is:

5
Se Hct » k.Z.,
: Lt
t=1
where

101.08708 Z, = (a7}/2 i 5071/2)

~
pan
iT]

=
ie)
i]

122.04763 Zp eo (ATT 028508 ee)

16

k3; = 0.14082397x10 /4 Z3 = H. (A> = 50")
Kat 15),0400717 Zit =e Hot > = S007)
ke=~ ©, 13492191K10"* Zc = HOGA 9550 °).

For this equation the standard error of estimate is 0.90 foot.

These equations are valid for ages from 20 to 200 years and for site indices from
10 to 110. The site index of trees older than 200 years should be estimated as if age
were 200.

GRAND FIR!

Site Index

Information on productivity of grand fir has been developed using a direct index
of relative productivity (Q) rather than site index. Procedures for calculating Q from
stand age and height have been published (10). Figure 10 shows a FORTRAN program RPGF.

SUBRCUTINE RPGF( CyHyT,yCR,TER )

CCUBLE PRECISION AyByCyDyX,BIyLET, TOL, VALyBRAC,ALGCR
DIMENSTON C5)

CATASC /—10.29862_ 203 OLOLSLE Ts —2ef -» 260 /

GIVv = 0.5
TcR = O
[GT = 2
GOR. = Ge
I = 0

BY =-Ck3)4(1 + CC4))
DEC Se I06. Oyo, 92
1 TOL = O.0CO

GE FE 95

92 TOL = ALCG( O.8*H/T )

95, ALGCR = ALGOOU(H—-4.5) = CALOG(CR)—3.747474-8. 218305/1)*18.7867T
1S = TEL

99 Tet VEU «67. 20.) SE TO 101
DET = BT*DEXP( TCL )
Pe ( GET ) 99 100; 100
991 BRAC =) LeOCO}——DEXP (DET )
IF (BRAC) 100,100,992
S992 1S Ctl) + C(2)*(ALGCR — C{(5)*CLOG(BRAC) )
I= 1 + 1
996 .COMMOMY 214 225:2 3')'s TCT
21K = AO:
= Xi = oS

(con. next page)

a : ' : Bhcsead
The section on grand fir was contributed by Albert R. Stage, Principal
Mensurationist, Intermountain Forest and Range Experiment Station.

17

Se) en
IGT = 2
CUNTE-99
22 VAL = X -— TOL
START IFERATICGCN LOOP
DPC VAL) el sho
EQUATION IS NOT SATISFIED BY x
1 #8 = B/VAL - 1.000
TF(B) 29892
LTERATICNS? TS: POSS EBIZE

22h e= AB
xX = X + A
G&G = VAL
TOL = X
IGT = 3
COTTE 99

23 VAL = X - TOL
TEST CN SATISFACTORY ACCURACY
D = DABS(X)
TOL = 0.C008*D
IF ( DABS(A)-TOL) 59596
IF (DABS (VAL)-L.LOLO*TCL) 7796
LE 25206) 141500
ENG CF I[TERATICN LOOP
7 © = CEXP(X)
RETURN
ERROR RETURN IN CASE OF ZERC DIVISOR
8 IER = -2
RETURN
NC CGNVERGENCE AFTER 20 ITERATION LOOPS. ERROR RETURN

10 TER = I
RETURN
ERRCR CCNCITION ON INITIAL ITERATION
2 LER = == 3
RETURIN
14 IER = -4
RETURN
100 GO TG (Ci2s127¢102),16T.
101 GG TC ( 149145102), 1GT
102 X¥ =X - A
A = CIV*¥A/8
DIV = DIV*CIV
= 1+ 1

aout

Figure 10.--FORTRAN subroutine to calculate relative productivity.

The variables in the calling sequence are:
Q = the measure of relative productivity

H

height in feet

=
iT]

age in rings at 4.5 feet above ground

CR = ratio of live crown to total height in percent

IER = an integer used to flag possible errors in calculation.
(Should be zero if no errors.)

If required, site index can be approximated from Q by the FORTRAN equation:

Ole= 455 +2SORT(Q))* 172.2352 * (2. = EXP(=.6905*Q) )**2.

Yield Capability

The yield equation also already has been published (11). The maximum m.a.i. for
average stocking corresponding to this equation can be solved by the FORTRAN function

CUPGF illustrated in figure ll.
the previous subroutine RPGF in figure 10.

The single argument to the function is Q obtained from

Figure 12 shows the linear approximation

of the relationship between yield capability and site index.

Figure 11.--The functton
subprogram for grand fir

yteld capability.

FUNCTIUN CUPGF(S)

oo = (896 (OLS S = 220 1G)

A= B e(O25 + SORTCC.25 + 12./1S*B)))
CUPGF = 9 22228.7EXPI(—B/AIS1A® 4 125/75.)
RETURN

ENC

GRAND FIR

Yield capability (cubic feet per acre per year)

Site index (height in feet at 50 years)

Figure 12.--Relationshtp between site index and yield capability.

19

QUAKING ASPEN

Site Index

A table published by Jones (5) gives site index of quaking aspen according to
stand age and dominant height. Age at breast height is used, and it is important that
average dominant height not be estimated from trees that are all of the same clone.

In Jones' table a site index base age of 80 years is used. Equations were fitted by
regression methods to the site index values in Jones' table, using stand age and height
as independent variables. The best of several equations fitted is:

10
S=H+t ” Diskia.s
VG
where
S = site index at an 80-year base age
A = stand age at breast height
H = average height of the dominant stand
by, = -8.7810841 Xp COO / Ae = te2s]
b, = -5.1824560 X= (in A ein 80)
b3 = -40.260849 X,; = H[((1n A)/A) - ((1n 80)/80)]
b, = 1.8589039 X,, = H[(100/A) = 1.25]
b, = 0.34567436X10 X, = H[AS - 3,276,800,000]
be = -0.16454828X10"° X, = H[A? - 512,000]
bz = -11,647,641.0 X, = H[exp (-A) - exp (-80)]
b, = 0.19235397X10"° X, = H2[ VA - 80]
by = 0.30310790x10 *" Xg = H2[AS - 3,276,800,000]
by = -0.84256893X10 Xi o= H2[ (10 ,000/A2) = 15625 1h

The standard error of estimate for this equation is 0.55 foot. The equation is valid
for breast high ages between 20 and 160 years, and site indices (base age 80) between
20 and 90. Figure 13 shows how this equation might be programed in FORTRAN IV (360).
This equation will not be valid for ages beyond 160 years. Older trees should be
treated as if age were 160 years.

20

SURROUTINE ASPSITT(AsH,S)

DIMENSION X(10),8(10)

DATA B/—Be 781084_—-50 1824564 -4N 6 26085 910 85 RON 4905 3456744F-11,-00,164
15482 8E—6_- 10) 64764E1 91 0 92354F—4,y 200310 79E K-15 9-90 8425628 VF -3/
S=C.20

Q=A

P=H

P2=Px*P

RECQ=10029/Q

Q2=Q0*Q

Q2=Q*Q2

QLN=ALOG(Q)

X{T)=RECOQ-1225

X(2)=OLN—40 382927

X( 3) =P*(QLN/Q-Ce. 5477533 E-1)

X( 4) =P*K( 1)

X(5)=P *(02 *0 3-32 76820E5 )

X(6)=P *(93-512-CE3)

X(7)=P*( CC LCOEZ SEXP (0/250) )*¥*2 )—-1e 804351E-29)
X(8)=P2*(SORT(0)—-8., 944277)

X(9)=P*X{(5)

X( 19) =P2*(1.9F4/02-1.54625)

DO 1 T=1,10

S=S+BC 1) *X( 1)

S=S+P

RE TURN

END

Figure 13.--A site index estimatton progran for quaking aspen.

21

10.

Ll.

22

LITERATURE CITED

Alexander, Robert R.
1966. Site indexes for lodgepole pine with corrections for stand density
Instructions for field use. U.S. Forest Serv. Res. Pap. RM=24;
7 aDe sa llelus®.

Brickell, James. Ee
1966. Site index curves for Engelmann spruce in the northern and central Rocky
Mountains. U.S. Forest Serv. Res. Note INI=42, -8*p.j-dus.

1968. A method for constructing site index curves from measurements of tota}
tree age and height--its application to inland Douglas-fir. USS:
Forest Serv. Res:-Pap. INT=47, 2375... aiduse

Haig, Irvine T.
1932. Second-growth yield, stand, and volume tables for the western white pine
type: U.S. Dep. Agr. Tech. Bulll.- $25,267 spej. alms:

Jones, John R.
1966. A site index table for aspen in the southern and central Rocky Mountains.
U.S. Forest Serv. Res. Note RM-68, 2 p.

Krajicek, John E., Kenneth A. Brinkman, and Samuel F. Gingrich.
1961. Crown competition factor, a measure of density. Forest Sci. 7: 35-42.

Lynch, Donald W.
1958. Effects of stocking on site measurements and yield of second-growth
ponderosa pine in the Inland Empire. Intermountain Forest and Range
Exp. ota. Res. Papes56; 56 p. 5 acblus:

Meyer, Walter H.
1938. Yield of even-aged stands of ponderosa pine. U.S. Dep. Agr. Tech. Bulle
630, 59 pi, 1Llus.

Myers, Clifford A.
1967. Yield tables for managed stands of lodgepole pine in Colorado and Wyoming.
U.S. Forest Serv. Res. Pap. RM-26, 20 p., illus.

Stage, Albert R.

1966. Simultaneous derivation of site-curve and productivity rating procedures.
Soc. Amer. Foresters Proc. 1966: 134-136.

1969. Computing procedure for grand fir site evaluation and productivity
estimation. USDA Forest Serv. Res. Note INT-98, 6 p., illus.

AFLC/HAFB, Ogden

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fee ae eee

Headquarters for the Intermountain Forest and
Range Experiment Station are in Ogden, Utah.
Field Research Work Units are maintained in:

Boise, Idaho

Bozeman, Montana (in cooperation with
Montana State University)

Logan, Utah (in cooperation with Utah
State University)

Missoula, Montana (in cooperation with
University of Montana)

Moscow, Idaho (in cooperation with the
University of Idaho)

Provo, Utah (in cooperation with

_ Brigham Young University)

ABOUT THE FOREST SERVICE ...

As our Nation grows, people expect and need more from their forests—more
wood; more water, fish, and wildlife; more recreation and natural beauty; more
special forest products and forage. The Forest Service of the U. S. Department

of Agriculture helps to fulfill these expectations and needs through three major
activities:

Conducting forest and range research at over
75 locations ranging from Puerto Rico to
Alaska to Hawaii.

Participating with all State Forestry agen-
cies in cooperative programs to protect, im-
prove, and wisely use our Country’s 395
million acres of State, local, and private
forest lands.

Managing and protecting the 187-million
acre National Forest System.

The Forest Service does this by encouraging use of the new knowledge
that research scientists develop; by setting an example in managing, under
sustained yield, the National Forests and Grasslands for multiple use purposes;
and by cooperating with all States and with private citizens in their efforts to
achieve better management, protection, and use of forest resources.

Traditionally, Forest Service people have been active members of the com-
munities and towns in which they live and work. They strive to secure for all,
continuous benefits from the Country’s forest resources.

For more than 60 years, the Forest Service has been serving the Nation as a
leading natural resource conservation agency.