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ELSEVIER
Forest Ecology and Management 98 (1997)49-60
Comparison of diameter-distribution-prediction,
stand-table-projection, and individual-tree-growth modeling
approaches for young red alder plantations
Steven A. Knowe
•
a .*
,
Glenn R. Ahrens
a
,
1
Dean S. DeBell
b
Depanment of Forest Science. Oregon State UniL'ersiry. Cvn·allis. OR 97331. USA
USDA Forest Sen·ice. Forestry Sciences Laboratory. Olympia. WA 98502, USA
b
.-\ccepted 4 io.larch 1997
Abstract
A red alder planting spacing srudy was used to compare three modeling approaches that have been successfully used for
other tree species. These three approaches predict stand strucrure and dynamics in plamations that are 7 to i 6 years old. with
planting densities of 976 to 13 399 trees/ha. The diameter-distribution-prediction approach tended to \)ver-predict the
diameter at breast height (dbh) for larger trees in stands planted at low density and to under-predict dbh for smaller trees in
stands planted at high density. This approach may be useful for comparing planting densities when a tree list is not available.
The stand-table-projection approach tended to under-predict dbh for smaller trees in young stands planted at low density and
to over-predict dbh for smaller trees in young stands planted at high density. This approach. however, provided consistency
between stand- and tree-level growth projections. and should be useful for comparing planting densities when a tree list is
available. The individual-tree-growth approach provided the best representations of observed diameter distributions at all
planting densities, stand ages, and growth intervals. This approach may be best suited for stands that have been thinned.
stands with mixrures of species, and stands with heterogeneous size classes. © 1997 Elsevier Science B. V.
Keywords: Alnus ntbra: Weibull function: Relative size; Distance-independent: Simultaneous regression: Compatible equations
1. Introduction
Red alder ( Alnus rubra Bong.) is the most abun­
dant hardwood species in the Pacific Northwest.
Recently, and for a variety of reasons. there has been
an increased interest in managing red alder as a
timber resource (Hibbs et a!., 1989, 1994). The
Corresponding author. Tel: 904-225-5393: Fax: 904-2250370; E-mail: steve.knowe@ravonier.com.
1 Paper 3047. Forest Res
Laboratory, Oregon State Uni­
versity, Corvallis. OR . USA.
•
e:uclt
hardwood industry has begun to manage alder in
plantations to ensure future availability of alder saw­
timber (Hibbs et al.. t 994: Raettig et a!., t 995).
Also, because alder is not susceptible to laminated
root rot ( Phellinus weirii), it is increasingly planted
on sites infected with the disease. When n\anaged in
mixture with conifers on suitable sites, alder can
increase nitrogen tixation and. therefore. productivity
(Tarrant and Trappe. 1971; Tarrant et al.. 1983:
Binkley et al.. 1994).
Very little is known about the growth and devel­
opment of alder in plantations. however. Much of the
0378-1127/97j$17.00
1997 Elsevier Science B.V. .\II rights reserved.
Pfl S0378-ll27(9 7 )00075-3
S.A.
50
Knowe eta/./ Forest Ecology and Management 98 ( 1997) 49-60
growth and yield research conducted on red alder has
sizes and number of surviving trees equals projected
focused on natural stands and size-density relation­
stand basal area and stand survival, respectively.
1987; Hibbs and Carlton, 1989;
Puettmann et al., 1992), and on thinning effects
(Hibbs et al., 1989) and height-age curves (Harring­
ton and Curtis, 1986). Silvicultural regimes proposed
Diameter distributions obtained for each modeling
for crop trees often rely on relatively low planting
limited long-term plantation data were available. the
ships
(Hibbs,
method were compared to the observed diameter
distribution across a range of planting densities and
1-
for
or 2-year and 5-year growth intervals. Because
densities (compared with typically dense natural re­
equations presented here represent a framework for
generation) in order to maximize diameter growth
additional modeling as more data become available.
during the alder's rapid juvenile growth phase (Hibbs
Despite limitations in the quantity of data. we feel
and DeBell,
1994).
Preliminary results of plantation
trials indicate that spacings in the range of 2 to
may produce a dominant alder stand within
2
3
m
to
3
the assessment of modeling approaches was not oth­
erwise limited.
years, while providing adequate space for rapid di­
ameter growth (Ahrens et al.,
Bell,
1992;
Hibbs and De­
1994).
There is a critical need for detailed quantitative
information about the performance of alder at differ­
2. Methods
2.1. Data
The red alder spacing study site was established
ent planting densities. Toward that end, a network of
1974
near Apiary. Oregon. Six planting densities
new test plantations has been established to provide
in
a high-quality database for modeling growth and
were included in the study: 976 (3.2 X 3.2 m), 1600
(2.5 X 2.5 m), 3086 (1.8 X 1.8 m), 4630 (1.2 X 1.8
m), 6944 (1.2 X 1.2 m). and 13 889 (0.6 X 1.2 m)
yield at managed-stand densities (Hibbs et al.,
1993).
In anticipation of this database, an appropriate mod­
eling framework needs to be developed to ensure
treesjha. respectively. These planting densities are
that the future database will include the appropriate
higher than those currently being considered for
v ariables. As a first step, modeling methods that
operational plantations (Hibbs and DeBell.
have been applied to other species should be evalu­
Densities of
ated. Such an evaluation may be applicable to exist­
by one plot each, and the remaining planting densi­
976
1994).
and 3086 treesjha were represented
1).
ing red alder management systems-specifically, to
ties were represented by two plots each (Table
yield systems.
With few exceptions. all plots were measured annu­
The purpose of this paper was to develop and then
ally from plantation ages 7 to
12
and at ages
14
and
compare modeling systems that predict stand struc­
16
ture and dynamics in red alder plantations. Though
vations. Diameters of all surviving trees on each plot
years. The resulting damset consisted of 7'2 obser­
long-term data is limited, an indication of the long­
term performance of alder over a range of spacings
was available from one dataset (described below).
Useful as a test of model forms, this existing dataset
contains a variety of initial planting spacings and
ages of rapid stand dynamics.
Modeling approaches included
(1)
for stand-level
data only, a diameter-distribution-prediction method;
(2)
for both stand- and tree-level data, a stand-table­
projection method; and
(3)
Table I
Summary of data used to develop and compare mode Iing ap.
proaches for red alder plantations
Density
(treesjha)
Spacing
(m)
Plot size
(ha)
Number of plots and
measurement ages
976
3.2X3.2
0.0809
1600
2.5X2.5
0.0809 3086
4630
6944
13 889
1.8X 1.8
1.2X 1.8
L2X 1.2
0.6X 1.2
I plot with ages 7 to
12 only
l plot with all ages
l plot with ages !4
and 16 only
I plot with all ages
2 plots with all ages
2 plots with all ages
2 plots with all ages
for tree-level data, an
individual-tree-growth modeL The stand-table-pro­
jection and individual-tree-growth methods are con­
ceptually similar in terms of generating future diame­
ters. The former does so, however, in a manner such
that the sum across diameter classes of projected tree
•
0.0187
0.0196
0.0180
0.0163
S.A.
Knowe et al. I Forest Ecology and Management 98 I 1997) 49-60
were measured to the nearest mm. and total height of
10 trees per plot was measured to the nearest 0.1 m.
51
Shiver (1986); this function is asymptotic with re­
spect to dominant height and survival. A compatible.
path-invariant basal area projection function (Eq.
2.2. Regression analysis
(1h)) was derived from the prediction function by
using an algebraic difference method. Parameters of
All measuremenc ages were included in the devel­
opment
of
the
diameter-distribution-prediction
method, but the predictions for age
7
were not
the Weibull diameter distribution function were ob­
tained by using a percentile-based parameter recov­
ery procedure as described in Section
2.3.
The four
considered in the comparison and assessment of
diameter distribution percentiles were predicted as
modeling approaches. For the stand-table-projection
functions of quadratic mean diameter at breast height
and individual-tree-growth methods, data were ar­
(dbh), which was obtained from basal area and sur­
ranged in nonoverlapping growth intervals ot' l to 2
viving number of trees at a given age, and plantation
years. According to Borders et al. (1988), there are
age ( B ailey et al., 1989;
fewer problems with serial correlation of real growth
1990; Knowe and Stein, 1995).
Borders and Patterson,
series-derived from either remeasured plots or trees
A survival function (Eq. (1f)) was selected for use
-when the data are arranged in nonoverlapping
in all three modeling approaches. The path-invariant
growth intervals rather than all possible intervals.
function developed by Clutter and Jones (1980) is
Error component models for stand-level equations
based on a differential equation which implies that
(Gregoire, 1987) partition residuals into plot errors.
the instantaneous rate of stand mortality is propor­
which remain constant for a given plot through time.
tional to age and density.
and measurement period errors. which remain con­
Mortality rate of other tree species may also be
stant for all plots at a given measurement age. Simi­
related to site index. with greater mortality occurring
larly, error component models for tree-level incre­
on the more productive sites. However. a preliminary
ment equations (Stage and Wykoff. 1993) include
model with age replaced by dominant height did not
stand-level errors tor plots and measurement periods
fit the observed data as well as the model with age.
plus the errors associated with trees within plots and
measurement errors.
Because the
individual-tree­
An
algebraic
difference
fonnulation
of
the
Richards (1959) equation (Eq. (1g)) was selected for
growth models are based on tixed projection inter­
describing dominant height growth patterns in red
vals, the 2-year growth intervals between 12 and 14
alder plantations. The resulting equation (Eq. ( 1g)) is
dividing periodic growth by 2 years.
These properties pennit the equation to be alge­
and between 14 and
16
years were annualized by
Stand-level models for the diameter-distribution­
prediction and stand-table-projection methods, in­
cluding prediction equations for basal area and diam­
anamorphic. base-age invariant, and path invariant.
braically rearranged for predicting either dominant
height ( H
age
(A 1 ),
1)
as
a function of site index ( H-c), current
and base age
(A )
or site index as a
eter distribution percentiles, and path-invariant pro­
function of dominant height. current age. and base
jection equations for survival, dominant height, and
age.
basal area, were fit as a nonlinear system of seem­
( 1 a)
ingly unrelated simultaneous equations (Eqs. (1a),
(1b), (1c),
(ld),
(le), (lf),
(lg)
and ( l h)). The path
invariant property results in the same final yield
regardless of whether several short growth intervals
or a single long growth interval are used. Fitting the
stand-level models as a system of simultaneous equa­
tions accounted for the contemporaneous correlation
Do= ,\10 +A.,, Dqt + A12 A,
(1b)
D25 = ,\20 + A21 Dqt
(lc)
Dso= A 3o + A31 Dqi
(ld)
( 1 e)
among equations (Borders, 1989).
The basal area prediction function selected for
young red alder plantations (Eq. ( l a)) was a nonlin­
ear variant of the model proposed by Pienaar and
( lf)
52
S.A.
Knowe et al. I Forest Ecology and Management 98 I 1997! 49-60
( ) (-) (-)
N
2
BA -=BA1N1
al
H2
a
H1
A2
A1
( lg)
aJ
( lh)
where A 1 =plantation age (years) at the start of the
growth period, A2=plantation age (years) at the end
of the growth period, BA 1=predicted basal area (m2 /ha) at the start of the growth period, BA2=
projected basal area (m2jha) at the end of the growth
period, D0 Oth percentile of diameter distribution,
D25 = 25th percentile of diameter distribution, D50=
50th percentile of diameter distribution, D95=95th
p ercentile of diam e te r distribution. D " '
= /BA1/(0.00007854 X N1), H1 = dominant
height (m) at the start of the growth period. H, =
projected dominant height (m) at the end of the
growth period, N1 =number of surviving treesjha at
the start of the growth period. and N2=projected
number of surviving treesjha at the end of the
growth period.
Using the approach applied to red alder planted in
Neider plots (Knowe and Hibbs. 1996), parameters
of the basal area prediction function and dominant
height projection function were fit to each planting
density. The resulting parameter estimates were ex­
amined for relationships with planting density; how­
ever, no consistent relationships were observed.
Parameter estimates for individual-tree equations
used in the stand-table-projection and individual-tree
diameter growth methods were obtained by using
nonlinear regression. Distance-independent tree-level
models included a relative basal area projection tunc­
tion (Pienaar and Harrison, 1988; Borders and Patter­
son, 1990; Knowe, 1994; Knowe and Stein, 1995;
Knowe and Hibbs, 1996) and a diameter growth
equation (Hester et al., 1989; Wykoff, 1990).
=
2.3. Diameter-distribution-prediction approach
The Weibull cumulative distribution function has
been widely used to model diameter distributions
since the early applications by Bailey and Dell (1973)
and others:
p( D)
I
{ (
1- exp -
D
a
n
(2)
where p( D)= cumulative probability of a diameter
(D) less than or equal to D, a= location parameter,
corresponding to the minimum dbh. b= scale pa­
rameter. and c=shape parameter.
This approach is most useful for comparing the effects of planting density when a tree list is not
available. Only stand-level information is needed to
generate a diameter distribution at any age. Compo­
nents of this approach include a basal-area-prediction function (Eq. (la)), the survival function (Eq. ( lf)),
diameter-distribution-percentile-prediction functions
(Eqs. (lb), ( l c), (ld) and (le)), and a parameter
recovery procedure for the three-parameter Weibull
distribution tunction. (The basal area function se­
lected for red alder plantations is asymptotic with
respect to the dominant height and survival tunc­
tions.)
Recently, parameter recovery procedures have
been developed to estimate the Weibull parameters
(Bailey et a!., 1989; Borders and Patterson. 1990;
Knowe and Stein. 1995). The main advantage of
these procedures is that diameter distribution charac­
teristics. such as minimum diameter or diameter
percentiles. can be predicted with more confidence
than the parameters, which are indirectly estimated
by analytical relationships. A percentile-based ap­
proach has been used to model fertilization effects
(Bailey et a!.. 1989), the impacts of interspecific
competition in conifer plantations (Knowe et al..
1992: Knowe and Stein, 1995), and mixtures of
eastern cottonwood (Populus deltoides L.) clones
(Knowe et a!., 1994).
The parameter recovery procedure is based on the
Oth. 25th. 50th, and 95th percentiles ( D0,D25,D50,
and D95, respectively). Assuming that c=3. the
location parameter, a, is obtained by using the mini­
mum ( D0) and median ( D50) diameters and sample
size (n):
o
n
t.J333 Do - Dso a=
0
ll .3333- 1
( 3)
If Eq. (3) results in a negative value, then a is set to
zero. The sample size ( n) is the product of the plot
size and the surviving number of trees. For predict­
ing a in Eq. (3), sample size may be estimated by
multiplying the number of surviving treesjha ob­
S.A.
53
Knowe et al.j Forest Ecology and ivfanagemem 98 I !9971·J9-t50
rained from the survival function by
0.032.
the aver­
age plot area in hectares.
The shape parameter is estimated by using the
estimate for the location parameter
(a)
and
D95 and
where bli = basal area of rh tree at the start of the
growth period. b:i =basal area of
of the growth period.
irh tree the at end
b 1 =average
basal area per
tree at the start of the growth period.
b:: = average
basal area per tree at the end of the growth period.
D2s:
2.343088
( 4)
c = -----In D95- a -In D25- a
(
(
)
)
The scale parameter, b, is obtained by solving the
second moment of the Weibull distribution for the
p ositive root using the estimates for
a.
c,
and
D :
A1 =age at the start of the growth period, A2 =age
at the end of the growth period, and {3 =parameter
to be estimated. The sign of {3 can be interpreted as
the future contribution of individual trees in the
projected stand. If {3 is positive, the relative contri­
bution to basal area in the future stand will increase
for trees larger than the average relative size and will
decrease for trees smaller than the average relative
(5)
where
T1 = T ( l
+
ljc).T" = T(l +
2/c).
T= the
gamma function, and other terms as previously de­
tined. This procedure ensures that
D4 in the pre­
dicted diameter distribution is the same as the D4
implied by prediction equations.
size. Conversely. if [3 is negative. then the relative
contribution will decrease for larger trees and in­
crease for smaller trees.
Inspection of the parameters of the relative size
projection function obtained at each planting density
suggested a quadratic relationship. with the greatest
parameter estimate at the 71 T2 planting density.
However. only a small improvement in explained
variation
2.4.
Stand-table-projection approach
was achieved by including a
ing projected diameter distributions were not as ac­
This approach produces a future stand table that is
consistent
with
tree-
(Pienaar and Harrison.
and
1988).
stand-level
functions
Therefore, a tree list
and stand information are needed. Components of
the stand-table-projection approach include a relative
size projection function. which contributes the tree­
level information, and the survival function and a
basal area projection function, which contribute to
the stand-level information. According to Borders
and Patterson
(1990),
the advantages of the stand-ta­
ble-projection method are as follows: the functional
form of the diameter distribution (e.g., Weibull) does
not have to be assumed: multimodal distributions can
be reproduced; and information on initial stand struc­
ture can be use�i.
The tree-level function of Pienaar and Harrison
( 1988)
( < 0.1 C:C)
quadratic function of planting density. and the result­
is based on relative tree size, defined as the
ratio of individual-tree basal area to average basal
curate as those obt:llned by using the overall parame­
ter estimate.
The second part of the stand-table-projection sys­
tem developed by Pienaar and Harrison
(1988)
is the
projected mean size obtained from stand-level basal
area and survival functions. The stand-basal area­
projection function (Eq. (l h)) is an implied growth
function that is functionally compatible with the
basal-area-prediction function (Eq.
( 1 a)).
The future diameter of individual trees can be
estimated from the product of projected relative basal
area and projected mean basal area per tree. which is
the quotient of the projected basal are
divided by
the number of surviving trees. The relative size
projection function for individual trees can be con­
strained to ensure consistency of the future stand
table with stand-level basal area and survival func­
tions.
area per tree in the stand, and changes in relative size
over time.
(6 ) (7) 54
S.A.
Knowe eta/./ Forest Ecology and Management 98 ( 1997) -/.9-60
where b2i =projected basal area (m2 /ha) of trees in
th dbh class, BA2 =projected basal area (m2 jha) of
the stand, ni =number of trees in ith dbh class, and
k = number of dbh classes. Stand mortality is allo­
cated to dbh classes, on the assumption that smaller
trees in a stand have a greater probability of dying
than do larger trees in the stand. Borders and Patter­
son (1990) found this stand-table-projection proce­
dure to be superior to a Weibull-based diameter-dis­
tribution-projection system and a percentile-based
projection system with an empirically defined distri­
bution. Variants of this approach have been used to
predict the effects of interspecific competition in
young Douglas-fir plantations (Knowe, 1994; Knowe
and Stein, 1995) and planting density in red alder
plantations (Knowe and Hibbs, 1996).
2.5. Individual-tree-growth approach
The distance-independent, individual-tree-diame­
ter growth function is similar to those used in
ORGAl ON (Hester et al., 1989) and PROGNOSIS
(Wykoff, 1990). A tree list is required to make
growth predictions for fixed (annual) growth inter­
vals. The function is applicable to stands that have
been thinned or to stands with multiple canopies.
(8)
where Ll Di =annual dbh growth (em) of j!11 tree,
Di =current dbh (em) of ith tree, BAL =current
basal area (m2 jha) of trees with larger dbh than the
ith tree, BA =current basal area (m2 jha) of stand,
and SI = site index (m).
The potential annual diameter growth of individ­
ual trees is predicted as a function of current diame­
ter. Smaller trees in a stand have less growth than the
larger trees, and trees in stands with low basal area
exhibit greater growth than trees in stands with
higher basal area. Stand mortality is allocated to dbh
classes, on the assumption that the smaller trees in a
stand have a greater probability of dying than do the
larger trees.
2.6. Evaluation of methods
The two-sample Kolmogorov-Smirnoff test
(Sakal and Rohlf, 1981) was used to determine
whether the obserVed and predicted or projected
diameter distributions are samples from the same
population. This test does not validate the prediction
system, but verifies that the observed and predicted
stand tables are similar for young red alder planta­
tions. The hypothesis of no difference between ob­
served and predicted diameter distributions was tested
by using a=0.05.
Beginning with a plantation of age 8, the ob­
served density, dominant height, and age were used
to predict the diameter distribution at each measure­
ment. The cumulative diameter distribution was ob­
tained directly from the Weibull distribution func­
tion. Beginning with a plantation of age 7, in the
stand-table-projection method, density, dominant
height. and age at the beginning of each 1- or 2-year
growth interval were used to predict the number of
trees and stand basal area at the end of the interval.
In the individual-tree-growth prediction method, dbh,
BAL, and stand basal area at the beginning of each
growth interval were used to predict growth, which
was added to current dbh. In both the stand-table­
projection method and the individual-tree-growth
prediction method, stand mortality was allocated pro­
portional to the inverse of relative diameter, and was
then used to compute the cumulative diameter distri­
bution at the end of each growth interval.
This procedure was also used to compare ob­
served and predicted diameter distributions for 5-year
growth intervals (ages 7-12 and 11-16 years) ob­
tained by using the stand-table-projection and indi­
vidual-tree growth methods. Because of difficulties
in allocating 5-year mortality in the stand-table-pro­
jection method, both projection methods were evalu­
ated by estimating annual growth for 5 consecutive
years. At the end of each 1-year. growth cycle, tree­
and stand-level vari bles were updated prior to per­
forming the next growth cycle. Visual inspections of
observed and predicted diameter distributions were
made for selected plots at each planting density.
S.A.
Knowe et at./ Forest Ecology and Management 98 ( 1997) 49-60
55
Table 3
Parameter estimates and standard errors for tree-level projection
equations for red alder plantations near Apiary, Oregon
3. Results
3.1. Auxiliary functions
The survival function (Eq. (lf)) accounted for
97.6% of the variation in observed N2 and the root
206.43 treesjha.
mean square error (RMSE) was
Survival trends depicted by this function indicate
that little mortality occurs over time in the 976
planting density. As planting density increases, how­
ever, mortality increases so that the
6944 and the
13 889 planting densities have about the same num­
ber of surviving treesjha at age 16. The height-age
function accounted for 97.1% of the variation in
observed H2 and the RMSE was 0.43 m.
3.2. Diameter-distribution-prediction approach
Using the parameter estimates in Table 1. the
Equation a
Parameter
Estimate
Standard error
Eqs. (6) and (7)
Eq. (8)
Eq. (8)
Eq. (8)
Eq. (8)
Eq. (8)
{3
-0.306601
0.298965
0.577319
-0.002186
-0.009640
-0.330934
0.021452
0.139387
0.067428
0.000258
0.0003 35
0.008471
a
s0
a,
<52
83
0
Corresponds to equation number used in text.
height growth (Eq. (lg)) indicate that stand basal
area begins to converge by age 15 for planting
3086 treesjha but still di­
verges for lower planting densities. The 6944 and the
13 889 planting densities have about the same basal
densities greater than
area at age 1 6. which retlects the combined effects of
basal area prediction function accounted for 95.5%
mortality and reduced-diameter growth at the highest
of the variation in observed basal area and the
planting density.
RMSE was 1 .2 1 m2'jha. Trends in basal area over
The red alder spacing study data indicated that
time (Eqs. (la) and (l h)) obtained by using the
age
surviving number of trees (Eq. (lf)) and dominant
diameter percentile. D0• The percentile prediction
was
important in predicting only the minimum
equations (Eqs. (!b). (lc), (1d) and (1e)) accounted
82.6 to 97.2% of the variation in observed per­
centiles and RMSE ranged from 0.42 em for D'25 to
0.86 em for D95. Increased planting density not only
for
Table 2
Parameter estimates and standard errors for system of stand-level
prediction and projection equations for red alder plantations near
Apiary, Oregon
Equation'
Parameter
Estimate
Standard error
Eq. ( Ia)
Eqs. (1 a) and(l h)
Eqs. (l a) and(Ih)
Eqs. (Ia) and ( lh)
Eq. (lb)
Eq. (!b)
Eq. (!b)
Eq. ( l c)
Eq. (lc),
Eq. (l d)
Eq.(ld)
Eq. (l e)
Eq. (!e)
Eq. (lf)
Eq. (lf)
Eq. (lf)
Eq. (l g)
Eq. (lg)
ao
a,
a2
aJ
0.053966
0.325035
0.903040
0.412420
-4.324698
0.285263
0.555794
-1.449393
0.917854
-0.518221
l . O l l046
1.940934
1.175086
-2.028169
7.282x w-6
2.257404
0;185226
2.574843
0.01019
0.01494
0.09392
0.07225
0.49295
0.06029
0.06234
0.20248
0.02065
0.16503
0.01653
0.36726
0.03657
0.51245
2.138X to-s
0.71305
0.01749
0.31157
,\10
A,,
,\12
A;o
,\21
AJo
,\3/
,\40
,\41
a,
82
1)3
{3,
{3
' Corresponds to equation number used in text.
reduced the mean diameter, but also increased the
degree of positive skewing of simulated diameter
distributions, and increased the coefficient of varia­
tion in diameters.
3.3. Stand-table-projection approach
The relative basal area projection functio n was
based on
4183 observations and accounted for 96.4%
of the variation in observed future relative size: the
R.\1SE was
0.10 m1jm1. Parameter estimates and
3.
standard errors are shown in Table
The second component of the stand-table-projec­
tion approach is the basal area projection function
(Eq. (1h)), which was based on
accounted for
62 observations: it
92A% of the variation in projected
1.46 m2jha.
stand basal area and the RMSE was
This projection function depicts similar trajectories
in basal area over time as the compatible prediction
function (Eq. (la)).
S.A.
56
Knowe eta/./ Forest Ecology and J\tlanagement 98 ( 1997) .J9-60
3.4. Individual-tree-growth approach
ing density at age 1 6, were significantly different.
For the individual-tree-growth approach,
The diameter growth function was based on
observations and accounted for
59.8%
of the varia­
tion in annual diameter growth: the RMSE was
the ob­
served and projected diameter distributions for one
3698
plot
(1.6%
of the total), corresponding to the
1600
treejha planting density at age 1 6 only, were signifi­
0.25
em. Site index (SI) was not included in the final
cantly different.
model because a negative coefficient was estimated
Visual inspections of diameter distributions for l ­
l)
in a preliminary model. Logically, diameter growth
or 2-year growth intervals (Fig.
should increase rather than decrease with increasing
diameter-distribution-prediction approach tended to
revealed that the
site productivity. The negative coefficient may have
over-predict dbh for larger trees at all ages in stands
been an artifact of the restricted range in SI and was
with the
possibly confounded with planting density as a con­
predict dbh for larger trees in stands with the
sequence of using data from only one location. Po­
planting density (Fig. lD), and to over-predict dbh
976
planting density (Fig. lA), to under­
6944
tential growth depicted by the diameter growth equa­
for smaller trees in stands with the 1 3 889 planting
tion indicates that the maximum annual growth of
density (Fig.
IE).
In the
976
planting density (Fig.
4.3 em/year is attained for trees with 1 1 -cm dbh.
1 A), the stand-table-projection approach tended to
Only
under-predict dbh for smaller trees at younger ages.
50%
of the potential diameter growth is ex­
pressed for trees with
7
to
11
trees in stands with more than
m:! /ha BAL or for
5
However, in the
m2jha basal area.
(Fig. ID and
6944
E).
over-predicted at age
3.5. Evaluation of methods
Observed
were
by
planting densities
13 889
16.
The individual-tree-growth
approach provided the best representation of ob­
served diameter distributions at all planting densities
and predicted diameter distributions
compared
and
diameters for smaller trees were
using
two-sample
and at all ages (Fig. 1 ).
Kol­
For the 5-year growth intervals. the predicted and
For the l - and
observed diameter distributions for three plots ( 17.6%
2-year growth intervals, the predicted and observed
of the total) obtained by using the stand-table-projec­
mogorov-Smirnoff test
the
(a = 0.05).
diameter distributions obtained by using the diame­
tion method were significantly different (Table 4).
ter-distribution-prediction approach for three plots
The rejected plots included one for the
(4.8%
of total) were significantly different (Table
These correspond to the
density at ages 1 4 ( 1 plot)
13 899
and 16
4).
density in projections from ages
treejha planting
one each for the
(2 plots). For the
4360
7
and 1 3
in projections from ages 1 1
976
planting
to 1 2 years and
planting densities
889
to 16
years. Significant
stand-table-projection approach. the observed and
differences between the predicted and observed di­
projected diameter distributions for one plot ( 1.6% of
ameter distributions were detected for only one plot
the total), corresponding to the
13 899 treejha plant­
(5.9%
of the total) obtained by using the individual-
Table+
Number of occurrences with a significant difference between the observed and predicted diameter distribution,
·
two-sample Kolmogorov-Smirnoti test
as
determined by the
( a= 0.05), for the diameter-distribution-prediction method, the stand-table-projection method, and
the individual-tree-growth method developed for red alder plantations near Apiary, Oregon
Number of rejected distributions (percent of rejected distributions
Method
1- or
Diameter-distribution-prediction
Stand-table-projection
Individual-tree-growth
•
Percentages
are
based on
2-year interval
3(+.8%)
1(1.6%)
I( 1.6%)
(n
=
62)
5-year interval
•)
(n = 17)
3(.17.6%)
1(5.9%)
62 observations with 1- or 2-year growth intervals and 17 observations with 5-year growth intervals.
S.A. Knowe era{. j Foresr Ecology and Management 98 ( 1997) -19-60
Visual inspections of predicted and observed di
tree growth method. This plot corresponded to the
1600 planting
16 years. density and projected from ages
11-12
9- 0
7-6
1.0
0.9
--- Observed
0.8
--- We1bull prediction
0.7
--- Stand Iailie
,.,
Stand Iallie c
1l
0
OrQJectfOO "
cr 0.6
------ lndiVIdual·tree growtn
"
g 0.4
3
(.)
"
E
"
0.3
0
0.2
4
2
6
10
I4
12
18
16
dbh (em} Projection
B. 0.8
--- Wetbud prediction
--- Stano :aOla :0 cr
"
0.6
------ lndivldual·tree "
0.5 "
;
'3
"
:;
(.)
.orojectlon
growth
0.4
0.3
0
>nterval 14-16
1.0,
t
0.91
growtn .
;
:;
0.3
(.)
0.2
--
14
16
18
20
22
24
16
Pro1ectton
9·10
18
20
interval
11-12
4-16
Observed
Wetbuil predlctton
Stand table
pTOJ8CtiOO
growth
0.4
0.3
0.2
0.1
12
14
------ !nlj1v1dual·lree
0.0 +-......
0
2
10
12
7·8
0.0
4
10
0.
0.1
2
a
6
dbh (em)
0.81l ---
" 0.5
0.5
'I>
4
2
j
,..
"
0.7 .... ___
c
D
::J
cr
pro1ection
0.4
0
14·16
------ lndivtdual·tree
0.5
0.0
22
---Observed 0.7
,..
20
11·12
9-10
7·8
c
<.J
interval
9·10 11-12
0.1
0.0 0
0.9
Projection
0.2
0.1 1.0 stand table projection
7-8
---Observed
0.5 the
c.
--- Weibull prediction cr 0.6
;::
that
0.8
,., --"
0.7
"'
"
3
indicated
0.9
c
2)
interval Projection
"'
>
ameter distributions for the 5 year growth intervals
(Fig.
A.
1.0 to
11
57
;._.:::;;,.:Je!...,.;::..Z.
:
6
a
--,..--10
12
------,
14
:6
18
dbh (em)
abh (em)
Projection
.01
0.9
0.8
]
--
Observed
---
Weibutl prediction
"jJ ---
interval
11-12
14·16
Stand Iallie
iJ'
0. 7
5-
0.61------ lndivtdual·
;
9·10
7·8
E.
P!Oiecttor\
i ::l' -···::
o.3
(.)
02
.
0.1
1
v' ·
f·'
0.0 +-��o:!!!:::,.....::::-.,--.:;...r--_..,�--,�-----'r�--,
6
0
s
10
12
14
16
18
20
abh (em)
Fig. 1. Comparison of observed and predicted diameter distributions obtained by
a
diameter-distribution-prediction method, a stand-table­
·(A) 976 planting density: (B) 1600 planting
(C) 3086 planting densil)'; (Dl 6944 planting density: :1nd (E) 13889 planting density. Note differences in X-uxis and projection
projection method. and an individual-tree-growth method and 1- or 2-year growth intervals:
density:
intervals.
S.A.
58
Knowe et a/./ Forest Ecology and Management 98 ( /997) 49-60
tree-growth method tended to over-predict dbh for
smaller trees in young stands and at high planting
densities, and to under-predict dbh for smaller trees
in older stands and at low or high planting densities.
method tended to under-predict dbh for smaller trees
in young stands and at low planting densities, and to
over-predict dbh for larger trees in older stands and
at intermediate planting densities. The individual-
Projection
A. 1. 0 0.9
--- ObseiVed 0.8
--- Stand table protection c: 0.7
"
------
}'
(I'
lndtvldual·tree grow<h
"
:J E
0.4
.<:
(.)
:J
0'
0.6
"
0.5
:;
E
0.4
)/
/)
I,J
:J
!I
j.,
0.3
(.)
:J
/l
0.2
0.2
_;
I
./_../
0.1
___
0. 0 0
2
4
6
j..../ ,...1
8
10
12
0.1
0.0
1 4
16
dbh (em)
18
20
Projection
B.
7-12
22
>.
0
"
>
· §
(.)
0.9
---Observed
--- Stand lal>le PIOiectJon
:;
0.7
-
Vi
'0
"
0.6
:g
r:r 0.6
0.5
0.4
16
18
Projection
20
interval
t 1-16
- - - Individual-tree growth
--
0.5
-
0.4
E
0.3
;;
:;
0.3
14
7-12
0.8
c:
-
:J 12
o.
1. 0
Stand table projection 10
8
6
interval
11-16
0.7
;;
4
dbh (eml
--- CbseiVed 0.8
2
0
1. 0 0.9
Individual-tree growth
>
,I I
0.3 0.8
0.7
c:
d
j}
--- Obs!!IVed
--- Stand table projedion
interval
11·16
7-12
0.9
"
"
>.
•
r:r 0.6
0.5
c.
1. 0
LJ
d
:J Projection
interval 7-12 (.)
:J
0.2 0.2
0.1 0.1
0. 0 0
0. 0
2
4
6
10
12
14
16
18
20
22
24
;'
0
4
2
6
8
10
12
14
16
18
dbh (em)
dbh (em)
Projection
E.
1. 0
7·12
0.9
--
Obs8M>d
0.8
-- Stand table pcojection
51
0.7
------ lndlviduaHree growth
r:r
0.6
>.
:;
:J
"
.<:
;;
:;
E
(.)
:J
interval
11·16
0.5
0.4
0.3
0.2
0.1
0. 0
0
2
4
6
a
10
12
14
16
18
20
dbh (em)
Fig. 2. Comparison of observed and predicted diameter distributions obtained by a diameter-distribution-prediction method, a stand-table­
projection method. and an individual-tree-growth method and 5-year growth intervals: (A) 976 planting densicy: (B) 1600 planting density:
(C) 3086 planting density; (D) 6944 planting density; and (E) 13 889 planting density. Note differences in X-axis.
S.A.
Knowe et al. /Forest Ecology and Management 98 (}997) 49-60
4. Discussion and conclusions
A
yield system for red alder was not previously
available because of lack of long-tenn data for vari­
ous planting densities. The predicted stand tables in
all three modeling methods depicted increases in
both mortality and reductions in tree diameters as
planting density increased. Because limited data were
available for fitting the equations, the system should
be viewed as a framework for improvement. The
diameter-distribution-prediction approach tends to
over-predict dbh for larger trees in stands planted at
low density and to under-predict dbh for smaller
trees planted at high density. Adding an expression
of density to the minimum diameter percentile ( D0)
and 95th percentile (D95) may improve the accuracy
of the diameter-distribution-prediction approach.
However, this approach does give a reasonable rep­
resentation of the smoothed diameter distribution.
and may be useful for comparing planting densities
when a tree list is not available. The stand-table-pro­
jection approach tends to under-predict dbh for
smaller trees in young stands planted at low density
and to over-predict dbh for smaller trees in older
stands planted at high density. This approach pro­
vides for consistency between stand- and tree-level
growth projections, and should be useful for compar­
ing planting densities when a tree list is available.
The individual-tree-growth approach provides the
best representation of observed diameter distribu­
tions at all planting densities. This approach may be
best suited for stands that have been thinned, stands
with mixtures of species, or stands with various size
classes. It should be noted that stand-level models
have been developed to incorporate the effects of
thinning (Pienaar and Shiver, 1986), site preparation
(Knowe and Stein, 1995; Pienaar and Rheney, 1995),
release from competition (Knowe et al., 1992;
Knowe, 1994), and clonal mixtures (Knowe et al.,
1994).
Our results were consistent with those reported
for similar studies in loblolly pine (Pinus taeda L. )
and young Douglas-fir (Pseudotsuga menziesii). In
comparisons of modeling approaches, Borders and
Patterson ( 1990) and Kno we (1994) obtained a better
representation of diameter distributions when using
the stand-table-projection approach than when using
diameter-distribution-projection and prediction meth­
59
ods. Knowe and Stein (1995) reported only a slight
improvement in representation of diameter distribu­
tions obtained by using the stand-table-projection
approach as compared with the diameter-distribu­
tion-prediction method.
As in the current study, Borders et a!. (1988)
noted decreasing precision with increasing growth
interval for stand-level projection models similar to
the functions developed for red alder plantations.
Also, Knowe (1994) reported similar decreases in
precision for a stand-table-projection method devel­
oped for young Douglas-fir plantations with interspe­
cific competition. However, Knowe and Stein ( 1995)
did not observe appreciable differences in precision
for 1-, 2-, and 3-year growth intervals in young
Douglas-fir. Knowe (1994) and Knowe and Stein
(1995) reported that more significant differences be­
tween predicted and observed diameter distributions
were detected at younger ages than at older ages
when the diameter distribution prediction method
was compared with the stand-table projection ap­
proach. In the present study, we found more signifi­
cant differences in distributions at older ages and at
extremes in planting density.
Knowe and Hibbs ( 1996) observed consistent re­
lationships between parameters of growth projection
models and planting density of young red alder in
Neider plots. We did not observe similar relation­
ships in the present study. The narrower range of
planting densities in the present study (976-13 889
trees/ha vs. 238-10 118 treesjha in the Neider plots)
may have obscured consistent relationships. Also,
the nonindependence of observations in Neider plots
may impart consistent and systematic variation in
parameters of growth models developed by using
data from Neider plots.
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.
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