GROWTH MODELS
The basic form of the growth models used was developed in earlier papers
by Wensel and Koehler (1985) and Wensel, Meerschaert, and Biging (1987).
Conceptually, they express tree growth as a product of two factors, the
first reflecting the potential of the tree on the site and the second
reflecting the inability of the tree to reach its potential growth rate.
The first factor is intended to reflect the physiological capacity of the
tree while the second factor is intended to reflect the competition on the
site.
For tree height, the basic form of the prediction for the 5-year
change in tree height, chgHt, is
chgHt = PH x CH [1]
where CH is the competition factor which ranges between 0 and 1 and PH
is the potential growth of the tree. The equation for potential tree height
growth is derived from the site index equation (Biging 1985; Wensel,
Meerschaert, and Biging 1987)
PH = [c0 * S * c1 + c2 * H * c3]/c3 - H [2]
where S is the site index and H is the total height of the tree. Noting
that trees with insufficient crown cannot reach this potential regardless of
the competition and, conversely, that trees with very large crowns may grow
more than the average potential, the potential growth is adjusted for live
crown ratio, LCR, as follows:
PH' = PH * (d1)/(1+exp (d0 - d2 * LCR)) [3]
Thereafter, PH' is substituted for PH in equation [1]. (A further adjustment
on the site index that was used in previous versions was dropped here.)
The height growth competition component, CH, is given by
CH = exp (d3 * CC66 * d4 * PBA * d5) [4]
where CC66 is the crown closure of the plot at 66% of the subject tree's height
and PBA is the percent of the basal area of the plot composed of that tree's
species. In order to compute CC66 one must model the crown shape. The procedures
and coefficients used here are developed by Biging and Wensel (1989) and reported
by Wensel, Meerschaert, and Biging (1987). The intuitive value of CC66 as a
measure of competition is based upon the presumption that the crown density at
two-thirds of a tree's height is a strong factor in the growth rate of the tree.
Thus, root competition from other trees and/or shrubs is not included and may be
a source of error in estimating competition.
The combined prediction equation is given as follows:
chgHt ={[c0*S*c1+c2*H*c3]/c3 -H}{d1/(1+exp(d0- d2*LCR))}{exp(d3*CC66*d4*PBA*d5)} [5]
The models used for estimating diameter growth are identical to those shown above for
tree height except that DBH2 is substituted for H in the above equations. This gives the
combined equation for the 5-year change (() in tree DBH2 as
chgDBH2 ={[c0*S*c1+c2*DBH2*c3]/c3-DBH2}{d1/(1+exp(d0-d2*LCR))}{exp(d3*CC66*d4*PBA*d5)} [6]
where the coefficients are computed separately for tree diameter and height growth for each species.
With 10 parameters to estimate in each equation, not all of which are independent, it is
clear that the prediction equations are over-parameterized in the usual sense. That is, if we were
to fit the entire equation to any of our data sets it clearly would find many of the coefficients to
be redundant and therefore not significant. However, each of the coefficients must be fitted in order
to maintain the structure of the model. Without this structure, the predicted growth rates that follow
simulated partial harvesting or thinning would misrepresent the response of the trees.
This problem is solved by fitting the model in stages. First, the coefficients for the potential
growth, the ci's in equations 5 and 6, are estimated while fixing the competition coefficients, the di's,
at the values reported previously. Since we are interested in fitting the potential growth of the trees
(equation 2) in the absence of competition, we select a subset of the trees where there is little or no
crown competition and with sufficient crown. Second, the crown adjustments (equation 3) are fitted to
the entire data set to adjust the potential down for trees with small crowns and up for trees with very
large crowns. Finally, the competition coefficients (equation 4) are computed from the entire data set
with the potential coefficients fixed.
Under-parameterizing a model could have some serious side effects. The overall model could fit
but the components could be confounded so that the model would not accurately predict the differences due
to changes in competition. In that case, using CACTOS to evaluate the effects of alternative thinning
trials could lead to misleading conclusions about the desirability of the cultural practices evaluated.
ANALYSIS
Previous estimates of the coefficients for estimating height and DBH growth are given for the STEM data
by Wensel and Koehler (1985) and for the combined STEM and PERM0 data by Wensel, Meerschaert, and Biging
(1987) . As shown in Figure 1, these estimates produced underestimates of the observed growth rates or
the observed difference between the two measurements of the permanent plots. This is in general agreement
with other reports received by CACTOS users.
The new coefficients, developed by nonlinear regression using the previous values as starting
points, are shown in Tables 3 and 4 for height and DBH2 growth, respectively. Only the data from regions 1, 2,
and 3 were used in this analysis as the model was unstable for region 41. The growth relationships for region
4 will be considered in a separate study with an expanded data base.
The growth estimates derived from these revised coefficients appear to be unbiased but with more
variation than that encountered when using the coefficients from the backdated initial measurement data. The
new model statistics are summarized in Appendix tables A1 and A2. Regional adjustments for the growth rates
computed using the revised coefficients are given in Tables 5 and 6 for height and DBH2 growth, respectively.
Table 3. Revised Coefficients for Height Growth Model
Coefficient PP SP IC DF WF RF
c0 0.30475 0.30475 0.20432 0.27076 0.27443 0.27443
c1 0.28170 0.28170 0.48943 0.30046 0.31810 0.31810
c2 0.94786 0.94786 0.88692 0.94904 0.94757 0.94757
c3 0.54992 0.54992 0.54992 0.54992 0.54992 0.54992
d0 0.76370 0.98431 4.0 1.53401 1.53592 4.0
d1 3.28585 2.26250 1.0 1.0 1.0 1.0
d2 1.27950 3.20291 20.000 9.75154 8.20475 20.000
d3 -0.55080-0.59041-0.70324 0.0 -0.25402-0.56146
d4 0.10562 0.18682 0.16483 1.0 1.20104 1.56660
d5 0.03335 0.0 0.0 0.0 0.0 0.0
Table 4. Revised Coefficients for Diameter-squared Growth Model.
Coefficient PP SP IC DF WF RF
c0 0.0522513 0.04808 0.04831 0.07181 0.22682 0.21689
c1 0.0300 0.08063 0.0300 0.07856 0.21230 0.21230
c2 0.95 0.95 0.95 0.95 0.95 0.95
c3 0.02027 -0.04024 0.01027 0.07793 0.27990 0.27990
d0 1.89658 1.49303 1.83549 2.96988 1.37157 2.60000
d1 3.48315 2.79942 1.54834 11.87146 1.13672 1.09156
d2 1.71132 1.37713 4.04275 1.01335 6.33080 8.50000
d3 -0.89686 -0.405455 -0.6093 -0.5770 -1.3907 -1.56166
d4 0.60626 0.875528 0.4112 0.7961 1.0394 1.73935
d5 1.05966 0.0000 0.0000 0.0000 0.0000 0.66029
Table 5. Regional proportional adjustments for revised height growth
estimates (with numbers of trees in PERM1 for each shown in Table 6).
Region no. PP SP IC DF WF RF
1 1.16 1.13 1.28 1.09 1.19 0.71
2 .92 .92 .64 .91 1.01 1.06
3 .87 .83 .50 1.09 0.92 1.20
4 .86 .73 .68 1.05 0.84 .82
Table 6. Regional proportional adjustments for revised diameter growth
estimates (with numbers of trees in PERM1 for each).
Region no. PP SP IC DF WF RF*
1 1.04 1.02 1.05 1.06 1.01 0.99
(375) (125) (448) (96) (530) (97)
2 1.03 1.08 0.95 1.08 0.99 1.11
(724) (170) (335) (186) (785) (141)
3 0.78 1.04 0.75 0.90 0.97 0.87
(162) (52) (37) (303) (149) (33)
4 0.90 0.87 0.71 0.80 0.63 0.61
(66) (21) (16) (58) (68) (6)
* The data for red fir was not split due to the small number of
observations.
DISCUSSION
It is not particularly surprising that the previous coefficients
lead to growth estimates that differ from those observed by remeasuring
the permanent plots. Differences can be expected because of the different
data acquisition procedures used, differences in the analysis procedures,
and differences in the growth periods studied. However, the magnitudes of
the differences are unexpected. Certainly they suggest that it will take
another measurement of the permanent plots to more accurately estimate
long-term average tree growth rates. More detailed discussion of the results
follow.
Height growth
The initial models of height growth and the crown models are based
upon the STEM data set as no height growth data were available from the
initial measurement of the permanent plots. The initial estimates were
reported by Wensel and Koehler (1985) and revised from a re-analysis with
revised crown models by Wensel, Meerschaert, and Biging (1987). However,
PERM1 represents the first opportunity to test the previous height growth model.
Even with a cursory analysis it was clear that the previous model was
significantly under-estimating total height growth. As a result, new model
coefficients were estimated for each of the 6 species using the previous
coefficients as starting points in the nonlinear analysis. The new model provides
an unbiased estimate of height growth based upon the measured height growth of a
larger and more widely distributed sample of trees.
Diameter growth
The diameter-squared estimates from the PERM0 data set were based upon a
"back dating" of the current tree sizes on the basis of DBH increments from
increment cores (Wensel and Koehler, 1985). Failure of the earlier coefficients to
fit could be a result actual differences in the growth rates for the periods or the
way in which the data were developed. For PERM1 , no backdating was necessary
because the trees were actually measured at the beginning and end of the growth period.
The PERM1 data for all 4 regions were used to compute the regional proportional
adjustments given above. The small sample sizes in some regions suggest that these
adjustments must be used with care. In fact, we suggest that serious users of CACTOS
develop adjustments for their own properties based upon a comparison of actual and
predicted growth rates for each species.
IN CONCLUSION
Sampling error is inherent in any forest sampling procedure. These errors are
exacerbated by the relatively short remeasurement periods (nominally 5 years but varying
from 4 to 6 years). Also, we have two different techniques for estimating tree grow for
the two different growth periods. It will take further study to determine which is the
better indicator of long-term average growth rates.
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