A Study of Models for Height Growth - Kyushu University Library

A Study of Models for Height Growth
and Yuui
The objective of this study was to determine a function which could be applied to
tne estimation and prediction of height growth of Oyj)tomeria jajJonica, by comparing
the Mitcherlich, Gompertz, Logistics and modified Weibull functions which are commonly applied in stand growth analysis. An attempt was also made to determine an
additional function which could be used as a model for the changes occuring in the
variation of height with changes in time, and to construct models which could be
applied for estimation of the parameters of the most suitable among the four functions
The use of height growth data corresponding to age intervals from 5 to 53 years
resulted in the smallest value of residual mean square (R. M. S.), a measure of how well
the data fit to a function, is given by the Gompertz function and successively followed
by the modified Weibull, Logistics and Mitcherlich functions. The smallest variation
of estimated and predicted values for height growth with change in observation
numbers and their corresponding age intervals was given by the Gompertz function.
The analysis showed that the standard deviation of height (Sd) could be represented as a function of age (A), the relationship being written as Sd:::: IOn A).
The estimated growth curves for various site indices implied that the Mitcherlich
function results in overprediction of future growth, while conversely the Logistics
function results in underprediction. On the other hand, the Gompertz and modified
Weibull functions result in almost the same predicted values of future growth and are
thus more acceptable than the Mitcherlich and Logistics functions.
Further analysis regarding the models for the parameters of height growth based
on the Gompertz function revealed that each parameter has a close relationship to site
index. In other words, the parameters of the function could be estimated on the basis
of site index.
The height of a tree or stand is a factor which is not radically affected by
management activities. Because of this, height growth models can be used in basic
studies of tree or stand growth models. In other words, before carrying out a study
involving a stand growth model, the first step is finding a means of expressing the
change in tree height with change in time. A growth model for another tree parameters (such as diameter, crown dimension etc.) can then be contsructed on the basis
of the height growth model bp adding new stand variables such as the density and the
variation of the height itself. Consequei~tly,the applicability of a stand growth model
is considerably depeildent on the accuracy of fit of the functioil applied in height
growth modelling.
In this paper, ail attempt was 111ade to compare the accuracy of fit of variotls
functions, which have recently been applied in stand growth modelling, to the height
growth. The model for each parameter of the best function was also constructed.
General Review of Stand Growth Models
Models for growth of stand variables' mean
The growth of a stand variable is a function of time and increment, or the rate of
change of a forest state inay be represeiltecl as a function of time or a fuiictioi? of the
amount of a present state, or a coinbination of both. In general, the rate of change of
a system (increment) may form one of the foilowing digerential equations (LEAIIY,
1970) :
dY/cl/ = f ( 1 ) ,
dY/dl = g ( Y ) ,
dIf/d/ = h ( Y , t ) ,
wlnere d Y/dl is the rate of change and Y is the amouilt of a state at a given time (i).
Some of the most conlmonly used forms of g in ecl. 2, and the names associated
with them, arc :
; Mitchcrlich
; Gom~ertz
g = i7YZ+DY ,
; Logistics
g = aYc-4-61" ,
; Richards
where IT is the amouilt of a present state, A is an asyinlstotic value or maximum size
of Y,and a, h and c are parameters.
One of the forms of eq. 3 which has recently bee11 applied in stand growl11
et al., 1983) is :
/ = Bclc "(A- Y ) , ; moclilied llreibull
where I' is the amount of a state at a given time ( l ) , A is a11 asymptotic value of Y,
and B and c are parameters.
Integration of the differential equations above produces growth models which can
1981) (YAMAMO'L.~
et al., 1982) (YAKG
be presented ill the fo1lo.cving equations (M,~crl~zoo,
Y = Al(1-i-Bexp(--kt)} ,
Ii = R {l- B exp( - h/ ) } " ( 1 - 7 ' t )
Y = A { l - - c x ~ ~ ( - - B t ". ) }
; Logistics
; Richards
; rnodificd Weibull
(1968) put forward a hypothesis formulated for growth rate or increment
as follows :
l. Increment is proportional to the difference between the maximum size and the
attained size of Y itself. This version is based on the principle of the so-called "law
of effect of growth factors" by Alfred E. Mitcherlich.
2. Increment is proportional to the attained size ancl to future growth Ym,,,(Autocatalytic curve).
3. Ii~crementis logarithmic clirni~~ishing
at a coilstant rate (Gompertz).
Meantvhile, PII'NAR
et al. (1972) clarified that the Richards model (also called the
V011 Bertalanffy model) is Isased on a hypothesis which expressed that the rate of
volurne growtli of an organisln is difference between anabolic rate (constructive
metabolisn-i) ancl catabolic rate (destructive metabolisn-i). The anabolic rate is proportional to the surface area of the organism, ~vhilethe catabolic rate is proportional to
the value of the biornass.
With regard to the i~iodifiedMreibull function, it is actually an expansion of the
Weibull distributiorz function which is origil~allywritten as :
p = 1- e - ( x
where F is the probability of rnaterial failure, X is the size of a given material, and
o and A are paran~eters.
The function was developed by TVeibuIl in a study of the probability of material
failtire (WEIBULI*,
1951 in YANGet al., 1978). 170s the purpose of stancl growth modelling,
random variables X and F in eq. 14 are substituted by age and sland growth, cvhich
increases wit11 age, respectively. The value of 1; in eq. 19 expresses the probability
value betcveen 0 ancl 1. The actual value of growth can be obtained by multiplying an
expanding factor A into eq. 14, so that the growth function call then be written a s :
where 2' is the stand growtl~at a given time (l), R is an asyrnptotic value of stand
growth, and o and A are parameters.
By substituting (110) and A wit11 B and c, respectively, the function as presented
in eq. 13 is ohtainecl, which states that the growth rate or increl~-ientof an organisn~is
proportional to the attained age and to the future growth or to the difference between
the inaxilllum size A and the size Y itself.
(1980) has applied the Chapinan-Richards modificatioil ill the inodelliilg
of Pi~~zts
faedu plantation and concluded that the inodel is a very flexible fuilctioil for
fitting basal area growth. Various treatinents of the Richards function has also been
given by OSIJ~II
ef 01, (1983), who stated that the form of the Richards function curve
is determined by the value of parameter m ill eq. 12. If the value of m is equal to 0,
the fiui~ctionrepresents the Mitcherlich fuilction (eq. 9). If the value of paranleter m
is eclual to 2, on the other hand, the function represents the Logistics function (eq. 11)
and represents the Gompertz fu~lctioilif the value of nz approaches 1.
el (11. (1983),described that the form of the modified Weibull fuilctioil
is similar to the Mitcherlich function if the value of paranleter c in ec1. 13 is equal to
l . By integration of eq. 8, it can be seer1 that tile il~aximuinvalue of the g r o ~ t hrate
is attained at
and that therefore, the function has no inflection point if parameter c is less tllan 1.
Futhermore, it can be said that froin the viewpoint of the value of Y at the inflection
point of the curve, the function is similar to the Logistics lunctioil if the value of
parameter c is equal to 3.26 and similar to the Gompertz fu~lctionif the value of c is
eclual to 1.55.
Studies co~nparingthe applicability of the NIitcherlich, Gompertz, Logistics and
Richards functions Ilave been conducted by sevesal researchers. YAh4A~o.roet (11.
(1982), using data on 8ueml.s s]~f).,
described the Gompertz fu~lctioilas being the best
fit for all growth factors. Also, YANGet al. (1978),using dala on Pilzzls eluusa, clarified
that the performa~lceof the Go~npertzancl the Von Bertalallffy fuilctioil (Michards
function) is exellell1 for voltline clata, but only moderately useful for height data. The
nloclified Weibull functio~~
was said to be good for both volu~neand height data, while
the Logistics was poor in both cases and the Mitcherlich function is poor for height
data and failed to converge in parameterizing procedures for volume growth data.
The model for changes in variation of stand variables
In general, the variation of a giver1 variable for a stand increases with age.
'I'herefore, the assumption that variatioil is a function of time call be applied in
modelling such cllanges in variation. I-Iawever, if insteacl, the variation is directly
considered to be a function of time, some rescarchers col~siclerthat the coefficient of
variation, a relative inearls of ~neasuringthe variation of variables ~11ichis defined as
of time.
the ratio of standarc1 deviatioi~to the rneail value of the variable, is a fu~lctio~l
I-lere, coefficient of variation decreases with age. NISIIIZAWA
et al. (1981) applied the
futlction as presentecl in eq. 17 and its Iogarith~nicform as shwoll in eq. 18.
log C V = log 04-b logA ,
where CV and A are coefliciel1t of variation and age, respectively, and a and b are
The estimated standard deviation is found by multiplying the mean value of a
variable at a given time by the calculated coefficient of variation.
The above rnoclels are used for ul?thinned stands. They cannot be applied to
thinned stands because variation in the stand variables is affected by thinning activity.
Variailce l1or1tlallydecreases just after thinning and then illcreases again with age. In
this case, it can be assumed that variation is a function of tree population number and/
~ ~ applied the
or the mean value of a variable at a given age for a given site. L A A(1980)
regressior~equatioi~as shown in eq. 19 in order to estimate the standard deviation of
dia~netersof Pi~zzis~adiata.
where Scl is the standard deviation of diameters, N is the tree population numbers, and
cr and b are parameters.
Data and Methods of Analysis
Data used in this study .were stern analysis data obtained from Kagoshilna
University Forest in Taltaltuina district. This stem ai~alysishas been done for 3G trees
which are selected from various height classes at an age of 53 years. Recapitulation
of thc data is presented in Table 1.
'Table 1
Recapitulalioil of stein analysis data of Cty/~tan~e?i(/
jnponicu obtained from Kagoshiina
University Forest in T;~lcaltumadistrict.
1 Mean
St;iiidanl deviation of height Coelbcicnt of var~ationof height
Methods of analysis
2 . 1 , Analysis of the mean height growth
I11 order to deterilli~lethe most applicable height growth model for C7.~,Dlome~ia
[email protected]?z;ca, coinpai-isoil tests were undertalten for four growth f u ~ ~ c t i owhich
~ ~ s are
norn~allyapplied in stand growth studies, namely the Mitchei.lich, Gompertz, Logistics
and modified Weilsull functions. These functions were described in the previous
The fitting of the growth models to data was based on Deming's least squares
et al., 1983).
method (S~IIRAISI~I
Residual mean square, a measure of how well the data fits to the function, was
used as a standard in detern~ii~ing
the accuracy of fit of the function, The constancy
of the estimated values of para~netersand of the estimated values based 011 a given
functioi~with changes in observation lumbers also indicated that the fu~~ction
could be
used in growth prediction.
Construction of site index curves
The difierence among functions is more clearly indicated by their predicted values
for various site indices.
Prediction values of height for various site indices are calculated by the followit~g
ht = H t - l - [{SI-H.~o}/Sd~olSdt
where II* and Sdt are the estimated values for meal1 and stai~darcldeviation of heights,
at I-year-old, and Ni10
respectively, lzt is the estimated height for a given site index (SZ)
asid Sddoare the estiinated values for meail and stai~darddeviation of heights at an age
of 40 years, respectively.
The stai~clarddeviation of heights is calculated 1sy the regressioi? equation which
is forrned on the basis of the assumption that the variation is function of time or age.
Other possilsle lunctions for variation were tested.
Fitting of ~nodelsfor the parameters of height growth
The for111 of any growth curve for a given sta~ldvariable will vary according to
the condition of sites. III other words, the forin of the height growth curve is affected
by site conditions, which are norn~allyexpressed by the site index. Since the form of
the growth curve is determined by its parameters, all attempt was made to coi~struct
a model which cotlld be used for estiinatioil and predictio~iof changes in parameters
wit11 changes in site conditio~ls. A Aow chart of inodel fitting for the parameter of
height growth is given in Fig. 1.
/ Calculation of
lieiglit for \.arious
s i t e indices by formula
Parameterizing for meal] height
Fig. l
Flow chart of models Fitting for parameters of height growth iunction.
Results and Discussion
Growth ~nodelsfor height
The estimated values of parameters for every function are given in Table 2.
Table 2 shows that by using all data on height from an age of 5 until 53 years, the
largest residual mean square (R. M. S.) is given by the Mitcherlich function, followed
successively by the Logistics, modified Mreibull and Gompertz function. The values of
R. M. S. are 0.404, 0.076, 0.061 and 0.024, respectively. Furthermore, Table 2 also
indicates that the values of parameters for the Mitcherlich function are largely
affected by changes in observation numbers and their corresponding age intervals. The
fitting of functions using data froin an age of l 5 until 53 years and fro111 25 until 53
years resulted in the R. M. S. of all functions being small and similar in value.
The degree of accuracy of fit of the height data to each function is more clearly
shown by Fig. 2. Fig. 2 (2a) suggests that the data from an age of 5 until 53 years as
shown by the middle curve, fits well to the Gompertz function. The modified Weibull
fu~ictionas shown by the middle curve in Fig. 2 (Zd), also gives a good fit to the data.
Fig. 2 (2b) irtdicates that the h4itcherlich function overestimates and the Logistics
f~inction,as shown by the middle curve in Fig. 2 (24, underestimates for height at ages
of 50 and 53 years.
Fig. 2 (2a) also shows that the predicted values of height growth based on the
Mitcherlich function are inf-luenced by changes in observation numbers and their
corresponding age intervals, while Fig. 2 (213) indicates that predicted values of height
growth based on the Gompertz function are relativelji unaffected by such changes. Fig.
Table 2
Parameters' value and residual 1neat1 squares for functions fitted to mean height growth
Fitting based on stem analysis data for 36 trees.
data of C ~ f ~ l o m e rjaho?lim.
25 --53
5 -- 53
4 .g173
0. 00116
1 .g964
GF* ; growth function (R4 : Milcherlich function, G : Gornl~ertzfunction, 1- : Logistics function,
W : rrlodified Weibull lunct~on)
RR/IS>) ; residual mean squares
C M * " : corresporiding age interval
; the values of ~ ~ a r a m e t ec r for the inodified \47eibuli ful~ction
M ; obseriration numbers
2 (2d) indicates that the predicted values based on the modified TVeibull functio~~,
clata for ages between 5 and 40 years, results in overpredictioi~of future growth with
a large deviation. This means that extrapolation based on t l ~ emodified Weibull
functio~lfitted to the height data where growth rate is still increasillg with age, results
in coilsidesable overprediction. Overpredictioll also results from using the Gompertz
function, as shown in Fig. 2b, but here the deviations are smaller.
Based on the value of R. M. S. and variation of estimated and predicted values
with changes in observation numbers and their corresponding age intervals, the best
growl11 fui~ctionfor height growth is the Gompertz function. The Logistics function
can be fitted to height growth data for a stand where the growth rate increases with
age, but underpredictioll can occur if it is used in future growth prediction.
The modified Weibull frrilction also gives a small R. M. S. for such data as that
above, but its estimated curve overpredicts future growth, while the Mitcherlich
Age ( y c v ~ r s )
Fig. 2
Age (ycnl-S)
Cttrves of the estimated values oT height growth based an the Mitcherlich, Gompertz,
Logistics and modified Weibull fttnctions.
fuxlctioii has a large R. M.S. All fu~ictionsfit well to the height data of heights after
the maxil11um growth rate has beell passed whereas the growth rate decreases with
If it is col~sideredthat the intercept at X = O should equal 0, it can theoretically be
used as an indicative factor to verify whether a given function can be applied for fitting
a fiinctio11to a given piece of data. By the i1-1troductio11of this third indicative factor
as ~nentionedabove, the modified Weibull function was shown to be the best function
for estimation of past growtl-1.
Moclels for variation in height and construction of site index curve
I11 order to clarify the fuilctio11which could be used in the estitl~atioi~
and preclic-
tion of tree height variation, several determi~iisticregressioll ~nodelswere tested. Four
~nodelsare presented below and their curves are given in Fig. 3. T h e functions are :
a. CV = 1.48A-O" ,
11. Sd = 0.961-0.029A ,
(Y = 0.931 and
c. Sd"
0Q.9-i-0.102A ,
d. Sd = -0.444-1-0.718 l n A ,
= 0.983 and F ratio = 287.11)
F ratio = 64.79)
( r = 0.963 and F ratio = 126.45)
= 0.988 and F ratio = 361.84)
where CV and Scl are the coefficient of variation and the standard deviation of
heights, respectively, and /l is the stand age.
et 01. (1981), has a large
'The first function, which has been applied by NISHIZA\IIA
Age (years)
I;ig. 3
Curves of the estil~latedvalues of variance (Sd"),standartl deviation (S(/) and coefficient of
variation (Cl') for height.
correlation coefficient ( r ) of 0.983. It also gives a large F ratio of 287.11. K o ~ ~ e v e r ,
estimated values based on the fui~ctionresulted in a quadratic curve for the standard
deviation, which is calculated by formula S d = CV X lz. This means that, unustially,
the standard deviation increases up to a certain maximum point and then decreases, as
shown in Fig. 3.
The function of CV ==/(A)
can only be applied if it is fitted without values of the
coefficient of variation ( C V ) at ages of 5 and 10 years. The values of CV at both of
these ages are relatively large and result in underprecliction of the future coefficient of
The model of Sd= -0.4444-0.715 InA, gives the largest values of correlation
coefficient and F ratio, which are equal to 0.988 and 361.84, respectively, This model
is the best one to be applied in the estimation of height variation, and it has been
applied it1 the calculation of height growth for several site indices.
Fig. [l Curves of thc estimated values of height growth for site indices 13, 16 and 19.
Calculated values of height growth for site indices of 13, 16 and 19 based on the
Mitcherlich, Gompertz, Logistics and inodifiecl Weibull functions are presented in
Table 3 and corresponding curves are given in Fig. 4.
Table 3 shows that these functions resulted in differences in the estimated values
for height growth for the above mentioned site indices. The differences in the
estimated values obtained among the four functiolls are shown most clearly by the
estiillated and predicted values after an age of 40 years. The Mitcherlich function
resulted in the largest of the estimated values and the Logistics fimction resulted in the
smallest estimated values after this age, while the Goillpertz and tnoclified Weibull
functions resulted in values which approached closely, the difference being no larger
than 0.25 meters. The estimated values for a site index of 13 showed that the
differeilces in the final value for height growtlr occur with every function. This
indicates that for the Logistics function, the final value of height growth for a site
index of 13 is 13.74 meters, attained at an age of 60 years. For the Gompertz fuilction
the final value for the samc site itldex is 15.18 meters, attained at an age of 85 years
and for the modified Weibull function the final value is 15.21 meters, also attained at
80 years. On the ohher hand, the estimated values based on the Mitcherlich function
indicated that: the final value of height growtil is reached at an age of more than 100
'I'able 3
The calriilatecl values of height growth for sile illdices 13, 16, 19 based or1 the Mitcherlich
(M), Gonlpertz (G), Logistics (L) and ~nodifiedWeibt~llf~i~lclion
Site index 13
Site index 1(
- .-.------p
1 10.34
' 11.73
/ 10.46
I 13.06
11.90 14.60 14.81
13.00 16.00 1 16.00
Site index 19
The difference in estiillated and predictecl values described above is more clearly
shown by Fig. 4. The figure indicates that the height growth curve for a site index of
13 based on the Mitcherlich functioil crosses the curve of height growth for a site index
of 16 based on the Go~npertzand ~nodifiedTnieibull functions at an age of 70 years,
while it crosses the height growth curve for a site index of L9 based on the Logistics
function at an age of 90 years. This means that the Mitcherlich function results in
predicted values for a site index of 13 which are larger than the values predicted on the
basis of the Gompertz and modified Weibull for a site index of 16 at age of 70 years
and over. At 90 years and over, the value is larger than that predicted for the height
growth for a site index of l9 based on the Logistics function. These results indicate
that the Mitcherlich function results ill overprediction of future growth and that,
conversely, the Logistics function presents underprediction of future growth as shown
by Fig. 4, while the Gonlpertz and modified Weibull functions result in site index curves
which are almost the same and more acceptable than those of the Mitcherlich and
Logistics functions.
Models for parameters of the height growth function
The analysis of models for parameters only were undertaken for the Go~npertz
The estinlated values of height for various site indices calculated by forillula :
lz~=IIt'r[ ( ( S / -lf~o)/Sd+o]Sdc
Table 4
Age iI Sd
Site indices
Sd ; esti~nateclvalues of sta~lclarcldevialiotl,
111 : estimated values of height based on Gompertz furlction, Stte iildices,
S1 ( \-alucs of height at age of 40 years.
Un11 ; (meter)
function, the best function for the height growth of Crypfomeria jufionica, as mentioned
in the previous section. The steps involved in the analysis were given in section 3. 2.
3, Fig. 1. The calculated values of standard deviation based on the regression equation
Sd= -0.4444-0.718 In A,together with the calculated values of height for various site
indices using the forlnula hl = Ht -C [{S]
- H r o } / S d t o J S dare
t , given in Table 4. The
values of paralneters resulting from fitting of the function to the calculated values of
height for various site indices are given in Table 5. Based on these values, a model for
each paran~eterwas analyzed, resulting it1 the following equations.
where A, B, and k are parameters of the Gompertz function for height growth, S1 is
the site index and r is the correlation coefficient.
The correlation coefficient values indicate that these parameters can be estimated
on the basis of site index. The curves correspondi~lgto the above regressio~~
are given in Fig. 5.
Based on the above equations, the height growth for every individual tree can be
'Table 5
'The estimated values of parameters for various site ii~ciicesbased on the Gotnpertz
Site index
- --- - --
1) : the cleterlninistic values of site inclices applied for height growth estimation a s presenteci
in Table 4.
2) : The refitted values of site irldices
Fig. 5 Curves of the estirnated values far parameters A, B ancl k of tlre Gomperlz fui?ction.
estirnatecl. Then the mean height growth of the stand is calculated on the basis of the
estimated and predicted values of the height growth of every individual tree. By using
these calculated values, the change occuri~lgin height growth with decrease in tree
nuii~bers,either by thinning or mortality, can be clearly revealed. Such changes callnot
be determined if the mean height gro~x7this estimated on the basis of a model resulting
from the fitting of mean height growth data. Furthermore, the latter estimation
method results in misestimation, with a deviation dependent on thinning grade aild
thinning frequency. The larger the thinning grade and the greater the thinnii~g
frequency, the larger the deviation that will result.
Basecl on the values of residual mean square and variation of estimated and
predicted values with changes in observation numbers and their corresponding age
intervals, the best growth ftui~ctionfor the height growth of C~yplo?neriujufionica was
found to be the Gompertz function. The inoclified Weibull function is also good, while
the Mitcherlich and Logistic functions are less satisfactory.
All functions can be applied for the estimation of height growth corresponding to
agc after the maximt~rngrowth rate has been passed and where growth rate decreases
with age. I-lowever, the extrapolation of estimated curves to past resulted in different
values aInong the functions and the modifiecl Weibull function giving the most accepta-
ble values. This means that if the applicable data are limited to a recent period and
estrapolatio~ito past growth is needed, the modified Weibull function is the best one
to be applied.
Standard deviatioii of height can be estiiliated 011 the basis of age. Estimatecl
curves of site indices, constructed on the basis of the estimated values of standard
deviatio~lof height and mean height growth, clarified that the Gompertz f.unctio11gives
the most acceptable predicted values for future growth, followed by the modified
Weibull function. Application of the Mitcherlich functior~for future growth prediction
results in overpredicted values, while conversely, the Logistics fuilction presents
The changes in values of the parameters for height growth functio~lwith changes
in site conditions, caii be estimated on the basis of site index.
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A. ii~ldS l r i ~ . rJ.~ ,If. G. (1978) : The potential of Weibull.type functions as flexible
growth curves. Can. J. For. Res, 8 : 424-431
九大演 粥
75 7891
樹高生長 モデルの研究
マ ラマ ッサ ム
ダウ ド 弓関 屋
本研究 は,林分生長 の解析 に適用 され る ミッチ ャア リッヒ, ゴムベルツ, ロジステ ィッ
お よび修正 ワイプルの関数式 を比較 す る ことによ り, スギ林分の樹高生長の推定 と予測
に適用 で きる関数 を見出す ことにある. また,時 1制 勺変化 に伴 う樹 霜の変動の変化 を示す
lデル として用 いられ る関数 を決定 す ることも意図 している.
35 年 までの 5 年間隔の年齢 に対応 す る樹 高生長 の資料 を]
削 1て上記各式 を比較 した結
果,関数の適合度の尺度 すなわ ち残差平均平方の最小値 はゴムベル ツ式 によって与 えられ,
ついて修正 ワイプル, ロジス ッテ ィツク, ミッチ ャア リッヒの順 に大 きくな る.
樹高変動 は,Sd--O
4によって推定 され る. ただ し,Sd,A はそれぞ
れ樹商の標準偏差,年齢である.平均樹商 とその標準偏差の推定値 によ り,樹高生長 に対
す る様 々の指標 が推定 され る. これ らの指標 に対 して推定 された生長 曲線 は, ミッチ ャア
リッヒ式 は将来の生還 に対 して過大推定 とな り,逆 にロジステ ィック式 は過小推定 とな る
ことを意味す る. -・
方, ゴムペ )
,ツ と修正 ワイプルの式 は将来推定 に対 してほ とん ど同 じ
値 とな り, ミッチ ャア リッヒ, ロジステ ィックの式 よ り適用 可能 である.
ゴムベル ツ式 に基づいた樹商生長のパ ラメ- 夕に対 す る分析 は,各パ ラメ- 夕が地位指
数 に密接 な関係 を もつ ことを明 らかに した.