109 A Study of Models for Height Growth Daud MALAMASSAM and Yuui SEKIYA Abstract 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 studied. 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. Introduction 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) : 1. dY/cl/ = f ( 1 ) , (1 dY/dl = g ( Y ) , (2) dIf/d/ = h ( Y , t ) , (3) 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 : g=a(A-Y), ; Mitchcrlich g=aYln(A--Y), ; 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 ~nodellii~g (h/lnrsuhru~i~t et al., 1983) is : / = Bclc "(A- Y ) , ; moclilied llreibull (8 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 (11) ; Richards (12) ; rnodificd Weibull (13) PRODAN (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. Y 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 m* (14 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 : Y =; R{~-~-(IIC)* 1, (15) 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. M1-icr~,\uo (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. MKL'SU~~LJRA 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. 2. log C V = log 04-b logA , (18) where CV and A are coefliciel1t of variation and age, respectively, and a and b are parameters. 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 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. 1. 'Table 1 1 1 Recapitulalioil of stein analysis data of Cty/~tan~e?i(/ jnponicu obtained from Kagoshiina University Forest in T;~lcaltumadistrict. Aac (pars) 1 Mean / height (m) St;iiidanl deviation of height Coelbcicnt of var~ationof height 1 (in) 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 chapter. 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. 2. 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 for~nula: 2.2. ht = H t - l - [{SI-H.~o}/Sd~olSdt , (20) 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. 2.3. / Calculation of $4 lieiglit for \.arious s i t e indices by formula I 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. 1. Table 2 Parameters' value and residual 1neat1 squares for functions fitted to mean height growth ia Fitting based on stem analysis data for 36 trees. data of C ~ f ~ l o m e rjaho?lim. N GP* CAI** 1 l M G 1 1 i l1 10 8 11 10 8 f) 7 L I W 10 8 11 10 8 9 1 / 1 / 5--53 5-50 5-40 15-53 25 --53 5-53 5-50 5-40 15-53 25-53 5 -- 53 5-50 5-40 15-53 25-53 5-50 5-40 15-53 25-53 1 / 1 l I Parameters A 1 B 25.07 31.79 139.58 17.94 16.93 1 1.0692 0483 1.(l072 1.6058 2.5105 1 1 17.11 17.29 17.88 16.76 16.66 15.89 15.60 15.02 16.25 16.48 17.03 17.35 20.11 16.32 16.41 3.9481 3.9059 4.6018 4 .g173 , / 20.2291 20.1628 21.0947 14.6487 9.7821 0.0044 0.0044 0. 00116 0.0024 0.0028 - / 1 l 1 1 ~ RMSl) K 0.0178 0.0160 0.0038 0.0521 0.0724 0.494 084 0.276 0.045 0.003 0.0726 0.0874 0.0904 0.024 0.030 0.023 0.005 0.005 0.1414 0.1416 0.1514 0.1245 0.1086 1.6428(i') 1.6368 1.5297 1 .g964 1.8124 l 1 C 1 0.076 0.058 0.044 0.020 0.009 0.061 0.085 0.075 0.009 0.009 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~~, using 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 age. 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. 2. 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" , (Y 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) (7, = 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 20 40 60 80 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 variation. 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 years. '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 (\V). Site index 13 Site index 1( i i - .-.------p 1 30 35 1 10.34 ' 11.73 10.50 11.94 1 1 10.64 12.14 13.00 13.44 / 10.46 I 13.06 13.22 11.90 14.60 14.81 13.00 16.00 1 16.00 13.80i17.28116.87 13.36 15.01 16.00 16.56 Site index 19 G p . 1 rM-T--. 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. 3. 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 l Age iI Sd (years) I / Site indices I l 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 equations 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 ftlnction. Site index - --- - -- I) I 2) 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. Conclusion 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 i~iiderpredicteclvalues. 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. References Lsnlc, A. van, (1981) : Spacing trial in IJi???a~crdio/cr. Proc., XVII IUFRO WORLD CONGRESS, Icpoto, Session ii : 67-74 L e ~ l t u l<. , A. (1970): System identificatio~~ principles in studies of forest dynamics. USDA For. Ser. Pap. NC-45 : 38 MACIIADO, S. d. A. (1981) : The use of a flexible biological no del for basal area growtlt and yield studies of Pilztrs loecio pla~ltatioliin the state of Panama-Brazil. I'roc., XVII IUFRO WOIiLD CONGRESS, Kyoto, Sessio~l4 : 75-91 MA'~SUX~U N. Rand A , Y,I;iti\nio'ro, M. (1983) : The growth curve of Weibull type. J. Jap. For. Statis. 8 : 37-42 (in Japanese) Nlsr~~z;i!r~\, M. and TAICESIIITA, 1C. (1981) : Report of the ci~ltivationfund system of water resource forest 1 : 62 (in Japanese) Osr 111, S. and ISIIII<.\WA, Y. (1983) : Applicability of the RICIIARDS growth function to a ~ ~ a l y sof is growth of tree. Sci. Rep. ICpoto Pref. Univ., Agr. 35 : 49-76 (in Japanese with English summary) I'II.NA~II~, L.V. and T L I ~ N I ~ U 1C.J. L I . ,(1972): The Chapmatl Iiichards generalization of V011 Bert:~lanffy's gro\vtlt 111ode1 for basal area growl11 and yield in even-aged stands. For. Sci. 19: 2 -22 I-'~cou.\r,h.1. (1968) : Forest Bio~net~ics. Fergussorl Press. Oxford : 417 (3111-391) S I I I I ~ A I ~N.I I iltld I , ISIIIIIASIII, Al. (1!>83): Fitting program of Mitcherlich, Gompertz and Logistic functioil based on Basic, J. PC-For., 1 (1) : 4-7 (in Japanese) Y~ali\h!olo,i\lI., Y,ISUI, H. and AI(IYARIB, F. (1982) : A ~ ~ a l y s of i s growth curve. Bull, of Fac. Agr., Shimane Ulziv. 16 : 48-52 (in Japanese with Er~giishsummary) Y,\NG,R. C., I<OLAI<, 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 】 52 . 75 7891 樹高生長 モデルの研究 マ ラマ ッサ ム ダウ ド 弓関 屋 要 雄 健 旨 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 ク. 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 こ L 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 A4+0. 871l l lJ 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 ) i ,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.

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