Forest Ecology and Management, 22 (1987) 173-183 173 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Simple Linear Model Reliably Predicts Bark Thickness of Radiata Pine in the Australian Capital Territory

T.S. JOHNSON and G.B. WOOD

Department of Forestry, Australian National University, Canberra, A.C.T. 2601 (Australia)

(Accepted 16 March 1987)

ABSTRACT

Johnson, T.S. and Wood, G.B., 1987. Simple linear model reliably predicts bark thickness of ra- diata pine in the Australian Capital Territory. For. Ecol. Manage., 22: 173-183.

A series of linear and nonlinear models was tested for predicting bark thickness of radiata pine (Pinus radiata D. Don ) growing in the Australian Capital Territory. A simple linear model proved superior to all other models. This model predicts bark thickness at any point along the bole, relative to bark thickness at breast height, from the fourth power of relative diameter. Relative diameter is defined as the ratio of diameter overbark at a point on the bole to the diameter overbark at breast height. The model is independent of site and age and appears to be unaffected by forest location. Using the model, the underbark volume of 109 trees was estimated to be 54.6 m 3 com- pared with the actual volume of 54.9 m 3. This represents an error of -0.5%.

INTRODUCTION

High-performance optical dendrometers are being used more extensively in forestry to estimate the upper stem dimensions and other characteristics of standing forest trees. Because estimates of diameter are overbark, an adjust- ment for bark thickness must be made to derive underbark diameter and, sub- sequently, underbark volume. Faced with this problem, Grosenbaugh (1967) incorporated three options (equations) in his computer program STX for es- t imating bark thickness along a tree stem given measurements of bark thick- ness and diameter overbark at breast height and diameters overbark along the bole. One advantage of these options is that they do not require different coef- ficients for different species. They were designed to cater for the three broad patterns of behaviour observed in the ratio of diameter underbark (DUB) to diameter overbark (DOB) along tree boles, namely: O p t i o n I - - constant at all levels (applies to many hardwoods; Wiant and Koch, 1974; Castaneda, 1979; Wingerd and Wiant, 1982; Biggs et al., 1985) ; O p t i o n 2 - - increases curvilin-

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174

TABLE 1

The main data set

Forest Age (years) Number of observations

Total Avg. per tree

Kowen

Pierce's Creek

2 18 16 8.0 5 26 40 8.0

18 41 164 9.1 90 43 1028 11.4

93 42 999 10.7 47 43 452 9.6 32 44 327 10.2 22 45 222 10.1

309 3248

early above breast height (BH) and decreases curvilinearly below BH { applies to many softwoods and some hardwoods; Mesavage, 1969; Wingerd and Wiant, 1982) ; and Option 3 - - decreases curvilinearly above BH and increases curvi- linearly below BH ( a p p l i e s to some hardwoods; Boehmer and Rennie, 1976; Wingerd and Wiant, 1982). Subsequent tests of the second of these options, or a variation of it, confirmed that it gave reasonably accurate predictions of bark thickness for a range of softwood species in the U.S.A. (Mesavage, 1969; Brickell, 1970).

The pattern of varition in the DUB : DOB ratio along stems of radiata pine (P inus radiata D. Don) in the A.C.T. fits the pattern implied in Option 2 reasonably well in the lower half of the bole but not in the upper half, where the ratio decreases towards the tip. This is the case also for radiata pine in New Zealand (Gordon, 1983). Because a high proportion of the wood volume in trees whose main boles are continuous from ground to tip occurs in the lower half of the bole, this discrepancy may have little effect on estimates of mer- chantable volume derived, using STX and Option 2, from dendrometer mea- surements of radiata pine trees.

The purpose of the study reported here was to develop a bark correction equation for plantation-grown radiata pine in the A.C.T. and compare its pre- dictive ability with that of Option 2 of Grosenbaugh.

T H E D A T A B A S E

The data comprised measurements made by diameter tape on felled trees firstly overbark, and then underbark after bark removal. They derived from two sources. The main data set was supplied by the Forests Section of the

TABLE 2

The data set used for verification

175

Forest n Age (years) Number of observations

Total Avg. per tree

Kowen

Uriarra

Stromto

17 14 24 16 6 3

12

3 7

7

109

21 563 33.1 27 449 32.1 28 687 28.6 29 478 29.9 32 79 13.2 33 41 13.7 40 435 36.3

22 46 15.3 23 116 16.6

45 144 20.6

3038

Depar tment of Territories, A.C.T. This set comprised 3248 observations from 309 trees located in Kowen and Pierce's Creek forests, A.C.T. The second set comprised 3038 observations made by staff and students of the Depar tment of Forestry, A.N.U. on 109 trees drawn from Kowen, Stromlo and Uriarra forests in the A.C.T. The main data set was used for est imating the bark functions and the second set for verification. Tables 1 and 2 and Figs. 1 and 2 summarise the data sets and show their characteristics.

METHOD

The linear model

A plot of relative bark thickness (RBT m i.e., bark thickness expressed as a ratio of tha t at breast height) against relative diameter ( RD - - i.e., diameter overbark expressed as a ratio of tha t at breast height) for trees selected at random from the main data set indicated a curvilinear relationship (Fig. 3a). To select a t ransformat ion tha t would provide a linear relationship, R B T was plotted against various functions of RD, v i z . , natural logarithm, square root, inverse, square, cube and fourth power. The natural logarithm ( In ) Of RBT also was plotted against In (RD). The fourth power t ransformat ion of RD was ap- proximately linear for all trees (Fig. 3b). Thus, the model:

RBTi = a + bRDi (1)

176

o

>

o

D z

3000

2500

2000

1500

1000

50O

0 0 io 20 30

A g e ( y e a n s )

40

q 50

(A)

Z

50 t 40

30-

20-

:tO-

O

(B) + .~-

+ ÷ + + ~ ÷ +

÷

+

I i i i I I I

10 20 30 40 50 60 70

EIBHOB (cm)

Fig. 1. Characteristics of the main data set: (a) age distribution; (b) height/diameter distribution.

was fitted to the data in the main data set using GENSTAT 4.04 (N.A.G., 1983; Table 3 ).

The effects of Forest ( Kowen and Pierce's Creek) and site index ( top height in metres at age 20 years) on a and b were tested by examining the following model:

TABLE 3

Regression coefficients of the simple linear model (Eqn. 1 )

Parameter Estimate Standard error Student 's-t +

a 0.12265 0.00155 79.05*** b 0.84116 0.00347 242.48***

+Student 's- tvalue:***indicatessignificance at P<0.001.

177

0

0 > L ®

.CI

E

2000

1500

1000

500

0 0 10

m

20 30

Rge ( y e a n s )

4O 5O

(A)

o~

L. h-

50

4 0

30 -

2 0 -

l O -

0

(B)

+ ÷ ÷ ÷

+ +

+

] I ] ~.0 ~10 310 4 0 5 0 6 0

DBHOB ( cm )

Fig. 2. Characteristics of the data set used for verification: (a) age distribution; (b) height/di- ameter distribution.

RBTi=a+alF+a2S+a3F'S+ (b+blF+b2S+b3F'S)RDi (2)

where RBTi is the relative bark thickness at the ith measure, F is a dummy variable (0 = Kowen, 1 = Pierce's Creek), S is site index, and RDi is the relative diameter at the ith measured point raised to the fourth power.

Coefficients of all terms in Eqn. ( 2 ) involving the dummy variable for Forest had Student's-t values that were not significant at the 5% level of probability (Table 4 ). Also, when the function was plotted for any site index in the range 17-24 m, which roughly covers the variation in site within the data base, the lines for Kowen and Pierce's Creek Forests coincided. This showed conclu- sively that the same function applied to both forests. Equation (2) was then plotted for Kowen Forest at site indices 17 and 24 m, together with Eqn. (1) and the 95% confidence limits associated with a prediction of relative bark thickness from it ( Fig. 4 ). The plotted lines for the two site indices were little different from one another and well within the 95% confidence limits. Clearly, site index did not significantly improve the prediction of relative bark thick- ness from relative diameter. Thus, Eqn. (1) was chosen as the linear model.

178

i 1.2

.~ o .e

X

i 0.4

2 . o . (A)

1.6

0.0 0.0

2 .0 '

1.6

1.2

°

j ' o:~ ' o:6 ~.'2

ReJotlve d l a B o t e r over- -bar-k

(B)

. .t

~:6 ' 2:o

°

t

x v

0.8 4"° ,~, ~

_ o . • - , : . ~ o • . .

- - • 4 ~ o • "G o.4j r . ..~... o , , , •4

0 . 0

0 . 0 0 . 4 02B 1 ; 2 1 ;6 2 ; 0

R e l a t l v a d l a m e t l r - a v e r - b a r k r a i s e d to t he f o u r t h powmr-

Fig. 3. Scatte~lot of relative bark thickness against: (a) relative diameter overbark; (b) relative diameter overbark raised to the fourth power.

The nonlinear model

Grosenbaugh's (1967) three equations for predicting bark thickness can be generalised to the nonlinear equation:

179

TABLE 4

Regression coefficients of the full linear model (Eqn. 2 )

Parameter Estimate Standard error Student's- t +

a 0.2635 0.0260 10.14"** al - 0.0050 0.0307 - 0.16 n~ a2 - 0.00633 0.00123 - 5.15"** a:~ 0.00025 0.00141 0.18 "s b 0.7432 0.0657 11.30"** bl 0.0719 0.0738 0.97 "~ b2 0.00485 0.00312 1.55 "s b:~ - 0.00380 0.00344 - 1.11 ns

+Student's-t value: *** indicates significance at P<0.001; ns indicates not significant at P < 0.05.

RBTi-----RDi P( ( Q - 1 ) / ( Q-RDi r)}t (3)

w h e r e RBTi a n d RDi a re as d e f i n e d p r e v i o u s l y , a n d p, Q, r a n d t a re p a r a m e t e r s

to be e s t i m a t e d .

A n a t t e m p t to use t h e c o m p u t e r p r o g r a m NLPE ( B a r d , 1967) to e s t i m a t e t h e

p a r a m e t e r s in E q n . (3 ) for t h e m a i n d a t a se t was u n s u c c e s s f u l as t h e

G a u s s - N e w t o n m e t h o d d id n o t c o n v e r g e to a s o l u t i o n . T h i s s u g g e s t e d t h a t t h e

m o d e l was o v e r s p e c i f i e d a n d t h a t o n e o f t h e se t s o f m o d e l s w i t h one , t w o or

/ 1 . 0 . gS~, C o n £ 1 d e n c e I l m l t o /

/ The e lmp le l i n e a r model / f ~

- S l t e l n d e x 1"7

- - x - - x S i t e l n d e x 24

0 . 8 .

e c

~ 0 . 6 .

o

~ 0 . 4 . >

Q

~ : 0 , 2 .

0 . 0 o.o o:2 0.4 o16 o[e 11o

R e l a t l v e d l a m e t e r

Fig. 4. Effect of site on the linear model.

180

three parameters might adequately define the model. If the parameter Q is removed (i.e., set Q=O) then Eqn. (3) reduces to the equation RBTI=RD p. This equation is equivalent to the linear model ln(RBTi)=p ln(aDi) which was considered when choosing the transformation for the linear model. Thus, the models tested were:

Three-parameter:

RBTi =RDiP( Q_ I ) / ( Q_RDi r) (4)

RBTi =RDi P{ ( Q - 1)/(Q--RDi)}t (5)

RBTi=RDi{ ( Q _ I ) / ( Q_RD i r) }t (6

Two-parameter:

RBTi =RDi P(Q--1)/(Q-RDi) (7

RBTi = (Q--1)/(Q--RDi r) (8

RBTi={ (Q--1)/( Q--RDi) } t (9)

One-parameter:

RBTi = ( Q - 1 ) / ( Q - RDi) (10)

The three-parameter models ( Eqns. 4, 5 and 6 ) had residual sums of squares of 11.46, 11.25 and 11.32, respectively. For each of these three models, at least one parameter had a Student's-t value that was not significant at the 5% prob- ability level indicating that an equation with only two parameters should ad- equately define the model.

The residual sums of squares for the two-parameter models were almost identical (11.33, 11.32 and 11.28 for Eqns. 7, 8 and 9, respectively) and the residual sum of squares for the one-parameter model (Eqn. 10) was only slightly higher at 12.70.

The four equations (7-10) were then plotted together with the linear model (Eqn. 1 ) and the 95 % confidence limits for a prediction from it (Fig. 5). Clearly, the four models are similar and, at the 5% probability level, they are not sig- nificantly different from the linear model for predicting relative bark thickness and hence bark thickness.

VERIFICATION

When applied to the 3038 measurements of diameter for the 109 trees in the verification data set, Eqn (1) returned a mean error of 0.4 mm in bark thick- ness and showed good precision with absolute errors less than 1 mm for 49% of the measurements and less than 2 mm for 76% of the measurements. Only 4% of the estimates for bark thickness were more than 5 mm from the meas- ured bark thickness.

181

1 . 0

0,0

9 . 0

0 . 8

o ~: O,G

>~ 0.4_

o

0,2.

/ 95Z C o n ? l d o n c e I l m l t e

The wimple I l n e a P model / v

. . . . The one parameter model

-- x - - x T w o p a r a m l t e r e - F a n d 0

. . . . . . . Two parameters - R and O

x x x x x x ~ T w o p a r a m e t e r s - T a n d 0

0:2 01, 0:s o]8 i:0

Relatlve dlameter

Fig. 5. Comparison of the simple linear and nonlinear models.

A histogram of the errors with a normal distribution curve superimposed showed that the errors were normally distributed with a mean of 0.4 and a variance of 0.06. The errors were then grouped according to the height decile from which they originated and the mean error for each decile was calculated. There was a t rend of overestimating RBT in the lower half of the tree bole and underestimating it in the upper half. The largest mean overestimate was 1.1 mm in the 20-30 decile and the largest mean underest imate was 0.7 mm in the 60-70 decile. Histograms of the errors for each forest showed no discernible differences between the forests, and scattergrams of the errors in the bark thickness estimates against height of tree, diameter overbark at breast height, and age also showed no trends.

Functions to est imate bark thickness along a tree bole are used mainly to estimate underbark volume from overbark diameters without having to mea- sure the underbark diameters. Thus, for each of the 109 trees in the verification data set, the 'actual' underbark volume was calculated using the measured un- derbark diameters and Smalian's formula. Then, Eqn. (1) and Smalian's for- mula were used to estimate the underbark volume of each tree from the measured bark thickness at breast height and the measured overbark diame- ters. The total underbark volume of the 109 trees was est imated as 54.6 m 3 using Eqn. (1) compared with the 'actual ' total underbark volume of 54.9 m 3, a difference of - 0 . 5 % .

The error in the underbark volume estimate for each tree was then expressed as a percentage of the 'actual' underbark volume. The percentage errors ranged from - 5 . 5 % to + 8.0% with a mean error of - 0 . 7 % . The errors in the volume

182

estimates were less than 3% for 84% of the trees and less than 5% for 97% of the trees. For comparison, the total underbark volume of the 109 trees was determined based on estimates of bark thickness using Option 2 of Grosen- baugh's STX program. The result, 50.7 m 3, represents an error of -7.6% (cf. -0.5% for Eqn. 1).

When the bark thickness estimates derived using both the linear model and Option 2 of Grosenbaugh were plotted alongside the actual bark thicknesses of several trees randomly selected from the verification data set, the linear model overestimated bark thickness in the lower half of the tree bole whereas Option 2 overestimated it over most of the bole. Also, estimates above breast height using the linear model were always better than those derived using Op- tion 2. Clearly, the linear model (Eqn. 1) is superior to the nonlinear model (Option 2 of Grosenbaugh) for predicting underbark volume of radiata pine trees growing in the A.C.T.

DISCUSSION

Because of intra-tree correlation in the data, the variances derived by ordi- nary least-squares analysis will be underestimated. The estimates of the pa- rameters should be robust and reflect the true parameter estimates but the confidence intervals in Figs. 4 and 5 should be interpreted cautiously. Alter- native estimators and procedures will be investigated when more extensive sample tree data for radiata pine in the A.C.T. become available (see below). Similar problems of serial correlation in data (e.g., from repetitive measure- ments) have been discussed by Sullivan and Reynolds (1976) and Ferguson and Leech (1978).

The results presented here of tests on the generalised non-linear model (Eqn. 3) are consistent with those reported by Brickell (1970). Brickell tested the equation on ten coniferous species in western Montana and found that the parameters r and t had no significance for estimating bark thickness of those species. He suggested also that the term RDi p was superfluous as the parameter p was nearly zero.

For the A.C.T. data, the performance of the one and two-parameter nonlin- ear models was similar to that of the simple linear model (Fig. 5 ), but the latter model is preferred because of its simplicity, ease of solution and robustness. The model is unaffected by age and site, and indications are that it is also unaffected by forest location. This is supported by Tables 1 and 2 and Figs. la and 2a which indicate that the frequency/age distributions and forest sources of the main data set and the verification data set are quite different. Clearly, there is opportunity for improving the model by sampling more widely across forest sources and within the 10-40 year age classes. This work is now in prog- ress. Meanwhile, we are incorporating Eqn. (1) (Table 3) in STX as a user-

183

suppl ied func t ion for p rocess ing da t a der ived f rom d e n d r o m e t r y of P. radiata in the A.C.T.

ACKNOWLEDGEMENTS

We t h a n k the Fores t Sect ion , D e p a r t m e n t of Ter r i to r ies , A.C.T. , for access to the i r s amp le t ree da t a base for r ad i a t a pine; Mess r s G. Mitchel l , P. Gr i f f i ths and J. M a r s h of the D e p a r t m e n t of Fores t ry , A.N.U. , for a s s i s t ance wi th field m e a s u r e m e n t s ; and D r B.J . T u r n e r , D e p a r t m e n t of Fores t ry , A.N.U. , for his c r i t i c i sm of the m a n u s c r i p t .

REFERENCES

Bard, Y., 1967. Nonlinear parameter estimation and programming. IBM Contributed Program Library No. 360D, 13.6.003. IBM, New York, 145 pp.

Biggs, P.H., Wood, G.B., Schreuder, H.T. and Brink, G.E., 1985. Comparison of point-model based and point-Poisson sampling for timber inventory in jarrah forest. Aust. For. Res., 15: 481-493.

Boehmer, W.D. and Rennie, J.C., 1976. Predicting diameter inside bark for some hardwoods in west Tennessee. Wood Sci., 8: 209-212.

Brickell, J.E., 1970. Test of an equation for predicting bark thickness of western Montana species. USDA For. Serv., Intermount. For. Range Exp. Stn., Res. Note INT-107, 7 pp.

Castaneda, F., 1979. Converting outside bark to inside bark diameters. Resour. Inv. Notes BLM- 19: 9-12.

Cochran, P.H., 1982. Estimating wood volumes for Douglas-fir and white fir from outside bark measurements. For. Sci., 28: 172-174.

Ferguson, I.S. and Leech, J.W., 1978. Generalised least squares estimation of yield functions. For. Sci., 24: 27-42.

Gordon, A., 1983. Estimating bark thickness of Pinus radiata. N.Z.J. For. Sci., 13: 340-353. Grosenbaugh, L.R., 1967. STX-FORTRAN-4 PROGRAM for estimates of tree populations from 3P

sample-tree-measurements. USDA For. Serv., Pac. S.W. For. Range Exp. Stn., Res. Pap. PSW- 13, 76 pp.

Mesavage, C., 1969. Converting dendrometer estimates of outside-bark diameters to wood diam- eters on major southern pines. USDA For. Serv. Res. Note SO-93, 4 pp.

N.A.G., 1983. GENSTAT 4.04 - - A General Statistical Program. The Numerical Algorithms Group Ltd., Oxford.

Sullivan, A.D. and Reynolds, M.R., 1976. Regression problems from repeated measurements. For. Sci., 22: 382-385.

Wiant, H.V. Jr. and Koch, C.B., 1974. Predicting diameters inside bark from outside bark mea- surements on some Appalachian hardwoods. J. For., 72: 775.

Wingerd, D.E. and Wiant, H.V. Jr., 1982. Variables for predicting inside-bark diameters of upper stems of Appalachian hardwoods. J. For., 80: 791-792.