Of the equations with the form (1) used by the Department, one of the most satisfactory is the volume equation 010273' ...... N c D @ - N = f l r ) ( o @ c D C D N -.
N0.35
RESEARCHNOTE
F OPTII'UM SAMPLING OF SAI'PLETREES F0R V0LUMEEQUATt0NS
BY J. K. VANCLAY
OPTIMUM SAMPLING OF FOR
VOLUME
SAMPLE TREES
EQUATIONS
BY J. K. VANCLAY
rssN 0481 €219
1.
ABSTRACT Th is not e deals w i th th e c h a ra c te ri s ti c so f a sati sfactory predomi nanthei ght vol ume equati on,and su gges t sa s am p l i n g i n te n s i ty w h i c h s h o u l d achi eve an equati onw i th such characteri sti cs. S ampl i ng sh ould at t em ptt o i n c l u d e a t l e a s t tw o tre e s per 1 cm d.b.h.o.b. x 2 m predomi nanthei ght cel l , i n every ce! l f or whic h s a mp l etre e s c a n b e fo u n d . l t i s parti cul arl y i mportantto sampl efor extremeval ues of d a t a.
INTRODUCTION
Vo lum e equat ion su s e d to p re d i c t m e rc h a n tabl evol umesof pl antati onthi nni ngs, for both researchand co m m er c ialpur p o s e s ,a re p re p a re dfro m a n a nal ysi s of sampl etree data. The current vol umeequati ons for the major plantation species, hoop pine ( Araucaria cunninghamii Ait. ex D. Don), slash pine (Pinus elliottii Engelm"var ellionii) and Honduras Caribbean pine lP. caribaea Mor. var. hondurensis Ba r r . et G olf . ) , w e re d e ri v e d a fte r a n a l y s i s o f over 1000 sampl etrees of each speci es. M eas ur ings am p l e tre e s i s a n e x p e n s i v ee x e r ci se, and i t i s expedi entto determi nethe opti mumnumbe rof sam plet r ees r eq u i re dto m a i n ta i n a re a s o n a b l ysmal l confi dencei nterval about the equati on,w hi l e keeping the num berof s a m p l etre e s to a m i n i m u m . Sa m plet r ees ar e c u rre n ti y o b ta i n e da s re q u i redw hen sui tabl e researchpl ots are thi nned, and duri ng rout ine t hinning o p e ra ti o n s . T a b l e s i n d i c a ti n g the sampl i ng i ntensi ty and di stri buti on of sampl etrees (Table 1) ar e pr o d u c e dp e ri o d i c a l l y to d e te rm i new herefurther sampl i ngi s requi red. N o prescri pti onfor sam plingex is t s , b u t s u g g e s ti o n sv a ry i n g fro m 2to10 trees per cel l r, have been madefor a sui tabl e s a m p l i n gi n t e n s i t y .
BACKGROUND Merchantable volumeunderbarkto7 cm top diameter(t.d.u.b.)is estimatedby an equation: Vr:a+bA+cH+dAH wh e re d , b , c a n d d a re c o n s tants A i s s e c ti o n a l a re a b re ast hi gh over b a rk (s q m ) and H i s p r e d o m i n a nht e i g h t ( m ) .
(1)
T h i s e q u a t i o ni s u s e d f o r c o m m e r c i a pl u l p s a l e s , a n d i s t h e b a s i s f o r e q u a t i o n sw h i c h e s t i m a t ev o l u m e to 10, 12 or 15 c m t.d . u .b . T h e s e e q u a ti o n sa re determi ned as the di fferencebetw eenequati on (1) and e q u a t i o n sw h i c h e s t i m a t et h e v o l u m eb e t w e e n7 c m a n d 1 0 , 1 2 o r 1 5 c m t . d . u . b . :
Vr-" = a' + e/(A + g) + fHl(A + g) w h e r e x = 1 0 ,1 2 o r 1 5 c mt . d . u . b . and d' , e, f and g are constants.
(2\
Ta ble 2 illus t r at e s th e c l o s e re l a ti o n s h i pb e t w eenthe confi dencel i mi tsof the V , and deri ved V ,, e q uat ions . lt is a s s u m e dth a t th e b e h a v i o u ro f the V ,o, V ,rand V * equati onsw i l l fol l ow that of the bas e V , equa ti o n . Of the equations with the form (1) used by the Department,one of the most satisfactory is the volume e q u a t i o n0 1 0 2 7 3 ' , w h i c he s t i m a t e st h e m e r c h a n t a b l ve o l u m eu n d e rb a r k , t o 7 c m t . d . u . b . a n de x c l u d i n ga 1 5 c m st um p, ofhoop pi n e i n th e l m b i l re g i o n . D u ring the ni ne years thi s vol umeequati onhas been i n use, a n u m berof s t udie s h a v e fo u n d i t to b e e n ti re l y sati sfactory. l ts characteri sti csare exami nedi n Tabl e s 3 and 4. Table 3 is derived from part of Table 2, and shows the volumes as well as the percentagesin the co nf idenc er egio n . T a b l e 4 c o n ta i n s d a ta fro m a saw i ng study i n hoop pi ne (D .K . Gough,unpubl i shedr epor t ) whic h f ound no d e fi c i e n c i e s i n th e v o l u me e q uati ons. L
Throughout this paper, a cell refers to a I cm d"b"h. x 2 m predontinant height region of the data space as tabulated in Table 7. D"b.h" denotes diameter breast height overbark and predominant height is the mean height of the tallest 5O stems/ha measured on the basis of I stem per 0"02 ha unit.
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T he at t r ibute s w h i c h d e te rmi n eth e a c c e ptabi l i ty of thi s equati onare not readi l y determi ned. N o s ingle par a m e te ri s s u ffi c i e n t to d e s c ri be the behavi ourof the confi denceregi on about the equati o n. A total of 13 parametersrare required to fully define the confidence region about an equation of the form (1). Perhapsthe most useful way to express the requirementfor a satisfactory volume equation is t o s pec if y th e m a x i m u mc o n fi d e n c ei n terval permi ttedovera gi ven range of tree si zes. Conf idenc ere g i o n s m a y re fe r to a n i n d i vi dual , the meanof a sampl eof a numberof i ndi vi dual s, or t o t he populat i o n . T h e re i s n o re q u i re m e n that t a predomi nanthei ght vol umeequati onshoul d yi el d an ac c ur at ees ti ma te o f v o l u me fo r a s i n g l e tree. Indi vi dual tree vol umescan be better esti matedby t ot al heigf t tv olu me e q u a ti o n s . T h e p re d o m i nanthei ght vol ume equati onsare more correctl y appl i ed to t he m eanof a n u m b e ro f tre e s a s i n c o rn me rci alti mber sal es. l t i s conveni entto nomi natea sampl e size of 100 as a s ta n d a rdfo r c o n fi d e n c ei n te rval studi es. A l though 100 stems i s a commonsampl e si ze in r es ear c hexp e ri m e n ts ,i t i s a t l e a s t a n o rder of magni tudesmal l er than any commerci alsal e of ti m ber . T he m inim u mm e rc h a n ta b l ed i a me te rb re a st hei ght overbark(d.b.h.) di ffers for pul p and saw l og sales, but t he s m a l l e s t me a nd .b .h . e n c o u n te re dfor:a sampl eof 100 trees i s l i kel y to be 20 cm d.b.h. S im ilar ly, 40 c m d. b. h . i s a re a s o n a b l ee s ti ma teo f a maxi mummean di ameterfora sampl e of 100 trees fel l ed dur ing a r out ine t h i n n i n g o p e ra ti o n . An i n s p e cti onof Tabl es 3 and 4 reveal s that thi s range of di ameter sis likely t o enc om pa s smo s t ro u ti n e fe l l i n g o p e rati ons. l nspecti on of the D epartment' sLog B ank (a compu t erf ile of dat a f r om c o m m e rc i a lth i n n i n g o p e ra ti o n s)reveal s that appropri atepredomi nanthei ghts are 1B m fo r t he - . 20 c m d. b. h . l i mi t a n d 3 5 m2 fo r th e 4 0 c m d.b.h. l i mi t. These val ues of predomi nanthei ght do not pr oduce t he s m alles t c o n fi d e n c e i n te rv a l s a b o u t the regressi onsurface at the speci fi ed d.b.h., but repres entt ypical pr edom ina n h t e i g h ts fo r th e s p e c i fi e d d i a metersi n recent sal es l Tabl e 5). Tabfe 5.
Behaviourof the 95 per cent confidence interval (C.1.) about volume equation O1OZ7}
D.b.h" (cm) Min. pre. ht Me a n p re . h t M a x . p re . h t M i n .C . l .
40
95% C.l. for 100 stems
Predominant height (m)
Min. pre. ht Mean pre. ht M a x . p re . h t M i n .C . l .
Volume (cu ni)
(cu ml
%
14 17 38 23
0.159 0 . 18 5 0.364 0.236
0.0147 0.0143 0.0158 0.0139
9.2 7.7 4.3 5.9
30 37 40 29
1.454 1.77..0 1.905 1.409
0.0194 0.0270 0.0326 0.0193
1.3 1.5 '1.7 1.4
B as edon t h e p re c e d i n gd i s c u s s i o n , a s a ti sfactoryvol ume equati onmay now be defi ned as an equa t ionf or w h i c h t h e 9 5 p e r c e n t c o n f i d e n c ei n t e r v a la b o u tt h e m e a no f 1 0 0 i n d i v i d u a l sl i e s w i t h i n 1 0 p e r c e n t o r 0. 020 c u m , w h i c h e v e r i s th e l e s s e r, o f the regressi onsurface,for both sampl esw i th 20 cm d.b.h. and 1B m pr edomi n a n ht e i g h t, a n d s a mp l e sw i th 40 cm d.b.h. and 35 m predomi nanthei ght Thi s requir em ent is r igor ous,a n d T a b l e 6 s h o w s th a t v o l u meequati on010273,al thoughgeneral l ysati sfactory,fai l s t o m eet t h i s s t a n d a r da t 4 0 c m d . b . h . T h e c o n f i d e n c er e g i o na s s o c i a t e dw i t h a v o l u m e e s t i m a t ef o r a c o m m e r c i a l t h i n n i n go p e r a t i o nw i l l b e s u b s t a n t i a l l ys m a l l e rt h a n t h e r e g i o ns p e c i f i e da b o v e . t
6 parameters in the upper right sguares and products 3 parameters I parameter
representing
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T, n and AV
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means)
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I parameter f or the number of sets of data I parameter
f or the number
I parameter for student's degrees
of indiv idua ls under cons ideration t at the reguired
level of significance,
with the appropriate
of freedom.
35 m predominant height is lower than the mean value for 40 cm d.b.h. hoop pine at lmbil, and higher than the mean for Yarraman hoop pine and for slash and Caribbean pine, lt is a good approximation of overall mean predominant height.
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