Tree volume equations for slash pine

Tbee volume equations for slash pine by J. K. Vanclay rssN1035-9788 Tree volume equations for slash pine by K. Vanclay J. ABSTRACT New Eee volume equationsare given for slash pine in south-eastQueensland- The equationsestimate merchantable volume from diameter at breast height and sand predominant height, and total and merchantablevolumes from total tree height and diametersat breastheight and at five metres. INTRODUCTION Vanclay and Anderson (1982) identffied shorrcomingsin the current volume equations for slash pine (Pinrc elliottii Engelm. var. elliottii) and recommended that new equations be developed. They identified a need for two equations:one for commercial use, and a morc detailed equation for research use. Equations should provide a good fit to the data result in reasonableestimateswhen extrapolated o extreme sizes (bottt small and large), ?trd fulfill the statistical requirementsin having residuals which exhibit a normal distribution with zero mean and constant variance. The Australian equation (as described by Spurr 1952, p. 94)t has traditionally been used by the QueenslandForest Service to estimate stem volumes (Vanclay and Shepherd1983). It provides a tolerably good fit o the data, is easy to comput€, and can be interpolatedbetween(or exrapolated beyond) abulated values. However, it may yield poor estimatesof volume in small stems (Vanclay 1980), and frequently fails to meet the statistical requirementin linear regressionof independentlyand identically distributed residualswith zero mean and constantvariance. Equations are required only to predict total and merchantablevolumesf. Volumes to other utilization standards(e.g. 10, 12 and 15 cm t.d.u.b.)$ may be predictedusing volume ratio equations(Vanclay 1982b). METHODS Data from the Department's sample uee library were used in the sordy. The library contains detailed measurementsof trees of various species from diverse locations within Queensland(Vanclay and Shepherd 1983). It includes over 4000 slash pine sample trees, collected from plantations and provenancefials on state fcests at Beerburrum in south-eastQueensland. Data collection procedures have resulted in a data basewhich containsa diqproputionate number of sampleuees from experiments, and so does not reflect the composition of the forest estate. Stem form of rees from experimentsmay differ ftom the norm in plantations. This occurs becauseof different seed sourcesand different site preparation, sprcing, tending, fertilising, thinning and pruning histmies. Sample rees derived from experimentsmay be used to provide daa for volume equations. When such equations are applied o routine planation thinnings, biased estimatesof volume may result. The problem is aggravatedby the large numbersof sample rees of similar dimensionsfrom many experiments. For example 756 sample uees of similar size, frorn a single location and seed source, were derived from one qpacing rial (Vanclay and Andenon 1982). The (so-called)Australiancquationhas thc form V=a+M+cfil+dr{If, where Vir tree volunre,A is trec basalare! ard H is rtand heighr It can algobccxpressedac a stsndvolume cquation: V=rAI+M+drtlI+&{H, whereA is stsnd brcd ttct, /V is the numbcr of ctcmr in the stand,ud trI, 1b,c rnd d are thc lrmc ls ebove. Totel volumc rcfers to total ctsm volume cxcluding r 15 crn $rmp. MerchantaHevohmrc is defined as st€rr vohmre to 7 cm top diamcter under bart cxcluding e 15 crn filrnp. t"d.u.b.ir topend diametcr urder borlc Vanclay (1982a) observed that the sample nee library generally contains more sample trees of any speciesthan is necessaryto constructa reliable volume equation. Thus one solution to the problem is to select a suitable subsetfrom the library, using sratified random sampling to samplethe full range of the regressff variables while restricting the number of stems in each stnahrmto an ryproximately equal number. This should ensure that the data set used in regressionanalysis more accurately reflects the composition of the planation estate. Poor performance of the Australian equation in estimating volumes of small trees has been noted (Vanclay 1980). In order to povi& improved estimatesfor Eeeswith merchantablevolumes near zero, the regressionmay be constrainedto passthrough some qpecifiedpoint near the uigin. There has been considerabledebate conceming the validity of defining such a point fu volume tariff syslemsf (e.g. Hummel 1955; Prodan 1965; Carron l97l; Rennolls and Tee 1979). However, argumentspresentedby the various authors do not apply to equationsassumingcurvilinear relationships benveen volume and basal area An efficient way to constrain the regression is o explicitly identify the tree of zero merchantable volume. Sample tr€es smaller than l0 cm diameterbreasthigh over bark (DBHOB) were selectedfrom the library, oDd a relationship was establishedbetweenDBHOB and diameter under bark at a stump height (DUBSfi of 15 cm aboveground: Ln(D UBSI{) = 0.9792 Ln(DBH OB). The regressionon 9l data points resulted in a multiple correlation coefficient (r) of 0.88. The standard error of the estimatedparameteris 0.003705. From this relationship,it was deducedthat the theoretical volume has a diameterof 7.3 cm (d.b.h.o.b.)and a basal area of 0.00418 uee of zero merichantable squarcmetres(over bark at breastheight). RESULTS Commercial Volume Equation hactical considerations dictate that merchantablevolume equations for commercial use must be a function of predominant heighg and d.b.h. However, Vanclay and Anderson (1982) found that these two parameterswere insufficient to accurat€lyestimatethe volume of individual trees, which may also be influenced by spacing, site preparation and genetic make-up. Thus it is convenient to produce a volume equation to be used only for stoms from standsconforming to the curent managementpractice (e.g. current seedsourses,site preparationand spacing). To ensurea rcpresentativeset of datrytreeswere selectedfrom two subsetsof the sampleuee library: . sampletreesfrom standsinitially plantedat spacingsin the range2.4 x 2.4 m to 3.0 x 3.0 m, which encompassesthose currently encounteredin plantation sales;and . the moe recently rcquired sampletrees. Of the 644 trees meeting theserequirements,a random selectionwas made to yield tttree stemsin each cell of one centimetre d.b.h.o.b.by two meres predominantheight (Vanclay 1982a). Two outliers were discarded,leaving 642 sampleEeesfor subsequentanalyses. f Volume tariff systemsare volume lines of the form V= a * M, ard inclu& the Aucnlim equatim (Canon l97l). t kedominant height is deftred ac the meenhcight of the rellest 50 cterns/hecurc,semplcdet thc ntc of qrc stcflr pcr 0.@ ha. The generalisedform of the model was Vtl(BA-0.00418)=f(DBH,PI{)i V7 is stem volurne to 7 cm t.d.u.b., BA is basal area in square meEes, and f(DBH,PI, implies some linear combination of tenns involving diameter and p'redominant heighr The mo&l resulted in a satisfactory distribution of residuals and explicitly identifid the intercept at zero volume. This model performed better than a number of popular models, irrcluding the Australian equation and the 'Schumacher' equation (Schumrcher and llall 1933). Equation (l) was adopted. Resulting volume estimates and their associated95Vocnnfidenceregions for a parcel of 100 stemsare given in Table l. vt = @A - 0.00418)(0.02326D8H+ 0.3469pH + rr.96lDBH - 0.00l692pHlBA) ..............(1) Teble l. Nlnety.five pcr cent con0dencc reglon rbout estlmrtcs of Orc volume d r percel of 100 stcmg ec crk{haed from cquetlon (1). Mern d.bft.o.b. (cm) Meen stend predcnlnenl helght (metrcs) 10 15 lot 0.010 o.(m 0.012 0.0(B 2 t% 0.015 0.0c8 22% 0.149 0.011 7% 0.189 0.012 7% o.229 0.014 6% o.268 0.016 6% 0.308 0.01E 0.395 o.429 7% 0.5u} 0.081 6% 0.610 0.034 6% o.7t7 o.a37 5% 0.825 0.041 5% 0.960 0.ffi7 7% 1.162 0.flo 6% 1.365 o.m4 5% t.567 0.080 5% 1.895 0.130 7% 2.220 0.134 6% 2.545 0.1,CI 6% 3.293 0.2,4 7% 3.768 20% 20 0.109 0.010 9% 30 40 50 35 @ t 6lo o.230 6% CEll entries are: top lirc - estimlted volume (cu m); middle line - 95% cqrfidenoe intcrvd about 100 stems (cu m); ard lower line - confidencc intcrvd expmred rs per cent of estimatc. Vanclay (1982a) defines a satisfactory volume equation as 'an equation for which the 95Voconfidence interval about the mean of 100 individuals lies within l0% or 0.020 cu m (whichever is the lesser) of the regressionsurface, for bottr sampleswith 20 cm d.b.h. and 18 m predominant height and samples wittr 40 cm d.b.h. and 35 m predominant height'. However, as that study took no account of the heteroscedasticdisribution of residuals,the confidenceregion about the larger stems (exceeding40 cm dbh) would have been sulstantially underestimated. Thus while this equation fails to meet Vanclay's (1982a) criteria, it should yield volume estimates within sevenper cent of the tnre value in 95 per cent of cases. This woutd apply for salesas small as 100 stems. Iarger sales will be predicted even more accurately. Information necessaryfor computing confiderrceintervals about the regressionsurfaceis given in the Appendix. I I :: i ' l Two-way Equation for Merchantable Tree Volume When measurementsof total tree height are available, a more accurat€estimateof tree volume may be obained using equationswith oal height rartrer than predominantheight as a regrcssu variable. Such equationsare of lirle commencialimportance,but are useful in evaluatingresearchresults. Howevetr,as two-way volume equationsdo not embracevariables such as stem form and bark thickrpss, they cannot accurately predict volume under all circumstances,and use must be restricted to stems from stands conforming to the cunent man4ge,ment practices. A subsetof 546 sampletrees was selectedusing the samecriteria outlined earlier, but using cells based on total ree height rather than stand predominant height. Several models were investigated, and a model similar to equation (2) was adopted" (IIl is total height and V7, DBH and BA are as previously defined). Volume estimatesand their associated95% corfidenceregions are given in Table 2. Vz = @A - 0.00418)(0.4258TH - O.A2$7DBr{) Trble 2. Meen d.b.h.o.b. (cm) rot 20 ......e) Nlncty-five per cent conffdencereglon ebout estlmetes of the volume of I percel of 100 stemg es crlculeted fran cquetlon (2). Meen totel trec helght (metres) l0 15 0.015 0.ml 4% o.v23 0.001 3% 0.(R0 0.001 3% 0.lq} 0.161 0.06 4% 0.2t9 0.006 3% o.n7 o.376 0.017 5% 0.m5 5% 30 30 35 0.ffi/ 3% 0.335 0.offi 2% 0.393 0.009 2% 0.51t 0.01t 3% 0.659 0.019 3% 0.801 0.021 3% 0.943 0.02,3 3% 0.916 0.q,9 4% t.t75 0.040 3% t.434 o.u2 t.692 0.045 3% 1.812 50 o.w4 4% 3% 2.221 o.w6 3% 2.630 3.151 0.125 4% 3.744 0.128 3% o.w9 3% t Cell entries are: top line - estimatcdvolunr (cu m); middb linc - 95% cmfidcncc intcrval ebout 100 stcnrs(cu m); rnd lower line - srfidcnce inrcrvd expreued ar per cent of estimate. Confidenceinterrralsmay be computedusing the daa given in the Appendix. The confidenceregions in Table 2 are smaller than those for the commercial volume equation (equation (l), Table l). Although the trees are not directly comparable,as they are based on different daut sets, the difference may be attributed in part to the use of total tree height rather than sand predominantheight. Two-way Equation for Total Tree Volume Equations to predict total Eee volume are useful in assessingearly volume increment in experimental plots. The data subsetused in developing the two-way equationfor merchantabletree volume Clable 2) was again employed, and various Eansformationswere consideredin an attempt to stabilise the variance. The most suitable transformationenrployedWl(BAxTm as the dependentvariable. The selectedmodel (equation (3)) is in effect a refinement of the constant form factq equation (Spun 1952). Volume estimatesand their associatedconfidenceregions are given in Table 3. W=(-0.4341D8H+r.357fTH+0.l056LnQn)(BAxTm Teble 3. Nlncty-fivc pcn cenl confldcncc rcglon rbout ccdmeter of the volume of r percel d lm *arg rr celcuhted ?rmr cquetlotr (3). Mern totel tree hdght (metres) Meen d.b.ho.b. (cm) t lof 20 l0 30 15 o.CI26 o.qn 2% 0.039 0.001 2% 0.054 0.001 2% 0.112 0.(xI2 o.t6l 0.(m o.D3 0.m5 2% 2% o.228 0.fix 2% 0.384 0.006 2% 30 /rc 50 35 2% 0.361 0.005 2% 0.432 0.006 t% 0.523 0.00E 2% 0.671 0.010 2% o.8n 0.012 t% 0.989 0.014 t% 0.938 0.015 2% t.2M 0.01E 2% 1.484 O.UZL t% 1.774 0.025 t% t.892 0.02,8 t% 2.331 0.(}34 t% 2.787 3.3@ 0.049 l% 4.428 0.057 l% @ t ...............(3) 0.(Xo r% CeU entries lre: top line - estimated volurnc (o m); middle line - 95% cmfidencc intcrvd ebotrt 100 stems (cu m); and lower line - cqrfidomce intervd expressed EEper cent of estimetc. The confidence regions about the total tree volume equation (Table 3) are smaller than those of the merchantable tree volume equation (Iable 2). As the same daa set was used in both analyses, confiderrceregions should be directly comparable. Improvement in the confidenceregions implies ttrat total volume can be pedicted more reliably than can merchantablevolume. The improvement also highlights the inadequaciesof usilg the coefficientof determination(t2) o discriminate-models, as the toal volume equation yields an 12 of only 18 per cent while the merchantablevolume equation (Iable 3) yields 93 per cent (seeAppendix). I I Three- and Four-way Equations for Merchantable Tree Yolume Vanclay and Andenon (1982) recommendedttre adoption of a volume equation incorporating an upper stem diarnet€r as a regressorvariable. The purpose of a&pting such an equation is o enable more accurateestimation of stem volumes in standsexposedto non-standardtneatmarts(e.g. prcvenarrcetrials, spacing experiments). Both upper stern diamet€rsand bark thicknessat breast height were investigated as possible regressorvariablesin ttree- and four-way volume equations. Upper stem diameten at heights of two and five metreswere considered,as theseheightsconform to the curent presaiption for sample uees (Vanclay and Shepherd 1983). Heights great€r than five metres were consideredimprrctical. Prreliminaryanalysesindicated that diameter at two metres (d.2) offered little improvement on the basic model. Diameter at five metres (d.5) resulted in a substantial improvementin a model feauring d.b.h. and otal heighr Stem taper (botr d.5 - d.b.h. and d.2 - d.b.h.) was also examined, but offered rn improyement over d.5 as a rcgressorvariable. Although it may be difficult to measur€d.5 using a ginh tape and ladder, measurementshould be practical using height poles and an optical instrument such as the pentaprismcaliper ffieeler 1962). Bark thicknessat breastheight contributedrelatively little to the regression. Becauseof the difficulty of obtaining an accurate and repeatablenondestnrctive measurementof bark thiclness (Carron 1968, p. 46), equationsare given incorpoating both d.b.h.o.b.ard diameterbreasthigh under bark (d.b.h.u.b.). A subsetof data was selectedftom the sample ree library, sampling the range of d.b.h., d.5 and otal height measu€ments. In selecting data used for equations(l), (2) and (3), an anay in the computer's memory was rccessed. In this case, a different approach was required. The toal and merchantable volumes were comprted for erch tree, and the d.b.h.o.b.,d.b.h.u.b.,d.5, otal height, merchantable volume, toal volume, and a random number were wriuen to a file. This file was then sorted according tJothe random number. Finally, a subsetof 1479 trees was producedby selecting the first three nees encounteredin each cell. Cell sizes were one centimetre d.b.h. by one centimetre d.5 by two metres toal heighr Of the several variance-sabilising functions considered,the logarithmic transformation produced the most desirabledisribution of rcsiduals. However, this transformationresulted in severallarge residuals amongstthe very small stemsQessthan l0 cm d.b.h.). To overcomethis, stemswith d.b.h.u.b.lessthan l0 cm wer€ discarde4 leaving 1383 rees for furttrer analysis. The final models are: Ln(Vr)=-9.016+0.l4O8TH+12.468H-0.001512TH2+ l.768Ln(D5)-8.297tD&H+0.002........,(4) and llr(v?)=-8.509 +O.L?9.. 9TH + rr3rlrr - 0.001307THz + l.675Ln(D5)+ l4.96lDBH-zO.rTlDUB + 0.001 ............(5) where D5 is diameter over bark at five meEes,DUB is d.b.h.u.b.,and other terms are as previously defined. The final tenns (0.002 ard 0.001 respectively) representa correction eqrul to half the error mean square (Wuner 1985). The ccrection is included to adjust for the logarithmic transformation discrepancy(Husch et al. 1982), the small bias introducedby the transformation. Details for computing confiderrceregions about the equationsarc presentedin 0re Appendix, Innritively, one might expect a better result ftom a variation on the Schumrcher eqgation such as the logarithmic-fsm diameter formula (describedby Spun 1952,p. %); or from equationssuch as: Ln (V ) = a + bLn(DBIt) + c In(?If) + d Ln(SF); Ln (V)= a + bIn(DUB) + c II(TI/) + dIn(SF); and Ln (V7) = a + bLn(DBI{) + c In(TIl) + d Ln(SF)+ eLn(RB?). (Sf is stem form expressodas DSIDBH; RBTis relative bark thictness expressedas DUBIDBII). However, there are tlvo problemswith theseformulations: ' estimated merchantablevolume (Iz7) approrcheszero as d.5 approacheszero - in practice, a Eee five metreshigh may have a significant volume exceedingsevencentimetnesdiametec and ' the correliationbetweenstem form (DSDB$ and d"b.h. meansOru the parameter'd' (preceding the Ln(SF) term in eachof the ttree equations)cannot be reliably estimated" Both theseproblems could be overcomeby defining stem fsnr as the ratio of fumquotient diameter to d.b.h. (Spun t952, p. 95, defines form-quotient diameteras diameterover bart at height (rH - B)fD. This is more difficult to measure,ard is thus of less utility. Three. and Four-way Equations for Total Tree Volume The same data set, and the same variance-stabilisingfunction wer€ also used for otal volume. The models are: Ln(W)=-9.2?5 + 0.1359TH + r3.37fTH- 0.m1307TH2 + 1.785Ln (D5) - 4.3BB!DBH + 0.002........(6) and Ln(W)=-8.814 + O.l2|lTH + t2.$frH - O.ml t4tTH2 + t.7WLn (D5) + t4.52lDBH- t6.40tDUB CONCLUSIONS Several new equations for merchantableand total volume of slash pine in south-eastQueenslandare presented. These equatiurs r€presenta developmenton previously publistred work (Vanclay and Shepherd 1983). Volumes to utilisation standardsother than sevencentimetresLd.u.b. can be obtairrcdgsing the merchantrable volume equationsand a volume ratio equation(Vanclay 1982b). ACKNOWLEDGEMENTS The work reported here forms part of the forest researchprogamme of the QueenslandForest Service, and was originally canied out in 1984. I am indebted to the officers of ttp Service who have been involved, over the years, in collecting the sample trees and maintaining the sample tnee library. I am also grateful to N.B. Henry, MR. Nester and G.B. Wood fs commentson the draft manucript. REFERENCES 'An Outline of Faest Mensuration with Special Reference to Australia'. Carron, L.T. (196t). (AN.U. Press:Canberra). 224 W. Carron, L.T. (191). Volume taritr systems.Forestry 44Q): 145-150. Hummel, F.C. (1955). The volume - basal area line. Forestry Commission Bulletin No. Z: &4 pp. (H.M.S.O.:Iondon). 'Forest Mensuration'. 3rd edn. (Wiley: New YorD. Husch, 8., Miller, C.I. and Beerc, T.\lY. (19t2). 4V2W. Prodan, M. (1950. A simplification of the volume tarifr systems. SecondConferenceof the Advisory Group of Forest S6tisticians of IUFRO, Stockholm. Secr 25. Rapp. Uppsats. Instn. Skoglig Mat. Statist.Skogshogst:No. 9. Rennolls, K. and Tee, V. (1979). Estimation of the volume of a stand using a tariff procedure. /n 'Planning, Performanceand Evaluation of Growth and Yield Snrdies', pp. Wright, HL. (Ed.). 9l-99. (CommonwealthForesry Instiurte: Oxford). OccasionalPaperNo. 20. Schumacher,F.X. and Hall, F.dS. (1933). Iogarithmic expressionof timber-treevolume. fowrcl of AgriculturalResearch4il: 7 l9-7y. 'ForestInventofy'. (RonaldPressCo.: New YorD. xii + 476 W. Spurr, S.H. (1952). Vanclay, JJ(. (1980). Small tree stem volume equatiurs for three plantation species. Qld. Dep. For. Res. Note No. 32: 5 pp. Vanclay, J.K. (19t2a). Optimum sampling of sample trees for volume equations. Qld. Dep. For. Res. Note No. 35: 15 pp. Vanclay, JJ(. (19t2b). Volume to any utilization standardfor planation conifers in Queensland. Qld. Dep. For. Res. Note No. 36: 8 pp. Vanclay, J.K. and Anderson, T.M. (19t2). Initial sprcing efrecs on thinned stem volumes of slash pine in south-eastQueensland. Qld. Dep. Fm. Res. Note No. 34: 9 pp. Vanclay, J.K. and Shepherd, PJ. (19E3). Compendiumof volume equationsfor plantation species usedby the QueenslandDepartmentof Forestry. Qld. Dep. Fu. Tech. PaperNo. 36: 2l W. In \ilarner, AJ. (19t5). Developmentof a radiaa pine tree volume equation for APPM plantations. 'Modelling (Eds.). B.M. R.D. Spencer, urd P.W., Spencer, R.E., West, McMurtrie, Leech,J.W., Tr€es,Sands and Faests', 1ry.A5-29. Univ. of MelbourneSch. of Fa. Bull. No. 5: 559 pp. Wheeler, P.R. (1962). Pentaprismcaliper for upper stem diameter measuements. J. For. 50(12): 877-887. APPENDIX Parameter Estimates and Standard Errors for SevenVolume Equations Details of parameter estimates and standard erronl of estimatesare gven below for all the volume equations. Confidenceintenrals may be computedusing the formula V f t r s l +n. i i EiE where c' ; i x ; x i I is the estimatedvalue of the dependentvariable; t is Student'st; S is the squareroot of rhe ResidualMean Square(RMS); n is the number of stems in the parcel for which the confiderrceinlerval is o be computed; cij is an elementfrom the inverseof the uncorrectedcrcss productsmatrix; ri, ri are the valuesof the regressc variables(e.g. DBH, PH, TH, llDBH, etc.); and p is the numberof variables(including constailts)in the regrcssion. Merchantable Volume from Predominant Height Equation(l): N = 642, R2 = 0.897. v/(BA - 0.00418) = Estimate o.oz3zffiDBH + 0.3469rPH + rr.9&IDBH - 0.m169r7 PHIBA Shndard Error (0.008931) (0.01092) (2.986) (0.m02343) RMS = 0.7265. C matrix DBH PH \IDBH PHIBA 7.9758 -9.5370 -1.8414 r.6725 DBH -5 -5 -2 { r.20y 4 2.M2 -2 -2.m23 4 PH 8.9170 4.5594 4 IIDBH 5.4889 -8 PHIBA l0 Merchantable Tree Volume fFom Total Height Equation(2): N = 546, R2 = 0.930. v/@A - 0.00418) = Estimate 0.425837H StandardError (0.Wt299) - o.aa373DBH Q.m5772) RMS= 0.5076. C matrix 53n5 -5 4.r2r2 -5 TH DBH TH 3.3312 -5 DBH Total Tree Volume from Total Height Equation(3): N = 554, R2 = 0.179. wl(BAxTm = Estimate - O.4Y36|DBH + r.3567lrH + 0.10561LnQI{) Error Standard (0.1004) (0.07180) (0.0008081) RMS = 0.0008689. C matrix \IDBH rtrH Ln(TA 1.0080 --2 -6.5639 -3 -3.3626 -5 IIDBH 5.1557-3 4.2519 4 rftH 6.5295 -7 LnQA ll Three- and Four-way Equations for Merchantable Tree volume Three-way equation Equation(4): N = 1383,R2 = O.gn. Ln(V) = Estimate -9.0160 + 0.14078TH Sandard Error (0.1834) (0.m6783) .]3:ff1f.{",* (0.6827) (0.wr227) :'r'lffiE;:;'(0.w4%) (0.5527') RMS= 0.003786. C matrix 3.3652-2 -r.0793 -3 I TH VTH rH2 Ln(D5) UDBH -{.1l120 1.9639 -5 -3.6519 -3 -7.861l. -2 I 4.6003 -5 4.4652 -3 -8.2510 -7 6.8628 -5 r.4620 -3 TH 0.4661 -7.8546 -5 7.7395 -3 0.1584 VTH 1.5067 -8 -r.3673 4 -2.7917 -5 TH2 6.2207 4 r.3M7 .-2 Ln(D5) 0.3053 UDBH 4.2331 4 5.5500 -3 2.617 -3 Ln(D5) 0.96520 4.6r,2W 0.57410 \IDBH rDaB Four-way equation Equation(5): N = 1383,R2 = 0.998. Ln(V) = Estimate -8.509.1 + O.lD9lTH Standard Error (0.1503) (0.m5528) (0.5566) (0.0001m1) (0.02057) (0.e825) (0.7s77) .]'o:.Xrffior,r, + 1.6747Ln(DS) + l4.958lDBH - 20.r6lDUB RMS= 0.002501. C matrix I TH rfTn rH2 Ln(D5) IIDBH IIDUB 2.2594 -2 -7.2079 4 -7.4283 -2 r.3r2r -5 -2.4795 -3 -3.5304 -2 -r.421 -2 I 3.0558 -5 2.9676 -3 -5.4823 -7 4.6773 -5 6.W02 4 3.@6 4 TH 0.30980 -5.2223 -5 5.265r -3 6.682n -2 3.2803 -2 VTH r.0013-8 -9.3030 -7 -l.l73t -5 -5.8207 4 r* 12 Three- and Four-way Equations for Total Thee Volume Three-way equation Equation(6): N = 1383,R2 = 0.996. Ln(W) = Estimate -9.2259 + 0.r3594TH + l3.374fTH SandardError (0.1835) (0.00678t (0.682e) - o.ffir3o7orriz (0.000r228) + 1.7845Ln(DS) 4.38771D8H (0.02res) (0.552e) RMS = 0.003788. C matrix 3.3675 -2 -r.0800 -3 -o.r l l30 1.9653 -5 -3.6y5 -3 -7.8ffi -2 I I TH VrH TII2 Ln(D5) UDBII 4.ffi35 -5 4.463 -3 -8.2568 -7 6.8676 -5 1.4630 -3 TH 0.4ffi -7.8600 -5 7.749 -3 0.15850 VTH 1.5078 -8 -1.3683 4 -2.7936 -5 TH2 6.2251 4 1.3056 -2 Ln(D5) 0.30570 UDBH 4.9762 4 6.5U3 -3 3.1289 -3 Ln(DS) 1.1350 4.77830 0.674n \IDBH UDUB Four-unayequation Equation(7): N = 1383,R2 = O.gg7. Ln(W) = Estimate -8.8141 + O.lTllOTH StandadError (0.1630) (0.005e94) (0.603s) (0.0001085) + 1.7085Ln(DS) (0.02231) + l4.Sr9lDBH (1.065) - r6.396lDUB (0.8215) .]'o.ff{{{*,,,' RMS= 0.002940. C matrix I TH VTH rH2 Ln(D5) IIDBH IIDUB 2.656r .-2 -84733 4 -8.7323 -2 r.yu -5 -2.9147 -3 4.1502 -2 -r.6952 -2 I 3.5923 -5 3.4885 -3 4,4448 -7 5.4984 -5 7.1593 4 3.6379 4 TH 036/,?n {.1391 -5 6.18% -3 7.8551 -2 3.8562 --2 VTH r.r77l -8 -1.0936 -6 -1.3790 -5 4.8/,25 4 r*