FAO ForestForest volumvolume FORESTRYFORESTRY PAPER estimationestim tion andand 22/122/1 yield predictionprediction

Vol.Vol. 1i- - VolumeVolume estimationestimatbn by F. CailliezCailliez Centre technique forestier tropical Nogent-sur-Marne, FranceFrance

FOOD AND AGRICULTURE ORGANIZATION OFTHEOF THE UNITED NATIONSNATIONS Rome, 19801980 RepriReprintednted 1992

The designations employed and the presentation of material in thisthis publication do not imply the expressionexpression of any opinionopinion whatsoeverwhatsoever onon thethe part of thethe FoodFood andand AgricultureAgriculture OrganizationOrganization of thethe UnitedUnited Nations concerning the legal status of any country, territory, citycity or area or of its authorities,authorities, oror concerningconcerning the delimitation of itsits frontiers or boundaries.

M-35M-35 ISBN 92-5-100923-692-5-100923-6

All rights reserved. No part of this publicationpublication maymay bebe reproduced,reproduced, stored in aa retrieval system, or transmittedtransmined inin any formform or by any means, electronic, mechani-mechani­ cal, photocopying oror otherwise,otherwise. withoutwithout thethe priorprior permissionpermission ofof thethe copyright owner. Applications forfor suchsuch permission,permission, withwith aa statement statement of of the the purpose purpose and and extent extent of of the the reproduction, shouldshould bebe addressedaddressed to to thethe Director,Director, PublicationsPublications Division, Division, Food Food and and Agriculture OrganizationOrganization of the United Nations,Nations, VialeViale delledelle TermeTerme didi Caracalla,Caracalla, 00100 Rome, Italy.lIaly.

© FAO FAO 1980 1980 Mr. F. Cailliez acknowledges with thanks the assistanceassistanoe provided by his colleaguesoolleagues in writing this manual, namely J. Bouchon, P,P. Duplat,Duplat, F.F. Guinaudeau,Guinaudeau, andand N.N. Ogaya.Ogaya. Thanks also go to Miss C. Gueguen who did the drawings and sOIDesome typing in French and English. FOREWORD

There is probably little argument amongamong forestforest managersmanagers thatthat thethe abilityability toto estimateestimate the volume of trees andand standsstands and toto predictpredict whatwhat thethe forestforest willwill produce,produce, onon differentdifferent sites, in response to particular typestypes ofof silviculturalsilvicultural treatment,treatment, isis centralcentral toto allall rational planning processes connectedconnected withwith forestry.forestry. There is, however, a considerableconsiderable diversity of opinions over what constitutes "yield", andand howhow itit maymay bebe estimatedestimated andand proprojected jec'~ ed into the future.

ThiThis s manual is an attempt to codify current praoticespractices in the fieldfield of tree andand standstand vovolemel'.une estimationestimation andand forestforest yieldyield predictionprediotion inin a" wayway thatthat isis practicablepracticable andand usefuluseful toto the person who is charged with the responsibility of producing volume estimations andand yield forecasts, butbut perhapsperhaps hashas notnot hadhad thethe benefitbenefit ofof extensivoextensive experienceexperience inin thisthis field.field.

It mustmuet be appreciated, however, that this isis aa fieldfield ofof humanhuman endeavourendeavour thatthat isis currentlycurrently in a state of rapidrapid evolution, especially withwith regal~regard toto forestsforests growinggrowing inin tropical environments. Consequently, all that isis saidsaid inin this manual must bebe regardedregarded asas provisional and subjectBubject toto futurefuture refinementrefinement forfor particularparticular situationssituations thatthat cancan arise,arise, oror new techniques that cancan be developed,developed, whilstWhilst otherother techniquestechniques maymay existexist whichwhich areare notnot referred to in this text andand which",hich maymay bebe superiorsuperior forfor particularparticular purposes.purposes.

Thus(rhuB itit isis notnot aa manualmanual inin thethe truetrue sense;sense; itit isis ratherrather aa setset ofof guidelinesguidelines forfor thethe choice of procedure combinedcombined withwith moremore precisepreCise instructionsinstructions concerningconcerning calculationcalculation tech-tech­ nique for some80me specified cases.cases.

The manual isis donedone withwith specialspecial referencereference toto thethe tropicstropics andand appliesapplies toto naturalnatural asas well as man mademade forests.forests. Because of the great difficulties in assessing growth and yyieldield of natural mixed and uneven aged forests,foreets, the methods given to construct growth models, however,however, mainlymainly applyapply toto even aged forests. For mixed forests nono specificspecific instructionsinstruotions are giveng~ven but ratherrather somesome examplesexamples ofof possible waysway. of dealing with the problem.

The manual consistsconsiste ofof twotwo volumes.volumes. The firstfirst volumevolume describesdescribes techniquestechniques ofof measurinmeasuringg trees and thethe assessment of volume of trees and stands,stands, and the second volume deals with growth and yield prediction.prediction. DeSCriptionsDescriptions ofof statisticalstatistical andand mathematical techniques, selected statisticalstati.stical tables,tab1 ee, blank copiescopies ofof calculationcalculation and data recording forms and an annotatedennotated bibliography are includedincluded inin a seriesseries of appendices.

Volume II of the manual has been written by FrancisFranois Cailliez, Centre Technique FoForestierrestier TropicalTropical (CTFT), Nogent-sur-Marne,NogentsurMarne, France,Franoe, andand VolumeVolume IIII byby DenisDenis Alder,Alder, Commonwealth Forestry Institute (CFI),(eFI) , Oxford, GreatGreat Britain,BrHain, who also compiled the appendies. The work of the two authors has been coordinated by JUranJBran Fries, Swedish University ofof AgriculturalAgricultural Sciences,Sciences, Uppsala,Uppsala, Sweden.Sweden. The work was formulated andand guidedguided b~'by JeanPaulJean-Paul LanlyLan1y and and Kern Karn Deo Deo SinghSingh ofof thethe ForestForest ResourcesResources DivisionDivision ofof FAO.FAD. Jean Clement (CTFT)(CTFT) was asuociatedassociated at the initial stage of the study.

1'heThe first draft ofof thethe manual was preeentedpresented atat thethe meetingmeeting ofof thethe TUFROIUFRO SubjectSubject Group 54.0134.01 (Mensuration,(Mensuration, GrowthGrowth andand Yield)Yield) heldheld inin OxfordOxford inin SeptemberSeptember 1979, and was discussed for one full day inin detail. Among the participants therethere werewere tropicaltropical forestforest mensurationietsmensurationists especiallyespecially invitedinvited byby FAOFAD toto makemake aa thoroughthorough andand criticalcritioal reviewreview ofof thethe contents ofof thethe manual.manual. In addition, the manual was also sent to a number .of specialists for comments. Based on these remarks, a revised version of the manual was prepared by the authors concerned.concerned.

This manual, being the firstfirat of itsits kind inin thethe fieldfield ofof tropicaltropioa1 forestry, hashas con-con­ siderable scope for further improvements and additions. Particularly inin thethe casecase ofof mixedmixed uneven aged stands further complementary studiesstudies are immediatelyimmediately needed,needed. All suggestionssuggestions in this respect will be very much appreciated. M.M.A. A. FloresFl ore s Rodas Assistant Direoto~GeneralDirectorGeneral Forestry DepartmentDepartment

CONCONTENTS TEN T S -:-:--:-:-:-:-:-:-:-:-:-:- :-:-

PARPARTT I Pages o0 INTRODUCTION 1

11 THE DIFFEREDIFFERENTNT VOLUMES THAT CAN BE DEFINEDDEFI NED IN A TREE ...... 2

1111 WWhathat is the physicalphysical objectobject concernedconcerned? ? ...... 2 1212 In whatwhat part of this object isis oneone interestedinterested ...... 2 o 121 Dimension crosscuts ...... 2 122 Form crosscuts --E Examplesxamples...... 3 122.1 The stump ...... 3 122122.2. 2 The base of the crowncrown ...... 3 123 Quality crosscuts ...... 3 1313 Some eexamplesxam ples of grossgros s volumesvolumes ...... 3 14 Concerning usable volumesvolume s ...... 5

2 DIRECT MEASUREMENTMEASUREMENT OF THE VOLUME OF A TREE ...... 6

2211 Measurements onon standingstanding treestrees ...... 6 211 MeasurementsMeasurements of size (diameter(diameter oror girth)girt h) ...... 6 211.1 Definition of the referencereference diameterdi ameter andand of the reference circumferencecircumferen ce ...... 6 211.2 Practice of diameter measurementsmea surements on on stan-st an- ding trees ...... 8 211.21211 .2 1 Diameter measurement withwith aa cali-ca l i - per ...... 8 2l1.211211.211 The usual calipercal iper ...... 8 2211.21211.212 The FinnishFinnish calipercal iper ...... 9 211.22 Girth measurement with aa tapet ape . . . . . 10 211.23 Method of the rulerruler forfor diameterdi ameter measurement atat smallsmall heightsheights ...... 11 211211.231.231 ConstructionConstroction of thethe ruler ruler. 11 211.232 Operating method ...... 12 211.24211 . 24 The Wheeler pentaprismpentapri sm calipercaliper ...... 1133 211.25 Diameter measurement withwith thethe Bit-Bit- terlich Relascope ...... 14 212 Height measurements ...... 18 212.1 Definitions ...... 18 212.2 Height measurements onon standingstanding treestrees .. . .. 1188 212.21212 . 21 Some dendrometersdendrometers ...... 1188 212.211212 . 211 Principle of twotwo instru-instru­ ments easyeasy toto constructconstruct . . 19 212.21221 2.212 Five commercomercialcial dendro-dendro - meters ...... 22 212.22 Some practicalpractical remarksremarks ...... 25 213 Measurement of barkbark thicknessthickness ...... 27 22 Measurements onon felledfelled treestrees ...... 2828 221 Length measurementsmeasurements ...... 2828 222 Size measurements ...... 2828 - I I - Pages

223 Accurate measurement ofof thethe diametersdiameters underunder barkbark or under sapwood ...... 29 224 Mensuration of stackedstacked woodwood...... 29 23 Direct calculation of the volume ofof aa treetree fromfrom mea-mea- surements taken on the tree ...... 31 231 Calculating procedures ...... 31 232 Recommendations for thethe measurementsmeasurements toto bebe takentaken with regard to the requiredrequired volumesvolumes ...... 34 233 ExampExamples 1es ...... 35 24 Study of tree form ...... 39 241 Measure of stem form with a coefficient ...... 39 241.1 Definition ...... 39 241.2 How to calculatecalculate ffor or f'f' ...... 41 242242 Description of stem formform byby thethe equationequation ofof thethe taper curve ...... 43 242.1 The two types of curve -- ProblemsProblems ofof scaling...... scaling 43 242.2 Fitting a taper curve byby calculationcalculation ...... 45 242.21 PrinciplePri nci p1 e ...... 45 242.22 Examples ...... 49 242.221 Four measuramentmeasurrent pointspoints -- Fitting a 3'3 degree poly- nomial ...... 49 242.222 Four measurement points.points. Division into two logs and fitting a curve to each of them ...... 49 242.223 Description of the mean profile of 33 stemsstems with aa 3d3d degree polynomialpol ynomi a 1 ..... 51 243 Crown measurements ...... 55

3 3 INDIRECT MEASUREMENT OF AA STANDSTAND VOLUMEVOLUME:: THE TARIFFSTARIFFS ..... : 59 31 Principle and definitions ...... 59 32 Choice of the entries ...... 61 33 Procedure to estimate thethe volumevolume ofof aa standstand withwith aa tariffta ri ff ...... ;...... 62 34 Sample choice to construct aa tarifftariff ...... 63 341 Tree-tariTree-tariff ff ...... 63 342 Stand-tariStand-tariff ff ...... 64 35 Different ways to construct the tarifftariff with collectedcollected data ...... 65 351 Direct method...... method ...... 65 352 Graph;Graphical ca 1 methods ...... 66

353 The statistical method:method : regression analysisanalysis ..... 66 353.1 Concerning thethe choicechoice ofof thethe regressionregression model ...... 66 353.11 Simplicity ofof thethe modelmodel ...... 67 353.12 Concerning models where aa functionfunction of V andand notnot VV itselfitself isis estimated.estimated. 68 353.13 Piecewise fitting ofof aa modelmodel ...... 69 353.14 Weighted oror unweightedunweighted regression?regression? 72 353.15 How to appreciate the quality of a regressionregression...... 7474 353353.2.2 ExampleExample...... 76 - III -

Pages

36 Concerning underbarkunder bark volumesvolumes...... 86 361 Bark thickness and diameterdiameter...... 86 362 Overbark volume -- UnderbarkUnderbark volumevolume ...... 87 362.1 Bark quotientquoti ent ...... 87 362.2 Conversion of volumevolume overbarkover bark intointo volumevolume underbark ...... 88

4 ESTIMATION OF USABLE VOLUMES ...... '" ... . 90 41 An example of method applied in tropical high forest ... 90 411 Gathering of data ...... 90 412 Analysis of datadata...... 93 42 Estimation of the usable volume by a tariff ...... 94

SHORT BIBLIOGRAPHY 96

-oO() 00- - I - oO INTRODUCTION

How to measure the volume of trees and of stands ?? The first part of the mqnualmanual intendsintends to answer thisthis question.question.

It is, in effect, worth treatingtreating thisthis subject forfor twotwo essential reasons :: the problemproblem in itself is neither as simple, nor as clearly

posed as itit seems ; it is necessary toto define asas preciselyprecisely

as possible the nature of thethe required volume : is one interested in the ligneous biomass, inin the volume of thethe large stems, inin thethe volumevolume ofof sawablesawable timber,timber, etc...etc ... ?

. once the required volume or volumes have been specified,

the way of measuring themthem hashas toto bebe defineddefined ; in this field,field, forest practices are very old and varied and it isis importantimportant to try to unify them to be able toto make valid comparisons between estimates made by differentdifferent personspersons inin differentdifferent countries.

Being a manual, thethe accentaccent hashas beenbeen putput onon thethe mostmost robustrobust methods; sophisticated techniques whichwhich cancan onlyonly bebe appliedapplied byby weZZwell equipped research institutes areare ·not mentionedmentioned (utilization ofof expensive dendrometers, measurement of volumes on aerial photographs,photographs, etcetc...). . .• ).

Furthermore, thisthis manualmanual isis addressedaddressed mainlymainly toto fbrestersforesters of tropical countries, where thethe majormajor problemsproblems concernconcern thethe utili-utili­ zation of the wood as fuel and as prime material forfbr the supply of sawmills and of veneer units or01' forfor the production of pulp.pulp. This is lJhywhy the otherother usesuses ofof the forest (harvest(harvest of minor productsproducts suchsuch as cork, utilization of fodder of the forest,forest, etc...)etc . .. ) which posepose specific problems as regards thethe measurementmeasurement ooff these products, havehave not been considered.considered.

After having defined the most important types of volume,volume, a description is given of the procedures toto be followedfollowed toto coZZectcollect the data and for the calculations toto bebe applied.applied.

This first part of the manual can be considered as an introduction to dendrometry and could as well be placed in a manual on forest inventoryinventory; ; it can be read easily by every person having to work in thisthis field.field. - 2 -

1 THE DIFFERENTDIFFERnIT VOLUMESVOLUMES THATTHAT CANCAN BEBE DEFINEDDEFINED ININ AA TREETREE

The volumevoZ ume one is taZkingtaZking abouaboutt aZwaysalways has to be defined very pprecisely.reciseZy. An answer hashas thereforetherefor e toto bebe givengiven toto thethe followingfonowing two questions :

- WhatHhat isis thethe physicalphysical objectobject concernedconcerned ?? - In wwhathat ppartart ooff thisthis objectob jec t is oneone interestedinteres t ed ??

Furthermore,Fur thermore, it is advisableadvisabZe to specify . how-thehow · the voZumevolume of this object was calculatedcaZcuZated; ; thisthis questionquestion arisesarises becausebecause thethe realreaZ voZumevolume is seZdomseldom known exactZyexactly (it wouZdwould be the voZumevolume of watewaterr which the object displacesdispZaces when immersedimmersed inin aa vat).vat). TheThe proceduresprocedures for the estimation of thisthis exactexact volumevoZume areare describeddescribed inin §f 23.

Let us return, forfor thethe moment,moment, toto thethe twotwo questions.questions.

11 vJHATWHAT ISIS THE PHYSICALPHYSICAL OBJECTOBJECT CONCERNEDCONCERNED ??

It can bebe : - thet he stem : The partpar·t of thethe treetree which isis f011owedfo nowed going fromfrom thethe footfoot of the treetree toto thethe terminalterminaZ bud.bud. For ramifiedramified treestrees oneone considers conventionaZZyconventionally thatthat thethe terminalterminaZ bud isis thethe mostmost elevatedeZevated bud.bud. - thet he branbranchesches - ththee rrootsoots

- ththee treetree : Stem + branches ++ rootsroots

Specify if the bark is incZuded oror not.not.

12 IN ~WHATI HAT PART OF THIS OBJECT ISIS ONE INTERESTEDINTERESTED

TheThe Zimits of the object aarere a lowerZowe r crosscut (at the largerZarger end) and an upper crosscut (at(at thethe smaZlersmaZZe r end).end) . EachEac h of these cross-cross­ cuts can be defined inin severalseveraZ ways.ways.

121 Dimension crosscuts

The followingfoZZowing aarere threethr ee examplesexampZe s of upperupper crosscutscrosscuts of this typetype : - the 0 cm diameter crosscutcrosscut mmeanseans that the limitZimit is the physicalphysicaZ extremityextremity of the stemst em oror of the bbranch.ranch . One then speaks of "t"total"otal" volume.voZ ume .

_ the ?7 cm diametediameterr ccrosscutrosscut is tthehe mmostost often uusedsed one toto make a limitZimit withwi th thethe twigs,twigs, whichwhi ch areare veryver y numerous,numerous, difficuZtdifficult to measure and of ZittZelittle iinterest.nterest. It is the upper ccrosscutrosscut of thethe volumevoZ ume of "big"big wood".wood". - 3 -

- other crosscuts areare possiblepossible : the 5 cm diameter crosscut is,is, forfor example, often accepted as the upper crosscut of the pulpwood volume.

112222 FFormorm crosscuts - Examples

122.1 Ib~_~!~~~:The stume : can be defined as the base of the part of the stem which is extracted fromfrom thethe fbrestforest under optimal exploi-exploi­ tation conditions (losing(losing the leastleast possible utilizable volume). For trees without buttresses, this leveZlevel is at a height fromfrom the ground of between 1010 and 50 cm in general ; if thisthis leveZlevel is not specified, it isis presumed toto be at a distance fromfrom thethe groundground equal to a hundredthhundredth of the total height of thethe tree. For treestrees with buttresses or with aerial roots,roots, thethe stumpstump is thethe toptop ofof thethe buttressesbuttresses oror of the roots (a level which is,is, in general, higherhigher thanthan thethe fellingfelling height).height).

122.2 The base of the crowncrown:: it is the place where the stem clearly ramifies.ramifies ~------

These two crosscuts aZZowallow one toto define

the botebole : the part of the stem situated between the stump and thethe base of the crown.crown. Baseof Crown the crown Chet he low branches : the branches inserted on the bole.

thethe crowncrown : the partpan of the Bole stem situated above Low branchbrunch the base of the crown 4-+ the branches inserted above thethe Stlll7f>Stump base of the crown.crown.

123 Quality crosscuts

One mentions fbrfor example slicingslicing crosscut,crosscut, sawingsawing crosscut,crosscut, post or pole crosscut, etc...etc .. . These notionsnotions areare evidentlyevidently ratherrather delicate to appreciate because theythey comprisecomprise notionsnotions of fbrmform and of dimension and are closely connectedconnected toto technologicaltechnological andand com-com­ mercial practices of the moment.

1133 SOME EXAMPLES OF GROSS VOLUMESVOLUMES

The fbregoingforegoing shows that forfor a givengiven treetree oneone cancan definedefine anan almost infinity of volumes. Here are some examples, given in order of increasing complexity of theirtheir measurement. TheThe firstfirst threethree areaPe thethe mostmost used.used. - 4 -

Physical Lower crosscut Upper crosscut Name ofof object (at the larger end)end) (at the smaller end)end) volume

Bole Stump Base of the crown Bole volume

,-:-.1 (4 Stem StStumpump Crosscut D = 7cm Big wood stem o ''o N (0 volume N `'. Stem Stump Physical extremity Total stemstem ,z co volume 03 P.2 o ------o- Stem + low For thethe stemstem : Crosscut D = 7 cm Bigiiig -;:;ood-~ol~mewood volume 2. branches stumps t ump for the stem andand stem + low o. each low branch branches Q (t) N t. Z Stem + branches Stump Physical extremity Total volume o- of thethe stemstem andand ofof above groundgrol1nd % N . each branch .-CD

O.'_ Stem + branches Stump Crosscut D = 7cm Big wood volume N . .' for the stem andand ababoveove groundg.round ce, each branch _ o. o.

Branches For each branch : Physical extremityextremity Total branch o insertion on thethe of each branch volume t stem or on thethe z . -. branch on which o. it is inserted.inserted. o- ______,l.. ______o Branches - idem - Crosscut D = 7cm Big wood o- for each branch branch volumevolume C:74 ...s A-4 Crown For the stemstem Physical extremity Total crown § crosscut D = 7cm7cm of the stemstem andand volume (lower(lower __ for each branch: of each branch crosscut D=7cm o. insertion on thethe for the stem)stem) o. stestem.m. Q - P, c-i- Stem + branches GGroundround levellevel Physical extremity Total ligneousligneous (0 of the stemstem andand biomass aboveabove o.3-4 of each branch ground o- o. Tree Physical extremity of stem, branches, Total treetree co rootsroot s ligneous biomass - 55 -

14 CONCERNING USA3LE VOLUMESVOLUMES

VoLumesVolumes cited above-above areare grossgross volumes.volumes. How to estimateestimate thethe corresponding usable volumes wouLdwouZd require lengthyLengthy explanationsexpLanations which cannot bebe tackledtackled inin thisthis manual.manuaL. TheThe subjectsubject isis indeedindeed difficultdifficult and is farfar fromfrom beingbeing solved.soLved.

The difficulties areare of severalseveraL types :

- the current and futurefuture usesuses ofof thethe woodwood havehave toto bebe known.known (veneer, sliced wood,wood, sawn wood, telegraphic poles,poles, pulp wood,wo ed, fuel wood, chip wood,...)wood, ... )

- for each use, thethe chain of transformationstransformations toto whichwhich thethe woodwo od will bebe submitted hashas toto bebe knownknown in detail (felling,(felling, skiddingskidding and transportation systems, industrialindustrial processing,...).processing, ... ).

- the different constraints imposedimposed byby thesethese transformationstransformations have to be expressed inin termsterms ofof measurable datadata (length,(length, diameter, cylindricity, heart eccentricity,eccentricity, stemstem bending,bending, admissible defects...).defects ... ),

TheThe procedure toto bebe followedfollowed inin orderorder toto con!!ertcor;,)!e rt grossgross volumesvolw.:e!! into usable volumes is therefore vveryery muchmuch subsubordinaeordina \q toto locall ocal condi-~ondi­ tions and available resources. fieWe ccanan onlyonZy givgivee aa fewfew*deas; 'deas andand sendsend thethe readreaderer back toto specializedspeciaLized handbooks.handbooks. •

- Data collected by forestersforesters during inventoriesinventories (observations(observat i ons on standing or felled trees) can only provide volumes presupresumedmed suitable for such and such use. Inquiries among logging ccompa-ompa­ nies and processing industries are essential to determindeterminee the exact transformation coefficients from gross volume to ususeded volume. An exampleexample willwill bebe givengiven inin paragraphparagraph 41,41,

even when a volumevolume estimationestimation isis undertakenundertaken with with a awell well definec,rlefine(~ purpose (for instance, pulpmill supply), datada ta shouldshould bebe collected in order to be able to estimate other volumes bbecauseecause the final destination of wood and/or loggers and manufamanufacturersc turers requirements might change inin future,future ,

- give priority to gross volume estimation and consider usuableusuable volume estimation asas aa specializedspec ialized task.task. -6-- 6 -

2 DDIRECTIRECT ~'EASURmENTMEASUREMENT OF THE VOLUME OF A TREE

According to the type of the required volume, the measurements will be momorere or lessless numerous. As the variousvarious parts of the tree ((stem,stem, branohes)branches) nevernever are solidssoUds of a perfectly knownknoum geometricalgeometriaal form,f orm, suchsuah as cylinders,conescylinders,aones etc...,eta ... , thethe principle is tot o measure on eacheaah of them the di~neterdiameter at diffdifferenterent heights and to calaulatecalculate the volumevolume fromfrom thesethese measurementsmeasurements ;; of course, this volume will be the more exact as thethe numbernumber ofof measuredmeasured diametersdi~eters willwill bebe Zarge.large . It is obvious that these measurements are easier and more accurate on felledfelled thanthan on standingstanding trees,trees, which expZainsexplains thethe laylayout out of this para-papa·' graph.gmph. § 21 Measurements onon standing tretreeses Ikalculation Calaulation of the volumevo l ume § 22 Measurements11easurements onon felledfelled treestrees \ based on these measurements:§23measurements: § 23

21 ~lEASUREMENTSMEASUREMENTS ON STANDING TREES

211 Measurements~'easurements ofof sizesize (diameter(diameter oror girth)girth)

The sizesize of a treetree is traditionally describeddesc1'ibed by thethe followingfollowing values : refe1'enaereference diameter, referencerefe1'ence circumference,circumference, basalbasal area.area. OneOne measures thethe diameterdiameter or01' thethe circumferenceciraumference andand thethe basaZbasal areaa1'ea isis deducteddeduated by the formula corresponding toto thethe circleci1'ale

BasBasalal area = iT1 (reference diameter)2

= 4~ (reference circumference)2 4.7.

The basalbasaZ area is thus a conventional value whichwhiah gives an approximationapp1'oximation of thethe areaarea of thethe reference1'eference section.seation. TheThe knowledgeknOWledge of the exactexaat value of thethe area of thisthis sectionseation isis indeedindeed practicallypraatiaally impossible on a standing tree and, onon a felled t1'ee,tree, itit requiresrequires ttei,e use of aa planimeter.planimete1'.

211.1 Q~fi~i~iQ~_Qf_~~~_r~f~r~~~~_~i2~~~~r_2~~_Qf_!~~Definition of the reference diameter and of the referencer~f~r~~~~_~ir~~~f~r~~~~ circumference

Between aZZall the diameters andand aZZall thethe circumferencesthatai1'aumferenaestha't can be measu~d,measured, the referencereference diameterdiameter andand thethe referencereference circumfe-circumfe­ rencer ence play an essentiaZessential role.role.

On a standing tree,t1'ee, this diameterdi~ete1' (or(01' thisthis circumference)airav.mfel'erwe) isin r.rearruredmeaoured : - at 1.301. 30 m from the ground (4'3")(4'3") for trees without butt1'essesbuttresses 01'or with buttresses or aeriaZaerial rootsroots Zessless thanthan 11 m. high.hioh. The reference1'efe1'enae diameter is then, traditionally,t1'aditionally, calledaalled diameterdiameter at~t breastbreast heightheight . It is recommendedreaommended toto avoid thisthis ambiguous expressionexp1'ession and toto take aarecare that the height of measurement aoesdoes not depend upon the height ofof the operator.ope1'ator. - at 30 ecmm above the end of the buttress or thethe aerial rootsroots if these are higherhighe1' thanthan 11 meter.meteI'.

WhenWhen the height fromfrom the ground is not equal to 1.31. 3 m,m, it should be recorded.reao1'ded.

The following page illustratesillust1'ates somesome casesaases whichwhiah occuroaau1' in practicep1'aatiae fbrfop the definitidefinitionon of the rreerencee~c1'e nae diameter.di~ete1'. - 7 -

REFERENCE DIAMETER

FlatF1 at terrainterrai n Sloping terrain StStraightr4ight tree without buttresses or VertVertical ieal treetree

with buttresses leesless than I1 meter or As a ru/e,rule, the base of the tree iais the Hithwith aerial roots lessle8s than 1 metermeter.. level markedmarked ...••• (/ocation(location of thethe seed).seed). For practicalprHctical reasonsreasons thethe measurementmeasurement isis taken at 1.301.30 m atat thethe uphilluphill side.side.

1.30 m

1.3Om

Leaning trees

ThThee 1.30 m lengthLength hashas toto be measuredmeasu~ed parallelp~Ltet toto thethe tree,t~e, notnot vertically.ve~icaLty . The mea8u~edmeasured section has toto bebe perpendicularpBrpendiaul~ toto thathe axisi2:::-ia of ti7etlJe tr>etree,e , noti-zot hor'i hori-­ >:ant;)!.ont21. Flat terrain Sloping terrain

1.30 mm measured on 1.30 m measured the side where \30matat up hill tree is leaning side

1.30m

Trees with aerialaerial Trees with buttresses higher Forked treestrees -- roots higherhiOer thanthan i1 m.m. than I1 meter

PorFo~ a good estimate of level (A)rAJ Bottom of thethe forkfo~k viebJview the treet~e from a distance /' \ LessLe88 than HigherHighe~ thanthan 1.30 m 1.30 m - - -.--measurement-measurement _ _ 0.30.30 AJ --measurement-- mea8urement 1 _ _ end of 0.30 bubuttresstt~ss " end of --aerial- aeriaL roots \ I - ~ - measure-one v ment ! rnm / twotbJO I _30 1- or more d imeasu-remonteL. \ in general,generaL, hh isis smallersmaller ConsiderConside~ therethe~ than 66 m.m. are twotoo trees.tree8.

Anomaly atat_1N 1.30, m mLi.(not_i_sw_ellin9, (knot, swelling, deformationdeformation...) . • • )

The measurema-ntameasW'\!Wlm ts n.n>ehave to be takentaJum outsideouts ide thethe --,Meaeu­ -t Nment aeu-t deformed part.p~. ----I~M"""",n.. s a/ - 00,Do, ififpassible, possibte, 22 measu-measu­ - --- rementsl'6ments at eqULllequal diet

Remarks :

If diameter has to be remeasured in the future for increment, the level ofof measurementmeasurement has to be materialized (painted(painted mark,...)mark, ... ) Such a markmark can induce a reaationreaction of thethe tree ; it is thus advisabadvisablele to put the mark at a fixed distanaedistance (for(for instanceinstanae 10 cm)cm) from the measurementmeasureMent levellevel and toto recordreaord thethe heightheight ofof thethe markmaI·k inin caseaase itit should disappear. In inventory or permanent sample plot mensurationaZmensurational wwork,ork, referencereferenae diameter is in general measured onlyonly forfor treestrees of a minimum size.size. InIn most cases,aases, referencereferenae diameter is measured if it is more than 5 emcm (on smallersmaller trees, height is measured)measured) but on studies focusedfbcused on regeneration, diameterdiameter measurement isis ofof interestinterest andand requiresrequires special instruments (mini(mini calipers).calipers).

211.2 Practice~~~£!i£~_2f_Qi~~~!~~_~~~~~~~~~~!~_2~_~!~~Qi~g_!~~~~ of diameter measurements on standing trees The fbllowingfollowing schemescheme indicates thethe lay-outlay-out ofof thethe paragraph.paragraph.

c, I/Usual Usual calipercaliper -- Finnish. Finnish calipercaliper 1:' _Tape The height ofof 1'\ measurement isis \For"For measurement ofof bigbig diametersdiameters : ruler manually accessible ~ (eventually( eventually byby v!heelerWheeler climbingc limbing or with 4 / ppentapri.men tapr i ~j :l poles) 0 Commercial instrumentsinstruments . 2 :2 examples\ for optical measurements'measurements examrleS\ Bitterlir: h , relascope

211.21211 . 21 Diameter measurement with a caliper

211.211 The usual calipercaliper

- Do prefer aa metallic calipercaliper toto aa wooden caliper (climatic(climatic stabilitystability -- easy toto clean).clean). - HoZdHold horizontally.horizontally. - Do nnotot press the arms too much against thethe treetree (soft(soft bark).bark).

- Verify frequently the parallelid~paralleliJm of the arms.arms . FixedFind IIobileMobile of -.III'III anoarm _ -- - Take at leastleast one measurement,measuloement, without choosing the direction. For a better precision oorr for f or a flatflat tree, do a secondsecond measurement inin thethe perpendicular direction and taketake the 111,1.1111 • ti '--~--- arithmeticari thmetic average.average . 1f - Carry out the measurement with thethe graduations maximum precisionprecision allowed by the in mm 1111 graduation (in(in general to thethe nearest cm, if possiblepossible tot o thethe nearestnearest mm).mm). - 9 -

VapiousVarious imppovementsimprovements can be made to the instrument,instpument, of which the aboveabove dpawingdrawing shows the most simple type : additional graduationsgpaduations (~i(,,firth,p th , basal basal apea), area), adjustment adjustment by by a a scpewscrew ofof thethe finalfinal positionposition of the mobilemobile aPmarm,, movement on rollerspolleps of the mobilemobile aPm,arm, addition of a system of automatic registrationpegistpation of the measupemmeasurementent onon punched . papeppaper tape orOP mini cassette,...cassette, . ..

211.212211 .212 TheThe FinniFinnishsh cacaliperli pe r

st ruight a1'f7!

cUI'Ved a1'f7! \

The grotiuatigraduationsons are parallelpcrraUe l tot o tthehe insiinsidede edgeedge ofof the straightstreight arm.ann. ,

handhandlel e

J I I I ~

GGrasppasp the handle with leftleft handhand (the(the leftleft armapm shouldshould bebe stpetchedstretched out as farfap as possible), appZyapply thethe calipercalipep againstagainst ¿he'i he tree,tpee, opthogonalorthogonal to stem axis. The diameterdiametep isis obtainedobtained byby sightingsighting parallelpapallel toto thethe graduationgpaduation marks.mapks.

AdvantagesAdv antages onon thethe usualusual calipercalipe r

. no movablemovab le part,papt , when fixedfixed on telescopictelescopic poles,poles, thethe caZipercalipep allowsallows toto measuremeasupe diametediametersps up to approximatelyapppoximately 88 m fromfpom thethe ground,gpound, andand eveneven up to a dodozenzen metemetersps if binocularsbinoculaps areape usedused forfop thethe reading,peading, instpumentinstrument easeasilyi ly selfMadeselfmade with plywood (7(7 layer,layep, 99 mmmm thick)thick) ; graduategpaduate both facesfaces toto bebe usedused withwith leftleft orOP rightpight handhand; ; varnishvaPnish the whole instrument.instpument. - 10 -

I The curved formfo~ of the gpaduatedgraduated I armqpm is suchsuch thatthat thethe distancedistance aa / does not depend on treetpee size, which / guaranteesguapantees the same degpeedegree of ac­ac- I curacycupacy forfop treestpees of diffepentdifferent sizes ( aa == 55.5.5 cm in the instrumentinstPUment of previousppevious page).page).

211.22 Girth measurenentmeasurement withwith aa tapetape

The use of a tape is indispensable forfop largelapge treestpees becausebecause the calipepcaliper is impracticable.imppacticable. Also forfop smallsma ll trees,tpees, a tape is ppefe­prefe- pablerable to a calipecaliperp : - the tapetape estimaestimatestes a size called girth (it(it isis inin factfact thethe perimeterperimeter of the convex hull of the section), thethe definition of which isis not ambiguous whereas there exists an infinityinfinity of diameters. By reference

to aa cipcle,circle, the quotient of the measupedmeasured girthgipth by TT~ is taken as the diametepdiameter (certain(ceptain tapes comprisecompPise aa diameterdiametep graduation).gpaduation). A mathe-mathe­ matical propertyppopepty adds aa supplementarysupplementapy justificationjustification toto thisth,:s practiceppactice the measupedmeasured girthgipth divideddivided byby TT~ is equaZequal toto thethe averageavepage of the infi-infi- . nity of diametersdiameteps thatthat could bebe measuredmeasuped withwith aa caliper.calipep. - measurements with aa tapetape areare moremore reliablereliable thanthan measurementsmeasurements withwith

a calipercaliper : the tape, providedppovided that it does not extend,extend, is strongerstpongep and the riskpisk fOrfop compressioncomppession ofof thethe barkbapk isis lessless thanthan withwith aa caliper.calipep. It is essentiaZZyessentially forfop thisthis reasonpeason thatthat oneone cancan hearheap saysay thatthat thethe diametepdiameter measuredmeasuped with a tapetape isis systematicallysystematically Zargerlapgep thanthan thethe diametepdiameter measuredmeasuped ·with a calipercalipep andand itit isis toto correctcoppect forfop thisthis biasbias that the frenchfpench standardstandapd forfop exampleexample definesdefines thethe girthgipth atat 1.501.50 mm asas the referencepefepence fbrfop the sizesize ofof aa tree.tpee. ThisThis definitiondefinition isis notnot toto bebe recommendedpecommended inin orderopdep toto unifyunify andand alsoalso becausebecause detaileddetailed studies,studies, practical as well theoretical, have shown that the difference ppactical as well asas1ltheopetical, have shown that the diffepence between D1D . and ;Tri: C .C1.30mis,is, in in genepal, general, smallsmall andand does not have I .30m30m I 30m a realZypeally systematicsystematic character.chapactep. - The main ·point isis toto hoZdhold thethe tapetape inin aa planeplane perpendicularpeppendiculap toto thethe stem axis, aftera[tep having removedpemoved lianas,lianas, mosses,...mosses, ... (but(but capecare mustmust be taken not to removepemove barkbapk inadvertently). Prefer tapes providedppovided withwith a hookhook atat thethe extremityextpemity toto fixfix in the bark, which allows a single person to measure aa largelarge tree.tree. LinenLinen tapestapes stretchstretch andand wear.wear. Metal tapestapes areape better but kink.kink. WithWith recentpecent matePialmaterialss such as fiberglass,fibepglass, thesethese disadvantagesdisadvantages disappear.disappear. - CarryCarpy out thethe measurement with thethe maximum precision alZowedallowed by the graduationgpaduation ;; inin generalgeneral toto the the nearest neapest cm cm (82.4 (82.4 -4- ~ 8282 ;; 82.6 ~ 83), if possible to the nearest mmmm.. However,Howevep, . forfor rapid measupmeasurementsements undepunder difficult conditions andand withwith unskilledunskilled labour,labour, wholewhole unitunit measuremeasure (82.4(82.4 ~ 82 ; 82.682.6 -,-~ 82), 82), though though biased, biased, maymay bebe mopemore accupateaccurate because of the reduced riskpisk of misunderstanding. - 11II -

211.23 Method of the ruler for diameter measure­measure- ment at smallsmall heightsheights (5(5 toto 66 mm maximum) maximum)

211.231 Construction of the rulerruler

a) Take a board of 150 cm xx 1010 cmcm xxl 1 cmcm andand paintpaint white.white. b) Attach in the middle a rod of 1 m long and on thisthis rod a detachable handZehandle of 2 m long.

1m j 1 2m j c)e) Mark with bZackblack paintpaint thethe limitslimits andand thethe numbersnumbers of the classes as given beZow.below.

Lower limitslimits ofof thethe classesclasses Classes Exact limitslimits Positionspositions ofof limiLsl imi ts on rulerruler D (cm)(em) d (cm)(em) __ - , 2 15 1414.9.9 3 25 24.7 4 35 34.4 5 45 44.0 6 55 53.6 7 65 63.0 8 75 72.4 9 85 81.7 1010 95 90.9 II11 105105 99.9 1212 115 108.9 1313 125125 117.9117 .9 14 135 126.7 1515 145 135.5 - 1212 -

Justification : to avoid parallax errors, the cZassclass limitslimits onon thethe ruler are corrected :

OMOne canoan provepl'Ot1e easily that

dd. D + ~ I + ;a a a • Distance fromfrom thethe eye of thethe observer toto thethe centre of thethe ruler.ruler.

The operating method which follows supposes that the observer is at a horizontalhorizontal distance ofof 1010 m fromfrom thethe tree.tree. IfIf the ruler is at his eye level,level, a =~ 1010 m If not, the maximum heightheight at whichwhich the ruler can be placed being appro- ximateZyximately 55 m and adMittingadmitting thatthat thethe terrain is flat and that the eye of the observer is at 1.51.5 m fromfrom thethe ground, the distance will thenthen bebe : ~ I 5m ~1024102 + (5(5 - 1.5)21.5)2 = 10.610.6 m 1.5 m I

10m One can therefore estimate thatthat aa isis equivalentequivalent toto 10.310.3 mm on an average. TheThe correctedcorrected diametersdiameters dd whichwhich areare to bebe marked on the ruler have been calculatedcalculated withwith thisthis value.value.

211.232 Operating method - The observer places himself at aa horizontalhorizontal distancedistance ofof 1010 mm fromfrom the side of thethe tree.tree.

Ruler __ - - ~ - Tree Tree - - - - -101 0 mm ------~ Reading of the nr.Dnbernumber of the diamaterdiameter claseelasa - A helperheZper places the ruZerruler against thethe tree at thethe heightheight ofof measurement.measurement . It is important that the ruZerruler shouldshould be perpendicular on the Zineline of sight. The left edge of the ruZerruler has to be in a lineline withwith thethe left edge of the stem in relationrelation toto thethe obobserver.server. - Read the number of thethe diameterdiameter classclass on the rightright handhand partpart of the ruZer.ruler. - 1313 --

211.24 The l~heelerWheeler pentaprism calipercal iper

This optical instrument has two advantages _ the observer cancan standstand atat anyany distance fromfrom the tree and this distance has not toto be known, _ it allows diameter measurements at any height.height.

between But it requiresrequires aa good visibility : a strong contrast between ttreeree and background is necessary.

Diagram showing pathways takentaken byby the sightings throughthroug~ thethe instrumentinstrument

Direct lineLine of sight above fixed prism ~~~T------~~~~~~-=~~- eye Fixed pentaprism

,

9- - - ReflectedRej7.ected Zi;;e-;'jline of sight sight- throughV;;;;ugh - prismsp;.,;sm; _ . Movable pentappentaprismrism

Two vertical Wl--_-----LeftLeft bark edge of tree guidelines (direct sighting)

Prism rereflected j7.ected image of right bark edge - 1414 --

Hold the caliperoaliper 8 or 10 cmom inin frontfront ofof thethe eyeeye withwith thethe graduated scalesoale up. LookLook intointo andand throughthrough thethe · viewingviewing slot.slot. Most operatorsoperators keepkeep bothboth eyeseyes open.open.

Through the upper part of thethe sZot,slot, thethe leftleft bark ofof thethe tree is seen direotlydirectly.. In the lowerlower part, appears thethe rightright barkbark edge reflectedrefleoted throughthrough the twotwo prisms.prisms. SlideSlide thethe movablemovable prismprism with the right handhand untiZuntil thethe rightright barkbark edgeedge reflectionrefleotion isis brought intointo directdireot verticalvertioal alignmentalignment withwith thethe leftleft barkbark edge,edge, midwaymidWay betweenbetween thethe twotwo verticalvertical guidelines.guidelines. ReadRead thethe diameterdiameter onon the scale.soale.

'i'heThe instrument exists with 33 Zengthslengths

44 cmom 4-+ ma~~mummaximum diameterdiameter 36 cm

69 cm + maximum diameter 6262 cmom

95 cm + maximum diameter 8686 cm.om.

To checkcheok the caliper forfor accuracy,acouraoy, measuremeasure aa targettarget ofof knownknown width and adjust thethe pointer positionposition toto getget thethe correctoorrect value.value. Verify also that the measurement does not dependdepend uponupon thethe distancedistanoe (some instruments are defective).defective).

211.25 Diameter measurementmeasurement with with the the Bitterlich Bitterlich RelascopeRelas cope

The Bitterlich relascope is an instrument quite universally used by foresters,foresters, which permits thethe followingfollowing principalprinoipal measurementsmeasurements a) diameter of the tree at any height b) tree height c) basal area of standstand d) certain horizontal lengths e) slope of a terrain.terrain. Its desoription,description, thethe principleprinciple ofof itsits functioning,functioning, andand itsits handling forfor thethe measurements c)c) d)d) e)e) , are not givengiven; ; thethe follo-follo­ wing are thethe directions forfor useuse ofof thethe instrumentinstrument (of(of thethe widewide scalesoale model, more adapted to the measurement of Zargelarge trees than the narrow scale model) fbrfor the measurement of diameters at various heightsheights.of.of the bole.bole. --1515 -

Diameter measurements with the wide scale Bitterlich

Relascope for the'the calculation of a bolebole volume

1/ StandStand atat a horizontaLhorizontal distance D from the centre of the tree equaLequal to at leastLeast 2/3 of its height (( D can be equalequaL to 4, 6, 8,8, 10, 12,12, 14,14, 16, 18,18, 2020 meters).

2/The boleboLe must be seenseen completely.compLeteLy. CZearCLear vegetation if necessary.

3/3/Measure Measure heightheight h of buttress, reference diameter (with(with aa tapetape oror aa ruler),ruLer), and barkbark thickness.thicknes. ' 4/4/H H beingbeing thethe firstfirst entireentire heightheight (read(read onon thethe scaZescale correspondingcorresponding toto the distance D) situated above the end of the buttressbuttress," measure : - diameter at end of buttress - height h between H and end of buttress - diameter at heightsheights H, H+1,H+I, H+3,H+3, H+5, H+5, etc etc......

DonDon't't measuremeasure the diameter if there is an anomaly.anomaLy.

5/5/Estimate Estimate thethe heightheight L between lastLast measurement and upper crosscut (base of the crown) (L << 2m).2m).

6/6/Measure Measure diameterdiameter atat upperupper crosscut.crosscut.

7/7/Indicate Indicate thethe parts of the boLebole which cannot be used (defect,(defect, insertion of a big branch...)branch •.. )

8/8/Do Do thethe quaZitativequaUtative observationsobservations (see(see § 41)

------.- Cl'OLIncrown point

2m

1

2m

1stt snti~entire height above E 1m ~lS

- ReReference ference diameterdiamsul' h O.3Om0.30m ! E : end of buttress / he \ - 1616 -

The measurements are gathered in the following form which is provided fbrfor thethe calculationcalculation of volume with a programming calculator no place is therefbretherefore prepared forfor thethe volume ooff each log.log. Modify consequently the fbrmform if calculation is toto bebe perfbrmedperformed byby hand.hand.

To transform the relascope-units into thethe realreal values,values, useuse the relationship :

one scale inin cmem =~ 22 Xx D inin meters (ex.(ex. DD = 10m,10m, oneone scalescale == 20cm)20cm)

Remark

For measurementsmeasurements by tree-climbing method,method, useuse aa similarsimilar fbrm.form. In the two columns fbrfor relascope-units, place overbark and underbark diametdiameters.ers.

~~~~2!~Example : The following form is forfor a tree which has been measured in the fbllowingfo llowing conditionscondi tiol1s :

Heights in the relascope

__o ao

H ==-1 -1 reference diameter measured 0.9m with the ruler (§211.23) 0.30m with the ruler (§21 I .23)

h -= 2.52. 5mm hcc

12m12 m I LE 11.-1.. I

Volume calculations havehaye been made with SmaZian'sSmalian's formulaformula (see § 231) forfbr each log limited by two measuremeasurements.ments. For the bottom log (0.9 m high) the cylinder formula has been used. For the loglog with the broken branchbranch,, the length of the defective part has been estimated 1.8 mm -4. ~ lengthlength of good part == 2.22.2 m.m.

Bark thickness hashas been assumedassumed constantconstant fromfrom bottombottom toto top.top. - 1717- - Form forfor volumevolume estimationestimation ofof thethe bolebole of aa standing treetree with with Bitterlich Bitterlich Relascope Relascope !lockBlOCK :1NW-: ...... -: Plot"'O~ Bark scd:.ecE I ~ Code Height I Code Location Species : II I I I " I above COmCIEm F H 1/W I I I I ground I-IensurationistMensurationist : Date : Qualitative Bark OJ observations thickness ~ Horizontal distance : mrn rnI I I ~ on radius I r~arkMark Relative RelascopeRelas~~p~)units Units unusa- ~ heights (R U)

ble paridpa~ 1I 11/4/ 4 ble \ Relative Relascope Units - scale scale r.elas(ope Units - scale 1j scale heights ( R U)U) 1 1 1/4 --0 0 seascale Ie ; scale;cale 1 CD v; 1 1 --0 0 1 CD ---0 0 1 rn 1 1 1 ---0 0 1 OJ --0 0, rn 1 1 1 --0 IT] --0 0 1 CD 0 1' 1 1 --CJ 01 CD --0 0 ' CD 1 1 1 1 ---CJ 0 CD --cJ 1 1 0 rn = == 1 1 1 ---cJ 0 1 CO 1 1 R U scale;scale;1/4 1/4 scalesca le 1 reference ---0 0 1 rn diameter I I f I I I I t F" or 1 1 reference -0 0 CD circumference I I I I }:. 1 1 bole heightheight -- -0 0 , 1 rn I I ! In 1 -0 0 1 CD overbark volumevolume I I I I I I f3 1 --c::J D I EECO underbark volume I I I I I F3 2m I I ) heightheight 1st1st entireentire_::i!. 0 0 1 rn I I ! F 1mlm height aboveabove H-O 1 usable' h [}J", 0 1 rn E .• end of V overbark 1 I I I I I I'" buttress 0 rn part buttress/ hc 1 ITJm\ V underbark 1 I I1 I} I I IIIIJ - 18 -

212 HeightHeight measurements

212.1 g~fi~i!iQ~~Definitions

The total height of a tree is the Zengthlength of the straight lineline connecting the foot of the tree (ground level) with the extremity of thethe terminal budbud of the stem. For forked trees, there is a total height if the fork is above 1.30 m and as many heights as therethere are stemsstems if the fork isi8 below 1.301.30 m.m.

In the sanesame way as for volumes, the heights at certain crosscuts are defineddefined: : the height "big wood"wood" for example will be the Zength ofof the Zineline connecting the foot of the tree with the 7? cm dia­dia- meter crosscut of the stem.

Remarks :: •. fOrfor very badlybadZy fOrmedformed treestrees oror fbrfor shrubsshrubs withwith multiplemultiple stemsstems as can be encountered in savannahs,thesavannahs, the termterm diameterdiameter hashas littlelittle practicalpractical sense ; total height then becomes thethe essential characteristic.characteristic.

. total height ha.3has littlelittle concreteconcrete sensesense forfor treestrees withwith aa brokenbroken or dead crown. Avoid toto useuse suchsuch treestrees toto constructconstruct aa volumevolume table.table.

212.2 Height measurements on standing trees

Height measurements taketake more timetime andand areare moremore delicatedelicate thanthan diameter measurements. TheyThey areare sometimessometimes impossibleimpossible (Zack(lack ofof visibility).visibility).

A height is measured : either using a systemsystem ofof graduatedgraduated telescopictelescopic polespoles whichwhich is put against the tree.tree. This is possible onlyonly forfor smallsmall heights (ranging(ranging aboutabout tenten meters),meters), - or, most frequently, by optical procedure,procedure, usingusing aa dendrometer.dendrorneter.

212212.21. 21 Some dendrometers

There exists a greatgreat varietyvariety of dendrometers. WeWe will1Jill only describe the principle of two instruments easy toto constructconstruct and quotequote some examples ofof commerciaZcommercial instruments.instruments. - 19 --

2212.21112.211 PrinciplePrinciple of two instruments easeasyy to construct -- The------The dendrometricdendrometric------ruler It is a ruZerruler- equipedequi ped with a plumb-lineplumb- line attachedatt ached toto oneone corner.corner .

/ / / / / c ----

------

ABAB isis graduated in centimeterscenti meters startingstarting fromfrom B on both faces.faces .. n n l1 n22 H h +h+h 'Jithwith : hhi = DD and h2 == DD 7 Htott o t = h1l 2 1 a a

If DD = 1010m m andaanda = 1010 cm,em, HH in metemetersr s is the sum n + n in centimeters. tott ot n1l + n22

It is difficultdifficul t toto readread thethe resultr esult withI.lith aa precisionprecision betterbetter than half centimetercentimeter; ; thethe errorer ror onon thet he sumsum of the two measurements isi s thethereforerefore 11 cm maximummaximum ;; the error on the height is thus about 1 metermeter if Do == 1010 meters,me t e r s. about 22 meters if Do = 20 metersmeters ,, etc..etc .. . The use of thiSthis instrumentins trwnent isis thusthus notnot recommendedr ecommended atat moremore thanthan 1010 meters fromf r om thethe treetree ;; the maximum heightheight which cancan bebe measuredmeasured isis then about 1010 meters.meters.

This instrumeinstrumentnt requiresrequires thethe measurementmeasurement of the distance fromf r om the tree.tree . This is an instrumentinstr ument whichwhich avoidsavoids thisthis measurementmeasurement : - 2020- -

- The CHRISTEN hypsometer

o

5 c

A'C'A'e' =- 30cm30cm 10 ASAB iiss a ppoleo1- e of known lenZengthgth

15 20 30 ~ The observerobserver stands atat aa distancedistance suchsuch 100 that the height to bebe measuredmeasured isis seenseen between A' and C'.c' . The instrumentinstrument hashas to be held looselyloosely soso thatthat itit takestakes itsits verticalvertical equilibequilibriumrium position but shouldshould graduations for not move ;the; the heightheight isis readl'ead atat B'BI on a pole lengthlength AB the scale.scale. of 4 meters

The Zongerlonger A'C' , thethe shortershorter the distance fromfrom thethe treetree but it becomes more difficult to control simultaneously the C'Cc'C , B'BIBB and A'AA'A aZignments.alignments.

InIn general,gene ral, the chosen lengthlength for the instrument isis A'C'A'C' = 30 cm, which leads oonene tot o stand at a distancedi stance fromf rom the tree approximapproximatelyately equal toto thethe measuredmeasured height.height . AB A,c,. A'B'A'B' = AB A'C ' The scale is graduatedgraduated accordingaccordi ng toto thethe formulafOI'fTIula AC .

HereHere areare somesome valuesval ues ofof the AA'B'' B' ocalescale as a function of pole Zengthlength ABAB and treetree heightheight AC,AC, fbrfor anan instrumentinstrument of length A'C'A'C' = 30cm.30cm. -21- 21 --

AB .. . 3m3 m 44m m 55m m 6m 6 m 7m 7 m AC 4-+ 55m m 180 mmmm 240 mmmm 300 mmmm 66m m 150 mmmm 200 mmmm 250 mmrom 300 mmmm 10 mm 90 mmmm 120 mmmm 150 mmmm 180 mmmm 210 mmmm 11 mm 82 mmmm 109 mmmm 136 mmmm 164 mm10m 191 mmmm 15 mm 60 mmmm 80 mmmm 100 mmmm 120 mmmm 140 mmmm 16 mm ! 56 mmmm 75 mmmm 94 mmmm 113 mmmm 131 mmmm 2020m m 45 mmmm 60 mmmm 75 mmmm 90 mmmm 105 mmmm 21 mm 43 mmmm 57 mmmm 71 mmmm 86 mmmm 100 mmmm 30m30 m 30 mmmm 40 mmmm 50 mmmm 60 mmmm 70 mmrom 31 mm 29 mmmm 39 mmmm 48 mmmm 58 mmmID 68 mmmm

40m40 m 23 mmmm 30 mmmm 38 mmmm 45 mmmm 53 mmII1III 41 mm 22 mmmm 29 mmrom 37 mmrom 44 mmmm 51 mmrom .

This table shows that the precisionprec.s.on of the measurement decreases if the measured height increases and if pole Zengthlength ABAB decreases.decreases .

In practicepractice,, this instrument is used oonZyn ly for heights Zowerlower thanthan about 20 meters, which often is sufficientsufficient forfor measurementsmeasurements of bole heights inin tropicaltropical highhigh forest,forest, becausececause beyond~eyond 0,at, t"at , thethe t-rye t~ee has littlelittle chance of being entirely visible andand thethe distancedistance .fromfrom thethe observer to the tree becomes too largelarge as can be seen with the followingfollowing calculus : a pprecisionrecision better thanthan x == 33 mm mm on thethe BB' reading seems hard to get.get. Let us impose that such an incertitude induces a yy incertitude of no more thanthan one meter onon the AC result.

TheThe following relatireZationon between x and y

AC = VAB x A'C'A'C' x -31~

shows thatthat ifif A'CA'C'' = 3030 cmem and x = 33 mm, mm, the condition y 5_~ 1m i.sis satisfied if

AC ~5 2020 mm for aa polepole ABAB of 44 metersmete rs

AC ~ 5 22.4m22.4m for for aa polepole ABAB ofof 55 metersmeters

AC ~ 5 24.5m24.5m for for aa ppoleole ABAB ofof 6 meters. - 2222 -

212.21221 2.2 12 Five commercial dendrometersdend rometers

These instrumentsinstJ?wnents are,ar e, of coupse,Jcourse, moremope precise.precise.

- ThThee BLUME-LEISSBLUME-LEI SS de dendrometerndrome t er isis composed of a clinometerclinomet ep with pendulum which can be blocked at thethe moment of taking a sight iinn ffrontpont ofof fourfoup scales graduatedgpaduated in heightsheights and aa fifthfifth oneone inin angles.angles. TheThe heightheight scales coppespondcorrespond at a distance fromf pom the treetpee to be measuredmeasuped ofof 15,15, 20, 30 and 40 m. These distances can be measuredmeasuped with the aid of a dioptepdiopter which gives 22 shifted images ofof a~ small,smalZ, foldablefoldable targettapget boapdboard which is hooked to thethe treetpee ;; onon this targettapget areape 33 lineslines A5&5 ecmm aapartpa rt on one face and of 60 cm on the othepother,, which corresponds,coppesponds, when the images of two lineslines comecome toto col:ncide,tocofncide, to distances ofof 15,15, 2020,, 30 opor 40 meters.meteps .

11 - too far

2 - correct distance

3 - ttoooo closeclose

1 2 3

One takes a sight and when the pendulum has reachedpeached its posiposi-­ tion of equilibpumequilibrum,, it is immobilized by aa knob.knob. TheThe heightheight isis given directlydipectly on thethe scalescale whichwhich correspondscoppesponds toto thethe distance.distance.

On slopesslopes,, whewherepe the sight on the mamarkpk is iinclined,nclined, tthehe incliincli-­ nation is measuredmeasuped on the scaZescale graduatedgpaduated in angles and the cor-cop­ rectionpection to be made to the height readpead is calculated accoaccordingpding tot o a table which is engravedengpaved on thethe instrument.instpument. The markmapk isi s seen obliquely underundep an angZeangle ii , the interceptedintepcepted lengthlength of the markmapk is thus divided by cos ii ; on the otherothep hand,hand, a measuremeasupe isis takentaken

of the oblique OA and not ofof thethe horizontalhorizontaZ OC == OA cos i . The truet pue height is therefbrethepefope equal to the measumeasuredped height multiplied by cos2co s 2 1. - 23 -

2 . true height = measured height x cos 1i

= measured height - measured height xX sin2S1n. 2 il' plane of the mark

o

. L .. A table carved on the instrument givesgives thethe valuesvalues ofof sin2S1n 1.

- The HAGA dendrometer is practically identicalidentical butbut offersoffers thethe advantageadvantage that onZyonly the scale corresponding toto thethe chosenchosen distancedistanoe isis visible,visible, which eliminates the risk forfor error. A delicatedelicate pointpoint asas regardsregards thesethese two instpumentsinstruments : makemake suresure thatthat thethe actionaction onon thethe knobknob doesdoes notnot locklock the pendulum in a popositionsition slightly different fromfrom itsits exactexact position.position.

- The SUUNTO dendrometerdendrometer

actualaotual sizesize

For establishing thethe measuringmeasuring distancedistance (15,(15, 20,20, 30 or 4040 meters)meters) the instrument has a double-refractingdouble-refracting prismprism andand aa separableseparable calibratedcalibrated target board made of reinforced plastic (the(the samesame thatthat isis usedused withwith the Blume-Leiss dendrometer). TheThe target is fixed verticallyvertically onon the treetree trunk at thethe eyeeye levellevel;; sight it throughthrough thethe prismprism andand movemove backwards or forwardsforwards untiluntil thethe lineslines coincide.coincide.

Place the instrument toto the eye and move it inin a verticalver tical arc until the horizontalhorizontal index line,line, viewed throughthrough the lens,lens, is alignedaZigned with thethe desireddesired object.object. •

- 24 -

Look simultaneously,simultaneously, with bothboth eyeseyes open,open, throughthrough thethe lens andandalongside alongside the instrument.instrwnent. TheThereading reading obtained is the height above thethe eyeeye leveZ.level.

---,

+ • r­ --, + .. I I I index I I Zineline I I I I I I I I I I L __ -/--t------~ scale scale 1:2011:20 1:15: 15

Advantage overover thethe Blume-Leiss dendrometer : sighting and reading are simuZtaneous.simultaneous.

Drawback : sighting is more difficult.difficult.

- Any instrument which measuresmeasures angZes ("cliseometer"(" c liseorneter" oror "clinometer")"clinomete r") can be used, thethe height in relation toto thethe horizontalhorizontal beingbeing thethe product of the horizontalhor izontal distance tot o thethe treetree byby thethe tangenttangent 'ofof the angle. Instruments where thethe reading isis donedone onon thethe moment of sighting are preferablepreferab le to avoid thethe inconvenienceinconvenience mentioned causedcaused by the knob. This is the case of the SUUNTO clinometer andand thethe Bitterlich Relascope,Relascope, instruments which are much usedused by foresters.forester s. The first one comprises a graduation ofof thethe angles in tangent ;; one can stand at any distance fromfrom the treetree but thethe product of the dis-dis­ tance and the tangent has tot o be made. The relascoperelascope gives the heightheight automatically if oonene standsstands at 20,20, 2525 or 3030 m fromfrom the tree (model(model with narrownarr ow scales)scales) or at aa distancedistance equalequal toto anan eveneven numbernumber of - 25 -

metersmeter>s between 44 andand 2020 (model(model withwith widewide scales).scales).

- All these instrumentsinstr>uments givegive aa maximummaximum precisionpr>ecision whenwhen oneone standsstands at a distance fromfr>om the treetr>ee perceptibZyper>ceptibly equalequal toto itsits height.height. ThisThis precisionpr>ecision is, underunder> optimumoptimum useuse conditionsconditions ofof eacheach instrument,instr>ument, in the or>der>order of a few percent.per>cent .

Recommendation : calibr>atecaZibrate each instrumentinstPument as socrsoen asas itit arrivesar>r>ives ;; itit is not rarer>ar>e indeed toto findfind out divergencesdiver>gences asas highhigh asas 3 % between two instrumentsinstr>uments ofof thethe samesame mark.mar>k.

212.22 Some practical practical remarksremarks

212.221 Even if it is illus7:veillusive to trytr>y to measuremeasu~e a total height with a precisionpr>ecision better>better than the d~cimeter>decimeter for>fbr "mallsmall treestmeg (a(a fewfe),) meters)meter>s) oro~ thanthan thethe metermete~ forfor talltall trees,tr>ees, itit isis advisable,adl1isable, inin orderor>deY' toto -loselose thethe leagtleast possiblepossible pr>ecision,precision, to do the measurement with the maximum precisprecision io/! permitedpeY'mited byby thethe instrumentingtr>ument used,used, let us say tentatively : - to the nearest cmem forfor treestrees ofof lessl ess thanthan 22 metersmeters high,high, - to the nearest dm for trees of height between 2 and 5 m, 1 - to the nearnearestest 1 m for trees of height between 5 and 1010 m, "22 - tot o the nearest meter forfor treestrees higherhigher thanthan 1010 m.m.

212.222 Measur>eMeasure a total height only ifif thethe top of the crown can be seen ; if an apparentappar>ent top is viewed,viewed, thethe mea-mea­ suredsur>ed height overestimatesover>estimates thethe realr>eal height.height. ThisThis overestimateover>estimate cancan bebe ver/peru important1:r:rportant ; it1;t is about 2020 %% inin thisthis case.case .

Apparent toptop of the crowncrown--~~~ ., .c ., ....,"" -;, .c ...., ...,., ....c ...... , .,.." ... I! - 2626 -

212.223 HeightHeight measurementmeasurement onon aa Zeaningleaning tree.tree. I CC/ -

A B AC = = total total heightheight c

FromFl'Om above ::

B Observer

The observer should not be in thethe verticalvertical planeplane defineddefined by the tree but perpendicularly to this plane, at equalequal distancedistance fromfrom AA and B.

With the CHRISTEN dendrometer (or(or any other instrument which does not necessariZynecessarily measure verticalvertical distances),distances), thethe referencereference polepole is put alongside AC and the observer takes a sight on AC,AC, which gives the exact height.height.

With a dendrometer which measures vertical distances only (Blume-Leiss,(Blume - Leiss, BitterZichBittertich relascope,relascope, S(1UHTOSUIINTO dendr-ometer·,dendrometer,...) . .. ) a co>:'cor-­ rection is theoreticallytheoretically necessarynecessary becausebecause thethe measuredme'1.sured heightheight isis BC and not AC

2 BC AC == /AB2I AB2 + BC2BC = BC cos a but this correction is generally smallsmall :

AC - BC Relative error - =- I1 - cos a)= 1.5I .5 %% forfor a0. == 10°10° AC - cos ~ ): = 3.43 .4 % for a~ = 15°15°

212.224 Dendrometers which measuremeasure heightheight aboveabove obser-vep'sobserver's eye.eye.

From or 'to the measurement of the top, it is necessary to sub-sub­ stract or add the measurement ofof thethe basebase dependingdepending onon whetherwhether thethe eyeeye is beneath or above thethe footfoot of the tree :

hl hl

totaltotal height height sm - total heightheight - hih1 - h2h2 hihI + h2 On lwrizontalhorizontal terrain, the measurement of the footfoot of the tree need not be done because h2 is then known (distance(distance fromfrom thethe eyeeye toto thethe ground).ground). h2 - 2727 -

212.225 The definition of totalt otal height involves the extremity of the terminal .bud of the stemstem;3 this is not necessarilynecessariZy the highest point of the tretree.e . The dis-dis­ tinction has a practical consequence total (which can be important) only forfor thethe height small trees the tiptip ooff which can be reachedreached by hand.hand.

"

Example of of a youngyoun~ pinepine

213 Measurement of bark thickness

To know the volumevolume under bark isis aa necessitynecessity if it is the utiliutilizablezable volume that one wants toto knowknow becausebecause thethe barkbark isis generallygenerally not utilized.uti Zized.

The proportion of bark volumevolume overover thethe volumevolume withwith barkbark variesvaries fromfrom a fewfew percentpe rcent toto approximately twentytwenty percentperaent forfor the majority of species. This proportion is all the more important if the tree is young,young, if the altitude increases and,and, inin aa generalgeneral way,way, if the growing condi-condi­ tions are more difficuZt.difficult.

Instruments have been specially designed to measuremeasure thethe barkbark thickness. They measure thickness on thethe radiusradius (maximum(maximum capacitycapacity

approximately 5 cm) but be careful : some instruments are graduatedgraduated toto give double barkbark thickness.thickness. a/ The barkbark gaugegauge

N '" .. . / 1 Cutting and sliding direct reading of part. . bark thickness

Place the instrument perpendicularly against the tree and pushpush the handle until the whole of the bark (but(but only the bark !! this is the delicate part of the operation) has been traversed. Do not use a - 2828 -- maLLetmallet to make the job easier.easier. ItIt isis betterbetter toto dodo twotwo measurements,measurements, inin two points diametricallydiametrioaLLy opposed, and toto take thethe arithmeticarithmetio mean.mean.

b/ ThisThis isis an other instrument, the "boring hammer", becausebeoause it is used as a hammer, thethe cuttingoutting tubetube havinghaving toto hithit thethe treetree atat aa rightright angZe.angle.

___ PusherPUsher to get out the coreOON

HollowHolLow cuttingoutting 44__ ---1 tube

This instrument has been conceivedoonoeived to take rapidly smaZZsmall wood oorescores but it is sometimes used forfor rapid measurements of thinthin barksbarks (appro(approximatelyximately 2 omcm thioknessthickness maximum). This is not to be recommendedreoommended because measurements cancan bebe veryvery inaccurate.inaccurate.

22 MEASUREMENTSf1EASUREf1ENTS ON FELLED TREES

Important remarkremark:: Whatever thethe measurements may be thatthat are carriedcarried out on a felled tree, its referencereference diameter has toto be known. If possible,possible, the reference diameter has to be measured before felling, if not, reconstitute, examining thethe stump,stump, which was thethe height of the reference diameter and taketake thethe measurement there.there.

221 Length measurements

Length measurements are carriedoarried out with a decameterdeoameter' tapetape and are given in meters with at leastleast oneone decimaldeoimal placeplaoe (round(round offoff toto the neal'estnearest din,dm, oorr toto the nearest cm)om) ; sometimes also a graduated ruler is used of one meter long,long, eequippedwithq',ipped with aa steelsteeL pinpin onon eacheaoh edgeedge,': streohingstreching aZternatipelyalternatively eaoheach pin, a single person canoan rapidZyrapidly take the measurement.

222 SSizeize measurements

Size measurements are alsoalso carriedoarried outout withwith aa tape,tape, goodgood careoare being taken that the tape is put perpendicularlyperpendioularly on the axis of the stem and oloselyclosely fitted on the whoLewhole of the periphery. IfIf itit isis diffi-diffi­ oultcult to slide the taper under the tree, even when using a ourvedcurved steel needle attaohedattached toto thethe tape,tape, thethe diameterdiameter isis measuredmeasured withwith aa caliper.oaliper. - 9929 -

223 Accurate measurement of the diameters underunder bark or under sapwood

Using a bark measurement instrument is of course possible,possible, but one can taketake advantageadvantage of the fact that the tree is felled to debark the tree or to cut of the sapwood and to take measurement of the diameters under bark and/or under sapwood at variousvarious heights.heights.

224 r'lensurationMensuration of stacked wood

The volumevoZwne obtained is expressed in "stacked cubic metersmeters" II with one decimal place.pla(Je.

A "stacked cubic meter" is the bulk volumevolume occupiedoccupied by piecespieces of wood one meter longlong piled on oneone meter width, oneone metermeter high.high.

1m

It is thus a volumevolume which containscontains airoail' andand woodwood inin variablevariable proportions according to the "OWlor 0 " tJ'e pieces. The pilingpiling coefficient proportions according to the of0 P'e pieces. The is the volume of wood expressedexpressed inin m3m3 containedcontained inin aa "stacked"stacked cubiccubic meter".meter" if all pieces were cylindrical and of the same diameter, the piling 0 coefficient would be : = 0.785 . In practice it varies between 0.45 coefficient would be: i4 ~ In practice it varies between 0.45 (small branches of bad form) and 0.80 (split(split cordwood piZedpiled smaZZsmall end to large end).end).

It is difficult to estimate precisely a piling coefficient. Here are some indications toto estimate thethe volumevolume of wood in a parallelepipedic pile. For furtherfurther details, seesee bibliography,references 9 and 1 ;,. - IfIf thethe pieces are not tootoo small, take on each of them the followingfollowing measurementsmeasurements-?------diameter at each end and in the middle . and appapply ly Newton'sNewton's formulaformula (see(see §§ 231).231).

This is tedioustedious andand forcesforces toto pullpull downdown thethe pile.pile. MoreMore simply,simply, measure diameters of every piece on both faces of thethe pile (don't(don't trytry toto associate the two measurements of a piece) : Smalian's formula (see(see § 231)2.,1) gives :: - 30-

Volume of woodwood n =2 L (ED2) + (ED2)(E02) I in the pile = 88 face 1 face 22 ~ 4- length of pieces

Carry out this operationoperation on some similarsimilar pilespiles and taketake fbrfor pilingpiling coefficient :

Total volume of wood contained in thethe piles p P = Sum of the volumes of the piles inin "staked"staked cubic meters"

A confidence interval forfor pp cancan be estimated.estimated. SeeSee aa manualmanual onon

sampling techniques, chapter "ratio estimates". For instance : ref. 7.?

It is possible alsalsoo to weigh,~e igh thethe ;pilespiles (the(the woodwood is,:s sometimessometimes sold by weight),weight), whichwhich allowsallows dens-stydenSity estimation.est"mation.

- If!f_!~~_Ei~£~~_~~Y~_~~~!!_~!~~~!~E~' the pieces have small diameters, weightingweighting onlyonly isis practicablepracticable butbut the problem of volume estimation remainsremains if the piledpiZed wood mustmust bebe expexpressedressed in cubic metersmeters :: ImmerseIr,rnerse thethe woodwood andand measuremeasure thethe volumevolume of water displaced...displaced ...

RemarkRenark : when stacked wood isis cutcut byby axeaxe leavingleaving "pencil"pencil poine'ends,point" ends, taketake fbrfor lengthlength of the piece thethe lengthlength withoutwithout thethe points.points.

Attention : the shorter straighterstraighte oror fatterfatter thethe piledpiled pieces,pieces, thethe higher thepilingthejOiling coefficient.coe-fficient. IfIf forfor examplee~~le aa pilepile consists of-piecesor pieces of-wood0; Wood 22 mm long,long, aa coejT2csentaoeJJ"wient whschwhich has been calculated fbrfor the samesame piecespieces butbut of only 11 m long cannot be usedused; ; thethe difference betweenbetween thethe twotwo pilingpiling coefficients is oftenoften importantimportant (in(in thethe orderorder ofof 2020 percentpercent mmoreore forfor pieces of 1 m than fbrfor pieces of 22 m, but thisthis has to be verifiedverified inin eacheach case).case). -31- 3' -

23 DIRECT CALCULATION OF THE VOLUME OF A TREE FROMFROM MEASUREMENTSt1EASUREtlENTS TAKEN ON THE TREE

231 Calculating proceduresprocedures

The volume soughtsought wiZZwill be obtainedobtained byby addingadding volumesvolume·, of its components.components . The basic calculationcalculation thereforetherefore consistsconsists inin calculatingcalculating the volume of a loglog (stem(stem oror branch).branch) .

Volume of a log of length L

L c, and C are the girths at the • )11• CI C22 extremities.

c is the girth at mid-lengthmid- length C2 Cmm Cm D"D ,D are thethe correspondingcorresponding 1JI,D2,Dm2 m diameters.

1 , . . .. • L L 2" 2" Various calculation methods are possible CC22 m n7 2 (1) If Cm C isis known V == -L=L= Tt-D Dm LL m 4n47 4 m Huber 2 21 2 2 , DI+ D2 [CI+ C, (2)( 2) If cl and CC2 aarere known V - 4'Tr L L = n c, 2 V = 4if 2 4 2 Smalian 2 CI +C,1 [DI+ D2 1 ii V= L - (3(3) ) 4u 2 4 2

1 (C12 + C22 + C1C2) L , V == '~ n (C~ + C~ + C, C2) L (4) . fonnulformula a forfor 2 2 = ;27(DI (D~ + D2+ D~ + +D1D2) D,D ) L L a truncated 2 ( cone

2 2 if C1,C2and Cm are knownV = -c2 + 4C +C]2 24 1 (5) Newton -- L[D2 4D2+D21 Simpson = IT 24 I m 2

(2) -(4) Remark that (3)(3) < (4) < (2) and that (4)(4) -- (3)(3) - 71L? (no)2

Let us see what these .formulas give in some classical casescases : - 32 -

formula formula formula formulaformula formula If the Zoglog isis :: (1) (2) (3) (4)(4 ) (5) A cylinder is exact

2 1 2 22] volume with 3e3£ volumevolume withwith 2e2£ Cmc = =-2-r 1. l2+c rn 2 1+C22 lJ (*) (*)

A conic r-----, underestimates overestimates underestimates is exact is exact log L.----J the real the real the realreal ,

I volumeolume with e£ volume withwith 2e2£ volume with e( c = 1. l +C ] Cmrn 2-2 lJC1+02] 2 (*) (*) (*)

(*) ,\,*\ (*) . A neloidicneloidicri C:=, f1Ad~restirnatesukl4restimates overestimatesWrestimates iSñAerestimatesunBerestimates overestimates Wrestimates is exact log 1...... 1,-~- thethe real the real the realreal the realreal

1 volume with volume with C2/3=2/3 L2/3+' 2/3]2/1 volume with volume with volume with volume with = 1. +C ~olurne C3 c 8e e£ 2e Cmm 2-2 l IJ 2 p£-£'3e-e' > 4£ >4 • 2£ I > 2e' >4E4e'-> 3e-e'3£-( >> 8; :>8€. '' c'£' <<-- c-e'(-(' >>""""3 > 2e:: ' 2 3 £ --j.-)8e 3 3

2 1/3 rL £ • 71, 2/3 2/3 with : e = (D - D2)2and E'= D D' D1/3I D2 48 16 1

( e£ >> 36'3£' )

Formula (5)(5) is exaotexact fórfor eacheaoh solidsolid (not(not necessarilyneoessarily of revolution) for whiohwhich the area of the 'sectionseotion isis a cubicoubio of the distanoedistance of thisthis seotionsection to a sectionseotion origin.origin, o

2 3 s • a + bx ++ excx2 + dxdx3 x Sxx

x-axis - 33-33

This is thethe case forfor thethe cylinder,cyZinder, thethe paraboZoid, thethe cone and the neZoid,neloid, because these solidssoZids are obtained by rotationrotation around b the x-axis of a curve y = axbax withwsth :

2 cylinder : b = 0 which gives S = lTa cylinder b = 0 which gives Sxx = 78.2 1 2 paraboloid : b = 1-which givesgives S 1[arra2x x =z2 Sxx = Tra2x22 2 cone : = 1 which gives S 1[a x b = which gives Sxx = 3 2 3 neloid : b = which givesgives S == Ira2x3na x 2 Sxx =

One wouZdwould thus think that,that, toto calculatecaZcuZate thethe volumevoZ ume of a stem ofofwhich which thethe girthsgirths atat aa spacingspacing ofof lengthZeng t h L are known, itit wouZdwould be preferabZepreferable to appZy formulaformuZa .(5)(5) fOrfor logslogs ofof lengthZength 2 L L instead of one of the formulas (1)(1) to (4)(4) forfor logsZogs of lengthZength L .• This is not necessariZynecessarily so since the conditions of vaZidvalidityity of Newton'sNewton's formula,formuZa, though ratherrather general,general, areare notnot necessarilynecessariZy fulfilledfuZfiZled byby eacheach Zog.log.

As a matter of fact,fact, formulaformuZa (5)(5) beingbeing somewhatsomewhat lessZess easyeasy to use, one rather uses the others. WhichWhicn one is thethe bestbest? ? OneOne cannot answer thisthis question which,which, moreover,moreover, isis ofof smallsmalZ importanceimportance because ththee precisionprec ision ofof the the estimation estimati on of of the the volume vo lume depends depends more more from the diameterdiameter measurementsmeasurements (p(precisionr ecision andand number)number) thanthan fromf rom thethe calculating methodmethod used.used. C . C22 Remark :The 5 formulasformuZas can be considered to give simiZarsimilar resuZtsresults ifsf c > 0.82 1 because for each case marked * in the previous table,table, the relativrelativee error is thenthen ZessZess thanthan 11 %.%.

Whichever way way a avolume volume has has been been cal calculated,culated, itit shouldshould bebe expresseexpressed·d inin cubic meters,meters t withwith 33 oror 44 decimaldecimal places.places. 232 Recommendations for thethe measurements toto bebe takentaken withwith regardregard to the required volumesvolumes

The followingfoZlowing recommendationsrecommendations apeare made about measurements toto bebe takentaken onon aa treetree forfor aa directdirect estimation ooff itsits volume.volume .

----- __ ~~9~!~~~_!~!~~~Required volume __ {~!!~_~~c~2(with bark) ____ _ TotalTotal stem volume Stem volume at a fixed cross cutcut TotalTotal and big ,Ioodwood stem + (for exampleexample : big wood volume)vOlume) branchesbrooches volumevn 1limp.

rn Reference diameter DR . Reference diameter D DR rn Reference diameter D . Stump. diameter D,DS . Stump diameter D5DS R

Length between D;DR and DDS . Length between DR and DDS S A diameter atat aa iiigherhigher . H = height at crosscross cutcut

level than DR' fo~for' exam­exam . ACaiameterAc5iameter at a higher Compulsory Compulsory ( ple,pie, diameter atat aboutabout level thanthan DR' for exam-exam­ 1/4114 H or 1/2 H Htottot or 1/2 Htottot ple diameter at about This volume cancan onlyonly 1/4- H!i ' 'or 1/2ilz " H " (preferable). cr Crcr STANDING (preferable).(preféiable). HHtot = totaltotal height.height. be measured on aa tot felled tree.tree. w TREE Diameter at othero ther heights~~eights .. ,asas provided for on thethe (optical form for the mensuration of standing treestrees with Diamet~rDiameter at thethe crosscross cutcut measure- measure- I If possible { the Bitterlich relascope If possible2possible \ and ((§ 211.25) :: diameter (diametersatdiametersat other heights.heights. ments) every 2 m with an inter-inter- calated measurmeasurementement inin the lower part.part. ------~------~------QJL3J LC ' Reference diameter D0 rn . Reference diameter D Stem : see 31 and DR QJ [!! Stump diameter DD R . Stump diameterdiameter DD R 5S . -S·s Compulsory . Length between DaDR andand DsDS Compulsory Length between DRDR andand DeD~ Branches : two possibilities HtotH = totaltotal heightheigh t H = heightheight atat crosscross cutcut tot orcr - proceed forfor eacheach largelarge branchbranch . Diameter atat HHandand FELLED . Diameter atat 1/2 1/2 .11tot H H Cr as for the stemstem andand tot 1/211112 . or cror stackstack thethe smallsmall branches Diameter every meter TREE Diameter every meter Diameter everyevery metermeter I If possible l or every twott..1O meters If possible oro r every twotwo meters II !i....P.ossible - moremore simplysimply : stack all thethe from the stump from thet he stumpstump branchesbranches --3535 --

Whatever the volume ~~~uired,re uired, it is good to take always thethe measurements enumeratedenumerated under LUI ; they allow, by interpolation, thethe calculation of the stem volume comprised between any lowerlower and any upper crosscrOSB cut.cut.

Remarks

1 - The volume underunder bark can onlyonly bebe obtainedobtaingd accuratelyaccurately onon aa felledfelled tree because the bark thickness cancan bebe measuredmeasured atat anyany height.height. On a standing tree thethe bark isis measuredmeasured atat referencereference heightheight andand some assumption is made on thethe decreasedecrease of bark thickness taking into account data collected onon felledfelled trees.trees. WithoutfVithout anyany suchsuch .. information an assumption hashas toto bebe mademade (fbr,{for instanceinstance == constantconstant bark thickness).thickness}. See paragraph 36.

2 - The measurements given in thethe tabletable allowallow calculationcalculation ofof grossgross volumes. To get net volumes,volumes, thethe additionaladditional measurementsmeasurements toto bebe taken depend on the type of net volume required. The formform givengiven in paragraph 211.25211.25 forfor volumevolume calculationcalculation onon standingstanding treestrees withwith Bitterlich relascoperelascope showsshows aa simplesimple wayway toto recordrecord thethe defectivedefective parts of the bole of a high-foresthigh-forest tree.tree.

3 -- ProblemProblem raisedraised byby thethe stump.stump. The fellingfelling levellevel depends on species,species, tree size,sise, local habits and changes with sawing equipment. Thus, a volume calculated on a standinstandingg tree contains an uncertainty due toto thethe unknownunknown stumpstump height.height. InIn order toto be able toto calculatecalculate easilyeasily thethe volume.volume inin functionfunction of dif-dif­ ferent hypothesis on stump height, aa simplesir,'p,Le wayOIay (for{for treestrees withoutwithout buttress only}only) is to give a volumevolume inin whiChwhich thethe partpart underunder referencereference diameter DR is the cylindercyZinder betweenbetween DR andand the groundground an~and withwith diameter DR' From this volume which contains the stump,stump, it is easy, diameter DR. From this volume which contains the it is easy, knowing the felling level,level, toto derivederive thethe felledfelled volume.volume. SeeSee exampleexample 233.2.

233 Examples

233.1 Let us take a felled tree on which have been measured overbark diameters every meter starting from the stump. Let us ·calculate the total volume and the "big wood" volume of the stem.stem.

Here are 5 different calculating methods for the tree divided into 11 m Zogs.logs.

" ------. . ,i heigheghtshts VVolume l in "I : 3iets Diameter o ume 'l.n 1,:' o FormulaFOr'mula used Hei'l.hts Diameter i m mma emcm Method 11 Method 22 Method 33 Method 44 MethodMet hod 55 9.51- X 1199.5. 5

GJ, 1 0,00270 . 0027 m4 0.00270,0027 m141 0.0027 m141 0.0027 m4 0.0027 - - 7.4 I----+t--- 7.4 12j2 1 0.00090,0009 31 00,0009, 0009 2 1 0.0009 121 0.0009 121 0.0009 ; : 7 .22 I-----Hr---- 8~ II II 20,0009 7.2~ 210.0028 6.77 1-----+-+-- x - 99 [IIpi 0.00640.0064 rn31 0.0064 m121 0.0064 m 0.0028 6.7

10 6.2 1----+-+-- 10 1 mfi 0,01660.0166 CD1 0.0079 [2[IT] 0.0106 rn31 0.0104 f------I-I-lIS. 57 7

5.2 1-1---+--\--- 13 1mE 0.0279 0.0279 --"=""---1 CD 0.0133

m 0.0180 rn31 0.0177 f------t-I-il 4.7 1 ..., m w .., '""Cl os C> '" .,., g 4.2 1-1---+---+-- 17 I 1m 0.0452 1[]J.0.02270.0227 I 13 . C> 377 J! .,., 0.0271 131 0.0269 .<:l IT] 0.0271 rn 0.0269 N 20 ~ ~ 3.21 20 I 1m 0,07000.0700 ---Ii CD 0,03140.0314 I 12.7 C> 2.7 ,;; .,., IT] 0.04650,0465 7rn 0,0452 0.0452 .%l 28 2.2 f---+----+- 51 0.1194 CD1 0,06160.0616 1.7 d = 31 0,0710 '3rn 0.0707 .2 32 0,1598.1598 ril0,0804 0.7 7 y= 33,5 2J 0,0883 131 0.0881 0.0883 p-- 0.0461 - 35 0.2 L'y're Total stem volume m3 0.27150,2715 0.2690 0.26770,2677 0.2731 0,26980.2698

Big wood stemstem volumevolume m3 0.2688 0,26630.2663 0.26500,2650 0,27040.2704 0.2671

Note : d =~ reference diameter = 31 cmcm ;!~ estimated diameter_(bydiameter. (bw average)avergge) used in methodmethod 5 5 - 3737- -

The 5 methods give very similarsimiZar resultsresuZts (this(this is alwaysaZways the caseaase when thethe logsZogs areare short).short). Of course,aourse, itit cannotaannot bebe saidsaid whichwhiah oneone is the nearest to thethe truth.truth.

233.2233 . 2 Measurements on aa standingstanding treetree forfor totaltotaZ 3tem.1tem volume.voZume. Case where onlyonZy compulsoryaompuZsory measurements are avaiZabZeavailable (c:oelieU 71rn of tabletabZe of paragraph 232).

For thethe upper log,Zog, anan hypothesis isis neededneeded onon thethe form.f orm. LetLet usus suppose a conicaonia form.form.

Heights in meters 9.5 ------( w 2 . X0 1 Cone formula 12 XO • 1662 x5.5 =- 0.03686 12 " \ 4 - - -- 1- - -DD = I6cm Smalian'sSmalian I s 2 2 : w (0.162+0.312 -"<'j formula [0 . 16 ; 0.31 ) 2.7 - 0.129040.12904 T 2

- -- -_DR -= 31cm31em 1.3 -- R Cylinder with r 2 " xX 0.31 x 1.31.3 -= 0.09812 0.2 -- - - diameter 3Icm31em T4 - -DDS -= 35cm35em\ 0 0.2640

2 total stem volume.volume * [0.26400.2640 -- -4- x 0.312x H m3 1 0.31 x HS)S m3 ~ stump height in meters

To estimate the big wood volumevoZume of"the stem without knowingknowing thethe height of crosscutarossaut DD == 77 cmem ,, this height has firstfirst toto bebe estimated,estimated, whiahwhich neaessitatesnecessitates an hypothesis on the formform of the lastZast ZogZog ;; letZet usus suppose again the conicaonia form

Ix 7 cm 7 5.5m > 2.4 m and y =- 3.1 m Y 5.5 n'r 16 ". \QJ~ ::D = 16,: :cm - 38 -

Calculation...... of .big.... wood...... volume...... of thethe stemstem

Heights inin meters

7.17 • 1 ---- D" 7 cmcm-- - · Formula forfor a T2W (0.072+0.162+0.07x0.16)3.1( 0.07 2 +0.16 2 +0.07~0.16 13.1 =a 0.033840 .03384 truncated cone 1112

4 ---- D =16 cm- -- SmaSmalian's I ian's ir [0.162+0.3121 2.7 =- 0.129040 .12904 formula 4 2

1.3 ----Oa31R=31 cm-cm--- R --I Cylinder withwi th 2 cm .17;" x~ 0.31 2 x~ 1.31.3 =a 0.09812 0.20 . 2 ----OS=35DS= 35 em \ diameter 31cm3Jcm

0.26100

TI 2 ~ =---'> Big wood volume of thethe stemstem : (0.2610 -- 7;;It xX 0.3120.31 xX HS)Es) m3 • stumpstwnp heighthe-ight in meters

IIIIn these twotwo volwnevolume calculations,calculations, thethe DSDmeasurement measurement isis notnot S used as the tree below DR is assumed to be a cylinder. However, the used as the tree below DR is asswned to be a cylinder. However, the

data DR ', DSDs ', length between DROR andand DS , are useful to study thethe DS ' form of the stem basis. --3939 -

24 STUDY OF TREE FORM

Refer·enceReference diameter and total height cannot suffice to describe completely the fbrmform of a tree.t ree . We shaZZshall restrict ourselves to the study ooff the formfom of the stem because it is impossible toto recommend fbrfor every speciesspecies a singlesingle metmethodhod tot o describedescribe thet he morphology of everyever y constitutive papartr t of a treet ree ; moreovermoreover,, such observationsobsel'Vations (fbr(for instance :: number, in-in­ sertionsertion angZeangle and straightnessstraightness ofof branches,branches, etc...)etc . .. ) areare mainlymainly useful,useful for geneticists and this is out of the scope of thisthis manual.manual.

241 Measure of stem form with a coefficientcoeffi cient

241241.1.1 Definitions

The simplest coefficientcoeffici ent isis thethe form factorfactor f

VoVolumelume f -~ ~(Refere(Reference~----~--nce b basalasa~l ~area area)~~) xx~ ttotalotal~,,~~ heightheight

Thus, every volume which can be defined in a tree has a corcor-­ responding fbrmfom factor.factor. The most common isis relatedrelated toto thethe totaltotal volumevolume of the stem but the formform factor oorrespondingcorresponding to the volume ooff the stem until a given upperupper limitlimit maymay howeverhowever bebe considered.considered.

The form factor f is not a characteristic of stem formform :;.

(a) : two stems with same f do not have nnecessariZyecessarily the same form and mainly:mainly

(b) : two stems of same fbrmform do not havehave thethe samesame ff ; let us indeed consider twotwo stemsstems of same formfom (similar(similar stems).stems). Stem 22

/ / / / / / / / / / / H2 = totalt otal StemStem I1 / height / / ///iiii . ,_ 2 /. ..-- __ / /' ,, . /' ./ ..--_, - 2 /' Reference H= total 11 ' diameter 1 , . level ne-taht / /' level hei.ght / ..-- 1 .., / ..-- o /' 0 L"-----_ I stlonps - 40 -

2 The similitude with 0 as center and k = as ratio trans- 171 forms stem 1 into stem 2. 1

'i'heThe stem-volumes are VVI and V (stump volume i8is included or V2z not;not ; this doesdoes notnot affectaffect thethe result)resul~and and thethe correspondingcorresponding form factors are :

17 1 V2 f- and f - 1 gliii 2 g2H2

3 Z The relationships: V2V == kk3 VVI , H2 = k 111 and g'g = k2g1 z I HZ k HI Z k gl f I gz 1 g2 show that : =-T = gz t;"f2 g2

Since gz > g' , is less than fl. Since g2 > g Z fZf2 is fl·

Hohenadl removed drawback (b)(b) by definingdefining aa coefficient,coefficient, called stem-natural form factor(as opposedopposed to f which is sometimes calledcaZZed artificial formform factor)factor) stem volume f' =- ==c...:.~=.=. g H x H 10

where : H = totaltotal height H g H = area of stem sectionsection atat height TO10 Td10 However, the difficultydifficulty of measuring a diameter at a relative height on a standing tree limits thethe useuse ofof thethe f'coefficiCcoefficient.ent.

Remark : the natural form factor corresponding toto thethe total volumevolume of a stem is generally between 0.30.3 andand 0.6.0.6. ItIt isis equalequal toto : 5 "9 = 0.56 forfor aa parabolo'idparaboloid

100 = 0.41for a cone 243 = 0.41 for a cone 250 = 0.34for a neloid. 569 = 0.34 for a neloid. -~ 4141 -

241,2241.2 How to to calculate'Vf calculatévf oror f'

_ The fo1'TT1form factor of a stem cannot bebemeaBured.directly measured.directly ·;; thethe 'volume'volli6 has to be calculated first.first. It has been saidsaid in paragraph 2323 that the principle i8is to divide the stem into Zogslogs and to add the volumes of these Zogs.logs. ForFor instance,instance, if the stem isi8 divided intointo 55 logs of equal lengthlength and if the diameter isi8 measured at the middle of eacheach:

Stump j I I I I I I I : I I I I I I ~ I I I : I I D D D D DH 3H 5H 7H 9H 10 10 10 TO10 TO

H

the application of Huber-fo1'TT1UlaHuber-formula to each loglog gives 2 2 2 2 2 D2 +D2 +D2 +D2 +D2 D H + D3H3H + DSR5H + D7H7H + D9H9H f 1 TO10 TO10 TO10 TO10 TO10 5 (reference diameter)2 2 2 . 2 2 D2 + + + H + D3HD3H + D5HDSH ++ D7HD7H + D9HD9H

1 10 10 10 10 10 f'f TO TO TO TO TO - 55 D2 D2H 1010

In order to calculate thethe averageaverage formform faatorfactor of aa set of n trees, several calculus are possible.possible. HereHere areare threethre. of themthem: : --4242 --

Mean ofof naturalnatural formforr.! factorfactor ~{eanMean of artificial formform factorfactor

n 9 2 n n 9 2 n 2 y f: 11 1 V.V. [D2 HI LnX ff.. [2[1)-OR 1.1Hf LVV.. [1)2 HI i=,. I I f:i1 CHD'[0 H Hfi. i.,L 1i [02HH H].. i R H] i 1= TO 1 i=1 i i=1i=l 1 i i=1 1 [O~R i 0 _ 10 1 (I)1 = =- 2 n 9 2 9 2 n 2 1[ 1[ [D2 H I;2 ; /rD - H [1)- HI I; L [DR H] it H]: 4 iU02HH H]:1 } [O~ Hf [OR H i=1 [o:~1110 i i=1 i i=11=1 i1 i=11=1 i 10 l: 10 n n n n 1 f!iD2 H 1 V.V. 1 ft2H 1 V. L fii .[02H Hl. L 1 L 1 R H]i L i1 i=1 1=1i=1 i=1 fi[O~ 1i i=1 10 J 1i _ (2)(2 .= = n n 9 n 9 n 9 1[ X rD2li HI ; X [1)- H [D1-1 HI I[02[D- H L [O~ H]. I;4 H 1 } [o~ H]. ~4 R H] i=1 IT) i i=1 Hti 1=11.=1X 1 ;i4 i=1 R i 10 1 1 [o~o , 10 n n V.V. n n V. I I 1i I I 1 -I X f. = -I X i- IX nf! == -I IX n L 1 n I (3) n 1i n n i=1 n i=1 71[ I 2 ' i=1 i=1 r1[ H w [p211DH H I;T DR 4 [02H H]. [O~ H] i 10 / 1i

1 D = diameter at height H DR = referencereference diameter °HH It10 Htottot OR 10 H = total height H = totaltotal height

Each of these 3 formulasformuZas is a weighted meanmean of the individualindividuaZ formform factors, the weight given to a tree being (D2H)2(02H)2 in (1) and (D2H)(02H) in (2)(2) ;3 in (3),(3), thethe treestrees areare givengiven equalequaZ weights.weights. TheThe formulaformuZa toto bebe taken depends on the reZationship between the variance of V and D2H02H , with the f011owingfoZZowing ruleruZe :: if the variance of V17 isis proportionaZproportional to 2 2 (D 2H)H)a , the variancevarianc~ of the form factorfactor is thenthen proportionalproportionaZ toto (D2H)a-2(D H)a-2, 9 2-a the weight to be givengiven toto aa treetree isis (D-H)(02H) . - 4343- -

. _ , 20 F"omrulaFbrmula (1)r1.) isis thus ··validvalsd if the variance · of V'V.does does not depend onon DD`H H (a(" = 0); formulafOl'mula (2)(2) is validvatid if the variance of V is proportionalproportional to D2HD2H (a(" = 1)I) and formulafOl'mUla (3)(3) is valid if the variance of V isis proportional to (D2H)2 (a(,,' = 2)2) .

1-7/In practice, it is thusthus necessary toto startstart by estimatingestimating "a ; the same problem occurs before fittingfitting a volumevolume equation by regressionregression asas will be explained in paragraphs 353.14353.14 andand 353.2.353.2. TheThe" a coefficientcoefficient isis oftenoft en fOundfound toto lietie between 11 and 22 : fOl'mUlasformulas (2)(2) 01'or (3)(3) are therefbretherefore the most used ; in absence 'of no ppreciserecise knowledge about the lawlaw of variationvariation ofof the variance of VV with a" , it is recommended to take formulaformula (2) ; moreover, thisthis formulafOl'mUla is thethe most practicalpractical sincesince thethe totaltotal n volume of the set of trees, IX V. , appears explicitlyexplicitly inin it.it. i=1 V.'1

242 Description of stem form by the equation of the tapertaper curve

242242.1. 1 The two types of curve - Problems of scalingsca ling

Having measured diameter at different heights,heights, datadata can be represented by two types of diagram .:

Diameter Sectim area Diagram 1 Diagram 2

w 2 SH • 7; DH . ------1

DH - - -­ i Height He ight .0 i o H. 1 tot - 44 -

Section area

Hatchuredfiatchured area == V . VH.H 1 DiagDiagrampam 1 representspeppesents SHSH. . -¥A'/~."":/7'" ,,:~} ..... the stem asas it is seen ; diadia-­ 1 gramgpam 2 offepsoffers the advantage ooff showing ththee volume vVHiH. to a 1 given heightheight Hi.H.. 1 hheighte ight

H·1

In orderopdep that two treestpees of same formfoPm be representedpe ppesented byby thethe same curvecupve and that twotwo treestrees with thethe samesame curvecUPVe havehave thethe samesame form,foPm , the scales have to be changed.changed,

HereHepe areape twotwo possibilities forfop typetype 11 diagramdiagpam

H

0;Diagram agram 1l' ' DaHtot 0lagram, 1"

H 1 o a HIH/ 0121.aH aH TI--fl . tututL tot Osa<1Osa< 1 Example

I I u = = diameter at TO H a = TO10 .• DaH 10 tottot tott o t - 45 --

and thethe twotwo correspondingcoppesponding possibilities fbrfop typetype 22 diagramdiagpam :

Diagram 2'2' DiaQram 2" SH SH S H2 tot aHtot

ao H 1 oO H/DaHHID" H tot Htot~ < 0O S5.: a" < 1 tot Example : .~ 1 HaH = diameter at 10 Htot 1 \ Ua H = diameter at TIT Htot 1 tot "(1 = TO -, I 1 = area of section at TIT1 H \ s area of section at IG Htot 7SaHtot=aH tot

Whateverflhatevep the heights ofof diameterdiametep measurementsmeasupements are,ape, diagramsdiagpams 1'l' and 2'2' can be donedone; ; diagramsdiagpams 1"1 " andand 2"2" cancan bebe donedone onlyonly when diameters havehave been takentaken at thethe samesame relativepelative heightsheig~ts onon eacheach tpee.tree.

242.2 Fitting a taper curve by calculation

242.21 Principle

The intepestinterest in having a formulafopmula forfop thethe tapertapep curvecupve isis to allowaZZow an easy volume calculation of the stem portionpoption between two heightaheights HI and HH2.• 2 It is natupalnatural to considerconsidep firstfirst thethe modeZmodel :

(see example 242.221) SH = b0 + b1H +b2H2 + b3H3 because we have seen in paragraphpapagpaph 231231 thatthat thisthis relationpelation isis satisfiedsatisfied by most of the simple geometricalgeometricaZ solids to which a stem can be compa-compa­ red. If this fórmulafopmula describesdescpibes thethe formfoPln badZy,badly, herehepe areape twotwo possible solutionssolutions ::

" - 4641i -

(i) try a polynomdalpolynomial ooff higherhigher degreedegree ;; tthehe model is thust hus writtenwritten in the general form p ,S=b0 +bH+b2H2 + +bHP = L b Hk H I k=0k=Olbk k (since this modelmodel hashas pp+I+1 paparameters,rameters, atat leastleast (p+l)(p+1) measure-measure­ ments (H(H,, SH) SH) are necessarynecessary tot o fitfit it).it) .

( i i) divide the stem into logslogs and fitfit aa model toto each ofof them,them, with constraints on the coefficientscoeffi cients toto forceforce thethe curvescurves to join correctlycorrectly.. There are numerousnumerous ways toto proceed,proceed, ac-ac­ cording to the number of measurementsmeasurements available,available, thethe number of Zogslogs consideredconsidered,, the models chosenchosen forfor each Zoglog and thethe conditions imposed for the junction of the curves.curves.

TheThe exampleexample ofof § 242.222242 . 222 gives a case where 4 measurement points are available and 2 logs are considered. TheThe methodmethod of "cubic"cubic splinespline functions" f unctions " (see BONEBONEVA VA et al.at. ref.ref. 1) is the extreme case ofof functions ofof thist his typetype becausebecause thethe Zogslogs areare defineddefined byby twotwo successivesuccessive diameter measurements ::

SH = ---- log i logl og 1+1i+1

x _ r--- x. x = i I1 "i '-:'"L H ~ --- measurementsm~9urements~

To each log a cubic is fittedfit ted 2 3 logl og ii : y a. + b . x + c.x2c .X + cl.x3d . x 1 1 1 1

2 , 3 llogog i+Ii+1 : y = ai+1a + + b . IX + +l + d. x i 1 1+ + ccixi+I X + ni+lx1+1

The coefficientscoeff icients of the cubiccubic are obtainedobt ained by wwritingriting that,tha~ , ata~ each measurementmeasureme~t point,point, 9y equals y and its first and second derivativesder.va~.ves equal the firstf.rst and second derivatives of the cubic of thet he ad,]adjacent acen t loglog : --4747 -

2 3 aa., ++ b.x.b.x. ++ c.x.c.x. ++ d.x.d . x. == y.y. 1 1~~1 1~~ 1 1~l1 1. 2 3 aa.· ++ b.x.b .x. I + C.X.c.x. I + d.x.d. x. I = v.Y· I 1 11-1-1 11-1-1 11-1-1 '1-11-

2 2 + 2c.x. + 3d.x. = b. + 2c.x. + 3d. xb.. 1 11 11 1+1 1+1 1+1

c. ++ 3d.x.3d.x. == c.c. I + 3d,3d. IX.X. 1. 1.1 11. 1+1+1 1+11+ 1.

Knowing the coefficientsooeffioients of the cubicoubio forfor loglog 1+1,i +I , this system of 4 equations canoan be solved, whichwhioh givesgives values of a.,a ., b.,b., c.,c ., d.. 1 1 1 d .. 1 The solution is thus obtained step by step :: the formform of the last loglog is imposed (conic(oonio formform forfor instance)instanoe) ;; the oubiocubic of the precedingpreoeding log is derived and so on untiluntii the first loglog (the(the symmetricsymmetrio procedureprooedure canoan be followed, fixing the form of the first Zoglog and deri-deri­ ving the oubioscubics step by step fromfrom bottom toto toptop of the tree).tree).

Remarks

- A taper curveourve canoan be fittedfitted toto thethe bolebole onlyonly; ; whatwhat hashas beenbeen saidsaid isis stillstiZZ valid provided that bole heightheight takestakes placeplaoe of total height in the models;models ; a simple modeZmodel (cubic)(oubio) isis sufficientsuffioient inin general.general.

- Instead of the area of the seatsection, ion, the diameter canoan be expressed as a functionjUnotion of height. In general, thethe modeZmodel whichwhioh isis usedused isis alsoalso aa polynomial

DH = X ckHk k=0

Taking aooountaccount of thethe ' previousprevious paragraph,paragraph, the models are transformedtransformed in order to get coefficientsooeffioients whiohwhich are ohacharacteristicraoteristio of ththee form :: S H k y= b' xxk ; Yy == x • Y r k kk=0=O H29 H tot Htottot

H k SH Y = b" xkx x = :;---- Y = ~/ ; Y = D k=0k=O k SaH DaHa H tot tot

( 0O ,;5- aa << 1)I) H c;( xkk DH c' x x =- Y I k ; Y = k=0k=O Htot HtOt - 4848 -

H u k DHH Y = e"ck Xx ; y x -= r k Y -D D D k=0k=O aH DaHaH tot tot

(0 S a << I)1)

Variants can be brought to these modeZsmodels (see(see ref. 6 - 9 - 13).

- FormuZaFormula to calculatecaZcuZate aa volumevoZume knowingknowing aa tapertaper curve.curve.

Let us suppose that the foyaulaformuZa of the curve isis

S == bb + b H +.•••. + bb HP • 0 1 p area of section at heightheight H.H. I H2 HP+ The integral of SS is g(H) = boH + bi + ••• + bpb-­ 13+1 P p+1 The volumevoZume of the section of stem between heightsheights HI and H2 isis g(Hg(H2)) -- g(H1).g(H )· 2 I

If the taper curve is given by a functionfunction reZatingrelating diameter to height D = co + c1FI + + c HP

u 2 the expression of SS == *D2D has to be written firstfirst ; thethe required volumevoZume isis g(Hg(1-12)) -- g(H1)g(H ) where g(H)g.(H) is the integraZintegral of S ; calcula-caZcuZa- 2 I is of S ; tions are more numerousnumerous and lessZess preciseprecise ;; thisthis isis whywhy itit isis betterbetter to express a tapertaper curvecurve as S = f(H)f(H) instead of DD == f(H)f(H) .

To caZculatecalculate the volumevoZume untiZuntil a given crosscut, the samesame method is followedfollowed afterafter havinghaving calculatedcalculated thethe heightheight of this crosscut.crosscut. This is easy by lookingZooking at thethe plottedpZotted curvecurve but,but, withwith aa computer,computer, thethe calcuZationcalculation raises problemsprobZems which cannotcannot bebe treatedtreated here.here.

- It is necessary to verify that thethe tapertaper curvecurve whichwhich isis obtainedobtained is reaZrealisticistic :: y shouldshouZd not get negative andand shouZdshouZd decreasedecrease asas x increases. --4949 -

242.22 Examples d 242.221 Four measurement pointspoints - Fitting aa 3d3 degreedeg ree polynomial

Let us take again thethe treetree of example 233.2233.2 and try to 2 3 SuSH H fitfit thethe modelmodel: : y = a + + bx ++ cxcx2 + dxdx3 ; Y =-­ x -= - - y- H H2 HtOttot tot

The solution of the following system

2 3 2 (a b 0.2 [0.2] d[0.21 a + --0.2 + c d 7 (0.35) 9.5 (~l9.5 + (~:;)9.5 = ~4 (~l9.59 .5 2 3 2 1.3 [1.3] (1.3] 7 [0.311 a + b + c + dd = L b 9.5 + c (~l9.5 (~l9.5 ~4 (~l9.59 .5

2 4 [ 12+d[ 4 137 (0.161 a ++b- b - +c, c ~)2 9.5 [9\)2+94.5 d (9\(9.5 *4 [ 9.5 j

Ill\a +b++b+c+d=c +d 00

3 gives : a = I.1.1035 1035 xx 10-10-3 3 b = -1.7385-1.7385 xx 10-310- 3 c == -1-1.9100.9100 xx 110-30- 3 d = 2.5450 x 10-310-

This curve cannot be retained since y is negativenegative when x is between 0.55 andand 1.1.

242.222 Four measurement points.points. DivisionDivision intointo two logs and fitting a curve toto eacheach ofof them

Let us take again the previous tree ~dnd suppose thatthat thethe upper half of the stem is aa conecone ;; let us fit a 3 degree polynomialpolynomial to the lowerlower half, with the constraint that the two curves are tangent at thethe junction-point.junction-point. --- 50 -

SH Y H2 tot 2 n [0.35 49.5)

7n [Q:l!.)2(0.31 2 - A, B,B, C, D : measurement pointspoints 7;4 9.5 ]

2 3 y - a + bx + cxcx2+ dx3dx

N 0.1612 l. ______4 9:5)2 I I 92 ' (x-I)(x-1)-

1 1

, o0 4_-+_____ +-If- ____-==~_~D:..-. ___ x _ _ H_ H 0.2 1.3 4 0.5 1 tot 9.5 9.5 9.5

The system to be soZvedsolved is

[o.21 2 (0.21 3 (0.35]2 a + b 0.2 + c + d = 11-- ll- : passing throughthrough AA 9.5 (Hr9.5 (}~r9.5 4 (~29.5

(1.3) 2 (1.31 3 7f0.31) 2 passing through B a + b~++ c +d+ d (~]3 = ~_ (Q:l!.]2 : passing through B 9.5 (~r9.59.5)9.5) 9.5 4 l9.59.5 ) 2 r )3 7[0.16] : passing through C a + b _4_ + c (9\J 2 + d = ~ [~] passlng through C 9.5 c [94.512+ d [9~J(.94.5J 4 9.5

(0.5_1)2 a + b 0.5 + c (0.5)2 + d (0.5)3 = a'a' (0.5-1)2 in the point with abs- ccissaissa 0.5 the two cur- b + 2c 0.5 + 3d(0.5)23d(0.5)2 = 2a'2a'(0.5-1) (0.5-1) ~ yesves join and are tangent

3 10-3 one gets a = 1.0976 x 10- Xd = 7.4226d = 7.4226 Xx 10-3 3 3 b = -1.4004 Xx 10-10-3 a'= 0.5671 xX10-3 10- 3 cC = -4.7336 xX10-3 10- The corresponding curvecurve isis drawndm"m above;theabove; the fitfit isis correct.aOY'l'ect. - 5151 -

242.223242.223 DescriptdonDescriptlon of thethe mean profileprofile ofof 33 stemsstems with a 3° 3 degreedegree polynomialpolynomial

Diameters at different heights have been measured on 3 trees (( HH in meters anand.a °D in centimeters) :

------Tree I1 ------Tree 2 ------Tree 3 H i: DH H : DH H :------DHD : DH : DH : H 0.2 44 0.3 57 0.3 : 66 1.2 39 1.31.3 : 52 1.3 61 1.3 38 2.3 ~ 47 2.3 56 2.22.2 34 3.3 43 3.3 52 3.23.2 31 4.3 39 4.3 48 4.2 27 5.3 36 5.3 44 5.2 25 6.3 33 6.3 41 6.2 22 7.3 31 7.3 38 7.2 20 8.3 29 8.3 36 8.2 17 9.3 26 9.3 34 9.2 12 10.310.3 23 10.3 32 9.8 7 11.311.3 19 11.3 29 10 0 12.3 13 12.3 25 12.8 7 13.313 .3 21 13 0 14.3 14 14.8 7 15 0

Do these stems have the samesame fbrmform ?? -52- 52 -

2 H DHH]2 The values of HH-H- -= xx and "4rr4 [--il == yy areape calculatedcalculated withwith Htot, heights and diametepsdiameters inotmeteps.iPtmeters. Htot

Tree I1 Tree 22 TreeTr ee 33 ------4------4------~------y; I04-- x ; : yx164yx IO x 1 : yx104yxlO x i yx104 0.02 15.205 0.023 15.09915 . 099 0.0200.020 15.20515.205 0.12 11.946 0.1000 . 100 12.56612.566 0.0870.087 12.989 0.13 11.341 0.1770 . 177 10.266 0.1530.153 10.947 0.22 9.079 0.254 8.593 0.2200.220 9.439 0.32 7.548 0.3310 . 331 7.069 0.2870.287 8.042 0.42 5.726 0.408 6.023 0.3530.353 6.758 0.520 . 52 4.909 0.485 5.0615.06 1 0.420 5.868 0.62 3.801 0.562 4.466 0.4870.487 5.041 0.72 3.142 0.638 3.908 0.5530.553 4.524 0.82 2.270 0.715 3.142 0.620 4.035 0.920. 92 1.131 0.792 2.458 0.6870.687 3.574 0.98 0.385 0.869 1.678 0.753 2.936

I1 0 0.946 0.785 0.8200 . 820 2.182 0.984 0.228 0.887 11.539.539

I1 0 0.953 0.684 0.987 0.171

1I 0

Plotting these values shows that the relationpelation between y and x is ppacticallypractically the same in each tree.tpee. The 3 treestpees can thusthus be consideredconsideped ooff same form.fonn. Let us see ifif the modelmodal 2 3 y == aoa ++ aixa x ++ aa2x2x + aa3x3x O 1 2 3 ddescribesescpibes well the commonCOmmon form.fonn.

The coefficients have been calculated by the method descPibeddescribed in appendix A,A, (§ A 1.4) : solution of the system of 4 equations with 4 unknowns obtained by fbrcingfopcing the curvecupve to pass throughthpough thethe fourfoup fol-fol­ lowing points

x ¡ 0 0.3 0.675 1 x 0.3 I 0.675 I y ! 0.0016 00.000766.000766 0.000350 0

-44 The resuZtpesuU is : 16 x 1100- The is aaoO = 16 x 4 aal= - 40.40.14341434 xx 110-40- l 4 = 48.4348.431010 xx 1010-4- aa22 = 4 = - 24.2824.287676 xx 1010-4- aa33 = -

The gpaphgraph ofof the coppespondingcorresponding curvecupve showsshows thatthat thethe fitfit isis good.good. - 553 3 -

4 Y x 10

16

9

B -

7

6 • .. ~

5 - "

4 ..

3 ~

2

o 0.1 0.2 0.3 04 0.5 00.6.6 0.7 0.8O.B 0.9 1 x --5454 -

Exam2le of volume calculation

The formulaf01'llrU la of the tapertaper curve.eJurve isis

S _H_+H . HZH2 H3 -a + + a -_. + a -3- a1 a2 a33 3 H2 H 2 H2HZ H . tot tot tot tot ·

al 1 a 3 3 ====>> SS = a HZ + a H H ++ a H2HZ +--+ · HH' a0Htot2tot +a1l Htottot a2Z H o tot

The integral of S is

z a Z2 u HH2 H3 a33 H4 g(H) = a H H + a J Htot + a -+-- a0O Htottot alHtot T2 -I-a2Z 3 H 4 tottot

al 2 aa2Z 3 a3a3 4]41 u 3 H - --->> g(H)g(H) -a o x +2x +-x +-x withwith x ==-- 4- 3 x -I- 4 x -tot H [ tot

The volume of the stem between heightsheights HI and HZH2 isis g(H ) - g(H ) ; forfor example, thethe totaltotal volumevolume of the stem is g(H2)Z - g(H1)I ;

V = g(H ) - g (H ) Vtottot =g(Htot)tot g (Hstumpstump) [a al a2 a3 3 4 g(H ) = lao + ad + a; +a; ] 6.00007 x 10-410- x H8H3 tot 0 2 + 3 +4 Htot tot H stump I Let us suppose thatt h at H = 100 Htottot 100

10-2+al10-4+ 10-6+-al10-8 H8 g(Hstump) =la0 2 3 4 tot

6 = 15.8008915.80089 xx 10-10-6x H3 tot

4 . J Thus : VtotV = 5.842065.84Z06 xx 10-10-4x H~otH3 with H in meters and V in m3m . tot tot Htottot Vtottot 7.n

The total volumevolume of the stem of a tree with totaltotal heightheight 1414 m.m. is theretherefore fore 4 3 V = 5.84Z06 x 10-10-4x (14)3(14)3 = 1.60311.6031 m3m Vtottot = 5.84206 what is the "bigwood""bigwood" volumevolume ofof thatthat stemstem ?? TheThe heightheight H7 where isis - 5555- -

located the 7? cm diameter crosscutcrosscut isis suchsuch thatthat

w(0.07)2 2 3 H7 ao + a1x7 + a2x7 + a3x7 = = 14 142

By successive approximations startingstarting fromfrom thethe valuevalue xx == 0.9870.987 read on the curve,curve, we getget x7x == 0.9876080.987608 ;; if we replace x7x byby thisthis 7 if 7 valuevaZue and H byby 1414 in thethe expressionexpression fbrf or g(H)g(H) , we getge t : Htottot 3 g(H7) = 1.646051 .64605 m3.m

6 3 Now, g(H ) = - 15.8008915.80089 xx 1010-6- x (14)3( 14)3 = 0.04336 m3m Now, g(Hstumpstump 3 the "bigwood""bigwood" stemstem volumevolume isis therefóretherefore: : 1.646051.64605 -- 0.043360 . 04336 == 1.60271.6027 m3.m •

243 Crown measurements

A complete description of treetree fbrmform includesincludes measurementsmeasurements on crowncrown :these measurements areare possiblepossible onlyo11ly if the crown isis entireZyentirely visiblevisible..

Height : distancedistance betweenbetween the end of the bole and the tip of the treetree ;; it is measured IJithwith a dendrometer,dendrometer, as the difference betwee11between tlJOtwo measurements.

Measurements onon thethe horizontalhorizontal projectionprojection: :To To describedescYlibe correctlycoppectly thethe pro-pr'o- ...... jectionjection of the croo'"crown on a horizontalhorizontal plane, thethe number ofof radiiradii should be more than oneone as this projection differs fromfrom a circlecircle: : at Teastleast 4, preferably 8,8, radiimdii areare measured,measured, in directionsdirections fbrmingforming equalequal anglesangles :

Example

N-E N

E cC =~ centre of thethe stem o

S-0 - 56 --

Procedure to measure the radius in one direction :

graduatedgroduated tape,tape, fixedfi:lled /inin A Awith Lri.th aa stringstt>ing JJ which"'hieh surroundssUJ'J'OUllds thethe treetree occ~ ______at 1.3 m from the {ll'Ound +-:: ______l' at 1.3 m from the ground A Rod setset inin thethe I radius x=. 2 reference direetiondirection readroead on the """'Passcompass diameter +AB +AB Apparent Ieoncontour tour of the crowne1'OlJn

WaLkWalk forwards and backwards onon AJAJ lineLine andand locateLocate pointpoint B with an instrument. Here are twotwo examplesexampLes of suchsuch instruments :

a/ A mirror-type instrument

Description :: The instrument is balancedbalanced andand remainsremains vertical.verticaL. It contains a planepLane mirrormirror makingmaking anan angleangLe of 45045° with the horizontaLhorizontal and two panes of gLassglass in the middLemiddle of which areare twotwo linesLines : marks 11 andand 2.2. The observer stands soso thatthat thethe twotwo marksmarks comecome toto coincide. It remains toto havehave thethe pointpoint ofof thethe contour of thethe crowncrown inin coincidencecoincidence withwith thesethese twotwo marks ; thethe projectionprojection of this point on the ground is givengiven by the plumbedpLumbed lineLine which is fixed on the instrument.

crin

Marks 11--_ ___1 'ng toto holdhoLd and 22 the instrumentinst1'!#rlllnt

mirror - 5757 -

This instrument isis difficultdifficult toto useuse whenwhen thethe treetree touchestouches a neighbour because thethe mirror allowsallows toto seesee aa smallsmall partpart onlyonly of the crown and it is difficult toto differentiatedifferentiate thethe branchesbranches of a treetree ftomfrom the branches of thethe other. ItsIts useuse cancan be veryvery tiringtiring andand timetime consu-consu­ ming.

b/The PUN-CHUN crown-meter (*)

Description : The . components ofof thisthis instrumentinstrument areare aa holding-holding­ rod, aa stringstring andand aa see-throughsee- through plumb-bob.plumb-bob. Contrary to the mirrormirror-type-type instrument, thethe crown-crown­ memeter ter allowsaZZows toto seesee aa large part of thethe crowncrown andand therefore to locatelocate far more quickly pointpoint A. ItIt is alsoalso very simple and can be constructed very cheaply.

IA v I I I

I I

: I 1..3i- 3 .7cria.i.7cm.l ,I

I L1__.56.777_15cm-, II L-7 .. 3crnI3"",-'

Plumb-bob seen from aboveabove - StM-nq - - -- ] e See-through 41em41cm _ jplumb-hob.1'!:~:'~~ __

heholding-rod lding-1'Od /

(*)(*) WAHEEDWAHEED KHAN KHAN M.A. M.A. (1971) (1971) -- Pun-Chun crown meter -- IndianIndian Forester 96 -- nOn° 6 - pages 332-337 - 58 -

QuantitiesQua...... ntities calculatedcalculat...... ed...... withwith...... thesethese measurements...... n 2 X rir. r.= radius inin direction ii r 1 i i=i=1i (i)( i ) S = n crown n n = 4 or 88 : numbernumb er of radiiradii

..4. measured areaQPea of crown's horizontalhopizontal projectionppojection

2 r r. 4 i= I 1 ( i i ) D S 2 crown 7 crown [JJn c rown

..4, diameterdiametep of the cpowncrown

i ( iii) VB = sS x H c rown 3 crowcrownn Hcrownccrownrown 4... .. bulk volumevolwne height ooff of the crowncpown the crowncpown

Remark 1 : These characteristicschapactepistics of cpowncrown apeare importantimpoptant in growthgpowth studies hutbut they apeare rarelypapely taken intointo accountaccount because of the difficulties ooff fieZdfield measurements.measupements .

V = volume of tY'oodwood containedcontained inIn thethe crowncrown ccrownrown Remark 2 : ------Rema r k 2 The ratiopatio VB = bulk volume of thethe crownc r own crown is a nwnbepnumber Zessless than 1,1, similarsimi lap toto thethe pilingpiling coefficient,coefficient, which can be usedused to estimate V fromfpom VB fbrfop a crown VBcrownc rown standing tree.tpee. It is recommended,pecommended, eacheach timetime aa treetpee isis felledfelled toto measuremeasupe V , to measuremeasupe befbrebefope fellingf elling Vcrowncrown , H andsand S and to derive VB crown crownc rown crown -59_59 -

3 INDIRECT MEASUREMENT OF A STAND VOLUMEVOLUME : THE TARIFFS (= VolumeVolume tables) tables)

31 PRINCIPLE AND DEFINITIONS

To estimate thethe volumevolume of a stand, one canoan measure directlydireotly the volume of eaoheach tree and add all these figures. On large stands this is unacceptable.unaooeptable.

~_!~E~f!A tariff is a table, a formula or a graph, whiohwhich gives an estimate of thethe volumevolume of a tree or of a oolleotioncollection of trees from variables called the entries of the tariff.tariff.

The entries of thethe tarifftariff are measurements ofof thethe treetree (reference - diameter~-totaL-height,diameter, total height,...) ... ) oror of the standstand (basal(basal area perper ha, ha, meanmean height,...)height, . .. ) moremore easilyeasily obtainableobtainable thanthan thethe volumevolume itself·

A tree-tariff£!££:£~E~ff gives the volume of a tree from the entries relating to the tree. A stand-tariffstand-tariff gives the volumevolume of a stand directly fromfrom thethe entriesentries-reLating-to reZating to thethe standstand itself.itself·

A tree tariff cannot estimate the volumevolume of a single tree with a goodgood precision.precision. SuchSuch aa tarifftariff isis mainlymainly usedused toto estimateestimate thethe volume of a collectioncolleotion of treestrees as.theas' the sumsum ofof i)olumes,)olumes ooff inindividualdividual trees. Examples of tree-tariffstree-tariffs 2 3 - one entry (D)(D) (i)(i) V == aa ++ bD + cD2cD ++ dDdD3 2 particular cases : V = a +,+ bD2bD , 2 V = a ++ bDbO ++ cD2co 2 3 V = a ++ bD2bD + cDcD3,... , . ..

(ii)( ii) V = aa DbOb

- two entries (D(0 and H)H) (i)(0 V = aa ++ bH ++ cqD2HdD2H + dD2H

particular cases : V = a +bD2H 2 V = a + bH ++ cDcD2H,... H, ... (ii)V = aDb He

Db HcC D12 - three entries (0,(D, H'H, DH/2)DH/ 2) V = a Db H 0~/2

In these tariffs,tariffs,

V = stem volume (or(or volume of thethe stemstem toto a diameterdiameter limit)limit)

D = reference-diameter

H = total height = diameter at height H/2 ILDH/2 --6060 --

ExamplesExamples ofof stand-tariffss t and-tariffs :

- 2 entries (G(G and and H)H) : V = a + bG + cH + dGH + eeGH2GH2

particularparticular casescases : V = a + bGHbGH V =a= a ++bG+cGH, bG + cGH,......

V =aGb Hc

V = stem-volume/hastem- volume/ha (or(or volume/havolume/ha of stems toto aa diameterdiameter limit)limit) whewherere G = basal area/ha H = mean height or dominant height.height.

- 3 entries :V = a1N1 + + (NN2'N3) a2N2 a3N3

V = over bark fuel wood/ha N = number perper haha of polespoles of total height << 22 m J where N2 = number per ha of polespoles of total height between 22and6 and 6 m N3 = number per ha of poles of total height >> 6 m

This Zastlast modelmodeZ is well adapted to stands in which the measu-measu­ rement of diameter isis more difficultdifficult thatthat thethe measurement of height (multiple stems trees, treestrees ofof badbad form,...).form, ... ).

RemarksRem arks :

(0( i ) . Some tariffs are of an intermediate type between tree-tariffstree- tariffs and standstand-tariffs- tariffs :: they give the volume of a tree as a function ofof variables rreZatingelating to the tree and variables reZatingrelating to the standstand where it is located.located. They are tree-tariffstree- tariffs forfor which thethe coef-coef­ ficients aarere knowwknown functions of variables relating toto thethe stand.stand. 2 EExamplexampl e : V = (a(a ++ bHdom)bHd ) + (c + dHd ) D om (c dHdom)om D2 wherewhere V is thethe volumevolume ooff a treetree of diameterdiameter D.D.

Such a tarifftariff isis calledcalled aa parametrizedparametrized tree-tarifft ree- tariff becausebecause itit can be considered as a family of tree- tariffs V = a,. + b.D2, •, Hd being thethe parameterparameter whichwhich indicatesindicates thethe tarifftariff toto bebe usedused for f or Hdomom the trees of a given stand.stand. To construct suchsuch a tariff,tariff, oneone cancan

fit directly V to the 3 variables Hd • D2 •, D2HdomD2Hd oror starts tart byby Hdom'om om esestabZishingt ablishing a tariff forfor eacheach class of Hd and deduce thethe finalfinal Hdomom . equationequation subsequently.subsequently. OnOn thisthis subject,subject, seesee appendixappendix AA (§(§ AA 1.51 . 5 andand A 1.6).1 . 6) . - 61 _-

(ii) . A two entries-tariffentries- tariff is more precise than a one entry-tariff but it is more difficult to use. Therefore,Therefbre, we sometimes derive a one entry tariff from a two entries tariff%tariff. The fol-fol­ lowing procedure can be indicated :

- suppose a two entries tarifftariff V == ff (D, H) isis available. Measure D andand HH onon approximativelyapproximatively 3030 trees,trees, - then, calculate theirtheir volume byby V == ff (D, H) and construct with these 3030 treestrees aa tarifftariff withwith onlyonly oneone entryentry (D)(D)

or : on thesethese 3030 trees,trees, establishestablish aa lawlaw HH == gg (D)(D) and taketake VV == ff (D,(0, g(O))g(D)) as thethe oneone entryentry tariff.tariff .

(iii)(iii) . A tariff must be considered asas aa tooltooL toto bebe wellwell maintained.maintained. ForFor example,example, it should be updated with additionaladditional data as plantationsplantations growgrob' and extend.extend.

32 CHOICECHOICE OFOF THETHE ENTRIESENTRIES

The entries of a tarifftariff shouldshould bebe :

- few and easy to measure inin order thatthat thethe tarifftariff will\.;d 11 havehave aa widewide range of application andand bebe easyeasy toto use,use, - strongly correlated with thethe volume,volume, - weakly correlatedcorrelated toto each other inin orderorder thatthat thethe interestinterest in kee- pingPing a variable inin thethe model remainsremains when thethe othersothers areare inin it.it.

In genemgeneral, l, no more than twotwo entriesentl'ies areare usedused :: the firstfirst one is always reference diameter, thethe secondsecond oneone beingbeing diameterdiamet e ~ at a fixed height (5(5 m forfor instance) or at a reZativerelative height,height, orOl' bolebole height, or total height,height, or crown diameter ... Among ththeseese variables, bboleole height and diameter at a fixedfixed heightheight are moremOl'e easyeasy tot o measure,mea.gure, diameter atat aa fixedfixed heightheight beingbeing oftenoften moremore usedused thanthan height.he ight . - 62 -

33 PROCEDURE TO ESTIMATEESTlr~ATE THE VOLUME OF A STAND wITHWITH A TARIFF

- Take a sample of n treesintreroin the stand (see(see § 34) and measure directly thethe volumevolume ofof each.each. EstablishEstablish thethe tariff.tariff.

- Measure thethe variables which are the entries of the tariff With"With a treetree on thethe N-n treestrees which werewere notnot usedused toto constructcons truct thethe tariff.tariff. tarifftariff.· - Estimate thethe volume ofof thethe standstand byby (Sum of volumes] Tariff estimate of the volume vV = (sum of VOlumes] + (Tariff esti~a~e of the VOlume] of thethe n treestrees of the remaining trees

- In stands similar to the stand under study, measuremeasure V (volume per ha) and other characteristics (entries(entries ofof With a stand the tariff) more easily measurable. EstablishEstablish thethe tariff.tariff. tariff·tariff. - In the stand under study, measure thethe variablesvariables whichwhich areare the entries of the tarifftariff and apply thethe formula.formula.

In fact,fact, thisthis procedureprocedure hashas oftenoften toto bebe changedchanged becausebecause itit isis noinot possiblepossibZe toto estabZishestabZish aa tarifftariff forfor everyevery stand.stand. InIn practicepractice oneone veryvery often uses a tariff which has not been establish2destablish,d with a samplesampZe coming fromf"!'Om the standstand underunder study.s.tudy.

This is ,iujustifiedstified if the reZation between the volumevoZume and the entries of the tariff is approximatelyapproximateZy the so:unesame in the stand and in the sampZesample used to estabUshestabZish the tariff, so "

. for a tree-tarifftree-tariff ,, the variability of tree forms must be the same in the stand and in the sampZe.sample. One should there-there­ fore ensure that factorsfactors which influenceinfZuence the formf orm of treestrees have the same variabilityvariabiZity in the standstand and in thethe samplesampZe (variabiZity(variability ofof genetics, environmental factors,factors, silviculturalsiZvicuZtural treatments, age and tree-size). The more the domain of vaZi­vaZi- dity of a tariff is intended toto be Zarge,Zarge, the more thethe samplesampZe used to construct itit hashas toto bebe diversified,diversified,

the same occurs for a stand-tariffstand-tariff : ensure that the varia-varia­ bilitybiZity of environmentalenvironmentaZ factors,factors, densitiesdensities ofof trees,trees, silvi-siZvi­ cuZturaZcultural treatments are similarsimiZar in the stand under studystudy and in ·the samplesampZe plotspZots usedused toto constructconstruct thethe tariff.tariff. - 6363 --

34 SAMPLE CHOICE TO CONSTRUCTCONSTRUCT A A TARIFF

341 Tree-tariffTree-tariff

For a singZesingle species homogeneous stand, one can consider that between 50 and 100100 treestrees (one(one entry tariff)tariff) and between 80 and 150150 trees (2 entries tariff) are needed. For a Zarge heterogeneous region a separate tariff is estabZished fbrfor each homogeneous sub- region. Comparison of these tariffs can leadZead toto poolingpooZing some of them :so, exampZesexamples are given in literatureZiterature of tariffs constructed with severalseveraZ thousands ofof trees.

TheThe numbernumber of trees is not the onlyonZy criteriacriteria toto considerconsider ;; the stands where these trees wiZZwill be taken and the samplesampZe trees in these stands havehave toto bebe chosenchosen ;; herehere areare some recommendations :

divide the region for which the tariff isis toto be established into homogeneous compartments (with(with regard toto site condi-condi­ tions, silviculturalsilvicultural status,...)status, ... )

divide the compartments into age-classes and follow the rules :

Number of sample trees Area ~fof the age in the age class of class ofof thethe the compartment compartment ( I ) (I) Total number of samplesample Area of thethe trees region

In an age class of a compartment, taketake thethe samesame number of sample treestrees inin eacheach basalbasal areaarea class (2)

In practice, thesethese principlesprincipZes cancan bebe hardhard toto applyappZy becausebecause the repartition in age cZassescZasses cancan bebe impossibleimpossibZe (planted(pZanted fbrestsforests of badlybadZy knownknown history,history, naturalnaturaZ untreated fbrests,...).forests, ... ). The fa-lowingfoZZowing ruZesrules wiZZ then bebe usedused as substitutes.substitutes.

Number of samplesample treestrees AreaArea ofof thethe compartmentcompartment in the compartment (I ' ) Total number ofof samplesample Area ofof thethe regionregion trees

In a compartment, take thethe samesame number ofof samplesample "treestrees inin (2') each basal areaarea class. - 6464- -

1 Comment on rules £21 and £211

What is needed is the meanmean volume of the trees which havehave a given valuevalue ofof thethe entriesentries (D 1,3 ' H , ... ):). the volume variability (D.1 ' Htottot raisingraosong in generalgeneral with treetree size,size, itit isis moremore usefuluseful toto measuremeasure aa bigbig tree than a small one.one . Rules (2)(2) and (2')(2') aim at avoiding that the majority of treestrees beZongbelong toto a smallsmall numbernumber ofof sizesize classes.classes. ReaZizeRealize that a random sample taken according to the rule "one"one tree takentaken at rrandomand om over nn "" isis not what isis wanted.wanted.

For example, a tariff is wanted forfor highhigh fbrestforest treestrees between diameter 2020 cmcm andand 11 meter.meter. TheThe rangerange ofof basaZbasal areasareas isis divideddivided intointo ten equal classes ; the limitslimits of thethe corresponding diameter classes are 200 - 369 - 482 - 573 - 65165 1 - 721 - 785785 -- 844844 -- 899899 - 951951 - 10001000 (mm)(mm) In each of thesethese classes, a samplesample of about ten trees willwiZZ be taken according toto a samplingsampling design which coverscovers thethe wholewhole area.area.

Remark------onon mixedmi xed forestsforests

The number of species in natural fbrestsforests is oftenoften such that it is impossible toto establishestablish aa tarifftariff for f or everyevery species.species. TariffsTariffs forfor groups of speciesspecies areare thusthus necessary.necessary. How to group species ?? TheThe sim-sim­ plest way isis toto pZotplot datadata (V(V andand D2D2 oror D2H)D2H) andand decidedecide withwith thesethese diagrams.diagrams .

342 Stand-tariff

These tariffs are untiluntil now lessless used thanthan tree-tariffstree- tariffs : the experimental knowZedgeknowledge is not sufficientsufficient toto allowallow reliablereliable recommenda-recommenda­ tions. Consider what followsfo llows as a Zooseloose guidanceguidance :

taktakee at leastleast thirty plots,plo ts, area of a plotplot in ares = HdomHd inin metersmeters with aa minimum om thrthresholdeshold ofof 10 ares. (Hdom(H = 20 m ->- plot ofo f 2020 aresares (0.2(0.2 ha),hal, d om

H ~ 9 m .4-->- plot plot of of 1010 aresares (0.1(0. I ha)...)ha) .•. ) Hdomdom =

Example : A tariff"Cariff is wanted.,anted toto givegive thethe fuelfuel woodwood volumevolume ofof aa savannah,savannah, the mean heigh;ofheight of which is aboutabout 66 meters.meters. TheThe secondsecond modelmodel ofof para-para­ graph 3131 isis chosenchosen

Vv=a11,714.,321,12+,331,13whereN.is a lN + a N + a N3 where N. is thethe number/hanumber/ha ofof treestrees I 2 2 3 1 of height cZassclass ii (observe(observe that coefficient aia. is here interpreted 1 as the mean volumevolume of a tree of height classclass i).i). -65- 65 --

ProcedureProcedure

- take atat randomrandom 3030 plotsplots 3030 mm x 3030 mm

- beforebefore felling,felling , inventoryinventory eacheach plotplo t byby heightheight class.class. IdentiIdentificationf ication ofoJ speciesspec i e s isis notnot compulsory,compulsory,

- fell and stack eacheach plot,plot,

- fit the mmodelodel onon data da ta (V,(V, N1'N , N2'N , N3)N ) ofof plots.plots. 1 2 3

35 DIFFERENT WAYS,JAYS TO CONSTRUCT THE TARIFF WITH(,ITH COLLECTEDCOLLECTED DATADATA

351 Direct method

This method seemsseems thethe mostmost naturalnatural atat firstfi r st sightsight : each entry of the -tarifftariff is divided intointo cclasses.lasses .

ForFoY' a one entryent l'Y tarify,taY'iff, calculatecalculate meanmean volumevolwne inin eacheach class.class .

ForFor aa twotwo entriesentY'ies tariff,taY'if!, setset aa tablet abie byby crossingcros siYl g thethe classes of the two entries,entries, aZZocateallocate the treest rees (tree-tariff)(tree-tariff) orOY' thethe plotsplots (stand-tariff)(Mand- tariff') to the cells,cens, calculatecalculat e meanmean volumevolwne inin eacheach cell.ce/.l.

. . . etc

The series of volwnevolume cOY'respondingcorresponding to thethe combinationscombinations ooff explainingexplaiYling variablesvariables constitutesconstitutes thethe tariff.tal'iff. IfIf necessary,necessar y, aa formulafo rmula can thenthen be adjusted toto thesethese vaZuesvalues (see(see appendixappendix A).A). - 66-

Advantage : CatcuCalculationtation easy.easy.

Drawbacks : . the tariff isis veryvery impimpreciserecise forfbr the combinations of exptainingexplaining variablesvariabtes where scarcescarce datadata areare available.avaitabte. The lawtaw of variation of the volumevotume can be very irregular.irregutar.

. It is impossibteimpossible to estimate the precision with which the tariff estimates thethe volumevotume ooff a stand.stand.

352 Graphical methodsmethods

In practice,practice, they are easyeasy tot o applyappty onlyonty forfor one entry tariffs (see(see appendixappendix A)A) : trees (tree-tarifp(tree- tariff) oror plotsptots (stand-tariff)(stand-tariff) are plottedptotted with volumevotume ini n y-axisy-axis andand thethe explainingexplaining variablevariabte (entry(entry ofof the tariff)tariff) inin x-axis.x-axis . ForFor somesome xx values,values, aa pointpoint isis placedptaced atat thethe mean value of thethe volumesvotumes which correspondcorrespond toto thatthat xx andand aa continuouscontinuous curvecurve is drawn byby handhand acrossacross thesethese pointspoints (see(see appendixappendix A).A).

Advantage : . PracticallyPractically nono calculation.catcutation.

. The tariff is "smooth""s moo th" (superiority(superiority upon thethe direct method).method) .

. The graphicalgraphicat representationrepresentation callscalts attentionattention toto outliers.outtiers. Anomalies likelike negative volumesvotumes areare instinctivelyinstinctively avoided.avoided.

Drawbacks : . The resuUresult depends on plotter's skilfulness and on hishis intuitive knowZedgeknowtedge of the lawlaw variationvariation of volume with explainingexplaining variables.variables.

. Here also,also, there is no means toto estimateestimate withwith what precisionpreeinion thethe tariff estimateseshmates thethe volumevotume ofof a stand.stand. This is thethe main drawback.drawback.

353 The statistical method : regression analysisanalysis

This method is maintymainZy used,used, the inconvenience of calculations having diminished with thethe ,developmentdevetopment ofof computers.conputers.

353.1 Concerning the choice of the regression model

Somes exampexamp lestes of modeZsmodets have been given in paragraph 31 and numerous otherother onesones areare usedused; ; it is impossibteimpossible toto recommendrecommend a unique model (for(for tree-tariffst ree- tariffs for instance, this woutdwould mean that for eveeveryry species in whatever condition, thethe formform factorfactor variesvaries inin thethe samesame wayway with DD and H)H).,. Let us indicate some important points.points. - 67 -

353,11353,11- Simplicity ofof the the model model

Always try to have the most simple model as far as i) . Always try to have the ·most simple model as far as possible, ·that is'1-8 the one with thethe fewestfewest number of coefficients.ooefficients. TheThe more numePousnumerous thethe coefficientsooeffioients the more likely VV isis to vary illogicallyiZZogioaZZy with thethe entries.entries.

Example

, v , ,,

,,

range of data D2

2 The curveourve is'l-S the Zineline V == a + b0bD2 . The model : 2 4 6 8 V -= aa + bD2bO ++ cD4cO ++ dD6dO ++ eD8eO can bebe representedrepresented byby aa curvecurve asas strangestra>lge as thethe ------curvecurve: : the two curves willwiZZ be very close inin thethe usableusable part of the tariff but a very smallsmall extrapolationextrapolation willwill bebe muchmuch moremore dangerous with thethe complicatedcomplicated model.model.

In practice, the following models with two coefficientscoefficients

2 b 2 . vV === a + bD2bD ,V, V = aDaDb ., V = a + bObD2H H , V = a(D2 H)b givegz.ve oftenoften ggoodood resultsr esults and have toto bebe triedtried first.first .

First, plot the data : . 2 for a one entry tarifftariff : V and D2D 2 for a tariff with several entriesentries:: V andand Dn H ; ifif data areare numerous,numerous, plot also V and D202 for each class of H (or(or for each class of an other entry). These diagrams allow aa firstfirst choiceohoioe ofof thethe regressionregression modelmodel and studystudy of relationship between volume variance and enentries,tries, in order to decidedeoide with which weightingweighting functionfunction thethe modeZmodel willwill bebe fitted.fitted.

.,. , /. - 6858- -

353.12353 . 12 Concerning models models where where a afunction function ofof VV andand not VV itself is is estimated estimated

Let us take the followingfoZZowing modeZmodel which is very oftenoften used ("logarithmic tariff")tariff") :

V = a Db

Least squares estiestimationmation of the coefficients consists in seeking a and b which minimize the quantity : n b 2 LY. (V(V.. - - aaD. D1?i)2) i =1 1 1 This calculation is possiblepossibZe but difficult because the model isis notnot a linearZi near combinationcombination of the unknown coefficients.coefficients. There-The re­ fore one takestakes logarithmslogarithms and comescomes downdown toto a linearUnear modelmodel: log V = logl og a + b log l og D and it is this modelmodeZ which isis estimatedestimated byby leastZeast squaressquares; ; so,so, thethe variable predicted is logl og V and not V.V. The a aandnd b coefficients obtained are such thatthat logl og aa ++ bb loglog DD isis an estimateestimate ooff the mean of logarithmslogarithms of the volumes ooff trees ooff diameterd7:ameter DD .•

TheThe quanhtyquantity a Db is therefbretherefore an estimate of the geometric (and not arUhmetic)arithmetic) meanmean ofof the volumesvoZumes of trees of diameter D.D. Now, logarithmicZogarithnric mean is systematicallysystematicaUy belowbelow arithmetical'ithmetic meanmean (example(exampZe : the 4 numbersnumbers 3)3, 4, 77,, 10 havehave a logarithmicZogarithmic mean (3x4x5x10)1/4(3x4 x5xI0)1 / 4 = 5.38 and an arithmetiarithmeticc meanmean- ± (3+4+7+10)( 3+4+7+ 10 ) == 6)6) ; so : 4 a logarithmiclogarithmic tarifftariff systematicallyoys·tematically underestimatesunderestimates thethe volume.volume.

The disadvantage can be partlypar tZy correetedcorrected (see(see appendixappendix AA -- § A 2.3)2. 3) but therethere isis anotheranother disadvantagedisadvantage :: if one estimates with the tariff thethe volumevolume ofof a setset ooff N trees by : N a LX Db i=1 1 the precisionprec"s"on ooff this estimate cannot be known because regressionregression theory gives the precision ooff N X llogog V.V. L 1 i=1 It seems therefore better toto useuse logarithmiclogarithmic tariffstariffs onlyonZy whenwhen itit isis difficdifficultult to fitfit a simplesimpZe model wherewhere VV appearsappears untransformed.untransformed. - 6969 -

353.13 PiecewiPiecewisese fitting ofof aa modelmodel

If it is difficult to fitfit a singlesingle model coveringcove'f'ing the wwholehole range of data (for(fo'f' example, if it iei~ impossible to eliminate bias forfo'f' small trees and/orand/o'f' big trees),trees), a solutionsolution is toto divide thethe ranrangege of data and to fitfit a model inin eacheach part.pa'f't.

Let us indicateindiaate two ways of actingaating :

(i) Method with severalseveral regressionre gression analysisanalysis

In orde'f'order that the models linklink well, overlappingove'f'lapping sub-rangessub-ranges aarere taken,taken, with anan overlapoverlap ofof aboutabout 2020 %.%.

examplexamplee : diamete'f'sdiameters 'f'angerange fromf'f'om 20 to 120120 cm plotting data showsshows thatthat :

. - a modeZmodel V a + b D2 is good forfo'f' treest'f'ees withwith 2020 < D < 40 cmam V=a1l +b1D2I 1.s . - a model V = a + b2D2b D2 2,sis good forfo'f' trees with 40 < D < 80 cmam a22 2 - a model V = a + b D2 is. good forfo'f' treest'f'ees withwith 8080 << DD <120<120 cm.cm. a33 +b3D23 The first model willwiZZ be fittedJoitted on treest'f'ees with 20

v

I

I I

I I

I 1 2 I I 2. D In cm 202 362 402 442 762 802842 1202

the Zineline then representsrep'f'esents thethe tariff.tariff.

This procedu'f'eprocedure offersoffe'f's the advantage ofof comingcoming downdown toto aa seriesseries of fittingsfittings of simplesimple curvescurves butbut itit hashas thethe drawbackd'f'awback of not aZ-al­ Zowinglowing the exact calculus of thethe residualresidual variancevariance :: it is thereforetherefo'f'e impossible toto estimate withwith whichwhich precisionprecision thethe tarifftariff estimatesestimates the volume of a setset of trees. That isis thethe reason'f'eason whywhy thethe secondsecond procedure,procedure, althoughalthough a littlelittle more complicatedcomplicated as regard'f'egard the calcu­calcu- lations, is rather'f'ather recommended'f'ecommended :: - 7070 - (ii) Method with with onlyonly one regression analysisana lys is

Let us take again thethe above example :

y ml• a"a " + buxb"x yY

X0 "

The valuesvalues xoXo == 4040 and xlXI = 80 being ohosenchosen,, the following model with 4 parameters is fittedfitted: = xif x 5-x y = a + bz, + b'z2 + b"z3with z1 if X ,;; Xo0 = xo ifif xx ~ xoXo

z2= 00 if x ,;; x Z2 if x Xo0 = x - xoXo if xo.xOs x_ xS xixl = xxl- - xo if xx ?.~ x I Xo xII

= 0 if xx S z3 x1xI l:= xx -- xIxi if xx ;:?: xxI the following relations give the a' and a"a" coefficientsooeffiaients a' = a + (b-b')(b-b') xoXo alla" = a'at + (b'-b")(h '-hl!) xiXl

------Remarks : . Fitting of only two lineolineD

y

x "0 - 71 -

the value Xoxo is chosen and the following model with 3 pa~amete~sparameters is fitted

y = a ++ bzibZ ++ b'z2b'z2 with z] =x if xx :55 I if Xoxo

= xo ifif xx ~ Xoxo

z2 .--0 if x :5 x if x Xo0 x - xo if x ?. xo

a' is given by : a' == aa + (b-b') xoXo

. To fitfit threeth~ee parabopa~aboZas Zas whichwhich a~eare tangent at the junction-pointsjunction- points

y

L------+------+------~X Xo xI

fix the Xoxo and xlx I values and fitfit thethe followingfollowing modelmodel withwith 55 parameterspa~amete~s

y = a' ++ biz1biz ) ++ c'z2c 'z2 ++ CZ + C"Z with zzJ =x cz33 + c"z44 1 2 = iiff x x x z2 2xx02xxO --xxO0 :5 Xo0 2.2 = xx if xoXo 5~ xx 5~ xlxI 22. = 2xx -x ~ 2xx1-x1l l if x1xI 5 xx 2 . = (x0 -x) if x ~ z3 (xO- x) if x 5 xXo0

= 0 if xoXo 5~ xx

z4= 00 ifif x ~Sx1 xI 2 f ~ = (x-x1)(x-x I) iif x1xI xx the otherothe ~ parameterspa~amete~s are~e givengiven byby : 2 a = a ' + (c-c') x(2)x ; = a = a' + (c-c') o b = b' -- 2(c-c')2(c-c') xoXo ; (c,-c,,)x2 bu=b"= b'b' ++ 2(c'-c")x2(c'-c")x . a"= a' ' ' 1I au= a' - (c -c I - 7272 -

Model to fitfit only two parabolas which areare tangenttangent atat thethe junction-junction­ pointpoint:

y

y'_a'+b'x+c t X2

X0 x

y = aa ++ bz1bZ + CZ + c' z3 withwi.th = x I cz22 + c'z3 Z I

2 . x if z2 =2ifx 5x0~ Xo0

(2x - if x ~ x0 ~~ Xo (2x - xxO) ) if x Xo0

z 0 si x x z)3 )\ = 0 si x S Xo0 2 = (x-x ) if x ~ x0 I = (x-x0)2O if Xo

a' and b' are given byby : 2 aa' = aa -- (c-c')(c-c') x2x ; b' = bb ++ 2(c-c')2(c-c') xo 0o

353.14353 . 14 WeightedI,e; ghted oror unweightedunwe; ghted regressionregress; on ??

The regression must be fittedfitted withwith weights when thethe volumevolume variance depends on the entries. WithoutWithout enteringentering mathematicalmathematical expla-expla­ nations, letlet us say that this isis necessarynecessary inin orderorder toto bebe ableable toto estimate correctly. the precision with whichwhich thethe tarifftariff willwill estimateestimate the volume of a standstand; ; if the calculation of this precision is not judged necessary and ififonly only aa goodgood fitfit isis wantedwanted (that(that isis toto saysay without bias and with small residuals), weightingweighting isis notnot essential.essential. -73-- 73 -

It is not easy to know the relationshipre~ationship between the variancevariance of vo~umevolume and thethe entriesentries ;; this necessitates a large~arge amountamount of data, far greater than is usuallyusua~~y available.avai~ab~e. However, each timetime veryvery large samp~essamples have been collected, thethe variancevariance of volume has been fbundfound to vary, often very much, with the size of the trees. This impliesimp lies recommending thethe systematicsystematic useuse of weighted regression.regression.Wth With few data, it is imposs,:bimpossible le to estimate preciseZyprecise ly the weighting function and the followingfo~lowing hypothesis are generallygeneral~y takentaken

(a(a1 ) )volume volume variancevariance proportionalproportional toto (D2) l one entry tariff (a2) volume variance proportional to (D2)(D2)2 2 (a2) volume variance proportional to I volume variance proportional to (D2H)2 (b l ) volume varIance proportional to (D H) two entries tariff volume variance proportional to (D2H)2(D 2H)2 (b2) volume var1ance proportional to I

The modeZmode~ isis fittedfitted withwith eacheach hhypothesisypothesis successivelysuccessively and the best fitfit is chosen (the oneone which rea~izesreaZizes the best compromisecompromise between the following requirements : no biasbias,, small residual standard deviation, simplicity of model). ThisThis isis done by thethe program describeddescribed in the bibliographic reference n°nO 33 ; itit fits the 44 models : 2 V = a + bD2bD under hypothesis (a1)(a ) andand (a2)(a ) l 2 2 V = a + bD ++ cDcD2 ! V = a + bD2H under hypothesis (b1)(b ) andand (b7)(b ) under l 2 2 ! V = a + b VD2H + c(D2H)c(D H) - 74-74 -

353.15353 .15 How to to appreciate appreciate the the qualityquality ofof aa regressionregression

Never judge of thethe qualityquality ofof aa regressionregression oniyonly by the numericalnwnerical valuevalue of thethe multiple correlation'lorrelation coefficient R (correlation(cor'relation coefficient betweenbetween VV andand VV == VV adjustedadjusted). )

The fit can be bad and RR highhigh : here are three typical situations of such a casecase . V V V A .. t , B " , .. ,, , ' , ,.. • • , • '. • . • ~ ~ ...... , ,. , " .. • • • • .. ' • • .• , " • V V VV (I)( I ) (2) (3)

(I) biased model (2) non homogeneous samplesample (3) presence of "abnormal""abnormal" treestrees (without(without treestrees AA andand B,B, RR wouldwould bebe low)low)

It can also happenhappen thatthat thethe fitfit isis good and R lowlow; ; it can be so forf or instanceinstance when someSOme treest; rees area'l"e "abnormal"."abnormal".

V ' A / ForPor trees other than A and B, / the fit is goodgood.. Withoutr-lithout A and B, rP. would bebe high.high. / K B .."' .' ~

4-

--~·-vv

NumerousNwnerou$ quantities quantities other than other than R can be consideredconsidered :the most use are the residual standardstandard deviation

E(r.-17)2 - I where r. V. -V.- V. , r =1= - EL r. , n - c 1 1 n 1 s c = number of coefficients inin thethe modelmodel n = number ofof datadata (trees)(trees) and the residual coefficient of uo.Y'iatvariationion :

s - I1 Wherewhere V == - rv. = meanmean ofof measuredmeasured volumesvolumes n 1. V (for aa modelmodel withwith aa constant ter'mterm ",hewherere yV is untransformed,untransformed, thethe nwneratornumerator of s takes the simpsimplele form :

- \I 2-2 - -2 E y.:%7 EV -EV because then : EV = EV and 1: vv=V V= EV ) • --7575 --

The "aggregate deviation"

rvEV - rvEV rvEV

oror the "average deviationdeviation"ll

rlvEIV - vi rE V

are sometimes used also. What has been said about R can be repeatedpepeated fbrfop each of them : they do not aZZowallow one to appreciateapppeciate completely the quality ofof thethe fit.fit.

To judge effectively of the quality of a regressionpegpession :

(i) plot on the same graphgpaph thethe data and thethe fittedfitted curve.cUPve. ForFop a one entpyentry tariff,tariff, it wiilwill bebe thethe graphgpaph of V against D0 ; fbrfop a two entpiesentries tariffs,tapiffs, it is recommended,pecommended, whatevepwhatever the model is, to plot V against D2H.02H.

(ii) calculate and plotpZot the residualspesiduals V - V ; threethpee typestypes of plot-plot­ ting apeare possible :

V - V V - V V --V- V

1------__ 0 ol--~~--~o 01------o V V · or H or an other entry

( J ) (2) (3)

When therethepe is no biasbia s , diagpamdiagram (1)(1) is wewell II scatteredsoatteped _ aroundapound an upwardupwapd lineline (the(the sZopeslope of the regressionpegpession lineline ofof VV -- VV inin function[unotion of V is IJ - R2R2 )) ; on diagramdiagpam (2),(2) , data areape scatteredsoatteped aroundapound v-axis.V-axis . When~,en therethepe isis aa bias,bias, diagpamsdiagrams (3)(3) provideppovide anan aidaid toto decidedecide howhow toto correctooppeot thethe modelmodel (see(see appendixappendix A,A, § AA 2.6). 2. 6) . -76 -

353.2Example

The referencepefepenae diametep,diameter, the total height and the big wood stem volume havehave beenbeen measuredmeasured onon 5050 treestpees ::

0 n° Tree n :D0 Hm Vm1V3 Tree D0 HHm V3 _111-_ - m m - Vm3

1I 0.095 112.902.90 0.037 26 0.150 17.00 0.135 2 00.105. 105 12.00 00.040. 040 27 0.150 16.90 0.108 3 O.0.111 III 114.004.00 0.052 28 0.153 17.5017 . 50 0.132 4 O.0.111 III 15.015.000 0.060 29 0.156 17.00 0.138 5 O.0.115 115 1616.50.50 00.067. 067 30 0.156 17.0017.00 0.148 6 0.118 15.9015 . 90 0.075 31 0.156 16.30 0.123 7 0.124 15.4015.40 0.070 32 0.162 17.1617. 16 0.135 8 0.124 15.3015.30 0.080 33 0.162 17.50 0.165 9 0.127 16.50 0.090 34 00.162. 162 17.50 0.1500 . 150 10 0.127 13.00 0.075 35 0.169 18.5018.50 0.1800 .1 80 11II 0.127 16.05 0.093 36 0.169 17.30 0.160 12 0.131 14.50 0.074 37 0.169 16.30 0.140 1313 0.134 17.40 0.102 38 0.169 16.50 0.165 14 0,1340.134 17.10 0.107 39 : 0.172 15.70 0.148 15 0.137 18.00 0.100 40 0.175 18.50 0.156 16 00.134.1 34 15.00 0.080 41 0.175 17.70 0.184 17 0.137 14.50 0.090 42 0.175 17.30 0.191 18 0.131 17.717.700 0.090 43 0.175 17.40 0.1620 . 162 1919 0.140 1515.50. 50 0.103 44 0.188 18.50 0.225 20 0.143 1818.20.20 0.127 45 0.188 18.50 0.235 21 0.143 I18.50S.50 0.120 46 0.191 17.50 0.197 22 0.1430. 143 16.50 0.1170. 11 7 47 0.194 18.50 0.256 23 0.146 15.80 O.0.10n 101) 48 00.197. 197 16.50 0.230 24 0.146 116.006.00 00.105.1 05 49 0.197 16.60 0.210 25 0.150 17.00 00.122. 122 50 0.204 18.6018.60 0.254 :

A one entryentpy (0)(D) taPifftariff and aand two entriesa two entpies (0(D and H)H) tapifftariff 2 apeare wanted. One startsstapts by plottingplotting thethe datadata inin functionfunation of V V and D0 and in functionfunation of VV and 02HD2H : the numbepsnumbers plotted areape thethe treetpee numbepsnumbers.. Av

49Z'O0.254 j 47 50

0.241LI7 4594

0.227L ...vt,

0.214L 496V 0.201LOZ'O /~ 0.188110 88 42Zt7 4f 15 0.174O

0.191.01611 .A. +, ot,40 " 0.148j841.0 ,9 " ,7 19 1."0134 VE ~ 0 . 1 " " to 0.1211210 11K ,1 "U 0.1088001 ,. /, 17 " ".. 0.09.)0 1760 11 9 ...... 17 0.08,]180O " *0 890 '/oil 5 4 0'0. 0551.590 ~ 1.170'0 ~

0.028r8Z0'0 0.009116000 0.0117Ll1.00 0.0143Et71.00 0.01690'0 69l 0.0195L0'0 96 0.02210 ZO' LZ 0.0247LbL0'0 0ZOO.0273 £L 0.0298ZO'0 86 0.0324E0.0 VZ 09E000.0350 0.03769LE0'0 0o .0402Ovo 0çj v

0.254 4741 ·50

0.241 45 4& 0.227 44

0.214 49 0.201 ..46 0.188 42-

0.174

0.161 43 .... (J) 404()

0.148 ~o30

0.134 2.6 ~

0.1210 . 121

0.108

0.094 "9 1/1 0.081 ,.

0 . 068

0.055

0.0410.041.

0.0281 i f ;1 '3 1 4 5 5 f If ? • 0.02~0.118 0:1700.170 0.222 0./75O. 75 0.1327O. 27 O.0:i74 79 0.4320 . 4~ 2 · O.0.484 84 0.36O. 36 0.589O. 89 0.641 0.693 0.746 D'HD-H --7979 -

The graphs show thathatt the following simple models are valid

V == a ++ bD2bD2 and V = aa ++ bD2H bD2H

Fitting these models bb~ regression requires knowingknowing how the variance of VV varies with nD and with D2H : the variance of VV is calculated for some groups of trees chosen suchsuch thatthat nD22 (or(or D211) n2H) is approximately constantconstant inin eacheach group.group. 2 Study of the relationship between var VV and 0D2

0 D2= mean 2 Group n ° Trees n°n log nP Variance of

A 3 - 44 -- 55 0.012652 - 4.3704.370 0.0000563 - 9.7849.784

99- - 1010 - II11 -- B 0.016540 - 4. 102 0.0000833 - 9.393 1122 - 18 - - 4.102 - 9.393 1133 - -1414 - 15-15 - C 0.018217 - 4.005 0.0001162 - 9.060 16 - 17 - - 4.005 - 9.060 20 -- 21 - -22-22- nD 0.020886 - 3.869 0.0001230.000 123 - 9.006 23 -- 2424 - - 3.869 - 9.006

E 25 -- 26 -- 27-27 - 0.022382 -- 3.800J .800 0.00018230.000182J - 8.6108 . 61 0

F 29 - 3030 -- 3131 -;- 0.0243270.024327 - 3.7163.716 0.00015800.000 1580 - 8.7518.75 1 I-- I G 32 -- 3333 -- 34-34 - 0.026354 -- 3.6363.6J6 0.000225 - 8.3998.399

J'i - J6 - J7 - H 35 -36 - 37' 0.028461 - 3.559 0.0002729 - 8.206 38 - - 3.559 0.0002729 - 8.206

I 47 -- 48 -- 49-49- 0.0385J20.038532 -- 3.2573.257 0.000532 - 7.5397.539

J 44 -- 45 -- 46-46 - 0.OJ56720.03567288 -- 3.3333.333 0.0003880 . 000388 - 7.8557.855

40 --414 1 - 42-42- K 0.030650 - 3.485 0.00023490.0002849 - 8.1638.163 43 - - 3.485 _ 80_80 _

Study of thethe relationshiprelationsh ip betweenbetween var VV and D2H02H

~ -- 0 0 GroupGr oup n°n TreesTrees n°n ! D2H=mean02H~mean loglog D2H 02H VarianceVariance ofof log var VV of 02D HH V = var V

~- - A 5 - 6 - 1010 - 0.215987 - 1.5331.533 0.0000213 - 10.75510.755

B 7 - 8 - ·1212 - 0.240027 - 1.427I .427 0.0000253 - 10.58310.583

9 -11- II -- 16-16 - C 0.266856 - 1.321I .32 1 0.00003230.0000323 - 10.34210. 342 1717 - 1133 -1414 - 18-18 - D0 0.305533 - 1.1861. 186 0.0000537 - 9.833 19 - 1155 -2222 - 23-23 E 0.339384 - 1.081 0.00006430 .0000643 - 9.651 24 - 20 -2121 - 2525- F 0.378446 - 0.972 0.00009830.0000983 - 9.2279 .227 26 - 2727 - 28 --2929 - 30-30 G - 0.408047 - 0.8960.896 0.0001100.000110 - 9.113 31 - - 32 --3333 -- 3434 - H 0.461998 - 0.772 0.0000.00015615 6 - 8.769 3737- - 3838 -- 3939 -- 35 --4141 - 4242- I1 0.533143 - 0.629 0.00015290 .000 15 29 - 8.786 43 - 44 --4545 - 4646-- J 0.646497 - 0.4360 . 436 0.0002443 - 8.317 48 - 4949

2 2 Diagrams (logClog varvar V,V, logl og D2)0 ) and Clog(log var V, logl og D2H)D H) are given inin next page.page. - 8181 - log varvar VV

-77 line ofof slopeslope 22

% •

-88

•• 'F

-99 c· 'D •• ,.. -10 2 -5 -4 -3 log D20

log varvar VV

- 8 line ofof slopeslope 22

-9

~. -10

-II-11 ao. 02H _~2;----J.----r------:r--""'10glog D2H - 8282 --

These graphsgraphs show that one can asswneassume that 2" log varvar VV = a + 2 log DD2 -2- log varvar VV = a' + 2 log DD2H H

Thas it to say

2 the variancevariance ofof VV is proportional toto (D2) 2 2 the variance of V is proportionalproportional toto (D2H)(D H) .

FortheoneentrYtanW,theweightoftreeiisthusw.For the one entry tariff, the weight of tree i is thus D. for the 2 entries tarify.,it 41) 1 for the 2 entries tariff, it is w. 1 2 D.D~ H.H~ 1 1

RESULTS 2 One entry entry tarif tarif : V = aa ++ bD2b0

a and b are the numbers which minimize thethe quantity :

n 9 2 2 ) S = w. (V. - a - bD ) where w· S= Xc..T.(V.-a-W)wheresq.=j-4-L i1 1i 1i i1 i=1i=) D.D~ V. 1 n i1 1 2 If = and x.x . = , S can be written :: S = LX (y.-ax.-b)2(y.-ax.-b) yiYi -T 1 -2 ' .) i.1 13. D~ D~ ~= , 1 1

The problem is thus to fit the modelmodeZ y = axax ++ bb byby thethe usual leastleast squares method :: oneone getsgets :

(Ex)(Ey)(LX)(Ll) LXYExy - n a = = - 0.02464 2 (LX)(Ex)2 2 Ex2LX - n

_ - b = y -- axax = 6.59166 . 59 16 -83- 83 -

(Ex)(Ey)' VR (residual (residual .variance)variance) =- -­I v2 (EY)2 - - n-2n-2 E- n - aHy - 0.27501

VR var a - ---'-"'--""2 = 0.0000169 2- 00000169 2 (Ex) LXE2 -~ n

VR -2 var bb = -71- ++ x var a 0.04483 n var a =0.04483

covcoy (a,b) == -- Xx varvar aa = - 0.00081530.0008153

The volumevo~ume of a stand of NN trees on each of which DD hashas beenbeen measu:t'edmeasured willwi~~ be estimated byby : N 2 V = N a + b ~X D~D. TOT 1 VTOT=Na+b i=1i=1

The confidence intervalinterva~ of is, at 0.95 level~evel of VT0TVTaT ,' is~ V ± 2 Ivar/liar V Vtottot VTOTTaT 2 2 with : var V N2 var a + b + 2Na cQv(a,b) VTOTTaT = N var a + aa2 var b + 2Na cov(a,b) ++ 8(VR) 8(VR) NN, N where : a = D~D4: and 86 = X D~ ~ 1 i=1I 1 i=1 1 i=1

Two entries entries tarif: tarif: V = aa ++ bD2H bD2H

a and b are the numbers which minimize the quantity n 2 1 sS = 1 w. (v.(V. - aa -- bD.bD~ H.)2H.)2 where w.W. - I 1i 1 1i 1 1 i1 2 2 i=1 (D.(D~ H.)H.)2 1i 1 . V. 1 1I If Yi = and x. == Ss can be written : If Y i = D~2 H. 1i 2 D. H. D.D~ H.H. 1i 1i i1 1i

n \ ( _ b)2 sS =1(y.-ax.-b)2L Yi - aXi i=1I 1

Thus, the modelmode~ yY == ax + bb is adjusted by the usualusua~ Zeast~east squares method ; one gets : -84- 84 -

a =~ -- 0.0036090.003609 b = 0.33677

var aa =~ 0.000004922 var bb = 0.00005569 coyCOy (a,b)=(a,b)~ -- 0.0000149U.UUUOI49 VR = 0.000533

The volume of a stand of N trees on each of which D and H have been measured isis estimatedestimated byby N VVTOT ~= Na ++ b I D~D.2 HH.. TOT 1 1 i~1i=1

The confidence interval of VT0T V is, at 0.95 levellevel TOT is, V ± 2 /varIvar V VTOTTOT ± VTOTTOT 2 2 with : var VTOTV = NN2 var a + a2a var b + 2Na COy (a,b) + S(VR) TOT ~ var a + var b + 2Na coy (a,b) + 8(VR)

N N 4 T whewherere : a ~= I D.2D~ H. and B8 = X D7D~ Er:H~ . 1 1 I 1 1 i=1i~1 i~1i=1

The 3 diagransof next page concern thethe 22 entriesentries tarifftariff; ; ttheyhey show that thethe tariff_istariff~is correctcorrect :: the residuals do not tendtend toto vary systematically with V , D and H. - 8585 - v-v

0.=

0.010 , , , , o~----~~~--~~----~~.~~----~------~----~~ 0.05 0.10. 0.15, 0.20 V(mJ) -0.010 • , ,

-0.020

v-vV-V

0.020

0.010

0~~0+.m~'~~~~--~~+-~0~.=5,~-r---+---T--~---,~02~O~,------~D~(m-)~ 0.10 0.15. .020. D (m)

--0.0100·010

·-0.020·0.020

v-v

0 .020

0 .010

O~-T--~~~-T~~~~-+_,~, +-__~-+ ______~~ 11 15 : 20 H(m)

-0.010

-0.0 - 86-

3636 CONCERNING UNDERBARK VOLUtlESVOLUMES

361361 Bark thickness and diameter

The bark factor k is the quotient of overbark diameter over underbark diameterdiameter :

D D = diameter over ovov 2B 1 ov kk= = 1 + -- = bark D D 2B bark Dunun Dunun I- D ov D diameter under Dunun = bark where BB is singlesingte bark thickness.thickness.

Bark thickness tends to decrease from the bottom to the tip of the tree but it is not possiblepossibte to give a generalgenerat formulaformuta forfor' thisth-is trendtr'end whichlJhich hashas toto bebe studiedstudied inin eacheach case.case.

k is sometimes constant from the bottom toto the tip of the tree : barkbark thickness is then proportionalpr'oportionat toto DD (and conse­conse- ov quently to Dun)) but it can also happen tha~thatfromfrom bottom to tip, quentty to Dun but it can atso happen bottom to tip, k decreases first,first, thenthen remainsremains constantconstant and thenthen increases.increases.

28913 In general,gene rat, 0-- varies between 6 and 10 % ; k varies ov therefbretherefore betweenbetween 1.061.06 andand 1.12.1.12.

In most cases (optical(opticat measurements onon standingstanding trees)trees) ontyonly one bark thickness (at(at referencer'efer'ence height)height) isis availableavaitabte perper tree.tree. To see if bark thickness at thethe referencer'eference levellevel variesvaries systematically with the reference diameter,diameter, startstar't byby plottingplotting datadata :

B a= single bark thickness,thickness, at referencereference diameterdiameter levellevel

S S'

reference L-______~------... D • reference about diameter 20cm20c:m

The cloudcloud ofof pointspoints often hashas anan S shapeshape :: barkbark thicknessthickness increases curvilinearilycurvitinearity fbrfor smallsmaH diametersdiameters andcznd thenthen moremore slowlystowty ; sometimes, bark thicknessthickness is practicallypracticalty constant forfor diametersdiameter's bigger -87- 87. --

than some value (( S'S' curve),cupve), but thethe relationshippelationship isis seldomseldom strong.stpong. ThisThis relationshippelationship is describeddescPibed byby formulasfOPmUlas suchsuch asas :

( ) ) B= a +a D B = singlesingle bark thicknessthickness at reference diameter D (2) B=B lelevelvel a + a1 D aO + a ) D D = reference diameterdiame ter aaI) (3) B = a D (a] < 1)) ) a0O (a)

.' RealizeRealize that it is a priorippioPi incorrectincoppect toto useuse suchsuch aa formulafOPmUla to estimateestimate inin a treetpee barkbapk thicknessthickness atat differentdiffepent heights,heights, knowing the diameterdiametep at thesethese heightsheights becausebecause thisthis wouldwould givegive the vapiablesvariables ofof thethe formulafopmula aa meaningmeaning differentdiffepent toto theirtheip definition.definition.

362 Overbark volumevo lume -- Underbark volumevolume

362.1 ~~~~_g~Q!i~~!Bark guotient is the quotient of the bapkbark volume ovepover the overbarkovepbapk volume.volume . v ovOV - ) V V V Vbb un un P = = ) - V V V ov ov ov V Vunun

(V =V= V + ) +ovov ,unun Vb -1- -I, Volume Volume volume of overbarkove pbapk undepbapkunderbark bapkbark

ForFop each volume therethepe is a correspondingcoppesponding bark quotient.quotient. The barkbapk quotient relatedpelated toto thethe bolebole volumevolume isis thethe mostmost used.used.

In a given tree,tpee, thethe relationshippelationship betweenbetween PP andand kk dependsdepends on the volume which is consideredconsideped and on the relationship between k and heightheight of measurement.measupement.

To get an idea on the formfoPm of this relationship,pelationship, it seemsseems reasonablepeasonable toto supposesuppose thatthat thethe formfoPm factorfactop isis thethe samesame underbarkundepbapk and overbark,ovepbapk, whichwhich givesgives V V ov Vunun (g = basal area) g xU x H ov gun Thus

_ gun = 2B 2B ) 2B (4) P = 1 - = 1) - 1 - [2- 29 D Bovov k-k ov [2 - i:vov 1

12)132B (Example : = 8 %% .. k == 1.087) .087 4-.. P = 15.4) 5.4 %)%) D ovOV - 88 -

362.2 Conversion of volume overbark into volume underbark

If oveoverbarkrbark and underbark volumes have been measured on a set of trees, a firstfirst method is toto calculatecalculate twotwo tariffstariffs :

V = ff (D(D ) or f (D, H ) ov ov (Dov '

VunV = gg (D ) or g (D0v,(D • HH )) un ov ov ensuring that VunV isis alwaysalways less than VVov because the regressionregpession un ov lines can crossCT'OSS eacheach other.other.

The following method is more often usedused - calculate the overbark tariff :

V = ff (Dov)(D ) or V = f (Dov,(D • H)H) ov ov ov ov - fitfit a formulaformula givinggiving barkbark quotientquotient fromfrom thethe tariff'stariff's entries.entries. In general,general, bark quotient isis reZatedrelated toto referencereference diameterdiameter only :

P The following models

a1 a2 P + P = a0 D2 ov ov

(consequence of (1)(1) andand (4)(4) )

-a- a D I ov pP == a0a O ee Dov are often adequate.adequate .

- taketake thethe followingfoZZow,:ng expressionexpression fbrfor underunder barkbark tarifftariff :

V = ((I 1 - P)P) VV un ov - 89-89-

Examples :

2 V = ao ++ a1D8 D ov = aO. 1 a _ 8a33 8a44 V. =_ (I __. a2...__ .__ ] 8a3 8 un (a0 + a1D2) 3 a44 2 D D2D21 P == 8 + -+-+ a22 2 (DD is D DD2 written

2 for DD ) V = 8 + 8 D ov ov =a00 + a1D1 -a3D ==> VunV = 1 -8 D un - a2e (a0 +a1D2) -a3D3 P = 8 e a22 --9090 --

4 ESTIMATION OFOF USASLEUSABLE VOLUMESVOLUf-1ES

The measurements which can be done in thethe fieZdfield on standing or felled treest rees provide mainly gross volumes ;; supple-supple­ mentary data which can be collectedcollected (borings(borings toto detectdetect holZows,hollows, observation of apparent defects...)de fects ... ) cancan onlyonly givegive indicationsindications onon useusefulful volumes.volumes . The knowledge ofof usablevolumesusable volumes requiresrequires toto carrycarry on observationsobservations up toto thethe placesp laces where thethe woodwood isis manufactured.manufactured.

ThisThis is an example of procedure followedfollowed inin tropicaltropical high forest (ref.(ref. 1?)17) whichwhich allows, providedprovided itit isis carriedcarried onon completelycompletely,, to convert bole gross volumesvolumes intointo usedused volumes.volumes .

41 AN EXAMPLE OF t1ETHOOMETHOD APPLIED ININ TROPICAL HIGH FORESTFOREST

411 GatGatheringhering ofof datadata

'lheThe procedure isis inin twotwo phases.phas es .

411.1 In the inventoried region,region, oneone makesmakes qualitativequalitative observationsob servations on standingstanding treest,'ees inin orderorder toto splitsplit thethe bolebole grossgross volume in different fractions correspocorrespondingnding eacheach toto aa qualityquality ooff standing wood.wood.

The bole of eacheach tree is virtually divideddivided intointo 3J parts ooff equal length and each part receives threethree notesnotes (these(these notesnotes rangerange fromfrom 11 to 5 - seesee next page) ,,)hichwhich de,qcribdePcribee l'espectiveZyr.rwectively : thethe form of the bolebole,, it:.;itu health and the appearanceappeaY'(Ulc:e ofof " the l.,;ood ood.. The ththreeree nnotesotes given to each third ooff bole are pooledpooled inin oneone notenote ranrangingging fromfroom 1 to 5 accoaccordingrding to the fofollowingz/'ou,ing corcorrespondencerespondence gridgrid

, ~~----~--~~~----r------n~~----~--~~~---r------NotesNotes givengiven tot o the Notes given tot o thethe -'~ Pooled Pooled tthirdhird of bole third of bolebo l e ----.6 note note e.).-8",'8 '"u /j0 given C.)u ao given c.--2 .(.? ::: :3 zc: ~:-.., to thethe 03 to thethe , .s71 f- 0.1 1, zi ~ '"~ '"~ t'\l .!:'-',...." ...... third of '-'. J-J .- e<,or-..C.::" ...... ,- third of E ~ 1~ s::-r= wa.>.w:;! ",.-J .--,x 0.J~ u4-) 3:''' 5-~ "--~ e - -- o.EZo. .~ .., bole s.. ---- .,._-, ca,a. ,...~ bole ~ ca '"o~ a. 4-"" o 0 a.u..:a.~ ==""'===_1- =;r;====..._ =""=

1 I I I I1 3 I 2 I I 2 II 3 2 I 2 1 1I 2 3 I

2 I 2 I1 3 3 3 1 I 2 1 3 3 I ? 2 2 1I 2 3 2

I 2 2 2 3 3 2 2 2 3 3 2

I1 1I 3 3 3 3

I 2 3 2 a 44 in the thirdthird 7 2 I 3 column 2 2 31 Every set with one or

3 I1 1I several 4 (except(except oneone 4 3 I1 2 4 in thethe thirdthird

3 2 I1 column)

3 I 3 Every set with one or 3 2 2 several 55 5 3 2 3 VALUE OF THE QUALIFICATIONS

I Qua~ifi-Qualifi- -> II --o. I L2 catlonscations 3 4 5

1 1 sligslightht bend I1 pronounced bendbend 1 pronounced bend fluted section +1 extended buttress ribbed section Conical shapeshape Conical shapeshape ++ (2 ribs or more) or + I1 groove 2 m OvaOval 1 sectionsec t ion or + 2 or 3 flat sur-sur­ faces

Oval section 2 or 3 flat sur- 1 2 or 3 flat sur­ I Straight extended buttressbuttress elbow faces +1 groove 22 m or

I 1 flat surface overover +2 or 33 flatflat surfacessurfaces I 2 slightslight bendsbends "bayonet" the who!~_!~~gthwhole length and FORM II slightslight groove I extended buttress I groovegr oove 22 mm I deep groove of +2 or 3 flat surfaces (F) +2 or 3 f!~~_~urfaces 2 mmeterse ters Cylindrical 2 or 3 flatflat surfacessurfaces 2 slightslight groovesgrooves 2 bends (pronounced)(pronounced) above thethe buttresses

I pronounced bend +1 slightslig h£_~~!!£ bend

I rib ""

Sound (neither (neither I large off-shoot 2 largelarge off-shootsoff-shoots More thanthan 22 largelarge Visible decaydecay atat HEALTH off-shoots nor nor off-shootsoff -shoe~~ __ _ foot covered knots)knots) 1I black tracetrace I brokenbroken branch 1 decayed knot (H)

1 woodpecker hole Hollow soundingsounding stem Irregular graingrain (very(very Slight twisting15°>15 0 Straight grain slight ribs inin allall + slight swelling ddirections)irections) 2 or 3 swellingsswellings oror Swelled surface and no defect Swelled surface 4PPEARANCEPPEARANCE large healedh~~l~~_knots knots I healed wound mark (no thorns,thorns. OF THETHE splinters, "grains II slight swelling various wound marks MoreHore thanthan 33 swellingsswellings WOOD d'orge"d'orge",, wound in large healed knots f"OOD Visible thornsthorns (W) marks, excres-excres­ Slight localisedl ocalised twis-twis- cences,etccences,etc...) ... ) tingLing Raised barkbark inin many places - 9292 -

On anan aveaverage,rage, the bolebole volume is distributed amongst the height thirds according to thethe fractionsfractions : 44% forfbr the bottom third,third, 33% forfbr the middlemiddZe third, 23%23% fbrfor thethe upperupper third.third. WithWith thesethese figures,figures, the bolebole grossgross volume can bebe distributed amongst the 5 classescZasses of apparent quality.

411.2411.2 Being determined from external defects visible on standing trees, the 5 classescZasses which result from the first phase give only an approximate evaluationevaZuation of wood quality.quality. A secondsecond phasephase

takes place on logging companies near the inventoried regionregion :: the qualitative observations are made on treestrees befbrebefore fellingfelling; ; after felling, the evolution ofof eet-611ltch third third of logZog is fbllowedfollowed: :

a/ Measurement of the parts eventually leftleft in fbrestforest before tractor Zogginglogging :: stump-waste, wastewaste duedue toto pollarding,pollarding, waste in the central part of the bole to eliminate a big defect, if any.

end ofof buttressbuttress

st d 11st third I 2d third 33d third I / base ofof I the crowncrol'm / I Naittaa/ A\ I / ~ waste

b/ measurement of waste leftleft onon Zoadingloading area afterafter logging.logging . c/ ifpossible,if possible, measurementmeasurement ofof partsparts leftleft onon thethe sawmillsawmill park or befbrebefore thethe embarkation inin casecase ofof export.export.

It is easy toto realize thatthat thesethese operationsoperations areare difficultdifficult and require fullfull timetime employees duringduring severalseveral months.months. OperationOperation c/ is often impossible.impossible. - 9393 _.

412 AnalysisAna lys; s ofdf datadata

412.1 If only operations a/ and b/ have been done, it is possible toto estimate,estimate, forfor eacheach apparentapparent qualityquality class,theclass, the proportionproportion of volume which comes out of fbrest.forest.

Example~~~~E!~ : SIBITI ZANAGA region (République(Republique PopulairePopulaire du Congo).Congo ). OkoumeOkoum6 (Aucoumea(Aucoumea klaineana) exploited forf or veneer inin PointePo i nte

Noire:Noire : 76 observed trees.trees.

Apparent qualityquality 1 2 3 4 5 Total

Distribution ofof bolebole 42.9% 35.1% 12.12.9%9% 1.71.7%% 7.4% 100% volume of thethe 76 trees

Proportion ofof volume 73.6% 56.7% 10.6%10.6% 9 % 25.8% comingcoming outout of forestforest

Volume coming out of forest 31.6% 20 % 1.4%1. 4% 0.2% 1.91.9%% 55% StaStandingnding bole gross vovolume,lume, all qualitiesqualities togethertoge ther

. fr. 0.4290.429x0.736xO.736

55% is the global merchantable coefficient : it represrepresentsents the ratio : vovolumelume comingcoming outout ofof thethe forestforest standingstanding volumevolume

Suppose that qualitative observations on N trees during the inventoryinventory of a fbrestforest givegive the fbllowingfollowing figuresfigures :

ApparentApparent qualityquality 1 2 3 4 5 Total I Distribution of bole 50 %% 30 %7, 8 %% 2 %% 10 %% 1100%00% volumevolume of thethe N treestrees I If this forest is to be exploited in the same purpose and in the same way,way, its merchantable coefficient can be estimated by (0.50 xx 0.736)0. 736) ++ ..• + (0.10(0.10 xx 0.258)0 . 258) == 57.457.4 Z.%. 412.2412 . 2 IIff operation c/c/ is done, it is possible to share this volume between different categoriescategories (log-export,(log- export, locallocal sawing,..).sawing, .. ). -94- 94 --

42 EESTIMATIONSTIMATION OF THE USABLEUSABLE VOLUMEVOLU r·1 E BBYY A TARIFFTARIFF

The direct procedureprocedwoe isis :: fellfeU a samplesample of trees, measuremeaswoe the useful volume VV and construct a tariff V = ff (D) or VVu = ff (D0-1).(D,H). u u u

This can rarely be done because the necessary samplesample size is far Zargerlarger than for a tarifftariff giving thethe grossgross volumevolume becausebecause the variability of internal defects adds toto thethe variabilityvcriability of tree forma.forms.

An indirect procedureprocedure is thereforetherefore foZlowedfollowed inin practicepractice :

(a) withoJith a samplesample of trees,trees, construct a tarifftariff givinggiv'ing thethe gross volumevolume V.v.

(b) fellfell a set of trees (belonging if possible toto thethe sample),sample), measure thethe grossgross volumevolume VV and thethe usableusable volume V and V u . Vu uu calculate fbr each tree the ratio : k = -- calculate for each tree the ratio : IIV

(c)(e) withoJith these ratios,l>atioe, fitfit a model givinggW1-ng kk inin functionfunction ofof tariff entries ; in general a function of D alone is taken as model. 2 Z~=_1= examEle~"'~~'" : k = a + bD ++ cD2cD ; this parabola starts fromf!'Om a pointpo"int oi4erelJhere thethe refer-encereference ddiameter-iarfletcr eqequalsuals thethe minimumminimwn diameterdiameter of a merchantable log,log, DD == DminDmin

b2- 4ac - b b D ,. = min 2c

b ForPor D = - 2c ' the parabola reachreacheses its maximum. The desdes-- ' cending part of the curve is not usedused; ; it isi s replaced by the horizontal Zine.iine.

k

2 4ac - bb2 ------?T~----- 4c4c , , \ \

b D = reference diameterdiameter Dmln-:' - 2c - 9595 -

D -D Dmmmin . b- D 1 nd 2 examle : k = a 1 - e k

a -1- / , / I

L-~--+------· D = referencereference didiameterame ter D . D .• +b OD.o min mInmin

(d ) take for "usabl"usablee vvolume-tariff"o lume-ta riff" VuV == kVkV where V is the (d) take for u function estabZishedestablished in ((a)a) and k the functionfunction esesta-t a­ blished inin (c).(c) . - 96 -

SHORT BIBLIOGRAPHY OFOF PARTPART I -:-:-:-:-:-:-:-:-:-:-:-:-:-:

1 1 BONEVA L.I.- 1971 - SplineSpline transformationstransformations:: three newne~1 diagnosticdiagnostic aidsaids KENDALL D.D. for the statistical data-analyst -- STEFANOV I.I. JournalJo urna~ of thethe RoyalRoya~ StatisticalStatistica~ SocietySociety Vol.33Vo~.33 p1-71.pl-71.

2 BOUCHON J. - 19741974 -- Les Tarifs de Cubage -- EcoleEco~e NationaleNationa~e dudu GgnieGenie Rura~Rural des Eaux et ForétsForets - 19, avenue du MaineMa ine -- 75732 PARIS CEDEX 1515 (French(French texttext only)on~y) Very full study of the subject with a rich biblio-biblio­ graphy - Few references toto thethe tropics.tropics.

3 CAILLIEZ F.F. - 1979 -- Description du programme de calcul de tarifs de BLANC N.N. cubage d'arbres -- NoteNote n°n° 1717 -- CentreCentre TechniqueTechnique Forestier Tropica~Tropical - 45Bis, avenue de la~a Belle Gabrielle - 9413941300 NOGENT SUR MARNE France Internal document which cancan bebe sentsent freefree ofof chargecharge on request - French text only - Detailed description of a program ususeded by CTFT. Mathematical accountaccount of the subjesubjectc t - LiListingss tings FORTRAN andand useruser guideguide - Example.

4 DRAPER N.R.N.R. - 1966 - Applied Regression Analysis -- Edit.Edit . J. WILEYWILEY SMITHS~1lTH H.H. Full mathmathematicalematical statement ofof high level.lev~l.

5 FREESE F.F. - 1964 -- Linear Regression Methodstlethods forfor ForestForest ResearchResearch -- USDA Forest Service Research Paper -- FLP17 Short and clear manual onon regressionregression technique.technique.

6 HUSCH B.B. - 1971 -- Forest Mensuration - The Ronald Press Co., NEW-YORKNEW-YORK MILLER C.I.I. t-1lLLER C. TextbookTextbook ofof dendrometry containingcontaining 44 chapterschapters onon BEERS T.W.T.W . Forest inventory.inventory.

7 LANLY J.P.J.P. - 1977 -- Manual of forest inventoryinventory - FAO - ROME English and French versions.versions. Mainly devoted toto inventoryinventory ofof heterogeneousheterogene ous tropical forests, useful book forfor thethe dendrometrydendrometry of thesethese forests.forests.

8 LETOUZEY R.R. - 1969 -- Manuelt·lanuel de BotaniqueBotanique ForestièreForestiere -- AfriqueAfrique Tropicale-Tropicale- 3 fascicules. Centre Technique ForestierForestier TropicalTropical -- 45Bis,45Bis, avenue de lala BelleBelle Gabrielle -- 9413094130 NOGENTNOGENT SUR MARNE France.France.

FasciculeFascícule 1 contains a full description (in(in quali-quali­ tative terms)terms) ofof thethe morphology ofof trees.trees . - 97 -

9 LOETSCH F. - 19641964 -- Forest InventoryInventory -- Vol.Vol. i1 HALLER K.E.K.E.

LOETSCH F.F. - 1973 - Forest InventoryInventory -- Vol.Vol. 22 - BLV VerlagsgesellschaftVerlagsgesellschaft -- ZOHRERZÖHRER F.F. - MiinchenManchen - Bern - Wienf·/ien HALLER K.E.K.E. ThoughThough mainlymainly devoted to forest inventory, these ttwowo very complecompletet e booksbooks containcontain veryvery importantimportant developments onon dendrometrydendrometry problemsproblems -- NumerousNumerous refreferenceserences to the tropics.tropics.

10 rMACKAY·1ACKAY E.E. - 1964 -- Dasometria - Escuela Superior de Ingenieros de Montes - Madrid (In spanish)wpanish) Numerous geometric studies, bbutut no reference to the tropics and no bibliography.bibliography.

11 MATERN B.B. - 1956 -- On the geometry of theth e Cross-SectionCros s-Section of a Stem - MeddeZandenMeddelanden FreinFran -- StatensStatens Skogsforskning,:,wtitutSkogsforskninginstitut Band 46 - NRNR 1111 Paper on the mathematicalmathematical study of stem sections - PracticaPracticall consequences.

12 PARDE J.J. - 1961 -- DendrometrieDendrométrie - Ecole NationaZeNationale desdes EauxEaux etet For'etsForets -- 14, ruruee GirardetGirardet - 54032 NANCY - France (French(French text only) ThougThoughh ratherr ather old,o l d, veryvery usefuluseful andand practicalpractical text-text­ book -- EasyEasy toto readread -- FewFew referencesreferences toto thethe tropics.tropics.

13 PRODAN M.M. - 1965 -- HolzmesslehreHolzme ss lehre - J.D.J . D. Sauerldnder'sSauerlande r's Verlag -- Frankfurt amam MainMain (in(in german)german) Very fullful l book onon dendrometry,dendrometry, tarifftariff constructionconstruct i on and incrementincrement estimationestimation -- NumerousNumerous bibliographicalbibl iographical referencereferences.s.

14 PRODAN M.n. - 19681968 -- Forest Biometrics - PergamonPergamon Press (Translation(Translation ofof ForstZicheForstliche BiometrieBiometrie -- -1961).1961). Illustration on forestryforestry problemsproblp.ms of numerousnumerous statisticalsta tistical techniquestec hniques -- ManyMany numericalnume rical explanationsexplanations

1515 SNEDECOR G.W.G.W . - 1967-1967 - Statistical Methods (Sixth(Sixth edition)edi tion ) COCHRAN W.G.t·1. G. (English and French versions)versions) ReferenceReference manual -- NoNo specialspecial referencereference tot o forestry.forestry. - 9898- -

ESTIMATION OFOF WOODl~OOD QUALITYQUALITY

Besides the manuals onon dendrometrydendrometpy citedaited above,above, canaan bebe mentionnedmentionned :

16 IUFRO, Section 25 - 19691969 -- ProceedingsProceedings ofof thethe Reinbek meeting by thethe workingworking groupgroup "Mensurational"Mensurational problems inin tropicaltropical forestforest inventory"inventori' Mitt. Bundesforsch.Bundesfopsah. anst.anst. f.f. Forst -- undund Holsw.Holsw. Komm. verZ.vepl. MaxMax Wiedebusch,Wiedebusah, HAMBURG.HAMBURG. A record of thethe discussions held andand background papers submitted at thisthis meeting which focusedfocused principally onon qualityquality appraisalappraisal andand recoveryrecovery studies.studies .

17 Centre Technique Forestier Tropical -- Revue BoisBois et ForetsFordts des Tropiques -- n°n° 129129 -- 19701970 "Estimation des volumes commercialisablescommercialisables dansdans lesles inventaires forestiers tropicauxtropicaux par sondages" -- LANLY J.PJ.P.. - LEPITRELEPITRE C.C. -

18 C.F.I. - University ofof OxfordOxford -- DepartmentDepartment ofof Forestry -- 19771977 Appendices toto "A"A manualmanual onon speciesspecies andand provenancesprovenances research with particularparticular referencereference toto thethe tropics"tropics" -- Tropical ForestryForestry PapersPapers n°n° 1010 A.A.

-0--()- FAO TECHNICAL PAPERSPAPERS

FAO FORESTRY PAPERSPAPERS

1 ForsstForest utilizatiot:'!utilization contracts on public land, 1977 30 Tropical forest resources,resources. 19821982 (E (E FF S)5) (E F S)5) 31 Appropriate technology in foforestry,restry. 19821982 (E) (E) 2 Planning forestforest roads and harvesting systems, 32 Classification and defdefinitionsinitions of forest products,products. 1977 (E(E FF S)5) 1982 (Ar/E/F/S)(ArIEIFIS) 33 World list of forestry schools,schools. 19771977 (E/F/S) (E/F/S) 33 Logging ofof mountain forests, 19821982 (E (E FF S)5) 3 Rev.Rev. 1. World list of forestryforestry schools, schools, 19811981 (E/F/S)(E/F/S) 34 Fruit-beeringFruit-bearing forestforest treestrees,, 1982 (E(E FF S)5) 33 Rev.Rev. 2. World list of forestryforestry schools,schools, 19861986 (E/F/S) (E/F/S) 35 Forestry inin China.China, 1982 (C(C E)E) 4/1 World pulp and paper demand, aupplysupply and trade 36 Basic technology in forest operations,operations, 19821982 (E (E FF S)51 -- Vol.Vol. 1,1. 19771977 (E (E FF S)5) 37 Conservation and development of tropical forest 4124/2 World pulp and paper demand, supply and trade resources, 1982 (E (E FF S)5) -- Vol.Vol. 2,2. 19771977 (E (E FF S) 38 Forest products prices 1962-1981, 19821982 (E/F/S) (E /F/S) 5 The marketing of tropical wood,wood. 19761976 (E IE S)5) 39 Frame Sewsaw manual, 19821982 (E) IE) 6 National parksparks planning,planning, 19761976 IE(E FF 5··) S**) 40 Circular saw manual, 19831983 (E) (E) 7 Forestry forfor locallocal community development, 1978 41 Simple technologies for charcoal making,making. 1983 (Ar E F S) (E F SIS) 8 Establishment techniquestechniques for forest plantations,plantations, 42 Fuelwood supplies in thetha developingdeveloping countries,countries. 1978 (Ar C E"E' FF S)5) 1983 (Ar(Ar EE FF S)S) 9 Wood chips - production,production, handling,handling, transport, 43 Forest revenue systems in developingdeveloping countries,countries, 1976 (C(C EE S)SI 1983 (E(E FF S)S) 101110/1 Assessment ofof logginglogging costscosts fromfrom forest 441144/1 Food and fruit-bearing foforestrest speciesspecies inventories in the tropicstropics - 1.1. Examples Examples frornfrom easterneastern Africa,Africa. 19831983 (E (E FF S)5) - 1.1. Principles Principles andand methodology,methodology, 19781978 (E IE FF S)51 441244/2 Food and fruit-bearing forest speciesspecies 1012 Assessment ofof logginglogging costscosts fromfrom forest - 2.2 .Examples Examples fromfrom southeasternsoutheastern inventories inin thethe tropics Asie.Asia, 19841984 (E (E FF S)S) - 2.2. Data Data collectioncollection and and calculations,calculations, 19781978 (E IE FF S)51 441344/3 Food and fruit-bearingfruit-bearing forestforest species -- 3. ExamplesExamples 1111 Savanna afforestationafforestation in Africa, 19771977 (E{E F)FI from Latinlatin America, 19861986 (E (E S)5) 12 China: forestryforestry support for agriculture, 19781978 (E)(E) 45 Establishing pulp endand paper mills, 19831983 (E)(E) 13 Forest productsproducts prices 1960-1977,19791960-1977, 1979 (E/F/S) (E/F/S) 46 Forest productsproducts prices 1963-1982.19831963-1982, 1983 (E/F/S) (E/F/S) 14 Mountain forest roadsroads and harvesting, 19791979 (E)(E) 47 Technical forestry education - designdesign andand 14 Rev.Rev. 1. loggingLogging andand transporttransport in steep terrain, 1985 (E)(E) implementation,implementation, 19841984 (E (E FF S)51 15 AGRIS f forestryorastry -- worldworld cataloguecatalogue of information 48 Landland evaluation for forestry, 19841984 (C IC EE F F S)S) and documentation services,services, 19791979 (E/F/S) (E /F/S) 49 Wood extraction withwith oxen oxen and and agriculturalagricultural 16 China: integrated wood processing industries, 1979 tractors,tractors, 19861986 (E IE FF S)5) (E F S) 50 Changes inin shiftingshifting cultivation in Africa, 19841984 (E (E F)F) 17 Economic analysisanalysis of of forestryforestry projects,projects, 1979 501150/1 Changes inin shifting cultivation inin AfricaAfrica (E F S)SI - sevenseven case-studies, 1985 (E)(E) 17 Sup. 1. Economic analysis of forestry projects:projects: 511151/1 Studies on the volume and yield of tropical forest case studiesstudies,. 1979 (E(E S)5) standsstands - 1.1. Dry Dry forest forest formations, formations. 19891989 (E (E FIFI 17·17 Sup.Sup. 2. EconomicEconomir. enalysisanalysis of forestry projects:projects: readings,readings, 52/15211 Cost estimating inin sawmillingsawmilling industries:industries: guidelines,guidelines, 1980 (CE)(C EI 1984 (E) (E) 18 Forest productsproducts pricesprices 1960·1978.1960-1978, 19801980 (E/F/S) (EIFIS) 52/25212 Field manual on cost estimation in sawmillingsawmilling 191119/1 Pulping andand paper-makingpaper-making propertiesproperties ofof industries,industries, 19851985 (E) (E) fast-growing plantationplantation woodwood speciesspecies 53 IntensiveIntensive multiple-usemultiple·use forestforest managementmanagement inin Kerala,Kerale. - Vol.Vol. 1,1. 19801980 (E) (E) 1984 (E (E FF S)5) 191219/2 Pulping and paper-making properties of 54 PlanificaciónP1anificaci6n del desarrollo forestar,forestal, 19841984 (S) (S) fast-growing plantationplantation woodwood speciesspecies 55 IntensiveIntensive multiple-use forestforest managementmanagement inin thethe - Vol.Vol. 2,2. 19801980 (E) (E) tropics,tropics. 19851985 (E (E FF S)5) 20 Forest treetree improvemimprovement,ent, 1985 (C(C EE FF S)51 56 Breeding poplars for diseasedisease resistance, 19851985 (E) (E) 20/22012 A guide to forest seedseed handling,hendling, 19851985 (E IE S5 ) ) 57 Coconut woodwood -- processingprocessing and and use,use, 19851985 (E {E S)51 21 ImpactImpact on soils of fast-growing speciesspecies inin lowlandlowland 58 Sawdoctoring manual,manual, 19851985 (E (E S)SI humid tropics, 19801980 (EIE FF S)5) 59 The ecological effects of eucalyptus, 1985 22/12211 Forest volume estimation andend yield prediction (C E F S)51 - Vol.Vol. 1.1. Volume Volume estimation, estimation, 19801980 (C {C E E F F S)51 60 · Monitoring andand evaluationevaluation of participatoryparticipatory forestryforestry 22/22212 Forest volume estimation and yield prediction projects,projects. 19851985 (E (E FF S)51 - Vol.Vol. 2.2. Yield Yield prediction,prediction, 1980 1980 (C {C E E F F S) 51 61 Forest products prices 1965-19841965-1984,. 19851985 (E/F/S) (E/F/S) 23 Forest products prices 1961-1980, 19811981 (E/F/S)(E/F/S) 62 World list ofof institutionsinstitutions engagedengaged in forestryforestry andand 24 Cable logging systems.systems, 19811981 (C{C E)EI forestforest productsproducts research,research, 19851985 (E/F/S) (E/F/S) 25 Public forestry administrations inin Latinlatin America,America, 63 IndustrialIndustrial charcoalcharcoal making, 19851985 (E) IE) 1981 (E)(EI 64 TreeTree growing by ruralrural people, 1985 (Ar(Ar E E FF S)5) 26 Forestry and rural development, 19811981 (E(E FF 5) 65 Forest "legislation in selectedselected AfricanAfrican countries,countries, 27 Manual of forest inventory, 19811981 (E(E F)F) 1986 (E (E F)F) 28 Small and medium sawmills inin developingdeveloping countries,countries, 66 Forestry extension organization,organization. 19861986 (C (C EE S)5) 1981 (E(ES) SI 67 Some medicinal forest plants of AfricaAfrica andand Latinlatin 29 World forest products, demanddemand and supply 1990 America, 19861986 (E) (E) endand 2000.2000, 19821982 (E (E FF S)5) 68 Appropriate forest industries,industries, 19861986 (E) (E) 69 Management of forest industries,industries, 19861986 (E) (E) 92 Forestry policies in Europe -- anan analysis,analysis, 19891989 (E) IE} 70 Wildland fire management terminology, 1986 93 Energy conservationconservation inin the mechanical forestforest (E/F/S) industries.industries, 1990 (E(E 9)SI 71 World compendium of forestry and forest products 94 Manual on sawmill operationaloperational maintenance,maintenance, researchrasearch institutions,institutions, 19861986 (E/F/S) IE/F/S) 1990 (E) (E) 72 Wood gas as88 engine fuel, 1986 (EIE S)5) 95 Forest products prices 1969-1988, 19901990 (E/F/S) IE/F/S) 73 Forest products: worldworld outlookoutlook projectionsprojections 96 Planning and managingmanaging forestryforestry research: guidelines 19851985-2000,-2000, 19861986 (E/F/S) (E/F/S) for managers, 1990 (E)(E) 74 Guidelines forfor forestry information processing,processing, 97 Non·woodNon-wood forest products: the wayway ahead, ahead. 1986 (E) (E) 1991 (ES)(E 5) 75 An operational guide to the monitoring andand 98 Les plantationsplantations à1\ vocation dede boisbois d'ceuvred'csuvre enen evaluation of social forestry inin India,India, 19861986 (E) (E) Afrique intertropicale humide, 1991 (F)(F) 76 Wood preservationpreservation manual,manual, 19861986 (E)IE) 99 Cost control in forest harvestingharvesting andand roadroad 77 Databook on endangeredendangered tree andend shrub speciesspecies construction, 19921992 (E) (E) and provenances, 1986 (E)IE) 100 Introduction to ergonomicsergonomics in forestry inin developingdeveloping 78 Appropriate wood harvesting in plantation forests, countries, 19921992 (E) (E) 1987 (E)(E) 101 AmtinagementAmiMagement et conservation des for6tsforks densesdenses enen 79 Small-scale forest·besedforest-based processing enterprises, AmériqueAmtirique tropicale,tropicale. 19921992 (F) IF) 1987 (E(E FF S)5) 102 Research managementmanagement inin forestry,forestry, 1991 (E)IE) 80 Forestry extension methods, 1987 (E)IE) 103 Mixed and pure forestforest plantations in the tropics andand 81 Guidelines forfor forestforest policy formulation, 1987 (C(C E)E) subtropics, 19921992 (E) IE) 82 Forest products prices 1967·1986,1967-1986, 19881988 (E/F/S) IE/F/S) 104 Forest productsproducts pricesprices 1971·1990,19921971-1990, 1992 (E)(E) 83 Trade in forest products:products: aa study ofof thethe barriersbarriers 105 Compendium of pulp andand paperpaper trainingtraining andand faced by thethe developingdeveloping countries,countries, 19881988 (E) IE) researchresearch institutions,institutions, 19921992 (E) IE) 84 Forest products: world outlookoutlook projectionsprojections 106 Economic assessment ofof forestry project impacts,impacts, 1987-20001987·2000 -- productproduct and and countrycountry tables,tables, 19881988 1992 (E)(E) (E/F/S)(ElF IS) 107 Conservation of genetic resources inin tropical forest 85 Forestry extension curricula,curricula, 19881988 (E/F/S) (E /F/S) management: principles and concepts, 19931993 (E)(E) 86 Forestry policies inin Europe,Europe, 19881988 (E) IE) 108.108' Conservation of genetic resources inin tropical foreforestst 87 Small-scaleSmall· scale harvesting operations of woodwood andand managementmanagement· - Principles and concepts,concepts, 1993 (E)IE) non-woodnon· wood forestforest productsproducts involvinginvolving rural rural people,people, 1988 (E(E FF S)5) Availability: February 1993 88 Management of tropical moist forests in Africa, 1989(EFP)1989 (E F P) Ar Arabic Multi! -- MultilingualMuttmngual 89 Review of forest managementmanagement systems ofof tropicaltropical C Chinese Out of print * Asia, 19891989 (E) (E) E English " In preparation 90 Forestry and food security, 19891989 (ArIAr EE S)5) F French 91 Design manual on basic wood harvestingharvesting P Portuguese technology, 19891989 (E (E FF S)5) S Spanish (Published only asas FAOFAO Training Series,Series, No.No.1 18) 8) The FADFAO TechniClJITechnical Papers lJreare lJvlJilableavailable through thethelJuthorized authorized FAOFAD SafesSales AgtmtsAgents or directly from Distribution andlJnd SalesSafes Section,Section, FAO,FAD, VialeVia/e delle Terme d;di CiJflJc811lJ,Caracalla, 0010000100 Rome, ItalyItaly..