Quicker/Cheaper Stand Assessments 1

Transcription

1 Quicker/Cheaper tad Assessmets 1 by Thomas E. Burk Jue 004 Origially Published October 1999 taff Paper eries No. 139 Revisio Departmet of Forest Resources College of Food, Agricultural ad Natural Resource cieces Uiversity of Miesota t. Paul, Miesota 1 Research supported by Miesota Agricultural Experimet tatio uder project MN Published as paper o of the Miesota Agricultural Experimet tatio. The author is Professor, Departmet of Forest Resources, College of Food, Agricultural ad Natural Resource cieces, Uiversity of Miesota, t. Paul, MN.

2 For more iformatio about the Departmet of Forest Resources ad its teachig, research, ad outreach programs, cotact the departmet at: Departmet of Forest Resources Uiversity of Miesota 115 Gree Hall 1530 Clevelad Aveue North t. Paul, MN Ph: Fax: The Uiversity of Miesota is committed to the policy that all persos shall have equal access to its programs, facilities, ad employmet without regard to race, color, creed, religio, atioal origi, sex, age, marital status, disability, public assistace status, vetera status, or sexual orietatio.

3 Table of Cotets Page Abstract 1 amplig for timber attributes with variable-radius plots A review Why variable-radius plots? Tree tally at a sample locatio Relatioship betwee plot size (for a tree) ad tree size Per acre timber characteristics 4 Computig a samplig error 4 What BAF to use? 5 How may sample locatios are eeded? 6 Is it ok to systematically spread out sample locatios? 7 Field implemetatio 8 Data summary example 9 Double samplig for timber attributes with variable-radius plots 11 Motivatio 11 amplig pla 11 Tree tally ad measuremet 1 Volume per acre estimatio 1 Estimatio for categories (breakdows) of iterest 13 What BAF to use? 14 What should L ad be? 14 Aother look at the double samplig ratio-of-meas estimator 15 Estimatio for categories whe extra data are collected o tally plots 16 Tally sheets 17 Data summary example 17 The Big BAF method 18 Motivatio 18 amplig pla 18 Volume per acre estimatio 18 Number of sample locatios, umber of measure trees, ad implied BAF B 0 Additioal commets Tally sheets 3 Data summary example 3 A cruise exercise: Coduct ad review 4 i

4 Role of volume tables/equatios 5 Costat form factor volume equatios 5 Volume table variatio i VBARs 6 The origi of shortcut formulas 7 Elimiatig tree measuremet from timber stad assessmets 7 Local (localized) volume tables 8 A look at samplig efficiecies 30 Variable- versus fixed-radius plots 30 Double samplig with the ratio-of-meas estimator 30 The Big BAF method 31 ummary 3 Refereces 34 Appedix 1 ample sizes for double samplig with fixed cost 36 Appedix ample sizes for ad efficiecy of the Big BAF method 38 Optimal umber of plots ad umber of trees (fixed precisio) 38 Re-expressig costs 39 Cost compariso with simple radom samplig 40 Optimal umber of plots ad umber of trees (fixed cost) 40 ii

5 Abstract amplig ad measuremet techiques are available for reducig the effort/cost associated with stadard timber stad assessmets depedig o the quatities of iterest as well as the volume table(s) beig used. This paper will itroduce ad cosider the applicatio of stadard variable-radius plot samplig, double (two-phase) samplig, a ew techique called Big-BAF samplig, as well as assorted shortcut measuremet ad observatio techiques as they apply to Lake tates forestry practice. The relative efficiecies of the techiques from both cost ad precisio viewpoits are also discussed. The paper is writte for idividuals who supervise or pla or coduct timber stad assessmets for their orgaizatio. A basic uderstadig of the coduct of timber stad assessmets usig stadard methods (e.g. lie-plot cruises) is assumed. The paper could be the basis for a workshop; hads-o activities, icludig a field exercise, are suggested. preadsheet code implemetig may of the calculatios preseted i the paper is also available at <

6 Copyright 004 T.E. Burk amplig for timber attributes with variable-radius plots A review Why variable-radius plots? Theory proves that samplig error will be miimized by selectig elemets for iclusio i the sample with probability proportioal to the variable of iterest. For the variable-radius plot type we ll cosider, trees at a sample locatio are selected i proportio to tree basal area. Basal area is typically closely related to volume (at least more so tha is tree umber), the primary variable of iterest i timber stad assessmets. Tree tally at a sample locatio A agle gauge, projectig a fixed, horizotal agle, is used to determie which trees are i at a sample locatio. The tally rule is illustrated below; the schematic is a view from overhead with tree cross-sectios at breast height show. The dashed lies represet the fixed, horizotal agle from the sample locatio. A prism works similarly by refractig light rays through a agle equal to that of the fixed horizotal agle; i trees are those where the true ad false (as see through the prism) images of the tree at breast height overlap. Relatioship betwee plot size (for a tree) ad tree size The schematic below shows a tree cross-sectio at breast height for a tree that is at a distace from a sample locatio such that it is o the border of its plot; agai Part 1 amplig for timber attributes with variable-radius Page

7 Copyright 004 T.E. Burk the dashed lies represet the fixed, horizotal agle. The distace R (feet) is thus the radius of the plot for a tree of DBH D (iches). Plot border Agle θ/ R - p lot rad ius ample locatio Tree of DBH D Takig q = si (θ/), a costat for a give horizotal agle, ad watchig uits gives D R = 1 q which meas that plot size i acres for ay tree i is a i π Di = q Bi = q [1-1] Bi = BAF where B is tree basal area (square feet DBH ) ad BAF is basal area factor. We will use BAF rather tha θ to idetify a istrumet s costat agle. BAF is the umber of square feet per acre that each i tree represets at a sample locatio. Part 1 amplig for timber attributes with variable-radius Page 3

8 Copyright 004 T.E. Burk Per acre timber characteristics I a timber stad assessmet we are ormally iterested i stad attributes like volume per acre, basal area per acre, ad stems per acre. To get per acre estimates we eed to expad tree characteristics accordig to a tree s plot size. If there are t trees o a plot ( i at a sample locatio) ad v represets the volume of a tree (from a table or equatio), the volume per acre would be t t vi vi = BAF i= 1 ai i= 1 Bi The volume to basal area ratio, VBAR, plays a key role i variable-radius plot samplig. If there are sample locatios total with t j i trees at the jth sample locatio the t j BAF vi volume per acre = j= 1 i= 1 Bi [1-] ad likewise t j BAF 1 stems per acre = [1-3] B j= 1 i= 1 i BAF BAF basal area per acre = t j = T [1-4] j=1 where T is the total umber of i trees at all sample locatios. If iterest lies i a estimate for some class of tree, the same formulae ca be used with the secod sum oly take across trees i the class of iterest. To fid the stock table etry for 8-ich DBH aspe oe would sum the volume to basal area ratios for ay such trees tallied, multiply by BAF ad divide by. To fid the stad table etry for the same trees oe would multiply.865 (iverse basal area for a 8-ich DBH tree) by the umber of such trees tallied ad agai multiply by BAF ad divide by. Computig a samplig error From a statistical perspective, it is the variability betwee observatios at sample locatios that is relevat. To compute a samplig error requires separate tally by sample locatio ad computatio of the variable(s) of iterest sample uit by sample uit. For sample uits take at radom where V j is the volume per acre observatio at locatio j, the variace of the observatios is estimated by Part 1 amplig for timber attributes with variable-radius Page 4

9 Copyright 004 T.E. Burk V j j= 1 V j j= 1 sv = [1-5] 1 The coefficiet of variatio of the observatios is estimated by CV V sv = 100 [1-6] V / j= 1 j The samplig error, or stadard error of the estimate, if the estimator is the sample average (as suggested above), is estimated by s = s [1-7] V V / What BAF to use? Choosig a BAF is like choosig a plot size for fixed-radius plots, though BAF ad plot size are iversely related. Geerally, if variatio i the stad is high, BAF should be low; that way variability is captured at a sample locatio ad variability betwee sample locatios (which drives samplig error) is made smaller. A good approximatio to the relatioship betwee BAF ad variability is (Freese 196, page 7) CVBAF BAF = [1-8] CVBAF1 BAF1 The variatio, as measured by coefficiet of variatio, for 0 BAF is thus about 0 percet greater tha that for 10 BAF. For a fixed cost samplig effort there will be a tradeoff betwee BAF ad umber of sample uits take. The smaller the BAF, ad hece the more trees tallied at each sample locatio, the fewer plots that ca be take. A large umber of studies have bee coducted addressig the issue of optimal BAF. Though study results are varied, it ca be show that, uder a reasoable set of assumptios, a optimal BAF (oe that results i miimal time/effort to achieve a desired precisio) will result i equal amouts of time beig spet o travel betwee plots ad measuremet at a plot (Zeide 1980). This result holds for radom or systematic placemet of sample locatios, a commo practice for timber stad assessmets. Ufortuately, traditio ofte dictates choice of BAF ad this ofte leads to iefficiecies. A BAF of 10 for Lake tates forestry practice is likely too small o average; too much time is spet o a idividual plot. Part 1 amplig for timber attributes with variable-radius Page 5

10 Copyright 004 T.E. Burk The ability to see trees from a sample locatio should also ifluece specificatio of BAF. Large trees ca be a log distace from a sample locatio ad still be i with a small BAF. Heavy uderstory coditios also makes esurig that all i trees are sighted difficult. It has bee commoly foud that usig a BAF that is too small results i sigificatly egative measuremet bias (too few trees couted). everal studies have show that it is efficiet to tally 5 to 8 trees at a sample locatio (Wesel et al. 1980). To tally 7 trees, o average, at a sample locatio i a stad estimated to have 10 square feet per acre would imply use of a 17 BAF istrumet. The 5 to 8 tally guidelie has led some to employ a scheme where a differet BAF is used at each sample locatio i a tract. Oe approach is to start with a large BAF ad try successively smaller BAFs util the desired tree cout is reached. This is ot a good rule to follow, as it ca lead to cofusio i estimatio, though there is some argumet as to whether it produces biased results (Iles ad Wilso 1988). A agle gauge is ay object of kow width held at a kow distace from the cruiser s eye. Maipulatig equatio [1-1] shows that distace from eye (l) ad agle gauge width (w), both i iches, are related by w l = / ta( arcsi BAF / 43560) [1-9] To use a 1-ich wide gauge for 0 BAF would require holdig it 3.33 iches from your eye. You ca also calibrate a agle gauge by backig away from a tree of DBH D (iches) util the tree is just o the plot border ad the measurig the horizotal distace R (feet) from the eye to tree ceter. D BAF = [1-10] R This would be a coveiet calibratio method if you tallied trees usig your thumb at arms legth as your agle gauge. May suggest employig such a calibratio as it accouts, i some sese, for the ellipticity of trees i your cruise area. There is little reaso to use a BAF that is iteger-valued. How may sample locatios are eeded? ample size is ofte ecessarily set by time or cost costraits. till, it is good to have a uderstadig of the relatio betwee umber of sample uits, populatio variability betwee sample uits, ad samplig error. tudy of a particular situatio may lead oe to coclude that cost costraits lead to a uecessarily large sample or a sample that is of little value for the purposes to which the derived iformatio is to be put. Part 1 amplig for timber attributes with variable-radius Page 6

11 Copyright 004 T.E. Burk For a simple radom sample, the umber of uits required to obtai a samplig error that is P% of the mea, usig sample uits with variability (betwee uits) CV is 1 CV e = [1-11] P For example, if a samplig error of 10% is acceptable ad CV is 50% (based o BAF 10 plots, say) oe should establish 5 sample locatios. With BAF 0, equatio [1-8] suggests a CV of 59.5% givig a sample size of 36. If the umber of uits required to obtai a desired samplig error appears out of lie, it is likely that effort should be put ito stratifyig the tract to be sampled; stratificatio is the most powerful tool available to the sampler. It may also be the case that the specified desired samplig error is ureasoable. Related to this, ote that tract size does ot appear i the sample size formula. This implies that small tracts with large variability will require what could be cosidered a ridiculously large umber of sample uits; or that desired samplig error for small tracts will eed to be relaxed. imilar reasoig suggests that precisely estimatig summary breakdows, such as volume of a particular species-size combiatio, will require large umbers of sample uits. Is it ok to systematically spread out sample locatios? Commo sese would suggest placig sample locatios o a systematic grid, ala lie-plot cruises, throughout the tract of iterest i a attempt to obtai as represetative a sample as possible. ystematically traversig a tract also expedites mappig, a importat task i forest lad assessmet. tatistical iferece, o the other had, dictates some sort of radomizatio be used i selectig sample locatios. Experiece ad umerous studies of timber stad assessmet applicatios reach the coclusio that ratioal applicatio of systematic samplig used i cojuctio with stadard statistical formulas will produce good poit estimates with coservative precisio estimates. Ratioal applicatio meas that kow treds i the tract, with respect to a variable of iterest, must be represeted appropriately i the sample. A commo guidelie is to ru cruise lies perpedicular to kow treds. Tract edge areas ca be particularly difficult to represet properly, a special cocer i log, arrow tracts. Aerial photography, existig maps, etc. will usually be sufficiet to lead the forester to a ratioal systematic layout whose results ca be summarized usig stadard statistical formulas. The figure below illustrates a lie-plot cruise with six sample lies ad a total of 3 sample locatios. 1 Note that the sample size developmet here differs from may textbooks where a sample size is computed that gives a certai cofidece iterval half width. Part 1 amplig for timber attributes with variable-radius Page 7

12 Copyright 004 T.E. Burk Plots are ormally located by compass ad pace i timber stad assessmets so that pipoitig exact radom locatios would be impractical. Of greater importace is that sample locatios be placed i a ubiased fashio as will be discussed below. Field implemetatio Istrumet usage Agle gauges must be held at the proper distace from the eye, perpedicular to the lie of sight, ad ot tilted alog their vertical axis. The cruisers eye must costatly be over the sample locatio whe viewig trees; doig otherwise will result i tally of too may trees. imilar care must be applied whe usig a prism. The prism itself is the vertex of the critical agle ad must therefore always be held over the sample locatio. A commo tedecy with a prism is to push the poit (Oderwald ad Gregoire 1995), the bias of which ca be sigificat for large BAFs ad small trees. The lie of sight for either istrumet is assumed to be horizotal from the sample locatio to tree breast height. light icliatios are ot of cocer; istrumets ca be adjusted for slope or a questioable tree ca be checked (see below). Trees hidde by other trees or obscured by brush will likely eed to be checked to determie whether they should be tallied. Movig off the sample locatio should oly be doe with extreme cautio. ome will view the tree above breast height, where brush obstructios are likely less severe, to observe whether the tree is obviously i ; recall, however, the ecessity of proper istrumet orietatio. Clearly, missig a etire tree or icludig a tree that is off plot are errors that should ad ca be avoided with little extra effort. Checkig questioable trees Difficult coditios ad iexperiece will make it difficult to judge whether a tree is i with a istrumet. The DBH (to 0.1-ich) Part 1 amplig for timber attributes with variable-radius Page 8

13 Copyright 004 T.E. Burk ad distace from sample locatio (horizotal to tree ceter) must be measured for such trees. DBH of the tree i questio should be plugged ito the formula R = DBH [1-1] BAF If the measured distace is greater tha the computed distace, the tree should ot be tallied. The radical term is ofte called the plot radius factor ad is the umber of feet per ich of DBH that a tree ca be from the sample locatio ad still be i. ome suggest applyig the tally every other oe rule to questioable trees. If iterest lies solely i basal area per acre of all trees ad several plots are beig istalled, such a rule will be adequate. I all other situatios the rule should be avoided (Iles ad Fall 1988). I their study ivolvig experieced cruisers Iles ad Fall (1988) foud that oly about percet of trees are borderlie. Establishig sample locatios i a ubiased fashio Accurate placemet of sample locatios is icosistet with the methods commoly used i timber stad assessmets. Of far greater importace is ubiased placemet. If the locatio is iside a tree, so be it; this is of special cocer for large BAFs. Overly ope or dese areas must ot be avoided because they appear urepresetative. Plots ear the tract edge If the tract of iterest cotais a moderate to high proportio of edge ad edge areas are suspected to differ from iterior areas with respect to variables of iterest, it is critical that edge areas be appropriately represeted i the samplig effort. Plots must ot be moved away from tract boudaries. I extreme cases it may be advisable to stratify the tract ito edge ad o-edge strata ad sample the two strata separately. More ofte, edge coditios ca be appropriately treated by takig partial plots or usig the mirage method of boudary overlap correctio (Avery ad Burkhart 1994, page 8). With the mirage method a sample locatio ear the tract edge is tallied twice: oce at the ormal locatio ad oce from outside the tract. The positio outside the tract is foud by proceedig alog the cruise lie a distace outside the tract boudary equal to the distace the ormal locatio lies from the tract boudary. Trees i edge will be tallied twice with the mirage method. However, the locatio ca be treated like ay other sigle sample locatio as cocers data summary. Data summary example The Excel workbook iglephase.xls summarizes data from a variable-radius plot timber stad assessmet. Data are etered o the Data worksheet; this icludes BAF, umber of plots, ad for each tree: plot umber, tree umber, species (user chose labels, up to seve for oe cruise), DBH (iteger), ad umber of 8-foot bolts (iteger). Tree volumes i rough cords are computed usig Table 6 of Gevorkiatz ad Olse (1955) (worksheet VolTable). ummaries appear o the Results worksheet ad iclude cordwood volume per acre, its stadard error, ad per acre stock ad stad tables. You ca also compute the umber of plots eeded Part 1 amplig for timber attributes with variable-radius Page 9

14 Copyright 004 T.E. Burk to achieve a specified samplig error expressed as a percet (equatio [1-11]). Istructios for usig the workbook appear o the Results ad Data worksheets. Part 1 amplig for timber attributes with variable-radius Page 10

15 Copyright 004 T.E. Burk Double samplig for timber attributes with variable-radius plots Motivatio o-called double or two-phase samplig ivolves the observatio of a additioal auxiliary variable to improve o the samplig efficiecy for estimatig some primary variable of iterest. To achieve efficiecy gais, the auxiliary variable must be closely related to the variable of iterest ad must be cheaper to observe tha the variable of iterest. For the case we ll cosider, related meas liearly related so that the correlatio coefficiet betwee the auxiliary variable ad the variable of iterest will be a coveiet measure of relatioship closeess. For timber stad assessmets the variable of primary iterest will be some measure of volume per acre. A cadidate auxiliary variable is therefore basal area per acre. Volume ad basal area per acre will typically have a correlatio above 0.9 or Basal area per acre will be much cheaper to observe tha volume whe we use variable-radius plots as it requires oly a tally of trees i at the sample locatio. amplig pla With double samplig, the auxiliary variable is observed o some large set of plots, L, while the primary variable of iterest is also observed o a subset,, of the larger set of plots. Whe usig variable-radius plots for a timber stad assessmet this meas simply tallyig trees o the L - plots while tree characteristics required to obtai tree volume (DBH, height, ) are measured for i trees o the plots. The plots are ofte referred to as measure plots while the L plots ca be called tally plots. Note that the auxiliary variable, basal area per acre, is free o the plots. A double sample ca be laid out much like a lie-plot cruise. The L total plots are placed o a systematic grid i the tract of iterest, with attetio paid to ratioal placemet of the sample lies. ome fractio (every secod, third, fourth, fifth, etc.) of the total umber of plots is idetified for tree measuremet: the plots. The followig schematic illustrates for a total of 3 plots where every fourth plot ivolves tree measuremet (solid circles). Part Double samplig for timber attributes Page 11

16 Copyright 004 T.E. Burk Tree tally ad measuremet The subset of plots that ivolve tree measuremet,, should be tallied i the same maer as a stadard variable-radius plot cruise. Measuremets will ormally be those ecessary to compute idividual tree volumes. Tallies must be kept separate by plot if statistical precisio estimates are to be computed. For the remaiig L plots it is oly ecessary to accumulate the umber of i trees across all plots, eve if precisio estimates are required. I multi-species stads where estimates of volume per acre by species are of high iterest, the cumulative tree tallies o the L plots should be further broke dow by species. imilarly, if estimates by tree DBH class are a high priority, oe might cosider ocularly estimatig DBH class for i trees o the L plots ad completig the cumulative tally by DBH class, or possibly species ad DBH class. The extra cost associated with implemetig such refiemets must be cosidered ad be ratioalized to be outweighed by the gais i precisio for importat estimate breakdows. This is likely to be the case if experieced cruisers are ivolved (where species idetificatio ad DBH class, 1-ich or - ich classes, assigmet are automatic). Volume per acre estimatio everal alteratives are available for summarizig timber stad assessmets coducted usig double samplig. We will cosider the ratio-of-meas estimator. The ratio-of-meas estimator is appropriate if the variable of iterest ad auxiliary variable are liearly related with the relatioship passig through zero ad the variace of the variable of iterest is proportioal to the magitude of the auxiliary variable. These assumptios are reasoable for most volume per acre basal area per acre combiatios cosidered i practice. However, the assumptios should be evaluated ad a alterative estimator applied if ecessary (cf. Freese 1960, page 39). A example of a istace where the ratio-of-meas Part Double samplig for timber attributes Page 1

17 Copyright 004 T.E. Burk estimator is iappropriate would be usig the auxiliary variable basal area per acre of all trees whe the variable of iterest is board foot volume per acre of trees 10-iches DBH ad larger; the itercept of the relatioship betwee these two variables will be sigificatly egative ad the ratio-of-meas estimator, cosequetly, biased. Equatio [1-] ca be applied to the plots upo which tree measuremets are take to obtai a volume per acre value, V. Equatio [1-4] ca be applied separately to the ad L plots to get two basal area per acre values, The improved estimator for volume per acre is the V G G ad V RMD = L [-1] G The estimator ituitively icreases or decreases the volume per acre estimate based o the plots a fractio depedig o how the larger sample estimate of basal area per acre compares to the smaller sample estimate. A estimator for stadard error of [-1] is G L. 1 1 V V sv s + + V RMD = sv svg sg [-] G G L L where sv is equatio [1-5] applied to the volume per acre values ad s G is foud usig the same equatio applied to the basal area per acre values. The pairs of volume per acre basal area per acre values are used to fid the covariace betwee volume per acre ad basal area per acre V j G j j= 1 j= 1 V jg j j= 1 svg = [-3] 1 1/ Estimatio for categories (breakdows) of iterest Equatio [-1] ca be applied to categories of volume per acre. Volume per acre for a particular species is estimated by fidig volume per acre for the species usig the measure plots (equatio [1-]) ad the multiplyig by the ratio G L / G. The same could be doe for volume of a species-dbh category ad thus a etire stock table (Oderwald 1994). Basal area per acre ca be broke dow i the same maer. Applyig this approach to stems per acre by species-dbh categories (first usig equatio [1-3], for example) produces a stad table cosistet with the overall Part Double samplig for timber attributes Page 13

18 Copyright 004 T.E. Burk estimate of basal area per acre. This is the usual suggestio though it implicitly assumes stems per acre ad basal area per acre are liearly related. What BAF to use? The same cosideratios hold i choosig a BAF for double samplig with variable-radius plots as did with stadard samplig with variable-radius plots (discussed at legth earlier). The 5 to 8 trees per sample locatio guidelie is the simplest to follow. Optimal BAF has yet to be worked out. Normally a sigle BAF is used for all plots (both measure ad tally plots). What should L ad be? Oderwald ad Joes (199) preset formulas for choosig L ad for timber stad assessmets usig double samplig with the ratio-of-meas estimator. They assume specificatio of the umber of stadard plots required to meet a precisio specificatio (computed, e.g., usig equatio [1-11]) ad compute L ad such that the same precisio is achieved ad cost (time) is miimized. The resultig formulas are complex fuctios of ratio of time to observe volume per acre to time to observe basal area per acre at a sample locatio correlatio coefficiet betwee volume per acre ad basal area per acre ratio of the coefficiet of variatio of volume per acre to the coefficiet of variatio of basal area per acre The worksheet amizep i the Excel workbook TwoPhase.xls implemets the formulas. For example if we assume the time for observig volume per acre is four times that of observig basal area per acre, a correlatio betwee volume per acre ad basal area per acre of 0.9 (coservative), ad that the ratio of coefficiets of variatio is 1.5, the optimal double sample would measure trees o oly oehalf the plots of a stadard sample but observe a total of 50 percet more plots ( L ) for basal area per acre. Oderwald ad Joes cite cosiderable experiece idicatig that the ratio of coefficiets of variatio will be aroud 1.0 whe iterest lies i pulpwood-sized timber to less tha 1.5 for sawtimber-sized timber. The series of tables i worksheet amizep show how sesitive the results are to typical values of the cost ad coefficiet of variatio ratio; the degree of sesitivity is small to moderate. The double samplig sample sizes that give miimum variace for a fixed cost (time) are derived i Appedix 1. They are agai fuctios of the three factors idetified above. The worksheet amizec i the Excel workbook TwoPhase.xls implemets the formulas. For the example above the solutio is L = e with measuremet o every third plot. It is also show i Appedix 1 that the ratio to L is the same for both the equal precisio ad equal cost sample size solutios. If data are compiled separately by sample locatio uder existig lie-plot cruise procedures, those data ca be examied ad aalyzed to obtai some uderstadig of the gais possible from double samplig. It should be kept i Part Double samplig for timber attributes Page 14

19 Copyright 004 T.E. Burk mid though that the total umber of plots tallied uder double samplig will typically be larger tha that of a stadard lie-plot cruise. Aother look at the double samplig ratio-of-meas estimator The ratio-of-meas estimator we have bee cosiderig (equatio [-1]) ca be expaded usig equatios [1-] ad [1-4] as follows V VRMD = GL G = BAF j= 1 i= 1 = VBAR G t j BAF T L v i B i G L [-4] where VBAR is the average of the idividual tree volume to basal area ratios for the T trees o the measure plots. The estimator for sigle phase samplig (equatio [1-]) ca be writte i this form as well. Volume per acre estimatio whe samplig with variable-radius plots is thus, geerally, see to be a matter of estimatig a average tree VBAR ad a average basal area per acre ad formig their product. It is critical to uderstad that the average tree VBAR i [-4] (or its sigle phase equivalet) is obtaied with trees selected accordig to their basal area, i.e. via variable-radius plots. This average would, of course, be biased high if iterest lies i mea volume to basal area ratio for all trees. Coversely, obtaiig the average tree VBAR i [-4] via equal probability samplig would result i a volume per acre estimate that was biased low. The relative (to the mea) variaces of the two compoets of a product cotribute equally to the relative variace of the product (cf. Freese 196, p. 17). Idividual tree volume to basal area ratios typically exhibit lower variability tha plot-to-plot basal area per acre values. Plus the samplig error of average VBAR is cotrolled by the umber of trees measured, a umber that is usually larger tha the umber of sample locatios observed. Equatio [-4] clearly poits out the advatage of the double samplig approach for timber stad assessmets: a greater umber of plots are take observig basal area per acre thus reducig the variability i G ; a fewer umber of plots, actually fewer measure trees, are L required to cotrol the variability i V BAR. Equatio [-4] suggests a geeral approach to volume estimatio for double samplig with a ratio-of-meas estimator ad variable-radius plots (or sigle phase samplig with variable-radius plots as well): fid the best estimate of basal area per acre correspodig to the volume of iterest ad multiply by the appropriate volume to basal area ratio. Part Double samplig for timber attributes Page 15

20 Copyright 004 T.E. Burk Estimatio for categories whe extra data are collected o tally plots It was suggested above that istead of observig oly a tree cout o the L plots, tree couts by species or species ad DBH class (ocularly) might be practical. uch iformatio ca be used to improve estimates of volume per acre breakdows. We ca apply the rule gleaed from the previous sectio: fid the best estimate of basal area per acre correspodig to the volume of iterest ad multiply by the appropriate volume to basal area ratio. If tree couts are available by species for all plots, our best estimate of basal area per acre for a species comes from all the plots (the total umber of trees tallied of that species * BAF / L ). That quatity should be multiplied by the species average of VBARs from all measure trees of that species. The same approach could be applied if tree couts were available by tree size (DBH) or product class (as examples). If tree couts are available by species ad DBH for all plots, we ca obtai a good estimate of basal area per acre from all L plots for ay species-dbh combiatio. Multiplyig that quatity by the average of VBARs from all measure trees of that species-dbh combiatio gives the best estimate of volume per acre for that species-dbh combiatio. It should be poited out that the sum of volume per acre estimates across categories obtaied i the suggested maer will ot be equal to the estimate of volume per acre foud usig equatio [-1] uless all the category VBAR averages are the same: a simple matter of algebra. However, the differece will ormally be small. The fact that VBARs typically have low variability is importat here. We may have few VBAR values for a rare species or for ay species-dbh combiatio. This suggests that if strog iterest lies i volume breakdows we may eed to take a larger tha optimal (optimal from the viewpoit of estimatig total volume per acre) measure sample ( plots) to esure we obtai sufficiet VBAR values for estimatig breakdows. The alterative is to borrow VBAR iformatio from similar species or sizes of trees. It should be poited out that VBARs will vary most as a fuctio of tree height if we cosider oly merchatable size, commercially-importat Lake tates tree species. pecies estimates of basal area per acre ca be used to costruct improved stad table estimates as well. For a particular species, usually defiig a colum of a stad table, the ratio G L / G computed for the species should be multiplied by each estimate of trees per acre (obtaied from the measure plots oly). The example below illustrates for a species where basal area per acre was o the measure plots but for all plots. If both species ad DBH class (ocularly) are observed o tally plots, a stad table ca be costructed directly from those data; Part Double samplig for timber attributes Page 16

21 Copyright 004 T.E. Burk this assumes ocular DBH estimates have the same iformatio as measure plot DBH measuremets. DBH tems per acre (uadjusted) Basal area per acre (uadj.) tems per acre (adjusted) Basal area per acre (adjusted) Total Tally sheets A tally form for double samplig with variable-radius plots should track tree tally (possibly by species) for the L plots ad tree measuremet (species, DBH, height, ) o the plots. If samplig error is to be computed, separate sheets are required for each of the plots. Data summary example The Excel workbook TwoPhase.xls summarizes data from a double sample with variable-radius plots timber stad assessmet usig the ratio-of-meas estimator. Data are etered o the Data worksheet; this icludes BAF, L,, umber of tally trees by species across ALL ( L ) plots ad for each measure tree: plot umber, tree umber, species (user chose labels, up to seve for oe cruise), DBH (iteger), ad umber of 8-foot bolts (iteger). Tree volumes i rough cords are computed usig Table 6 of Gevorkiatz ad Olse (1955) (worksheet VolTable). ummaries appear o the Results worksheet ad iclude cordwood volume per acre ad its stadard error ad per acre stock ad stad tables. Istructios for usig the workbook appear o the Results ad Data worksheets. This workbook is ot as automatic as iglephase.xls, as you are required to maually produce the stock ad stad tables that are improved usig the species basal area data assumed to be collected o all plots. Part Double samplig for timber attributes Page 17

22 Copyright 004 T.E. Burk The Big BAF method Motivatio Formulatio [-4], which we have see is geeral for volume per acre estimatio with variable-radius plots, ad its associated relative variace suggest the average VBAR ad basal area per acre estimates that costitute the variable-radius plot volume per acre estimator eed ot be liked. Oe should strive to obtai as good a estimate (low variace) of each as is possible. I double samplig, both average VBAR ad basal area per acre are obtaied from the same set of plots. Trees measured for volume, ad hece those that will cotribute to a average VBAR, are obtaied o a subset of the total umber of plots, ad hece a subset of the total tract area: they are clustered. Number of measure trees drives the samplig error of average VBAR. I double samplig with variable-radius plots the umber of measure trees is ot directly cotrolled but rather dictated by the BAF applied to all plots. It is likely that this umber is sub-optimal ad may forest samplig experts would suggest it is geerally too high: ofte way too high. o, how do we spread the measure trees aroud the tract, ituitively appealig, ad cotrol their umber, statistically appealig? Oe method is to use a secod, possibly much larger, BAF at every sample locatio to select measure trees (recall that the VBAR trees must be selected i proportio to their basal areas). This method was suggested about 15 years ago by Dr. Kim Iles (persoal commuicatio) ad has come to be kow as the Big BAF method. The method is beig successfully applied i the wester Uited tates ad Caada (cf. guest articles by Cori ad Crowther at There seems o reaso why the method would ot work equally well i Lake tates timber stad assessmets. amplig pla The Big BAF method requires specificatio of a umber of sample locatios,, ad two BAFs. The sample locatios will ormally be laid out i a lie-plot cruise fashio presumably usig the same umber of sample lies as a equivalet stadard cruise. At each sample locatio, a tally of trees is made usig a stadard, labeled smaller, BAF (deoted BAF ), ad measure trees, those for which VBAR will be determied, are selected usig a secod, larger BAF (deoted BAF B ). Just as suggested with double samplig, tally trees may have species ad ocular DBHs recorded as well. Volume per acre estimatio A simple average VBAR ca be computed usig the measure trees selected with BAF B. For clarity we will deote that V BAR. The best estimate of basal area per acre is obtaied usig the tallies based o BAF (oly): agai equatio [1-4]. We will deote that G. Trees tallied with BAF B will, of course, be tallied with BAF as well. Takig the product of the two estimators gives the Big BAF estimate of volume per acre Part 3 The Big BAF method Page 18

23 Copyright 004 T.E. Burk V = VBAR G [3-1] Note that, applyig the reverse of the logic used to arrive at equatio [-4], V BAR is equivalet to the ratio of average volume per acre to average basal area per acre usig stadard formulas [1-] ad [1-4] based o the BAF B tallies. Usig E%(X) to deote stadard error of the estimator X as a percet of the estimator X, the suggested estimator for stadard error of [3-1] is where ( V ) E% ( VBAR ) E% ( G ) E = + [3-] E% % ( VBAR ) = T v B T k = 1 k k ( T 1) T v k = 1 B T VBAR k k 100 G j j= 1 G j j= 1 E% ( G ) = 100 ( 1) G ad T is the total umber of measure trees across the sample locatios usig BAF B ad G j is basal acre per acre at the jth sample locatio tallied usig BAF. The stadard error for V BAR assumes the tree VBARs are i.i.d. which will be reasoable for large BAF B s. Formulatio [3-] has bee foud to give reasoable results i practice (Iles, persoal commuicatio), givig coservative samplig error estimates (overestimates) if ideed the two compoets of the product are idepedet. Curret research suggests the idepedece assumptio is reasoable (cf. figures below). BAF VBAR G Part 3 The Big BAF method Page 19

24 Copyright 004 T.E. Burk BAF VBAR G Number of sample locatios, umber of measure trees, ad implied BAF B The total umber of sample locatios,, ad measure trees, T, required to achieve, at miimum cost, a samplig error equivalet to the simple radom sample e ca be foud by applyig the approach of Oderwald ad Joes (199). The derivatio is preseted i Appedix. The resultig formulas are ot ituitive ad ivolve the cost ratio c + c where c c c B m c m B = cost of tallyig a plot with BAF = cost of tallyig a plot with BAF = cost of measurig a tree oce it is determied "i" These costs (times) are best expressed i terms of expected umber of measure trees per plot with BAF, BAF B, ad B k = cost of measurig trees idetified to be "i"at a locatio cost of tallyig trees at a locatio The solutios for ad T are implemeted i worksheet amizep of workbook BigBAF.xls. To use the worksheet requires specifyig e, k, BAF, expected basal area per acre, ad CVV B = CV G CV U = CV VBAR G Typical values for k are { 5} depedig o the tree measuremets take. B was itroduced with double samplig where it was suggested that values ragig from 1 to 1.5 are commo; it is required here that the two CVs i B are determied with Part 3 The Big BAF method Page 0

25 Copyright 004 T.E. Burk the same BAF (BAF ). BAF will be chose usig the 5 to 8 trees per sample locatio guidelie. Computatio of U from early 6000 stad cruises resulted i the followig frequecy distributio Frequecy U More The mea value of U was approximately 0.5, though its distributio is quite skewed to the right. till, early 75% of the values of U were betwee 0. ad 0.7, while almost 90% of the U values were less tha 1.0. ice measure trees are selected with probability proportioal to tree basal area, variability i tree VBAR closely parallels variability i tree height. tads where U exceeds 1.0 are therefore those where the (relative) variability i tree height exceeds the spatial variability i basal area per acre. A iterative solutio is required for ad T with a guess for BAF B used as a startig poit. The Excel add-i olver ca be used. The fial BAF B is optimal for the supplied iputs ad cosistet with the solutio for the optimal, T pair. It is ulikely that you will be able to purchase a prism or agle gauge correspodig to a optimal BAF B. A good solutio is to costruct a rectagular agle gauge whose short side width correspods to BAF ad whose log side width correspods to BAF B, with the istrumet held a comfortable distace from Part 3 The Big BAF method Page 1

26 Copyright 004 T.E. Burk the eye. Equatio [1-9] ca be used to determie the dimesios of the rectagle ad is coded ito worksheet amizep i Excel workbook BigBAF.xls. Exactig costructio of the agle gauge is ot required as the value of BAF B itself is ot used i ay calculatio. ice T is expected total umber of measure trees, it is possible that the actual total umber of measure trees, upo completio of the cruise, will be too low. BAF B ca be decreased somewhat to help esure this does ot happe; results will be suboptimal i terms of miimizig cost. Also see related discussio below. Values for ad T that result i miimum variace for a cost equal to simple radom samplig are also derived i Appedix ad are implemeted i worksheet amizec of workbook BigBAF.xls. The optimal results for umber of tally plots ad measure trees (ad implicitly BAF B ) for Big BAF samplig will seldom be strictly applied. The sample size worksheets i workbook BigBAF.xls should be experimeted with usig coditios of iterest. It is likely that a set of four or five BAF B agle gauges (perhaps 80, 10, 160, 00, ad 40 for Lake tates forestry practice) will adequately cover the rage of circumstaces a cruiser will ecouter. Additioal commets Proper tree tally Whe BAF B is large, as it will be uder circumstaces where the Big BAF method works best, it is importat that all the details of proper istrumet usage are give appropriate attetio. The istrumet must be held at the proper locatio ad i proper orietatio. Questioable trees must be carefully checked. The effort ivolved is lesseed by the fact that questioable trees will be close to the sample locatio. ice BAF B does ot explicitly appear i ay calculatio formula it is oly importat that measure trees are cosistetly selected, regardless of coditios. Equatio [1-1] has bee coded ito worksheet amizep i Excel workbook BigBAF.xls to aid i checkig questioable trees. ample locatios must be placed i a ubiased fashio, iside of trees if ecessary, ad edge coditios must be properly accouted for. It might seem that use of BAF B for measure trees would result i oly large trees beig measured for VBAR. Equatio [1-1] clearly shows this is ot the case. The odds of measurig a tree whose basal area is x-times that of aother tree is x:1, idepedet of BAF. ufficiet umbers of measure trees As metioed with double samplig, if categorizatios (breakdows) of volume per acre of iterest, e.g. volume per acre by species or a stock table, it is ecessary to have a estimate of average VBAR for each category (or to borrow VBAR iformatio). If the expected umber of measure trees, T, is too small to adequately fill all categories, it could be adjusted upwards ad a ew, sub-optimal i terms of expected costs, Part 3 The Big BAF method Page

27 Copyright 004 T.E. Burk computed. Worksheet amizep of workbook BigBAF.xls allows this. Aother suggestio is to measure represetative trees of rare categories as they are ecoutered, though cautio should be exercised to avoid selectio bias. If a particularly rare category is of high importace a separate, smaller, BAF might best be used for selectig measure trees for the category. Estimatio for categorizatios of volume per acre, as well as other attributes of iterest, ca proceed as outlied with double samplig, icludig use of extra data collected o tally plots. Volume per acre for a category is agai the product of basal area per acre for the category multiplied by average VBAR for the category. Volume per acre categories (volume for a species ad product class for example) will typically exhibit lower tree-to-tree variability i VBAR tha what will be foud across categories. However, plot-to-plot variability i basal area per acre will likely be greater for a category tha for all categories combied. This agai emphasizes the importace of tallyig more plots (gettig more observatios of basal area per acre), a outcome the Big BAF method promotes. till, most cruisers would be ueasy if o trees were measured for a particular category of iterest, requirig the borrowig of VBAR iformatio from aother category. Ideally, two to five measure trees per category should be obtaied. The Big BAF advatage The Big BAF method trades off tree measuremet for tree tally. Focusig more of the samplig effort o tree tally at a larger (tha simple radom samplig) umber of sample locatios results i improved basal area per acre estimates, the major source of variability i estimatig volume per acre. If oe oly reduces tree measuremet itesity by selectig measure trees with a larger BAF, the advatage of the Big BAF method will ot be realized. Tally sheets A tally form for Big BAF samplig should track tree tally (possibly by species) for the plots ad tree measuremet (species, DBH, height, ) of trees selected with BAF B. If samplig error is to be computed, separate tree tallies are required for each of the plots. Data summary example The Excel workbook BigBAF.xls summarizes data from a timber stad assessmet usig variable-radius plots ad the Big BAF method. Data are etered o the Data worksheet; this icludes BAF, BAF B,, umber of BAF tally trees by species for each plot, ad for each measure tree: tree umber, species (user chose labels, up to seve for oe cruise), DBH (iteger), ad umber of 8- foot bolts (iteger). Tree volumes i rough cords are computed usig Table 6 of Gevorkiatz ad Olse (1955) (worksheet VolTable). ummaries appear o the Results worksheet ad iclude cordwood volume per acre ad its stadard error ad per acre stock ad stad tables. Istructios for usig the workbook appear o the Results ad Data worksheets. BigBAF.xls is used much like TwoPhase.xls; you are required to maually produce the fial stock ad stad tables. Part 3 The Big BAF method Page 3

28 Copyright 004 T.E. Burk A cruise exercise: Coduct ad review A test stad has bee laid out for participats to gai experiece usig the double samplig ad Big BAF methods discussed above. The istructor has obtaied data from a cruise of the stad usig a BAF istrumet, measurig all tally trees o plots. The results obtaied were Item Descriptio 1. Volume per acre (cords) (V). tadard error of estimate (%) 3. Basal area per acre (G) 4. Average tree VBAR 5. CV V 6. CV G 7. CV VBAR (trees) 8. Number of trees measured 9. Correlatio (G, V) Value Each crew of two idividuals should ru sample lies through the test stad takig plots as follows a. tally i trees at every sample locatio usig a BAF b. measure i trees at every sample locatio usig a BAF c. measure trees that are i with a BAF at every locatio Use the sheets provided by the istructor to record the results. Measurig a tree meas obtaiig its species, DBH, ad umber of 8-foot bolts to a pulpwood top. Be sure to check questioable trees. Also attempt to collect approximate average tally/measure times as follows Item Descriptio 1. Time to tally trees usig BAF. Time to tally (idetify i ) trees usig BAF 3. Time to tally ad measure trees usig BAF Value What should the desig elemets of a double sample be for this stad? What should the desig elemets of a Big BAF sample be for this stad? Part 4 A cruise exercise: Coduct Page 4

29 Copyright 004 T.E. Burk Role of volume tables/equatios Assumig iterest lies i volume per acre estimates, direct measuremets of tree dimesio will eed to be coverted to appropriate volume estimates. A wide variety of volume tables have bee developed for this purpose. Volume tables differ i terms of iputs required, output uits, merchatability stadards, ad the uderlyig fuctioal relatioship betwee iputs ad outputs. Choice of a appropriate volume table is importat as the values obtaied from it are most ofte assumed to be without error from a statistical samplig error calculatio perspective. For timber stad assessmet applicatios i Lake tates forestry practice, the volume tables of Gevorkiatz ad Olse (1955) have foud wide acceptace; for example they are the basis for the popular cumulative volume tally sheet. The tables are ot explicitly species-specific, though adjustmet factors based o species ad stad characteristics are provided for circumstaces that ca justify the extra expese ivolved i their applicatio. Table 6 of Gevorkiatz ad Olse presets estimates of rough cords as a fuctio of DBH ad umber of 8-foot bolts. Table 6 s popularity is evideced by several attempts at formulatig its implied fuctioal relatioship; examples are tott (196) ad toe (as reported by Hah 1984). However, Ek ad Droessler (1988) have poited out that cautio should be exercised i applyig Table 6 as it implies the use of variable top diameters that may be icompatible with preset-day merchatability stadards. till, the taper table (Table 8) i Gevorkiatz ad Olse, upo which all other tables are based, remais valid. The taper table ca be used to geerate a volume table of ay merchatability specificatios. Burk ad Ek (1999) preseted a approach to usig the taper table i automated processig systems ad thoroughly aalyzed the tables i Gevorkiatz ad Olse. Costat form factor volume equatios A costat form factor volume equatio relates some measure of tree volume to tree DBH (D) ad some measure of tree height (H) as follows v = f D H [5-1] The appropriateess of usig a volume equatio of the form [5-1] depeds o the particular volume measure ad particular height measure beig used. If a volume equatio of the form [5-1] is beig used, volume per acre estimatio usig variable-radius plots requires measuremet of height oly. ubstitutig [5-1] ito [1-] gives BAF volume per acre = j j= 1 i= 1 f BAF = t fd t i D j j= 1 i= 1 H H i i i [5-] Table 3 of Gevorkiatz ad Olse (1955) reports total iside bark volume i cubic feet as a fuctio of DBH ad total height. The table (for trees greater tha 30 feet tall) is equivalet to a costat form factor volume equatio with f = If Part 5 Role of volume tables/equatios Page 5