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United States Department of Agriculture
Forest Service
General Technical Re WO-54
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~A Primer on Stand and Forest Inventory Designs
USDA, National Agriouttural Library
NAL Bldg 40301 Baltimore Biva Reksviis, MD 90705-2381
United States Department of Agriculture
Forest Service
General Technical Report WO-54
September 1989
A Primer on Stand and Forest Inventory Designs
H. Gyde Lund and Charles E. Thomas
The authors, both with the USDA Forest Service, are respectively Forester, Timber Management, Washington, DC, and Research Forester, Southern Forest Experiment Station, Institute for Quantitative Studies, New Orleans, LA.
Acknowledgments
The authors extend special thanks to Jim Brickell, Forester, USDA Forest Service (Region 1), Dr. Albert Stage (Inter- mountain Forest and Range Experiment Station), and Dr. Pieter De Vries, Wageningen Agricultural University, the Netherlands, who provided considerable statistical assis- tance in the development of this publication. The authors also gratefully acknowledge the many individuals who reviewed the manuscript and made valuable suggestions.
Contents
Page
WVIRROMUMICEIONI erence cre cere cetera e cee e ooo oa rnc canes cost enbusucesinenteboncabesdee 6 BSTC OME cneenseeietetett - bead cr ee es aie PeP ocr RE eee a a 6 DAGPIDIM Ratertectcsnat eters cnetea stasis teen areeinee canst eta. avesscaranseacteswacwtanecuveveraesezanccctonss 11 COLASSILICALION rr cee eee aE eee e ear eis ee oe ecaoee ri eeabeas 11 REMOLerSelisI Qareessscrstccarstesssarctecssstesestretanssecscnerccecnbentnchetssantaccobepocesseony 11 RESOUICERINVEMLONES recrnct it csec ccs rccereoeeec ce teo cs cree eee e toe cao b a otee eens elivesundeaeeasi 13 RiGee atamCOlCCHOM rescore ere cree cece reseed ahtcs Pi lene caustnniiow 13 ENMUMOLatlOMpeccrrer reese ccoccrccreae scorn neces cee cence ce races eae eee ane eeen tens een nee eeaes 13 SAMI Saerraserersecee eet ce rere ee cecn: sere ks tes totese tener tee eden seer eine bones see enoey 13 BIO cm PUraUlOMercae strc crste- tect occ eros soeere cease en seesenehekeoenceh enaccaereeeceess 15 SAMIDIERSIZemeetcetrcstatecastaarerartectc rev ccueesestcesocgndea seule peeraao tio aas Tons beasaehovatoeen’ 15 OSES eee Ne ee eee et rie eee SEE 16 STAM IMVENTOG Verret ance ee aiace canes s a seebanraies soudsubonsles tedussteaseeesobatecebeeseeatetie 17 PINOLE SUNN) SETITOL UN (cence nan ans anette etree oc irp e are ernst eer Bee arerrie Ht apie eeie ir 19 RaAlGOla DA SciUE OMe ss mere eee re nese aeee bene ne eh es Re aoe baer ne 19 STALISELC AM ESEITIALESTE ee eee cere oa oe eh sear es cates cou eane eae es 19 GOSUESTIIM ALCS ers ecrerer cee eee ee ree renee a ease ehe tes eueootdaeens iteab este eaatnt 20 DIGEST iasscacceaonoucceBeeedseebase Gad ECHEL OE ROR ECE Dac EE Seen C EERE ES eae sere ee racrs 20 EiMeR I rAMSCC EAD ISUMOUTLON es ceeetecscetteeccsetacceiccne eases <neesasecentcansesevevsceevereeton: 20 STATISHICAIMESCITI ALE See rere rere esterase creer eee Ee ae teem 21 (COE SCTE EG OSE REBORN PACS nee 21 DISCUSSION :cacecbsédscodsoenotasconcicHeGucHeC CHORE BEC ROBO CREE O LEC EEE SanCaeas corn tin ECE? 21 RiCOCMetH OL LOCAL OMeestsersseosaes ees ne Pee ne eae nea aooscee So neuanctehoak vecenacceedacsentes 21
SS EALISEUCAIMESUIMI AL Sys oe core ccm nnn sea ce cota ossasacecsctccuedssceccusnuans seveduuestonee 22, GOSTHEStIIMATCS Ite eerrar se cere ene eee cone ds eoa ba ce Uosdeibooersccnbascecueetsee 22 DISCUSSION eeee cee crcae et ere eee naan sds sd bak seniussduusandaceck coteatensoeen D2. Systematic Distribution with a Random Start ............::cesseeeesseeeereeeeeseeeees 22 STAtiSCiGAlREStibnaleSisessee- ates seece eee teeny -cosesesctvcoosseasnevsedossiod netacotensenees 23 GOSISESTIT ALS oreee rene sene teat Scan ehcctoss eds te tens ta leensobeeccecesseredbedscciscetetenssant 23
| DTSOUSST ONT H: caaseuthce ead tice Sonos ec acct Saree eee eee see eee ere bah 23 Sime leplotmecceseee meses wen rere oe cnse nce h a we tSaE Les S02 oe, t Zecdestlacnees suasenuenerss 23 SUSU TISE ECTH DEES ee cee eR Ree 23 GOSTHESTITTI ALS tetera Nee eA SAS Fat ey ae sl Pe 23 DISCUSSION eee error te rae area eee oe rn M La ers J orebed caved deacested soe eaee te 23
NOS ROMMAGL Stenson ae orn Seman weet na ate 05 be rer oy sce RE ees 24 StatiStiGalBEStinmatesne sie cet ae forsee cceoe boss so cso catudocsesandedenesees Memeo et eee 25 GOSISEStINMALSS cme cre erce ee re ce carer ce ee ec sere ces bona teeesecssccdensssthantacaesasteestes 25
PD ISCUISSIOT eer e ee eae e IG 2 Oe en ES ea eta euaesceteeaieen 26 SUBD[ECHVERS Anrapl Inisereeeentssetstas hae sae sync os Saari acted ass 2 Ses onsed eas a saeseesene 26 STATISTICA IMESUITM Ate Smesert eer rests e ec ccac ones coco ss voseaiesceconan caasvadesiocuecteteneeeee 26 GOSHESTINM ALES eeretertrtert a eer corsa ce eens ee acenec roe eetbecesssentcessoeenicdt deen et dete buanes 26 DISGLUISSTION eesasshaadsaestsboosdoce coc cp Ca SE TCBE EEA Heo BARC OSCE SES ee Uae nEe RE eC ee aaa 27 GOMPICTEHEMUNMETAN OM res erecte eer cie recto eka sto teeneactaeuncusasbebaesecccececseatteets 28 StAGti CalmEStnmales perenne mee ns Snes Stee Na lls Sa lusbuchkaee edeoaceoueeeeceees 28 GOSTHESTIITI ALS are rere rene os soccer sade eteccesveccncosencevesasswiarseuacseleevedseotesest 28 DDUSUISSI ON mercer erat cern ec a toe caus ade See aves soe wcusvesssecbaueesvccseeunsuscaseedeustivees 28 SUIT RV AO IM INACTINOUS comers coe sean ete ceat ee acest ew tas ns cae Seai ns eteasltes sue Meassvensess ts 28 STaliSti@d MESHIMALES rete saaeeiec te oo. Seserae none seca ci cavaussttns boot saSvacteuecbasaececeeasses 28 GOSTHESIM ALES erste teeter eee cede e ans ss cocodeseccancsoueleceesiasusnuccaciuecsscaceevsee 28
REV MOGI TOMS es eee creer ere oe non aaa aeas c coc owes ts coaealus Secouuess stehecdusteshaeaeSenceee 29
Forest Inventory: (0.0) ccoe eee ek ies dead adele 1a ease oa ee 30
Subjective: Sampling tite cg eo eos oneal nae 2 eke Cy Rinne de ee nae 31 Inventories Without Prior Stand Mapping ...............csccesccesscesceseceeeceneceeceees, 31 Systematic 'Sampley sce eee ee ces Ee meu ais Riek ieee ete en eae 31 Statistical! Estimatesy..c3). cote. la secrete ek oe OR ME ire ee 32 Mapping and Unmeasured Area Estimates ...............sscccssccceseceeeseeeneeens 33 Cost Estimates) jth eee ene SUA isi ae ee Reet, Ener ee 35 DISCUSSION Hiss :2anse8 eoavesetesteses soseaseauee eaede ueeee ee cet a eee 35 Stratified Samplimg’ ..ciascecet. casssscidestcoyaeesdece acca eee ee toca See 36 Poststratifi ati Omijeccccrs. fissuc rece cessee seer eee eee dene ote ee 3Y/ Systematic Sample ic ..cso.taveacs cooks the ves sacesueeee reece ncereee cs oe ee 37 Statistical Estimates: ois. cess. cats sc ccd en emma ate oo coset Oe 37 Mapping and Unmeasured Area Estimates ................:ccs:cceseseeeeeees 38 Cost/Estimates bei sc5 sieectccetecunssnsscutthscose eee eee esac se 38 DISCUSSION cv. cio, sx smccccacossscesesesaeaseaescoedeter tae twee c Secor eee 38
Strip Grist give cates ceeceelccc ts .oe ca scuscssdoeeseemeeet ete tee es oe uacte re eaten oe 39 Statistical: Estimates: .:'.sc2ssassseestacaelcecsesseeseet re es iee reece ee 39 Mapping and Unmeasured Area Estimates .............sc00cccceeessseeeeeeees 41
Gost ‘Estimates: ssusov.i. sae Si coccee cess ucose canes tieeee See eee 41 DISCUSSION os.ccsecdeecc cotseseeseeceotov reer ee eee 41 Prestratifi cation: 2:30.) asa\c.sscacsassboven -mecscen cee .leesboassuseseneeaeeaee oe eee 42 Stratified: Double-Sampling: feces. sseescee eee ee 43 Statistical, Estimates: 360235 ns 2 een el ae e e 43 Mapping and Unmeasured Area Estimates ..............s::cccceeecseeeeeeeees 45 GOStEStIMATES. j55.55<5.cccsadeescncsbondececetsceee ter eee on eee tere Ce 45 DISCUSSION csecescca ease ces saeco eae eee Eee aoa EL ea 47
Wse of Satellite Imagery scc5 6. hes osieccoscoe eet ee eee 48 StatisticalWEstinmmates ccs. scoscereccccnce scceeocctenseesoeec eee eee 49 Mapping and Unmeasured Area Estimates ...............:ssccceeseereeeeeeee: 52 COStEESH MATES asa oie ase eue ake o ecu SOU OEE e USO U ESE CST SSSR 52 DISCUSSION ges cdhscctcees accesso eee Ones oo oe cea OuUa Re Oa TO STRE SO Nee OSES 53 Inventories With Prior Stand Mapping ...............:.cccsscccessceceeseeceeesceeeeseeerenee: 53 Wnstratified Sam pling. .ccc:ccateeescocescceces cscaveassccocscunescscsensussoncee eeeetecneeees 55 Equal! ProbabilitysSamplinig)(@xpis:)) <ccssc-cosceccescscecccccrstsccesccsscsseecessseenceel 55 Statistical! Estimates... chck cos Ws acces acesaac sarees eae ect Sences oe aee tee eee 55 Stand) EStu ates i scsc6 sccck seco sce aoc eca cause tec snmene teen codecs teas poate eeenCeneee 57 GCOSHIESH Mates: cicoskocesisaceess ches arseen reese eee ee ene eae REE eee 58 DISCUSSION ics oc cseisseibexecniev aoe ewe woe Saecia nec enaN SCR RO OSS E Re Ee Su aT RESTS ONE rece CEES 59 Probability Proportional to Size (p.p.s.) Sampling .............::cceeeeeeeeeeeee: 59 StatisticalWEStimates) <cccc. coscccs.coetessaces sesseseuseceeuecnoswesct cece ceceee a cueeneneee 60 Stama! Estimates iecccicccccscccse sos teas ecco nc cu tac eaves nace ateeneeeeen Cot nec teeeeees 61 Gost: Estinmatesicscc..ccccscssecscuacoewonetcccaws sc cuces suoctneteecscosbecenioe sNacecuewameccue 61 DISCUSSION: oc0oscn ccs ccsowewes socnoeduwsatens sua ceeweesnces Sonne on saceausseeenemce rece etcenee nce 61 Stratified, Samnplimpaceoscscsccc onto tikoiehs co csscososatectecccccteces tow meee ceeere menccesece 61 Poststratifi GathOMs vcic.5: scoses sssvccsuseseeecamectenec os ce conaccwemeeeee enero ereenrmer ce 62 Statistical) Estinmatesy:ccaieccccscicsoicc cctes sre careneccwcusceacemenceee ser encueaceccenens 63 Stand VEStimMatesy oocs ccs cs sec ecos co oece fe ware te sansa ee ee Oe Re 65 COST IEStIEM ALES) ca 525 cece cae Sona cere Cerner aac cee eeenee eae cneeeereee 65 PISCUSSIOMN: arscec cesses oseae aes eee eee o at SSN aa Se Sac cUanae eae eere ocee ew etenees 65 PrestratifiGattn).cieccccaccacse sess ceases ovscsacseceas concanenanteuseceeens caeeasseeenceeeeeneemeee 65
Statistical GEStiinateSrasss eerie erect sees ee Le 66
STAMCUEStI ALCS Mens see ee tcotree eter e cc steve atenaek ccaneate seaese ste badd sescaets 68 GOSMESLIMIALGS ss ceecece reece tear eeeee eee esta ictads haste Sie aude senaneeues 69 PVISCUSSIOMN -sietcctsccsssere rors s state nese icte tocconcees aster ttesvecccaddbaeedey cca boneteatt 69 Probability Proportional to Size (p.p.s.) Sampling ..............0::ccccceeeees 70 SLAUISCICAIMESEIMALES IO. Me. cc cctactseo ees e ati ne teosiuteeccealusscrsos sete tbereaane 70 STAMCRESUIIMALCS 0: Socvss ste ssdesiceovoss cotta tartrate one eee Sore 72 EOSTHESEIINIALCS peer ica sso cs caveccoss cetaceans sauce decccadesseluprveseesoveunes dacsvacvesewes 72 PVISCUSSIOMeseresdeosscissscscecee cesses eee a craccs tdvaseaieeleatesectae Mae nootbbaemeeees 73 Inventories Using Existing Stand Information .............:ccccccesccceessseeeeessceeeeees 73 WsevontherstandrasalSamplinerlWMitcccs..:ces0-cste-.scceeceiceseccesseeossecctaeeovsess 74 Gombiming INVvVentoniesiti ss: Lack tees. te ooo ete etree eu sovoteonenees 74 StatisticalsEStimatesieciscess ai sccceot eee odeten cca tecoe see tee aceteate ites dasdetatiads ID Stamaddestimatesterrcrn rst rcrsncct cece tseceree oe ccseeessesestateieene eae eaten te 78 DISCUSSION prestee eee eines bosec echo toseeueccncane docnevedseccavbutsubesdinecetinaversicoecoureoeites 78 GOMMPBIECENEMUNME PATON eset ee veses nese cvsseotes cdectsstecsesucesssesscussccenosesovscvoceesossees 78 StatistiGalHEStinidtesencseesstses sos. teres A cuskeesececteceoteeen sett deve Sees ae 79 GOSTHESTIITALCSgserrcree ee eae coat tea cece ence tees cata c ane betect ate aeeanied asseratneee 80 SUIMIIMAanyVAO ts FOFEStINVEMtONiGSis..ceeccssesccecessccecestescceceeceosoccecaseeeteduoceee sosneences 80 StatisticalpEstimatonsteetss sec coos cote as eee ee 80 FOTESIAESUIIM ALS ie taco rare once cece che ceec cr eeetevee ok nue sessecaseiigsdecsbvsscsuctasianes 80 INGIVidlaleStanaPEstimmatesteecerss:s tet. secechectececnssessacesctesseeeoaees eee poeeeeeee 81 GOSHESTIMMALES reser rec sete core cece ce cca atencddecaatebestotuattemeevedl panes 82 REVitOy CMI ONS pecs cows tee scee Aer tewaesasssctestetecevesestacenslssenessteccccncarenseronres 83 EOMIGIISIGIS eee ssh eee ere ee Need acaaddl uucalee coteeveh sceetoetes 84 RElErenCes: ClteG a, Fare See renee soe eee eae area eeceissdbconcacseoseausvantesdice Devotee 85 Additional Selected References .................cccccssccceesssceceeesseeeeeessceceessaceeensaeees 88 Mapping, Classification, Remote Sensing .............::ccccsssecceeessseeceeseeeeesseeees 88 StatiStiGstanGSaMmplingmr nies tre erckce eee catehae owcaksvedesosorscceaterteccsencconctes 88 PlOt Com PUrainie crcteetcs serene sic clec casera ten coc cecpestecseteetoroae cevactesbee nice 88 FOTESUIMVEMNOMV acces cesteccters estate raocaceneet cece ceo saceneetteccencoctdos orscstesdostsdettecsees 88 SPECAlMIMVENMCONI@S eer ere ee reese crc cteeetnocee te esse soeeoeccecna tbe etes theses 88 INMMONTONM B@rsaces setae een orcanoce cs eaeee tere ceraseoabactetcck oes nescocsscuececsoteeeettntceseuatieeaee 89 Appendix 1: Equations and Formulas Used in Text ............:ccccccccseesseeeeenteeeees 90 StatiSticalMEstinnatonsreses easter cress eeceneae ene ee ease eet ete: 90 GOStHESCIIMALONS eeterrrrree ce etote ceo te cect hentyececectocaschoootevsotsesctenvsccuceasastrestsstere 91 Appendix 2: Stand Characteristics of the Enchanted Forest .............:::::eceeeee 93
Appendix 3: Stand Estimates by Various Inventory Designs .............::eceeeeee 95
Introduction
Land managers need to know the location, the extent, quantity, and condition of the natural resources that they manage and how those resources are changing over time. Stand inventories provide this kind of information for areas generally 100 acres or less as a prelude to treatment. Forest inventories provide similar information over large areas for resource planning purposes.
A recent analysis of inventory expenditures in the USDA Forest Service National Forest System (Lund 1987) showed that up to 76 percent of the total costs of doing an inventory may be spent in data collection. Such costs are a function of the inventory design. Today’s resource man- agers need statistically valid, cost-effective, and defensible inventories (Laux et al. 1984).
Stand and forest inventories consist of at least three phases: mapping, sampling, and analysis. Mapping alone cannot provide the information usually required by the decision maker. Sampling in the field is also needed.
Mapping shows the locations of the resources and their extent and may be done before sampling, during the course of sampling, or after sampling. Sampling is used to obtain detailed data about part of the inventory unit. Analysis implies the calculation of estimates of certain parameters, variance, and confidence intervals. Analysis of the data is also used to make decisions regarding the management of the inventory unit.
There are many excellent references available covering forest inventories. The material in Avery and Burkhart (1983), in Section 7 of the Society of American Foresters’ Forestry Handbook (Wenger 1984), and in De Vries (1986) are among the most recent and the best. In addition, a recent contribution to computer simulation of inventory design may yield useful insights into application of various designs for given forest conditions (Arvenitis and Reich 1988). This report concentrates on commonly used and available options for doing stand and forest inventories. The major portion of the report is devoted to examining the more common sampling designs available to the Federal land management agencies and their costs.
The objective of this report is to provide resource manag- ers and beginning inventory designers with an understand- ing of the range of options available and the costs to:
e Sample within mapped entities.
e Use mapped polygons as sampling units.
e Generate maps and inventory statistics from sample data.
Sampling within individual mapped stands and among mapped stands, and sampling and creating spatial infor- mation where no stand maps exist are discussed.
Example inventories of a mythical Enchanted Forest using many of the designs in use by the Forest Service are presented. Detailed illustrations and step-by-step instruc- tions for each design are given to help beginning inventory specialists understand how the designs are implemented and how the statistical estimators are generated.
English units of measure are used in this text unless otherwise specified. Metric conversions are as follows:
1 inch = 25.4 millimeters (mm).
1 inch = 0.0254 meter (m). 1 foot = 0.3048 meter (m). 1 mile = 1.6093 kilometers (km).
1 acre = 0.4047 hectare (ha).
100 cubic feet (1 ccf) = 2.83 cubic meters (m*).
100 cubic feet per acre (1 ccf/acre) = 7.00 cubic meters per hectare.
This report incorporates much recent literature and sum- marizes many of the techniques being used by the USDA Forest Service National Forest System (NFS) Regions and by Research Forest Inventory and Analysis (FIA) Units (fig. 1). The proceedings from the In-place Resource Invento- ries Workshop (Brann, House, and Lund 1982), and the Forest Land Inventory Workshop (Lund 1984) were the principal documents reviewed. Even though timber situa- tions are discussed and illustrated, the options presented are equally useful for the inventory of other vegetative resources such as wildlife habitat or range production.
Definitions The following definitions will be helpful in using this report:
Accuracy: The closeness of a set of observations to the quantity intended to be observed (Kendall and Buckland 1971). The degree of accuracy is calculated by statistical inference.
Allowable error: Also called the allowable sampling error or tolerance specification. The largest acceptable size of the standard error of the estimate usually specified before a sample is drawn to determine sample size.
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Attributes (properties): The differentiating characteristics that must be discovered, measured, described, delineated, or derived to fulfill the objectives of the inventory (Valen- tine 1984). Species, height, and diameter are examples of tree attributes that are usually measured or observed. Volume per acre, basal area per acre, forest type, and stand size are stand attributes that are usually derived.
Bias: A systematic difference between a statistical result and a parameter that is being estimated. (Kendall and Buckiand 1971).
Classification: The process of describing categories for mapped or sampled objects (such as species, organisms, stands, sites, landforms, pedons, structures, or geographi- cal units). The categories may be based on natural affini- ties of the objects for one another with respect to charac- ters of interest, such as potentials, productivities, or inherent qualities or structures. Often the inherent quality or discrete structure of an object describes the category. The grouping of stands into forest types or trees into size classes is an example of a classification based on contin- uous quantitative characters. These two types of classifi- cations are not strictly the same, and care must be taken in constructing the classes in the latter case.
Cluster: A sampling unit comprising two or more elements or subunits (Canadian Forest Inventory Committee 1978). The subunits are observed as part of the single primary sampling unit (Scott 1982).
Coefficient of variation: The ratio of the standard deviation to the mean.
Confidence interval: The range of values within which one might expect to find the parameter with some degree of assurance.
Cost effective: Achieving specified objectives under given conditions for the least cost.
Delineators: Attributes used to locate or define an inven- tory unit or a stand’s boundary on a map or aerial photo, usually based on vegetative changes or topographic fea- tures (Mehl 1984).
Double sampling: See multistage sampling.
Estimate: The particular value yielded by an estimator in a given set of circumstances (Kendall and Buckland 1971).
Estimator: The rule or method of estimating a constant of a parent population. It is generally expressed as a function of sample values (Kendall and Buckland 1971).
Extrapolate: To estimate the value of a variable or descrip- tor outside its tabulated or observed range. To infer an unknown from something that is known.
Forest compartment: A basic territorial unit of a forest permanently defined for purposes of location, description, and record, and a basis for management (Ford-Robertson 1971). A compartment consists of a grouping of forest stands.
Forest inventory: A generally periodic survey covering all the forested land base used to support land and resource management planning and implementation.
Forest stand: A community of forest vegetation possessing sufficient uniformity in regard to composition, constitu- tion, age, spatial arrangement, or condition, to be distin- guishable from adjacent communities, so forming a man- agement entity (Ford-Robertson 1971).
Identifier (label): A code, symbol, letter, or number that links a mapped delineation to a legend, text, or data base.
Integrated inventory: An inventory or group of inventories designed to meet multilocation, multidecision level, mul- tiresource, or monitoring needs (Lund 1986a).
Interpolate: To insert, estimate, or find an intermediate term in a sequence or matrix.
Inventory: To account quantitatively for goods on hand or provide a descriptive list of articles giving, at a minimum, the quantity or quality of each, such as the number of trees in a stand or volume of timber within a forest.
Inventory (survey) unit: The land unit containing the population for which information will be summarized and analyzed. The unit may consist of any area of land such as grazing allotments; compartments; watersheds; 40-acre parcels; stands; National, State, or private forests; coun- ties; States; or even nations.
Isoline: A line representing equality with respect to a given single variable—used to relate points on a map such as elevation contours, precipitation amounts, and tempera- ture regimes.
Mapped stand (site delineation): An area delineated on a map or imagery containing at least one symbol (color, alpha, or numeric) and bounded by a continuous line (Valentine 1984). The delineation represents an area of land possessing some degree of internal homogeneity of attributes with respect to characteristics defined by a particular system. The mapped stand is identified by a set of delineators and is described by a set of descriptors.
Mapping: The identification of selected features, the de- termination of their boundaries, and the delineation of those boundaries on a suitable base using predefined criteria (Shiflet and Snyder 1982).
Measuring: Ascertaining the extent, characteristics, dimen- sion, or quantity of a population element such as the number of trees within a stand or compartment.
Monitoring: The process of observing and measuring over a period of time to detect change or to predict trends.
Multiphase sampling: A design in which some informa- tion collected from all of the units of a sample and additional and usually more detailed information is gath- ered from a subsample of the units constituting the original sample. Generally the sampling frame consists of a list of sampling units that remain the same size and at the same location in each phase of the sample. A sample that selects a 1-acre plot on an aerial photograph from a list of such plots and that also selects the same colocated, 1-acre field plot is an example of multiphase sampling.
Multistage sampling: A design in which the sampling frame consists of a list of sampling units (primary sampling units) that in turn are made up of smaller units (secondary sampling units) that in turn may again be made up of smaller sampling units (Nichols 1979). An inventory that selects stands from a forest or compartment, 1-acre plots within the sample stands, and 0.1-acre subplots within the sample plots is an example of a three-stage sampling design.
Parameter: A variable entering into the mathematical form of any distribution such that the possible values of the variable describe or yield different distributions.
Permanent plot: A sampling unit established and docu- mented so as to permit repeated measurements of the same variables at the same exact places but at different times.
Pixel: Contraction for picture element. The smallest, most elementary areal unit considered by an investigator in digital image (also called a resolution cell). Pixels may be represented digitally by shades of gray, colors, or alphanu- meric characters and are comparable to one of the many dots making up the picture on a television screen.
Plot configuration: The size and shape of the sampling unit (plot) and the spatial arrangement of subplots within that unit in the case of a cluster of plots.
Population: Any finite or infinite collection of individuals such as trees in a stand or stands within a forest (Cochran 1977). A sample frame must relate to its population.
Precision: A measure of the way in which repeated observations conform to themselves. In general the preci- sion of an estimator varies with the square root of the number of observations upon which it is based (Kendall and Buckland 1971). It is a reflection of the sample size and the care taken and techniques used when measuring inventory attributes.
Primary sampling unit (psu): The sampling units chosen in the first stage of a multistage sampling design.
Probability limits (levels): Upper and lower limits assigned to an estimated value for the purpose of indicating the range within which the true value is supposed to lie according to some statement of probabilistic character (Kendall and Buckland 1971).
Probability (random) sampling: Any method of selection of a sample based on the theory of probability (degree of belief); at any stage of the operation of selection the probability of any set of units being selected must be known. It is the only general method known that can provide a measure of precision of the estimate (Kendall and Buckland 1971).
Resource inventory: The collection of data for description and analysis of the status, quantity, quality, or productivity of a resource. Such inventories usually include some descriptive data, numeric data, and at times, maps show- ing the extent of the inventory unit and location of sample units.
Sample plot: A sampling unit or element of known area and shape such as a one-acre circular plot (Canadian Forest Inventory Committee 1978).
Sample size: The number of sampling units that are to be included in the sample. In the case of a multistage sample, this number refers to the number of units in the final stage of the sampling (Kendall and Buckland 1971).
Sampling: The act or process of selecting a subset from a population (a stand from all stands, or a plot from all possible plots) for estimating, analyzing, classifying, or characterizing.
Sampling (inventory) design: The specification of a con- figuration of sampling units and the method used to determine which sampling units will be measured, such as systematic sampling, stratified sampling, and multistage sampling.
Sampling error: That part of the difference between a population value and an estimate thereof, derived from a random sample, which is due to the fact that only a sample of values is observed. The totality of sample estimates in all possible samples of the same size gener- ates the sampling distribution of the statistic which is being used to estimate the parent value (Kendall and Buckland 1971).
Sampling frame: (See also population). The complete aggregate or list of sampling units from which the samples will be drawn, such as all possible plots within a stand or a listing of all stands within a compartment or forest.
Sampling intensity: The number of sampling units estab- lished per unit area (e.g., 1 plot per 3,000 acres) or the percentage of the population sampled.
Sampling unit (plot): One of the units into which an aggregate is divided or regarded as divided for the pur- poses of sampling, each unit being regarded as individual and indivisible when the selection is made (Kendall and Buckland 1971). A specified part (such as a fixed-area plot, or mapped stand unit) of the inventory unit.
Sampling with replacement: The return of a sampling unit, drawn from a finite population, to that population after its characteristics have been recorded and before the next unit is drawn (Kendall and Buckland 1971).
Sampling without replacement: The failure to return a
sampling unit, drawn from a finite population, to that population.
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Secondary sampling unit: The sampling unit chosen dur- ing the second stage or step of a multistage sampling design.
Stand descriptors: Attributes that provide information about the stand but do not determine a stand’s boundary (Mehl 1984).
Stand (silvicultural) examination (stand inventory): The collecting of data within a given stand to determine treatment needs or, if after treatment, to verify results.
Standard deviation: The measure of dispersion of a fre- quency distribution equal to the positive square root of the variance (Kendall and Buckland 1971).
Standard error: The positive square root of the variance of the sampling distribution of a statistic (Kendall and Buck- land 1971).
Statistically valid design: A design that permits inferences based on logical analysis, the premises, and the data to a well defined population. An inventory design that pro- vides samples that permit the calculation of estimates of population parameters and of their respective variances and standard errors.
Stratification: The division of an inventory unit into more homogeneous sub-units to improve the efficiency of the inventory.
Stratified sample: A sample selected from a population which has been divided into parts. Stratification is done by dividing the general population into homogeneous sub- populations so that more sampling effort can be put into heterogeneous strata or in strata of more interest than others, or to reduce the error by minimizing variation within. Stratification may be undertaken on a geographical basis by dividing the survey area into subareas on a map or through interpretation and classification of points from remote sensing imagery.
Stratum: Any division of the population for which a separate estimate is desired (Kendall and Buckland 1971). Plural of stratum is strata.
Systematic sample: A sample obtained by making obser- vations at equally spaced intervals, such as taking a sample every 100 feet along a transect line.
Timber cruise: A survey to estimate the quantity and value of timber on a given area according to species, size, quality, and possible products prior to the harvesting of the timber or before land is exchanged or acquired.
Update: A method used to make current inventory esti- mates by manipulation of the inventory data base through accounting procedures, projection models, or by adjust- ment of a base inventory by subsampled data.
Variance: The measure of dispersion of individual unit values about their mean (Freese 1962).
Mapping
Mapping is used to define and show: inventory unit and stand boundaries; location of resource; and the results of an inventory keyed to a legend or data base. Forest stands are usually mapped according to forest type, average size of trees (stand size), volume classes, and density of the overstory.
Maps and inventories are used to answer two questions: 1) where are stands having certain attributes located? and 2) what are the attributes at a given location? \n the first case, the attributes sought are known, but the locations are unknown. In the second case, the locations are known, but attributes are unknown. Both situations are considered in this report.
Classification—Two classification approaches may be used to develop the maps to answer the questions (Valen- tine 1984). A unifying system such as Daubenmire’s (1968) habitat types is useful for answering the first question. Most of the information is displayed on the map or in the legend. Such a system requires multiple catego- ries and numerous classes. All classes have to be fully defined. The classification process is an integral part of the mapping program. Some information may be lost when generalized into broader classes.
Descriptive systems are often applied through multilayer- ing of diverse maps and are better at answering the second question. A number of single-attribute classifications may be used to describe a piece of land.
In a descriptive system all information is attached to a polygon, pixel, or acre through an identifier or label. The identifier provides access to a data base. This system is simple, fast, and direct but is difficult to use to compare one area with another without rigorous analysis. Table 1 shows the kinds of timber-related data than can be stored
Table 1—Fxamples of stand data stored in a descriptive classification system (adapted from Mehl 1984)
Location attributes
Vegetation attributes
Other attributes
Stand number Ecological type Elevation
Region Range type and Aspect
State condition Slope percent
County Forest type Landform class
Administrative Forest Percent crown cover Mineral status
Proclaimed Forest Stand structure Soil series
Management area Stand age Rock, litter, duff number Average tree height percent
Analysis area Average tree d.b.h. Fuel load number Basal area per acre Dominant wildlife
Capability area Number of trees per species number acre Recreation
Land use class Net cubic foot opportunity
Land cover class volume spectrum
Year of last inventory Dominant shrub Average annual
Source of inventory species precipitation
Dominant grass Site index species Planned treatments
using a descriptive system. The mapping of such detailed information through a unifying system is difficult.
Maps should be constructed using uniform standards and conventions including: the specification of features to be displayed; minimum size of area to be delineated; and the handling of special situations such as stringers of vegeta- tion along riparian zones, road rights-of-way, etc.
Remote Sensing—Mapping may be done in the field, from aerial photos or other types of imagery, or automatically by satellite reconnaissance systems. If done in the field, the inventory or collection of data within the typed polygons may be done at the same time as the delineation of boundaries. A stratified sampling system is often used to collect the field data, after using aerial photography or satellite reconnaissance imagery for land classification.
For forest management, map units are usually based on delineations of potential vegetation, current vegetation on phoiographs, or by aggregating pixels having common band characteristics. Inventory data may be stored by these three kinds of mapping units. Table 2 presents advantages and disadvantages of each. The mapping of potential vegetation, such as habitat type, requires considerable field expertise and effort. Mapping of current vegetation, such as timber stand mapping, is usually done by interpreting aerial pho- tography with some field checking.
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Table 2—FEvaluation of possible common data collection and map storage units
Common storage units
Strengths
Ecological (habitat) type Stable. Reflect long term opportunities, neutral units
Weaknesses
High skill level required for delineation. Criteria variables
map units (potential vege- tation).
for competing resources. Visible to land manager. Allow variable sampling intensity.
Existing vegetation. Widely used, well understood, large, flexible data base system. Focus on primary resource for managment activity. Easily recognizable on the ground.
Cells, pixels. Good sampling design possible by using groups of like pixels. Spatial display. Minimal inter-resource coordi- nation needed. Stable over time. Automated.
may be subjective. Difficult to automate. Lack complete coverage. Require strong inter-resource coordination.
Criteria for delineation change geographically over time depending on many variables. Excessive focus on com- modity resources.
Require geographical information system (GIS). Groups of cells may not reflect resource pattern. Synthesis may not be recognizable on-the-ground. Pixels provide graphic displays of areas that have been digitized for computer storage, which may include such mixed enti- ties as stands and ecological types or any other defined
areas.
Generally, aerial photography at a scale of 1:20,000 or larger is suitable for the detection of most variables of interest for forestry purposes. These variables include stand size, tree heights, crown diameter, and number of trees. Measurements of these same variables from aerial photography generally requires imagery twice as large (i.e., 1:10,000 or larger) (Lund 1986a).
Cover type and overstory closure can be detected and measured from scales as small as 1:125,000. Satellite imagery has been used to measure these two items. Digital information, such as that from satellite reconnaissance, also maps the current situation.
Satellite imagery provides wall-to-wall coverage with res- olution available down to pixel size (1.1 acres for Landsat). For large areas, however, coverage of a selected forest type region may require several satellite scenes. Pixels may overlap traditional stand boundaries and often must be grouped to give map displays similar to those obtained through photo interpretation. Satellite imagery and to some extent aerial photography may be interpreted auto- matically as well as visually.
Table 3 (adapted from Hoffer 1982) lists some consider- ations to determine if computer classification or visual interpretations should be used for mapping. Lachowski (1984) provides guidance on selecting the appropriate kind and scale of aerial photography to use. Regardless of the type of mapping performed, consideration should be given to entering the data into a geographic information system (GIS). Hanson (1979) provides guidance on prep- aration of maps for manual digitizing.
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Table 3—Considerations in determining computer-assisted or manual mapping procedures
Computer classification’ po- tentially suitable if:
Geographic area of interest is very large (state, country).
Informational categories of in- terest are spectrally separable.
Spectral characteristics of data are relatively simple (spectral classes are reasonably homo- geneous and relatively few in number).
Spatial relationships are not required to achieve identifica- tion.
Manual interpretation? or field ex- amination probably more suitable if:
Geographic area is relatively small (town, county).
Spectral characteristics of data are complex and difficult to characterize.
Requirement for detailed spatial infor- mation exceeds capabilities of multi- spectral scanning systems, thereby in- dicating a need for geographical data and manual interpretation.
Required information is contained in spatial characteristics of the data (lin- eaments, circular features, etc.).
Convergence of evidence principle or correlation analysis is required to identify features of interest.
1 — Computer classification of multispectral scanner data, probably
from spacecraft altitudes.
2 — Manual interpretation might involve aerial photography, multispec- tral scanner data (individual, combined wavelength bands, or
enhanced), or radar data.
Resource Inventories—|nventories can be conducted with and without stand mapping. Inventories without stand mapping are usually a prelude to further data collection and in the past were called Stage | inventories in the Forest Service. Follow-up inventories using stand mapping were called Stage II (USDA Forest Service 1962). Common data collection forms and reports were and still are used to tie
the inventories together (Costello and Lund 1978).
Because stand boundaries change over time, the use of stands for sampling units has been avoided in the past. Grosenbaugh (1955) recommended that instead of stands, operating areas with meaningful permanent boundaries and convenient size should be established as the mini- mum unit for forest management records.
Changing technology, particularly in the form of geo- graphic information systems (GIS), now allows inventory specialists to create maps based on sample data and to use stand maps more effectively for resource inventories. Over time, spatial data and descriptive data in a GIS can be updated for use in continuous management programs (Langley 1983). Because of this emerging technology, more and more Forest Service Regions are combining the sampling efficiencies of the Stage | inventory with the mapping of the Stage II inventories. Similarly, some FIA units are exploring ways of using their existing inventories to provide mapped information across the entire survey unit.
Various techniques are shown in this report to illustrate how sample data can be spatially extrapolated to the whole inventory unit.
Field Data Collection Any inventory requires either a complete enumeration or a sample of the population of interest.
Enumeration—Complete enumeration requires visiting and observing all individuals or data items in a popula- tion, such as measuring each and every tree in a stand or all stands within a compartment or forest.
Complete enumeration of trees within a stand is generally restricted to areas of very high value. Complete enumera- tion of stands within a compartment or a forest is Common in areas that have been under intensive management for quite some time such as the Southern (Belcher 1984) and Eastern Regions (Johnson 1984). In these two cases, even though all stands are visited, sampling within the stands is used to obtain estimates.
Questions to be asked when considering complete enu- meration include (Cunia 1982); (1) is the real objective of the survey such that it would require the examination of each individual of the population? (2) must the results of the survey be presented separately for each individual of the population or is their presentation as averages or totals sufficiently satisfactory? (3) will complete enumeration provide unbiased results or will there be falloff due to measurement errors and lower quality of work that may occur when large numbers of individuals have to be measured?
Complete enumeration is costly and time consuming. It is used only when a small population is involved or when the population has a very high value. Surprisingly, the results from a sample may be more accurate than the results from complete enumeration. Measurements of many characteristics may not be error-free and the collec- tion of data on a large number of objects may be subject to measurement errors.
Because complete enumeration requires considerable time, different people may have to be used, some of who may have less training or dedication than others. One person having to make the same measurements over and over again tends to become negligent in quality control. Units may be missed and measurements made haphaz- ardly. The errors tend to accumulate. Because fewer observations are required in sampling, quality control is easier to maintain and the measurement errors decrease (Cunia 1982).
Sampling—When one cannot afford a census of a popu- lation, a subset of the population is sampled (some of the trees in a stand, some plots within a stand, or some of the stands in a compartment or forest are measured). The sampling units (trees, plots, or stands) are measured for the characteristics of interest and the resulting measurements are analyzed. The conclusions drawn are representative of and applicable to the entire population and are based on inferential logic.
Sampling may take on one of the two forms: subjective sampling or probability sampling. Subjective sampling is also referred to as purposive, nonprobability, or judgement sampling.
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Subjective sampling may be accomplished with or without preconceived bias (Mueller-Dombois and _ Ellenberg 1974). Preconceived bias indicates the investigator con- sciously avoids certain nonconformities or deviations in vegetation cover. The Southern Region (Belcher 1984) selects sample plots that appear to have average stand conditions.
“Without preconceived bias” indicates that the investiga- tor measures stand conditions that attempt to include the extremes of the population of interest. Foresters have employed this technique as a sort of stratification. It is often used to construct tree volume tables to ensure a wide range of diameter classes are measured. In a forest inven- tory, stands representing the extreme conditions are visited rather than areas representing average condition of the compartment or forest. Norton et al. (1982) used this technique to inventory riparian vegetation.
Subjective sampling is justified when (Cunia 1982):
e The variation between the elements of a population is very large and the sampling is extremely expensive.
e The needs for information about some population of interest are immediate and decisions are not sensitive to biases in estimates.
e The available funds to do a complete enumeration or statistical sample are very low.
e The subjective sample is to be used for planning a Statistical sample.
e Estimates of precision are not needed.
e The results are not likely to be contested in a court of law.
Subjective samples should be limited to a single purpose and for a short term.
For resource inventories serving multiple decisions over long periods of time, statistically valid samples are re- quired. Therefore, probability sampling is preferred for most inventories. The reliability of the sample estimators can be calculated and expressed in probability terms when statistical sampling is used, whereas the reliability of subjective sampling cannot be determined.
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It will often be useful to distinguish two levels of numerical estimates from sample surveys, those used in computation and final reported values. Computations often carry many additional digits reflecting the accuracy of the computer procedures used to obtain the estimates. Final reported figures tabulate the estimate, rounded to a reasonable number of digits, which should be based on the accuracy of the sampled data. A secondary consideration is that the reported figures should have a consistent number of decimal places and/or be nearly additive. In this report we often carry additional computational digits in the text. Figures reported in the tables are rounded to two or three decimal places or truncated to the last whole number. Rounding will follow the rule that the digit prior to the level of rounding is considered the last reliable digit. Ignore succeeding digits and round up if the rounding digit is greater than 5; round down if it is less than 5. If the rounding digit is exactly 5, round to the nearest even digit; thus 2.25 = 2.2, and 2.15 = 2.2.
Statistically valid data can often be used for purposes other than those originally intended and at different points in time. Even if resource management objectives change, statistically valid inventories retain much of their value. This is usually not the case with subjective inventories.
The basic assumptions for probability sampling are that the sampling procedure has been clearly defined in simple terms and if repeatedly applied to the same population, the following conditions are satisfied (Cunia 1982): (1) the set of all possible samples that may arise, as well as the particular individuals that enter in each of these samples, are known or can be known; (2) the population of individuals is completely covered by this set of samples, that is, each individual of the population must belong to at least one of these samples; (3) each sample in this set has a probability of occurrence that is known and nonzero; and (4) unique estimates of the population parameters of interest can be calculated from the data of each sample.
Note that it is not usually necessary to explicitly write down the set of all possible samples and the associated set of probabilities; it may suffice to know that it can theoret- ically be done.
In the sampling process, data are collected from a sample of the population and the results expanded to the inven- tory unit. The sampling process may vary from visiting all trees in the inventory unit and making measurements on a few to visiting only a portion of the inventory unit, establishing sample plots, and measuring some of the trees on those plots.
In 3P sampling (i.e., sampling with probability propor- tional to a predicted value), all trees in a stand are visited and only that portion is measured that is selected with probability proportional to predicted volumes. Wiant (1976) gives an excellent discussion of the use of 3P sampling.
Sample tree selection methods include the use of such devices as colored or numbered marbles, dice, or random number tables. When visiting every individual in a popu- lation is not desirable, data are collected from sample plots.
Plot Configuration—Sizes and shapes of field plots are commonly determined on the basis of custom, tradition, and experience. The most efficient configuration is one which has the smallest size in relation to the variability produced (Avery 1975). Simple plots having a fixed size are usually rectangular or circular in shape. Rectangular or strip plots are often oriented to cross maximum variation, to reduce between plot variance, and to increase within plot variation.
Myers and Shelton (1980) consider practicality in locating plot boundaries and taking measurements, edge bias, and the balance of effort between measuring a few large plots or many small plots as the primary considerations for choosing a plot design. Fixed-area plots are particularly useful for measuring change.
Combined plots (a grouping of subplots) are often used for sampling different types of vegetation in the same inven- tory. For example, a 1/10-acre plot may be used to tally sawtimber trees and a 1/300-acre plot established at the same plot center may be used to count seedlings. These two plots are often described as nested, concentric, or collocated.
A variable radius (Bitterlich or point sample) is commonly used in timber inventories. This is an example of a combined plot composed of nested circular plots with the plot radius of each being a constant multiple of tree diameter (Bitterlich 1959).
A cluster of plots is also a form of a combined plot in which the area of the sampling unit is spread out in some fixed geometric pattern around an initial plot reference point. The 10-point cluster commonly used in the USDA Forest Service is an example.
Strip transects (a series of long, narrow rectangular plots placed end to end), are also examples of combined plots common in vegetation inventories. Strips are more likely to sample variation in nonuniform vegetation. The disad- vantage of strip transects is the large amount of edge in relation to area, which introduces a strong possibility of edge bias (Myers and Shelton 1980).
Line plot sampling (Loetch and Haller 1964) may also be considered a form of combined plots. A line transect defines a common centerline where sample plots are spaced at regular intervals along the line.
In general, when using combined plots, the sub- components must be placed in the same geometric rela- tionship to one another each time a plot is established. In addition, because the combined plot is an integral unit, the subplots are combined, rather than treated separately, to produce a composite plot estimate (Myers and Shelton 1980).
Francis (1978), Morris (1982), Scott (1982), and Conant et al. (1983) provide good reviews of various plot configura- tions in use for inventories of the vegetative resources.
Subsampling should be used to observe time consuming attributes such as number of logs per tree, tree quality, age, and defect. Some plots should be monumented and remeasured for monitoring and modeling (Scott 1984). For future use or for monitoring, plot coordinates should be determined and stored in a data base. Plot configuration, subsampling, and monumenting are important but are not discussed further in this report.
Sample Size—The question of sample size must be con- sidered. At least two sample units per inventory unit (stand, compartment, or forest) are required if one wants to compute the reliability of the inventory. The need for additional plots varies with the objective. Traditionally 10 to 20 plots are established within a stand or 10 percent of a compartment or forest is inventoried. Ten to 20 plots per stand may be excessive if no immediate treatments are planned or if all that is needed is an estimate of what is in a particular area (Ek, Rose, and Gregerson 1984).
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Freese (1962), Hamilton (1979), and LaBau (1981) provide excellent discussions of determining sample size. Factors to consider include the consequences of errors in inven- tory estimates and the inventory costs. The resource manager should define and quantify the consequences of errors in the estimates. The inventory specialist should calculate the costs and determine the estimate of optimal precision.
Costs
The costs and methods of calculation presented in this report are given principally to illustrate the examples, but also to show the relative expenses of conducting invento-
16
ries by various methods. They are typical of those incurred by the USDA Forest Service Regions (Lund 1987).
The cost estimates include, where applicable, purchase of imagery, interpretation and mapping, and establishing and measuring field plots. These vary by design and size of the inventory unit. Additional expenses one needs to consider when developing an inventory are planning costs, over- head costs, costs of purchasing and maintaining equip- ment, per diem, data entry, editing, processing, data base, inventory documentation, and maintenance and storage. These may also vary by the design of the inventory and size of the inventory unit.
Stand Inventory
Inventories are often required of the smallest management unit a landowner has. These units may be stands, pastures, woodlots or parcels and could be termed mapped poly- gons in a geographic information system (GIS). Inventories within these mapped polygons are needed to determine what, how, and when specific treatments are to be made (Lund 1985). The treatments may be timber harvest, pasture improvement, precommercial thinning, etc.
The mapped polygon may also have been selected as part of a broader, more extensive inventory, such as the inven- tory of a compartment or a forest.
For the following discussions of collecting data within polygons or stands, assume: mapping is up to date and correct; that either complete enumeration or sampling is required; and if statistical sampling is used, at least two sample units will be observed. Remember, however, that serious objections can be raised concerning these assump- tions and that they need to be addressed in some organ- ized and orderly fashion for each inventory and not merely dismissed out of hand. Results for the examples are summarized to illustrate the advantages and disadvantages of the various options and to demonstrate how different techniques can yield different estimates of the same population.
This section describes methods of locating sample plots within a polygon using stand number 97 of the Enchanted Forest as an example (fig. 2). Options range from subjec- tive sampling to statistical sampling to complete enumer- ation. Each option is discussed through the use of exam- ples and simulated plot data to determine volume in hundreds of cubic feet (ccf). Many of the other attributes listed in table 1 could be obtained by the sampling options presented here. A summary of techniques is presented.
Statistical estimators for the various methods of sampling stand 97 are given for each option where (Freese 1962):
A = The total area of the inventory unit in acres.
a = The area of a sampling unit or a plot in acres.
n = The number of sampling units or plots established.
y; = The value for item of interest, such as volume per acre (ccf), observed or observed at each plot location.
N = The total number of possible sampling units in the entire population where:
N=A/a (1)
Y = The estimated mean value of interest such as volume per acre (ccf) where:
y= (2 y)/n (2) sf = The estimate variance of individual values of y
where:
sy? = {Zy, - (2 y)”/n}/(n-1) (3)
s, = The estimated standard deviation of y where: Sy = (Sayi4 (4)
s; = The estimated standard error of the mean for a simple random sample. For sampling without replacement (*):
s5* = {(s/7/n)*[1-(n/N)]}} "7 (5) or for where sampling is with replacement:
Sy = (s,7/n) "7? (6)
The expression [1—(n/N)] in equation (5) is the finite population correction or f.p.c. If (n/N) is less than 0.05, it is commonly ignored and equation (6) is used (Freese 1962).
S. = The estimated sampling error of the mean value such as mean volume per acre (ccf) where:
s. = S,/Y (7)
%S. = The estimated sampling error of the mean value (such as mean ccf volume per acre) expressed as a percent where:
%S, = (s.) * 100 (8)
Y = The estimated total value (such as total ccf volume) in the population where:
Y=y*A (9)
Cost estimators are determined as follows. It is assumed that the inventory unit (the stand) is already mapped. Therefore the cost of acquisition of remote sensing, inter- pretation and mapping is not included. Only field costs are considered. Stand 97, consisting of 75 acres, is the inventory unit.
Field costs are a function of: the size of the crew (C): the hourly wage (W) per person; the time per crew to measure
LEA
each sampling unit (M), the number of sampling units to be measured (n); travel time between sampling units (L), and the daily travel time to and from the inventory unit (D). To compute cost estimates, the following assumptions are made:
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Size of crew (C) = 1 person for subjective samples; 2 persons for statistical samples and complete enumeration.
Hourly wage (W) = $9.00 per person.
Plot measurement time (M) in hours = 0.167 hour for subjective samples; 0.5 hour for statistical samples; and 1 hour for complete enumeration.
Number of sampling units (n) = varies with design.
Time (in hours) traveling between sampling units (L) varies with distance or interval between plots (I) or (i) and number of sampling units (N). It is assumed that a crew travels at a speed of 10,560 feet per hour through the woods. For statistical sampling:
L = [(n—-1)iJ/10,560 (a)
where i = interval in feet between sample plots or points. Daily travel time to and from the inventory unit in hours (D) must be added to the between plot travel for each crew. For simplicity it is assumed that for each 8 hours spent within the inventory unit, 1 hour is spent in total travel time to and from the inventory unit.
Thus D = [L + n (M)/8 (b)
Forest Boundary
Figure 2—Location of stand number 97 (shaded) in the Enchanted Forest. Total area of the stand is 75 acres.
The general equation for field cost (F) is
F = CW {[L + n(M)] + D} = 1.25 * CW * [L + n * (M)] (c)
A listing of all equations and formulas used in this report is found in Appendix 1.
Probability Sampling
Probability sampling depends on some form of random- ization of the location of observations to be made. In the simplest form equal probability is assigned to each poten- tial plot location and one or more of these selected by some random process. A certain amount of care should be exercised so that selection is indeed equally probable for each location. Throwing a dart at the aerial photo or map might be an option, but it might tend to exclude plots at the edge of the map. Another selection process may be random initial plot choice by dropping a grid over the mapped stand and then using random numbers to select a coordinate or grid intersection.
Random processes for making selections in the field are also possible, but they must be rigorously enforced so that field personnel understand the importance of the random- ization process and do not substitute their “feeling of representativeness” for the random process consciously or unconsciously. One possible method of making the selec- tion in the field is to enter the stand at a point, choose a random azimuth and distance into the stand using a table
Stand 97 Enchanted Forest
O Sampling Unit
1000
Figure 3—Location of sampling units using randomly selected azimuths and fixed distances (614 feet).
of random numbers, and establish the initial plot at the distance and direction.
Random Distribution—\n this example, an initial, 1-acre plot is randomly located within the stand. Additional 1-acre plot centers are located at fixed distances (614 feet) but at random directions from one another (fig. 3). If the random direction makes the next plot fall outside the stand, the azimuth is rotated back into the stand according to some previously established rule or another angle is randomly picked. Figure 3 suggests that this method may be slow to disperse over an entire area.
Statistical Estimates—This example provides a random sample for which the sampling formulae 2 to 9 are applicable. The results of an inventory of stand 97 using 10 randomly selected plots are as follows:
Plot ccf per acre
17 15 LS 16 WE Zo 24 16 Ne 16
COCO ON DU BWH
—_
1500
19
yY = (17 + 15+... 16)/10 = 17.6 ccf per acre.
sy? = {(177 + 15? +... 167) — (17+15+ ... 16)7/ 10}/(10-1) = 13.8222.
sz = (13.822/10)'” = 1.1757 ccf per acre. The equa-
tion for sampling with replacement is appropri- ate as the same 1-acre plot may be chosen more than once.
%S. = (1.757/17.6)*100 = + 6.68%.
Y = 17.6*75 = 1,320 ccf for the stand.
Cost Estimates—Numbers of plots and intervals (n = 10, i = 614 feet) provide estimates of costs as before:
L = [(10—1)614]/10,560 = 0.523 hour.
D = [0.523 + 10(0.5)J/8 = 0.6904 hour.
F = 2(9) [0.523 + 10(0.5) + 0.6904] = $111.84, or $1.491 per acre.
These results portray a useful estimate of the cost of inventorying the current stand and might be used to estimate sampling other stands in the forest.
Discussion—Variations in sample selection may include selecting random distances as well as directions. Because it is difficult to implement in the field, this technique is seldom used in stand or forest inventories.
Line Transect Distribution—Line transects can be laid out using any of the following rules:
e Run the line through a randomly selected direction and point.
e Run the line through a randomly selected point and along the contours.
e Run the line through a randomly selected point and across the contours.
e Run the line through a randomly selected point and along the longest axis of the stand (fig. 4).
A line is drawn through a randomly selected point on the aerial photograph or stand map and the length measured. The length is divided by the number of plots to be established. The result is the interval between plot centers. The randomly selected point is usually plot 1. Additional plots are established along the transect line at the calcu- lated intervals on either side of plot 1. Sampling is without replacement.
Figure 4 shows the distribution of one-acre plots estab- lished at random azimuth through plot 1. The length of the transect line is 2,785 feet. Plots are established at 278-foot intervals.
Enchanted Forest © Sampling Unit
1000 1500
Figure 4—Location of sampling units along a randomly selected line transect. Plots are 278 feet apart.
20
Statistical Estimates—The results of an inventory of stand 97 using a line transect are as follows:
Plot ccf per acre
W7 16 1s) 15 19 19 22 21 23 24
GOO ONDUBWH
—_
Equations 1 to 9 are applied to obtain the statistical estimates.
N = 75/1 = 75 possible 1-acre plots. Y = (7 + 16 +... 24)/10 = 19.10 ccf per acre. sy? = {(177 + 167+ ...247)-(17 + 16+... 24)*/ 10}/(10-1) = 10.99. sy = {10.99/10 * [1-(10/75)]}}"7 = 0.9759 ccf per acre. %5_ = (0.9759/19.10) * 100 = + 5.11%.
Y = 19.1*75 = 1,432.5 ccf for the stand.
Cost Estimates—Applying equations a through c and using n = 10, i = 278 feet.
Stand 97 Enchanted Forest
O Sampling Unit
1000
FEET
Figure 5—Location of sampling units within stand 97 using the ricochet method. Plots are located at distances of 614 feet.
L = [(10—1)278]/10,560 = 0.237 hour.
D = [0.237 + 10(0.5)//8 = 0.655 hour.
F = 2(9) [0.237 + 10(0.5) + 0.655] = $106.06 or $1.414 per acre.
Discussion—Both the U.S. Department of Interior Bureau of Land Management (Baker 1982) and the U.S. Depart- ment of Agriculture Soil Conservation Service (Steers and Hajek 1979) have employed this technique.
Ricochet Plot Location—The ricochet technique was de- veloped in 1978 by the USDA Forest Service Resource Evaluation Techniques program and is a modification of the line transect. The length of the line transect varies within the size of the stand and the chosen direction, while with the ricochet method the length is fixed. The ricochet transect starts at a randomly selected point and is run in a cardinal direction until the line hits the stand boundary. At that point, the line is rotated back into the stand at 45-degree increments. Plot centers are established at fixed intervals along the line with the initial random point being plot 1. The rules for establishment do not allow crossing a previous transect line unless there is no alternative. Sampling is considered to be without replacement.
Figure 5 shows the distribution of 10 1-acre plots laid out using the ricochet technique. Plots were established at
614-foot intervals; distribution of plots is improved over random azimuth or distance method.
1500
21
Statistical Estimates—The results of an inventory of stand 97 using the ricochet technique are as follows:
Plot ccf per acre
17 16 21 V7, 19 19 18 20 22 23
COON AU BWN
—_
The sample design allows for all plots to be selected, hence, N = 75.
yY = (17 + 16 +... 23)/10 = 19.20 ccf per acre.
ae 2e NOs) ta. 232) (liz ey ty 23) 10}/(10-1) = 5.2889.
sy* = {(5.2889/10) * [1-(10/75)}}"7 = 0.6770 ccf
per acre.
(0.6770/19.20) * 100 = + 3.53%.
19.2 * 75 = 1,440 ccf for the stand.
Sy
%oSe
Y
W/ i
Carey,
\{
rp
Cost Estimates—Finally estimates of the cost for the inven- tory are obtained from equations a through c, for n = 10, i = 614 feet.
L = [(10-—1)614]/10,560 = 0.523 hour.
D = [0.523 + 10(0.5)/8 = 0.6904 hour.
F= 2(9) [0.523 + 10(0.5) + 0.6904] = $111.84 or $1.491 per acre.
Discussion—While this technique has not been employed, except for the initial development phase, it appears to have properties that might be preferred where it is important to measure edge or ecotone conditions as part of the stand inventory.
Systematic Distribution With a Random Start—Starting at a randomly selected point, plots are located at fixed directions and distances throughout the stand. The dis- tance between plot centers varies according to the size of the stand, the number of plots to be established, and layout. Variations in layout include use of squares, parallel line transects (Stage and Alley 1972), and equilateral triangles as shown in figure 6. An equation (Lund 1979) for computing the interval between points using equilat- eral triangles is:
Enchanted Forest © Sampling Unit ~~ 1000 1500 —
Figure 6—Location of sampling units using a grid or systematic distribu- tion. Plots are located at 60 degrees and 614 feet from one another.
224.272*(A/n)"/? (d)
interval between plot centers in feet. area of the stand in acres. n = number of plots to be established.
= =e ® ® > — i od
The area of stand 97 is 75 acres. Assuming 10 plots are to be established, the interval between plot centers would be 614 feet located at 60 degrees to one another.
The metric equivalent is: | = 107.456*(A/n)"? (e)
where | is expressed in meters and A in hectares.
Statistical Estimates—The results of an inventory of stand 97 using a systematic distribution of sample plots are as follows:
Plot ccf per acre
17 16 21 20 18 20 23 22 19 15
COON DU BWNH —
—
Estimates are obtained as usual using equations 1-9, N = 75 1-acre plots.
y = (17 + 16+... 15)/10 = 19.10 ccf per acre. ze N67 +)... 157) = (17 + 160+... 15)2/, 10}/(10-1) = 6.7667. s* = {(6.7667/10) * [1-(10/75)]}'2 = per acre. %S. = (0.7658/19.10) * 100 = + 4.01%.
Y = 19.1 * 75 = 1,432.50 ccf for the stand.
Sy
0.7658 ccf
Cost Estimates—Using the standard equations with n = 10, i = 614 feet.
L = [((10-1)614)/10,560 = 0.523 hour.
D = [0.523 + 10(0.5))/8 = 0.6904 hour.
F = 2(9) [0.523 + 10(0.5) + 0.6904] = $111.84, or $1.491 per acre.
Discussion—Simple random sampling formulae are al- most always used to compute statistics for a single random start. When random sampling equations are used with a systematic sample, the calculated variance for forestry examples frequently overestimates the variance in the population. Therefore the estimate can be considered somewhat conservative, to err on the safe side. However, the inventory analyst must be alert for unusual spatial patterns that might coincide with the distribution of samples. Property lines, superhighway rights-of-way, and township lines could coincide with systematic samples to the detriment of a systematic sample.
The systematic sampling technique is without replace- ment and provides the most uniform distribution of plots throughout the stand. The disadvantage is that, assuming all other things are equal, it may be one of the most costly to do. The distance between plot centers is often the greatest (Matern 1960), hence travel time is the highest. The Intermountain (Myers 1984) and Northern Regions (Brickell 1984) employ this sampling design within stands.
Single Plot—A single point does not an inventory sample make. However, a single plot will occasionally be the only sample for a stratum that was judged to be unimportant prior to the inventory and then still requires an estimate for the sake of completeness. Figure 7 shows a single plot; its location was selected by a random process. The observed volume is 17 ccf per acre.
Statistical Estimates—This can hardly be considered a statistical estimate, but for this plot, Y = 17 * 75 = 1,275 ccf total volume in the stand. Unless additional random samples are added it is not possible to compute the sampling error. However, if this is the result of a single point strata in a larger inventory, for which estimates of similar strata are available, a statistician might conjure up a Stein or simulation type estimate of the variance (Thomas 1986).
Cost Estimates—Selection of a single plot for an inventory is highly unlikely, hence a cost estimate for a single plot is not likely to yield relevant information.
Discussion—Often a cluster of subplots is used in lieu of a single plot. One or more subplot centers are systematically distributed near a randomly selected starting point. The random point is usually the first plot in the cluster or the center. A cluster is used in situations where travel costs are high or accessibility is limited, it is difficult to build a sampling frame, or where interest lies in a primary sam-
23
pling unit that is expensive to observe in total (Ek et al. 1984). It is very efficient if variability within the cluster is high and when the variation within the inventory unit (in this case, the stand) is expected be low.
10-Point Cluster—A 10-point cluster (fig. 8) is a common layout in the USDA Forest Service. Subplots are located at 70 feet and 60 degrees from one another. A 5-point,
L-shaped cluster is now used in California (Bowlin 1984) and a 19-point cluster is being used in a survey of Alaska (Larson 1984). Other configurations are outlined by Scott (1982). The pattern for the cluster is established prior to the inventory. In general, one should observe as many subplots as widely spaced as possible as time would allow to get maximum dispersion on measure of variation across the stand (Ek et al. 1984).
Stand 97 Enchanted Forest
O Sampling Unit 1000 1500
Figure 7—Location of a single sampling unit within stand 97 using
random selection.
Stand 97 Enchanted Forest
O Sampling Unit 1000 1500
Figure 8—Location of a randoml)-located, 10 point cluster. Subplots are located at 70 feet from one another.
24
If the initial point falls close to the edge of the stand, some subplots may fall into adjoining stands. When validation of the mapping or classification is important, substitute plots established according to previously defined rules are normally used (fig. 9).
In the inventory of stand 97, a 10-point cluster is estab- lished as shown in figure 8. Each subplot represents about 0.10 acre.
Statistical Estimates—A good deal of care must be taken in the consideration of the statistical nature of the estimates. For some sample clusters it is possible that a mean and variance for the stand would be meaningful. This is probably not the case for the example. The results of the inventory of stand 97 using a single 10-point cluster are as follows:
Subplot ccf per acre
17 17 17 iz, 16 15 IZ U7, IZ/ 17
CMO ON DU BWH —
—
Stand 97 Enchanted Forest
O Sampling Unit
1000
FEET
Figure 9—Location of substitute subplots in a 10 point cluster.
There are N = 75 1-acre plots if each cluster occupies about 1 acre. Y = (17+17+...17)/10 = 16.7 ccf per acre. sy? = {(1774+177+ ...177)-(174 174 ... 17)7/10}/ (10-1) = 0.4556. sy* = {(0.4556/10)*[1 —(10/750)}}"2 = 0.212 ccf per acre.
As this represents a single cluster the sampling formula without replacement of points is applied. Values for estimated percent error and total stand volume are com- puted as:
%sS_ = (0.212/16.7) * 100 = + 1.27 %. Y = 16.7*75 = 1,252.5 ccf for the stand.
This estimate should carry very little more weight in terms of actual information about the stand than a single point, and you may notice that it is not a very accurate estimate of the stand’s true volume.
Cost Estimates—Like the single plot estimate for a stand, the cost can be computed, but its informational value to an inventory forester is minimal. For a single cluster, n = 10, i = 70 feet.
L = [(10 — 1)70]/10,560 = 0.060 hour.
D = [0.060 + 10(0.5)/8 = 0.6325 hour.
F = 2(9) [0.060 + 10(0.5) + 0.6325] = $102.46, or $1.366 per acre.
1500
25
Discussion—As applied by the USDA Forest Service FIA units, the variation among the 10 subplots is not calcu- lated because it has been found to be small compared to the variation between clusters. The values from the sub- plots are simply averaged. It should be noted that the variability of a single cluster may be significantly more than a sample from a single point and the variability is not always insignificant. The cluster sample should not be routinely treated as a single plot as it often is. For our examples, the results are considered as being from a single plot in further computations.
Because the 10 subplots within a cluster are located close to one another, the subplots are frequently similar to each other. The cluster therefore may provide less information than randomly located plots that are truly independent. This reduces the precision of estimates, but cost savings from clustering frequently yield the most cost effective inventories because the total travel time is reduced.
The 10-point cluster plot is frequently used in extensive forest inventories and where maps of stands are not available. When the population of interest is highly vari- able, then a sampling system that provides the opportunity for a greater distribution of plots throughout the inventory unit is preferred. The following are some other options for
sampling within a stand. When used as a part of a forest inventory, each of the options may be considered a form of cluster sampling.
Subjective Sampling
Subjective sampling is not recommended in this primer. There may be occasions in which such a sample has been acquired and the analyst must make do with it. Our recommendation is to attempt to relate the sample to a probability sample.
Statistical Estimates—Figure 10 shows plot locations based upon extremes observed within the stand (sampling without preconceived bias). Observations yield estimates of 14 and 25 ccf per acre. The results of the inventory using the observation of extreme values are:
(14 + 25)/2 = 19.5 ccf per acre. = 19.5 * 75 = 1,462.5 ccf total volume in the stand.
<< !
Cost Estimates—To develop an equation to compute (L), assume the stand to be square and that the crew would traverse it diagonally 3.0 times for sampling without preconceived bias. The diagonal distance of a 1-acre square is 295.16 feet. The diagonal distance across a 75-acre square stand is 295.16 (75) = 22,137 feet.
Stand 97 Enchanted Forest
O Sampling Unit 1000 1500
Figure 10—Location of sampling units within stand 97 using subjective sampling without preconceived bias. Plots were located in portions of the stand representing extremes in volume per acre.
26
Without preconceived bias, two plots (n) are established and equations a through c are employed to obtain costs.
L = [3(22,137)]/10,560 = 6.289 hours.
D = [6.289 + 2(0.167)]/8 = 0.828 hour.
F = 1(9) [6.289 + 2(0.167) + 0.828] = $67.06 or $0.894 per acre.
Discussion—A justification often given for this type of sampling is that it is quicker than statistical sampling. This may be the case, but to select the average condition or the extremes it may be necessary to traverse the entire stand several times. Travel time within the stand may be equal to or greater than that required for probability based sam- pling, but time needed to establish plots may be substan- tially reduced particularly if the observations are also subjectively made.
One cannot compute a sampling error to evaluate the reliability of an estimate based on the extremes. The only way to determine how good the estimates are is by comparing the results with total enumeration.
While it is seldom advisable to use subjective sampling without preconceived bias, the method may be useful for obtaining a rough estimate of the variation within the stand or inventory unit which, in turn, could be used to determine the sampling intensity necessary to achieve a desired precision. The following illustrates how this may be done.
Snedecor and Cochran (1974) provide an equation for determining sample size where:
n = [(t*s,)/(s.* YI? (10)
where:
n = The estimated number of sampling units necessary to sample within certain prescribed precision and confidence limits.
t = Student's “t;’ which is a value establishing a level of probability. The values of “t’ have been tabulated and are available in most statistical textbooks, including those referenced in this report.
Where past inventory information is lacking, the standard deviation (s,) may be estimated by:
Syp = B/3 (11)
where: B = the estimated range from the smallest to the largest value likely to be encountered in sampling.
Note: Statistical arguments for using either 3 or 4 in equation (11) may be made and both have been suggested by respected sources.
Using the subjective samples of 14 and 25 ccf per acre and:
y = (14 + 25)/2 s, = (25 - 14)/3
19.5 ccf per acre. 3.67 ccf per acre.
To compute the number of samples (n) required to be within + 15 percent sampling error (%s,) at the 95 percent probability level, we need to estimate the degrees of freedom so that we can obtain a value for “t.” Past experience has shown that in similar stands we can use 10 plots and still be within the allowable sampling error. The degrees of freedom are 10 — 1 = 9. Using a table showing the distribution of Student's “t’ we find that for the 95-percent probability level and 9 degrees of freedom, “t’ = 2.262. Using equation (10) and the information given above:
n = [(2.262 * 3.67)/(0.15 * 19.5)]* = 8.055, or 9 samples.
If the number of plots had changed by an order of magnitude, we might need to recompute the number of plots allowing for the change in ‘n.’ That is, if 20 plots had been required our degrees of freedom would change from about 9 to 19 and it would be advisable to solve (10) again using the new value for “t.” More information regarding sample intensity formulation is found in Freese (1962) and in the statistics and sampling references listed at the end of this report.
27
Complete Enumeration Figure 11 shows the mapped results of a complete enu- meration. All trees were measured in the stand.
Statistical Estimates—Stand 97 has a total volume of 1,425 ccf, or 19 ccf per acre.
Cost Estimates—n = area of stand or 75 acres, i = O feet.
L = O hours.
D = [0 + 75(1)/8 = 9.375 hours.
F = 2(9) [0 + 75(1) + 9.375] = $1,518.75, or $20.25 per acre.
Discussion—The complete enumeration will serve as ground truth for summarizing the results obtained by the other options.
Summary of Methods
Table 4 shows the results of sampling stand 97 by the methods discussed. There is no sampling error for the complete enumeration because each stand in the popula- tion was sampled. There may be error within a stand, but for this discussion, we have chosen to ignore it (as do most introductory treatises). A sampling error for the subjective technique cannot be computed because the plots were not chosen randomly. A sampling error cannot be computed
for the single plot because at least two samples or prior information are needed. Similarly, the 10-point cluster is considered a single sample sc the variance for the stand cannot be estimated, although the variance for the plot can be and was computed.
Statistical Estimates—A direct comparison is not possible because the data are replicated only once for each tech- nique. However, some general comments or observations can be made.
As may be expected, the 10-point cluster design, being considered a single plot, had the least variation. Intuitively the systematic distribution should best represent the stand and the single plot or cluster sample should be the least representative. A systematically distributed sample usually encompasses more variation than does a randomly distrib- uted sample. This will usually be the case unless some cyclic variation in the population chances to coincide with the periodicity in the sample.
Cost Estimates—Table 4 also shows the costs encountered for each simulated option and the cost ($S.p) that would be required to achieve a desired percent sampling error (%Sep). This is computed by the following equation:
$S.p = $ * (%S,/ %S.p)" (f)
PSS cca oGRs Soave oes Haul Roag stecoaatn
PRR ON CPL ance
20 19
21
Stand 97 Enchanted Forest
1000 1500 FEET
Figure 11—Complete enumeration and ground truth for stand 97 show- ing the approximate volume (ccf) per acre distribution. The total volume for the stand is 1,425 ccf, or 19 ccf per acre.
28
Table 4—Volume per acre (ccf) and costs for various sampling methods for stand 97
Line
Statistic Random transect Ricochet Y (ccf) 1,320.00 1,432.50 1,440.00 y (ccf) 17.60 19.10 19.20 Se 13.82 10.99 5.29 $5 1.18 0.98 0.68 %S, 6.68 ot 3.53 Total cost ($) 111.84 106.06 111.84 Cost per acre ($) 1.49 1.41 1.49 Total cost @ 5% sample error © 199.62 110.78 55.75
a—Sampling errors are not knowable for subjectively drawn samples. b—A sampling error cannot be computed for only one sample. c—Computed by using equation (f).
where $ = total cost in dollars for a particular option.
Assume we would like to know what it would cost to achieve a desired sampling error of +5 percent (“oSep = 5). Using the random sample technique, we achieved a + 6.68 percent sampling error at a total cost of $102.47. While direct comparisons of the costs are prob- ably not advisable, we have scaled the data to a common sampling error base for each method and adjusted the costs accordingly. Thus for the random sample we find that:
$Sep = 102.47 * (6.68/5)? = $199.62.
In other words, we would have to put more plots into the stand using a random distribution technique to lower the sampling error to 5 percent. The total cost for doing so would have been $199.62.
When considering the actual costs, the single plot and 10-point cluster samples may be the cheapest to establish because the travel distance between points may be the least. The line transect may be the next cheapest to do and the ricochet, random, and systematic distribution should be increasingly time consuming (Matern 1960).
When considering the costs required to achieve a +5 percent sampling error in this particular stand, and disre- garding the cluster sample, the ricochet distribution was
Subjective without Complete Systematic Single plot Cluster bias enumeration
1,432.50 1,275.00 1,252.50 1,462.50 1,425.00 19.10 17.00 16.70 19.50 19.00 6.77 0.46 0.00 0.77 0.21 0.00 4.01 p 1.27 2 0.00 111.84 102.46 67.06 1,518.75 1.49 1.37 0.89 20.25
71.94 6.61
the least costly, the systematic sample next, followed by the line transect. The random distribution for the example given is the most costly to do. If it is laid out properly, the field crews can end close to the starting point at the end of the stand inventory when using the systematic distribu- tion. This may result in further travel cost savings in many inventory situations.
Key to Options—The following is offered as a rough guide and key to the selection of a plot distribution technique.
1. Variation tends to be in the middle of the stand or
polygon. a. Yes. Use line transect. b. No. Go to 2.
2. Variation is greater at the edges of the polygon. a. Yes. Use the ricochet technique. b. No. Use the systematic distribution.
While the random distribution can be used in any situa- tion, it is seldom employed in stand inventories. The single plot or 10-point cluster may result from post stratification of inventoried forest stands. There are new techniques available for combining inventory information from differ- ent sampling, simulation, and growth model sources that might improve the value of a single plot (cluster) for an estimate of the polygon resource value. (Hansen and Hahn 1983, Hansen 1984, Green and Strawderman 1988).
29
Forest Inventory
Often inventory objectives are to obtain estimates for compartments or forests for planning purposes rather than for developing stand prescriptions. This section describes various design options that provide both forest and loca- tion estimates. Initially this discussion assumes minimal information and then progresses to the more complex designs using stratified mapped stands.
The objectives of these simulated inventories are to determine:
e The total wildlife area used by a fictitious creature, the red-spotted snaileater (Lund 1986b).
e The total volume (ccf) of standing timber.
e The area and timber volume by vegetation type in the 15,300 acre Enchanted Forest.
We also wish to show the spatial distribution of timber volume in the Forest.
The calculation of forest-wide statistics is discussed first, followed by a discussion of the expansion of sample data to all areas within the Forest.
We will now issue a brief caveat, which will be repeated. Selection of a sampling design and associated statistical estimators are not independent. Nor are they independent of the population for which the estimates are required. Computation of estimates for several different sampling designs on a single simulated forest is not justified. The comparisons do not imply that one method is better than another. The results may inform us as to designs that are more appropriate for the Enchanted Forest, given its peculiar characteristics. However, the techniques pre- sented do not lend themselves to adequate comparisons directly. Statistical simulations (a statistical forest sampling simulator has been recently prepared by Arvanitis and Reich 1988) of sampling designs and computational algo- rithms would be required to make realistic evaluations of the appropriateness of the various techniques. Still, as an example of the application of the various techniques, the Enchanted Forest is a useful tool.
The options and equations for obtaining compartment or forest estimates are similar to those for obtaining and generating information within a stand. The sampling strategy options are somewhat similar to those used in the stand inventory and range from subjective sampling to statistical sampling and on to complete enumeration and vary with the amount of available stand information. The
30
reader is cautioned that all the designs in this section are simulated. Many of the inventory schemes were purposely laid out in such a manner that the same field locations were chosen to show the effect that different designs could have on the estimates generated on a particular stand.
Mapping and unmeasured area estimates for any attributes measured or observed on sample plots or stands can be extrapolated to other areas within the inventory unit. Extrapolation of sample information to the remaining stands in an inventory is usually quite simple. However, care must be taken especially when recombination of sample units or restratification of the population may occur. It is usually advisable to maintain identification of the measured units as being different from those units that are strictly predicted in a database.
Note that in the figures associated with the various stratified sampling schemes discussed in this report, both sampled volumes and stratum averages are displayed. This has been done primarily to show the source of the estimates. In actual practice, only stratum averages are usually displayed in the mapping process.
Cost estimates given in this section are similar to those incurred by the Forest Service Regions (Lund 1987). Cost assumptions are as follows:
Remote sensing acquisition is assumed to be 1:15,840 color aerial photography and digital tapes of satellite imagery of the Enchanted Forest. A coordinated pur- chase of complete coverage for several million acres is assumed. These assumptions make costs per acre, sim- ilar to those found for a real National Forest.
Aerial photography = $0.032 per acre, or $489.60 for the Enchanted Forest.
Digital satellite imagery = $0.004 per acre, or $61.20 for the Forest. Interpretation including aerial photography or satellite imagery.
Aerial photography = $0.02 per point or acre, or $306 for the Forest.
Satellite imagery = $0.025 per acre, or $382.50 for the Forest.
Mapping, including stand delineation, transfer to stable base, and determination of stand area.
Using aerial photography = $0.075 per acre, or $1,147.50 for the Forest.
Using satellite imagery = $0.02 per acre, or $306.00 for the Forest.
Field cost assumptions are the same as discussed under the stand inventory section.
Subjective Sampling
The disadvantages of using subjective sampling have been discussed previously. The danger of bias in the various subjective sampling methods cannot be overemphasized. For an individual stand, a mistaken prescription might not be a calamity, but a consistent bias over all (or most) stands in acompartment or forest could lead to some unworkable management plans. Therefore, subjective sampling should not be used for forest-wide inventories except possibly to estimate the coefficient of variation prior to probability sampling.
Assume that subjective sampling without preconceived bias of the Enchanted Forest yields estimates of 0 ccf and 30 ccf per acre. Further assume that 15 percent is an acceptable sampling error at the 20-percent probability level (t = 1.325 at 19 degrees of freedom). An estimate of the number of samples (n) needed to inventory the forest may be determined using the subjective sample data as follows:
y = (0 + 30)/2 = 15 ccf per acre. s, = (30 — 0)/3 = 10 ccf per acre.
ry, n
Sample intensity equations for specific designs are found in the references listed at the end of this report.
Inventories Without Prior Stand Mapping
Inventories without stand mapping are usually conducted for broad area assessments where location information is not needed or as a prelude to the further development of the resource. Such inventories focus on the resource stock and the land’s capability to produce on a sustained yield basis. The examples given in this report include the use of systematic sampling, poststratification, strip cruises, strat- ified double sampling, and stratification of satellite imag- ery. The inventory units are usually based upon political or administrative boundaries. Broad management goals and objectives and financial plans for the organization are the eventual products (Lund 1985).
= [(1.325 * 10)/(0.20 * 15)]? = 19.51 or 20 samples.
If the inventories are conducted so that the field plots are established and documented for remeasurement, it may be considered a continuous forest inventory or CFI. These inventories are typical of the Forest Inventory and Analysis units.
While stand maps may not be available at the start of a forest inventory, there are several tools available to pro- duce forest volume and area estimates and location maps until stand mapping can be done. These techniques are also illustrated in this section. Sampled volumes, stratum averages, and predicted values are often shown in the figures in this report. In actual practice, however, only stratum averages or predicted values are usually displayed in the mapping process. When using such maps, resource managers need to be aware of the source of the informa- tion and the sampling errors associated with averages and predictions.
Costs include purchase and interpretation of remote sens- ing, where appropriate, plus field costs. For field costs, a plot consists of 10 subplots located 70 feet apart. It takes 0.5 hours for the two-person crew to measure 1 subplot. A total of 20 plots are established in each of the following examples.
The time to measure (M) 1 plot (includes subplots) is: M = {[(n—1) (]/10,560} + n(0.5) (g)
where n is the number of subplots and i is the interval in feet between subplots.
Solving for M = {[(10—1)(70)/10,560} + 10(0.5) = 5.06 hours.
The plots are located systematically through the Forest at 60 degrees from one another. The interval between plots (I) is:
224.272 [(15,300/20)"”] = 6,203 feet.
= [(20—1)6,203]/10,560 = 11.161 hours.
= [11.161 + 20(5.06))/8 = 14.045 hours.
= 2(9) [11.161 + 20(5.06) + 14.045] = $2,275.31, or $0.149 per acre.
| E D F
Systematic Sample—This is a simple, intuitively appealing inventory design. A grid is superimposed across the forest
31
(fig. 12). Plots are established at the grid intersections. Once the initial plot is established or the grid is fixed on the area, the remaining design is fixed. For this reason it is not truly a random sample. Traditionally, random sampling formulae have been applied to this type of sample alloca- tion, assuming that a random process is associated with distribution of the forest variates of interest. Experience indicates that in most cases estimates of variance will be conservative. Hence the practicing forester is usually safe in applying this type of design, unless there is some regular variation in the forest that is correlated with the sample placement. A second consideration is that there is no possibility that a plot once chosen will recur in the sample, hence, this is sampling without replacement. For
=
Figure 12—Location of sampling units using a systematic grid over the Enchanted Forest. Sampling unit centers are located at 60 degrees and 6,203 feet from one another.
a Q FEET 5000
32
Stott 3 ODOR
this example, the field plots are assumed to be 10-point clusters covering an area one acre in size. The within-plot variance is not considered.
Statistical Estimates—Assume we want to compute the acreage of lands having a particular wildlife use such as that of the red-spotted, snaileater. To compute the estimate for areas having wildlife use, all plots classed as having evidence of snaileater use in the field are assigned a value of 1 and all other plots are given a value of zero. The results are shown in table 5.
N = 15,300/1 = 15,300 possible 1 acre plots. y=(0 + 1 +...0)/20 = 0.55 or 55% of the area shows wildlife use.
Forest Boundary
O Field Plot
Table 5—Results of an inventory of the Enchanted Forest using a systematic sample; volumes are in ccf
Vegetation Wildlife Wildlife Volume/ Volume
Plot type use estimators acre estimators
1 Hardwood 0 7
2 Conifer 1 31
3 Hardwood 1 8
4 Hardwood 0 3
5 Brush/open 1 3
6 Hardwood 1 7
7 Hardwood 1 10
8 Conifer 1 19
9 Hardwood 0 10 10 Conifer 0 34 11 Conifer 1 29 12 Conifer 0 14 13 Hardwood 1 17 14 Conifer 0 8 15 Hardwood 1 13 16 Hardwood 1 21 17 Brush/open 1 6 18 Conifer 0 18 19 Hardwood 0 0 20 Hardwood 0 20 In 11 278 y 0.550 13.900 si 0.260 92.305 SS 0.114 2.148 %S, + 20.75 + 15.46 Y 8,415. 212,670.
sy = {7 + 17+...07) — (0 + 1+...0)7/20}/ (20-1) = 0.2605.
s; = (0.2605/20)"? = 0.1141 is the standard error of mean wildlife use (this is a proportion and as such it is difficult to refer to it in terms that are completely clear in meaning).
The equation for sampling without replacement is appro- priate because the plots do not have a chance of being selected again.
%oS. = (0.1141/0.55) * 100 = + 20.75 percent.
a
Y = 0.5500 * 15,300 = 8,415 acres of wildlife use in the Forest.
Total volume estimators are calculated as follows:
N = 15,300 possible 1 acre plots. y = (7 + 31 +... 20)/20 = 13.90 ccf per acre. Sy (Zt 3124.6.%207)—[(7 +. 31+4.... 20)7/
20]}/20-1) = 92.3053.
s; = (92.3053/20) = 2.1483 ccf per acre. %S. = (2.1483/13.90)*100 = +15.46%.
Y = 13.90 * 15,300 = 212,670 ccf for the Forest.
Area and total volume estimates by vegetation type are computed similarly. For the conifer type, all plots not classed as conifer are assigned a value of 0 for volume and area. Table 6 lists the estimates for the conifer vegetation type. The areas are: for conifer type 5,355 acres, for hardwoods 8,415 acres, and for brush/open 1,530 acres.
The volume of timber in the conifer type is 117,045 ccf; in the hardwood type the volume is 88,740 ccf; and in the brush/open type, the estimated volume is 6,885 ccf.
Mapping and Unmeasured Area Estimates—Maps show- ing the approximate location or distribution of the re sources can be generated by the use of cells, isolines, or by partial field mapping.
Table 6—Results and estimates for conifer vegetation type using a systematic sample of the Enchanted Forest; volumes are in ccf
Plot Volume/acre
Volume estimators Type ‘Type estimators
1 0 0 2 31 1 3 0 0 4 0) 0 5 0) 0 6 ) 0 7 0 0 8 19 1 9 0 0) 10 34 1 11 29 1 12 14 1 13 0 0 14 8 1 15 0) 0 16 0 0 17 0 0) 18 18 1 19 0) ) 20 ft) 0) mn 153 Ti y 7.650 0.350 si? 143.818 0.239 Sy 2.682 0.109 %s, + 35.05 + 31.26 Y 117,045. 5,355.
33
Figure 13 illustrates the use of cells. Each field plot has an area expansion factor (EF) where:
EF = A/n (12) The expansion factor for the plots established in this sample design is:
EF = 15,300/20 = 765 acres per plot. The 765 acres surrounding each plot center are character- ized as the same as the plot itself. Because the plots in the
example above were established using a grid of equilateral triangles, the cells take on the form of hexagons. Areas
Lr) 0 FEET 5000
outside the plot locations are assumed to have the average characteristics of the Forest as a whole. Thus those areas are assumed to have 13.90 ccf per acre. Mapping suitable red-spotted snail eater habitat is a challenging task. The presence or absence of indicators of habitat even in sampled stands may not represent the proportion of the area actually utilized. Because the sampling fraction is quite small it may be most informative simply to associate the estimated proportion with each stand instead of indicating the presence or absence of habitat for the sampled stands. An alternative is to employ an advanced multivariate classification procedure to the sampled stands and predict presence or absence based on these findings.
Forest Boundary
Figure 13—Mapping showing measured (*) and Forest average (a) Conifer
volume (ccf) per acre based on a systematic sample. Each hexagon O) Hardwood
represents 765 acres. Brush/Open O Field Plot
Figure 14 illustrates the use of isolines for mapping volume distribution. The map is produced by considering the total volumes or volumes per acre at each sample point as elevations points. Isolines or contours of equal volumes are then developed through interpolation (Wiant and Knight 1982). Heine (1986) presents a total of four other ways of interpolation.
Partial mapping involves mapping stands or parts of stands during the course of establishing field plots. In the exam- ple shown in figure 15, the field crew delineated the stand boundaries on aerial photography as the sample plots were established using ground referencing. This informa-
Forest Boundary
tion was transferred to a base map. This technique is used by the Rocky Mountain Region (Mehl 1984).
Cost Estimates—The only cost is the field cost.
F = $2,275.31, or $0.149 per acre.
Discussion—Systematic sampling was very common in the Forest Service from about 1930 to the mid-1960’s (Stott 1968). It is useful where remote sensing is lacking or where data on multiple resources are desired. Statistical estimates from systematic samples are usually computed from simple random sampling formulae. Experience has
=z
Figure 14—An isoline map of the Enchanted Forest showing contours of O Field Plot
volume (ccf) per acre based upon interpolation of the systematic sample.
35
usually validated the application of these mixed design- computation-procedures. If the reliability of the estimates is extremely important as might be the case for an inventory that was being disputed in court, a systematic sample such as is described could be modified by includ- ing some sort of randomization process for subsets of the plots to assure multiple random starts within the area of interest. Computations would then follow the procedures for systematic sampling plans as described in Cochran (1977) or other statistical sampling texts.
Stratified Sampling—Often the forest is heterogeneous with respect to forest type, maturity, or site class. It may be
=
< Shorter,
>
=
worthwhile to consider stratified sampling if these character- istics are of interest or if the variance within the categories is more homogeneous than the overall forest. In stratified sampling, units of population are grouped together on the basis of similar characteristics. These groups are called strata. Total variance can be reduced by the amount of variance that can be attributed to the difference between the strata. For instance, suppose that a large tract of land had considerable merchantable volume interspersed with recently regenerated stands. Estimates of overall volume and the associated stan- dard errors will be considerably reduced if the forest is partitioned into merchantable and nonmerchantable strata. Stratification may be made after plots are established (post-
Forest Boundary
Soe
Figure 15—Mapping of stands around the sample plots showing meas- §3 Conifer ured (*) and Forest average (a) volume (ccf) per acre. C) Hardwood Brush/Open O Field Plot (Initial)
36
Stratification), or it may be done before plots are selected (prestratification).
Poststratification—In a poststratification design, plots are simply grouped by similar characteristics and the variance is computed for each stratum and then pooled for the forest as a whole. The number of samples in a given stratum is not predetermined, so there is a random com- ponent to the estimates of standard error and confidence limits. Some strata may have been poorly represented in the sample and estimates for these strata may be highly variable. In some ways post stratification can be likened to establishment of classification strata in remotely sensed multispectral imagery.
Systematic Sample—Table 7 shows the results of an inventory of the Enchanted Forest using poststratification of the same plots established under the systematic sample described above (fig. 12). The plots were group based on vegetation type. There were 7 plots in the conifer type, 11 in the hardwood type, and 3 in the brush/open class.
Statistical Estimates—The estimators are computed first for each sampling strata and then combined for the
inventory unit as a whole.
The area in each stratum (A;) is computed as follows:
A; = A*(nj/N;) (13) Where n,; = number of plots in stratum. N; = total number of plots in inventory.
i=c, h, or b for conifers, hardwoods, or brush/open strata respectively A. = 15,300 * (7/20) = 5,355 acres of conifers.
For area and volume estimates in the conifer vegetation type, n. = 7, N. = 5,355 acres.
To compute the area estimators for wildlife or the snaileat- ers use within each stratum, all plots classed as having some sign of use are assigned a value of 1 and all other plots are given a value of zero.
Table 7—Results of poststratification of a systematic sample of the Enchanted Forest; volumes are in ccf
Stratum Conifer
Hardwood
Brush/open
Enchanted Forest
Plot 2
Wildlife use
wooo-oo°o--—
oooj--=+0++0+-0
Wildlife estimators
0.429 0.286 0.202 + 47.14 0.005 2,295.
Volume/ Volume acre estimators 31 19 34 29 14 8 18 153 21.857 93.143 3.648 + 16.68 1.628 117,045. 7 8 3 7 10 10 17 13 21 0 20 116 10.546 44.673 2.015 + 19.11 1.227 88,740 3 6 9 4.500 4.500 1.500 + 33.33 0.022 6,885. 212,670. 13.900 1.696 + 12.20
37,
For the conifer (c) stratum:
Yo. = (1 + 1 +...0)/7 = 0.4286 acres of wildlife use per acre of stratum or 42.9 percent of the conifer stratum showed evidence of use.
(Sea = (Sheila ate) Cou ( Tle sapere (7-1) = 0.2857.
(5;). = (0.2857/7)"? = 0.2020 standard error for wild- life use per acre. Note that the f.p.c. is much less than .5 percent and is omitted.
ec = (0.2020/0.4286) * 100 = + 47.14 %.
Yo = 0.4286 * 5,355 = 2,295 acres of wildlife use in the conifer stratum.
os o n
— |
Estimates of variance for a stratum are weighted by the
proportion of plots in the stratum in order to obtain
estimates for the Forest as a whole:
(s,7); = [s? * (N/N)?/n] * [1 - (n,/N)] (14) -[
(S57). (0:2857,"*1(6,355/15,300)4/71 = = (7 5,355)] = 0.0050
The same variance estimators are computed for the hard- wood and brush/open strata. The area estimates are com- bined for the Forest as follows where:
Y = The total value in the Enchanted Forest.
Y=(J, + fot... + V%) (15) Y = (2,295 + 4,590 + 1,530) = 8,415 acres of wildlife use in the Forest. Y = The mean value in the Enchanted Forest. Y = Y/A (16) Y = 8,415/15,300 = 0.5500 proportion of wildlife use.
Sy = The standard error of the mean for the Forest computed from the estimates for individual Strata (from Freese 1962) is:
: so)? (17) Sy = (0.005 + 0.0075 + 0)? = 0.1117 wildlife use per acre. %oSe = The estimated sampling error of the mean value for the Forest expressed as a percent where: %S, = ( Sy/Y¥) (18) %S_ = (0.1117/0.5500) * 100 = +20.31%.
2 2 Sy = [ji + So +.
38
. 0)7/7]}/
For total volume estimates:
Yo = (314+19+ ...18)/7 = 21.8571 ccf per acre. = teC(Bilie a> 1g) a1 Giller Oe eae) 77-1) = 93.1429. (3). = (93.1429/7)"? = 3.6477 ccf per acre. (3.6477/21.86) * 100 = + 16.68%. Yo = 21.86 * 5,355 = 117,045.00 ccf in conifer
stratum.
(577). = [93.1429 * (5,355/15,300)7/7] * [1 - (7/ 5,355)]* 0.0050.
_ o %,)
v)
— fe) ll
The same estimators are computed for the hardwood and brush/open strata. The estimates are combined for the Forest as follows where:
Y = (117,045 + 88,740 + 6,885) = 212,670 ccf in the Forest.
Y = 212,670/15,300 = 13.90 ccf per acre.
Sy = (1.6279 + 1.2269 + 0.0225)'"? = 1.6962 ccf
per acre. %S_ = (1.6962/13.90) * 100 = + 12.20%.
The estimated area and volume by vegetation type are computed similarly. To compute estimates for the conifer type, for example, all plots not classed as conifer are assigned a value of O for volume and area. The area of conifer type is 2,295 acres; the area of hardwoods is 4,590 acres; and the area of brush/open is 1,530 acres.
The estimated total volume for each vegetation type is 117,045 ccf for conifer, 88,740 ccf for hardwood, and 6,885 ccf for the brush/open type respectively.
Mapping and Unmeasured Area Estimates—The op- tions are the same as for systematic sampling.
Cost Estimates—The costs are the same as for sys- tematic sampling. The only cost is the field cost.
F = $2,275.31, or $0.149 per acre.
Discussion—There is little use of this technique in the USDA Forest Service. The procedure does offer the advantage of lowering the sampling error with no addi- tional field work or costs. It remains something of a mystery why the method has not been widely imple mented. Perhaps a combination of circumstances can be invoked for the apparent disuse. Where there is a large potential for stratification in the South, there is also the possibility of rapid forest type change. In much of the West where stratification could be applied, growth rates are
slow enough that inventories are seldom necessary and hence are performed only near the rotation age on National Forests.
Strip Cruising—These techniques are based on the traditional strip cruise. Plots are laid out in strips or on a grid. The inventory crew maps strips of the Forest as they travel from plot to plot. Under the technique shown in figure 16, a total of 15,833.33 feet of lines were run. During the course of the inventory, a tally was kept of the number of feet of transect line run in each vegetation type. There were 6,792 feet run in the conifer type, 8,208 feet run in the hardwood type, and 833 feet run in the
Forest Boundary
brush/open class. These figures provide the stratum weights.
Statistical Estimates—Table 8 shows the result of the inventory where:
A. = 15,300*(6,792 / 15,833) = 6,563 acres.
For area and volume estimates in the conifer vegetation type, n. = 7, N. = 6,563 acres. The area estimates for the red-spotted snaileater wildlife usage within the conifer stratum are computed as follows. As before, plots having signs of wildlife use are assigned a value of 1 and all other plots are given a value of zero.
=z
Figure 16—Location of strip cruise lines and field plots across the E] Conifer
Enchanted Forest. O) Hardwood Brush/Open O Field Plot
39
Table 8—Results of an inventory of the Enchanted Forest using strip cruising; volumes are in ccf
Wildlife Wildlife | Volume/ Volume Stratum Plot use estimators acre estimators Conifer 3 1 31 9 1 19 1 0 34 19 0 29 14 1 14 6 0 8 7 0 18 In, 3 153 y 0.429 21.857 S27 0.286 93.143 sy 0.202 3.648 %s, + 47.14 + 16.69 (S57). 0.008 2.446 Vc 2,813. 143,448. Hardwood 11 0 7 12 1 8 10 0 3 2 1 7 13 1 10 5 0 10 8 1 17 20 1 13 18 1 21 17 0 0 16 0 20 In, 6 116 y 0.546 10.545 s,? 0.273 44.673 Sy 0.158 2.015 %S, + 28.87 + 19.11 (Syn 0.007 1.090 Vn 4,326. 83,647. Brush/open 4 1 3 15 1 6 In, 2 9 y 1.000 4.500 So 0.000 4.500 0.000 1.500 %S, + 0.00 + 33.33 (S57)> 0. 0.006 Yb 805. 3,622. Enchanted Y 7,944. 230,717 Forest Y 0.519 15.079 Sy 0.119 1.882 %S_ + 22.91 + 12.48
40
For the conifer (c) stratum:
Yo = (1+1+...0)/7 = 0.4286, or 42.86 percent wildlife use of the stratum. (7). = {24 124 3. 02) —(ie tee 02/73/71) = 0.2857. (sy). = (0.2857/7)? = 0.2020. (%S.)- = (0.2020/0.4286) * 100 = + 47.14%. Y. = 0.4286*6,563 = 2,812.7 acres of wildlife use in the conifer stratum. (57). = [0.2857 * (6,563/15,300)7/7] * [1 — (7/ 6,563)] = 0.0075.
The same estimators are computed for the hardwood and brush/open strata. The area estimates are combined for the Forest as follows where:
Y = (2,813 + 4,326 + 805) = 7,944 acres of wildlife use in the Forest.
We repeat that expressing the acreage is simply a way of dealing with an important classification variable.
Y = 7,944.4/15,300 = 0.5192 proportion of wildlife use. Sy = (0.0075 + 0.0067 + 0)"? = 0.1190 wildlife use per acre. %S_ = (0.1190/0.5192) * 100 = + 22.91 %.
For total volume estimates:
VY. = (31+19 + ... 18)/7 = 21.8571 ccf per acre.
(5/7). = {(317+197+ ... 187)-(314+19 +... 18)?/ 7}(7-1) = 93.143. (s3). = (93.143/7)"? = 3.648 ccf per acre.
(%S.). = (3.6458/21.85) * 100 = + 16.69 %.
Yo = 21.8571 * 6,563 = 143,448.43 ccf in conifer stratum. [93.1429 * (6,563/15,300)7)/7] * [1 — (7/ 6,563)] = 2.4457.
(7),
The same estimators are computed for the hardwood and brush/open strata.
These estimates are then combined for the Forest as follows where:
Y = (143,448.4286 + 83,646.5455 + 3,622.5) = 230,717.474 ccf in the Forest. Y = 230,717.474/15,300 = 15.0796 ccf per acre. Sy = (2.4457 + 1.09 + 0.0062)" = 1.8820 ccf per acre %Se = (1.882 / 15.0796)*100 = + 12.48%.
The estimated area and volume by vegetation type are similarly computed. To compute estimates for the conifer type, for example, all plots not classed as conifer are assigned a value of 0 for volume and area. The area of conifer type is 6,563 acres, the area of hardwoods is 7,932 acres, and the area of brush/open is 805 acres.
The estimated total volume for each vegetation type is 143,448 ccf for conifer, 83,646 ccf for hardwood, and 3,623 ccf for the brush/open type respectively.
Mapping and Unmeasured Area Estimates—Areas
outside the sampled and mapped strips are assumed to have the average conditions of the inventory unit as a
Forest Boundary
Figure 17—Mapping showing measured (*), stratum average (s), and Forest average (a) volume (ccf) per acre based on the strip cruise inventory.
whole (15.08 ccf per acre). Within the mapped strips, unmeasured areas are assigned the average values for the stratum in which they fall. See figure 17.
Cost Estimates—The costs are the same as for sys- tematic sampling. The only cost is the field cost.
F = $2,275, or $0.149 per acre.
Discussion—This technique was used in the early days of the USDA Forest Service, but is seldom used today. Line intersect sampling (a hybrid probability sample and a line transect) has been used to sample downed woody material to evaluate fire hazard potential (Brown 1974, De Vries 1986).
=
41
Prestratification—By prestratifying, a heterogeneous inven- tory unit is divided into homogeneous subunits (strata). Each stratum is then sampled for additional attributes. The strata estimates are combined to give a population esti- mate. Stratification has provided satisfactory estimates of the inventory unit as a whole with less field work than if stratification had not been used (MacLean 1972).
The principle means of obtaining prestratification informa- tion is usually by interpreting remote sensing imagery. Strata may be formed along many lines, such as overstory density classes, vegetation types, or even administrative units (though this latter may not result in gains in inventory efficiency). Strata should: (1) be logically related to item or
=
| 0 FEET 5000
Figure 18—Location of systematic distribution of photo points in the Enchanted Forest. Points are located at 60 degrees and 3,101 feet from one another.
42
items of information sought; (2) exist in nature or be artificially established; (3) represent a relative homoge- neous condition with respect to the estimates that can be defined in specific terms; (4) have differentiating criteria easily recognizable from remote sensing, maps, and from the ground; (5) represent a grouping that the manager definitely wants sampled on the ground; and (6) be meaningful to the manger (Lund 1978a and b).
To eliminate potential biases and to keep calculations simple, the same plot configuration should be used throughout the sampling stratum. Plot configuration may be changed between strata but not within.
A Photo Point O Field Plot
Stratified Double Sampling—A grid of points is established across the inventory unit (fig. 18). These points are usually transferred to aerial photos, which are in turn interpreted for attributes to form sampling strata (in this case overstory crown cover or density class). The photo- interpreted points are the primary sampling units. These are stratified and subsampled in the field as secondary sampling units (fig. 19). The use of random numbers or a systematic system with a random start may be used to select the secondary sample within each stratum. At least two sample plots must be chosen in each stratum.
In this example, three density classes—low, medium, and high density—were formed. A total of 80 photo points
Forest Boundary
ES tn ae
eee neces
were established: 32 in the low density strata; 25 in the medium density class; and 23 in the high crown cover category. These photo points were subsampled with field plots. Ten photo points were measured in the field in the low density strata; 6 points were field measured in the medium strata; and 4 were measured in the high density class.
Statistical Estimates—The results of an inventory of the Enchanted Forest using the stratified double sample of photo points are shown in table 9, where the area and volume estimators are computed as follows:
=z
a | 0 FEET 5000
Figure 19—Location of stratified photo points and field samples based Density Classes
upon overstory density classes in the Enchanted Forest.
00 0-30% 31-60% 614+%
O Field Plot
43
Table 9—Results of an inventory of the Enchanted Forest using a stratified double sample of photo interpreted points; volumes are in ccf
Stratum Point Vegetation type Wildlife use Wildlife estimators Volume/acre Volume estimators
Low density 1 Hardwood 0 7
10 Hardwood 1 8
12 Hardwood 0 3
14 Brush/open 1 3
16 Hardwood 1 U
28 Hardwood 1 10
50 Conifer 0 8
60 Hardwood 1 13
64 Brush/open 1 6
75 Hardwood 0 0
rn, 6 65
y 0.600 6.500
si? 0.266 14.055
s;* 0.135 0.983
%S, + 22.57 + 15.12
(S57), 0.003. 0.155
vi 3,672. 39,780. Medium density 32 Hardwood 0 10
44 Conifer 1 29
46 Conifer 0 14
48 Hardwood 1 ZA
66 Conifer 0 18
77 Hardwood 0 20
one 2 108
y 0.333 18.000
s,? 0.267 41.200
s;* 0.184 2.284
%S, + 55.14 + 12.69
(S37)m 0.003 0.510
Vn 1,594. 86,062. High density 3 Conifer 1 31
30 Conifer 1 19
34 Conifer 0 34
62 Hardwood 1 21
rn, 3 105
y 0.750 26.250
s,? 0.250 54.250
Sa 0.227 3.347
%S, + 30.30 + 12.75
(S3)n 0.004 0.926
Vn 3,299. 115,467. Enchanted Forest Y 8,565. 241,310.
y 0.559 15.771
Sy 0.102 1.261
%S_ + 18.30 + 8.00
The area in each stratum (Aj,A,,,Ap, representing low, medium and high density, respectively) is computed as follows:
A; = 15,300*(32/80) = 6,120 acres.
For area and volume estimates in the low density type n, = 10, N,; = 32 plots.
To compute the area estimators for the snail-eater wildlife use within each stratum, all plots classed as having wildlife use are assigned a value of 1 and all other plots are given a value of 0. For the low density stratum:
(y), = (0 + 1 + wildlife use.
(5/7), = {(07 + 17+ ...07) - (0 + 1+... 0)7/10}/ (10 — 1) = 0.2667.
(s)*, = (0.2667/10) * [1-(10/32)]"7 = 0.1354 wild- life use per plot. Note that a large portion of the sample is drawn, hence the f.p.c. is neces- sary in the computation.
(%s,); = (0.1354/0.6000) * 100 = + 22.57%.
y; = 0.6000 * 6,120 = 3,672.0 acres of wildlife use in the low density stratum.
(5)? = [0.2667 * (32/80)7/10] * [1 — (10/32)] = 0.0029.
...0)/10 = 0.60 proportion of
The same estimators are computed for the medium and high density strata. The area estimates are combined for the Forest as follows where:
Y = (3,672 + 1,593.75 + 3,299.0625) = 8,565.8125 acres of wildlife use in the Forest. Y = 8,564.8125/15,300 = 0.5598 wildlife use per acre. Sy = (0.0029 + 0.0033 + 0.0043)? = proportion of wildlife use. %S_ = (0.1025/0.56) * 100 = + 18.30%.
0.1025
For total volume estimates:
(y), = (7 + 8+ ...0)/10 = 6.500 ccf per acre.
(s7), = {777 + 8? +...07)-[(7+8+ ...0)7/10]}/ (10-1) = 14.0556.
(5,)", = (14.0556/10) * [1 — (10/32)]"? = 0.983 ccf per acre.
(%5_), = (0.983/6.500) * 100 = + 15.12%.
y,; = 6.500 * 6,120 = 39,780 ccf in the low density
stratum.
(s;?), = [14.0556 * (32/80)7/10] * [1 - (10/32)] = 0.1546.
The same estimators are computed for the medium and high density strata.
The estimates are combined for the Forest as follows where:
Y = (39,780 + 86,062.5 + 115,467.1875) = 241,309.6875 ccf in the Forest. Y = 241,309.6875 / 15,300 = 15.7719 ccf per acre. Sy = (0.1546 + 0.5096 + 0.9261)? = 1.2611 ccf per acre. %S_ = (1.2611 / 15.7719) * 100 = + 8.00%.
The estimated area and volume by vegetation type are similarly computed. To compute estimates for the conifer type, for example, all plots not classed as conifer are assigned a value of O for volume and area. Table 10 shows the results for the conifer type. The area of conifer type is 6,301 acres, the area of hardwoods is 7,774 acres, and the area of brush/open is 1,224 acres. The estimated total volume for each vegetation type is 145,879 ccf for conifer, 89,922 ccf for hardwood, and 5,508 ccf for the brush/ open type respectively.
Mapping and Unmeasured Area Estimates—The op- tions for creating map displays are the same as those given under systematic sampling. Each photo plot has an expan- sion factor (EF) or represents an area of 191.25 acres. To illustrate the source of estimates, field sampled photo points retain their measured values. Other photo points take on the stratum averages. All other areas are assigned the average for the inventory unit. See figure 20.
Cost Estimates—The field costs are the same as for the systematic sample plus the cost of purchasing aerial photography of the Forest plus the costs of interpreting 80 points at $0.02 per point.
Field costs = $2,275 Aerial photography = 489 Photo interpretation = 2 Total costs = $2,766, or $0.181 per acre.
45
Table 10—Results and estimates for conifer vegetation type using the stratified double sampling of photo points; volumes
are in ccf
Stratum Low density
Medium density
High density
Enchanted Forest
Point Volume/acre 1 0
a (=) mooowmwooocae
bs Foko&
Volume estimators
145,879.125 9.534 2.442
Type
-ooo-oao0:0c0 000
oo}-o--_OA0
woo — —
Type estimators
6,301 .687 0.411 0.095
5 + 23.13
46
Discussion—This technique is very common in the United States, particularly by the USDA Forest Service Forest Inventory and Analysis Units (Beltz 1984, Cost 1984, Hahn 1984, Born 1984, and Ohmann 1984). Stratified double sampling is particularly useful where aerial photography exists and large areas must be covered in a short period of time. Some attention to the ages of photographs and to the classification of points versus areas of photographs could improve the estimates obtained from the technique. Unfortunately, stratified double sampling is too often applied as if there were no differences in ages of photography and the point classification. These problems are minor though they probably should be considered for
Forest Boundary
applications where areas are the most important factor in the inventory.
Lund (1974) also used this technique for forest inventories in the U.S. Department of Interior Bureau of Land Man- agement, and Lund and Kniesel (1975) used the same process to inventory multiresource values, including herb- age production, soil surface factors, soil cover, and deer- days use.
The Northeast FIA unit uses a modification of this tech- nique and sampling with partial replacement (Barnard 1984). Sampling with partial replacement (SPR) is very
=z
Figure 20—Mapping showing measured (*), stratum average (s), and Density Classes
Forest average (a) volume (ccf) per acre based on the stratified double
sample of photo points. Each hexagon approximately represents 191 acres.
0 0-30% 31-60% 614+%
47
effective for remeasurements or reinventories when stra- tum weights remain relatively stable over periods between inventories and the variables of interest are few. There are significant computational and data storage and retrieval costs associated with SPR that must be considered in a practical application of the method. Further discussion of SPR is beyond the scope of this report. The reader may consult the references given at the end of this publication for further details.
Use of Satellite Imagery—Earth-orbiting satellites, such as Landsat, have been applied to research inventories for broad mapping of forest resources and for forming
=z
sampling strata (Langley 1975, and Poso 1986). Figure 21 is a simulated satellite scene of the Enchanted Forest. The pixels are 15 acres in size. A sampling frame consisting of 1,020 possible cells is constructed for the Enchanted Forest. Three sampling strata based upon apparent vege- tation density or canopy cover are created through classi- fication of the scene. Classification techniques are becom- ing increasingly automated and produce increasingly repeatable results. While the earliest applications for forestry inventory were overly optimistic about the capa- bilities of satellite imagery acquisition, cost, interpreta- tion, continuous improvements have made the practical application on a large scale more and more realistic.
Forest Boundary
Figure 21—Raster mapping of satellite imagery showing vegetation Overstory Density
O 0-30% 31-60% 61+%
O Field Plot
density classes and location of field plots selected based on stratified sampling with probability proportional to size. Each square or pixel equals 15 acres approximately.
While our example forest is too small to warrant satellite imagery, it does serve to illustrate some possible uses of digital imagery.
Satellite imagery was used to classify pixels into vegetation density classes. Foresters might think of these as crown densities for forest stands older than the youngest seedling and sapling stands. Crown densities were classified thus: 6,240 acres (416 cells) at 0 to 30 percent; 5,115 acres (341 cells) at 31 to 60 percent; and 3,945 acres (263 cells) at 61 percent or greater. The number of cells are used to determine the strata weights.
A grid is superimposed across the classified satellite scene. Potential field plots are established at the grid intersections and grouped by the density classes resulting in a stratified sample of the forest. At least two samples are required per stratum. If the grid does not yield a sample of a particular stratum, the grid intensity can be increased or a special grid may be created for the unmeasured stratum.
Statistical Estimates—The results of the sample of the Enchanted Forest are given in table 11. For area and volume estimates in the low-density type, n, = 9, N; = 416 cells.
To compute the area estimators for snaileater use within each stratum, all plots classed as having the wildlife use are assigned a value of 1 and all other plots are given a value of zero. For the low-density stratum:
y, = (0+ 1+...0)/9 = 0.5556 proportion of wildlife use acres. (s/7), = {07 + 17+ ... 07) (9-1) = 0.2778. (ss)) = (0.2778/9)"? = 0.1757 standard error of esti- mate for proportion of wildlife use. (%S,); = (0.1757/0.5556) * 100 = +31.62 %. y, = 0.5556*6,240 = 3,466.667 acres of wildlife use in the low-density stratum. (S,)?, = [0.2778 + (416/1020)7/9] * [1 - (9/416)] = 0.005.
(0+1+ ...0)7/9}/
The same estimators are computed for the medium- and high-density strata. The area estimates are combined for the Forest as follows where:
Y= (3,466.6667 + 2,557.5 + 2,367) = 8,391.1667 acres of wildlife use in the Forest. Y = 8,391.1667/15,300 = 54.84 percent wildlife use in the area. Sy = (0.005 + 0.0055 + 0.0039)? = 0.1201 ccf per acre. %S_- = (0.1201/0.5484) * 100 = +21.90%.
For total volume estimates:
¥, = (7+ 8+...0)/9 = 6.5556 ccf per acre. (Sot); mut(Zon +) Bate. Og (7 + Bi. 1O)/ 9}/(9-1) = 15.7778. (s3), = (15.78/9)"? = 1.3240 ccf per acre.
(%s,), = (1.324/6.56) * 100 = + 20.20 %. y, = 6.5556 * 6,240 = 40,906.6667 ccf in the low-density stratum. (S5))7 = [15.7778 + 416/1020)7/9] * [1 — (9/416)] =
0.2853.
The same estimators are computed for the medium and high-density strata. The estimates are combined for the Forest as follows where:
Y = (40,906.6667 + 80,135 + 98,625) = 219,666.6667 ccf total volume in the Forest.
Y = 219,666.6667/15,300 = 14.3573 ccf per acre.
Sy = (0.2853 + 1.1468 + 0.6326)"? = 1.4369 ccf
per acre.
%S_ = (1.4369/14.3573) * 100 = + 10.01 %.
The estimated area and volume by vegetation type are similarly computed. To compute estimates for the conifer type, for example, all plots not classed as conifer are assigned a value of 0 for volume and area. Table 12 shows the estimates for the conifer type. The area of conifer type is 5,618 acres, the area of hardwoods is 8,136 acres, and the area of brush/open is 1,546 acres.
The estimated total volume is 123,825 ccf for conifer, 88,646 ccf for hardwood, and 7,195 ccf for the brush/
open type respectively.
49
Table 11—Results of an inventory of the Enchanted Forest using stratified satellite imagery; volumes are in ccf
Stratum Low density
Medium density
High density
Enchanted Forest
50
Plot Vegetation type 1 Hardwood 3 Hardwood 4 Hardwood 5 Brush/open 6 Hardwood 7 Hardwood 14 Conifer 15 Hardwood 19 Hardwood =n, y 2 Sy Sy %S_ (S57) vi 9 Hardwood 11 Conifer 12 Conifer 13 Hardwood 18 Conifer 17 Brush/open Inn y 2 Sy SS %S, (S57)m Ym 2 Conifer 8 Conifer 10 Conifer 16 Hardwood 20 Hardwood rn, y 2 Sy SS %S, (S57)n y Y Y Sy %S_
Wildlife use
o-OoO+-0+-0 ao -o--+" 00
woo-o—- —
Wildlife estimators
0.555 0.278 0.175 + 31.62 0.005 3,466.
0.500 0.300 0.224 + 44.72 0.006 2,557.
Volume/acre
OWWONWWON
Volume estimators
15.666 62.666 3.232 + 20.63 1.147 80,135.
Table 12—Results and estimates for the conifer vegetation type using stratified satellite imagery of the Enchanted Forest; volumes are in ccf
Stratum Plot Volume/acre Volume estimators Type Type estimators Low density 1 0 0 3 0 0 4 0 0 5 0 0 6 0 0 q 0 0 14 8 1 15 0 0 19 0 0 <n, 8 1 y 0.889 0.111 Ss 7.111 0.111 Sy 0.889 0.110 %S, + 100 + 100 (S;)*; 0.129 0.002 "7 5,546. 693. Medium density 9 0 0 11 29 1 12 14 1 13 0 0 18 18 1 17 0 0 Inn 61 3 y 10.167 0.500 37 148.167 0.300 Ss; 4.969 0.224 %S, + 48.88 + 44.33 (S3)?m 2.711 0.006 \he 52,002. 2,557. High density 2 31 1 8 19 1 10 34 1 16 0 0 20 0 0 In, 84 3 y 16.800 0.600 Si3 266.700 0.300 SS 7.303 0.245 %S, + 43.47 + 40.83 (Sn 3.479 0.004 Vn 66,276. 2,367. Enchanted Forest Y 123,825. 5,618. Y. 8.093 0.367 Sy 2.513 0.106 %S_ + 31.06 + 29.09
51
Mapping and Unmeasured Area Estimates—The sat- ellite scene also serves as a rough map of the resources. Areas classified and mapped as having low density, for example, can be assumed to be hardwood vegetation 78 percent of the time and have a volume of 15.78 ccf per acre + 19.98 percent based upon the field survey. Pixels containing the field plots may be assigned the measured values. All other pixels are assigned the stratum averages. See figure 22.
Note that both sampled volumes and stratum averages are shown. In actual practice, only stratum averages are usually displayed in the mapping process. In such in-
=
ben nen 0 FEET 5000
stances, resource managers need to be aware of the source of the information displayed and the associated sampling errors.
Cost Estimates—The cost estimates include costs incurred from purchases, interpretation of imagery, and field activities.
Field costs = $2,275 Imagery of Forest = 61 Interpretation of Forest = 382 Mapping = 306 Total costs = $3,024, or $0.198 per acre.
Forest Boundary
Werte. ep rcca ea eet Me, ~~
Figure 22—Mapping showing measured (*) and stratum average (s) volume (ccf) per acre based on the stratified satellite imagery sample. Each pixel represents 15 acres.
52
Discussion—The example given in this report used pixels 15 acres in size. In reality, pixels of 30 x 30 meters are common when using Landsat Thematic Mapper or SPOT image data. The California Region uses this tech- nique using Landsat imagery (Bowlin 1984). The Alaska FIA unit uses a technique employing two additional phases of high and low altitude aerial photography (Larson 1984).
The digital nature of many satellite systems creates an instant geographical information system that can be com- bined with other georeferenced data and used to produce various theme maps (Mattila 1984). A truly practical inventory based on available satellite imagery has been an elusive and illusory goal for nearly 20 years. It has been applied in both industry and forest survey, perhaps only on a show-me scale. There is still a hefty and expensive requirement for equipment and statistical as well as remote sensing expertise. There is increasing evidence that satellite imagery is now the design tool of the present instead of the dream of the future.
Inventories With Prior Stand Mapping
If all the stands in a compartment or forest have been mapped, the use of these mapped stands as primary sampling units offers several advantages over the use of sample plots. In addition to providing estimates for the inventory unit, mapped stands provide (Stage 1984):
¢ Opportunities to study spatial relations between stands.
e Easy coordination among resources. The stands can be used as common data sources and can be overlaid upon maps of streams, wildlife “edge” habitat, and nesting sites or other delineations that are represented by lines or points rather than areas.
e More appropriate prescription because the full range of variation within realistically-sized treatment units can be used in analysis.
e¢ Concentrated day-to-day work in nearby areas with potential savings in transportation by using larger crews or local “spike” camps.
e Dual use of the stand data to meet several objectives, such as quality control for stand examination proce- dures and pre-sale cruising. Estimates generated for stands that were not sampled can be used to establish priorities for more detailed silvicultural examinations or timber cruises thus helping to implement forest plans.
It should be noted that different sampling designs have been developed to deal with different populations. Char- acteristics of the population should be considered when selecting a sampling design. If we know nothing about the population, then simple random sampling is almost cer- tainly the first method to consider. Populations for which information exists concerning some characteristic such as the area from aerial survey, but not for those variables we wish estimates such as proportion of the area showing wildlife usage, should be sampled using a design that incorporates the known information. Even for these cases certain sampling designs may be appropriate for some mapped conditions and not for others. We will try to illustrate computational methods for a variety of sampling designs and at the same time note the characteristics of the Enchanted Forest that appear to support the use of that particular method or to negate its use.
Inventories using mapped polygons or stands are used to develop forest management plans. The resources and their condition and potential are generally described only in sufficient detail to direct the manager's attention to spe- cific locations within the inventory unit for more intensive planning. Area, volume, and production estimates are usually tied to each polygon or stand (Lund 1985). If plots are established for remeasurement, they are usually estab- lished to measure growth and monitor response to treat- ments. In both cases plot establishment should be done with a good deal of attention being paid to probable time sensitivity of the plots. Time interval before remeasure- ment should be planned well in advance and should consider the likely rotation of the species or forest type in which the plot is being established.
Stand mapping has been accomplished using satellite imagery (Hame and Tomppo 1987, Tomppo 1987) and this capability will certainly improve in the future. How- ever, for general purposes aerial photography is still ade- quate. Processing of aerial photographs might be done with electronic scanners in the future and this could well provide a relatively low cost transition technology leading to the use of satellite imagery.
This section examines sampling stands mapped using aerial photography to obtain a forest inventory as well as providing estimates for all stands in the forest. Even though the examples are for timber estimates using aerial photo- graphy on a small forest and a relatively few stands, the principles, techniques, and options apply to most natural resource surveys using all forms of remote sensing and any size area.
To compute statistical estimators, assume that the En- chanted Forest has been mapped into 200 stands (fig. 23), acreage has been determined for each stand, and the boundaries of the stands will not change upon field visitation.
Further assume that a total of 10 sample plots (n) within selected stands will form the secondary sample and will be systematically established by overlaying a grid of equilateral triangles, and plots require 0.5 hours to mea- sure (m). Plot results are expanded for the sample stands. The estimated values for these sampled stands then con-
Saas Q FEET 5000
stitute the basis for predicting the inventory for stands that were not sampled in the forest inventory unit. Finally, for simplicity and uniformity of presentation we have consis- tently computed the finite correction factor for all estima- tors. This is readily available with computer spreadsheets; however, it would not be necessary (the sampling fraction being much less than .05) for hand computation for some of the examples.
For cost estimators, assume the purchase of aerial photo- graphy ($489.60) and the cost of mapping of $1,147.50 for the Forest applies to all of the following designs. Stands are the primary sampling unit.
Forest Boundary
Figure 23—Location of previously mapped stands in the Enchanted Forest showing only stand boundaries and stand identification numbers.
Using equation (d), i = 224.272 (a/10)"” ori = 70.921 a"? (h)
where a is the area of the sample stand in acres.
The time (M,) to traverse and measure a sample stand is M, = {(n—1) 70.921a"7}/10,560 + n(0.5).
M, = { 9 (0.0067 a"”) } + 10(0.5) for 10 plots, (i) or 0.0603 a’? + 5.
Stand no. 10 for example, is 120 acres. Using equation (i), M, = {0.0603(120)"7} + 10(0.5) = 5.1 hours.
The daily travel time (D) is revised as follows: D =[L + =(M]/8 (j)
where L (M) = the sum of the time to measure all selected stands.
Similarly, total cost of field time (F) is adjusted to: F = CW[(L + 2 (M,) + D] (k)
Unstratified Sampling—Auxiliary information consists of the location, identification number, and acreage of each stand (see Appendix 2). Given this limited information, the options for selecting stands for sampling are unstratified, equal probability sampling (e.p.s.) and probability propor- tional to a measure of size (p.p.s.). Note the difference between allocation of a sample with probability propor- tional to size and to a measure of size. In the first case we must have information regarding size of the variable of interest itself. In the second case a variable that may have a relation to the variable of interest is used as a surrogate— in this case the acreage of each stand is used as the selection surrogate.
Equal Probability Sampling (e.p.s.)—Equal probability sampling means that each stand, regardless of acreage, has an equal chance of being selected for measurement. A simple random number generator can be used to select stands by identification number or, if these are not con- secutive, a unique whole number may be assigned to them. Figure 24 shows the stands selected for forest inventory. It should be noted that this is not the method that would be preferred for estimating most properties of interest in the Enchanted Forest. The occurrence of very large area units that are an order of magnitude larger than the average stand should be a red flag warning us to use a sampling system other than e.p.s.
However, this example gives us an opportunity to demon- strate the necessity for employing two different estimation techniques for parameters having different sampling distri- bution characteristics. Wildlife use is sampled by record- ing the presence or absence of evidence on a single plot in the stand; it is a binary variable. Plotting the data from a sample of 20 stands reveals no trend between this variable and the size of the stand. Consequently, wildlife use or any other variate sampled by a presence/absence indication should be estimated as a simple random sample obtained without replacement. In contrast volume estimates when plotted versus acreage do show a relation between volume and acreage. The relationship is not the classical linear relation through the origin for a ratio and gives some large values that have very low volumes associated with the acreage. This indicates that stratification would be ex- tremely useful, but for this example we are going to ignore the stratification protential in order to demonstrate the use of a weighted mean to compensate for the unequal sizes of stands acreage.
Statistical Estimates—To calculate the estimated mean volume per acre a weighted mean is employed: Yw = 2wiy; / 2w; (19)
where w is the stand acreage and y is the continuous variable of interest (volume per acre). A simple un- weighted mean is not appropriate when the unequal area stands are sampled with equal probability.
The variance of a weighted mean is equal to:
Sy? = Zwiy; - Yy)? / =w; (20)
The standard error is simply the square root of the weighted variance divided by the number of samples:
Sy = (Sy? / ny”? (21)
Finally the percent sampling error (%5,.,) is:
Sey = (Sy / ¥y)*100 (22)
Note that the above calculations do not account for the variation within the sampled stands from the secondary sampling units. In most cases of broad forest inventories, as illustrated in this section, such variation is rarely computed and is beyond the scope of this publication. An alternative treatment of these computations may be found in Loetsch and Haller (1964).
55
Table 13 shows the results of the equal probability sam- pling where area and volume estimators are calculated as follows.
For area of snaileater use, a proportion estimate is em- ployed: the value one is assigned to stands in which the sample showed evidence of snaileater use, otherwise stands are assigned value zero. Recalling that only a single plot per stand is assessed regardless of area and that the expected value for a proportion is given by:
2
a | 0 FEET 5000
Vp = Xnj_,)/n, (23)
where n, is the sample size and n,_, is the number of occurrences.
The variance of a proportion is:
s*(¥,) = Yp(1-Y,) (24) Yp = (0+ 1+... + 0)/20 = 0.500; a proportion expressing wildlife use per plot. s*(¥,) = 0.500 * (1 — 0.500) = 0.250. Sy = (0.250 / 20)? = 0.162.
%Seay = 0.162 / 0.500 * 100 = +32.40%.
Y = 0.500 * 15300= 7650.
Figure 24—Location of primary sampling units (shaded stands) based upon a random draw of stand numbers using equal probability sampling.
56
Table 13—Results of an inventory of the Enchanted Forest using equal probability sampling of mapped stands; area estimates are in acres and volume estimates are in ccf
Vegetation Wildlife Wildlife Volume/ Volume
Stand Area type use estimators acre estimators’
18 60 Conifer 0 21
25 60 Hardwood 1 10
98 45 Hardwood 1 30
74 60 Hardwood 0 15 187 30 Hardwood 1 14
26 30 Brush/open 0 0 154 75 Conifer 1 23 144 45 Hardwood 0 23
61 30 Hardwood 1 12
62 30 Brush/open 0 0 152 75 Hardwood 0 19
10 120 Conifer 1 30 172 45 Hardwood 1 21
53 30 Conifer 0 9
95 45 Hardwood 1 25
81 90 Conifer 0 v 116 30 Hardwood 0 2
92 45 Conifer 1 6 170 45 Conifer 0 31 153 45 Conifer 1 3 In 1,035 10 301 y 0.500 16.913 sy* 0.250 95.992 SS 0.162 2.191 %S, + 32.40 + 12.96 y 7,650 258,770.
1—Volume estimates are computed using weighted estimators (equations
19-22).
For per acre and total volume estimates weighted estima- tors (equations 19-22) are employed:
Yw = [(60*21) + (60*10) +... + (45 * 3)] / (60 + 60 +... + 45) = 16.913 ccf per acre.
Sy? = 60(21— 16.913)? + 60(10— 16.913)? +... +
45(3— 16.913)? / 1035 = 95.992.
(95.992 / 20)? = 2.191.
S = ase — oy 16:91 >. 100 =) 12.96%. Y = 16.913 * 15,300 = 258,770 ccf in the En-
chanted Forest.
The estimated area and volume by vegetation type are computed analogously. To compute estimates for the conifer type, for example, all plots not classed as conifer are assigned a value of 0 for volume and area. Table 14 shows the estimates for the conifer type. The area of conifer type is 7,096 acres, the area of hardwoods is 6,874 acres, and the area of brush/open is 887 acres.
Table 14—Results and estimates for the conifer vegetation type using equal probability sampling of mapped stands in the Enchanted Forest; area estimates are in acres and volume estimates are in ccf
Volume/ Stand Area acre
18 21
Volume Type estimators Type estimators
_
3k PEREBSASERISSRHLSSEBR Ne’e es Os? OOOH OOOO} OO OO FO
_
8.971 0.464 Ss 586,648.716 870.063
2.124 0.082 %s, + 23.67 + 17.64 a, 137,256. 7,096.
The estimated total volume is 137,256 ccf for conifer, 121,513 ccf for hardwood, and 0 ccf for the brush/open type, respectively. This last value is strong evidence that we have overlooked an advantageous stratification. It is reasonable to assume that if there is 0 ccf of timber in the brush/open category that we could improve our sampling efficiency by introducing strata that isolate this type.
Stand Estimates—Stands’ attributes are estimated from the area estimates. Measured stands can have their observed inventory estimate or they may be assigned as part of the general problem of predicting a volume for each stand in the forest. The former is done in this publication and the inventoried stands are identified with an asterisk. All other stands must be assigned the average for the appropriate stratum or inventory unit. An alterna- tive is to use two types of records—one for predicted values and one for measured. The alternative might be- come more valuable as the complexity of the inventory
57
increased. The estimated volume per acre for the unmea- sured stands is 16.9 ccf. Estimates for vegetation types are that 49 percent of the area is conifer, 45 percent is hardwood, and 6 percent is brush or open. See figure 25.
Cost Estimates—The sum of the approximate straight line distance between sample stands as measured from the map is 104,000 feet.
L = 104,000/10,560 = 9.85 hours. Z(M,) = 101 hours.
D = (9.8 + 101)/8 = 13.8 hours.
F = 2(9)[9.8 + 101 + 13.8] = $2,243.
=
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“Figure 25—Mapping showing measured (*) and Forest average (a)
Thus: Field costs = $2,243 Aerial photos = 490 Mapping =A 7, Total costs = $3,880, or $0.254 per acre.
Discussion—This technique is seldom used in forest inventories. When polygons or stands are mapped, useful correlated attributes are also noted. Attributes such as vegetation type, density, and height may then be used to
Forest Boundary
volume (ccf) per acre based on equal probability sampling of stand
numbers.
58
develop a stratified sampling plan. However, the proce- dure might be useful in sampling geo-political divisions— mapped counties within States for example. It might also reasonably be employed where an agency was exploring the forest, such as in large, unmapped tropical areas.
Probability Proportional to Size (p.p.s.) Sampling—There are several possible estimates of size that might be used to develop a sampling frame. Our example is acreage, but this is really a correlated measure of size. Recent sampling developments remind us that there should be a strong positive correlation between the measure of size and the
Forest Boundary
2) t 3 fe LE
Figure 26—Location of primary sampling units (shaded stands) and initial secondary sampling units using a superimposed grid. The secondary units are located at 60 degrees and 6,203 feet from one another. This provides a selection of stands based upon a probability proportional to their size.
actual variate of interest in order for p.p.s. sampling to provide improvements over simple random sampling. This is not strictly the case in the Enchanted Forest data, but for the purpose of this example there should be little loss in efficiency, and the acreage provides a convenient measure of size. When stands are sampled on the basis of proba- bility proportional to their area, several methods of select- ing samples are possible. An intuitively appealing method that yields a sample with probability proportional to area is to place a grid over the forest map. Only those stands in which a grid intersection is located are sampled in the field (fig. 26). Each grid intersection serves as the random
2a
=z
: (a | 0 Sot 0 FEET 5000 O Field Plot (Initial)
59
starting point for a cluster of sub-plots within the sampled stand. i
An alternative stand selection method is to list stands by area and then systematically select sample stands using a random start and a predetermined acre interval (Lund 1978b). Either method provides a systematic sample of the forest and a sample proportional to size of the stands (i.e., larger stands have a higher probability of being sampled). A further refinement for selection would be the prelimi- nary sorting of the a data base consisting of the measures of size followed by selection using pps. This refinement has some of the qualities of a systematic selection in that a representative distribution of the selected sample will usually be obtained (Stage 1971). Areas and volumes are developed by measuring the sampled stands in the field and expanding the sample to the forest.
Statistical Estimates—The results of a probability proportional to size inventory of the Enchanted Forest are shown in table 15. Note that stand 2 was selected twice (sampled with replacement). The stand is measured only once, but the stand values per acre are used twice in the calculations. Either grid intersection may be used as the initial starting point for the sample within stand 2. Forest estimates are computed as follows:
For snaileater area estimates, a value has to be assigned to each sampled stand. To compute the estimators for wild- life use, all sampled stands classed as having wildlife use are assigned a value of 1 and all other stands are given a value of 0.
Vp = (0+1+ ...0)/20 = 0.55 a proportion express- ing wildlife use per plot.
s*(¥,) = (0.55)(1-0.55) = 0.2475 s< = (0.2475/20)'”? 0.1112 (with replacement sampling equation). (0.1112/0.55)*100 = + 20.22%. 0.55 * 15,300 = 8,415 acres of wildlife use in the forest.
i}
° o » te. i]
< ll
For total volume estimates:
yY = (3 + 30 +... 19)/20 = 12.9 ccf per acre. sy = {(37 + 307+ ...197) — (3 + 30 +... 19)?/ 20}20 - 1) = 92.41. sy = (92.4105/20)"? = 2.15 ccf per acre. %S~ = (2.1495/12.90) * 100 = + 16.66 %.
Y = 12.90 * 15,300 = 197,370 ccf for the forest.
60
Table 15—Results of an inventory of the Enchanted Forest using selection of mapped stands based on probability proportional to their size; volume is expressed in ccf
Vegetation Wildlife Wildlife | Volume/ Volume Stand type use estimators acre estimators 2 Hardwood (0) 3 10 Conifer 1 30 67 Hardwood 1 6 2 Hardwood 0 3
16 Brush/open 1 3
33 Hardwood 1 7
70 Hardwood 1 Uf
30 Conifer 1 19
28 Hardwood 0 9
36 Conifer 0 32 175 Conifer 1 26
91 Conifer 0 8
97 Hardwood 1 19
53 Conifer 0 9 200 Hardwood 1 14 188 Hardwood 1 22 129 Brush/open 1 3 124 Conifer 0 18 166 Hardwood 0 1 152 Hardwood 0 19 rn 11 258 y 0.550 12.900 s,? 0.248 92.410 Sy 0.111 2.149 %s, + 20.21 + 16.66 Y 8,415. 197,370.
The estimated area and volume by vegetation type are computed similarly. To compute estimates for the conifer type, for example, all sampled stands not classed as conifer are assigned a value of 0 for volume and area. Table 16 shows the estimates for the conifer type. The area of conifer type is 5,355 acres, the area of hardwoods is 8,415 acres, and the area of brush/open is 1,530 acres.
The estimated total volume is 108,630 ccf for conifer, 84,150 ccf for hardwood, and 4,590 ccf for the brush/ open type respectively.
Stand Estimates—Measured stands can be assigned their actual measured values or in some cases it may be more appropriate to estimate their values as part of a
Table 16—Results and estimates of the conifer vegetation type from an inventory of the Enchanted Forest using mapped stands selected based on probability proportional to their area; volume estimates are expressed in ccf
Volume/ Stand acre
Volume Type estimators Type estimators
—
o
@ ooo
w@ o —_
_ Ni a hm w
—_
Le)
-
—_ Noor-;?oo0c0o-o0o0+-+-o0o+-o0o0cc0ce0=+-0
ive) N NOOWOOCWHCOAMNOCHCACCSO
“i =) = p-S
y 7.100 s° 127.463 SS 2.524 %S, + 35.55 108,630.
0.350
0.239
0.109 + 31.26 5,355.
general computational scheme. Stands that were not measured are assigned the average values of the inventory unit. The average volume per acre for the unmeasured stands is 12.50 ccf. Based on the sample, 35 percent of the area is conifer, 55 percent is hardwood and 10 percent is brush. See figure 27.
Cost Estimates—Selection was made by a grid. Each sample stand is approximately equidistant or 6,203 feet from each other. As stand no. 2 was selected twice but measured only once, n = 19 in this case.
L = [(19-1)*(6,203)]/10,560 = 10.57 hours. x (M,) = 96.5 hours.
D = (10.6 + 96.5)/8 = 13.4 hours.
F = 2(9)[10.6 + 96.5 + 13.4] = $2,169.
Thus:
Field costs = $2,169 Purchase aerial photos = 490 Mapping = 1,148 Total costs = $3,807, or $0.249 per acre.
Discussion—This technique is useful when only the boundaries and the area of the mapped polygon are known. In forestry, this is seldom the case. Characteristics of the stands are usually noted from remote sensing and are used for forming sampling strata. The advantage of the method over e.p.s. lies in the simplified computational formulae. All of the statistical estimates revert to the simplest expression of means and variances. The condi- tions under which these simplifications are justified have been noted above, but are worth reiterating. There must be a strong, positive correlation between the dependent variable and the auxiliary variable used as a measure of size. This may often be the case for timber volume and area of forest, though it is not really the case for our example. There is somewhat less likelihood that p.p.s. sampling formulae simplifications can be justified for the snail-eater. The dependent variable is a binary variable and can rarely have a strong correlation with a continuous variable-like area. Still, the method can be applied in a general inventory if interest in the snaileater is not a pressing wildlife issue that needs special attention. We would not suggest the application of this estimation- design combination if a high value were placed on the results.
Stratified Sampling—lf there is additional (auxiliary) infor- mation about each stand such as vegetation type (Ap- pendix 2), stratified sampling offers important advantages over almost any non-stratified method. Refer to the earlier exposition on stratification. Each and every stand must fall into one and only one sampling stratum. For timber inventories, Scott (1984) specifically recommends:
© Mapping stands based upon type, size, and density classes. Note that it may be preferable to stratify geographically because stand size often changes with intensity of management.
e Using these classes to form sampling strata. e Establishing plots systematically within selected stands.
e Developing means by stratum (vegetation type-size- density class) and expanding to total using stratum areas.
61
Data from sampled, mapped stands can be extrapolated to unmeasured areas assuming stands do not vary widely within sampling stratum or map classes such as forest type, stand size, and density (Ek, Rose, and Gregersen 1984). As previously discussed, poststratification or pre- stratification can be used.
Three sampling strata for the Enchanted Forest based upon apparent overstory vegetation type in each stand inter- preted from aerial photographs (see Appendix 2) are
=
a a | 0 FEET 5000
developed. The strata are conifer vegetation, hardwood vegetation, and brush/open. Figure 28 shows the stands mapped by each stratum. A total of 6,240 acres have been mapped as conifer, 7,965 acres have been mapped as hardwoods, and 1,095 acres mapped as brush/open veg- etation type.
Poststratification—Assume a grid of plots has been system- atically iocated as illustrated in figure 26. We can post- stratify by combining the grid and the mapping informa- tion, as illustrated in figure 29 and table 17. The mapped
Forest Boundary
Figure 27—Mapping showing measured (*) and Forest average (a) volume (ccf) per acre based on sampling proportional to stand size.
62
information provides the strata weights. We refer to this For the conifer stratum: process as poststratification with known weights.
Yo = (1+1+...0)/7 = 0.4286 a proportion ex- Statistical Estimates—For area and volume estimates , Bessie wildlife use per plot. ’ in the conifer vegetation type n. = 7, N. = 6,240, repre- Se= {141° +... O)-(+1+ ...0°/7}(7-1) = senting the same number of acres. 0.2857. (ss). = (0.2857/7)"? = 0.2020.
To compute the area estimators for the red-spotted (%S.)< = (0.2019/0.4286) * 100 = + 47.13 %. snaileater wildlife use within each stratum, all sample Vets 0.4286 ‘ 6,240 = 2,674 acres of wildlife use stands having evidence of wildlife use are assigned a value . in the conifer stratum. . of 1 and all other sample stands are given a value of zero. (S5“)_ = [0.2857 * (6,240/15,300)° /7] * [1 —(7/6,240]
= 0.006781.
Forest Boundary
Sa
= F 3
eg 7.
y Sz. ie es
Ge
ALE Lee “<
=z
Figure 28—Location of mapped stands in the Enchanted Forest showing £3 Conifer vegetation type classes. (0 Hardwood Brush/Open
63
The same estimators are computed for the hardwood and brush/open strata. The area estimates are combined for the Forest as follows where:
Y = (2,674.2857 + 4,344.5455 + 1,095) = 8,114 acres of wildlife use in the Forest.
Y = 8,114/15,300 * 100 = 53% of area sampled with wildlife use in the Forest.
Sy = (0.00678 + 0.00671)'? = 0.1162.
%S_e = (0.1162/0.5303) * 100 = + 21.90%.
Next we illustrate computation for total volume in the conifer stratum:
z
a 0 FEET 5000
Figure 29—Location of systematically located field plots and mapped stands in the Enchanted Forest. O Hardwood Brush/Open O Field Plot
Y. = (31 + 19 +... 18)/7 = 21.86 ccf per acre.
(s/7). = {317 + 197+ ... 18) — [(31+19 + .. . 34/77 -1) = 93.14. (5;). = (93.1429/7)"? = 3.6477 ccf per acre. (%S.)- = 3.6457/21.8571 * 100 = + 16.68%.
=< 0 ll
21.8571 * 6,240 = 136,389 ccf in conifer Stratum.
[93.1429 * (6,240/15,300)7/7 * [1 —- (7/ 6,240)] = 2.2108.
—= f oy) a) i]
The same stratum estimators are computed for the hard- wood and brush/open strata. The estimates are combined to yield totals for the Forest as follows:
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Table 17—Results of an inventory of the Enchanted Forest using post stratification of a systematic sample with mapped stands providing known stratum weights; volumes are expressed in ccf
Wildlife Wildlife | Volume/ Volume Stratum Plot use estimators acre estimators Conifer 2 1 31 8 1 19 10 0 34 11 0 29 12 1 14 14 0 8 18 0 18 rn, 3 153 y 0.429 21.857 SA 0.286 93.143 s;" 0.202 3.646 %s,* + 47.11 + 16.68 (S;*). 0.007 2.211 Y. 2,674. 136,388. Hardwood 1 0 1 3 1 8 4 0 3 6 1 7 if 1 10 9 0 10 13 1 17 15 1 13 16 1 21 19 0 0 20 0 20 rn, 6 116 y 0.545 10.545 si? 0.273 44.673 s;* 0.157 2.014 %s,* + 28.85 + 19.10 (S37)n 0.007 1.099 Vie 4,344 83,994. Brush/ 5 1 3 open 17 1 6 rn, 2 9 y 1.000 4.500 Sy 0.000 4.500 s;* 0.000 1.498 %s,* + 0.00 + 33.30 (S;)*b 0. 0.012 Ye 1,095. 4,927. Enchanted Y 8,114. 225,311. Forest Y 0.530 14.726 Sy 0.116 1.822 %S_* + 21.90 + 12.38
Y= (136,389 + 83,994 + 4,927) = 225,310 ccf in the Forest. Y = 225,310/15,300 = 14.73 ccf per acre. Sy = (2.211 + 1.099 + 0.012)'"7= 1.8225 ccf per acre. %S_ = 1.8225/14.7262 * 100 = + 12.38 %.
The estimated area and volume by vegetation type are computed in the same manner. To compute estimates for the conifer type, for example, all plots not classed as conifer are assigned a value of 0 for volume and area. The area of conifer type is 6,240 acres, the area of hardwoods is 7,965 acres, and the area of brush/open is 1,095 acres.
The estimated total volume is 136,388 ccf for conifer, 83,994 ccf for hardwood, and 4,928 ccf for the brush/ open type, respectively.
Stand Estimates—Stands containing the field plots may be assigned the values from the field measurements. We reiterate that provision be made to indicate the difference between an observed and predicted value. All other stands are assigned average values for the stratum in which they lie. See figure 30; observed values are indi- cated by asterisk.
Cost Estimates—The costs would be the same as for systematic sampling plus the purchase of aerial photogra- phy, interpretation of the entire Forest, and mapping.
Field costs = $2,275 Purchase aerial photos = 489 Mapping = 1,147 Interpretation = 306
Total costs $4,217, or $0.276 per acre.
Discussion—The National Forests in the Eastern (johnson 1984) and the Southern (Belcher 1984) Regions in conjunction with the North Central (Hahn 1984), Northeast, Southeast (Cost 1984), and Southern (Beltz 1984) Forest Inventory and Analysis Units of the Forest Service employ variations of this technique. FIA field plots serve as the basis for stratification and mapping by the National Forest Regions provides the stratum weights from stand mapping.
Prestratification—Assumptions regarding the known at-
tributes of the Enchanted forest remain the same as for nonstratified sampling examples.
65
66
=
¥
< SRO. = Sse, —s
Equal probability sampling (e.p.s..—Two or more stands are randomly chosen from each stratum (fig. 31). Consecutive stand numbers can be used as described in the previous section, or a unique stand number can be generated.
Statistical Estimates—Table 18 shows the results of the inventory, where, for the conifer (c) stratum, n, = 8 sample stands and N.. = 81 total stands. To compute the area of snaileater use, an indicator value is assigned to each sampled stand in the stratum. All sampled stands classed as having evidence of the wildlife use are assigned a value of 1 and all other sampled stands are given a value
Forest Boundary
Figure 30—Mapping showing measured (*) and stratum average (s) volume (ccf) per acre based on poststratification of systematically located
field plots.
The same area estimators are computed for the hardwood and brush/open strata. The estimates are combined for the
Forest as follows:
of zero. Computation follows examples presented earlier for a proportion estimator (equations 23 and 24):
Voc = ((60*0) + (75 * 1) + ... (45 * 1) (60'+ 75
+... 45) = 0.5588 wildlife use per plot. Y = (3,487 + 4,368+0) = 7,855 acres for which S*(V pe = 0.5588 * (1 — 0.5588) = 0.2465 wildlife use is estimated in the Enchanted Forest. (SV Je = (0.2465/8)'? = 0.1755. Y = 7,855 / 15,300 = 0.51 acres showing wildlife (%S.). = .1755/0.5588 * 100 = 31.41%. utilization.
Y. = 0.5588 * 6,240 = 3,487 acres of wildlife use Sy = [(977,021 + 637,985 + 0)/15,3007]'” =
in the Enchanted Forest. 0.0831. (S57). = [0.2465 * (6,240/15300)7/8] * [1 — 8/6,240] %S_ = (0.0831/0.5136)*100 = + 16.18 %.
= 0.00511.
Forest Boundary
RIP me 22
R Be,
Io
i ore
era 220
prea,
SALI ILE L ILS eeeee.
iene wintes
DEE Keres
“Ee
=
0 FEET 5000
€ Conifer aries) based upon a random drawing of stand numbers stratified by () Hardwood
Figure 31—Location of primary sampling units (stands with bold bouna-
vegetation type. Brush/Open
67
Table 18—Results of an inventory of the Enchanted Forest
Wildlife Wildlife Volume/ Volume
Stratum Conifer
Hardwood
Brush/ open
Enchanted Forest
Wildlife Wildlife Volume Stratum Stand Area use estimators acre _estimators_ 18 60 0 21 154 ae Al 23 10/8 at20) ot 30 53 30 O 9 81 90 0 7 92 45) nt 6 170 45 0 31 153 459 al 3 Ene 51091) 4 130 y 0.559 18.205 Sie 0.246 622,176. S; 0.176 3.436 %s + 31.41 + 18.87 Ve 3,487. 113,604. (S3)*c 00511 1.964 25 GOs 10 98 45 1 30 74 60 0 15 187 30) 1 14 144 45) 40 23 61 30) 44 12 152 7500 19 172 45 1 21 95 A5tae 25 116 30 O 2 In, 465 6 171 Vy 0.548 17.677 sy 0.248 111,780. Sy 0.150 1.299 %S, + 27.38 + 7.34 ve 4,367. 145,009. (S3)n .00627 0.457 26 30 O 0 62 30 O 0 rn, 60 0 0 Y 7,854.962 258,615. Y 0.513 16.902 Sy 0.109 1.556 %S_* + 21.20 + 9.20
using stratified mapped stands and equal probability sampling; area estimates are in acres and volume estimates are in ccf
—————— ee ee
68
For total volume estimates:
(Ye = [(60*21) + (75*23) +... (45*3)/(45 + 60 + ...45) = 18.21 ccf per acre. (547). = {[(60*21)? + (75*23)? +... (45*3)?) - [(2 * 18.2059) * ((60 * 21) + (75 * 23) +... + (45 * 3))] + [(18.20597) * (60? + 757 +... 45*))}/(8 — 1) = 622,176. (s,) = [1-(8/81)/8]"? * (15,300/200) * 622,176” = 0.1584 ccf per acre.
(%Sawe = (0.1584/18.2059) * 100 = + 28.35%. J. = 18.2059* 6,240 = 113,605 ccf for the conifer stratum. (S57). = [622176 * (6240/15300)7/8] * [1 — 8/6240] = 1.9644.
The same estimators are computed for the hardwood and brush/open strata. The estimates are combined for the Forest as follows where:
Y = (113,605 + 145,010 + 0) = 258,615 ccf in the Forest. Y = 258,615 / 15,300 = 16.90 ccf per acre. Sy = (1.964 + 0.457 + 0)”? = 1.556 ccf per acre. %Se = (1.5562/16.9029)*100 = +9.21%.
The estimated area and volume by vegetation type are similarly computed. To compute estimates for the conifer type, for example, all sampled stands not classed as conifer are assigned a value of 0 for volume and area. The area of conifer type is 6,240 acres, the area of hardwoods is 7,965 acres, and the area of brush/open is 1,095 acres.
The estimated total volume is 113,605 ccf for conifer, 145,010 ccf for hardwood, and 0 ccf for the brush/open type, respectively.
Stand Estimates—Sampled stand are assigned their observed values. Unmeasured stands are assigned the stratum averages. The average volume per acre for the unmeasured stands is 18.2 ccf for conifers, 17.7 ccf for hardwoods, and 0 ccf for brush. See figure 32.
Cost Estimates—The costs would be the same as Discussion—The USDI Bureau of Land Manage- unstratified equal probability sampling plus the cost of ment (Baker 1982) used this technique. This technique is
interpretation of aerial photos. easier to implement than probability proportional to size (see next paragraph), but the calculations are more com- Field costs = $2,244 plex. Purchase aerial photos = 490 Mapping = 1,148 Interpretation = 306
Total costs $4,188, or $0.274 per acre.
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=
Figure 32—Mapping showing measured (*) and stratum average (s) volume (ccf) per acre based on an equal probability sample of stratified stand numbers.
Probability Proportional to Size (p.p.s.) Sampling— A grid is superimposed over the stratified forest, as was done in figure 26. This provides a stratified sample of stands based on probability proportional to their acreage (fig. 33). At least two stands must be measured in each stratum. The initial plot within each selected stand is located at the grid intersection.
Statistical Estimates—Comments regarding the ap- plicability of p.p.s. methods to unstratified, whole forest are still valid. The relationship between the measure of
=
size and the objective variable needs to be positive and a strong correlation should exist; however, with the parti- tioning of the brush/open into a separate stratum, these assumptions are probably much more realistic. Still, the Enchanted Forest does not meet them as well as many real forests probably would (the method of generation for the stand volumes resulted in a distribution that does not strongly correlate with stand acreage). Table 19 shows the results of such an inventory.
For the area and volume estimates in the conifer vegeta- tion type, n. = 7, N. = 6,240.
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Bx?
Cz
70
Figure 33—Location of primary sampling units (stands with bold bound- Conifer
aries) and initial secondary sampling units using a superimposed grid C) Hardwood and stratification of the sample. The secondary units are located at 60 Brush/Open degrees and 6,203 feet from one another This provides a stratified O Field Plot sample based upon probability proportional to the area of the stratum. (Initial)
Table 19—Results of an inventory of the Enchanted Forest using stratified mapped stands and selection based upon probability proportional to their area; volume estimates are expressed in ccf
Wildlife Wildlife | Volume/ Volume Stratum Stand use estimators acre estimators Conifer 10 1 30 30 1 19 36 0 32 91 0 8 53 0 9 124 0 18 175 1 26 In, 3 142 y 0.429 20.287 Be 0.286 91.571 Sy 0.202 3.617 %S, + 47.14 + 17.83 (S;)*. 0.007 2.174 We 2,674. 126,582. Hardwood 2 0 3 2 0 3 67 1 6 33 1 7 70 1 7 28 0 9 97 1 19 200 1 14 188 1 22 166 0 1 152 0 19 rn, 6 110 y 0.545 10.000 s,? 0.273 53.600 0.157 2.207 %S, + 28.86 + 22.07 (S5)?h 0.007 1.319 vi 4,344. 79,650 Brush/open 16 1 3 129 1 3 rn, 2 6 1.000 3.000 SF 0.000 0.000 0.000 0.000 %S, + 0.000 + 0.000 (S3)*. 0.000 0.000 Vip 1,095. 3,285 Enchanted Y 8,114. 209,518. Forest Y 0.530 13.694 Sy 0.116 1.868 %S_* + 21.90 + 13.64
To compute the area estimators for snaileater use within each stratum, all sampled stands having evidence of wildlife use are assigned a value of 1 and all other sampled stands are given a value of 0. For the conifer stratum:
Yo = (14+1+...1)/7 = 0.4286 wildlife use per plot. (57). = {(17+17+ ...17) - (1 +14... 1)7/7}/(7 — 1) = 0.2857. (s). = {0.2857/7 * [1-(7/6,240)}}"7 0.2020
wildlife use per plot. (%5,). = (0.2020/0.4286) * 100 = + 47.14%. ¥. = 0.4286 * 6,240 = 2,674 acres of wildlife use in the conifer stratum. (S57). = [0.2857 * 6240/15300)7/7] * [1 —(7/6,240)] = 0.00678.
The same estimators are computed for the hardwood and brush/open strata. The area estimates are combined for the Forest as follows, where:
Y = (2,674 + 4,345 + 1,095) = 8,114 acres of wildlife use in the Forest.
Y = 8,113.8312/15,300 = 0.5303 wildlife use in the Forest. Sy = (0.00678 + 0.00671 +0)'” = 0.1162 wildlife
use per acre. %S_ = (0.1162/0.5303) * 100 = + 21.90%.
For total volume estimates:
Y. = (30+ 19 + ...26)/7 = 20.2857 ccf per acre. 6 7)W=i(307 + 197+ ... 267) — (30 + 19+... 26)?)/7}}/(7-—1) = 91.57. (5). = {(91.5714/7) * [1 — (7/6,240)]'* = 3.62 ccf per acre. (%S.). = (3.6169/20.2857) * 100 = + 17.83%. Y. = 20.2857 * 6,240 = 126,583 ccf in conifer stratum. (S37). = (91.5714 * 6,2407)/7] * [1 — (7/6,240)] = 508,796,000.
The same estimators are computed for the hardwood and brush/open strata. The estimates are combined for the Forest as follows where:
Y = (126,583 + 79,650 + 3,285) = 209,518 ccf in the Forest. Y = 209,518/15,300 = 13.694 ccf per acre. Sy = (2.173 + 1.319 + 0)'”= 1.8688 ccf per acre. %S_ = (1.8688/13.6940) * 100 = + 13.65%.
71
The estimated area and volume by vegetation type are similarly computed. To compute estimates for the conifer type, for example, all sampled stands not classed as conifer are assigned a value of 0 for volume and area. The area of conifer type is 6,240 acres, the area of hardwoods is 7,965 acres, and the area of brush/open is 1,095 acres.
The estimated total volume is 126,583 ccf for conifer, 79,650 ccf for hardwood, and 3,285 ccf for the brush/ open type, respectively.
Stand Estimates—Measured stands are assigned their actual values. Unmeasured stands are assigned the
=z
stratum averages. The average volume per acre for unmea- sured conifer stands is 19.5 ccf; 10.2 ccf for hardwoods; and 2.0 ccf for brush vegetation types. See figure 34.
Cost Estimates—The costs would be the same as unstratified probability proportional to size plus the cost of interpretation of aerial photos.
Field costs = $2,168 Purchase aerial photos = 490 Mapping = plats Interpretation = 306 Total cost = $4,112, or $0.269 per acre.
Forest Boundary
Figure 34—Mapping showing measured (*) and stratum average (s) volume (ccf) per acre based on stratified sampling of stands proportional
to their area.
7/2
Discussion—The Northern (Brickell 1984), Rocky Mountain (Mehl 1984), Intermountain (Myers 1984) and California Regions (Bowlin 1984) use this type of inven- tory design. As an alternative to a grid, Lund (1978b) uses accumulated acres within stands and strata. Stage (1971) provides a similar technique using a sorted list. The end result is nearly the same. The forest is systematically sampled; however, sorting the list assures that the sample will include a range of values for the auxiliary variable that approximates that for the population, an advantage that may well be needed if small numbers of plots are to be sampled. In both cases, the stratum and stands having the largest area have the higher probability of being sampled.
Forest Boundary
Inventories Using Existing Stand Information
In practice, some stand data may already exist from previous examinations or timber cruises. In such in- stances, provided the data are current and unbiased, the existing information may be used in addition to establish- ing new plots.
Assume data exist for stands 2, 10, 16, 28, 30, 33, 36, 53, 67, 70, 91, 97, 124, 129, 152, 166, 175, 188, and 200, as shown in figure 35.
=z
boundaries) in the Enchanted Forest.
0) Hardwood Brush/Open
73
The total volume and area by vegetation type in the stands that have already been measured are as follows:
Vegetation type Area (acres) Total volume (ccf)
Conifer 1,335 30,795 Hardwood 2,610 20,625 Brush/open 165 495 Total 3,360 51,915
The area of snaileater use is 2,280 acres. These estimates were obtained simply by summing the data from the stands that have been measured. This constitutes the existing inventory. Because all of the stands in the existing inventory were measured, there are no sampling errors.
There are two options for using existing stand data. These are: (1) to use the existing data as potential sampling unit information or (2) to combine the existing information with an inventory of the remaining areas. Example statis- tical calculations are shown for the second option. Cost estimates are not given because the options discussed are extensions of some of the above methods.
Use of the Stand as a Sampling Unit—If in the course of a new inventory, one of the stands for which data already exist is drawn for field sampling, then the data from that stand may be used to compile the inventory. For example, if stand 97 in the new inventory were selected, the stand would not have to be visited in the field, but the existing data from stand 97 would be used in the calculations for the forest. The Rocky Mountain Region (Mehl 1984) uses this procedure.
It is important to note that the stand must be selected in the course of the draw and not purposefully selected because the stand has already been inventoried. To do the latter can seriously bias the new inventory. Very often existing stand information results from recent stand exam- inations or timber cruises. Both types of inventories are usually conducted where access is good and timber volumes relatively high, and in areas that have just been or are about to be treated. Such areas may not be represen- tative of the forest as a whole. Hence, for this particular option, data from such sources should only be used when the locations are randomly selected.
This option has the advantage of reducing field costs, but
has the disadvantage of not making use of all the available information.
74
Combining Inventories—The second option is to use all of the existing data as a separate stratum, conduct an inven- tory of the remaining lands, and then add the figures together. »
To select a sample of the remaining stands, we will use stratified probability proportional to size sample selection. However, instead of using a grid to select the sample stands, we will use accumulated acres to illustrate another stand selection technique. This method is described in Lund (1978b).
To use the accumulated acres technique, list the unmea- sured stands and their area by strata, then, with the first stand listed, accumulate acres within the stratum as illustrated for the hardwood stratum in table 20.
Next, determine the number of sample stands to be visited in each stratum. For this exercise, select 10 samples in the hardwood stratum, 8 in the conifer stratum, and 2 in the brush/open vegetation type.
Next, determine the sampling interval (SI) and random start (RS) number for each stratum where:
SI = A; / n, (truncated to a whole number) (I)
Note the remainder (REM). For the hardwood stratum A, = 5,805 acres. SI = 5,805/10 = 580 acres with 5 remaining (REM).
The range (RG) from which we will choose a random start is RG = SI + REM (m)
For the hardwood stratum (A;), RG = 580 + 5 = 585 acres.
Lastly, choose a random number (RS) between 1 and REM. The stand having the accumulated acreage for that number is the first stand chosen for sampling. For the hardwood stratum, we randomly chose a number between 1 and 585. The random number we drew was 84. Stand no. 3 contains accumulated acres 16 through 90. Hence acre 84 falls within stand 3. This is our first sample.
Additional stands are selected at SI intervals until the desired number of samples have been drawn. Thus for the hardwood stratum, the sample 1 is acre 84, sample 2 is at 84+4580 or acre 664 (stand no. 23), sample 3 is at acre 1244 (stand 54), etc.
Table 20—Hardwood acres in uninventoried portions of the Enchanted Forest
Stand no.
Acres
Accumulated acres
Sample no.
Stand no.
116 119
Acres 30
45
60
Accumulated acres
3,285 3,315 3,375 3,465 3,510 3,540 3,615 3,660 3,705 3,750 3,810 3,885 3,915 3,975 4,035 4,080 4,170 4,215 4,260 4,320 4,365 4,425
Sample no.
10
7h
The same process is repeated for each stratum until all strata are sampled. There are a total of 5,805 acres in the hardwood type, 4,905 acres in the conifer type, and 930 acres in the brush/open class. A total of 10, 8, and 2 plots are chosen in each stratum respectively.
The stands sampled in this new inventory of the Enchanted Forest are shown in figure 36. Table 21 shows the inven-
tories combined using the existing data as a new stratum.
Statistical estimates—For area and volume estimates in the conifer vegetation type, n = 8 and N = 4,905 acres.
To compute the area estimators for the snaileater use
=
Dedede! Q FEET 5000
Figure 36—Location of previously measured stands (stands with bold boundaries) and new sampling units (n) in the Enchanted Forest.
76
within each stratum, all sampled stands having evidence of the wildlife use are assigned a value of 1 and all other sampled stands are given a value of 0.
Y¥.= (1+ 0+...0)/8 = 0.5000 wildlife use per
plot. (557), = {(17+07 +... 07)-[(1+0+ .. . 0)7/8]}/(8- 1) = 0.2857. (sy). = {0.2857/8 * [1 —(8/4,905)]}”? = 0.1890 wild- life use per plot. (%S~). = (0.1890/0.2857) * 100 = + 37.80%.
Y. = 0.2857 * 4,905 = 2,452 acres of wildlife use in the conifer stratum. (S57). = [0.2857 * (4905/15300)/8] * [1 — 8/4905] = 0.00366.
Forest Boundary
SSE SS
Wess
SCS ESSSS
Ao @
Conifer @ Hardwood SS Brush/Open
Table 21—Results of a combined inventory of the Enchanted Forest using stratified mapped stands and selection base upon probability proportional to their area; volumes are
expressed in ccf
Stratum Conifer 6
Hardwood 3
Stand
Wildlife
use
fFo-+-0+-00—
Net - + 0+++-0+0
Volume/ acre
16 12
Wildlife estimators
Volume estimators
14.875 70.696 2.973 + 19.98 0.906 72,961.
The same estimators are computed for the hardwood and brush/open strata as well as for the existing information. The snaileater area estimates are combined for the Forest
as follows where:
Nae (2,452 + 4,064 + 465 + 2,280) = 9,261 acres of wildlife use in the Forest. Y= 9,261/15,300
Forest.
0.6053 wildlife use in the
Sy = (0.00366 + 0.00335 + 0.00092 + 0)? = 0.0892 wildlife use per acre.
%oS_ = (0.0892/0.6053) * 100 = + 14.73%.
For total volume within the conifer stratum calculate the estimates as follows:
Yo = (16 + 12 +... 26)/8 = 14.88 ccf per acre. Gee lee e224 8 262) 10116) 124 .26)2/ 8}}/(8-—1) = 70.70.
Wildlife Wildlife Volume/ Stratum Stand use estimators acre
Hardwood y 0.700
si 0.233
Sy 0.152
%S, + 21.82
(S53)? 0.003
Y;, 4,063. Brush/open 20 1 0
75 0 3
rn, 1 3
y 0.500
Sia 0.500
Sy 0.500
%S, + 100.00
(S57), 0.001
Yb 465. Existing data Y, 0.623
Sf 0.000
Sy 0.000
%S, 0.00
(S3)*4 £0
We 2,280 Enchanted \/ 9,261. Forest Y 0.605
Sy 0.089
%S_* + 14.73
Volume estimators
19.100 89.211 2.987 + 15.63 1.282 110,875.
1.500 4.500 1.500 + 100.00 0.008 1,395.
14.184 0.000 0.000 0.00
+ 0. 51,915.
237,147. 15.499 1.482 + 9.56
(s;). = [(70.6964/8) * [1 — (8/4,905)]"? = 2.97 ccf
( % San Ve Pi
(S24)
y
per acre. (2.97/14.87) * 100 = + 19.97%.
14.8750 * 4,905 = 72,962 ccf in the conifer stratum. —
[70.696 * (4905/15300)7/8] * [1 — 8/4905] = 0.906.
The same estimators are computed for the hardwood and brush/open strata as for the existing information.
Y Y Sy
per acre.
% SE —
(1.4823/15.4998)*100 = + 9.56 %.
= (72,962 + 110,875 + 1,395 + 51,915) = 237,147 ccf in the remainder of the Forest. 237,147 /15,300 = 15.50 ccf per acre.
(0.906 + 1.282 + 0.008 + 0)? = 1.4823 ccf
The total estimates of area and volume by vegetation type are similarly computed. Combined estimates are 6,240 acres for the conifer type, 7,965 acres for the hardwood, and 1,095 acres for the brush/open type. Volume esti- mates are 103,757 ccf for the conifer type, 131,500 ccf for the hardwood, and 1,890 ccf for the brush/open type.
Stand Estimates—Previously measured and newly mea- sured stands are assigned their actual values. Unmeasured stands are assigned the stratum averages from the new inventory. The average volume per acre for unmeasured conifer stands is 14.9 ccf, 19.1 ccf for hardwoods and 1.5 ccf for brush vegetation types. See figure 37.
3*
=z
(oan
Discussion—this application makes use of all available information and permits the calculation of sampling errors in which a good deal of credibility can be placed. It is the method that would be preferred if there were no additional information on portions of the forest.
Complete Enumeration
Complete enumeration of stands requires visiting and mea- suring every stand in the compartment. Sample plots are virtually always established within the stands rather than measuring every tree. The Southern (Belcher 1984) and Eastern Regions johnson 1984) have used this technique.
Forest Boundary
Figure 37—Mapping showing measured (*) and stratum average (s) volume (ccf) per acre based on the combined inventory of the Enchanted Forest.
78
Statistical Estimates—The results of a complete enumera- tion of the Enchanted Forest are given appendix 2. The red-spotted snaileater usage is 7,905 acres. Volume results are shown in figure 38 and table 22.
Since all stands are represented in the sample, a sampling error ought not be computed from the equations presented in this publication. It may be possible to approximate a sampling error using a more sophisticated statistical ap- proach such as jackknifing or simulation if there is some way of obtaining the within-stand variation. Usually, we assume that within-stand variation is very much smaller than between-stand variation in the computation of sampling
Forest Boundary
Gi a Bos
Figure 38—Mapping showing volume (ccf) per acre using complete
enumeration of all stands in the Enchanted Forest.
es Bo
f
us
ct
Table 22—Results of complete enumeration of stands in the Enchanted Forest; volumes are expressed
in ccf Vegetation type Acres Total volume Volume per acre Conifer 6,240 106,515 17.05 Hardwood 7,965 112,170 14.08 Brush/open 1,095 1,410 1.29 Total 15,300 220,095 14.39
<
ae
ae
x
ee
es
79
error; however, this may not be the case and in the case of complete enumeration some estimate of within-stand vari- ability could provide us with valuable information on the remaining variability. These procedures might often be ap- propriate, but require substantial statistical computing and are beyond the scope of this publication.
Cost Estimates—Purchase of aerial photography and map- ping is required. Each stand is visited in the field and 10 plots are established in each stand. Assume that it takes 1 hour to move between stands and to start the measure- ment of each stand.
L = (200-1)(1) = 199 hours.
x (M,) = 1,011 hours.
D = (199 + 1,011)/8 = 151.3 hours.
F = 2(9)[199 + 1,011 + 151.2] = $24,503.4
Field costs = $24,503 Purchase aerial photos = 490 Mapping = 1,148 Total costs = $26,141, or $1.709 per acre.
Note that there is no photo interpretation, as all informa- tion will come from the field samples.
Summary of Forest Inventories
The inventory objectives were to estimate the area used by the red-spotted snaileater, the total volume of the forest by cover type, and the volume per acre for each stand.
We repeat the warning that no statistical comparison of the reliability of forest inventories is possible from the exam- ples presented. Additional replications in different situa- tions would be required. Nevertheless, some general observations can be made.
Table 23 shows the results and costs for estimating total ccf volume by the various designs described. Costs were recomputed to a common + 10 percent sampling error using equation (f).
Statistical Estimators—Estimates are reported for both the Forest as a whole and for individual stands.
Forest Estimates—Of the designs illustrated, the stratified double sample tended to give more precise total results and the systematic sample and unstratified probability proportional to size sampling the least. This must be attributed to the characteristics of the population being sampled. In the latter case the more sophisticated sam-
80
pling designs were in fact applied as if there was ignorance of the major differences between the volumes in the two forested strata and the essentially nonforested brush/open stratum.
Calculations, when probability-proportional-to-size sam- pling is used, are less complicated than when stands of unequal size are selected with equal probability, as previ- ously illustrated. Oderwald, Wellman, and Buhyoff (1979) confirm this observation. The simplicity of estimation procedures is a factor in favor of probability-proportional- to-size (area) sampling of stands. The actual calculation will likely be done with a computer in the context of a developed inventory system, but the simplicity of the estimation equations will aid understanding by users. Probability-proportional-to-size sampling is usually with replacement. Thus sampling probability is proportional to size, and the constant of proportionality does not vary between the selection of one unit and the next. In sampling small populations some precision is given up by sampling with replacement. The finite population correc- tion (fpc) cannot be used. Probability-proportional-to-size sampling without replacement is possible, but, because the constant of size proportionality changes each time a sampling unit is selected, the error estimation equation becomes more complicated. The characteristics of the population to be sampled and the distribution across subsampled elements of the population must be consid- ered when applying a computational method.
All options showed a reduction in sampling error when stratification was introduced. Even with the small area of the forest and relatively few stands, the benefits of sam- pling are easily seen.
Assuming the same variation and number of stands sam- pled, the sampling error would remain nearly constant for forests near the size of the Enchanted Forest and having a similar number of stands (i.e., measure only 20 stands of 1,000 stands using stratified probability proportional to size sampling and arrive at a sample error of + 15 percent for the forest). However, the area of the inventory should remain at approximately the same order of magnitude. Thus, if one had a forest of 1 million acres divided in 50,000 stands, one could not use the same technique and intensity to achieve the same sampling error. Several factors weigh on this. The population change from one scale to the next is probably the most significant factor. It is unlikely that an inventory performed on an area of 150,000 acres would retain the same variability. A second consideration is that it is unlikely that the objectives of a
Table 23—Summary of achieved sampling errors for estimating total volume (ccf) for various designs for the inventory of
the Enchanted Forest
Inventory design
Sampling error Field costs Field costs’
Imagery Interpretation Mapping Total costs’ Cost/acre’
% $ $ $ $ $ $ Systematic 15.4454 2,275.31 5,427.99 0.00 0.00 0.00 5,427.99 0.355 System. w/poststratification 12.2032 2,275.31 3,388.35 0.00 0.00 0.00 3,388.35 0.221 Strip cruise 12.4805 2,275.31 3,544.09 0.00 0.00 0.00 3,544.09 0.232 Double samp. w/estimated wts. 7.9957 2,275.31 1,454.63 489.60 1.60 0.00 1,945.83 0.127 Stratified satellite imagery 10.0082 2,275.31 2,279.04 61.20 382.50 306.00 3,025.75 0.198 Equal probability sampling 9.3388 2,243.59 1,956.71 489.60 0.00 1,147.50 3,593.81 0.235 Probability prop. to area 16.6631 2,168.43 6,020.84 489.60 0.00 1,147.50 7,657.94 0.501 Stratification w/known wts. 12.3757 2,275.31 3,484.82 489.60 306.00 1,147.50 5,427.92 0.355 Stratified e.p.s. 12.9600 2,243.59 3,768.37 489.60 306.00 1,147.50 5,711.47 0.373 Stratified p.p.s. 13.6466 2,168.43 4,038.26 489.60 306.00 1,147.50 5,981.36 0.391 Combined inventories 9.5630 2,168.43 1,983.05 489.60 232.80 1,147.50 3,852.95 0.252 Complete enumeration 0.0000 24,496.16 24,496.16* 489.60 0.00 1,147.50 26,141.40* 1.709*
' At 10-percent sampling error computed using equation f.
* Cost estimates for complete enumeration cannot be computed for a 10-percent sampling error.
much larger inventory would remain the same as that of the small area. Additional criteria for initiating an inven- tory of an area larger by an order of magnitude are very likely to increase the required sampling intensity in order that important resources receive sufficient samples to allow for meaningful results. As a general rule, as the size of an inventory increases the practical intensity may have to increase unless strong correlation to auxiliary data exists. Of course, by increasing the sample size (number of plots or stands visited) the sample error would decrease.
By using probability proportional to size sampling and stratification, one can reduce the sample error by using the same sample intensity or achieve the same sample error with less field work (i.e., measure fewer stands).
When all strata of interest are known, prestratification is preferred to poststratification because one is assured that all strata of interest can and will be sampled. The same cannot be said of poststratification.
Individual Stand Estimates—The ability to generate statis- tics for each stand depends on how much prior knowl- edge is available.
Appendix 3 shows side by side comparisons of volume estimates by various designs for the stands within the Enchanted Forest. As one might expect, there are consid- erable differences between the assigned values and the ground truth. Again stratification appears to improve the resolution of the inventory. While a sampling error cannot be computed for each stand, estimates of the sampling error are available for each stratum.
Where stratification is not used, the sampling error of the whole forest may be examined to give some hint as to the reliability of each stand estimate though prior knowledge is limited.
Stand estimates can be further refined using inventory data. If relationships can be established between variables that can be easily interpreted from aerial photos (such as height) and variables that are best measured or determined from field observations (such as volume), prediction equa- tions can be developed to assist in future stand mapping. For example, assume overstory heights (h) were estimated for each photo plot. A regression is developed between the heights and volume for photo points that were measured in the field (see Freese (1962) for formulation), where volume per acre (v) is:
v = —0.3271 + 0.317h (h) (25)
When the height (h) = zero, v is set to zero, and the coefficient of determination is 0.94.
The equation can be applied using the nonfield measured photo points to derive volume estimates for each photo point (see fig. 18). When the stands are eventually mapped, the overstory heights can be interpreted from aerial photos and volumes per acre can be predicted for each stand without further field work. Lund (1974) used this technique for forest inventories in the U.S. Depart- ment of Interior, Bureau of Land Management, and Lund and Kniesel (1975) used the same process to predict multiresource values such as deer-days use and forage production.
81
If the stands are already mapped and heights interpreted, as in the case of the stratified probability proportional to area sample, the predicted values can be directly applied and a volume map generated. The second-to-last column of Appendix 3 and figure 39 give the results of such an exercise using equation (24). This technique was used by Brickell (1984) and is an economical technique for obtain- ing stand estimates. Langley (1983) used a similar tech- nique and incorporates the results in a geographic infor- mation system.
Zz
[Eerie 0 FEET 5000
Cost Estimates—The comparison of inventory design costs involving no prior mapping versus prior mapping is clear. If it is the case that mapping is never to be undertaken, then our examples indicate that the costs do not justify if for inventory alone. Overall, the actual cost incurred for inventories without mapping was less. This result may be due primarily to assumptions. However, the eventual use of an inventory by a forest manager and resource specialist virtually always requires stand mapping. Thus, the cost of mapping is just delayed, not avoided. Prior mapping is
xy
82
Figure 39—Mapping showing measured (*) and predicted volumes (cc?) per acre based upon a stratified sample of stands proportional to their size and predictions base upon photo-interpreted heights of the overstory.
cheaper in the long run because the area information is collected only once. If mapping is delayed, the area information must first be derived from the sample plots and then be rectified when mapping is complete; this incurs additional time and cost that is avoided with prior stand mapping.
The comparison of costs necessary to achieve a +10 percent sampling error shows that the gains in sampling precision may outweigh the costs of mapping.
The cost of introducing randomization into an inventory design that provides the foundation for a statistical sample is minimal. The cost of an inventory of two stands that are selected for probability sampling is no greater than one based on a subjective sample with preconceived bias.
Inventory costs generally tend to increase with increasing complexity of the sampling design. Equal probability sampling may be more or less expensive than probability- proportional-to-size sampling. When using probability- proportional-to-size sampling, larger area stands are more likely to be selected than in equal probability sampling. Measurement of larger stands may require more field time than small stands because more area has to be traversed. It should be clear that only slightly more large area stands would be included; thus increases in costs could be minimal.
Stratified sampling also offers savings. In nearly all the cases demonstrated here, stratification offered lower total costs at equitable sampling errors even though the costs of forming the stratum were added.
MacLean (1972) confirms this relationship. In general, stratified sampling results in increased information for a given cost because (Mendenhall, Ott, and Scheaffer SA):
1. The data are usually more homogeneous within each stratum than in the population as a whole. Hence, fewer samples are usually needed.
2. The cost of conducting the actual sampling may be less for stratified random sampling than for simple random sampling because of administrative convenience. Plots falling in the brush stratum, for example, could possibly have been measured with a one person crew. Without that prior knowledge, a two person crew would have been sent to the plot location.
3. Separate estimates of population parameters can be obtained for each stratum without additional sampling. Estimates for each vegetation type were assured, where as without stratification this could not be guaranteed.
Key to Options—A rough key or guide to selection of inventory designs based on available information is as follows:
1. Stand mapping available? a. Yes. Go to 4. b. No. Go to 2.
2. Aerial photography available? a. Yes. Use stratified double sampling. b. No. Go to 3.
3. Satellite imagery available? a. Yes. Use stratified satellite technique. b. No. Use systematic sample.
4. Stand characteristics available? a. Yes. Use stratified equal-probability sampling or probability-proportional-to-size sampling. b. No. Use equal probability sampling or probability- proportional-to-size design.
In all cases, existing data should be incorporated appro- priately. In evaluating existing information, consider the age of the data and the definitions and standards, sample design, and control used in gathering the information (Lund and Schreuder 1980). Old information may be updated by accounting techniques (i.e., subtracting timber harvested) or by modeling techniques to “grow” the stands forward in time.
83
Conclusions
There are many ways of obtaining stand and forest infor- mation to produce spatially distributed resource informa- tion. They range from measuring every tree in every stand to the use of very light samples and multistage or mul- tiphase sampling, including satellite imagery at the highest level. Each method, indeed even repetitions of the same method, will produce slightly different results. This report has presented some of the most common designs in use by the USDA Forest Service and other agencies along with the statistics that allow us to judge the accuracy of these designs.
The objective of the inventory and the funds available will determine which technique to use. Complete enumera- tion is often impractical unless the benefits of increased precision outweigh the costs, as might be the case for an extremely valuable resource. Even then, the measurement error of a complete enumeration may be larger than the error obtained from sampling.
No matter which sampling technique is used, it is always assumed that the sampling unit represents the actual condition. This is a serious consideration that is often evaded in the preparation of an inventory. It is extremely important that the sample represent the population of interest. For example, a low-quality product of the forest, such as hardwood removals in a primarily softwood market area, might be poorly represented in a sample designed to monitor the flow of harvested softwoods.
Statistical sampling and subjective sampling should not be mixed. If, for example, one selects stands to be measured using probability proportional to the size of the mapped unit and then measures the stand using subjective sam- pling, inferences about the result, estimates of reliability, and other desirable characteristics of sampling will be compromised. If stands within a compartment are subjec- tively selected for sampling and plots are randomly estab- lished within the stand, one should not attempt to com- pute the variance and.sample errors for the compartment using the formulas presented here. Errors may be calcu- lated for the stand, however. There are valid techniques for arriving at an estimate of the variance, but they should be prepared by a statistician.
For stand inventories, the systematic distribution generally gives the best results, as long as there are no systematic regularities in the forest that correspond to the sample installations. For forest inventories, at least based on the examples given in this report, prior stand mapping is desirable and stratification reduces the sampling error. Stand estimates can be generated by prediction equations where correlations are relatively high (say greater than 0.6 to 0.7). Where correlations are low, stratification is also useful for providing rough stand estimates based upon stratum means.
Where stand mapping is not available, the use of cells, isolines, partial mapping, or digital satellite reconnais- sance may yield useful spatial information.
The examples presented in this chapter represent simple, straightforward application of basic statistical sampling formulae to increasingly complex inventory situations. Estimates of area, wildlife use, and wood volume may all be computed using these basic statistical formulae. While we are recommending the application of these design and computational formulae for many situations, we are also obliged to advise readers that there have been important advances in the area of survey (inventory) sampling. When a single attribute is the major interest, there are sampling strategies that can provide important gains in efficiency and cost that are beyond the scope of this primer (e.g., Schreuder and Wood 1986; Green 1987; Gregoire et al. 1987).
Even though the examples used in this report dealt gener- ally with a timber inventory situation, the options and techniques can be used for most resource inventories, such as surveys of wildlife habitat and range allotments.
Readers are encouraged to consult the publications listed in the sections on References Cited and Additional Se lected References. These provide the details not contained in this report.
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87
Additional Selected References
Mapping, Classification, Remote Sensing
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Avery, Thomas Eugene. 1978. Forester’s guide to aerial photo interpreta- tion. Agric. Handb. 308. Washington, DC: U.S. Department of Agri- culture. 41 p.
Bonnor, G. M. 1968. A comparison of photo and ground measurements of canopy density. Forestry Chronicle 44(3): 12-16.
Driscoll, Richard S.; Merkel, Daniel L.; Radloff, David L.; Snyder, Dale E. 1984. An ecological land classification framework for the United States. Misc. Publi. 1439. Washington, DC: U.S. Department of Agriculture. 56 p.
Greer, Jerry D., ed. 1988. Remote sensing for resource inventory, planning and monitoring: Proceedings of the Second Forest Service Remote Sensing Applications Conference; 1988 April 11-15; Slidell, LA and NSTL, MS. Falls Church, VA: American Society of Photogram- metry and Remote Sensing. 445 p.
Klock, G. O.; Jordan, L. E., Ill; Gum P. 1986. A forest vegetation inventory with Landsat imagery for use with fire management strategies. In: Proceedings of geographic information systems workshop; 1986 April 1-4; Atlanta, GA. Falls Church, VA: American Society for Photogram- metry and Remote Sensing: 293.
Jaakola, Sipi; Poso, Simo; Roiko-Jokela, Pentti. eds. 1987. Remote sensing-aided forest inventory. Seminars organized by SNS (Samarbet- snamden fur Nordisk Skogforskning) and Taksaattoriklubi (Forest Men- surationist Club). 1986 December 10-12; Hyytiala, Finland. Res. Notes. No. 19. Helsinki, Finland: University of Helsinki. 215 p.
Mybhre, Richard J. 1982. The role of aerial photography in multiresource inventories—techniques and tests in applications research. In: Techni- cal papers: 1982 ACSM-ASP Convention, American Society of Photo- grammetry; 1982 March 14-20; Denver, CO. Falls Church, VA: American Society of Photogrammetry: 585-594.
Paine, David P. 1981. Aerial photography and image interpretation for resource management. New York: John Wiley and Sons. 571 p.
Poso, Simo., ed. 1988. Satellite imageries for forest inventory and monitoring: experiences, methods, perspectives. Proceedings of the IUFRO Subject Group S 4.02.05 Meeting; 1988 August 29-September 2: Helsinki, Finland. Res. Notes. No. 21. Helsinki, Finland: University of Helsinki. 214 p.
Valentine, K. W. G. 1981. How soil map units and delineations change with survey intensity and map scale. Canadian Journal of Soil Science 61(4): 535-551.
Wertz, W. A.; Arnold, J. F. 1972. Land systems inventory. Ogden, UT: U.S. Department of Agriculture, Forest Service, Intermountain Region. 12 p.
Statistics and Sampling
Burk, T. E.; Ek, A. R. 1982. Application of empirical Bayes/James-Stein procedures to simultaneous estimation problems in forest inventory. Forest Science 28: 753-771.
Byrne, John C.; Stage, Albert R. 1988. A data structure for describing sampling designs to aid in compilation of stand attributes. Gen. Tech. Rep. INT-247. Ogden, UT: U.S. Department of Agriculture, Forest Service, Intermountain Research Station. 20 p.
Draper, N. R.; Smith, H. 1982. Applied regression analysis. 2d ed. New York: John Wiley and Sons. 407 p.
Freese, Frank. 1967. Elementary statistical methods for foresters. Agric. Handb. 317. Washington, DC: U.S. Department of Agriculture. 87 p.
Houseman, Earl E. 1975. Area frame sampling in agriculture. SRS No. 20. Washington, DC: U.S. Department of Agriculture, Statistical Reporting Service. 79 p.
Slonim, Morris James. 1960. Sampling. New York: Simon and Schuster. 144 p.
Stage, Albert R. 1974. Playing with probabilities: one key to meeting information objectives. In: Frayer, W. E.; Hartman, George B.; Bower, David R., eds. Inventory Design and Analysis: Proceedings of a workshop; 1974 July 23-25; Fort Collins, CO. Fort Collins, CO: Colorado State University: 18-37.
Plot Configuration
Beers, T. W.; Miller, C. |. 1964. Point sampling: research results, theory, and applications. Lafayette, IN: Purdue University Agricultural Experi- ment Station; Res. Bull. 786. 56 p.
Dilworth, J. R.; Bell, J. F 1963. Variable plot cruising. Corvallis, OR: Oregon State University Book Stores, Inc. 107 p.
Meeuwig, Richard O.; Budy, Jerry E. 1981. Point and lineintersect sampling in pinyon-juniper woodlands. Gen. Tech. Rep. INT—104. Ogden, UT: U.S. Department of Agriculture, Forest Service, Intermoun- tain Forest and Range Experiment Station. 38 p.
Pechanec, Joseph F; Stewart, George. 1940. Sagebrush-grass range sampling studies: size and structure of sampling unit. American Society of Agronomy Journal 32(9): 669-682.
Stage, A. R. 1958. An aid for comparing variable plot radius with fixed plot radius cruise designs. Journal of Forestry. 56(8): 593.
Wiant, Harry V., Jr 1987. When square plots are more efficient than circular ones. West Virginia Forestry Notes Circular 142: 8.
Wiant, Harry V., Jr; Yandle, David O. 1980. Optimum plot size for cruising sawtimber in eastern forests. Journal of Forestry 78: 642-643.
Zeide, Boris. 1980. Plot size optimization. Forest Science 26: 251—257.
Forest Inventory
Frayer, W. E., ed. 1979. Forest resource inventories—volumes | and II. Proceedings of a workshop; 1979 July 23—26; Fort Collins, CO. Fort Collins, CO: Colorado State University, Department of Forest and Wood Sciences. 1037 p.
Frayer, W. E.; Hartman, George B.; Bower, David R., eds. 1974. Inventory design and analysis. Proceedings of a workshop; 1974 July 23—25; Fort Collins, CO. Fort Collins, CO: Colorado State University. 368 p.
Husch, B. Planning a forest inventory. 1978. FAO Forestry Ser. 4, Forestry and Forest Product Studies 17. Rome: Food and Agriculture Organiza- tion. 121 p.
Husch, Bertram; Miller, Charles |.; Beers, Thomas W. 1972. Forest mensuration. 2d ed. New York: Ronald Press. 410 p.
Kuusela, Kullervo; Nyyssonen, Aarne, eds. 1983. Forest inventory for improved management. Proceedings of the IUFRO Subject Group 4.02 meeting; 1983 September 5—9; Helsinki, Finland. Re-earch Notes No. 17. Helsinki, Finland: University of Helsinki, Dept. of Forest Mensu- ration and Management. 207 p.
Loetsch, F.; Zoehrer, F.; Haller, K.E. Forest inventory volume II. Munchen, West Germany: Bayerischer Landwirtschaftsverlag Gmbh. 1973. 469 p.
Spurr, Stephen H. 1952. Forest inventory. New York: Ronald Press. 476 p.
Special Inventories
Costello, Thomas R.; Pettijohn, Linn. 1978. Intensive forest inventory—a selected bibliography. Tech. Note 319. Denver, CO: U.S. Department of Interior, Bureau of Land Management, Denver Service Center. 61 p
Food and Agriculture Organization of the United Nations. 1981. Manual of forest inventory with special reference to mixed tropical forests. Forestry Paper 27. Rome: Food and Agriculture Organization of the United Nations. 200 p.
Fox, D.G.; Bernabo, J.C., Hood, B. 1987. Guidelines for measuring the physical, chemical and biological condition of wilderness ecosystems. Gen. Tech. Rep. RM-—146. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky Mountain Forest and Range Experi- ment Station. 48 p.
LaBau, Vernon J.; Kerr, Calvin L., eds. 1984. Inventorying forest and other vegetation of the high latitude and high altitude regions. Proceedings of an international symposium; 1984 July 23-26; Fairbanks, AK. SAF 84-11. Bethesda, MD: Society of American Foresters. 296 p.
Lund, H. Gyde; Caballero, Miguel; Hamre, R. H.; Driscoll, Richard S.; Bonner, William, tech. coords. 1981. Arid land resource inventories: developing cost-efficient methods: Proceedings; 1980 November 30- December 6; La Paz, Mexico. Gen. Tech. Rep. WO-28. Washington, DC: U.S. Department of Agriculture, Forest Service. 620 p.
Lund, H. Gyde; Caballero-Deloya, Miguel; Villarreal-Canton, Raul, eds. 1987. Land and resource evaluation for national planning in the tropics: Proceedings of the international conference and workshop; 1987 January 25-31; Chetumal, Mexico. Gen. Tech. Rep. WO-39. Washington, DC: U.S. Department of Agriculture, Forest Service. 524 p.
Lund, H. Gyde; LaBau, Vernon J.; Ffolliott, Peter F.; Robinson, David W., tech. coords. 1978. Integrated inventories of renewable natural re- sources: Proceedings of the workshop; 1978 January 8-12; Tucson, AZ. Gen. Tech. Rep. RM—55. Fort Collins, CO: U.S. Department of Agriculture, Forest Service, Rocky Mountain Forest and Range Experi- ment Station. 482 p.
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Schlatterer, Ed; Lund, H. Gyde, eds. 1984. Proceedings of the inventory integration workshop; 1984 October 15-19; Portland, OR: Washing- ton, DC: U.S. Department of Agriculture, Forest Service, Range and Timber Management Staffs. 165 p.
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Monitoring
Bell, John F.; Atterbury, Toby, eds. 1983. Renewable resource inventories for monitoring changes and trends; 1983 August 15-19; Corvallis, OR. SAF 83-14. Corvallis, OR: Oregon State University; 1983. 737 p.
Cunia, Tiberius, ed. 1974. IUFRO proceedings, monitoring forest envi- ronment through successive sampling; 1974 June 24-26; Syracuse, NY. Syracuse, NY: State College of Environmental Sciences and Forestry, University of New York. 390 p.
Curtis, Robert O. 1983. Procedures for establishing and maintaining permanent plots for silvicultural and yield research. Gen. Tech. Rep. PNW-155. Portland, OR: U.S. Department of Agriculture, Forest Service, Pacific Northwest Forest and Range Experiment Station. 56 p.
Schmid-Haas, Paul., ed. 1986. Inventorying and monitoring endangered forests. Proceedings of IUFRO conference; 1985 August 19-24; Zurich, Switzerland. Birmensdorf, Switzerland: Eidgenossiche Anstalt fur das forstliche Versuchswesen. 404 p.
Schmid-Haas, Paul., ed. 1988. Inventorying and monitoring forests. Proceedings of the 18th IUFRO World Congress; 1986 September 7-21; Ljubjana, Yugoslavia. Nogent s/ Marne, France: Association Silva Arbres, Forets et Societes. Pages unnumbered.
Synnott, T. J. 1979. A manual of permanent plot procedures for tropical rainforest. Trop. For. Pap. 14. Oxford, England: University of Oxford, Commonwealth Forestry Institute. 67 p.
89
Appendix 1: Equations and Formulas Used in
Text
Statistical Estimators
A=
( yv2 =
90
The total area of the inventory unit in acres.
The area of a sampling unit or a plot in acres.
The number of sampling units or plots established.
The value for item of interest, such as volume per acre (ccf), measured or observed at each plot location.
The square root of the parenthetical expression.
The total number of possible sampling units in the entire population where:
N = A/a (1)
The estimated mean value of interest, such as volume per acre (ccf) where:
y = (2 ypin (2)
The estimated variance of individual values of y where:
sy? = {2 y-(@ y)7/n}/(n-1) (3)
The estimated standard deviation of y where:
S me (Sn (4)
The estimated standard error of the mean for a simple random sample. For sampling without replacement (*):
ss* = {s,7/ nm * [1-(n/N)]} 1? (5) Y) y
or for where sampling is with replacement:
Se = (s,7/n)"/? (6)
The expression [1—(n/N)] is known as the finite population correction or f.p.c. If (n/N) is less than 0.05, the f.p.c. is commonly ignored and equation (6) is used.
The estimated sampling error of the mean value such as mean volume per acre (ccf) where:
s.=5/¥Y (7)
The estimated sampling error of the mean value (such as mean ccf volume per acre) expressed as a percent where:
%S. = (s; / y)*100 (8)
The estimated total value (such as total ccf volume) in the population where:
a
Y=y*A (9)
The estimated number of sampling units necessary to sample within certain prescribed precision and confidence limits.
n= [(t* sis. * WP (10)
Students “t’ which is a_ value establishing a level of probability. The values of “t’ have been tabulated and are available in most statistical textbooks including those referenced in this report.
A first approximation guess of the standard deviation from a very small or preliminary survey.
sp = B/3 or B/4 (11) the estimated range from the smallest to the largest value likely to be encoun-
tered in sampling.
The field plot expansion factor. EF =A/n (12)
The area in stratum i. In the primer i = c,h,b for example.
A; = A = (n; / N)) (13) number of plots in stratum.
total number of plots in inventory.
=<
%o SE Mae
where a;
weighted stratum variance.
(S57); = [(s? * (N/N)?/n] * [1-(n/ Nj] (14)
The total value for a resource in the Enchanted Forest.
Y = X(Y;) (15)
The mean value in the Enchanted Forest.
Y=Y/A (16)
The standard error of the mean for the Forest.
Sy = [2(S;7)1"” (17)
The estimated sampling error of the mean value for the Forest expressed as a percent where:
%S,_ = (Sy / Y)*100 (18)
The estimated mean volume per acre for the Forest when using equal probability sampling of stands. An __ alternative description of the estimator, y,, is the weighted mean of stand per acre values, with the per acre value for each stand weighted by its acreage.
Vw = 2wiy; / 2w; (19) area in acres in sampled stand and
value per acre in sampled stand.
The weighted variance.
S,7? = Zw; (y; - Yy) / Zw; (20)
the weighted standard error of the mean value.
Sy = (S,7/n)"? (21)
the percent sampling error.
%s. = (S/Vy) * 100 (22)
expected vaiue for a proportion.
Yp = 2(n; _ /n, (23)
where n, is the sample size and n; _ ; is the number of
occurences.
s*(V,) = the variance of a proportion.
s*(V,) = ¥, (1 - ¥,) (24) v= The predicted volume per acre using
photo interpreted heights of the
overstory.
v = -—0.3271 + 0.317h (h) (25) where h = height of the overstory in feet. When the
height (h) = zero, v is set to zero.
Cost Estimators
C= The size of the field crew. Size of crew = 1 person for subjective samples; 2 persons for statistical samples and complete enumeration.
W = The hourly wage per person in dollars. Hourly wage = $9.00 per person.
M = The time per crew to measure each sampling unit in hours. Plot measurement time in hours = 0.167
hour for subjective samples; 0.5 hour for statistical samples; and 1 hour for complete enumeration.
nN = The number of sampling units to be measured. L= The travel time between sampling units
in hours. Time (in hours) traveling between sampling units (L) varies with distance or interval between plots (I) or (i) and number of sampling units (n). It is assumed that a crew travels at a speed of 10,560 feet per hour through the woods. For statistical sampling:
L = [(n—1)iJ/10,560 (a) where i = interval in feet between sample plots or points.
oi
D= The daily travel time to and from the ~inventory unit in hours (D). For simplicity it is assumed that for each 8 hours spent within the inventory unit, 1 hour is spent in total travel time to and from the inventory unit.
D = [L + n (M)/8 (b) F = The field cost in dollars. F = CW {[L + n(M)] + D} (c)
i= The interval between plot centers in feet based on equilateral triangles.
| = 224.272*(A/n)"? (d)
The metric equivalent is
1 = 107.456 * (A/n)'? (e) where | is expressed in meters and A in hectares. Field costs in dollars that would be
required to achieve a specified percent sampling error.
$Se, = $ (%S, / %Sop)” (f)
total field cost in dollars for a particular option.
$Sep =
where $ =
%Sep = the desired sampling error in percent.
M = The time to measure 1 plot (includes subplots).
M = {[(n—1) ()/10,560} + n(0.5)
92
Where n is the number of subplots and i is the interval in feet between subplots.
i = 70.921 (a)1/2 (h) where a is the area of the sample stand in acres. M, = The time to traverse and measure a
sample stand when each plot takes 0.5 hour to measure and there are 10 plots
to establish. M, = {(n—1)[0.0067 (a)? } + n(0.5) for 10 plots (i)
The daily travel time (D) is revised as follows: D = [L + =(M)V/8 1) where E(M) = the sum of the time to measure all selected stands.
Similarly, total cost of field time (F) is adjusted to: F = CW [(L + 2(M,. + D] (k)
Sl= The sampling interval for each stratum. SI = A; / n; (truncated to a whole number) = (I)
RG = The range from which a random start is chosen.
RG = SI + REM (m)
where REM = the remainder in the division performed using equation (I).
Appendix 2: Stand Characteristics of the Enchanted Forest
Stand data based on complete enumeration. For vegeta- tion type: 1 = hardwoods; 2 = conifers; and 3 = brush/open. For density: 1 = 0-30% canopy cover; 2 = 31-60% canopy cover; and 3 = 61+ % canopy cover. For wildlife use: 0 = no use; 1 = used.
Stand no. Wildlife use Vegetation type Acres Ccf/acre Density Stand no. Wildlife use Vegetation type Acres Ccflacre Density
1 0 1 45 15 2 46 1 2 75 28 3
2 0 1 720 3 1 47 0 2 30 16 2
3 0 1 45 28 3 48 1 2 60 15 1
4 1 2 45 17 2 49 1 2 75 16 2
5 0 2 75 6 1 50 0 2 75 9 1
6 1 2 135 16 2 51 0 1 45 30 3
7 0 2 15 20 2 52 0 2 120 20 2
8 0 2 135 16 2 53 0 2 30 9 1
9 1 1 165 8 1 54 0 1 75 3 1 10 1 2 120 30 3 55 1 1 240 13 2 11 0 1 150 7 1 56 1 2 60 16 2 12 1 1 60 22 3 57 0 1 75 10 2 13 0 3 45 0 1 58 1 2 30 23 3 14 0 2 330 Uz 2 59 0 1 15 28 3 15 0 2 45 9 1 60 1 2 15 21 3 16 1 3 120 3 1 61 1 1 30 12 2 Wz 0 2 90 12 2 62 0 3 30 0 1 18 0 2 60 21 3 63 1 1 15 12 2 ie) 0 2 15 12 2 64 1 2 30 27 3 20 1 3 180 0 1 65 1 1 30 29 3 21 0 1 60 1 1 66 0 2 30 28 3 22 0 1 90 13 2 67 1 1 6 6 1 23 1 1 75 27 3 68 1 2 165 8 1 24 0 1 45 4 1 69 1 1 90 13 2 25 1 1 60 10 2 70 1 1 150 U 1 26 1 3 30 0 1 71 0 3 105 0 1 27 0 1 60 18 2 72 1 1 165 15 2 28 0 1 60 9 2 73 1 2 15 23 3 29 1 2 105 10 2 74 0 1 60 15 2 30 1 2 675 19 3 75 0 3 135 3 1 31 1 2 150 10 1 76 1 1 75 27 3 32 0 2 45 29 3 UU. 1 1 285 21 2 33 1 1 90 7 1 78 1 1 120 30 3 34 0 2 75 6 1 79 1 1 15 13 2 35 1 1 165 9 1 80 1 3 45 3 1 36 0 2 360 32 3 81 0 2 90 7 1 37 0 2 75 15 2 82 1 2 90 13 2 38 0 2 30 4 1 83 1 1 30 24 3 39 0 1 30 4 1 84 1 3 75 2 1 40 1 1 15 21 3 85 0 1 45 11 2 41 1 2 30 12 2 86 0 2 60 19 2 42 0 2 60 2 1 87 1 1 60 19 2 43 0 3 60 0 1 88 0 2 60 5 2 44 1 1 60 9 1 89 0 2 60 4 1 45 0 3 60 1 1 90 0 2 60 30 3
93
Appendix 2—continued.
Stand no. Wildlife use Vegetation type Acres Ccf/acre Density Stand no. Wildlife use Vegetation type Acres Ccflacre Density
91 0 2 60 8 2 146 1 1 60 24 3 92 1 2 45 6 1 147 0 1 45 24 1 93 0 2 75 17 2 148 1 2 15 13 2 94 0 1 120 20 2 149 0 2 60 20 3 95 0 1 45 25 3 150 1 1 60 8 1 96 1 2 45 11 2 151 0 1 60 23 3 97 1 1 75 19 2 152 0 1 75 19 2 98 1 1 45 30 2 153 1 2 45 3 1 99 1 1 15 1 1 154 1 2 75 23 3 100 1 1 30 8 1 155 0 1 60 22 3 101 1 2 45 17 3 156 1 1 45 3 1 102 (0) 1 60 We 1 157 0 2 60 19 2 103 1 3 30 1 3 158 1 1 135 12 2 104 0 2 15 29 2 159 1 2 60 5 1 105 1 1 165 29 2 160 0 1 75 17 2 106 0 1 15 22 1 161 1 2 30 9 1 107 0 1 45 19 2 162 1 1 30 9 1 108 1 2 30 1 3 163 1 2 30 27 3 109 0 2 15 10 3 164 0 1 30 17 2 110 1 1 15 18 2 165 0 2 60 u 1 111 0 2 15 1 3 166 0 1 45 1 1 112 0 1 15 16 2 167 0 1 60 21 2 113 0 1 30 19 1 168 1 2 75 22 3 114 1 1 15 30 2 169 1 1 60 30 3 115 1 1 45 23 3 170 0) 2 45 31 3 116 1 1 30 2 1 171 1 2 60 17 2 117 0 3 30 0 1 172 1 1 45 21 3 118 1 2 30 3 1 173 1 2 75 11 2 119 1 1 30 12 1 174 1 1 45 18 2 120 0 1 60 14 2 175 1 2 60 26 2 121 0 2 60 18 2 176 0 2 30 30 3 122 0 1 90 23 3 177 0 1 45 11 2 123 (0) 1 45 3 1 178 1 1 45 8 1 124 0 2 30 18 2 179 0 2 420 26 3 125 1 2 45 4 1 180 0 2 45 15 2 126 0) 1 30 1 1 181 0 1 60 30 3 127 0 1 75 23 3 182 0 2 45 27 3 128 1 2 120 12 2 183 1 2 90 11 1 129 1 3 45 3 1 184 1 2 30 13 2 130 1 1 45 18 2 185 1 1 30 18 2 131 1 1 45 5 1 186 1 1 30 10 1 132 0 1 45 13 2 187 1 1 30 14 2 133 0 1 60 11 1 188 1 1 75 22 3 134 1 1 75 29 3 189 0 1 45 3 1 135 1 1 30 29 2 190 0 2 30 13 2 136 0 2 120 16 2 191 0 2 45 22 3 137 0 1 60 21 3 192 0 1 135 3 1 138 1 3 30 2 1 193 0 2 30 22 3 139 1 1 60 2 1 194 1 1 75 23 3 140 0 3 75 1 1 195 0 1 60 9 2 141 0 1 45 20 3 196 0 1 90 1 1 142 0 2 45 5 1 197 1 2 30 10 1 143 1 1 90 11 1 198 0 1 45 11 1 144 0 1 45 23 2 199 0 1 45 16 2 145 0 1 45 5 1 200 1 1 810 14 1
Appendix 3: Stand Estimates by Various Inventory Designs
Stand estimates of volume per acre by various inventory designs. U = unstratified. S = stratified. EPS = equal probability sampling. PPS = probability proportional to size. P| Height = height of overstory vegetation in feet as measured from aerial photography. Predicted CCF/AC =
Stand U-EPS- U-PPS- S-EPS- S-PPS- PI Predicted CCF/ no. CCF/AC CCF/AC CCF/AC CCF/AC height CCF/AC AC
16.9 12.9 17.7 10.2 45 14.0 15° 16.9 3.0* 17.7 3.0* 15 3.0* 3” 16.9 12.9 Utat/ 10.2 90 28.3 28* 16.9 12.9 18.2 19.5 55 17.1 Us 16.9 12.9 18.2 19.5 20 6.0 6*
16.9 12.9 18.2 19.5 50 15.5 16° 16.9 12.9 18.2 19.5 65 20.3 20* 16.9 12.9 18.2 19.5 55 17.1 16° 16.9 12.9 17.7 10.2 25 7.6 8
10 30.0* 3007) 23010") (30:07 95 SOO Fi SOr
11 16.9 12.9 17.7 10.2 25 7.6 Ue 12 16.9 12.9 17.7 10.2 7A! FAL) 227 13 16.9 12.9 0.0 3.0 0 0.0* 0* 14 16.9 12.9 18.2 19.5 60 18.7 Uns 15 16.9 12.9 18.2 19.5 30 9.2 9F
16 16.9 3.0* 0.0* 3.0* 10 3.0* 3* 17 16.9 12.9 18.2 19.5 40 12.4 125 Loe e205 12.9 PAldo)s 9 EHS) 65 20.3 Cae 19 16.9 12.9 18.2 19.5 45 14.0 WW? 20 16.9 12.9 0.0 3.0 0 0.0 0*
21 16.9 12.9 17.7 10.2 5 1.3 Ue 22 16.9 12.9 17.7 10.2 40 12.4 13* 23 16.9 12.9 UZet/ 10.2 8502657, Pas 24 16.9 12.9 UEoll 10.2 15 4.4 4* 25 10.0* 12.9 HOO aaa Or2 45 14.0 10*
GOON O|OASAN —
26 0.0* 12.9 0.0* 3.0 0 0.0 0* 27 16.9 12.9 17.7 10.2 55 17.1 18* 28 16.9 CHO Tear 9/05 35 9.0* OF
29 16.9 12.9 18.2 19.5 30 9:2 10* 30 16.9 19 Ops: 2 19.0* 70 NOLO eles
31 16.9 12.9 18.2 19:5 35 10.8 Oi 32 16.9 12.9 18.2 19.5 90 28.3 295 33 16.9 Udule 17.7 OKs 30 7.07 The 34 16.9 12.9 18.2 19.5 25 7.6 6* 35 16.9 12.9 UGE 10.2 35 10.8 95
36 16.9 32.0* 18.2 32. Oba a OO 32: Ona too 37 16.9 12.9 18.2 19.5 50 15.5 155 38 16.9 12.9 18.2 19.5 20 6.0 4* 39 16.9 12.9 ULL 10.2 15 4.4 4* 40 16.9 12.9 Wate 10.2 TAY | CAS) 21*
41 16.9 12.9 18.2 19.5 40 12.4 We 42 16.9 12.9 18.2 19.5 5 1.3 2a 43 16.9 12.9 0.0 3.0 0 0.0 0* 44 16.9 12.9 17.7 10.2 30 9.2 oe 45 16.9 12.9 0.0 3.0 10 2.8 lis
Calculated volume per acre using regression equation (25) and photo interpreted heights. Last column is ground truth ccf for the stand. * = Measured on the ground. All other values for volume per acre are either Forest or stratum averages or are predicted values.
Stand U-EPS- U-PPS- S-EPS- S-PPS- PI Predicted CCF/
no. CCF/IAC CCF/AC CCF/AC CCF/AC height CCF/AC AC
46 16.9 12:9 18.2 19.5 95 29.8 28* 47 16.9 12.9 18.2 19.5 60 18.7 16° 48 16.9 12:9 18.2 19.5 55 17.1 15* 49 16.9 12.9 18.2 19:5 50 15.5 16° 50 16.9 12.9 18.2 19.5 35 10.8 oF
51 16.9 12.9 17.7 10.2 95 29.8 30° 52 16.9 12.9 18.2 19.5 65 20.3 20° 53 0x S07 910% 9:05 30 905 9° 54 16.9 12.9 17.7 10.2 15 4.4 3* 55 16.9 12.9 17.7 10.2 50 15.5 13°
56 16.9 12.9 18.2 19.5 55 17.1 16° 57, 16:9 12.9 UCtet/ 10.2 40 12.4 10* BTS UO) 12.9 18.2 19.5 70 21.9 23* 59) 16:9 12.9 17.7 10.2 90 28.3 28° 60 16:9 12.9 18.2 19.5 75 23.5 ili
61 =12.0* 12.9 12.0* 10.2 45 14.0 25 62 0.0* 12.9 0.0* 3.0 0 0.0 0* 63 16.9 12.9 UUoth 10.2 40 12.4 125 64 16.9 12.9 18.2 19.5 90 28.3 7A fie 65 16.9 12.9 17.7 10.2 95 29.8 29"
66 16.9 209 18.2 19.5 95 29.8 28* 67, 16:9 6.0* 17.7 6.0* 35 6.0* 6* 68 16.9 12.9 18.2 19.5 25 7.6 8* 69/55 16:9 12:9 Uzoe/ 10.2 40 12.4 13° TAY UGS Oe 17.7 7.0* 30 Or Ue
TA 16:9 12.9 0.0 3.0 0 0.0 0* 26:9 12.9 Ute 10.2 45 14.0 155 T3e > N639 12.9 18.2 19.5 75 23.5 23° 74 15.0" 12.9 15.0* 10.2 60 79) 16:9. 12.9 0.0 3.0 10 2.8 3*
76 16:9 12:9). 17.7 10.2 85 26.7 ray ii 16:9 12.9 17.7 10.2 65 20.3 Zils 12.9 17.7 10.2 100 31.4 30* 79) 16:9 12.9 Wars 10.2 45 14.0 13° 80 16.9 12.9 0.0 3.0 15 4.4 3*
81 7.0* 12.9 Udue 19.5 25 7.6 Ve S25 16:9 219 18.2 19.5 50 15.5 13° 83 16.9 12.9 ULE, 10.2 75 23.5 24* 84 16.9 12.9 0.0 3.0 10 2.8 on 85, | 16:9 12.9 17.7 10.2 40 12.4 fits
86) 16:9 12.9 18.2 19.5 60 18.7 19% Sf 16:9 12:9 17.7 10.2 65 20.3 19° 88 16.9 12.9 18.2 19.5 20 6.0 5* 89°" 16:9 12.9 18.2 19.5 20 6.0 4* 90 16.9 12.9 18.2 19.5 95 29.8 30*
95
Appendix 3—continued.
Stand U-EPS- U-PPS- S-EPS- S-PPS- PI Predicted CCF/ Stand U-EPS- U-PPS- S-EPS- S-PPS- PI Predicted CCF/
no. CCF/AC CCF/AC CCF/AC CCFIAC height CCF/AC AC no. CCF/AC CCF/AC CCF/AC CCFI/AC height CCF/AC AC Sip 69 8.0° 18.2 8.0* 35 8.0° 8* 146 16.9 12.9 17.7 10.2 80 25m 24* 92 6.0* 12.9 6.0* 19.5 20 6.0 6* 147 =16.9 12.9 17.7 10.2 80 §=25.1 24* 93 16.9 12.9 18.2 19.5 60 18.7 wey 148 #8 16.9 12.9 18.2 19.5 40 12.4 13* 94 16.9 12.9 17.7 10.2 70 =«621.9 20* 149 16.9 12.9 18.2 19.5 65 20.3 20* 95 25.0" 129 25.0* 102 80 25.1 25° 150 16.9 12.9 17.7 10.2 30 9.2 8h 96 16.9 12.9 18.2 19.5 35 10.8 lithe 151 16.9 12.9 UC 10.2 80 25.1 23* 97 16.9 19.0* 17.7 19.0* 60 19.0" 19° 152 19.0* 19.0* Or VEO 70 VEO Ae 98 30.0° 12.9 30.0* 10.2 80 25.1 30* 153 3.0* 12.9 3:05 > 19!5 15 4.4 3* 99 16.9 12.9 17.7 10.2 10 2.8 We 154 23.0 12.9 23.0 19.5 75 23.5 23* 100 16.9 12.9 17.7 10.2 25 7.6 8* 155 16.9 12.9 ULets 10.2 75 23.5 22* 101 + 16.9 12.9 18.2 19.5 60 18.7 Whe 156 16.9 12.9 17.7 10.2 15 4.4 < fe 102. 16.9 12.9 17.7 10.2 55 17.1 vis 157 16.9 12:9 18.2 19.5 65 20.3 197 103. 16.9 12.9 0.0 3.0 10 2.8 Ul? 158 16.9 12.9 17.7 10.2 40 12.4 125 104 16.9 12.9 18.2 19.5 95 29.8 29* 159 16.9 12.9 18.2 19.5 20 6.0 5* 105 16.9 12.9 17.7 10.2 95 29.8 29* 160 16.9 12.9 UCL 10.2 55 17.1 Ue 106 16.9 12.9 17.7 10.2 75 23.5 22* 161 16.9 29 18.2 19.5 35 10.8 9 107. 16.9 12.9 17.7 10.2 65 20.3 19* 162 16.9 12.9 Certs 10.2 30 9.2 9° 108 16.9 12.9 18.2 19.5 10 2.8 iti 163 16.9 12.9 18.2 19.5 90 28.3 2te 109 16.9 12.9 18.2 19.5 40 12.4 10° 164 16.9 12.9 17.7 10.2 55 17.1 Un 110 16.9 12.9 17.7 10.2 55 17.1 18° 165 16.9 12.9 18.2 19.5 30 9.2 Ue 111 +16.9 12.9 18.2 19.5 10 2.8 Ne 166 16.9 12On UZAet/ Udde 5 1.0* ue 112 16.9 12.9 17.7 10.2 55 17.1 16* 167 16.9 12.9 17.7 10.2 75 23.5 Zils 113 «16.9 12.9 17.7 10.2 65 20.3 19° 168 16.9 12.9 18.2 19.5 70 21.9 22* 114 16.9 12.9 17.7 10.2 100 31.4 30* 169) 16:9 12.9 17.7 10.2 95 29.8 30* 115 16.9 12.9 17.7 10.2 80 25.1 23* ZO MSlLOn 12.9 3105 9:5 105 33.0 31* 116 2.0* 12.9 2.0* 10.2 15 4.4 2a WA Wew) 12.9 18.2 19.5 60 18.7 * aa 117. 16.9 12.9 0.0 3.0 0 0.0 0* Ue VeloH 12.9 21.0* 10.2 70 21.9 Ze 118 16.9 12.9 18.2 19.5 10 2.8 3* 173 16.9 12.9 18.2 19.5 40 12.4 uu 119 16.9 12.9 17.7 10.2 45 14.0 Wey 174 16.9 12.9 AIZ/ETA 10.2 55 ULL 18* 120 16.9 12.9 17.7 10.2 45 14.0 14* 175 16.9 26.0* 18.2 26.0* 85 26/07 cor 121 16.9 12.9 18.2 19.5 65 20.3 18° 176 16.9 12.9 18.2 19.5 100 31.4 30* 122 16.9 12.9 17.7 10.2 70 3=—.21.9 23* ite mlO:o 12.9 Nifet, 10.2 35 10.8 ny 123. «16.9 12.9 17.7 10.2 15 4.4 3* 178 16.9 12.9 17.7 10.2 35 10.8 8* 124 16.9 18.0* 18.2 18.0°* 60 18.0* 18° 179 16.9 12.9 18.2 19.5 90 28.3 26* 125 16.9 12.9 18.2 19.5 15 4.4 4* 180 16.9 12.9 18.2 19.5 55 17.1 15* 126 16.9 12.9 17.7 10.2 5 1.3 ue 181 16.9 12.9 UZEt/ 10.2 95 29.8 30* 127 =16.9 12.9 Weare 10.2 75). 23:0 23° 182 16:9 12.9 18.2 19.5 85 26.7 27- 128 16.9 12.9 18.2 19.5 45 14.0 Ue 183) 16:9 12.9 18.2 19.5 40 12.4 api 129 16.9 3.0* 0.0 3.0* 10 3.0* 3* 184 16.9 12.9 18.2 19.5 50 15.5 13* 130 16.9 12.9 WHat: 10.2 60 18.7 18* 185 16.9 12.9 17.7 10.2 65 20.3 18* 131 16.9 12.9 17.7 10.2 25 7.6 5° 186 16.9 12.9 17.7 10.2 40 12.4 10° 132 16.9 12.9 17.7 10.2 45 14.0 13* 187 = 14.0* 12.9 14:0" > (10!2 45 14.0 14* 133 «16.9 12.9 ULéare 10.2 40 12.4 Vials 188 16.9 22.0* UTéat/ 22.0* 65 22 0n eos 134 16.9 12.9 ULAT/ 10.2 100 31.4 29* 189) 9169 12.9 17.7 10.2 10 2.8 3* 135 16.9 12.9 17.7 10.2 95 29.8 eo 190)" 16:9 12.9 18.2 19.5 50 15:5 13* 136 16.9 12.9 18.2 19.5 60 18.7 16° 191 16:9 12.9 18.2 19.5 75 23.5 22* 137 16.9 12.9 17.7 10.2 70 21.9 21* 192)" / 16:9 12.9 17.7 10.2 15 4.4 3* 138 16.9 12.9 0.0 3.0 5 1.3 2* 193 16.9 12.9 18.2 19.5 70 21.9 22* 139 16.9 12.9 Weet/ 10.2 15 4.4 2e 194 16.9 12.9 UZArA 10.2 80 25.1 23* 140 16.9 12.9 0.0 3.0 10 2.8 ils 195) R16:9 12.9 17.7 10.2 30 9.2 2: 141 16.9 12.9 UC/st 10.2 70 21.9 20* 196 16.9 12.9 17.7 10.2 10 2.8 uly 142 16.9 12.9 18.2 19.5 20 6.0 Si 197 16.9 12.9 18.2 19.5 35 10.8 10* 143° «16.9 12.9 ULots 10.2 35 10.8 Ue 198 16.9 12.9 UATE 10.2 40 12.4 uth 144 8 23.0° 12.9 23.0* 10.2 65 20.3 23* 199 16.9 12.9 17.7 10.2 55 17.1 16* 145 16.9 12.9 Wears 10.2 25 7.6 5s 200 16.9 14.0* 17.7 14.0* 50 14.0* 14*
wii