Optimal plot design in a multipurpose forest inventory

RESEARCH Open Access Op imal plo design in a mul ipu pose o es in en o y Helena M. Hen onen 1 and Annika Kangas 2* Abs ac Backg ound: We explo e he ac o s a ec ing he op imal plo design (size and ype as well as he subsample ee selec ion s a egies wi hin a plo ) and hei ela i e impo ance in de ining he op imal plo design in amul ipu pose o es in en o y. The ac o s include ime used o lay ou he plo and o make he ee measu emen s wi hin he plo , he be ween-plo a ia ion o each o he a iables o in e es in he a ea, and he measu emen and model e o s o he di e en a iables. Me hods: We simula e di e en plo ypes and sizes and subsample ee selec ion s a egies on measu ed es a eas om No h Lapland. The plo ypes used a e ixed- adius, concen ic and elascope plo s. Weselec he op imal ype and size i s a plo le el using a cos -plus-loss app oach and hen a clus e le el byminimizing he weigh ed s anda d e o wi h ixed budge . Resul s: As elascope plo s a e e y e icien a he plo le el o olume and basal a ea, and ixed- adius plo s o s ems pe ha, he op imal plo ype s ongly depends on he ela i e impo ance o hese a iables. The concen ic plo seems o be a good comp omise be ween hese wo in many cases. The subsample ee selec ion s a egy was mo e impo an in selec ing op imal plo han many o he ac o s. In clus e le el, he mos impo an ac o is he ans e ime be ween plo s. Conclusions: While he op imal adius o plo s and o he pa ame e s we e sensi i e o he measu emen imes and o he cos ac o s, he concen ic plo ype was op imal in almos all s udied cases. Subsample ee measu emen s a egies need u he s udies, as hey we e an impo an cos ac o . Howe e , hei impo ance o he p ecision was no as clea . Keywo ds: Sample, Plo , Fo es in en o y, Measu emen , Cos , Loss Backg ound Op imal in en o y sampling design is a e y impo an goal in Na ional Fo es In en o ies (Mandallaz 2007). The in en o y design is op imized in a sense ha we wish o ha e he highes accu acy gi en a ixed budge o we wish o ha e he lowes cos o a gi en accu acy. Op imiza ion is possible, i we make assump ions con- ce ning he popula ion. In an analy ical se ing, we need o be able o an icipa e he popula ion a iance (Mandal- laz & Ye 1999). I is e en possible o op imize he mea- su emen s o ees in he plo s, o ins ance o de e mine how many subsample ees (i.e. second-phase sample ees) o measu e ou o he o al numbe o ally ees (i.e. i s -phase sample ees), i we can an icipa e he e o in he olume es ima es o he ally ees. De ining op imal sample plo size and ype analy ically would equi e ha we can an icipa e he e ec s o he plo size and ype on he popula ion (o be ween-plo ) a iance. I he expec ed be ween-plo a ia ion can be exp essed as a unc ion o plo size (see F eese 1961, Zeide 1980) he op imal plo size can be calcula ed ana- ly ically. Howe e , such a unc ion can only be an ap- p oxima ion o he be ween-plo a ia ion as he ela ionship depends on he cha ac e is ics o he popu- la ion such as spa ial pa e n o he ees, which canno ully be desc ibed wi h a model. In addi ion he expec ed cos s, measu ed wi h ime consump ion as a unc ion o plo size a e needed o op imiza ion. In ixed- adius plo s he numbe o ees in a plo is p opo ional o plo a ea, bu he ime needed o check he bo de line ees is p opo ional o he pe - ime e (Zeide 1980). In elascope plo s, ime consump- ion is in e sely p opo ional o he ixed angle de ined by he elascope ac o . Al hough Kulow (1966) and * Co espondence: [email p o ec ed] 2 Na u al Resou ces Ins i u e Finland (Luke), Economics and Socie y uni , Yliopis oka u 6, 80100 Joensuu, Finland Full lis o au ho in o ma ion is a ailable a he end o he a icle © 2015 Hen onen and Kangas. Open Access This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional License (h p://c ea i ecommons.o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and indica e i changes we e made. Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 DOI 10.1186/s40663-015-0055-2 G osenbaugh & S o e (1957) compa ed he coe icien o a ia ion using bo h ixed- adius and elascope plo s, hey did no compa e he o e all e iciencies o hese wo ypes o plo s ela ed o he ime spen . While many ac o s a ec ing he accu acy can be accoun ed o analy ically, some aspec s like he spa ial pa e n, a e mo e di icul . The analy ical calcula ions usually assume a andom pa e n (Mandallaz 2007). Likewise, he numbe o subsample ees and he selec- ion o measu emen s aken om each ee (e.g. heigh and/o uppe diame e ) can be di icul o accoun o in de ail in an analy ical se ing. The e o e, he op imal plo size and ype has mos o en been de ined by simula ing sampling in an accu a ely measu ed and mapped o es a ea. In he ea lies s udies, simula ion was ca ied ou by measu ing a g id o small cells and building la ge sample plo s as hei combina ion (Johnson & Hixon 1952, Mesa age & G osenbaugh 1956). In la e s udies, compu e simula ion based on mapped da a has been u ilized (e.g. Kulow 1966). In a simula ion based on eal da a, he op imal plo size is hea ily dependen on he o es condi ions on he a ea, which makes de ini e con- clusions di icul (Mesa age & G osenbaugh 1956). Op imal sampling design and op imal plo design (size and ype) depends highly on he pu pose o an in en o y. I is easy in p inciple o de ine an op imal in en o y o one a iable o in e es such as biomass o olume wi h ega d o measu emen cos s and accu acy. When he numbe o cha ac e is ics o in e es inc eases, he ask ge s mo e complica ed as he op imal plo numbe , size and ype a e likely o be di e en o each cha ac e is ic. Fo ins ance, class a iables such as land use and i s changes could be de e mined om a e y small plo o e en poin , bu olume and biomass equi e a la ge plo . Thus, p io i izing he o es cha ac e is ics is needed i an op imal plo is o be de e mined. The es ima ion me hod is also likely o ha e an e ec : i we assume a design ha is based pu ely on ield plo s, he op imal plo size and ype a e likely o be e y di e - en om a case whe e auxilia y in o ma ion such as e- mo e sensing in o ma ion is used in s a i ica ion (e.g. Tomppo e al. 2014), adi ional eg ession es ima ion, model-assis ed es ima ion o model-based es ima ion. In hese cases, he a ia ion be ween plo s may no be he decisi e ac o , bu a he he co ela ion be ween he o es cha ac e is ics and he emo e sensing da a. The esul s may also depend on he speci ic c i e ion used o de ining he op imum. One op ion is o minimize some c i e ion like s anda d e o o he es i- ma e o a gi en budge cons ain such as amoun o ime (Johnson & Hixon 1952, Mesa age & G osenbaugh 1956). Using his app oach, Johnson & Hixon (1952) concluded ha while long and na ow ec angula plo s ended o ha e smalle be ween-plo a ia ion, he ime needed o lay ou such plo s was la ge . Thus, he mos e icien plo s o a gi en amoun o ime we e compac plo s. Ano he way o de ine he op imal plo size is o use a cos -plus-loss (CPL) app oach (Hamil on 1978, S åhl 1994). I means ha he losses due o poo es ima es (possibly esul ing sub-op imal decisions) a e calcula ed as a unc ion o he unce ain y in ol ed and hese losses a e added o he measu emen cos s desc ibed as a unc- ion o measu emen ime. This c i e ion would be ideal, i he losses due o poo es ima es could be accu a ely de ined. O en he losses a e desc ibed as a unc ion o he s anda d e o o some o he c i e ion (Ba h & S åhl 2012), bu hey could also be calcula ed o an ac ual de- cision p oblem (Eid e al. 2004). When he in en o y is mul ipu pose, he cos -plus-loss me hod is mo e compli- ca ed (see Bu kha e al. 1978). I we we e able o de ine he losses due o he poo es ima es o each o he a i- ables o in e es (i.e. gi e ela i e weigh o he e o s o each a iable), i is possible. To al measu emen cos s can be calcula ed as a unc- ion o ime used o each sample plo . The ime de- pends on: 1) he ime equi ed o go o he plo and lay ou he plo ; 2) he o al numbe o ees o be measu ed and 3) he measu emen s ca ied ou o each ee. Laying ou he plo means de ining he plo cen e (o cen e o se e al sub-plo s) and de e mining which ees belong o he plo (s). Fo ci cula o elascope plo s ha means checking he dis ance o bo de line ees om he plo cen e wi h a measu ing ape o an (op ical) ange inde (e.g. Loe sch e al. 1973). The measu emen s needed o each ee depend on he cha ac e is ics o in e es (e.g. olume, biomass, s ems pe ha). Typically no all cha ac e is ics needed a e mea- su ed on all ees wi hin a plo . The diame e a b eas heigh (d1.3) is measu ed o all ally ees, bu heigh , uppe diame e s, age, and g ow h a e measu ed only o subsample ees. Thus, he measu emen ime also de- pends on he numbe o subsample ees wi hin each plo , and he numbe o measu emen s ca ied ou on each ee. As biomass and olume equi e addi ional subsample ee measu emen s compa ed o s ems pe ha, also he ime consump ion needs o be de ined sepa - a ely o each o he a iables. The p ecision o he sample plo measu emen s in de- sc ibing he o es s and can be measu ed using he s anda d e o s o he es ima o s o gi en o es cha ac- e is ics, which depends, in pa , on he spa ial a ia ion o he cha ac e is ics o in e es wi hin he o es . In gen- e al, he bigge he sample plo a ea, he la ge he p o- po ion o o al a ia ion ha alls wi hin he plo , and consequen ly he smalle he s anda d e o s (e.g. Loe sch e al. 1973, Koi uniemi 2003). Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 2 o 14 Measu emen and model e o s o he a iables used o calcula e he cha ac e is ics o in e es ha e also an e ec on p ecision (e.g. Päi inen 1987, S åhl e al. 2014). Thei combined e ec again depends on he numbe o subsample ee measu emen s and he models/me hods a ailable o gene alize he subsample ee measu emen s o he ally ees. I may be assumed ha he e o s in olume / biomass o subsample ees a e negligible, bu no o he ally ees. I is qui e possible ha he model which is mos e icien when all measu emen s a e as- sumed e o - ee is no he mos e icien when hese e - o s a e included (Eid 2003). The e o e, i would be bes o selec he models used o gene alizing he subsample ee cha ac e is ics o ally ees simul aneously wi h de- ciding he numbe o subsample ees and he a iables measu ed o each o hem. The aim o his s udy is o analyze op imal sample plo ype and size wi h a simula ion s udy and explo e he ela i e e ec s o di e en ac o s on op imal plo measu emen s a egy in he special condi ions o No h Lapland. The s udy egion is pa ially loca ed close o he no he n imbe line, whe e clus e ed spa ial pa e ns o ees challenge he planning o an e icien o es in- en o y. The s udied plo ypes we e ixed- adius plo s wi h a ying adii, a combina ion o wo concen ic plo s wi h a ying adii and a ying diame e limi s o he la - ge adius, and elascope plo s wi h a ying elascope ac o and maximum adii. The o es cha ac e is ics conce ned we e olume, basal a ea and s ems pe ha. The class a iables such as o es /non- o es we e ex- cluded om he s udy. Ma e ial Measu emen s o 50 m x 50 m es a eas we e ca ied ou in 2002 in Ina i, No h Finland (Figu e 1). The mea- su ed a eas we e sampled om he plo s o he 8 h Na- ional Fo es In en o y. In o al, 18 es a eas we e measu ed, oge he wi h he plana coo dina es o he Fig. 1 The loca ions o he mapped es a eas Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 3 o 14 ees (wi h d1.3 ≥2.5 cm) mapped wi h achyme e (SOKKIA SET 4C), as well as d1.3 (in wo pe pendicula di ec ions), heigh (h, m), and uppe diame e a a heigh o 6 m (d6, cm) ( o ees ≥8 m all). An example o ou da a is gi en in Fig. 2 illus a ing he spa ial dis ibu ion o ees and hei diame e s in a ma u e Sco s pine (Pinus syl es is L.) s and wi h bi ch (Be ula pubescens) unde g ow h. Volumes o he ees we e calcula ed using olume unc ions (Laasasenaho 1982). Fo ees wi h heigh s ≥8.1 m, olumes we e es ima ed as a unc- ion o d1.3, h, and d6. When he heigh o a ee was less han 8.1 m, olumes we e es ima ed as a unc ion o d1.3 and h. Me hods Analysis o he poin pa e ns o ees on he 50 m x 50 m a eas was ca ied ou using he R package spa s a (Baddeley and Tu ne 2005). Ou main in e es was in assessing whe he he poin pa e ns could be conside ed andom (Poisson). Fo his pu pose, we ca ied ou a simul aneous (simul aneous o di e en alues o he dis ance ) Mon e Ca lo es o Ripley’sK– unc ion and L- unc ion, which is a a iance s abilizing ans o ma ion o K. Inhomogenei y was aken in o accoun by model- ling ends as a unc ion o coo dina es. In a eas di ided in o wo di e en s ands, he s and was used as an indi- ca o a iable in modelling inhomogenei y. Analysis o he op imal plo design was ca ied ou a wo le els: plo le el and clus e le el. The clus e is in e p e ed he e as a combina ion o mplo s, bu any speci ic spa ial a angemen o he clus e is no de e - mined. I he spa ial a angemen we e speci ied, he clus e could also be in e p e ed as a plo wi h msub- plo s. In he plo le el analysis, he op imal plo design was de ined by minimizing he cos -plus-loss (CPL) de ined o he p a iables o in e es o one plo . The gene al unc ion o be op imized is OP ¼minCPL size; ype;s a egy g ¼min l1þ…þlPþcðÞ ð1Þ whe e he losses a e a unc ion o RMSE as lp¼wpRMSEpð2Þ whe e w p is he weigh gi en o he RMSE o a gi en a iable p. Cos s ca e de ined as a unc ion o he ime o ans e be ween he plo s (LT used a clus e le el, assumed o be 15 min/plo ), he ime needed o check he bo de line ees (BT assumed o be 0.5 min/ ee), ally each ee in he plo (TT assumed o be 0.5 min/ ee) and measu e he subsample ee cha ac e is ics om each he subsample ee (ST assumed o be 4.5 min/ ee) as c¼LT þBTn1þTTn2þSTn3ð3Þ whe e n 2 is he numbe o ally ees, n 1 is he numbe o bo de line ees and n 3 he numbe o subsample ees (Päi inen 1987). A ee was de ined as bo de line ee i i s dis ance om he plo cen e di e ed less han 0.5 m om he adius o a ee wi h a gi en size. The plo le el analysis was ca ied ou so ha wi h each plo ype and size, we simula ed N= 1000 andomly loca ed plo s (simula ed plo s) wi hin each mapped 50 m x 50 m a ea. The accu acy o each cha ac e is ic wi hin each a ea was analyzed as RMSEj¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X N i¼1 ^ yji−yj  2 N u u u ð4Þ whe e ŷ ji is he obse ed o es cha ac e is ics om he a ea jand simula ed plo iand y j is he ue alue o he cha ac e is ics calcula ed om he whole a ea j. Rela i e RMSE was calcula ed by di iding he RMSE wi h he mean ac oss he es a eas (Table 1). The plo -le el ana- lysis RMSE p (equa ion 2) was he a e age o hese RMSEs in he 18 es a eas. The a iables o in e es we e plo olume (V, m 3 /ha), basal a ea (G, m 2 /ha) and s ems pe ha (N). The simula ed plo s we e loca ed wi hin he es a eas so ha he cen e poin was a leas 11 m om he edge so ha edge co ec ions we e no used. The possible bias esul ing om his is included in he RMSE. Fig. 2 An example o a mapped es a ea wi h a clus e ed spa ial pa e n. The sizes o ci cles a e p opo ional o he diame e s o he ees Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 4 o 14 In he clus e le el analysis, he a e age MSE om he plo le el analysis was used as he “wi hin- es -a ea” a i- a ion (Va w ). In addi ion, he o al a ia ion included he “be ween- es -a eas” a ia ion among he 18 es a eas, de ined as Va b¼X 18 j¼1 yj− y  2 18 ð5Þ whe e ȳis he o e all mean o o es cha ac e is ic in hese 18 a eas (Table 1). We calcula ed he o al a ia ion as Va o al ¼Va bþVa wð6Þ whe e he wi hin-a ea a ia ion depends on he plo ype and size bu he be ween a ea a ia ion does no . The o al a ia ion can be used o simula e a si ua ion whe e bo h he op imal sample plo size and he op imal numbe o sample plo s a e selec ed. I can hus be used o analyse i i is mo e use ul o selec a la ge numbe o small plo s o a small numbe o la ge plo s. In his s udy, we analysed he op imal plo ype and size o one clus e consis ed o mplo s. The budge o measu ing one clus e was ixed o one day wo k, app oxima ely 420 min. We calcula ed he a o dable clus e plo numbe mas mcase ¼B ccase ð7Þ whe e he c case is he measu emen ime needed (Equa- ion 3) o one simula ed plo wi h a gi en ype and size and a gi en subsample ee measu emen s a egy, and B is he o al budge o measu emen s (in minu es pe day). The ela i e s anda d e o o he mean o clus e o olume (V, m 3 /ha), basal a ea (G m 2 /ha) and s ems pe ha (N) was calcula ed using he simple andom sam- pling (SRS) o mula wi h he numbe o plo s m case i - ing he budge and he o al a ia ion Va o al depending on he plo ype and size as SE ^  ycase  ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Va o al;y mcase ð8Þ Rela i e e o was calcula ed by di iding wi h he mean ac oss he a eas (Table 1) o s anda dize he uni s. The op imal plo size, ype and numbe o plo s in a clus e was de ined as he one wi h he minimum weigh ed mean o he SE%s as OP ¼min size; ype;s a egy;mcase g ðwVSE%VþwNSE%N þwBASE%BAÞ ð9Þ We examined h ee di e en plo ypes. The i s ype was a ixed- adius plo wi h adius a ying om 3 o 11 m. The second ype was a combina ion o wo con- cen ic plo s wi h he la ge adius a ying om 5 o 11 m and he smalle om 3 o 7 m. The diame e limi (DL) o ees included o he la ge plo s a ied om 5 cm o 15 cm. The hi d ype was a elascope plo whe e he elascope ac o (RF) a ied om 1 o 3 m 2 / ha. The elascope plo s we e es ic ed o maximum a- dius ( max) a ying om 6 m o 11 m. Thus, he adius o he inclusion zone o ees wi h diame e la ge han 100 max ffiffiffiffiffiffiffi RF p=50 was always equal o max. Fo he es- ima ion o mean alues pe hec a e using elascope plo wi h maximum adius see e.g. Tomppo e al. (2011). The speci ic plo designs es ed a e p esen ed in Table 2. We assumed ha diame e on ally ees is measu ed in wo di ec ions. He e we es ed wo subsample ee se- lec ion s a egies. In he i s one ( ixed s a egy o S1), ally ees wi h d1.3 > 25 cm we e measu ed as sub- sample ees, along wi h all ally ees close han 1 m o he plo cen e . The assump ion he e is ha la ge ees a e mo e impo an subsample ees han small ees, as hey con ibu e mo e o he plo olume and he a i- ance o hei olume es ima es is highe han ha o small ees (see discussion below). In he second s a egy ( elascope s a egy o S2) he subsample ees we e se- lec ed using a elascope ac o 5 m 2 /ha, also assuming ha la ge ees a e mo e impo an han small ees. We Table 1 Mean olumes, s em pe ha and basal a eas o he 50 m x 50 m mapped es a eas Tes a ea V,m 3 /ha N,/ha G,m 2 /ha 1 43.4 668 8.4 2 70.6 728 13.1 3 8.2 316 1.9 4 17.0 224 3.3 5 45.0 904 9.3 6 19.0 480 4.5 7 118.8 424 17.9 8 92.4 260 12.1 9 63.9 356 11.2 10 4.0 324 1.4 11 10.6 424 2.8 12 20.3 980 5.0 13 64.6 1276 12.5 14 25.9 1388 6.5 15 44.5 1412 9.8 16 21.0 960 4.7 17 92.9 984 12.6 18 24.8 692 5.1 Mean 43.7 711.1 7.9 Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 5 o 14 assumed ha he olume o he subsample ees could be measu ed e o ee (in ac he e is e o bu i is as- sumed negligible), while o ally ees we assumed an e o . Fo e e y ee in he 18 es a eas, he olume was cal- cula ed using d1.3, d6 and h as p edic o a iables (Laa- sasenaho 1982) and his was assumed o be he measu ed olume. Using hese olumes, a simple and less p ecise model ¼β0þβ1d1:3þβ2d1:32þεð10Þ was i ed using only d1.3 as a p edic o a iable. The i - ed model has R 2 = 0.9556 and RMSE = 0.02626 m 3 .The model e o s we e he e oscedas ic (Fig. 3), which is yp- ical o a olume model. In he analysis, he olumes o he ally ees we e es ima ed using his simple model (10), while o he subsample ees he abo e men ioned measu ed olumes we e used, o desc ibe he e ec o no measu ing he heigh and uppe diame e o each ee in he plo . Resul s The ixed- adius plo s included much mo e measu ed ees han he o he wo plo ypes. In ixed- adius plo wi h adius 11 m, he maximum numbe o ees (in he 1000 eplica ions and 18 es a eas) o be measu ed eached 80, while in elascope plo s he maximum num- be was below 40 and in concen ic below 60 (Fig. 4). The maximum numbe o bo de line ees was 2.39 on a e age o ixed- adius plo s and only 0.96 o elascope plo s. Wi h ixed- adius plo s, he p opo ion o bo de - line ees a ied om 8.9 o 30.4 %, o elascope plo s om 14.3 o 25.1 % and o concen ic plo s om 9.6 o 25.9 %. The a ia ion in he numbe o measu ed sub- sample ees be ween he plo ypes and sizes was qui e low (see also Fig. 6). When he a e age ela i e RMSE (Equa ion 4) was plo ed as a unc ion o measu emen imes (min) o di e en plo ypes and o es cha ac e is ics, i was clea ha he elascope plo ype was e y e icien o olume and basal a ea, while he ixed- adius plo was bes o s ems pe ha (Fig. 5). The concen ic sample plo s seemed o be a e y use ul comp omise, which was nea op imal o all cha ac e is ics. The s a egy o selec he subsample ees wi h an ela- scope ac o 5 (S2) was clea ly dis inguishable om he s a egy o selec all la ge ees and ees closes o plo cen e (S1) wi h longe measu emen imes o he plo (Fig. 5). The eason o his can be seen om he num- be o subsample ees measu ed in each case wi h hese wo s a egies: he elascope s a egy on a e age p o- duced abou 0.5 mo e subsample ees pe plo (Fig. 6). Wi h smalle adii he di e ence could be as much as 1.0 subsample ees, while he di e ences pe e ed ou wi h la ge plo sizes. Fo plo s wi h he smalles adii Table 2 The es ed combina ions o plo s Plo ype Radiusm Radius 2m Relascope ac o RF m 2 Diame e limi DLcm ixed- adius 3,4,5,6,7,8,9,10 and 11 - - - concen ic 11 7 - 5, 7.5,10,12.5,and 15 9 6 - 5, 7.5,10,12.5,and 15 7 5 - 5, 7.5,10,12.5,and 15 6 4 - 5, 7.5,10,12.5,and 15 5 3 - 5, 7.5,10,12.5,and 15 elascope 6,7,8,9,10 and 11 - 1 - 6,7,8,9,10 and 11 - 1.5 - 6,7,8,9,10 and 11 - 2 - 6,7,8,9,10 and 11 - 2.5 - 6,7,8,9,10 and 11 - 3 - Fig. 3 The esiduals o olume model (Equa ion 10) Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 6 o 14 he elascope s a egy (S2) seemed a li le mo e e icien wi h espec o he RMSE o olume while he ixed s a egy (S1) was mo e e icien o la ge adii plo s (Fig. 7). To selec an op imal plo ype and size o a ixed numbe o plo s (i.e. a plo le el), a cos -plus-loss ana- lysis was ca ied ou (Equa ion 1), wi h weigh w p = 0.08 o he RMSEs (Equa ion 2) o all h ee cha ac e is ics conside ed. Fo ixed- adius plo s he op imal s a egy o 6 m adius was e y clea (CPL 20.72). The elascope s a egy o selec ing subsample ees was clea ly less e - icien han he ixed s a egy (Fig. 8). Fo elascope plo s, he di e ences be ween he subsample ee selec- ion s a egies we e also e y clea , as well as he di e - ences be ween he elascope ac o s. On he o he hand, he CPL did no appea o depend on he maximum a- dius. The op imal elascope plo had a elascope ac o 1m 2 /ha and a maximum adius 7 m (CPL 21.42). Wi h concen ic plo s, he dependency on adius was simila bu less p onounced han wi h ixed- adius sample plo s. The op imal plo adius was a li le bi la ge (7 m), bu ees wi h d1.3 less han 15 cm we e only measu ed wi hin 5 m plo . This plo ype p oduced he smalles CPL (19.81). The e ec o a ying he subsample ee measu emen s a egy was much la ge han ha o he diame e limi . I he weigh o RMSE o s ems pe ha was ipled (ce e is pa ibus), he op imal adius o ixed- adius plo s was 7 m. In his case, he smalles CPL was ob ained wi h ixed- adius plo s (27.40). Thus, when s ems pe ha is impo an enough, he ixed- adius plo is he mos e - icien . Fo he concen ic plo he op imal diame e limi changed om 15 o 5 cm. I he weigh o he ol- ume RMSE was ipled (ce e is pa ibus), he maximum adius o elascope plo s inc eased o 8 m. The op imal plo was a concen ic sample plo wi h adii 9 / 6 m and diame e limi o 15 cm (CPL = 28.31). A mo e ma ked change occu ed when he ela i e im- po ance o losses compa ed o cos s was educed o 0.01 o all a iables (Equa ion 2). In ha case, he op i- mal ixed- adius plo adius was 3 m, he op imal ela- scope ac o 3 m 2 /ha wi h a maximum adius 6 m, and Fig. 4 The maximum numbe o ees in a plo , and a e age numbe o ally ees, bo de line ees and subsample ees as a unc ion o plo adius (m) wi h he wo subsample ee selec ion s a egies wi h ixed- adius (a), elascope (b) and concen ic (c) plo s. In elascope plo s he a ia ion wi hin each elascope ac o is due o a ying maximum adius and in concen ic plo s he a ia ion wi hin each adius is due o a ying diame e limi Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 7 o 14 o concen ic plo s he op imal adii we e 5 /3 m wi h he diame e limi o 15 cm. In his case, he concen ic plo had he smalles CPL (6.14) bu he elascope plo was e y close (6.18). Tha means ha o all plo ypes, he op imal plo size was he smalles conside ed. When he weigh o losses was inc eased compa ed o cos s (weigh 0.2 o all a iables), he op imal adius o he ixed- adius plo was 8 m, he op imal elascope ac o was 1 m 2 /ha wi h a maximum adius o 10 m, and he op imal concen ic plo had adii o 11 / 7 m wi h a diame e limi 15 cm. I also had he smalles CPL (36.69). When he budge was ixed o one day’swo ho measu ing minu es (420) and he sample plo numbe , size and ype wi hin a clus e could all be decided a hesame ime, heop imalcombina ionwas omeas- u e 19 concen ic sample plo s wi h adii 7/5 m wi h a diame e limi o 10 cm (Fig. 9). When he ime o ans e be ween he plo s (LT) was educed om 15 o 10 min, he op imal numbe o plo s inc eased om 19 o 25 and he diame e limi inc eased o 15 cm (Fig. 10). On he o he hand, when LT was in- c eased o 20 min, he op imal numbe o plo s e- duced o 14 and he op imal adii o he concen ic plo s inc eased o 9/6 m and he diame e limi in- c eased o 12.5 cm. When he ans e ime om plo o plo sho ens, i is be e o measu e a la ge num- be o smalle plo s and ice e sa. When he measu emen ime o one ally ee (in Equa ion 3) was inc eased o 0.7 min and ha o a sub- sample ee o 7 min, he op imal numbe o measu ed sample plo s pe clus e was educed o 18, he op imal plo adii o 6/4 m wi h a diame e limi o 7.5 cm (Fig. 11). Thus, he longe i akes o measu e one ee, he smalle he op imal plo size. Howe e , he numbe o plo s is a ec ed less han when he ans e ime is changed (Figs. 10 and 12). The analysis o he poin pa e ns showed ha 10 ou o 18 poin pa e ns could be conside ed andom (Pois- son). Howe e , six o hese a eas included pa s om Fig. 5 The a e age ela i e RMSE as a unc ion o measu emen imes (min pe plo ) o di e en plo ypes o basal a ea (a), s ems pe ha (b)and olume (c) wi h he wo subsample ee selec ion s a egies (S1 = ixed, S2 = elascope). In elascope plo s, he e ical a ia ion is due o a ying elascope ac o and he ho izon al due o a ying maximum adius. In ixed and concen ic plo s he e ical a ia ion is due o he a ying adii and he ho izon al due o he diame e limi Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 8 o 14 mo e han one s and, which may ha e a ec ed he spa ial pa e n. Se en o he poin pa e ns we e assessed as clus e ed. Howe e , he clus e s seemed o be qui e small (< 2 m) and hey we e p obably due mainly o bi ch (Be ula pubescens) clones. Only one a ea showed e i- dence o a egula pa e n. We compa ed he a e age ela i e RMSE’s (Equa ion 4) o he s ems pe ha (Fig. 13), olume, and basal a ea when he es a eas we e classi- ied in o he di e en poin pa e ns wi h he 6 a eas di- ided be ween di e en s ands excluded. Sampling was ca ied ou wi h a ixed- adius plo wi h plo adius a y- ing om 3 o 11 m. The di e ences be ween clus e ed and Poisson pa e ns we e in a e age small, bu he a i- a ion be ween a eas was highe in clus e ed pa e ns. The ela i e RMSE’s in he a ea wi h he egula poin pa e n seemed o be less sensi i e o he adius o he plo . Discussion In his s udy, we analyzed he e ec o plo ype ( ixed- adius plo s, a combina ion o wo concen ic plo s wi h a a ying diame e limi , and elascope plo s wi h a ying maximum adius), di e en plo size ( a ying adii o elascope ac o ) and wo di e - en s a egies o measu ing subsample ees wi hin plo s (ei he all ees wi h d1.3 > 25 cm and all ees wi hin 1 m om he plo cen e , o wi h elascope ac o 5 m 2 /ha). We examined h ee di e en a i- ables, olume, basal a ea and s ems pe ha in o de o each a comp omise solu ion ha would be sui - able o many o he a iables as well. We did no in- clude class a iables such as o es /non- o es classi ica ion o o es si e o ype classi ica ion, al- hough hese a e impo an a iables in o es in en- o y. In plo -le el conside a ions, a e y small plo o e en a poin would be op imal o many o hese a iables. Thus, including hese a iables o he cal- cula ions would make mo e sense i he whole design we e op imized a he han jus he plo ype and size. Relascope plo s we e mos e icien o olume and basal a ea, bu no as e icien o s ems pe ha. Fo s ems pe ha, ixed- adius plo we e op imal. When he weigh o s ems pe ha is inc eased enough, he ixed- adius plo becomes op imal o e all. I we con- side ed an in en o y pu ely o s ems pe ha o basal Fig. 6 Numbe o subsample ees as a unc ion o plo adius (m) wi h he wo subsample ee selec ion s a egies (S1 = ixed, S2 = elascope) o ixed- adius (a), elascope (b) and concen ic (c) plo s. In elascope plo s he a ia ion wi hin each maximum adius is due o a ying elascope ac o Hen onen and Kangas Fo es Ecosys ems (2015) 2:31 Page 9 o 14