Access to the published version may require journal subscription. Published with permission from: NRC Research Press.

Transcription

1 Ts s an autor roduced verson of a aer ublsed n Canadan Journal of Forest Researc. Ts aer as been eer-revewed and s roof-corrected but does not nclude te ournal anaton. Ctaton for te ublsed aer: öran tål ören Hol Toty. reore Tere obaen r Næsset & Ross Nelson. 0 Model-based nference for boass estaton n a LDR sale survey n Hedar County Norway. Canadan Journal of Forest Researc. Volue: 4 Nuber: tt://d.do.or/ 0.39/X0-6. ccess to te ublsed verson ay requre ournal subscrton. Publsed wt ersson fro: NRC Researc Press. tandard set stateent fro te ublser: Ts s an autor s verson of an artcle ublsed n Canadan Journal of Forest Researc Deceber 00. Te artcle s avalable onlne at: tt:// slon Oen rcve tt://eslon.slu.se

2 Model-based nference for boass estaton n a LDR sale survey n Hedar County Norway öran tål ören Hol Toty. reore Tere obaen 3 r Næsset 3 Ross Nelson 4 Det. of Forest Resource Manaeent weds Unversty of rcultural cences 9083 Ueå weden. -al: oran.tal@sr.slu.se; oren.hol@sr.slu.se cool of Forestry and nvronental tudes 360 Prosect t. ale Unversty New Haven CT U. -al: Toty.reore@yale.edu 3 Det. of coloy and Natural Resource Manaeent Norwean Unversty of Lfe cences PO Bo 5003 NO-43 Ås Norway. -al: Tere.obaen@ub.no; r.naesset@ub.no /Boserc cence Branc N-oddard ace Flt Center reenbelt MD 077 U. -al: Ross.F.Nelson@nasa.ov

3 bstract In forest nventores reresson odels are often aled to redct quanttes suc as boass at te level of saln unts. In ts aer we roose a odel-based nference fraewor for cobnn saln and odel errors n te varance estaton. It was aled to arborne laser LDR datasets fro Hedar County Norway were te odel error roorton of te total varance was found to be lare for bot scannn L; arborne laser scannn and rofln LDR wen boass was estated. Wt rofln LDR te odel error varance coonent for te entre county was as lare as 7% wereas for L t was 43% of te total varance. Partly ts reflects te better accuracy of te el-based reresson odels estated fro scanner data as coared to te odels estated fro rofler data. Te fraewor roosed n our study can be aled n all nds of sale surveys were odel-based redctons are ade at te level of ndvdual saln unts. secally t sould be useful n cases were odel-asssted nference cannot be aled due to te lac of a robablty sale fro te taret oulaton or due to robles of correctly atcn observatons of aulary and taret varables. Keywords: forest nventory odel-based nference reresson estaton scannn laser rofln laser varance

4 Introducton In forest nventores quanttes at te level of saln unts are often redcted usn reresson odels. Well nown eales nclude volue and boass odels based on easureents of daeter and et on ndvdual trees e.. Marlund 988. larly n nventores based on arborne lasers LDR; lt detecton and rann reresson odels are aled to redct er-ectare boass or volue based on easureents derved fro te lasers e.. Nelson et al a; Næsset 997. tandard saln teory assues tat te varables entern te estators are observed wtout error e.. reore & Valentne 008. If easureent errors or odel -related errors cannot be nored uncertanty estates suc as te varance tycally are underestated wen standard forulas are aled e.. ärndal et al. 99. Tus te reorted fures are less trustworty tan tey aear to be and as a result narorate decsons ay be ade based on te nforaton. In soe studes e.. ertner & Köl 99 ertner et al. 00 error budets are coled were te contrbuton of dfferent error sources to te total error often eressed as varance or ean square error s assessed. Te ortance of roer andln of odel-related errors also s onted out by te Interovernental Panel on Clate Cane IPCC 003 n ter ood ractce udance for reenouse as reortn for te land use land-use cane and forestry sector of te Unted Natons Clate Cane Conventon. However n any forest surveys odel related errors are nored and uncertanty estates are rovded as f redcted quanttes were true. LDR based forest surveys ave evolved radly over te last decade e.. Næsset 997 Means et al. 000 Holren 004. Te strent of te tecnque n te contet of forest nventory s tat very detaled nforaton about canoy et and cover can be obtaned trou easureents of te dfferences n te returns of laser ulses etted fro ar- or sacecrafts reflected fro te canoy and fro te round. Two dfferent aroaces usn so-called sall-footrnt laser data ave been develoed and deonstrated n oeratonal roects naely te use of arborne rofln lasers desned for saln-based nventores Nelson et al. 003a 004 and te use of arborne scannn lasers L for wall-to-wall an of forest stands for ractcal 3

5 forest lannn Næsset & Bernes 00 Næsset 00. Today t s coon ractce n any countres to aly L wen stand level nforaton for forest anaeent lans s coled e.. Næsset 004b 007. Te rofln syste develoed at N by Nelson et al. 003b labeled Portable rborne Laser yste PL s an neensve and sle devce wc can be oerated at low costs. rofln syste only collects a narrow lne of data on te round and does not rovde data for wall-towall an. In contrast to ts eac flt-lne of a scannn syste tycally as a swat wdt of say Tus scannn systes can rovde data for contnuous an of te forest and are terefore deal for estatn roertes of te forest at stand level. However L can also be used as a str saln tool to nventory tber volue and boass n lare areas. lcatons were LDR easureents are used n te contet of saln surveys currently are ann ncreased nterest e.. Nelson et al. 004 Parer & vans 004 ndersen & Bredenbac 007. In suc alcatons standard saln estators and varance estators are robleatc due to te cole structure of te surveys were lon lnes or belts ay etend over several strata e.. Nelson et al Furter any dfferent sources of errors are nvolved and odel-related errors need to be secfcally accounted for n te uncertanty analyss or odel-asssted estators ärndal et al. 99 be aled. Te latter aroac requres tat tere s a sound robablty sale of oulaton unts wt easured taret varables avalable fro te area of nterest wc s not always te case. Te reason ay be tat already estn boass odels are aled n new surveys or tat ood atcn of unts fro feld saln and reote sensn cannot be aceved. Te obectve of ts study was to develo and aly a eneral fraewor for odel-based estaton and error assessent accountn for bot saln and odel errors n cases were reresson odels are aled to redct te taret varables. secally te fraewor sould be useful n cases were odel-asssted estators cannot be aled. Te study was based on LDR data fro Hedar County Norway were bot scannn and rofln laser data ad been acqured n order to assess boass resources. Wle odel-based nference ay encoass any dfferent aroaces e.. 4

6 reore 998 McRoberts 00 n ts study we aled reresson odels at te level of saln unts tat ad been selected trou robablty saln. Materals and Metods Te Hedar County survey Te study area was Hedar County n souteastern Norway on te weds border F.. Te total area of te county s Tere are 309 eranent Natonal Forest Inventory NFI sale lots avalable n ts area dstrbuted systeatcally n a 33 rd; eac lot s crcular and as a sze of 50. Te easureent rotocol stulates tat 0% of te eranent lots are re-easured every year; ts selecton s ade accordn to a Latn square desn wtn a 4545 bloc of lots. On eac sale lot all trees wt daeter at breast et DBH 5 c were callered and tree ets were easured on an averae of 0 sale trees er lot. Trees wt DBH <5 c but taller tan.3 were counted and ter daeters estated by eans of odels Toter 998. Total above round dry boass of all trees taller tan.3 was ten estated accordn to seces-wse alloetrc odels wt DBH and et as redctor varables Marlund 988. Te coordnates of eac lot center were deterned wt an averae accuracy < usn dfferental lobal Postonn yste and lobal Navaton atellte yste easureents accordn to te rocedures suested by Næsset 00. Te county was stratfed nto et strata based on estn land use as and Landsat satellte aes. Te et strata ncluded four roductve forest classes.e. edu and 3 low roductvty forests and 4 youn forest. Te reann four strata were eter nonroductve forest or non-forest classes.e. 5 nonroductve forest 6 ountan areas >850 above sea level 7 develoed areas e.. resdental areas and nfrastructure and 8 oen water. Bot rofln and scannn laser data were collected durn te suer of 006. Te flt lnes were flown east-west and followed te NFI rd F. for ractcal reasons wtout any account for te stratfcaton. Observatons fro NFI sale lots easured n te erod fro 005 to 007 were used 5

7 as round trann data to construct odels of relatonss between te arborne laser data and aboveround boass as deterned fro feld easureents. Laser rofln In total 05 rofln flt lnes totaln 966 were flown; 763 NFI lots were overflown wtn 7.8 of lot center. Profln laser easureents of canoy et and crown densty slar n for to L varables descrbed by Næsset and obaen 008 ecton.4 were etracted alon fed-lent 7.8 seents closest to te center of te round lot and related to total aboveround boass. Te seent denson 7.8 was defned by te daeter of a 50 crcular round lot. Te et-class stratfcaton roved robleatc wt resect to analyss of te rofln LDR data. Due to te varablty n data t was found tat only rater aroate redctve boass odels could be develoed for eac of te four roductve forest classes. s a consequence one enerc lnear odel R = 0.59 RM = 39 t a - was develoed across all roductve forest classes. For te four nonroductve forest/ non-forest classes te R values raned between 0.46 and 0.64 wt RMs between.3 and 9.8 t a -. Profln results ten are reorted for roductve forest and te four nonroductve/non-forest classes based on estates ade on eac of te 7.8 seents on all flt lnes. In all te odels boass t a - was te deendent varable and ets to te 40 t 60 t and 90 t canoy cover decles te ndeendent varables toeter wt averae canoy et quadratc ean canoy et and standard devaton of canoy ets. tandard lnear reresson was aled. For eac odel te ndeendent varables rovdn te best ft were selected. Tycally two ndeendent varables were ncluded n eac odel: one varable related to decle et and one to eter averae et standard devaton of ets or quadratc ean et. Laser scannn Ffty-tree flt lnes were flown wt te scannn laser wt an nter-lne dstance of 6 wc eans tat aroately 50% of te avalable lots were covered by L 6

8 data F.. In total 4570 were flown covern 97 or 8.4% of te county s area. Te averae ulse densty for te scanner was.8 ulses -. Only frst ecoes were used n ts study. Non-lnear aboveround boass odels were estated for 7 of te 8 strata. Fro 30 to 5 sale lots were located n eac stratu ecet for water were no lots were easured. Te estated odels were bac-transfored and te calculated RM values ten raned between 9.6 and 3.9 t a -. Te squared Pearson correlaton coeffcents between observed and estated boass after bac-transfor raned between 0.79 and 0.9. ll te odels ncluded bot canoy et etrcs and canoy densty etrcs derved fro te L data. lttude was also ncluded n four of te odels. Te selected odels contaned fro two to four elanatory varables. Tycal canoy et etrcs were te uer decles of te canoy et dstrbutons. Te canoy densty etrcs were couted by dvdn te resectve canoy returns nto 0 dfferent vertcal layers of equal et. Te et of eac layer was defned as one tent of te dstance between te 95% ercentle and te lowest canoy et.3. Te canoy denstes were ten couted as te roorton of nuber of returns above a ven layer to total nuber of returns ncludn tose below.3 see Næsset 004a for furter detals. ac flt lne or str was dvded nto reular 50 rd cells and te cells were allocated to strata. Te laser ecoes tat beloned to eac rd cell were used to derve te sae canoy et and -densty etrcs as derved for te NFI lots. Te estated odels were used to redct boass for eac cell wtn 500 wde belts wt te center lnes of te L scans ben te center lnes of te belts. Te odel develoed for edu ste roductvty was used also for redctn boass for eac cell allocated to te stratu water. tatstcal etods In te follown sectons basc estators varances and varance estators wll be derved n stes. Frst we address a standard sle rando saln fraewor were 7

9 saln and odel errors are cobned. Ten ts s eanded to account also for cluster saln and stratfcaton wc were ortant features of te Hedar County LDR survey. Te follown basc setu of an nventory suc as te one n Hedar s assued: - frst sale s acqured by sle rando saln. Ts s te alcaton sale to wc we aly te reresson odels develoed based on te sale see below. Tus fro te saln unts roy varables are acqured and used as ndeendent varables n a reresson odel to redct te taret varable on eac saln unt. In Hedar te roy varables were te L or rofln laser easureents. - second sale s taen were easureents of bot taret and roy varables are ade. In Hedar our sale conssts of all round sale lots and te corresondn etrcs derved fro te rofln or scannn laser easureents. We assue tat tere s only one odel ste nvolved.e. tat boass s redcted drectly based on roy data at te level of sale lots. Moreover n te basc set-u we assue tat te sales and are ndeendent. ll dervatons are ade n a odel-based contet assun tat a oulaton odel s avalable. Here s te vector of ndeendent varables te vector of araeters and te devaton fro te true value. Te for of te eected value odel s assued to be nown. In ractce ts odel can be ftted usn lnear or non-lnear reresson ncludn bac-transforaton and correcton for transforaton bas n case te deendent varable was transfored. uc correctons are avalable for any coon transforatons e.. Mller Below te -vector varables are denoted =... were s te nuber of saln unts n. Te sale s assued to be acqured n a anner arorate for estatn te araeters n te vector. ccordn to te assutons of reresson analyss ts can be done n any dfferent ways rann fro urosve to rando saln [e.. Royall 970 Royall & Herson 973]. Te observatons are 8

10 denoted wt nde and te odel y s assued to be vald; te araeters are estated as. Trouout te dervatons we condton on te sale. True oulaton ean Te true fnte oulaton ean can be wrtten M ~ ~ M were M s te oulaton sze. For lare M ~ wll dffer only nelbly fro te oulaton odel ean M M obtaned by tan eectaton wt resect to. Trouout ts study we adot as defnton of te oulaton ean. staton stratforward odel-based estator of te oulaton ean follown sle rando saln of sze te sale sze of s 3 9

11 0 total s aled by ultlyn wt te nown oulaton sze. Our abton now s to derve a forula for te varance and a varance estator altou ottn fnte oulaton correcton. Varance and varance estaton We assue tat our -estates are accurate enou so tat we can lnearze te -odel n te neborood of te true value and use te frst and second oents of te lnear functon as roes for te true oents were ; s te nuber of araeters. ected values etc. are ten taen consdern bot te dstrbuton of -estates and te sale. Te estator of s ndeendent of all -values snce te sale s assued taen ndeendently of. ll are furter assued to be unbased or aroately so. ccordn to te assued odel. Furterore 5 D D were D s te ter wtn bracets and D te second su D and D are at least aroately uncorrelated and tus te varance for eac ter can be derved searately and te varances added. Te detals are rovded n end. Te resultn varance s 6 V

12 Te frst ter arses fro te saln error due to wle te second ter arses fro te effects of te uncertanty of te -estates wc s related to te selecton of on D. In equaton 6 s te oulaton varance of te -values and te ' -ters are te averae values of te frst order dervatves of te -functon. Here and elsewere te ndces and denote te sale wtn wc te oents are consdered. n estator of ts varance s see end : 7 V s In ts forula s s te sale based estate of te oulaton varance of te - values. Te covarance ter s estated fro te sale. Cluster saln We now eand te set-u to accoodate te ortant case were te odel s aled at te level of oulaton eleents selected trou cluster saln n te sale. Ts s te case wen LDR saln lnes rofler or strs L are dvded nto saller eces rofln 7.8 seents or L 50 cells n te Hedar case for wc boass s redcted usn a odel or wen odels for boass or volue are aled to trees on sale lots. We ntroduce te notaton for cluster totals. In te contet of LDR saln a cluster s a flt lne.e. te total of boass estates ade on all seents rofler or cells L n a ven laser flt lne. Te averae of te cluster totals ten can be eressed n a for slar to 3 as 8

13 T were s te su of te t t values for te T obects wtn te t cluster. Te averae fro 8 can easly be converted to total boass or averae boass er ectare for te reon of nterest. Note tat tere s a wetn ecans lct n te consderaton of cluster totals. Loner flt lnes contan ore seents or cells tereby contrbutn larer nubers of boass observatons to te flt lne or cluster totals. Varances can be derved follown te sae loc as n te forer secton and tus te follown varance forula s obtaned: 9 V Here s te varance of cluster totals n te oulaton and T t. Furter a varance estator can be derved alon te revous t lnes of dervaton usn te notaton t nstead of T nstead of t t t nstead of and T t t nstead of ĝ. Te resultn varance estator s: 0 V s tratfcaton and ost-stratfcaton

14 In case of ordnary stratfcaton e.. reore and Valentne 008 sales are selected ndeendently n dfferent strata. If ts s te case and f searate reresson odels are used for redctons n eac stratu stratfed saln can be andled easly by an searate estates of oulaton totals and varances for eac stratu and ten add te estates to obtan overall estates. However an ortant feature n any lare-scale forest surveys suc as te LDR surveys n Hedar s tat strata are fored based on avalable a data but saln unts are dstrbuted ndeendently of ts stratfcaton denoted oststratfcaton n our study. In ts case tere wll be deendences between te estates fro dfferent strata due to te fact tat soe cluster saln unts etend over several strata. n estator of te ean value n stratu s: In ts forula te suaton etends over all clusters ust le n te revous cases e.. q. 8 but te cluster totals are couted only based on te sub-unts belonn to stratu f a stratu s not resent n a cluster ts quantty s zero. Te varable s te area L or lent rofler of stratu wtn cluster. Te defne te reresson odel used n stratu. Te corresondn ean across all H strata wc can ten be ultled wt nown total area to obtan an overall total s: H W Here W s te nown area roorton of stratu fro for nstance te dtal a used to stratfy te entre study area. Te dervatons of a varance forula and a varance estator of are rovded n end. Te varance s aroately: 3

15 4 3 H H W W V H H W W ' ' Te frst art corresonds to te saln error and te second art to te errors ntroduced due to te uncertantes of te araeter estates of te reresson odel. n estator of ts varance s: 4 H H n W W V H H W W ' ' Te covarances of te araeter estates are ortant. If searate odels ave been derved for eac stratu based on ndeendent datasets ten all cross-stratu covarances are zero. If te sae odel s aled n several or all strata ten cross-stratu covarances sould be ncluded. For an ndvdual stratu te varance estator s: 5 V ' '

16 Results Te estaton fraewor descrbed above was aled to data fro te Hedar County survey usn bot L and PL data. Te results are suarzed n Table were bot overall and stratu-level results are rovded. ecfcally te total varance was searated nto odel and saln error coonents n order to llustrate te antude of te dfferent sources of varablty. Laser Profln We note te follown based on te results resented n Table :. Te estates based on rofler data are about 0% saller tan te corresondn round-based estates.. In all fve strata were searate odels were develoed ore tan 50% of te rofler varance s due to varablty assocated wt odel araeter estaton. Te roorton of odel varance s lare; n tree of te fve rofln strata odel varance accounts for ore tan 90% of te total varance. 3. In all fve strata were rofler versus L odel varance roortons can be coared te L odel varance coonent s consstently saller tan te rofler s. 4. Consdern te roductve forest class te rofler standard error s larer tan te round-based standard error. Pont seas to accuracy and onts -4 sea to te recson of te rofln LDR estates. Te larer roorton of odel error varance for te rofler reflects te cobned effects of te better laser ulse eolocaton accuracy of te scannn syste te fact tat te scanner acqures rann easureents across an entre round lot wereas te rofler easures only a lnear slce n te roty of a ven round lot and 3 te fact tat te rofler flew twce as any flt lnes as te 5

17 scanner. Factors and result n saller L odel errors; factor 3 reduces te rofler saln error. Te Table results quanttatvely descrbe te ltatons of te PL rofln LDR and suest two tes tat sould be addressed n order to rove rofln results. Frst te eolocaton accuracy of te ndvdual LDR ulses ust be roved see obaen and Næsset 009 for descrton of effects of L-round srestraton. econd alternatve odel fors sould be consdered e.. ln-ln odels e.. Næsset 00 or te square root odels utlzed by ndersen and Bredenbac 007 and Boudreau et al. 008 see also reore et al. 008 for an arorate bac-transforaton correcton. Model error s sly overweln te rofln error ter and addressn te two onts above ay decrease te odel error ter and te total varance. Laser cannn We note te follown based on te L-results resented n Table :. Te L based estates are sltly larer tan te corresondn round-based estates. Te dfferences rane between 0 and 0% n te dfferent roductve forest strata; n averae te dfference s about 5%. However larer dfferences were found n nonroductve forests and ountan areas were te L based estates were consderably saller tan te round-based estates resectvely. For te entre county te L based boass estate s very close to te corresondn round-based estate.. In total 43% of te L varance s due to varablty assocated wt odel araeter estaton. For te roductve forest te odel varance accounts for 4% and t vares between 58% and 85% for te four roductve forest strata. 3. ven f te odel varance coonent s consstently saller tan te rofler s te roorton of odel varance s. 6

18 4. In eneral te L standard errors are saller tan te round-based standard errors owever te dfferences are not very lare. Te LDR-based boass estates n ountan areas and n develoed areas were consderably saller tan te corresondn round-based estates. We arue tat te lac of corresondence for tese strata was due to te fact tat te NFI as been focusn on roductve forest and only easured sale lots located close to te roductve forest areas. Tus no reresentatve round-based sales were avalable for tese areas. Dscusson Te roosed odel-based fraewor for boass estaton based on LDR data s only one out of several ossble estaton aroaces. One alternatve would be a odel-asssted fraewor ärndal et al. 99 were te odels would be used to rovde roy values to wc adustents based on actual easureents are aled to ensure unbased estates; ts aroac was adoted by ndersen and Bredenbac 007 and furter by reore et al. 00. Te odel-asssted aroac as te advantae of stayn wtn te real of desn-based estaton but t reles on te avalablty of a sound robablty-based subsale of round lots wtn te taret area and ood eoracal atcn between round and LDR data. oe features of te Hedar study colcated usae of odel-asssted estaton e.. tat a rando sale of feld lots was not avalable n all reons and tat locaton errors n te PL data soetes ade t dffcult to atc feld data and LDR data. In revous studes wt rofln lasers Nelson et al. 003a; 004; 008 te and te sales ave been ndeendent; n suc cases a odel-based aroac would be te only stratforward estaton fraewor. 7

19 Te ercal fndns llustrate tat rofln LDR ay be less adequate for lare area saln tan L unless redctve odels assocated wt te rofler can be snfcantly roved. L systes see to be ore effcent wt resect to collectn LDR easureents over estn round lots as te wde swat of a scannn syste n ost cases wll ensure overlan easureent of te lots fro te ar and on te round. However we consder te furter develoent of saln desns and alcatons tat allow reonal estates based on rofln LDR observatons to be of reat ortance because for te net decade rofln LDRs wll be te only sace LDRs avalable caable of rovdn contnental ersectves; te N ICat II and DDynI are te only sace LDR ssons currently under consderaton for launc - bot are currently confured as roflers. Te resented estaton fraewor s alcable for rofln as well as scannn LDR alcatons and tus sould allow for tely and rad boass assessents at several eoracal scales fro reonal to contnental and even lobal levels. However tere are several ossbltes to furter rove te roosed etodolocal fraewor. Wle te current aroac accounts for te araeter estaton uncertanty n one odel ste cluster saln and stratfcaton/ost-stratfcaton t could be furter develoed to cover also redcton errors for ndvdual unts systeatc saln and addtonal odel stes. Incluson of redcton errors for ndvdual unts would be artcularly relevant for estates wtn saller reons. Tey would be stratforward to nclude wen rando saln of snle oulaton unts s aled altou wt cluster saln te deendences between eleents wtn clusters would lead to colcatons. ysteatc saln only would affect te saln error art of te varance and etods suc as successve dfferencn e.. Wolter 984 robably could be aled to accoodate ts effect. ddtonal odel stes also could be ncluded alon slar lnes as te frst odel ste. Tere are several forest nventory cases were te roosed fraewor would be stratforward to aly; one eale s feld-based nventores were volue or boass odels are used for tree-level redctons on sale lots. Tese odels are norally develoed ndeendently of te alcaton sale. lso te forulas allow for nvestatons nto te trade-offs between sale szes n alcatons and sale 8

20 szes for develon te reresson odels. Clearly oor odels aled to lare sales wll lead to sall saln error coonents and lare odel error coonents. It sould be noted tat te odel-based estators derved requre tat te colete estated varance-covarance atr of te estators of te reresson araeters s nown. Ts requreent ay be dffcult to eet n cases were estn reresson odels are aled and were tere s no access to data so tat te covarance atrces can be couted f tey are not reorted. ltou te results of te estaton fraewor aeared correct and local wen aled to LDR data a sall addtonal sulaton study was carred out n order to cec for te correctness of soe of te basc forulas. Te varance estator 7 was evaluated based on te odel: y e wt ln and 0.5 and dfferent dstrbutons for were 5 d e 45. lso two dfferent selecton etods reardn te sale sle rando saln and PP saln usn d as te sze varable were aled. Ts odel can be seen as a far aroaton of volue or boass y as a functon of daeter d. Te araeter estates were obtaned follown a lnearzaton trou loartc transforaton. In eac reetton of te sulaton new and sales were selected. Te dstrbutons for d were rectanular and alf-tranular wt d=5 nne tes as frequent as d=45. In all cases evaluated te ean of te varance estator corresonded closely to te sulated true varance ndcatn a sold erforance of te roosed varance estator. We conclude tat te roosed odel-based fraewor sould be very useful n nventory roras were reresson odels are used to redct te quanttes of nterest at te level of ndvdual saln unts. Te alcaton descrbed n ts aer LDR-based estaton of boass s an ortant eale wc deonstrated tat te odel error contrbuton to te total varance ay often be substantal. secally te roosed fraewor sould be useful n cases were odel-asssted nference cannot be aled; te reasons ay be eter tat odels ave been develoed fro an ndeendent dataset or tat atcn of te sales for odel develoent and alcaton cannot be fully aceved as was te case wen rofln LDR was aled n Hedar. 9

21 References ndersen H.-. and Bredenbac J tatstcal roertes of ean stand boass estators n a LIDR-based double saln forest survey desn. Proceedns of te IPR Worso Laser cannn -4 eteber 007 soo Fnland. IPR Volue XXXVI Part 3 / W Boudreau J. Nelson R.F. Marols H.. Beaudon. undon L. and Kes D Reonal aboveround forest boass usn arborne and saceborne LDR n Québec. Reote ens. nvron. : ertner.z. and Köl M. 99. n assessent of soe nonsaln errors n a natonal survey usn an error budet. For. c. 38: ertner.z. Wan. Fan. and nderson. 00. rror budet assessent of te effect of DM satal resoluton n redctn tooracal factor for sol loss estaton. Journal of ol and Water Conservaton 57: obaen T. and Næsset ssessn effects of ostonn errors and sale lot sze on boyscal stand roertes derved fro arborne Laser scanner data. Can. J. For. Res. 39: reore T Desn-based and odel-based nference n survey saln: arecatn te dfference. Can. J. For. Res. 8:

22 reore T.. Ln Q.F. Boudreau J. and Nelson R Reresson staton Follown te quare-root Transforaton of te Resonse. For. c. 54: reore T.. and Valentne H.T aln stratees for natural resources and te envronent. Caan & Hall Boca Raton reore T.. tål. Naesset. obaen T. Nelson R. and Hol. 00. Model-asssted estaton of boass n a LDR sale survey n Hedar County Norway. Can J. For. Res. n ress. Holren J Predcton of tree et basal area and ste volue n forest stands usn arborne laser scannn. cand. J. For. Res. 9: IPCC 003. ood ractce udance for land use land-use cane and forestry. IBN Marlund L Boass functons for ne sruce and brc n weden. weds Unversty of rcultural cences Deartent of Forest urvey.ueå. In weds.. McRoberts R Probablty- and odel-based aroaces to nference for roorton forest usn satellte aery as ancllary data. Reote ensn of nvronent 4: Means J.. cer.. Brandon J. F. Renslow M. erson L. & Hendr C. J Predctn forest stand caracterstcs wt arborne scannn ldar. Potoraetrc nneern and Reote ensn 66:

23 Mller R Beyond NOV: Bass of led tatstcs. Wley: New or. Næsset statn tber volue of forest stands usn arborne laser scanner data. Reote ens. nvron. 6: Næsset. 00. ffects of dfferental snle- and dual-frequency P and LON observatons on ont accuracy under forest canoes. Potora. n. Reote ensn 67: Næsset. 00. Predctn forest stand caracterstcs wt arborne scannn laser usn a ractcal two-stae rocedure and feld data. Reote ensn of nvronent 80: Næsset. 004a. Practcal lare-scale forest stand nventory usn a sall-footrnt arborne scannn laser. cand. J. For. Res.9: Næsset. 004b. ccuracy of forest nventory usn arborne laser-scannn: evaluatn te frst Nordc full-scale oeratonal roect. cand. J. For. Res. 9: Næsset rborne laser scannn as a etod n oeratonal forest nventory: tatus of accuracy assessents accolsed n candnava. cand. J. For. Res. : Næsset. & Bernes K.-O. 00. statn tree ets and nuber of stes n youn forest stands usn arborne laser scanner data. Reote ens. nvron. 78:

24 Næsset. & obaen T staton of above- and below-round boass across reons of te boreal forest zone usn arborne laser. Reote ens. nvron. : Nelson R. Krabll W. and Tonell J statn forest boass and volue usn arborne laser data. Reote ens. nvron. 4: Nelson R. Valent M.. ort. and Keller C. 003a. ultle resource nventory of Delaware usn arborne laser data. Bocence 53: Nelson R. Parer. & Ho M. 003b. ortable arborne laser syste for forest nventory. Potora. n. Reote ensn 69: Nelson R. ort. and Valent M Measurn boass and carbon n Delaware usn an arborne rofln LIDR. cand. J. For. Res. 9: [rratu : ] Nelson R.F. Næsset. obaen T. tål. and reore T Reonal forest nventory usn an arborne rofln LDR. Jour. Forest Plannn 3: Parer R.C. and vans D.L n alcaton of LDR n a double-saln forest nventory. West. J. l. For. 9: Royall R. M On fnte oulaton saln teory under certan lnear reresson odels. Boetra 57: Royall R. M. and Herson J Robust estaton n fnte oulatons. J.. tatst. ssoc. 68:

25 ärndal C.-. wensson B. and Wretan J. 99. Model asssted survey saln. rner New or Toter.M Berenn av volu de første år etter bestandsetablern. [Volue estaton durn te frst years after stand establsent]. Unublsed note. Norwean Insttute of Land Inventory.. In Norwean Wolter K.M n nvestaton of soe estators of varance for systeatc saln Journal of te ercan tatstcal ssocaton 79:

26 Land cover Table. Mean standard error standard error n ercent of ean and ercentae odel error of total varance for total above round dry boass usn rofln PL and scannn L LDR. Te corresondn round-based Norwean Natonal Forest Inventory estates are reorted n te two rtost coluns. No of lots Mean t a PL L round lots % odel No of Mean % odel No of Mean t a % a error lots t a t a % a error lots t a Productve forest: H Medu Low oun ll rod. forest t a Non rod. Forest/nonforest: Nonroductve forest Mountan areas Develoed areas no lots Water Total a %: standard error n ercent of ean above round dry boass. 5

27 Fure leends Fure. Te study area: Hedar County Norway. Fure. Ffty tree arborne laser scannn flt lnes sacn of 6 and 40 Natonal Forest Inventory round lots blac dots. Profln laser was flown at every 3.e. twce as any flt lnes as wt te scannn laser. 6

28 F.. 7

29 F. 8

30 end Te startn ont for te dervatons s te decooston We see to derve te varance of naely V. Te notatons D and D are ntroduced by D and D. Te stocastc nature of D s deterned by te sale and tat of D te estator by te sale. Ts s ndcated by te ndces and below. Now D s te devaton between te averae sale value f te true odel s nown and ts eected value and ts devaton s ndeendent of ; tus V D were te oulaton varance of te -values oulaton fro wc s taen. s Reardn D te Taylor aroaton leads to D Te varance of Condtoned on we ave D s obtaned byv D V D V D. V D were. Ts leads to te follown eresson for te uncondtonal 9

31 varance V D Note tat deends only on not on. Furter condtoned on te braceted ter n D s a constant. Hence D 0 or nearly so and tus also D D 0. Ts les tat bot D D 0 D D 0and tus D and D are at least aroately uncorrelated. Terefore we can sly add te varances of D and D to et te varance of V Te frst ter arses fro te saln error due to wle te second ter arses fro te uncertanty of te -estates wc s related to te selecton of. Varance estaton varance estator wll now be derved. To start wt we address te estaton of fro te -sale trou substtutn by and alyn te sale varance s as an estator. Condtonal on we obtan 30

32 3.... and ddn te ters we obtan / s and condtonal on s s were denotes te sale covarance of te varables and. Te uncondtonal eectaton tus equals

33 3 s s Dvdn ts wt and utlzn tat / we obtan s V / Te ter can be estated fro te sale wle te roduct robably s farly well estated by were. Ts les a certan bas aountn to wc would be n te order of / n relaton to te estate as suc In concluson a far varance estator can be eressed as s V In atr notaton te double su can be wrtten as T wt te radent as a row vector and T as a notaton for transoston. Varances for te ost-stratfed estator Te forulas for te varance and ts estator are n rncle derved by te sae etod as above but te detals are ore colcated snce a rato estator s nvolved. For te estator

34 33 H W were we ave te enerc forula H H W W V Furter alyn te custoary eresson for rato estators R R R R Now by a Taylor eanson R R By nsertn ts nto te nuerator covarance above and eandn we obtan

35 34 R R R R snce two of te four ters are at least alost zero due to te near unbasedness of. Te frst ter equals te covarance estator wt true data and ts eectaton wt resect to s tus equal to R R Te second ter equals by eanson and tan eectaton wt resect to By nsertn tese two eressons n te enerc forula we obtan te aroate forula 3 for te varance. Te aroate varance estator s obtaned by nsertn sale estates and observed values n te varance forula n analoy wt forula 7.