Estimation of bivariate diameter and height distributions using ALS

Transcription

1 SlvLaser 2008, Sept , 2008 Ednburgh, UK Estmaton of bvarate dameter and heght dstrbutons usng ALS Johannes Bredenbach 1, Chrstan Gläser 2 & Matthas Schmdt 3 1 Forstlche Versuchs- und Forschungsanstalt Baden-Württemberg, Abtelung Bometre und Informatk, Johannes.Bredenbach@forst.bwl.de 2 Georg-August-Unverstät Göttngen, Insttut für Statstk und Ökonometre, Chrstan_Glaeser@gmx.de 3 Nordwestdeutsche Forstlche Versuchsanstalt, Abtelung Waldwachstum, Matthas.Schmdt@nw-fva.de Abstract In ths paper, we present a method for estmatng a bvarate heght and dameter dstrbuton based on arborne laser scanner data (ALS). ALS are analyzed wth the area-based method. To construct a bvarate lkelhood functon, we assume that the dameters are Webull dstrbuted and the heghts are normal dstrbuted where the expectaton value s modelled wth the Näslund functon. Comparson of predcted and observed mean dameter dstrbutons result n an RMSE of 2.78 cm and a bas of 0.48 cm. Comparsons of observed and predcted heght dstrbutons are stll to come. Keywords Webull dstrbuton, Näslund functon, ldar, GLM 1. Introducton Hgh densty (.e., >1 return m -1, Magnusson et al. 2007) Arborne laser scannng data (ALS) can be used to estmate attrbutes of sngle trees (e.g., Persson et al. 2002, Popescu et al. 2003). Low densty ALS data are usually analyzed usng area-based methods (Næsset 2002) that result n plot-level estmates of, for example, tmber volume or basal area. The lower flyng heght requred to produce hgher densty ALS data causes longer flght tmes per area. Therefore, costs obvously ncrease wth the densty of the raw data. Ths may be one reason for why area-based methods are already used n operatonal forest nventores (Næsset 2004). Addtonally, some studes show, that the densty of the laser data may be reduced n a wde range wthout a sgnfcant lost of nformaton (Magnusson et al. 2007, Gobakken and Næsset 2007). Consequently, further cost reductons may be possble wth future generaton of laser scanners that allow larger flyng heghts. Whle plot level estmates of, for example volume, s an mportant pece of nformaton, for the actually wanted predcton of assortments, at least dameter, but also heght dstrbutons are needed. Magnussen & Boudewyn (1998) estmated heght dstrbutons usng ALS. Methods for estmatng dameter dstrbutons were shown, for example by Gobakken & Næsset (2004). However, heght and dameter dstrbutons cannot be combned f they are estmated ndependently. Whle Schreuder & Hafley (1977) and Zucchn et al. (2001) proposed functons to ft bvarate dstrbutons, only Mehtätalo et al. (2007) used a regresson model to ft a bvarate heght and dameter dstrbuton condtonal on ALS data. They frst used the method of moments to ft regresson models for the heght dstrbuton as a functon of the dameter and the dameter dstrbuton as a functon of ALS data. They then appled the method of parameter recovery to obtan the parameters of the bvarate dstrbuton. 366

2 SlvLaser 2008, Sept , 2008 Ednburgh, UK In ths study, we use the log-lkelhood method to estmate the bvarate dstrbuton of heght and dameter. The dameters are assumed to follow the Webull dstrbuton gven metrcs from ALS data. Opposed to Mehtätalo et al. (2007), we need to deal wth several tree speces. Therefore, tree heght s modelled usng the Näslund functon gven dameter and metrcs from ALS data. However, dfferent tree speces are not addressed separately. 2. Materal and Methods 2.1 Study area The tree speces composton of the 50 km² managed forest that served as study ste s domnated by Norway spruce (Pcea abes L. Karst.) wth a 70% proporton by area, beech (Fagus sylvatca L.) wth 11% and slver fr (Abes alba Mll.) wth 10%. More detals on the forest structure are gven n Table 1. Table 1: Forest characterstcs of the study ste Mnmum Medan Mean Maxmum Stem number [ha -1 ] Stem volume [m 3 ha -1 ] Basal area [m 2 ha -1 ] Basal area mean dameter [cm] Mean heght [m] Plot establshment In 2002, a permanent sample-plot nventory was carred out on a 100 m (eastng) by 200 m (northng) grd. Trees wth a dameter at breast heght (dbh) of at least 7 cm were measured on concentrc sample plots wth a maxmum dameter of 12 m. To ncrease the effcency of the nventory, trees wth a dbh <30 cm were sampled on plots wth smaller rad. Ths results n four possble plot szes of 2, 3, 6 and 12 m, where trees wth a mnmum dbh of 7, 10, 15 and 30 cm are measured. On each sample plot the heght of the two largest trees per speces were measured usng angle measurement nstruments. The heght of the other trees was estmated based on local dameter-heght curves calbrated wth the measured trees Laser data The laser scan data were collected wth an Optech ALTM 1225 laser scanner n wnter 2003/2004,.e. about one year after the nventory took place. A flght alttude of approx. 900 m above ground yelded an average dstance of 1 m between scan ponts on the ground. The frst as well as the last pulse data were automatcally classfed by the data provder nto vegetatonand ground ponts (reflecton from terran surface). A dgtal terran model (DTM) wth one meter pxel spacng was computed from the ground returns usng the average heght of returns f several reflectons were located wthn one pxel and blnear nterpolaton f no return was wthn the pxel. The value of the respectve DTM pxel was subtracted from the frst pulse vegetaton raw data to obtan vegetaton heghts. Vegetaton heght metrcs (e.g., percentles and mean) were derved for every sample plot (Næsset 2002). 367

3 SlvLaser 2008, Sept , 2008 Ednburgh, UK 2.2 Parameter estmaton If a s the locaton, b the scale and c the shape parameter, the densty of the Webull dstrbuton s denoted by c 1 c c y a y a f ( y a, b, c) = b for b, c > 0. b exp b To estmate the dameter dstrbuton, equaton 1 was extended to f ( d a, (1) where d s a dameter, to condtonal the scale and shape parameters on predctor varables. The locaton parameter (a) was set to the callper lmt (7 cm) because estmaton of ths parameter frequently causes numercal problems (Gobakken and Næsset 2004). Due to the concentrc sample plot desgn, we constructed four censored Webull dstrbutons for every possble plot rad by g R ( d a, = U (2) L f ( d a, f ( x a, d x where U and L are the upper and lower bounds of the dameters for the concentrc sample plot wth radus R, respectvely. Ths resulted n the functons g 2, g 3, g 6, g 12. The parameters are bound to the predctor varables (laser metrcs) wth lnk functons 1 h( P) ( x' ( P), β( P) ), where x (P) are the predctor varables, β are the coeffcents and P s ether b or c. The lnk functon for both parameters s the natural logarthm. The heght dstrbuton gven DBH and ALS data s assumed to follow a normal dstrbuton: h μ( Pμ ) f ( h ( ), ) = exp μ Pμ σ (3) 2π σ 2 σ where the expectaton value μ s an extended Näslund functon that ncludes metrcs from ALS and P μ are ts parameters: (,, 0 ( ), 1 ( ), 2 ( )) 1. 3 d μ d x β Näs β Näs β Näs = + (4) β0 ( Näs) + β1 ( Näs) x + β2 ( Näs) d where β 0 ( Näs), β 1 ( Näs) and β2 ( Näs) are coeffcents of the Näslund functon and x s a predctor varable derved from ALS data. The bvarate heght- and dameter dstrbuton s gven by g d a, b, c * f h μ ( d, x, β, β, β ), ) = (5) R ( ) f ( d, h x, β ). ( 0 ( Näs) 1( Näs) 2 ( Näs) σ The bvarate lkelhood functon can then be denoted as 3 368

4 SlvLaser 2008, Sept , 2008 Ednburgh, UK LLK = n g ln g 2 ( d a, b, c ) 12 ( y ) + g3( d a, b, c ) ( d a, 1 ( y ) + g ( d a, = ( y ) + 1 ( y + ) n = 1 ln ( f ( h ( P ), σ ) μ μ. (6) where 1U ( y ) are sze-class dependent ndcator functons. If U R then 1 y U 1 U ( y ) =. 0 y U The lkelhood functon was maxmzed usng the Nelder-Mead algorthm mplemented n the functon optm (Venables & Rpley 2002), wthn an R envronment (R Development Core Team 2007) On average, 12 trees were measured on a sample plot. The predcted dstrbuton can therefore not be compared wth observatons from one sample plot. Therefore, the observatons from plots smlar wth respect to the explanatory varables are aggregated to what we wll call vegetaton heght quartle classes for the remander of the text. Then, the predcted dstrbutons can be compared wth the hstogram of the observatons. 3. Results The frst and thrd quartle (Qu1 and Qu3) of the vegetaton heght and ther nteracton term (Qu1 * Qu3) were consdered as predctor varables for the Webull scale and shape parameters. The Qu3 was used as addtonal predctor varable for the heght dstrbutons besdes the DBH. The parameters of the Webull dstrbuton can be predcted by b = Qu Qu Qu1 Qu3 c = Qu Qu Qu1 Qu3 (7) Tree heght can be predcted by μ = d / ( Qu d ) wth standard devaton

5 SlvLaser 2008, Sept , 2008 Ednburgh, UK Densty Qu1: (24,26.8] Qu3: (28.9,32.7] Plots: 20 Trees: 242 Qu1: (15.4,18.2] Qu3: (21.5,25.2] Plots: 58 Trees: 725 Qu1: (15.4,18.2] Qu3: (17.7,21.5] Plots: 29 Trees: 430 Densty Qu1: (18.2,21.1] Qu3: (25.2,28.9] Plots: 45 Trees: 472 Qu1: (18.2,21.1] Qu3: (21.5,25.2] Plots: 38 Trees: 561 Qu1: (9.67,12.5] Qu3: (14,17.7] Plots: 40 Trees: 449 Densty Qu1: (21.1,24] Qu3: (25.2,28.9] Plots: 33 Trees: 436 Qu1: (12.5,15.4] Qu3: (17.7,21.5] Plots: 51 Trees: 676 Qu1: (6.81,9.67] Qu3: (10.3,14] Plots: 22 Trees: 228 Fgure 1: Probablty densty dstrbuton of observed DBH (hstogram) and predcted Webull dstrbutons (sold graph) for the 9 most densely populated laser-derved vegetaton heght quartle classes. The dashed curve marks the Webull dstrbuton whch has been drectly ftted to the observatons. Qu1 denotes the class wdth of the frst quartle (m) and Qu3 the class wdth of the thrd quartle (m). Plots and trees represent the number of sample plots and trees n the correspondng plot strata. The predcted dameter dstrbuton fts well to the observed dameter dstrbutons (Fgure ). Results for heght dstrbutons are not yet avalable, however the comparson of the expectaton value wth observed heghts s promsng (Fgure 2). For ths graphc, Qu3 was estmated gven DBH. (However, the regresson model s weak wth R²=0.3.) 370

6 SlvLaser 2008, Sept , 2008 Ednburgh, UK Heght (m) Fgure 2: Predcted and observed heght gven DBH. The means of the Webull and the observed dstrbuton was computed for the 20 most densely populated quartle classes (contanng at least 3 Plots). As the good conformty of the predcted dstrbuton wth the observed dstrbuton supposes, the dfference between the mean of the Webull dstrbuton and the mean of the observatons s rather small (Fgure 3). The RMSE s 2.78 cm wth a bas of 0.48 cm. Observed mean Predcted mean Fgure 3: Observed versus predcted mean DBH for the 20 most densely populated quartle classes (crcles) and 1:1 lne (sold lne). 4. Dscusson The proposed bvarate dstrbuton can be used to estmate dameter and heght dstrbutons. Snce we dd not use the parameter recovery method (Mehtätalo et al. 2007), we obtan densty dstrbutons. They need to be combned wth an estmaton of stem number or basal area to get 371

7 SlvLaser 2008, Sept , 2008 Ednburgh, UK stem numbers per dameter class. For the predcton of assortments, nformaton about tree speces are also requred. In ths study, we assumed the observatons to be ndependent of one another. One major drawback of the used data materal s that not all tree heghts were measured. Ths reduces the varance to be modelled. Acknowledgements We acknowledge the useful comments of two anonymous revewers. References Gobakken, T. and Næsset, E., Estmaton of dameter and basal area dstrbutons n conferous forest by means of arborne laser scanner data. Scandnavan Journal of Forest Research, 19, Gobakken, T. and Næsset, E., Assessng effects of laser pont densty on bophyscal stand propertes derved from arborne laser scanner data n mature forest In E. Rönnholm, P.; Hyyppä, H. & Hyyppä, J. (ed.), ISPRS Workshop on Laser Scannng 2007 and SlvLaser 2007, 200. Hafley, W. and Schreuder, H., Statstcal dstrbutons for fttng dameter and heght data n even-aged stands. Canadan Journal of Forest Research, NRC Research Press, 7, Magnussen, S. and Boudewyn, P., Dervatons of stand heghts from arborne laser scanner data wth canopy-based quantle estmators. Canadan Journal of Forest Research, 28, Magnusson, M.; Fransson, J. and Holmgren, J., Effects on Estmaton Accuracy of Forest Varables Usng Dfferent Pulse Densty of Laser Data. Forest Scence, 53, Mehtätalo, L., Maltamo, M. and Packalén, P., Recoverng plot-specfc dameter dstrbuton and heght-dameter curve usng als based stand characterstcs. In: P. Rönnholm, P.; Hyyppä, H. & Hyyppä, J. (Eds.). ISPRS Workshop on Laser Scannng 2007 and SlvLaser Næsset, E., Data acquston for forest plannng usng arborne scannng laser. Forestsat Symposum Næsset, E., Practcal large-scale forest stand nventory usng a small-footprnt arborne scannng laser. Scandnavan Journal of Forest Research, 19, Nlsson, M., Estmaton of tree heghts and stand volume usng an arborne ldar system. Remote Sensng of Envronment, 56, 1-7. Persson, A., Holmgren, J. and Söderman, U., Detectng and measurng ndvdual trees usng an arborne laser scanner. Photogrammetrc Engneerng and Remote Sensng, 68, Popescu, S. C.; Wynne, R. H. and Nelson, R. F. Measurng ndvdual tree crown dameter wth ldar and assessng ts nfluence on estmatng forest volume and bomass. Canadan Journal of Remote Sensng, 2003, 29, R Development Core Team. R., A Language and Envronment for Statstcal Computng. R Foundaton for Statstcal Computng. Venables, W. & Rpley, B., Modern Appled Statstcs wth S. Sprnger, New York. Zucchn, W.; Schmdt, M. and Gadow, K. v., A model for the dameter-heght dstrbuton n an uneven-aged beech forest and a method to assess the ft of such models Slva Fennca, 35,