A MIXED-EFFECTS MODEL TO ESTIMATE STAND VOLUME BY MEANS OF SMALL FOOTPRINT AIRBORNE LIDAR DATA FOR AN AMERICAN AND A GERMAN STUDY SITE

USDA Forest Service, Pacific Northwest Research StationAnchorage Forestry Sciences Laboratory, 3301 C Street, Suite 200, Anchorage, AK 99503, USAKEY WORDS: Mixed-effects models, random effects, lidar, forest inventory, stand volumeABSTRACT:Similar datasets (inventory plots, stand maps and lidar data) were available for study sites in the USA and Germany. These datasetsare grouped or hierarchical in that several sample plots are located within a stand and the stands are located within two study sites.Fixed-effects models and mixed-effects models with a random intercept on the stand level were fit to each dataset. Mean lidar rawdata return height and its interaction term with canopy cover as well as its interaction term with the coniferous proportion were foundto be the most influential predictor variables. The mixed-effects models significantly improved the estimates and especially reducedthe bias which was present for numerous stands in the estimates of the fixed-effects models. This resulted in a slight increase of thevariance within the stands. The RMSE for the German study site was higher (34.7% and 29.7% for fixed- and the mixed-effects modelrespectively) than on the US study site (19.2% and 16.8% for fixed- and the mixed-effects model respectively). A mixed-effects modelwith random effects on the study site and stand level was fit to the combined dataset. It showed almost the same errors as the localmixed-effects models (17.6% and 29.8% for the US and the German study site respectively). Hence a single model is sufficient tomake estimates for both datasets. The study shows the potential of mixed-effects models in this context. It illustrates that the commonpractise of fitting different models for different strata may be unnecessary.1 INTRODUCTIONHeight and density metrics, derived from lidar (light detectionand ranging) point clouds can be used as predictor variables instatisticalmodelstoestimateforestparametersatthestandorplotlevel (Naesset, 2004; Andersen et al., 2005, among others). Suchmodels are usually fit using sample plots where both lidar (co-variates) and ground-truth information (response) are available.To map the variable of interest, the entire lidar dataset is griddedinto tiles having the same size as a sample plot. Then the predic-tor variables are computed and the regression models are appliedto every tile. Compared to plot-based inventories, estimation er-rors can be significantly decreased for the area of interest (e.g.a single forest stand), since the number of observations (i.e. thetiles) is usually much higher than the number of sample plotswithin a stand.The predictor variables derived from lidar data are mainly relatedto the vegetation height and structure (e.g., height- and densitymetrics, crown cover). The vegetation cover can, under certaincircumstances, also be classified into broadleaf and coniferoustrees. However, information about the site quality or tree speciescannot be derived without additional data. Therefore, predictionsforstandswithraresiteindexclassesortreespeciescompositionsmight deviate from the mean model, resulting in a bias.Ifthegroupingstructure(i.e., thestandboundaries)isknown, thedeviation from the mean model of plot estimates within a standcan be utilized to reduce the bias using mixed-effects models(mixed models). From the statistical point of view, the group-ing structure has to be considered since the observations are notindependent. In a mixed model, the effects of the variable thatindicates the level of grouping (i.e. the stand-ID) are assumedto be a random sample of a larger population that vary randomlyaround a population mean. This is referred to as random effects.Mixed models with forestry application were discussed by Lappiand Bailey (1988). An in-depth description of mixed models isgiven for example by Pinheiro and Bates (2002).In a mixed model, the variance is split into within and betweengroup variance. The coefficients and standard errors for predic-tor variables that vary less within than between the groups aretherefore more accurate. Another advantage of a mixed model,compared to a fixed-effects model with the grouping level as adummy variable, is that predictions can also be made for individ-uals with grouping levels that did not exist in the dataset used tofit the model (e.g., in our case those stands without sample plots).In a forest inventory context, a mixed model provides an addi-tional advantage. A model can be fit to a large dataset (e.g., toa well inventoried public forest) and subsequently be calibratedwith just a few sample plots for a new forest area (e.g., a smallprivate forest). (A new model would need to be fit, if a fixed-effects model were used.)Publications regarding the estimation of volume and biomass ontheplotlevelincludetheseofNaesset(2002)whocreatedseparatemodels for different ages and site qualities and achieved R

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