Modeling height-diameter relationships for Norway spruce, Scots pine, and downy birch using Norwegian national forest inventory data

Abstract

We developed nonlinear mixed effects height-diameter models for three major tree species: Norway spruce (Picea abies [L.] Karst.); Scots pine (Pinus sylvestris L.); and downy birch (Betula pubescens [Ehrh.]) in Norway. We used data from four Norwegian national forest inventory (NFI) cycles (7th–10th NFI cycle) as model fitting data and data from the 6th NFI cycle as validation data. Among several bi-parametric functions tested as base functions in a preliminary analysis, the Näslund function showed the smallest residual variations, and therefore it was extended by incorporating stand variables as covariates that act as modifiers of the original parameters of the Näslund function. Sample plot-level random effects were also included in order to account for inter-plot variations within the populations. Unlike a basic mixed effects model, the extended mixed model described larger parts of variations in the height-diameter relationships and predicted heights without significant bias for validation data from the sample plots, where all measured heights of the focused species (species used for species-specific model) were used to predict random effects. For species independent models, when measured heights of other than focused species were used to predict random effects, a significant height prediction bias occurred. This bias could be reduced for certain diameter ranges by applying an extended ordinary least square model. We recommend using extended mixed effects models to estimate the missing heights on NFI sample plots and other sample plots, where measured tree heights of the focused species are available for prediction of random effects. When measured heights are not available, the extended ordinary least square model can be used.