Growth functions describe the change in size of an individual or population with time. The selection of appropriate growth functions for tree and stand modeling is an important aspect in the development of growth and yield models. Here we present information on the forms and characteristics of the more commonly-used growth functions for modeling forest development. When fitted to data, a number of these functions will give essentially equivalent results within the range of the observations used for estimating the equation's coefficients. However, their behavior when extrapolated may be quite different depending on the underlying mathematical properties involved. Hence, understanding these properties is helpful to modelers to determine which candidate functions to consider for specific applications.Unless the data available for modeling cover a very small range of time, there are certain properties that a growth function should exhibit to be consistent with the principles of biological growth (Fig. 6.1): (i) The curve is often limited by the value zero at a specific beginning (t D 0 or t D t 0 ), depending if the variable that is being modeled starts at t D 0, as is the case for the great majority of the tree and stand variables, or later on, as happens with tree diameter at breast height or stand basal area; (ii) The curve generally should exhibit a maximum value usually achieved at an older age (existence of an asymptote); (iii) The slope of the curve should increase with increasing growth rate in the initial phase and decrease in the final stages (show an inflection point).At this point it is important to understand the concepts of growth and yield. Growth is the increase in size of an individual or population per unit of time (for instance volume growth in m 3 ha 1 year 1 ) while yield is the size of the tree or population at a certain point in time (for instance total volume at age 50

Five families of competition indices were evaluated and compared on the basis of simple correlation with loblolly pine individual tree growth and multiple correlation with growth in the presence of other tree and stand attributes. The family of distance-independent indices included various relative size measures in the form of tree size to mean size ratios. Crown ratio was also included as a distance-independent measure. The four families of distance-dependent indices included various influence area overlap indices, distance-weighted size ratio indices, Spurr's point density, and Brown's point density or area potentially available (APA). All indices were significantly correlated with dbh and basal area growth. The relative size ratio indices, crown ratio, Spurr's point density, and several APA variations were judged best in simple correlations after accounting for tree size and stand density. The best distance-dependent indices had little if any advantage, either in simple or multiple correlation, over the best distance-independent indices. However, the point density index of Spurr and especially APA contributed significantly to growth prediction even in the presence of tree size, stand density, and the distance-independent size ratio and crown ratio indices. Further, APA had the highest partial correlation when all variables were included in this multiple correlation. It was concluded the APA would be a good index for growth prediction models when other tree and stand attributes are already known.

Abstract. Predicting how forest carbon cycling will change in response to climate change and management depends on the collective knowledge from measurements across environmental gradients, ecosystem manipulations of global change factors, and mathematical models. Formally integrating these sources of knowledge through data assimilation, or model-data fusion, allows the use of past observations to constrain model parameters and estimate prediction uncertainty. Data assimilation (DA) focused on the regional scale has the opportunity to integrate data from both environmental gradients and experimental studies to constrain model parameters. Here, we introduce a hierarchical Bayesian DA approach (Data Assimilation to Predict Productivity for Ecosystems and Regions, DAPPER) that uses observations of carbon stocks, carbon fluxes, water fluxes, and vegetation dynamics from loblolly pine plantation ecosystems across the southeastern US to constrain parameters in a modified version of the Physiological Principles Predicting Growth (3-PG) forest growth model. The observations included major experiments that manipulated atmospheric carbon dioxide (CO 2 ) concentration, water, and nutrients, along with nonexperimental surveys that spanned environmental gradients across an 8.6 × 10 5 km 2 region. We optimized regionally representative posterior distributions for model parameters, which dependably predicted data from plots withheld from the data assimilation. While the mean bias in predictions of nutrient fertilization experiments, irrigation experiments, and CO 2 enrichment experiments was low, future work needs to focus modifications to model structures that decrease the bias in predictions of drought experiments. Predictions of how growth responded to elevated CO 2 strongly depended on whether ecosystem experiments were assimilated and whether the assimilated field plots in the CO 2 study were allowed to have different mortality parameters than the other field plots in the region. We present predictions of stem biomass productivity under elevated CO 2 , decreased precipPublished by Copernicus Publications on behalf of the European Geosciences Union. itation, and increased nutrient availability that include estimates of uncertainty for the southeastern US. Overall, we (1) demonstrated how three decades of research in southeastern US planted pine forests can be used to develop DA techniques that use multiple locations, multiple data streams, and multiple ecosystem experiment types to optimize parameters and (2) developed a tool for the development of future predictions of forest productivity for natural resource managers that leverage a rich dataset of integrated ecosystem observations across a region.

A height-diameter mixed-effects model was developed for loblolly pine (Pinus taeda L.) plantations in the southeastern US. Data were obtained from a region-wide thinning study established by the Loblolly Pine Growth and Yield Research Cooperative at Virginia Tech. The height-diameter model was based on an allometric function, which was linearized to include both fixed-and random-effects parameters. A test of regionalspecific fixed-effects parameters indicated that separate equations were needed to estimate total tree heights in the Piedmont and Coastal Plain physiographic regions. The effect of sample size on the ability to estimate random-effects parameters in a new plot was analyzed. For both regions, an increase in the number of sample trees decreased the bias when the equation was applied to independent data. This investigation showed that the use of a calibrated response using one sample tree per plot makes the inclusion of additional predictor variables (e.g., stand density) unnecessary. A numerical example demonstrates the methodology used to predict random effects parameters, and thus, to estimate plot specific height-diameter relationships.

Measurements were made on loblolly pine (Pinus taeda L.) from permanent sample plots to study the effects of thinning on ring width distribution and to develop a ring width prediction model. Thinning significantly increased ring width over most of the tree bole, and its effects tended to persist over the 12 years since thinning. Significant regional variation in average ring width was evident, with average ring width tending to decrease from the Highlands, to the Coastal Plain, to the Piedmont. A ring width prediction model that accounts for position in tree, tree size, stand, and site factors as well as thinning effects was developed. While ring width showed considerable variation within trees and among trees, the model predicted a substantial portion of the variation in ring width. The model can be used to obtain reasonably good volume estimates in individual growth rings, thus enabling the direct evaluation of the effects of silvicultural treatments on volume production. The data used in this study consisted of correlated observations; hence, direct covariance modeling was used to address biases in the standard error of estimates and hypothesis tests.Résumé : Des mesures ont été effectuées sur des pins à encen (Pinus taeda L.) dans des parcelles permanentes pour étudier les effets de l'éclaircie sur la distribution de la largeur des cernes et pour développer un modèle de prédiction de la largeur des cernes. L'éclaircie a significativement augmenté la largeur des cernes presque partout dans la tige et ses effets avaient tendance à persister pendant les 12 ans après l'éclaircie. Il y avait des variations régionales importantes et évidentes dans la largeur moyenne des cernes qui avaient tendance à diminuer en passant des hauts plateaux à la plaine côtière et au piedmont. Le modèle de prédiction de la largeur des cernes qui a été développé tient compte de la position dans l'arbre, de la taille de l'arbre, des caractéristiques du peuplement et du site ainsi que des effets de l'éclaircie. Bien que la largeur des cernes variait considérablement dans le même arbre et d'un arbre à l'autre, le modèle a prédit une partie importante de la variation dans la largeur des cernes. On peut utiliser le modèle pour obtenir une estimation raisonnablement précise du volume individuellement dans chaque cerne, ce qui permet d'évaluer directement les effets des traitements sylvicoles sur l'accroissement en volume. Les données utilisées dans cette étude provenaient d'observations corrélées; la modélisation basée sur la covariance directe a donc été utilisée pour corriger les biais dans l'erreur standard des estimées et les tests d'hypothèse.[Traduit par la Rédaction]