ALBERT

Description: Both aggregative and disaggregative strategies were used to develop additive nonlinear biomass equations for slash pine (Pinus elliottii Engelm. var. elliottii) trees in the southeastern United States. In the aggregative approach, the total tree biomass equation was specified by aggregating the expectations of component biomass models, and their parameters were estimated by jointly fitting all component and total biomass equations using weighted nonlinear seemingly unrelated regression (NSUR) (SUR1) or by jointly fitting component biomass equations using weighted NSUR (SUR2). In an alternative disaggregative approach (DRM), the biomass component proportions were modeled using Dirichlet regression, and the estimated total tree biomass was disaggregated into biomass components based on their estimated proportions. There was no single system to predict biomass that was best for all components and total tree biomass. The ranking of the three systems based on an array of fit statistics followed the order of SUR2 〉 SUR1 〉 DRM. All three systems provided more accurate biomass predictions than previously published equations.

Description: The Chapman-Richards distribution is developed as a special case of the equilibrium solution to the McKendrick-Von Foerster equation. The Chapman-Richards distribution incorporates the vital rate assumptions o...

Description: The design bias in the sample mean obtained from sampling the trees nearest to points randomly and uniformly distributed over a forested area can be exactly quantified in terms of the Voronoi polygons (V polygons) surrounding each tree in the forest of interest. For this sampling method, the V polygon for a prospective sample tree is its inclusion zone. The sides of such polygons are perpendicular to a line joining adjacent trees and equidistant from these trees. For any individual tree attribute Y, the design bias in such a sample mean for estimating the population mean of Y will be equal to the covariance between Y and V-polygon area V divided by the mean V-polygon area. The bias as a percent of the population mean of Y is the product of the correlation coefficient between Y and V and the coefficients of variation for Y and V multiplied by 100. This implies that attempts to estimate the means of commonly measured individual tree variables such as DBH, basal area, and crown diameter or the area from sampling trees nearest to randomly located points will likely be positively biased, and the magnitude of that bias will depend on the strength of the linear relationship to the V-polygon area, as well as the variability among the V-polygon areas and the variable of interest. It is not obvious whether increment core data will be positively or negatively biased, because this depends on the characteristics of the forest of interest. The main conclusion of the study is that the bias formula derived for unweighted estimation from sampling the tree nearest to a point indicates that bias in the range of 5%–10% or greater can occur in many forest populations.

Description: A very common model for prediction of tree stem volumes to upper-stem height or diameter limits is the use of a merchantable to total volume ratio function multiplied by a total stem volume function. Many users of these prediction systems also desire taper equations that can predict heights to upper-stem diameters. While taper equations compatible with volume ratio equations have been used for many years, compatible taper equations from volume ratio equations that are functions of upper-stem height have been used infrequently. Yet many studies have indicated that height-based ratio equations perform well and frequently have statistics of fit that are comparable with diameter-based volume ratio equations. Compatible taper equations derived from height-based ratio equations are presented here. The methodology that uses height-based merchantable to total volume ratios does not require the solution of a differential equation after differentiating the height-based volume ratio, as is necessary when using the method of deriving taper equations from diameter-based merchantable to total volume ratios. This could be an advantage depending on the complexity of the ratio function. Example taper equations fitted to loblolly pine (Pinus taeda L.) data from the southeastern USA and the state of Oklahoma, USA, indicate good fit to these data, whether fitted directly to taper data or implicitly by using parameters fitted to volume ratio data.

Description: A study of the effects of measurement error was conducted on importance sampling and control variate sampling estimators of tree stem volume in which sample diameters are measured at randomly selected upper-stem heights. It was found that these estimators were unbiased in the presence of additive mean zero and multiplicative mean one measurement error applied to random samples of upper-stem diameter squared. However, biases due to measurement error are present if additive or multiplicative error is applied to upper-stem diameter rather than to upper-stem diameter squared. This is significant, as it appears that most of the previous studies on the magnitude of upper-stem diameter measurement error implicitly assume that the mean error is centered around the diameter rather than about the square of the diameter. Application of typical upper-stem measurement error obtained from previous studies to bias formulae derived here indicates that the bias could be a concern for small trees and with additive measurement error within ranges found in previous studies. Formulae for the variances of importance sampling and control variate sampling are derived, which include the contribution of both measurement error and sampling error. Results from previous studies of Monte Carlo integration estimator sampling error are combined with results from studies of upper-stem measurement error to obtain estimates of the typical magnitude of the contribution of measurement error to total estimator variability. Increases in upper-stem sample size may be warranted due to the impact of measurement error if precise estimates of stem volume at the individual-tree level are desired.

Description: The effects of measurement error on Monte Carlo (MC) integration estimators of individual-tree volume that sample upper-stem heights at randomly selected cross-sectional areas (termed vertical methods) were studied. These methods included critical height sampling (on an individual-tree basis), vertical importance sampling (VIS), and vertical control variate sampling (VCS). These estimators were unbiased in the presence of two error models: additive measurement error with mean zero and multiplicative measurement error with mean one. Exact mathematical expressions were derived for the variances of VIS and VCS that include additive components for sampling error and measurement error, which together comprise total variance. Previous studies of sampling error for MC integration estimators of tree volume were combined with estimates of upper-stem measurement error obtained from the mensurational literature to compute typical estimates of total standard errors for VIS and VCS. Through examples, it is shown that measurement error can substantially increase the total root mean square error of the volume estimate, especially for small trees.