Harold E. Burkhart

An integrated system of forest stand models

Abstract An integrated system of stand models has been developed in which models of different levels of resolution are related in a unified mathematical structure. Detailed models are specified, and from them a set of growth and survival functions is derived to produce models structurally compatible at lower stages of resolution. The most detailed model is a distance-dependent individual-tree model that simulates the growth and competitive interaction of trees in a stand. Tree basal area and height growth were modelled using a modified Chapman-Richards function in terms of potential growth, current size, relative size, crown ratio, and an index of competition. Potential growth was expressed as a function of site quality, age, and open-grown size. Tree survival probability was described using a logistic function in terms of age, crown ratio, and competition. The point density measure used was area potentially available (Ap), calculated as the area of the polygon constructed from lines which divide the distance between a tree and its neighbors. Mean Ap, or average area per tree, is estimated as the inverse of the number of trees per unit area, so that point density reduces to stand density and a distance-independent individual-tree model results. The distance-independent individual tree model was collapsed to consider trees grouped in size classes. Tree growth and survival equations were applied to the mean attributes of each size class, resulting in a size-class projection model. Following through, the dimensions of the model were collapsed to an ‘average’ tree. A stand-level projection model results from applying the tree growth and survival equations to the stands average tree attributes. At the stand level, the basal area growth function provides a transformation which, for a number of probability density functions (pdfs), will regenerate the initial pdf family. Considering a normal pdf to describe basal area distributions, a pdf-based size distribution model was developed, in which the projected parameters were expressed in terms of the growth function coefficients.

A Bivariate Distribution Approach to Modeling Forest Diameter Distributions at Two Points in Time

SUMMARY A procedure for predicting diameter distribution at two points in time in single-species forest stands was developed based on the bivariate SB distribution model. The future diameter distribution is determined from an initial distribution and a specified initial-future diameter relationship. The model developed is theoretically appealing and it performed as well as, and in some cases better than, the established methods of forest diameter distribution prediction such as parameter recovery, percentile prediction, and parameter prediction. Although developed for forest diameter distribution prediction, the methods presented here should be useful in a wide array of problems where initial and future distributions of some size characteristic are needed.

Regional mixed-effects height–diameter models for loblolly pine ( Pinus taeda L.) plantations

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 regional-specific 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.

Top height definition and its effect on site index determination in thinned and unthinned loblolly pine plantations

Abstract Seven definitions of stand (top) height: (1) mean height of dominant and codominant trees at the time of plot measurement, (2) mean height of dominant and codominant trees that have always been dominant or codominant over the life of the stand, (3) Lorey mean height, (4) average height of the 100 thickest trees per hectare at the time of measurement, (5) average height of the 100 thickest trees per hectare at the time of last measurement, (6) average height of the 20% thickest trees at the time of plot measurement, and (7) average height of the 20% thickest trees at each measurement that also survived to the last measurement were evaluated for use in site index (SI) determination. Data collected from permanent plots over a 15-year period in thinned and unthinned loblolly pine plantations were used to make comparisons. Except for a few cases at certain measurements, all seven definitions of top height were significantly different from each other. Site index was predicted based on these definitions to determine if one definition is superior. Magnitudes of the maximum range of residuals and the fit statistics indicated that SI estimated using mean height of dominant and codominant trees that have always been dominant or codominant over the life of the stand is more precise than site indices estimated using other definitions. Using this definition of top height, a new SI equation for these stands was developed.

Modeling tree growth in fertilized midrotation loblolly pine plantations

Diameter and height growth models for fertilized loblolly pine stands were developed using data from midrotation loblolly pine plantations across the southeastern United States. Tree growth in fertilized stands was predicted with a reference growth model multiplied by an equation predicting the relative growth response following fertilization. The temporal distribution of the growth response was modeled by the Weibull function. These equations for fertilizer growth response were developed to be compatible with individual-tree simulation models. Information about dose, nutrient elements, and time elapsed since fertilization are needed to predict the relative growth response following fertilization.

An application of mixed effects analysis to modeling thinning effects on stem profile of loblolly pine

Stem analysis data from trees in permanent sample plots were used to evaluate thinning effects on stem form using the form exponent. Thinning significantly increased the form exponent particularly in the lower bole leading to a more neiloid form. A stem profile model that accounts for changes in stem form due to thinning effects was developed. For testing hypothesis on thinning effects on stem form, direct covariance modeling with the first order autoregressive (AR(1)) specification was used to account for correlation among within-tree observations. The mixed effects analysis technique was used in stem profile modeling to indirectly account for correlation among observations. As judged by the Akaikes information criterion (AIC) and the likelihood ratio test (LRT), substantial improvements were achieved over techniques that ignore correlation thus addressing biases in the standard error of the regression parameter estimates.

Modeling individual tree growth for juvenile loblolly pine plantations

Abstract Modeling juvenile growth of loblolly pine plantations is important for a better understanding of the whole process of stand development and helping to schedule appropriate silvicultural treatments for young stands. This paper evaluates spacing treatment effects on young loblolly pine trees and presents individual tree growth models for loblolly pine plantations. Using data from spacing trials (plot size ranging from 1.22 × 1.22 to 3.66 × 3.66 m2) for loblolly pine plantations, a generalized randomized block design (GRBD) analysis was conducted to test spacing treatment and site quality effects on young tree growth. It is shown that planting density and site quality have significant influence, and spacing rectangularity is negligible. Based on the Bertalanffy differential equation and biologically reasonable assumptions regarding density and site quality effects on tree growth, a general projection model for individual tree diameter and height was developed. A stochastic model for generalizing tree height predictions within each diameter class was proposed. Models for crown ratio, and initial diameter distribution were also developed. The models were estimated with spacing trial data from young loblolly pine plantations. Validation results for these models showed that they are appropriate for predicting the juvenile growth of loblolly pine plantations where competing vegetation has been controlled in a similar manner to these spacing trials.

Modeling survival in juvenile and mature loblolly pine plantations

Abstract Data from a large, region-wide set of permanent plots were used to develop equations for predicting stand-level survival in thinned and unthinned loblolly pine plantations growing in the southeastern United States. A pre-competitive model predicts survival at young ages based on age and total number of trees surviving. A post-crown closure model predicts survival after the onset of intraspecific competition based on age, site index, thinning intensity, percent hardwood basal area and number of trees surviving. Analyses of the pre-competitive data indicated that survival trends differed by drainage class and site preparation method. Thus, separate sets of coefficients were estimated for two drainage class/site preparation groups. Analyses of the post-competitive data indicated that survival trends differed for thinned and unthinned stands and were affected by the amount of hardwood competition. Therefore, the post-crown closure model was developed and parameterized to reflect the changes in survival as a result of hardwood competition and thinning from below.

Conditioning a distance-dependent competition index to indicate the onset of inter-tree competition

Data from a loblolly pine (Pinus taeda L.) spacing trial were used to investigate relationships between a distance-dependent competition index (CI) and the inflection age of sigmoidal single-tree cumulative basal area curves. Inflection ages increased with increasing initial growing space (GS), consistent with the hypothesis of maximum usable GS. Competition intensity as measured by the CI was generally smallest at a given point in time for trees with large inflection ages, but the trend varied by planting density. CI values at the inflection age also varied with planting density. The CI was modified so that it gave a constant CI value at the inflection age, on average, across all planting densities. Effects of site quality were accounted for to a limited degree, but the range of sites was narrow in the spacing trial. The modified CI should be useful as an absolute measure of competition independent of spacing.

Consistent Estimation of Site Index Curves Fitted to Temporary Plot Data

SUMMARY Instrumental variable estimation is presented as an alternative to ordinary least squares when only temporary plot data are available to fit site index curves. With an instrument that is uncorrelated with site index but positively correlated with age, consistent coefflcient estimates can be obtained. Misspecification tests developed by Hausman (1978, Econometrica 46, 1251-1271) and Feldstein (1974, Journal of the American Statistical Association 69, 990-996) are presented which can be used to help decide when instrumental variable estimation is superior to ordinary least squares. Measures of stand density may provide effective instruments for curve fitting because site index is usually unaffected by density. But of five data sets examined, only one provided an effective instrument of this sort. In this one case, however, the instrumental variable estimates were significantly better than those obtained through the more conventional procedure.