Comparative evaluation of five height–diameter models developed for black spruce and jack pine stand-types in terms of goodness-of-fit, lack-of-fit and predictive ability

Abstract

The objective of this study was to comparatively evaluate five nonlinear models on their ability to describe the relationship between total height ( H; m) and diameter at breast-height ( D; cm) within six commercially-important boreal stand-types. Specifically, the five models evaluated were as follows: (1) the constrained (i.e., H = 1.3 when D = 0) Chapman–Richards-based model proposed by Peng, Zhang and Liu [Peng, C., Zhang, L, Liu, L., 2001. Developing and validating nonlinear height-diameter models for major species of Ontario's boreal forests. NJAF 18, 87–94; denoted Model 1] which excludes consideration of stand-level effects; (2 and 3) the constrained Chapman–Richards-based models proposed by Sharma and Zhang [Sharma, M., Zhang, S.Y., 2004. Height–diameter models using stand characteristics for Pinus banksiana and Picea mariana. Scand. J. For. Res. 19, 442–451; denoted Models 2 and 3] which includes consideration of stand-level density effects through the explicit inclusion of density-based predictor variables (density ( N; stems/ha) and basal area ( G; m 2/ha)); and (4 and 5) constrained and unconstrained allometric-based models (denoted Models 4 and 5, respectively) derived from a multivariate expansion of the equation of simple allometry which includes consideration of stand-level effects via the explicit inclusion of density and stand developmental predictor variables (relative density index ( R; %/100) and mean dominant height ( H D; m), respectively). The six stand-types considered were (1 and 2) natural (density unregulated) and managed (density regulated) upland black spruce ( Picea mariana (Mill.) B.S.P.) stands (denoted PIm UL(N) and PIm UL(M), respectively), (3 and 4) natural and managed jack pine ( Pinus banksiana Lamb.) stands (denoted PNb (N) and PNb (M), respectively), (5) natural black spruce–jack pine mixed stands (denoted PImPNb (N)), and (6) natural lowland black spruce stands (denoted PIm LL(N)). The full dataset, consisting of 26741 H– D measurements and associated stand-level variables ( N, G, R and H D) obtained from 985 sample plots situated throughout the central portion of the Canadian Boreal Forest Region, was randomly subdivided into calibration and validation subsets of approximately equal size by stand-type. Parameter estimates for each model were obtained using the calibration subsets in combination with nonlinear regression (Models 1–3) and multiple regression (Models 4 and 5) analyses. Employing the validation subsets, the calibrated models were evaluated using goodness-of-fit, lack-of-fit and prediction error indices at both the diameter class and stand levels. The results indicated that the best performing models were as follows: (1) Model 3 for PIm UL(N); (2) Model 5 for PIm UL(M); (3) Model 4 for PNb (N); (4) Model 5 for PNb (M); (5) Model 2 for PImPNb (N); and (6) Model 5 for PIm LL(N). Collectively, these results (1) reconfirms the utility of explicitly incorporating stand-level variables within the model specification when developing H– D models, (2) demonstrates the superiority of the newly introduced allometric-based H– D composite model which incorporates both stand density and developmental effects for four of the six stand-types assessed, and (3) provides a suite of calibrated functions and associated performance metrics for potential use in product recovery and value estimation, stand structural analyses, growth and yield projection systems, and carbon budgeting models. Furthermore, given the success of the allometric-based models derived from the multivariate expansion of the equation of simple allometry, suggests that this modeling approach may have wider applicability in the (1) development of prediction equations for other important dimensional relationships used in forest management (e.g., localizing stem taper, volume and biomass equations) and (2) study of allometry in general (e.g., provide analytical direction in the assessment of population-level effects on allometric scaling relationships).