Variable-top merchantable volume equations for Scots pine (Pinus sylvestris) and Sitka spruce (Picea sitchensis (Bong.) Carr.) in northern Britain

In this study, data collected from stands in Scotland and Northern England were used to develop variable-top merchantable volume equations for plantation-grown Scots pine (Pinus sylvestris) and Sitka spruce (Picea sitchensis). After fitting and evaluating a number of total volume and volume ratio equations, the Schumacher and Hall (1933) equation (Schumacher, F.X. and Hall, F.S. 1933 Logarithmic expression of timber tree volume. J. Agric. Res. 47, 719-734) gave the best results for total volume, while the Van Deusen et al. (1981) and Cao and Burkhart (1980) models (Van Deusen, P.C., Sullivan, A.D. and Matney, T.G. 1981 A prediction system for cubic foot volume of loblolly pine applicable through much of its range. South. J. Appl. For. 5, 186-189; Cao, Q.V. and Burkhart, H.E. 1980 Cubic-foot volume of loblolly pine to any height limit. South. J. Appl. For. 4, 166-168) gave the best fits for volume ratio equations to variable-top diameter and height limits, respectively. Variable-top merchantable volume equations were jointly estimated as a product of total volume and volume ratio equations. The final models, obtained using a non-linear mixed-effects modelling approach, showed considerable improvements compared with their fixed-effects counterparts. Even though further modelling of serial correlation using a banded toeplitz error structure improved model fits, it also led to an increase in the residual variance of the models and could degenerate the predictive ability of the models. We therefore recommend the use of mixed models with simple 'iid' error structure. However, in situations where additional data are not available to predict the random effects, we recommend the use of the models based on non-linear least squares.