Modelling the diameter distribution of Pinus sylvestris, Pinus nigra and Pinus halepensis forest stands in Catalonia using the truncated Weibull function

Parameter prediction models for the diameter distribution of Pinus sylvestris L., Pinus nigra Arn. and Pinus halepensis Mill. in Catalonia were developed using the truncated Weibull function as the theoretical distribution. The parameter models allow one to use individual-tree models in the simulation of stand development when only stand-level data are collected in forest inventories. Parameter models for the diameter distribution of stand basal area were developed. The data consisted of permanent sample plots from the Spanish National Forest Inventory in Catalonia. A total of 1780 empirical distributions of P. sylvestris, 1204 distributions of P. nigra and 1535 distributions of P. halepensis were used as modelling data. The empirical data represent left-truncated distributions, as the smallest diameter measured in the field was 7.5 cm. Two different approaches, namely, regression (two-step method) and optimization approach (one-step method), were used to find the coefficients of the parameter models. In the two-step modelling method, the Weibull parameters were first estimated separately for every empirical distribution by maximizing the log-likelihood function of the Weibull density function. In the second-step, regression analysis was used to find the relationship between Weibull parameters and stand basal area, number of trees per hectare and elevation of the site. The one-step method used optimization to find such coefficients for the parameter models, which minimized the mean of the squared differences between empirical and predicted cumulative tree frequencies in the whole modelling data. The one-step optimization method performed better than the two-step regression method for all tree species. The parameter prediction models developed in this study enable the prediction of the diameter distribution of P. sylvestris, P. nigra and P. halepensis in Catalonia from limited stand information.