# Stand Structure Impacts on Forest Modelling
## 6 Volume  
Volume is frequently modelled as function of diameter at least height and height. As a result, it is affected by both diameter at breast height and height measurement precision and variability in tree morphology and, thus, allometric relationships. The following examples illustrate the diversity of functions. Volume estimation functions have always played a primary role in the determination of productions and productivity. Many studies have developed tree level volume functions [35; 49; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105]. Additionally, volume equations have been compiled by several authors [106; 41]. Ref. [107] developed several volume functions for _Picea abies_ stands in Norway, with nonlinear regression. The functions were fitted for pure even-aged and for mixed unvenaged stands. The authors demonstrated that stand structure influenced function performance, with models fitted for pure even-aged stands attaining rather large errors when used for mixed and uneven-aged stands. Ref. [94] modelled the volume for _Picea abies_ in Sweden using destructive sampling and verified that soil influenced mean annual volume increment and that existing models underestimated volume because soil variables were not included in the model. Inversely, the models developed by Ref. [94], including site index and soil variables, resulted in estimations with high accuracy. Ref. [108], modelling the volume for five species in Canada using stem analysis data with nonlinear regression, demonstrated better performances with the functions that used diameter at breast height and total height when compared with those that used only the former variable. Ref. [103] modelled volume at national level for 25 most important species in Italy, with destructive sampling, using linear regression with a weighted function. The authors stated that volume estimates were unbiased. However, the authors pinpointed two problems: individual tree allometry variability caused by the geographical gradients coupled with the low number of sampled trees for some species might explain the error level, and, although weighted regression outcomes the residuals heteroscedasticity, error distribution did not meet the normality criterion; thus, in general, the intervals of confidence are not valid. Regarding _Pinus sylvestris_ and _Picea stichesis_, using destructive sampling in the United Kingdom, Ref. [91] modelled volume through several statistical methods (linear regression, nonlinear regression and mixed models). In general, the nonlinear mixed effect models' performance was better than that of the linear and nonlinear regression models. However, the former had a higher residual variance when compared with the latter, which might compromise its predictions. Ref. [98] studied volume for two species, _Pinus tada_ and _Eucaliptus_ spp., using data from the USA and Brazil, respectively. The authors incorporated the Hossfeldt diameter,Pressler height, the Kozak-type taper function and auxiliary variables in their developed formula. The developed volume equation was able to accommodate the variability in tree allometry and growth due to genetic improvement, site and silvicultural practices. Ref. [101] modelled the volume for _Pinus nigra_, _Fagus sylvatica_ and _Quercus pubescens_ with data from the national forest inventory of Greece, using three functions: Hohendl-Krenn (independent variable diameter at breast height), power (independent variables diameter at breast height and height) and generalised least squares (GLS). The authors denoted that GLS had the best performance when the residuals' variance was not constant; the diameter at breast height power equation and the combination of the diameter at breast height and height variance enabled the reduction of the residuals' heteroscedasticity, and the weighted regression residuals met the normality distribution criterion.