Study area

The study was conducted in the central macro climatic region of Chile, which has an annual rainfall ranging between 654 and 1,949 mm (^{Centro de Ciencia del Clima y la Resiliencia [CR2], 2015}), with a summer period of three to seven months depending on the location (^{Schlatter & Gerding, 1995}). The tree sample was collected in stands of Pinus radiata D. Don, located in 11 municipalities in the regions of Biobío and Araucanía, in three types of soils, i.e., volcanic sand, volcanic ash and marine sediments.

Volcanic sand (VS) soils have alluvial origin and come from andesitic and basaltic volcanic sand. This type of soil is locally known as Series Arenales. These are newly formed, poorly developed soils; with thick texture in the profile and moderately thick on the surface. They form what is called the alluvial fan of the Laja, whose topography is flat with excessive drainage and rapid to very rapid permeability, although the surface runoff is slow. From late autumn to mid-spring these soils have a temporary groundwater level at depths ranging from 70 to 120 cm, which disappears in the dry season. These soils have mild to moderate wind erosion. The stands sampled in this type of soils are located in the communes of Yumbel and Cabrero.

Volcanic ash (VA) soils were formed from recent volcanic ash deposited on glaciofluvial substrates or fluvial materials, hardly detectable by the depth in which they find themselves. Poorly developed soils, deep to very deep, well-drained, with medium loam texture or silt loam superficially and silt loam in the profile, good structure, with porosity without gravels the first 160 cm, moderate permeability and moderate runoff on slopes up to 3 %. They belong to the soils known locally as Santa Bárbara with topography of low ridges and hills, with less frequent occurrence of slightly undulating soils with 2-5 % slope. The stands sampled in this type of soils are located in the communes of Coihueco, Collipulli, El Carmen, Mulchén and Quilleco. Marine sediment (MS) soils are characterized by clay loam and clay texture in the profile. They are deep reddish brown soils on the surface, well-structured soils that facilitate root development and storage of rainwater. Locally they belong to the Asociación Merilupo, which has high marine terraces topography (dissected and undulated), with moderate to strong slopes. The stands sampled in these type of soils are located in the communes of Arauco, Curanilahue, Lebu and Los Álamos.

Sample selection of stands

The study considered a total of 27 properties: nine in volcanic sand, eight in volcanic ash and 10 in marine sediments. From each property, a pine stand at harvest age was selected from cluster analysis generated based on the number of trees per hectare (N) in contrast to the basal area (G), stand state variables obtained from historical data of post-harvest inventories provided by companies participating in the study (Forestal MININCO S.A., MASISA S.A. and Forestal ARAUCO S.A.). Conglomerates were generated by the hierarchical method of Ward (^{Hennig, 2003}, ^{Lozada, 2010}), which represent different site productivities and management intensities. A stand intended for harvest was selected from each of the most representative cluster. A sample of 10 trees in each stand was collected, covering the entire diametric spectrum recorded in the pre-harvest inventories. Overall, eliminating some trees that had errors in their records (Table 1), the sample was composed of 264 trees.

Destructive sampling of trees

At 1.30 m of each standing tree the diameter at breast height (DBH) was measured and marked. Once the tree was felled, we measured the total height and the stem was marked every 2 m from the dbh up to reaching the limit diameter of utilization (ldu) of 8 cm. From there on, the stem was marked every 1 m. Then a slice from the base of the tree (stump) was extracted, another at dbh and then every 2.44 m until reaching the height of ldu to facilitate the commercial use of the resulting logs. Slices were extracted every 1 m, from the ldu, up to a diameter not less than 2.54 cm. In the laboratory, the average diameter without bark was determined for each slice and, from this we estimate the volume without bark of each stem section between each pair of successive slices, using the formula of Smalian:

where:

The total volume of each tree was determined from the sum of the volumes of their respective sections. Measuring the diameters of the upper area of the stem above the ldu is very important for proper determination of the total volume (^{de Miguel, Mehtätalo, Shater, Kraid, & Pukkala, 2012}).

Stem Profile Model

A series of preliminary tests of trial and error allowed to determine the empirical model to represent adequately the dimensions in its entirety and that would have continuity along the stem, including the basal section and the upper end, depending on measures concerning the tree and stand state variables. Tests were performed to each of the taper model parameters obtained (including modeling of multiple linear regressions using stand state variables as predictors) and the model with the smallest index of the ^{Akaike information criterion (AIC) (Akaike, 1974)} was chosen between different alternative forms, which gave form to the following generalized model:

[1]

where:

However, some additional tests allow greater stability of the model in the basal region by replacing in the previous model the function that represents the stand state variables by a new parameter.

[2]

where:

By implementing this parameter, the initial model proposed is not only simplified, but in turn the parameters b₀, b₁ and b₂ are considered as constant values; based on such parameters that were adjusted in the generalized model [1], which considered in its adjustment the presence of the stand state variables. Thus, the following generalized and simplified model is obtained:

[3]

where:

From the linear expression of the generalized and simplified model [3] and then represented algebraically in the same way that the equation of Tartaglia-Cardano (^{Kichenassamy, 2015}), we can get any height according to their respective diameter through the following transformation of variables:

[4]

[5]

[6]

where:

The comparison of the variability of the parameters of the generalized model [1] according to the type of soil was performed using the F-test, of Fisher and Snedecor, where the variation of the sum of the squared error was evaluated under the following hypothesis:

H₀: The taper model does not differ among different types of soils.

H₁: The taper model differs among different types of soils, in at least some of its parameters.

where:

Fitting nonlinear models was performed by the method NLIN (^{SAS, 2009}). In the case of the assessment of the accuracy of fitting the taper model, we use the root mean square error (RMSE) and an indicator of predictive model efficiency

Quantifying the volume from the taper model

The taper model was integrated numerically using the IML procedure (^{SAS, 2009}). To quantify the volume only the generalized and simplified model [3] developed in this paper was considered. The procedure was used to estimate the volume of each tree from the database for validation from the parameters obtained in the fitting of the taper model, and was compared with the volume determined by the formula of Smalian through cross-validation between the estimated value and the observed value by a simple linear regression. Previously, 75 % of the samples were selected for fitting and, 25 % of the samples for validation. None of the samples selected for validation of model was used for model fitting, in order to assure the independence of the samples (^{de Miguel et al., 2012}).

Comparison of taper model

To compare the fitting of the generalized and simplified model [3], in contrast to the taper model with the same response variable published in the scientific literature by other authors (Table 2), the ^{Akaike information criterion (AIC) (Akaike, 1974)}, the root mean square error (RMSE) were used and the significant differences among the various models were compared through a Turkey’s multiple range test LSD (95 % confidence level) from the individual values observed when obtaining the mean absolute error (MAE).

where:

This study does not consider the model proposed by Munro, quoted by ^{Rojo, Perales, Sánchez, Álvarez and Von Gadow (2005)}, since when considering the stem profile as a straight line, underestimates the diameter both in the basal area as in the apical area. Likewise, the models of ^{Allen (1991)} and ^{Sharma and Oderwald (2001)} were excluded for analysis and comparison purposes due to its slow adjustment in the three different types of soil. Only those taper models with the highest accuracy, lowest bias, best graphic performance of the trendline, and proper random distribution of residuals versus predicted values were selected for comparative analysis.