Data for this research were obtained from two sites spanning a climatic range located at Santa Lucia (latitude 37° 12 41, longitude 71° 48′ 31″) and Santa Isabel (latitude 37° 24 45, longitude 72° 15′ 29″) in the Bio region (region VIII) of Chile. Santa Lucia is located in the Andean foothills at an elevation of 727 m. It has a relatively low mean annual temperature (10.9°C) and high mean annual rainfall (2,022 mm year−1). In contrast, Santa Isabel is located at a lower elevation (170 m) in the Central Valley and has higher mean annual temperature (12.9°C) and lower mean annual rainfall (1,166 mm year−1) than Santa Lucia. The soil types underlying the sites were volcanic ash and sandy, respectively, for Santa Isabel and Santa Lucia.
Plant material, treatments and stand history
Plant material at both sites were drawn from a half-sib family named Colicheu produced through the breeding programme of the Chilean company Forestal Mininco. Seedlings and one-year-old cuttings were used as treatments at both sites.
For each treatment, a permanent sample plot (PSP) of 630 m2 (30 m × 21 m) was established in which there were six rows of 15 plants. At both sites, the initial planting density was 1,428 trees ha−1 (2 × 3.5 m) and the PSPs were not thinned or pruned prior to the measurements.
Measurements were made during September 2012 on 17-year-old trees that were planted in 1995. A total of 40 sample trees at each site (20 per plant material treatment) were selected for measurement. These represented all diameter classes. Malformed trees (including leaning trees or those with forked trunks or with large branches) were not selected.
Following the felling of each tree, measurements of height and diameter at breast height (DBH) were taken. Stem slenderness was determined from these measurements as height/DBH. Each tree was cut into 3-m long sections until a diameter of 10 cm was reached. All branches were removed. Discs that were 30-mm thick were taken at the top and base of each section. Green density, ρg, (kg m−3) was determined on each sample disc following a standard protocol (ASTM D2395) from the following equation,
where Wg is green weight (kg), and Vg is green volume (m3) determined using the immersion method. All values of ρg reported in this paper use values of ρg averaged over the log from measurements made at the top and base.
Velocity was determined for each log without removing bark using a resonance tool (Hitman HM200, Fibre-Gen). Velocity (V) was determined from sample length (l) and the resonant frequency (f) as V = 2 fl.
Description of the modelling approach
Longitudinal measurements of velocity and ρg were used to calculate MOE from Equation (1). Although these calculated values were not direct measurements of MOE, as they were determined from V and ρg, they will be referred to hereafter as measured MOE for clarity. Predictions of MOE were then made by substituting measurements of ρg for values predicted by the following models (i) Model 1 - assuming a constant ρg of 1,000 kg m−3, (ii) Model 2 - using the mean ρg across all trees of 914 kg m−3, (iii) Model 3 - using a model with only fixed effects to account for the mean longitudinal variation in ρg, (iv) Model 4 - using a model with the previous fixed effects and random effects to account for variation in intercepts for ρg between trees and (v) Model 5 - using a model with the previous effects (in Model 4) and a random quadratic term to account for differences in the longitudinal pattern of variation in ρg among trees. A summary of the terms included within each model is given in Table 1.
Using SAS (SAS Institute Inc. ) a random-coefficients mixed-effects model was used to characterise variation in ρg at the log level for Models 3, 4 and 5. Mixed effects models can include both fixed and random effects. An effect is termed fixed if the levels in the study include all possible levels of the factor, or at least the levels about which inference is to be made. In contrast, a factor is considered random if its levels plausibly represent a larger population with a probability distribution. Only fixed effects (relative height) were used for predictions of ρg made by Model 3 while fixed effects and the random intercept were included for Model 4 (Table 1). The full random-coefficients model (Model 5) included fixed effects and random effects for both the intercept and quadratic terms (Table 1).
Relative height, which was defined as the ratio of the measurement or prediction height to the total tree height, was used as a predictive variable within the model rather than actual measurement height. This was because the Akaike Information criterion, which measures the relative quality of a statistical model (Littell et al. ), was lower (4673.4 vs. 4725.6) for the final model with the relative height term included. The fixed-effects part of the model included relative height as a third-order polynomial. Neither site nor treatment, nor the interaction of these two terms with any of the terms in the relative height equation, was significant at P = 0.05. Random effects included the intercept (P < 0.001), relative height (P < 0.001) and the square of relative height (P < 0.001). A full description of Model 5 is given in the next section of the methods.
Predictions of ρg against relative height using only the fixed effects in the model are shown as the black line while predictions of ρg using all random effects for individual trees (Model 5) are shown as the blue lines in Figure 1. Predictions of longitudinal variation in ρg for individual trees were markedly offset from the mean value. These results supported the significance of different intercepts for different trees but followed a similar longitudinal pattern to that of the mean, for most trees, although there were exceptions (Figure 1).
Differences between measured values (ym) and predictions (yp) of ρg and MOE using the five models were expressed at the mean tree level (by averaging across logs) and log level. These differences were expressed as both a percentage difference, D (D = ((ym-yp)/ym) ⋅ 100) and an absolute percentage difference, Da (Da = |D|). For a given model, both D and |D| were identical between ρg and MOE . Consequently results showing these differences from the measured values can be interpreted to represent either wood property. The distribution and mean values for D and Da were examined for each model to respectively assess model bias and precision. The relationship between relative height and both D and Da was also examined to determine within tree bias and precision for the five models.
Calibration of Model to predict longitudinal variation in ρg from a single disc
The methodology by which the most complete and accurate model (Model 5) can be used to predict longitudinal variation in ρg from measurements obtained from a single disc is demonstrated below. Use of the linear mixed-effects model described in Model 5 permitted the estimation of a mean response (population-specific) or a calibrated response (cluster-specific) for a new tree (Verbeke and Molenberghs ). A mean response can be obtained using only the fixed-effect components of Model 5, where the vector of random effects u
for a new k th individual is assumed to have expected value E(u
) = 0. In contrast, a calibrated response can be obtained when auxiliary information is available allowing for prediction of random parameter components. In this case, auxiliary information corresponds to a measure or prediction of ρg at a given stem height for a new tree. Using this information, the vector of random parameters can be predicted using an approximate Bayes estimator of u
(Vonesh and Chinchilli ; Rencher ).
whereand the error vector. The design matrixand the variance-covariance matrix for the random-effects D and residuals R
are defined for Model 5 as:
and, whereis an identity matrix of dimension (nk x nk) and nk is the number of observations used for calibration. Under this structure, random errors are assumed to be uncorrelated and have constant variance (σ2).
Longitudinal predictions of ρg can be made from Model 5 using ρg data obtained from a single disc. As an example, ρg measured at breast height (1.3 m) for a randomly selected sample tree in the dataset was 985.08 kg m−3. The relative height of the disc was 0.041 (disc height of 1.3 m divided by the total tree height of 31.6 m). Thus, the difference between the observed value and the estimated mean-response value from Model 5 is determined from the fixed effects component of the model as:
The design matrixand the variance-covariance matrices for the random coefficients and error estimated for Model 5 are determined as,
Now, replacing the matrices in Equation 3 gives the following predictions for the random parameters of this specific tree: u0k = 49.5454, u1k = −49.9722 and u2k = 18.8294 and the calibrated parameters for the model are given below in Equations 7, 8 and 9:
Thus, the calibrated model for the sample tree is
The accuracy of this prediction is assessed by comparing the predicted longitudinal pattern to that obtained through measurements.