Introduction

There are several categories of growth and forest production models. The most commonly used models are the stand-level models, diametric distribution models, and individual tree models (ITMs; Davis et al. 2005).

Individual trees are the basic modeling unit of an ITM that are used to simulate tree growth (diameter and height) and mortality (Campos and Leite 2017). Although these models vary in structures and utilities (Weiskittel et al. 2011), they are more complex than the models of total forest stand and diametric distribution because they use individual trees in each stand as the model input (Chassot et al. 2011) and depend on the adjustment of several submodels (Campos and Leite 2017). The most current and important studies related to this research topic were conducted by Castro et al. (2013a, b), Martins et al. (2014), Hevia et al. (2015), Vospernik et al. (2015), Orellana et al. (2016), Weiskittel et al. (2016), Miranda et al. (2017), and Moreno et al. (2017).

Forest plantations increased in area by approximately 0.5% between 2016 and 2017 in Brazil. Most of the planted forests are Eucalyptus spp, which currently encompass approximately 5.7 million hectares in Brazil (IBÁ 2017). Eucalyptus urophylla is the most planted species in forest stands in several South American countries because of its high rates of productivity and adaptation (Mendes et al. 2011; Rangel et al. 2017). This forest species shows high productive and economical potential for energy generation, panel production, and in pulp and paper industries (Reis et al. 2012; Guimarães Júnior et al. 2015; Jardim et al. 2017; Rangel et al. 2017).

However, advanced knowledge is required to improve forest management due to the silviculture complexity and product variety in this species. Growth and production models can help simulate tree growth (height and diameter) to predict forest productivity at different levels (total stand level, diameter classes, or ITM) of detail.

In the last few decades, improvements in planting techniques and silvicultural treatments have helped to improve forest production (Reiner et al. 2011; Tian et al. 2017). Currently, forest production have been improved based on scientific studies, especially those that have modeled short-cycle planted forests. Such advances have globally increased profits from plantation forestry, especially in Brazil (Pereira et al. 2016).

The objectives of the present study were to apply, adjust, and assess model accuracies to estimate the volume of E. urophylla trees. We also aimed to assess individual tree growth in permanent plots by applying an ITM.

Materials and methods

Study area

Our study site was in a forest clonal stand encompassing 320 ha of E. urophylla S.T. Blake, planted in 2010, spatially located in the Rio Verde municipality, state of Goiás, Brazil (central coordinates: 18°00′45″S and 50°53′15″W). According to the Köppen climatic classification system, the study site was an Aw climate region, and it was characterized by average annual precipitation of 1713 mm and average annual temperature of 24.6 °C with a wet and rainy summer and a dry winter (Alvares et al. 2014). The study region is mostly covered by dystrophic, deep, and well-drained red-yellow latosols (Embrapa 2013).

Dataset

We recorded field data from continuous forest inventory of trees measured at 24, 36, 48, 60, and 72 months of age. We measured trees in 29 permanent plots of 25 m length × 20 m width (500 m2) that were randomly distributed and demarcated over the study area. Trees were planted on a grid of 3 × 2 m in this forest stand, corresponding to a planting density of 1666.66 trees per hectare.

The diameter at breast height (DBH) and total height (Ht) of trees were measured using a diametric tape and vertex hypsometer, respectively, for every tree of DBH > 5 cm. Additionally, the dominant tree height (Hdom) for each plot was estimated using the Assmann (1970) definition. A total of 586 trees were strictly cubed during the study period and the measured volume of each cubed tree was determined using the Smalian equation (Spurr 1952).

Adjustment of volumetric models

Volumetric model adjustment was based on field measurements of 90% of the cubed trees using linear and nonlinear models (Table 1). Linear models were adjusted using Microsoft Excel® 2013 by applying the ordinary least squares method. The nonlinear models were adjusted by applying the Levenberg–Marquardt method (Levenberg 1944; Marquardt 1963) available in the Statistica® 7 software program (Statsoft 2007). The remaining trees (10%) were used for model validation.

Table 1 Candidate models for estimation of observed volume and structural components of individual tree modeling—ITM
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Site classification

The widely used guide curve method was used to classify the productive capacity of each sample plot (Petrou et al. 2015). Five sigmoidal site index (S) models (Table 2) were tested and adjusted during this analysis by applying the Levenberg–Marquardt method (Levenberg 1944; Marquardt 1963) available in the Statistica® 7 software program (Statsoft 2007). This type of index is efficient for describing the growth of living organisms and accurately estimates forest stand variables, thus enabling classification of the different productive units in forest plantations (Machado et al. 2010; Zlatanov et al. 2012; Retslaff et al. 2015).

Table 2 Models tested for Hdom estimation and classification of forest sites
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A total of 75% of the sample plots were randomly selected and used for the site index model adjustments. The remaining plots (25%) were used to validate and select the best adjusted model.

Competition indices and probability of mortality

Competition indices (CIs) are commonly used to estimate and simulate forest growth and production to improve forest management (McTague and Weiskittel 2016; Pothier 2017). In the present study, 10 distance-independent CIs were estimated (Table 3). Conceptually, indices that consider the tree spatial arrangement and further spatial information are expected to achieve better modeling results. However, studies comparing these two types of indices do not indicate large differences between them (Burkhart and Tomé 2012; Maleki et al. 2015). Thus, we assumed that there was no difference in the spatial arrangement and therefore in the area of the competitive influence of individual trees, following Campos and Leite (2017).

Table 3 Distance independent competition indices (CI) tested
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The optimum index was selected by analysis of correlation between the index and the annual variation of the number of individuals. The correlation test was applied between the CIs and the density of individuals per area unit (N/ha) that estimates the relationship between the CIs and stand mortality (Daniels et al. 1986; Tomé and Burkhart 1989). Based on this, the index showing the highest correlation with N/ha was defined as the optimum CI for the predictive probability of mortality (Pm) in the ITM. Subsequently, four models based on the CI results were adjusted to estimate Pm (Table 1).

For the adjustment of the Pm models, 75% of the sample plots were randomly selected and the remaining plots (25%) were used to validate and select the best adjusted model by applying the Levenberg–Marquardt method (Levenberg 1944; Marquardt 1963) available in the Statistica® 7 software program (Statsoft 2007).

Growth modeling of height and diameter

Two essential components for modeling individual trees are Ht and DBH future projections. The two variables were estimated using five statistical models (Table 1) by applying the Levenberg–Marquardt method (Levenberg 1944; Marquardt 1963) available in the Statistica® 7 software program (Statsoft 2007). The model adjustment was performed based on 75% randomly selected sampled plots while the remaining plots (25%) were used to validate each tested model.

Model selection and testing criteria

The selection of the best model to estimate volumetric, S, Pm, Ht, and DBH growth was based on the following statistics of fit and precision: residuals dispersion graph, standard error of estimate (Syx%), and coefficient of determination (R2) (Draper and Smith 1998). We then used the “observed-estimated” graph, the error classes graph, and the Akaike index (AIC) to select our overall optimum model. The coefficient of determination was replaced by the correlation coefficient (R) for the Hdom models because they are nonlinear models.

Individual tree modeling

The future survival (36, 48, 60, and 72 months) applied to the ITM growth and forest production was initially determined based on the field data acquired at 24 months of age. The estimated Pm of each sampled tree was compared to a random value simulated by the Monte Carlo method as described by Castro et al. (2013a). The tree was considered dead when the estimated Pm was greater than the random value (Pretzsch et al. 2002).

Ht and DBH of all surviving trees were estimated, which enabled estimation of their volumes. The total volume for each plot was estimated based on the sum of the individual tree volumes at the reference age (72 months) and extrapolated to volume per hectare.

Modeling using the ITM started from four different initial ages (24, 36, 48, and 60 months), each projecting the volume for forest stands at 72 months of age (the reference age). The projected volumes of 24, 36, 48, and 60 months of age and the observed forest stand volume at 72 months of age (control) were defined as the five treatments. For the forest stands aged 72 months, five different volume values were estimated for each sampled plot/hectare with four values projected, and an observed value determined. The ITM was performed using the Microsoft Excel® 2013 program and eight previously defined validation plots.

Statistical analysis

Projected volume data (treatments) and traditional inventory data (control) were tested using the Shapiro and Wilk (1965) normality test followed by the Bartlett (1947) homogeneity of variance test. By assuming data normality and a completely randomized block design (α = 0.05), analysis of variance (ANOVA) was performed. Different sites were considered as blocks, whereas the different time intervals of the volume projection up to the forest stands aged 72 months (reference age) were considered as treatments, where (T1) was the volume projection of forest stands aged 24 months, (T2) was for stands of 36 months, (T3) was for stands of 48 months, (T4) was for stands aged 60 months, and (T5) was the control, the observed volume based on field measurements (forest inventory) at the reference age (72 months).

Finally, a multiple mean comparison test was used to identify potential treatment differences using Tukey’s (1953) test at the 5% level of significance.

Results

Based on the Syx% results, most of the tested models of each analyzed variable showed satisfactory adjustment. The Schumacher and Hall model was selected for the volume estimation (Table 4). The adapted Hossfeld model and the adapted Lundqvist-Korf model (Table 5) were selected for the estimation of DBH and Ht at future age.

Table 4 Parameter estimators and precision statistics of volumetric models
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Table 5 Parameter estimators and precision statistics of DBH and total height at future age models
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The allometric model (Table 6) was selected for the estimation of Pm. The residual distribution patterns of the selected models for volume, DBH2, and Height2 estimations were acceptable (Fig. 1a1, a3, and a4), and the estimates of the dependent variable were accurate (Fig. 1b1, b3, and b4). Based on the frequency histogram (Fig. 1c1, c3, and c4), we observed a higher concentration of residuals in the central classes; therefore, the selected models proved unlikely to overestimate or underestimate the values of each of the analyzed variables.

Table 6 Parameter estimators and precision statistics of probability of the mortality models
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Fig. 1
figure 1

Percentage of the residual distribution (a), observed values used to predict tree volume (b), and slope distribution (c) of volume estimates (1), probability of tree mortality (2), breast height diameter (3) and height (4)

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However, the residuals of the probability model showed a greater degree of dispersion (Fig. 1a2), indicating weaker correlation between observed and predicted values (Fig. 1b2) and a stronger trend toward overestimation or underestimation (Fig. 1c2).

Based on the site classification, S (21), S (25), S (29), and S (33), four productivity curves were plotted. The generated curves accurately described the behavior of Hdom compared to age (Fig. 2b1). All models showed similar accuracy and degree of fit (Table 7), however, the logistic model was chosen because it yielded lower Syx%. The residual distribution was homogeneous without great bias and was stable in site classification (Fig. 2a1). We observed overestimation and underestimation of Hdom in the model estimation (Fig. 2c1).

Fig. 2
figure 2

Percentage of the residual distribution (a1), forest site curves (b1) and slope distribution (c1)

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Table 7 Parameter estimators and precision statistics of site index classification models
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CI was significantly correlated with the number of trees per hectare at different ages. Thus, the distance independent index proposed by Daniels et al. (1986) yielded better correlation results. The correlations between CI and stand density varied between − 0.9974 and 0.9615.

After the preliminary simulations and adjustment of the ITM calculation, parameter estimates varied by initial age (F (3, 4) = 6.55, p < 0.05) (Table 8), and mostly, the variation was explained by the treatments. Using the classification of productive units as an environmental gradient was important to achieve these results.

Table 8 Randomized block analysis of variance (ANOVA) results
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The mean test comparison results showed significant differences between volumes predicted by the ITM and the volume measured by traditional inventory (control) at the reference age (72 months). Differences were attributable to higher variation among estimates based on initial measurements of stands aged 24 and 36 months. Estimates based on longer duration treatments were indistinguishable from the control (Table 9). Measurements of stands aged 48 months were adequate to accurately estimate the volume of stands at 72 months. Therefore, the ITM accurately predicted individual tree volume for a future interval of up to 2 years.

Table 9 Mean volumes (m3 ha−1) estimated at 72 months, estimated based on the projection of field data collected at different ages of the stand (24, 36, 48 and 60 months) and mean observed volume measured at 72 months (control)
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Discussion

Adjustment of the various submodels proved suitable for the ITM. The models were selected from the best submodels adjusted to the dataset. The volumetric model of Schumacher and Hall that was selected to estimate tree volume is a classic model in the forest sector and is widely applied because of its statistical properties, which result in non-biased estimates (Campos and Leite 2017). This model has been used to estimate variables of planted forests (Leal et al. 2015; Sales et al. 2015; Carrijo et al. 2017) and natural forests (Colpini et al. 2009; Vibrans et al. 2015; Miguel et al. 2017).

Although the estimation of Hdom for site classification showed a standard error estimate greater than 15%, other studies using the guide curve method have estimated similar values of Syx% (Zlatanov et al. 2012; Miranda et al. 2014; Castro et al. 2015; Ribeiro et al. 2016). This result can be explained by the absence of the polymorphic curve construction in this method, which might not represent the different growth trends using the model (Binoti et al. 2012; Cosenza et al. 2017). However, such errors were considered satisfactory and did not compromise the accuracy of the modeling system when considered for different site classes.

The logistic model yielded the best results for forest site classification. This model is robust, is traditionally applied for classifying forest sites (Oliveira et al. 2008; Salles et al. 2012), is accurate, and is independent of tree species (Martínez-Zurimendi et al. 2015; Murillo-Brito et al. 2017). In addition, the logistic model was chosen based on the criterion of stability of the heights of dominant trees around the curves that express the forest site classes, which has been successfully applied in other studies (Scolforo 2006; Silva et al. 2018).

The selection of the height projection model followed Martins et al. (2014), who reported precision in the estimates obtained via the Lundqvist-Korf model for both height and DBH. This can be explained by the suitability of this model family to simulate the DBH-height relationship (Krisnawati et al. 2010; Lumbres et al. 2015). However, the result for the DBH projection differed from the results reported by Krisnawati et al. (2010) and Lumbres et al. (2015) that were based on adaptation of the linear model to estimate DBH.

Several studies have reported significant correlation between CI and population variables. Martins et al. (2011) and Castro et al. (2014) reported strong correlation between the Pm of plots and CI. Hui et al. (2018) reported that dead trees in native forests had experienced greater competition, which indicated that there was a strong relationship between Pm and CI.

CI is often correlated with growth indicator variables of trees (Chassot et al. 2011; Contreras et al. 2011; Pedersen et al. 2012; Castro et al. 2014; Maleki et al. 2015; Pothier 2017). Our results show that use of a variable that expresses competition among individual trees will significantly improve the prediction of the dimensions and dynamics of tree growth, as reported by many studies (Contreras et al. 2011; Oheimb et al. 2011; Castro et al. 2014; Maleki et al. 2015; McTague and Weiskittel 2016). The competition index is the most important variable of individual tree growth models (Maleki et al. 2015) and, therefore, a high correlation between CI and stand variables is crucial to ensure accurate results when modeling individual trees.

The model used to estimate Pm in the present study was tested by Castro (2011), who reported a good model fit for his study population. Our results showed better precision statistics of the Pm model than those reported by Miranda et al. (2017), who used exponential and Buchman models and estimated errors (Syx%) greater than 100%. This probably resulted from the difficulty of estimating mortality, which reflects the uncertainties of future forest stand conditions (Yang et al. 2003; González et al. 2006; Castro et al. 2014). Despite these difficulties and the number of known and unknown variables that may affect tree mortality (Strimbu et al. 2017), the adjusted models used to estimate this probability have yielded satisfactory results (Hevia et al. 2015; Ma and Lei 2015; Maleki and Kiviste 2016; Strimbu et al. 2017).

Thus, we conclude that the ITM was efficient considering the time interval adopted. Castro et al. (2013b) reported accurate estimates from ITM when using a dataset based on field measurements recorded 3 years before harvest. Similarly, Lhotka (2017) demonstrated the effectiveness of ITM in predicting tree growth expressed by DBH up to 5 years before harvest.

Models that predict future production by considering the density of the stands help to improve our understanding of tree growth in forest plantations (Lhotka 2017). ITMs are considered a precise method to estimate tree growth and mortality, and to predict forest production.

In the present study, all submodels showed statistically productive adjustments, which indicates a great potential for predicting timber volume at different ages in E. urophylla plantations. Our ITM proved effective for use in E. urophylla stands for up to 2 years between the actual age and the reference age (in this case, 48 and 72 months), and allowed us to accurately predict future volumes. This can help decision makers estimate forest production, contributing to reduce the cost of forest inventories. However, it is recommended for intervals up to 2 years because greater intervals may cause overestimation of the predicted forest volume when compared to the observed volume (Weiskittel et al. 2016).