1. Introduction
Forests play an important role in mitigating the effects of climate change [
1,
2,
3], contributing significantly to the uptake of atmospheric carbon dioxide [
4]. However, the large uncertainties usually associated with the estimation of forest biomass stock and stock change are an important limitation for the successful implementation of forest-based mitigation programs [
2]. As a result, there is an increasing interest from the international scientific community to reduce the uncertainties of forest biomass estimates [
5,
6].
Forest biomass estimates usually rely on allometric biomass models, which are regression models that predict individual tree aboveground biomass (
AGBi) as a function of easy-to-measure independent variable
xi (e.g., diameter at breast height, D, and/or tree height, H):
Allometric biomass models describe how tree biomass varies with the predictor(s). For trees of similar sizes, the variation in AGB is determined by the genotype, the environmental conditions, and their interaction [
7]. For example, the environmental conditions such as temperature and precipitation were shown to affect the allocation of biomass within trees [
8], whereas a change in tree biomass allocation due to different levels of tree competition was shown to significantly affect tree biomass allometry [
9]. It is well documented that allometric biomass models are species- and site-specific [
10,
11,
12,
13,
14], and, therefore, specific model parameters should be used for each species and each site. Developing species-specific allometric models, based on sample trees from multiple sites, requires the models to be applied in those same sites. This is required because the mean of site-effects would tend to zero and, therefore, the mean biomass per unit area is unbiased [
10]. However, applying a model calibrated for different sites to one single site, could yield biased biomass estimates [
13,
15]. As a result, calibration of allometric models at the site level becomes compulsory to obtain accurate estimates of biomass at the site level. Moreover, given the small forest property size, especially in the case of private owners [
16], estimates of forest biomass over small forest areas are often needed.
Development of unbiased allometric models for new sites requires measuring the biomass of a large enough number of trees from the total tree population [
17]. However, measuring the biomass requires extensive logistics and resources, limiting, therefore, the possibility to calibrate the model for each site. The most common fitting method for allometric models is the ordinary least squares regression [
18,
19,
20,
21]. However, because the residuals of allometric models are heteroscedastic, i.e., the variance increases with a predictor, a weighting approach should be used to account for the heteroscedasticity, which is usually the inverse of the predicted variance for a given value of the predictor. Since the variance is greatest for the largest trees, these trees will bring less effective information into the model (because the weights are calculated as the inverse of variance, large trees are weighted less compared to small trees) [
22]. These large trees are also the most difficult to measure (for biomass); therefore, the logistics limitations of measuring biomass make the calibration of these models at site level a very low cost-effective task. Because of that, much interest has been invested in finding ways to make local calibration more affordable. For example, [
23] proposed the “small sampling scheme” method, where the smallest two trees were used to detect the most appropriate parameters of the local allometric model from a database of allometric models.
When species-specific biomass observations from other sites are available, the mixed-effects models can be used to enhance the local information (of the site-specific sample). In a similar conceptual approach to [
23], just a few local trees can be used to calibrate the intercept of a random intercept model to local conditions, whereas the slopes remain similar for all locations. Therefore, the assumption here is that, at a species level, the allometric scaling is invariant. Although the assumption of invariant allometric scaling has been widely disputed, it was demonstrated that allometric scaling is species-specific [
12], but there is no information to support a site-specific allometric scaling. However, it has been shown that the intercept of allometric models vary by site [
10]. This approach can be very convenient since the intercept can be calibrated based on a few small trees, for which the biomass can more easily be measured.
Another approach extensively promoted lately is the Bayesian modelling, since it is considered an effective way to combine prior information with the local observations [
24,
25]. Compared to random intercept model, the Bayesian approach does not need the observations (biomass data) from other locations; it only needs the parameter values (it could be either theoretical or estimated) which can be integrated with new local information. Therefore, the Bayesian approach can be more versatile, as demonstrated by [
26].
The few remnant natural forests, located in remote mountain areas with steep slopes in Eastern and Southern Europe are often dominated by pure and mixed beech and coniferous forests [
27,
28,
29,
30]. European beech (
Fagus sylvatica L.) represents one of the widely distributed tree species across Europe [
31], in Romania accounting for a third of the Romanian growing stock [
32]. Allometric biomass models for European beech have been developed in numerous studies [
33,
34,
35,
36,
37,
38,
39]. In contrast, fewer studies reported allometric models for silver fir (
Abies alba Mill.), another species often present in the few remaining mixed virgin forests, sometimes, from the point of view of allometry, being assimilated to Norway spruce [
40,
41].
The following aims were set for this study: (i) developing allometric biomass models for European beech and silver fir to be used to estimate biomass of a European beech-silver fir virgin forest located in Southern Carpathians; (ii) investigating whether the local calibration of allometric models can be done appropriately based on a reduced sample of site-specific observations using a random intercept model or a Bayesian model; and (iii) assessing how these models and approaches perform in predicting AGB in a 1-ha sample plot of this virgin forest.
4. Discussion
In this paper, we developed site-specific allometric biomass models for European beech and silver fir to be used in Șinca virgin forest, Romania. The parameters of the site-specific allometric models, developed within this study, that use D to predict AGB (Equation (2)) differ slightly from those reported, for the same species, at European level [
35,
38,
52]. For example, for European beech, the slope (also called the scaling exponent) of the generic allometric model based on D was 2.36 [
38] and 2.45 [
52], whereas for our site-specific model was 2.64. The intercepts (of the models in log-scale), however, were bigger for the generic model (i.e., −1.66 and −2.07, respectively) and smaller for our site-specific sample (i.e., −2.66). For silver fir, the differences were as large as for European beech. The scaling exponent of the generic model was 2.45 and for the site-specific model was 2.70, whereas the intercepts were −2.39 and −3.41, respectively. For both species in this study, we reported slope values that were close to the theoretical scaling exponent value of 2.67 predicted by the “General model for the origin of allometric scaling laws” [
53]. For our study, the slope estimates indicate that an increase in D by 1.0% causes an increase in AGB, of 2.64% and 2.70%, for European beech and silver fir, respectively; a lower increase is documented at European level (2.36% for European beech and 2.45% for silver fir). These larger parameters for our site indicate that trees of similar D exhibit greater AGB in Șinca compared to European averages, which may have been caused by a larger H for any given D or by a larger crown for any given D and H in the virgin forest, with respect to the averages in European forests. However, more similar parameters to our models were reported for European beech in the Netherlands, where the reported slope was 2.60 and the intercept was −2.53 [
34].
When both D and H were used as predictors of AGB the effect of D on AGB was conditional on a constant H. Therefore, the parameter of D shows the increase in AGB produced by 1% increase in D while H was constant. For European beech, the scaling exponents were in line with those published in the literature; the scaling exponent of D was 2.15, which was similar to that reported by [
33] and very close to the value of 2.20 reported by [
36]. The scaling exponent of H (that shows the proportional increase in AGB caused by 1% increase in H, under constant D) was somehow different; a value of 0.69 was derived in this study, whereas [
36] reported a value of 0.56 and [
33] reported a value of 1.14. Therefore, compared to our sample of European beech trees from Șinca forest, the trees sampled from Czech Republic seems to have a greater effect of H on AGB. The increase in H by 1%, for trees of similar D, produced more AGB in Czech Republic than in our site.
For models based on D and H, the scaling exponent of D was approximately 2.0, whereas the scaling exponent of H was smaller than 1.0 for most species [
35]. Consequently, it has also been shown that the ratio between the parameter estimate of D and parameter estimate of H (i.e., the Q-ratio) is frequently larger than 2.0, usually between 3 and 4 [
54]. In the case of silver fir, the parameters showed quite a different pattern compared to European beech and other species. The scaling exponent of D (i.e., 1.33), surprisingly, was smaller than the scaling exponent of H (i.e., 1.45), showing a Q-ratio of 0.92 [
54]. That means, in the case of silver fir, a 1% increase in D (while H constant) produced a 1.33% increase in AGB, whereas a 1% increase in H (under constant D) produced a 1.45% increase in AGB. Although allometric biomass models for silver fir are rare, a similar anomaly was reported in a recent paper [
40]. Specifically, the parameter of D was 1.06 and the parameter of H was 1.40, which may support the hypothesis that these parameters may be influenced by an atypical biomass allocation pattern at the species level (for silver fir). Other authors, e.g., [
41], have also reported allometric biomass models for silver fir. However, the different model formulation does not allow us to compare the parameter estimates. Silver fir foliage tolerance under a shaded environment may be the starting point to speculate on the reported divergence. For small trees, the ratio between H and D should be higher compared to large trees, since the small trees often grow in a stronger competition. However, it is illustrated in
Figure 3a that the smallest silver fir trees have a H-D ratio that is comparable to that of largest trees and is much smaller compared to that of European beech trees (despite its similar shade tolerance behavior). This result could be a consequence of ungulate browsing, which for the studied area was reported to be more frequent on silver fir than on European beech [
27]. As a result, in the case of small trees, for similar D, the silver fir trees are much shorter compared to European beech (
Figure 3b). Therefore, the increase in H seems to be much more accelerated for silver fir trees compared to European beech, up to about D ≅ 35 cm, when the silver fir trees reach a maximum H-D ratio of about 0.8 (
Figure 3a). For large trees, the H-D ratios become comparable for the two species. The accelerated increase in H during early development stages resulted in a larger effect of H on AGB. Therefore, we argue that in the case of silver fir, the different pattern in H-D ratio is the reason for the anomalous parameter estimates.
The obtained results indicate that a RIM or a Bayesian model can be successfully used to combine measurements available from other locations with a site-specific reduced sample of small trees, in order to calibrate the allometric model at site level. These approaches would minimize the effort for biomass measurements, however, with a trade-off in prediction accuracy. The presented analysis takes a further step from some previous ideas in which a small sample of small trees is used to calibrate an allometric model for the entire covariate range in order to accurately predict the biomass [
23].
Although both the Bayesian approach and the RIM can combine the information from other locations with the site-specific observations, there is a crucial difference between these two approaches. To apply RIM the user needs access to the raw observations from other locations, however, the Bayesian approach uses only the model outputs resulted from these observations (i.e., no raw data is required). Therefore, if availability of raw observations data is limited, then Bayesian approach should be preferred. Since Bayesian model does not use the raw data from other locations, the model is fitted based on reduced sample only. Therefore, in the presented analysis, the Bayesian model was based on 7 observations only, whereas the corresponding RIM was based on 151 trees for European beech and 101 trees for silver fir, respectively (Dataset #4;
Table 4). Since Bayesian model was fitted on a smaller number of observations, it was also more difficult to accurately estimate the variance-covariance matrix, which is needed to propagate the errors from model parameters and residuals to plot level biomass prediction.
The main assumption of RIM is that the slopes of all sites are similar and only the intercepts differ between sites. Using just a few small trees to calibrate the model for the entire D range implies that the slope parameters are informed primarily by the observations from other locations, whereas the site-specific sample will inform mainly the intercept. In other words, the reduced sample of small trees is used mainly to calibrate the intercept, whereas the slope parameters are derived from the species-specific dataset (generic dataset).
A reduced sample of small trees can successfully be used to calibrate an allometric model locally, however, special attention should be given to:
- (a)
The H-D ratio, which should be checked in advance. As we observed in our analysis with silver fir trees, the H-D ratio can affect the parameter estimates, which affect further the performance of small trees sample approach. Therefore, the user should check whether the H-D ratio decreases relatively linearly with the increase in tree size.
- (b)
Either a random intercept model or a Bayesian model can be used with the reduced sample approach. Preference to one of the methods can be decided based on the raw data availability. Nevertheless, access to raw observations should not be an issue given the increasing trend in publication of biomass datasets, e.g., [
44,
55,
56,
57,
58].
- (c)
The generic biomass sample should contain as many species-specific observations as possible, including very large trees (D-range should match that of the local population for which the models are developed).
- (d)
The reduced sample of small trees should contain a large enough number of trees to calibrate mainly the intercept; at least 6–7 trees should be used (the greater the number, the better the result). It is recommended that trees with D < 5 cm should not be used with the reduced sample approach, since the allometry of very small trees can be affected by the competition with herbaceous plants.
- (e)
Using the reduced sample approach should always be performed using no less than D and H as predictors; other additional predictors can be used, because using both variables, the biomass estimates were more precise.