Allometric mixed-effects models for Dicksonia sellowiana Hook. and its contribution to biomass stocks of Araucaria forests in southern Brazil

Abstract

Predicting biomass of arborescent individuals on sample plots using allometric models is an essential step in the computation of design-based forest biomass estimates. Nonetheless, there are few models for species with different growth forms, such as tree ferns. This study, therefore, sought to construct models to predict {aboveground

Key words allometry; Atlantic forest; tree fern; Mata Atlântica; xaxim

Introduction

Forest ecosystems play a major role in the absorption of atmospheric carbon dioxide (Bar-On et al. 2018). As about half of plant dry mass consists of carbon (Ma et al. 2018), much efforts have been devoted to improving forest biomass estimates in recent decades (Duncanson et al. 2021). One of the first steps in the computation of such estimates is to predict the biomass of individual trees, palms, tree ferns or other arborescent individuals with the aid of allometric models using easy-to-measure forest inventory variables as inputs (Picard et al. 2012). Due to the scarcity of models constructed using large, regionally representative samples, generic models constructed based on aggregated datasets are often used to predict individual tree biomass (e.g., Feldpausch et al. 2012, Chave et al. 2014).

Nonetheless, generic models are typically constructed with data from a wide range of tree species and do not cover tree fern species (e.g., Chave et al. 2014, Fayolle et al. 2018), such as Dicksonia sellowiana Hook. (Dicksoniaceae). This species, popularly known as ‘xaxim-bugio’, occurs abundantly mainly in subtropical Brazilian Atlantic forests (Noben et al. 2018), more specifically in Araucaria forests (Sevegnani et al. 2019), upper hill evergreen rainforests (Maçaneiro et al. 2019), and cloud forests (Marcon et al. 2014). According to Hallé et al. (1978), D. sellowiana has the ‘Corner’ architectural model, which is featured by vegetative growth commanded by a single branching meristem. Generally, the species reaches 3 m height, exceptionally up to 10 m, and diameters of 15 to 20 cm (Noben et al. 2018), exceptionally up to 70 cm (Gasper et al. 2011). It has preference for cold and humid habitats (Schwartz & Gasper 2020), and typically exhibits an aggregated spatial distribution (Lerner et al. 2021).

Despite the abundance of D. sellowiana and its potential contribution to forest biomass pools, there are a few allometric model for this species. They were constructed by Ziemmer et al. (2016) based on dry biomass (quantified indirectly), diameter at ground level and total height of 40 individuals, half of D. sellowiana and half of Cyathea corcovadensis (Raddi) Domin, measured at a single site in southern Brazil. Although this species may represent a substantial portion of forest biomass in certain areas, some researchers assume a conservative approach and exclude it from estimates for biomass stocks due to the absence of a specific model for it (Vibrans et al. 2022). Others, in turn, have applied a model constructed by Tiepolo et al. (2002) for other abundant tree fern species, namely Cyathea spp., to predict biomass of D. sellowiana (Ribeiro et al. 2013, Souza & Longhi 2019). Uller et al. (2021) questioned this practice because Cyathea spp. and D. sellowiana have distinct morphological characteristics and thus may have different allometric relationships, despite having the same architectural model. Therefore, it is reasonable to assume that biomass predictions for D. sellowiana calculated using models for other tree fern species may carry non-trivial systematic errors (Uller et al. 2021), which thereby would affect estimates for forest biomass stocks (e.g., Oliveira & Vibrans 2022).

Given the above, the aims of this study were threefold: (i) to construct allometric models to predict the dry aboveground biomass and total height for D. sellowiana individuals; (ii) to assess the performance of allometric models constructed for different tree fern species in predicting biomass of D. sellowiana; and (iii) to assess the contribution of D. sellowiana to plot-level predictions and large-scale estimates for mean aboveground biomass per unit area for the Araucaria forest in southern Brazil. This study hypothesizes that models constructed for other tree fern species are not fully suitable for D. sellowiana, and that estimated aboveground biomass stocks of Araucaria forests are significantly greater when the latter species is included in the estimates.

MATERIALS AND METHODS

Study area

The study area was defined as the Araucaria forest (AF) in the state of Santa Catarina, southern Brazil, with a potential occurrence area of $\sim$56,000 km${}^2$ (Figure 1), of which $\sim$34% is forest land (Vibrans et al. 2021). The AF is a forest type of the Atlantic forest domain and is characterized by the mixture of Araucaria angustifolia (Bertol.) Kuntze with multiple broadleaved species, where the former typically occupies the highest forest layer. According to the Köppen climatic classification, the study area is influenced by the Cfb climate type — humid subtropical oceanic climate with temperate summers (Alvares et al. 2013). Long-term ($>$ 30 years) average annual temperature for the study area is $\sim$16.0 \(°\)C and average annual precipitation is $\sim$1,600 mm.

Figure 1
Study area, location of the FlorestaSC sample plots in the Araucaria forest, and sites where aboveground biomass and total height data for Dicksonia sellowiana were collected. SJNP: São Joaquim National Park.

Forest inventory data

The FlorestaSC program acquired data between 2007 and 2009 on 150 permanent sample plots, each with an area of 0.4 ha, located at the intersections of a 10 km $\times$ 10 km grid covering the AF (Figure 1). The plots met the forestland definition of Ribeiro et al. (2009): arborescent vegetation having regenerated for at least 15 years, with a canopy height $\ge$ 10 m and a basal area $\ge$ 10 m${}^2$ ha${}^{-1}$. The majority of the sampled stands consist of successional forests with age $\ge$ 35 years according to land use change detection based on Landsat time series imagery (Souza et al. 2020). All living trees, palms and tree ferns with diameter at breast height ($D$) $\ge$ 10 cm on the sample plots were identified, their $D$ was measured, and their total height (H) was visually assessed. The wood specific gravity ($\rho$) for each species record in the dataset was retrieved from a regional dataset (Oliveira et al. 2019). The elevation above sea level of the plots ranges from 514 to 1,560 m.

Biomass and total height data

The aboveground biomass (AGB), $D$ and $H$ of a total of 45 individuals of D. sellowiana were measured at six areas in the AF (Figure 1), in the year of 2022. Each sample individual was divided into the compartments (i) caudex and (ii) leaves. Biomass quantification initiated with separating the compartments, followed by complete weighing of each compartment in the field using a digital dynamometer mounted on a tripod. Two discs were taken from the caudex, one approximately at relative height of 25% and another at relative height of 75% to determine the moisture content of this compartment. A sample with a wet mass of $\sim$1 kg was taken from the leaves of every individual to estimate the moisture content of this compartment.

The wet mass of the samples was measured on a digital scale. The samples were dried in an oven with air circulation at 103 ±2 °C until they reached a constant mass. Their moisture content was calculated on the wet basis. The dry mass of the compartments was estimated based on wet mass and moisture content of the samples as $dm=wm\cdot (1-mc/100)$, where dm is the dry mass (g), wm is the wet mass (g), and mc is the moisture content (%). The moisture content of the caudex of each individual was computed as the average over the two discs taken from it. The total AGB (kg) of each individual was calculated as the sum of the dry mass of the caudex and leaves compartments.

Additional 160 individuals of D. sellowiana were measured for $D$ and $H$ with a tape and telescopic ruler, respectively, in the year of 2017 at the National Forest of Irati, located in the Brazilian state of Paraná (Figure 1). In total, data for 205 individuals were included in the dataset for modeling total height. Descriptive statistics for the variables in the datasets described above are given in Table I and relationships among them are presented in Figure 2. Only 7% of the individuals measured on the inventory plots had $D>$ 34.5 cm, which is the maximum value for this variable in the AGB model calibration dataset, and 0.17% of the individuals had $D>$ 60.2 cm, which is the maximum value for this variable in the $H$-$D$ model calibration dataset.

Figure 2
Relationships among variables in the datasets. PR refers to data collected in the state of Paraná, whereas SC refers to data collected in the state of Santa Catarina.

Table I
Descriptive statistics for the model calibration datasets and inventory data for Dicksonia sellowiana.

Statistic	$AGB$ dataset ($n$ = 45)				Total height dataset ($n$ = 205)			Inventory data ($n$ = 5,238)
2-9	$AGB$ (kg)	$D$ (cm)	$H$ (m)	$H$-$D$ ratio	$D$ (cm)	$H$ (m)	$H$-$D$ ratio	$D$ (cm)
Mean	13.1	18.7	2.16	11.5	23.8	3.0	12.5	23.2
Std. dev.	8.6	5.2	0.78	2.2	9.0	1.4	3.7	7.0
Min.	4.2	9.5	1.30	8.0	9.5	1.3	6.6	9.6
Max.	44.3	34.5	4.06	16.7	60.2	7.5	25.7	77.0
1${}^{st}$ Quartile	7.0	15.6	1.50	9.4	17.4	1.8	9.4	18.5
3${}^{rd}$ Quartile	16.1	20.6	2.63	13.1	29.0	4.0	14.4	26.1

Allometric models

Due to the potential correlation between individuals sampled at the same location, a mixed-effects modeling approach was used to accommodate the assumption of residual independence, thus ensuring more reliable estimates of model parameter uncertainty (McCulloch et al. 2008). In addition, mixed-effects models can be recalibrated to improve prediction accuracy through the prediction of random effects for new areas based on (a few tree) samples (e.g., Ciceu et al. 2020). The $AGB$ for D. sellowiana was modeled as the traditional power function used in volume/biomass allometry, with the following form

$$AG{B}_{ij}=[\alpha +a_i]\cdot D_{ij}^{{\beta }_1}\cdot H_{ij}^{{\beta }_2}+{\epsilon }_{ij},{\epsilon }_{ij}\sim 𝒩(0,𝚺)$$(1)

where $i$ ($i=1,2,\dots ,6$) and $j$ ($j=1,2,\dots ,45$) index sample locations and sample tree ferns, respectively; $\alpha$, ${\beta }_1$ and ${\beta }_2$ are fixed-effects parameters to be estimated; $a_i$ is a random effect to be predicted for each sample location; ${\epsilon }_{ij}$ is the random residual; and $𝚺={𝛔}_{\epsilon }^2𝐈$, being ${𝛔}_{\epsilon }^2$ a vector of heteroscedastic residual variances, with elements ${\sigma }_{{\epsilon }_{ij}}^2$ (Eq. (5) and (6)).

This model assumes that the relative growth rate of $AGB$ is a linear combination of the relative growth rates of $D$ and $H$: $\frac{\mathrm{d}AG{B}_{ij}}{AG{B}_{ij}}={\beta }_1\frac{\mathrm{d}{D}_{ij}}{D_{ij}}+{\beta }_2\frac{\mathrm{d}{H}_{ij}}{H_{ij}}$ (Picard et al. 2015). For the special case where the ratio (hereafter $Q$-ratio) between ${\beta }_1$ and ${\beta }_2$ is 2, Eq. (1) can be written as $AG{B}_{ij}=\alpha \cdot D_{ij}^{2\beta }\cdot H_{ij}^{\beta }+{\epsilon }_{ij}$, and thereby is equivalent to $AG{B}_{ij}=\alpha \cdot {(D_{ij}^2\cdot H_{ij})}^{\beta }+{\epsilon }_{ij}$ (Dutcă et al. 2019). However, this does not appear to hold for tree ferns. Available models suggest $Q$-ratios close to 1 for tree ferns growing in the Atlantic forest (Ziemmer et al. 2016, Uller et al. 2021). Hence, if the $Q$-ratio is assumed to be 1, Eq. (1) is equivalent to $AG{B}_{ij}=\alpha \cdot ({D_{ij}\cdot H_{ij})}^{\beta }+{\epsilon }_{ij}$. Therefore, a more parsimonious model with the following form is justified

$$AG{B}_{ij}=[\alpha +a_i]\cdot {(D_{ij}\cdot H_{ij})}^{\beta }+{\epsilon }_{ij},{\epsilon }_{ij}\sim 𝒩(0,𝚺)$$(2)

where $i$ ($i=1,2,\dots ,6$) and $j$ ($j=1,2,\dots ,45$) index sample locations and sample tree ferns, respectively; $\alpha$ and $\beta$ are fixed-effects parameters to be estimated; $a_i$ is a random effect to be predicted for each sample location; ${\epsilon }_{ij}$ is the random residual; and $𝚺={𝛔}_{\epsilon }^2𝐈$, being ${𝛔}_{\epsilon }^2$ a vector of heteroscedastic residual variances, with elements ${\sigma }_{{\epsilon }_{ij}}^2$ (Eq. (5) and (6)).

The $H$-$D$ allometry for D. sellowiana was modeled using two different three-parameter sigmoidal functions, known as the Chapman-Richards and Ratkowsky functions (Richards 1959, Chapman 1961, Ratkowsky 1990), respectively

$$H_{ij}=1.3+[\alpha +a_i]\cdot {(1-exp([-{\beta }_1+b_i^{\left(1\right)}]\cdot D_{ij}))}^{[{\beta }_2+b_i^{\left(2\right)}]}+{\epsilon }_{ij},{\epsilon }_{ij}\sim 𝒩(0,𝚺)$$(3)

$$H_{ij}=1.3+[\alpha +a_i]\cdot \exp \left(\frac{[-{\beta }_1+b_i^{(1)}]}{D_{ij}+[{\beta }_2+b_i^{(2)}]}\right)+{\epsilon }_{ij},{\epsilon }_{ij}\sim \mathcal{N}(0,\mathrm{\Sigma }\mathrm{\Sigma })$$(4)

where $i$ ($i=1,2,\dots ,7$) and $j$ ($j=1,2,\dots ,205$) index sample locations and sample tree ferns, respectively; $\alpha$, ${\beta }_1$ and ${\beta }_2$ are fixed-effects parameters to be estimated; $a_i$, $b_i^{\left(1\right)}$ and $b_i^{\left(2\right)}$ are random effects to be predicted for each sample location; ${\epsilon }_{ij}$ is the random residual; and $𝚺={𝛔}_{\epsilon }^2𝐈$, being ${𝛔}_{\epsilon }^2$ a vector of heteroscedastic residual variances, with elements ${\sigma }_{{\epsilon }_{ij}}^2$ (Eq. (5) and (6)). The asymptote of the models is defined as $1.3+[\alpha +a_i]$.

Accommodating heteroscedasticity in model residuals is necessary to obtain reliable estimates of the uncertainty associated with the estimation of model parameters (Kutner et al. 1996), which was assessed using the percent relative standard error (PRSE). It was calculated as the ratio between the standard error of the estimated parameter and the parameter estimate times one hundred. Heteroscedasticity in the residuals of all models was modeled using a power function with $D$ as the predictor variable

$${\sigma }_{{\epsilon }_{ij}}^2={(\sigma \cdot D_{ij}^{\delta })}^2$$(5)

where $\sigma$ is the scaling parameter and $\delta$ is the shape parameter of the function, both of which need to be estimated. When heteroscedasticity was not effectively accommodated using Eq. ((5)), an alternative function was employed

$${\sigma }_{{\epsilon }_{ij}}^2={(\sigma \cdot {\hat{y}}_{ij}^{\delta })}^2$$(6)

where ${\hat{y}}_{ij}$ is a prediction of an unweighted nonlinear mixed-effects model (with the same form of the model being estimated) for the response variable for the $j^{th}$ individual on the $i^{th}$ sample location.

The mixed-effects models were estimated using the Lindstrom and Bates algorithm (Lindstrom & Bates 1990), as implemented in nlme function of the nlme R package (Pinheiro et al. 2021). In summary, it iterates a penalized nonlinear least squares step and a linear mixed-effects step until convergence is reached; all models were estimated using the maximum likelihood technique. Model prediction accuracy was assessed through the metrics mean absolute percentage residual (MAPR), mean systematic percentage error (MSPE), and corrected Akaike Information Criterion ($\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{c}}$). The metrics can be evaluated as

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{R}=\sum _{i=1}^n\frac{|y_i-{\hat{y}}_i|}{{\hat{y}}_i}\cdot \frac{100}{n}$$(7)

$$\mathrm{M}\mathrm{S}\mathrm{P}\mathrm{E}=\sum _{i=1}^n\frac{{\hat{y}}_i-y_i}{y_i}\cdot \frac{100}{n}$$(8)

$$\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{c}}=-2ln\left(\widehat{ℒ}\right)+2k+\frac{2k(k+1)}{n-k-1}$$(8)

where $y_i$ is the observed aboveground biomass or total height for the $i^{th}$ individual in the model calibration dataset; ${\hat{y}}_i$ is the predicted aboveground biomass or total height for the $i^{th}$ individual in the model calibration dataset; $n$ is number of observations; $k$ is the number of model parameters; and $\widehat{ℒ}$ is the maximum value of the likelihood function for the model. The $\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{c}}$ gives a greater penalty to the number of model parameters compared to the standard AIC; the correction factor (last term of Eq. (9)) increases the penalty for additional parameters as $n$ decreases (Burnham & Anderson 2002).

It merits noting that random effects were evaluated at zero when assessing the prediction accuracy of the models, as they are expected to be applied primarily in areas beyond where data to estimate them were collected (i.e., areas for which no random effects were predicted). De-Miguel et al. (2013) suggested that fixed-effects models should also be reported along with mixed-effects models because the latter may yield predictions with strong systematic errors when random effects are evaluated at zero (e.g., Mehtätalo et al. 2015). Nonetheless, absolute relative differences among estimates for the fixed-effects parameters of the mixed-effects models constructed in this study and those for purely fixed-effects models were as small as 0.5%. Therefore, only mixed-effects models were reported here, and random effects can be evaluated at zero with no distinguishable effects on predictions.

Evaluating other models for tree ferns

The models constructed by Tiepolo et al. (2002) for Cyathea spp., Ziemmer et al. (2016) for C. corcovadensis and D. sellowiana, and Uller et al. (2021) for Cyathea delgadii Sternb. in the subtropical Brazilian Atlantic forest were selected:

$$\widehat{AGB}=\frac{-4266348}{1-(2792284\cdot exp(-0.3137\cdot H))}$$(10)

Tiepolo et al. (2002) – Cyathea spp. ($n$ = 22)

$$\widehat{AGB}=0.0701\cdot D_g^{1.3102}\cdot H^{1.0566}$$(11)

Ziemmer et al. (2016) – C. corcovadensis and D. sellowiana ($n$ = 40)

$$\widehat{AGB}=0.1523\cdot D^{1.1254}\cdot H^{1.0338}$$(12)

Uller et al. (2021) – C. delgadii ($n$ = 30)

where $\widehat{AGB}$ is the predicted dry aboveground biomass (kg); $D$ is the diameter at breast height (cm); $D_g$ is the diameter at ground level (cm); and $H$ is the total height (m). It should be noted that the model by Ziemmer et al. (2016) was fitted using ordinary least squares in its linearized form, but neither the estimated residual variance on the ln scale nor the bias correction factor for predictions on the original scale, as per Baskerville (1972), were reported. Hence, from the outset, this model under-predicts $AGB$ by a factor of $1/exp\left({\hat{\sigma }}_{\epsilon }^2/2\right)$, where ${\hat{\sigma }}_{\epsilon }^2$ is the estimated residual variance on the ln scale.

The accuracy of these models in predicting the $AGB$ of D. sellowiana was assessed using the MAPR and MSPE. Additionally, scatterplots with observed values against predicted values were built for visual inspection. The null hypothesis that the linear relationship between observed and predicted values has simultaneously intercept 0 and slope 1 was tested using the $F$-test described in Dent & Blackie (1979); this test provides approximate results because uncertainty in model predictions was not accounted for.

Estimating population mean AGB per unit area

The estimation of population mean AGB ha${}^{-1}$ for the AF using the FlorestaSC data had the following steps: (i) predict the $AGB$ for every tree on the $i^{th}$ sample plot ($i=1,2,\dots ,150$) using the pantropical model ${\widehat{AGB}}_{ij}=0.0567\cdot {(D_{ij}^2\cdot H_{ij}\cdot {\rho }_{ij})}^{0.989}$ (Feldpausch et al. 2012), where $\rho$ is the species wood specific gravity (g cm${}^{-3}$); (ii) predict the AGB for individuals of Cyathea spp. and Syagrus romanzoffiana on the $i^{th}$ sample plot ($i=1,2,\dots ,150$) using the species-specific models constructed by and Uller et al. (2021) and Moreira-Burger & Delitti (2010); (iii) predict the AGB for D. sellowiana individuals on the $i^{th}$ sample plot ($i=1,2,\dots ,150$) using Model 2 from this study and those constructed by Tiepolo et al. (2002), Ziemmer et al. (2016) and Uller et al. (2021), where $H$ was predicted using the Ratkowsky $H$-$D$ model from this study; (iv) scale the total AGB predicted for the $i^{th}$ sample plot to a per hectare basis; (v) estimate the population mean, $\mu$, over all plot predictions using simple expansion estimators given as

$$\hat{\mu }=\frac{1}{n}\sum _{i=1}^n{x}_i$$(13)

$$\widehat{\mathrm{\text{VAR}}}\left(\hat{\mu }\right)=\frac{1}{n(n-1)}\sum _{i=1}^n{(x_i-\hat{\mu })}^2$$(14)

where $\widehat{\mathrm{\text{VAR}}}\left(\hat{\mu }\right)$ is the variance of the estimator of $\mu$; $x_i$ is the $AGB$ ha${}^{-1}$ for the $i^{th}$ sample plot; and $n$ is the total number of sample plots. The standard error of the mean (SE) is obtained as the square root of $\widehat{\mathrm{\text{VAR}}}\left(\hat{\mu }\right)$.

Step (iii) was not conducted in the computation of estimates excluding D. sellowiana individuals with the aim of assessing its contribution to plot-level predictions and to large-scale estimates for mean $AGB$ ha${}^{-1}$. To assess the contribution of the species to local estimates for mean $AGB$ per unit area, the procedure described above was also conducted for a sub-sample of 18 plots located in counties adjacent to the São Joaquim National Park (Figure 1), where a great abundance of D. sellowiana has been reported (Gasper et al. 2011, Noben et al. 2018).

It should be stated that the treatment of a systematic sampling design based on the simple random sampling assumption is valid. It is assumed, however, that the sample variance may be overestimated, but never underestimated (Särndal et al. 1992, Magnussen et al. 2020). For simplifying the estimation procedure, uncertainty in individual-level model predictions was not accounted for; therefore, standard errors and confidence intervals for population means reported in this study should be construed cautiously, as they reflect only plot sampling variability. It is expected though that uncertainty in individual-level $AGB$ predictions for D. sellowiana would not affect standard errors substantially, as the main source of model-based uncertainty is the pantropical model, and the same model was applied in all cases.

Linear mixed-effects models were used to estimate the mean paired difference between $AGB$ ha${}^{-1}$ per plot predicted using each $AGB$ model, with the aid of the nlme R package; the models allowed for random intercepts for each plot, thus being equivalent to a repeated measures ANOVA. As the variances of paired differences were not equivalent across all contrasts, a variance structure was incorporated for each treatment (i.e., the $AGB$ model) using the varIdent function from the referenced R package for each treatment (i.e., $AGB$ model). Post-hoc Dunnett family-wise 95% confidence intervals were constructed to conduct pairwise comparisons of estimated mean paired differences among plot predictions (i) calculated using each evaluated model and plot predictions without D. sellowiana (control), and (ii) calculated using external models and plot predictions calculated using Model 2 (control). These intervals were constructed using the multicomp R package (Hothorn et al. 2008). This analysis provides approximate results because uncertainty in model predictions was not accounted for.

RESULTS

Biomass allocation per compartment

On average, 92.5% (sd = 5.0%) of the total $AGB$ of the sampled individuals was stored in the caudex compartment, while 7.5% (sd = 5.0%) was stored in the leaves compartment. Mean moisture contents were 64.2% (sd = 6.8%) and 71.2% (sd = 4.3%) for the caudex and leaves compartments, respectively.

Biomass models

The estimated parameters of Model 1 (Eq. 1) had PRSEs ranging from 17.1 to 39.3%, whereas those of Model 2 (Eq. 2) from 5.0 to 16.5% (Table II); all parameters were significantly different from zero at the significance level of 0.05. The residual variance models were effective to accommodate heteroscedasticity in both models according to the Breusch-Pagan test adapted by Dutcă et al. (2022) for nonlinear weighted models (Model 1: ${\chi }^2=0.81$, $p=0.67$; Model 2: ${\chi }^2$ = 0.03, $p=0.86$). The models estimated in this study had very similar prediction accuracy, with MAPRs around 13% and MSPEs between 2 to 3%. Nonetheless, Model 2 had a substantially smaller $\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{c}}$ than that for Model 1 (Table III), suggesting that the former better estimates the true underlying model.

Table II
Parameter estimates and standard errors for the AGB models.

Model	Component	Parameter	Estimate	$\widehat{\mathrm{S}\mathrm{E}}$	PRSE
1	Fixed effects	$\alpha$	0.3799	0.1491	39.3%
		${\beta }_1$	0.9566	0.1685	17.6%
		${\beta }_2$	0.8679	0.1487	17.1%
	Random effects	sd($a_i$)	5.56e–05	-	-
	Residual variance,	${\sigma }^2$	0.0036²	-	-
	${\hat{\sigma }}_{{\epsilon }_{ij}}^2=f(D_{ij},\widehat{𝛉})$	$\delta$	2.19	-	-
2	Fixed effects	$\alpha$	0.4166	0.0689	16.5%
		$\beta$	0.9106	0.0455	5.0%
	Random effects	sd($a_i$)	2.77e–06	-	-
	Residual variance,	${\sigma }^2$	0.1881²	-	-
	${\hat{\sigma }}_{{\epsilon }_{ij}}^2=f({\widehat{AGB}}_{ij},\widehat{𝛉})$	$\delta$	0.9757	-	-

In turn, the models constructed by Tiepolo et al. (2002), Ziemmer et al. (2016) and Uller et al. (2021) systematically under-predicted the $AGB$ for D. sellowiana individuals (Figure 3); on average, they under-predicted the $AGB$ by –71%, –39% and –24% of the observed values, respectively. Accordingly, the null hypothesis of the $F$-test for simultaneous intercept 0 and slope 1 was rejected for these three models (Table III).

Figure 3
Observed vs. predicted $𝐀𝐆𝐁$ for Dicksonia sellowiana individuals using the models estimated in this study and other models for tree ferns. Thick lines are 1:1 lines and thin lines represent linear regressions of observed against predicted values. Gray areas delineate approximate Working-Hotelling 95% confidence bands, as uncertainty in predictions was not accounted for.

Table III
Prediction accuracy metrics calculated for different AGB models for tree ferns.

Metric	Model 1	Model 2	Tiepolo et al. (2002)	Ziemmer et al. (2016)	Uller et al. (2021)
MAPR (%)	13.4	13.4	291.3	72.2	37.1
MSPE (%)	3.0	2.0	–70.8	–38.9	–24.1
$\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{c}}$	205.4	197.1	-	-	-
F(2,43); $p$	0.86; 0.42	0.88; 0.42	233.2; $<$0.001	65.8; $<$0.001	26.6; $<$0.001

H-D models

The estimated parameters of the Chapman-Richards $H$-$D$ model had PRSEs ranging from 12.4 to 41.5%, whereas those of the Ratkowsky $H$-$D$ model had PRSEs ranging from 21.7 to 33.0% (Table IV); all parameters were significantly different from zero at the significance level of 0.05. Both models converged with random effects associated with only two parameters (Table IV). The Breusch-Pagan test suggested that heteroscedasticity in model residuals was sufficiently accommodated (Chapman-Richards: ${\chi }^2=0.266$, $p=0.606$; Ratkowsky: ${\chi }^2=0.308$, $p=0.578$).

Although the Chapman-Richards model yielded a slightly smaller $\mathrm{A}\mathrm{I}{\mathrm{C}}_{\mathrm{c}}$ (506.1) than the Ratkowsky model (508.0), both models yielded an MAPR of 22.2% and an MSPE of 7.5%. The models produced very similar mean responses for $H$, except that the Chapman-Richards model under-predicted $H$ for the thickest individuals relative to the Ratkowsky model (Figure 4).

Figure 4
Mean response of the $𝐇$-$𝐃$ models estimated for Dicksonia sellowiana, and observed vs. predicted values, where thick lines are 1:1 lines and thin lines represent linear regressions of observed against predicted values. Gray areas delineate approximate Working-Hotelling 95% confidence bands, as uncertainty in predictions was not accounted for.

Table IV
Parameter estimates and standard errors for the H-D models.

Model	Component	Parameter	Estimate	$\widehat{\mathrm{S}\mathrm{E}}$	PRSE
Chapman-Richards	Fixed effects	$\alpha$	3.9853	0.4939	12.4%
		${\beta }_1$	0.1195	0.0243	20.4%
		${\beta }_2$	12.8291	5.3277	41.5%
	Random effects	sd($a_i$)	5.267e–05	-	-
		sd($b_i^{\left(1\right)}$)	3.809e–10	-	-
		cor($a_i,b_i^{\left(1\right)}$)	0.000	-	-
	Residual variance,	${\sigma }^2$	0.039²	-	-
	${\hat{\sigma }}_{{\epsilon }_{ij}}^2=f(D_{ij},\widehat{𝛉})$	$\delta$	0.974	-	-
Ratkowsky	Fixed effects	$\alpha$	7.7366	1.6799	21.7%
		${\beta }_1$	24.3671	7.3169	30.0%
		${\beta }_2$	–6.9888	2.3095	33.0%
	Random effects	sd($a_i$)	1.063e–07	-	-
		sd($b_i^{\left(1\right)}$)	1.895e–04	-	-
		cor($a_i,b_i^{\left(1\right)}$)	0.019	-	-
	Residual variance,	${\sigma }^2$	0.047²	-	-
	${\hat{\sigma }}_{{\epsilon }_{ij}}^2=f(D_{ij},\widehat{𝛉})$	$\delta$	0.913	-	-

Plot-level predictions and estimated population means for AGB per unit area

Due to the very similar prediction performances of Models 1 and 2, only estimates obtained with the latter were reported in this section. Predictions, considering only D. sellowiana, for the 100 inventory plots where the species was present ranged from 0.02 to 28.29 Mg ha${}^{-1}$, with its relative contribution to plot total AGB ha${}^{-1}$ varying between 0.02 to 40.35% (Figure 5). A map with spatialized plot $AGB$ predictions for the species is available as Supplementary Material (Figure S1).

Figure 5
Proportion of total AGB ha${}^{\mathbf{-}\mathrm{𝟏}}$ and predicted AGB ha${}^{\mathbf{-}\mathrm{𝟏}}$ for Dicksonia sellowiana per sample plot using Model 2 and other models for tree ferns. The sample plots were arranged in ascending order with respect to AGB ha${}^{\mathbf{-}\mathrm{𝟏}}$ predicted using Model 2.

The estimated population mean $AGB$ ha${}^{-1}$ for the AF in Santa Catarina without D. sellowiana was 82.97 Mg ha${}^{-1}$, while estimates including the species ranged from 83.34 to 84.92 Mg ha${}^{-1}$ depending on the $AGB$ model (Table V). Dunnett contrasts ranged from 0.37 to 1.95 Mg ha${}^{-1}$ and were significantly different from zero at the significance level of 0.05, except for the Tiepolo et al. (2002) model (Table V).

The estimated population mean $AGB$ ha${}^{-1}$ without D. sellowiana for the São Joaquim National Park region was 58.96 Mg ha${}^{-1}$, whereas estimates including the species ranged from 60.35 to 66.58 Mg ha${}^{-1}$ depending on the $AGB$ model (Table V). Dunnett contrasts ranged from 1.39 to 7.62 Mg ha${}^{-1}$ and were significantly different from zero at the significance level of 0.05, except for the Tiepolo et al. (2002) model (Table V).

Table V
Estimates for population mean AGB (Mg ha${}^{-1}$) based on individual-level predictions for AGB calculated using different models, and Dunnett contrasts comparing mean paired differences in plot predictions with (control) and without D. sellowiana.

Population	Without D. sellowiana		Model	With D. sellowiana		Mean diff.	95% CI	$p$
2-3	$\hat{\mu }$	$\widehat{\mathrm{S}\mathrm{E}}$		$\hat{\mu }$	$\widehat{\mathrm{S}\mathrm{E}}$
Araucaria forest ($n$ = 150)	82.97	3.56	(1)	84.92	3.55	1.95	[1.21, 2.69]	$<$0.001
			(2)	83.34	3.56	0.37	[–0.56, 1.31]	0.550
			(3)	84.42	3.55	1.45	[0.72, 2.19]	$<$0.001
			(4)	84.63	3.55	1.67	[0.94, 2.40]	$<$0.001
SJNP region ($n$ = 18)	58.96	6.17	(1)	66.58	6.63	7.62	[3.31, 11.93]	$<$0.001
			(2)	60.35	6.20	1.39	[–4.05, 6.82]	0.781
			(3)	64.82	6.49	5.86	[1.57, 10.15]	0.005
			(4)	65.59	6.55	6.64	[2.37, 10.89]	0.001

Dunnett contrasts revealed that the mean paired differences between $AGB$ ha${}^{-1}$ per sample plot predicted using the three external models and Model 2 were significantly different from zero for all contrasts, for both the regional and local populations (Table VI). The estimated mean differences between plot predictions calculated using Model 2 and the Uller et al. (2021) model were the smallest among all contrasts (Table VI). In relative values, the estimate using Uller et al. (2021) model was $\sim$15% smaller than that using Model 2 for the regional population, and $\sim$13% smaller for the local population.

Table VI
Dunnett contrasts for mean paired differences in plot predictions for AGB ha${}^{-1}$ calculated using different AGB models.

Population	Contrast	Mean diff.	$\widehat{\mathrm{S}\mathrm{E}}$	95% CI	$p$
Araucaria forest ($n$ = 150)	(2) – (1)	–1.58	0.27	[–2.20, –0.95]	$<$0.001
	(3) – (1)	–0.50	0.07	[–0.65, –0.34]	$<$0.001
	(4) – (1)	–0.29	0.05	[–0.41, –0.16]	$<$0.001
SJNP region ($n$ = 18)	(2) – (1)	–6.23	1.56	[–9.89, –2.58]	$<$0.001
	(3) – (1)	–1.76	0.38	[–2.64, –0.88]	$<$0.001
	(4) – (1)	–0.98	0.30	[–1.68, –0.29]	0.002

DISCUSSION

The two $AGB$ models estimated in this study had quite similar prediction performance. Nevertheless, Model 2 is preferred owing to the substantially smaller uncertainty in the estimates of its parameters compared to Model 1, and for being more parsimonious (Table II). These two advantages justify recommending Model 2 over Model 1, especially when models are considered intermediate tools in the estimation of mean $AGB$ per unit area, whose uncertainty (standard error of the mean) is predominantly affected by the magnitude of the uncertainty in model parameter estimates (McRoberts et al. 2015).

As generally expected for tree ferns growing in the Atlantic forest, the $Q$-ratio for Model 1 is close to 1 ($\hat{Q}=1.10$, $\widehat{\mathrm{S}\mathrm{E}}\left(\hat{Q}\right)=0.36$) and thereby justifies the form of Model 2. Dutcă et al. (2019) showed that when the $Q$-ratio strongly deviates from 2, the model $AGB=\alpha \cdot {(D^2\cdot H)}^{\beta }+\epsilon$ loses prediction accuracy. Accordingly, a $Q$-ratio strongly deviating from 1 would render the model $AGB=\alpha \cdot {(D\cdot H)}^{\beta }+\epsilon$ less accurate than the traditional three-parameter power model. $Q$-ratios close to 1 can be found for other species with different growth forms in the Atlantic forest, such as the palm Euterpe edulis Mart. and the hollow-trunk tree Cecropia glaziovii Snethl. (Uller et al. 2021). This indicates that $H$ has a greater effect than $D$ on $AGB$ for such species compared to ordinary tree species, which often produce models with $Q$-ratios greater than 2 (Fayolle et al. 2018, Dutcă et al. 2019). Uller et al. (2021) attributed the effect of $H$ on the $AGB$ of tree ferns to the fact that they do not undergo secondary growth, with the increase in diameter over time being primarily a result of overlapping adventitious roots.

Every other model evaluated in this study under-predicted the $AGB$ of sample D. sellowiana individuals, suggesting that it requires species-specific or multi-species models with dummy variables or random effects to account for the species effect, which appears to be primarily related to model parameter $\alpha$ (i.e., scaling factor). The model with $H$ as the sole predictor constructed by Tiepolo et al. (2002) for Cyathea spp. was the least suitable for D. sellowiana. Although not formally evaluated in this study, a model constructed by Uller et al. (2021) using $H$ as the sole predictor of $AGB$ for C. delgadii had a similar behavior to that of the Tiepolo et al. (2002) model when applied to D. sellowiana. This suggests that the $AGB$-$H$ allometry differs largely between these species. Therefore, models for Cyathea spp. using only $H$ as a predictor should be avoided for D. sellowiana.

Accordingly, estimates for mean $AGB$ per unit area for the species were strongly affected by the Tiepolo et al. (2002) model, to the extent that the contribution of the species to population means was statistically not significant. This was not the case when the other models were applied. Indeed, the model for C. delgadii by Uller et al. (2021) yielded an estimated mean $AGB$ per unit area closer to that yielded by Model 2 than that yielded by the Ziemmer et al. (2016) model, which included D. sellowiana individuals in the calibration dataset. The fact that a model for C. delgadii performed better than a model estimated using data from D. sellowiana individuals suggests that the behavior of the latter model when applied to our dataset is not related to sampling variability, but rather to other issues causing pronounced systematic deviations. One possible explanation is that Ziemmer et al. (2016) adopted $D_g$ instead of $D$ as a predictor variable and estimated $AGB$ indirectly (i.e., caudex volume multiplied by basic specific gravity), while Uller et al. (2021) used $D$ as a predictor variable and a quite similar $AGB$ quantification approach to that employed in this study.

Although tree ferns typically have considerably less dry mass than woody individuals, it was demonstrated that an abundant tree fern species can account for over one-third of the total stand $AGB$ ha${}^{-1}$ in some cases, and up to $\sim$11% of the estimated population mean $AGB$ ha${}^{-1}$ for a local population where the species is particularly abundant (Table VI). Further, it represented 2.3% of the estimated population mean $AGB$ ha${}^{-1}$ for the Araucaria forest in Santa Catarina. Therefore, to enhance the accuracy of estimates for mean $AGB$ per unit area, it appears reasonable to include abundant tree ferns in computations. Importantly, suitable prediction models should be employed, and biomass quantification methods should be standardized as much as possible so that model calibration datasets from multiple locations could be used to improve model representativeness and reduce uncertainty in model parameter estimates (Fu et al. 2017).

CONCLUSIONS

Four conclusions may be drawn from this study. First, the traditional double-entry, three-parameter allometric power model is suitable for D. sellowiana, but it can be simplified to a two-parameter model to reduce uncertainty in parameter estimates. Second, double-entry $AGB$ models (using both $D$ and $H$) constructed for tree fern species other than D. sellowiana should be applied with caution to this species, as they may systematically under-predict $AGB$ at the tree level, while models using only $H$ should be completely avoided. Third, the $H$-$D$ allometry for D. sellowiana was modeled using three-parameter sigmoidal functions; the Ratkowsky model had an estimated asymptote closer to the potential height reported for the species, which ranges from 8 to 10 m (Oliveira-Filho 2017, Noben et al. 2018); therefore, it generally appears more suitable than the Chapman-Richards model for practical applications. Fourth, D. sellowiana contributed approximately 2.0 Mg ha${}^{-1}$ to the estimated mean $AGB$ stock for a regional population, and approximately 7.0 Mg ha${}^{-1}$ to the estimated mean $AGB$ stock for a local population; at the plot level, the species accounted for over one-third of the total $AGB$ ha${}^{-1}$ in some cases. Therefore, the species should be included in estimates for mean $AGB$ per unit area, and allometric models could be further improved using larger and more geographically representative samples.

SUPPLEMENTARY MATERIAL

Figure S1.

ACKNOWLEDGMENTS

We are grateful to Adelar Mantovani, Alexandre Mariot, Jackson R. Eleotério, Heitor F. Uller, Instituto do Meio Ambiente de Santa Catarina (IMA-SC), and management staff of UHE São Roque for their valuable and indispensable support. The FlorestaSC program is currently funded by Secretaria de Estado do Meio Ambiente e da Economia Verde (SEMAE-SC, grant 2022TR001389) and Fundação de Amparo à Pesquisa e Inovação de Santa Catarina (FAPESC, grant 2023TR001432). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. LZO is supported by SEMAE-SC (grant 2022TR001389) and Universidade do Estado de Santa Catarina (PROMOP grant). ALN is supported by SEMAE-SC (grant 2022TR001389). ACV is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, grant 305199/2022-6).

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Publication Dates

• Publication in this collection
  16 Dec 2024
• Date of issue
  2024

History

• Received
  17 Feb 2023
• Accepted
  20 Aug 2024