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Article

Does It Matter Whether to Use Circular or Square Plots in Forest Inventories? A Multivariate Comparison

by
Efrain Velasco-Bautista
,
Antonio Gonzalez-Hernandez
,
Martin Enrique Romero-Sanchez
*,
Vidal Guerra-De La Cruz
and
Ramiro Perez-Miranda
National Institute of Forestry, Agriculture and Livestock Research, Ciudad de México 04010, Mexico
*
Author to whom correspondence should be addressed.
Forests 2024, 15(11), 1847; https://doi.org/10.3390/f15111847
Submission received: 27 August 2024 / Revised: 15 October 2024 / Accepted: 20 October 2024 / Published: 22 October 2024
(This article belongs to the Section Forest Inventory, Modeling and Remote Sensing)
Browse Figures
Figure 1
<p>Localisation of the study area.</p> "> Figure 2
<p>Localisation of the sampling plots and sampling design.</p> "> Figure 3
<p>Total height and diameter at breast height for all species recorded (blue dots: sacred fir, red dots: moctezuma pine; green dot: oaks, purple dots: ocote pine, brown dots: other species).</p> "> Figure 4
<p>Density (# tress/SSU), basal area (m<sup>2</sup>/SSU), and aboveground biomass (kg/SSU) in circular and square plots.</p> "> Figure 5
<p>Scatter plot for density (r = 0.9961), basal area (r = 0.9875), and aboveground biomass (r = 0.9811) from square and circular subplots (r = 0.9811). The colours identify the SSU for SPU (cluster).</p> "> Figure 6
<p>Empirical cumulative distribution function (blue) and modelled (Gamma, green; Burr, magenta; Weibull, pink) for density, (Gamma, green; Lognormal, red; Burr, magenta) for basal area and (Gamma, green; Weibull, red; Burr, magenta) for aboveground biomass. FDA: cumulative distribution function.</p> "> Figure 7
<p>Mean and confidence intervals for aboveground biomass, density, and basal area resulting from the joint statistical model.</p> "> Figure 8
<p>Histogram and normal distribution of residuals of multivariate statistical analysis of density, basal area, and aboveground biomass.</p> "> Figure 9
<p>Observed and normal percentiles of Pearson’s residuals.</p> ">
Versions Notes

Abstract

:
The design of a sampling unit, whether a simple plot or a subplot within a clustered structure, including shape and size, has received little attention in inferential forestry research. The use of auxiliary variables from remote sensing impacts the precision of estimators from both model-assisted and model-based inference perspectives. In both cases, model parameters are estimated from a sample of field plots and information from pixels corresponding to these units. In studies assisted by remote sensing, the shape of the plot used to fit regression models (typically circular) often differs from the shape of the population elements for prediction, where the area of interest is divided into equal tessellated parts. This raises interest in understanding the effect of the sampling unit shape on the mean of variables in forest stands of interest. Therefore, the objective of this study was to evaluate the effect of circular and square subplots, concentrically overlapped and arranged in an inverted Y cluster structure, over tree density, basal area, and aboveground biomass in a managed temperate forest in central Mexico. We used a Multivariate Generalised Linear Mixed Model, which considers the Gamma distribution of the variables and accounts for spatial correlation between Secondary Sampling Units nested within the Primary Sampling Unit. The main findings of this study indicate that the type of secondary sampling unit of the same area and centroid, whether circular or square, does not significantly affect the mean tree density (trees), basal area (m2), and aerial biomass.

1. Introduction

National forest inventories established in many countries represent the most extensive and reliable sources of data on forest status at national and regional levels [1,2]. These inventories employ various methods and definitions based on historical backgrounds and environmental conditions, leading to inconsistencies in international reporting [3]. The selection of appropriate plot shapes for forest inventory is a crucial consideration that can significantly impact the accuracy and efficiency of data collection [4,5]. Internationally, many countries implement their national forest inventories using sampling strategies that primarily employ three types of designs: systematic, stratified systematic, and stratified random [6,7,8].
For field data collection, these designs are often combined with two-stage cluster sampling, which involves Primary Sampling Units (PSUs) and Secondary Sampling Units (SSUs). PSUs can take various shapes, including square, rectangular, circular, cross-shaped, L-shaped, T-shaped, or inverted Y-shaped PSUs. SSUs, however, are typically circular, rectangular, or square [9,10]. One of the primary considerations in choosing a plot shape is the variance in the measured attributes, such as tree density, basal area, and timber volume [11,12]. However, the specific forest structure and terrain conditions can influence the relative performance of circular and square plots [13,14,15]. For instance, in areas with significant topographic variation or uneven tree distribution, square plots may be better suited to capture the heterogeneity of the forest stand [16].
In Mexico, The National Forest and Soil Inventory (INFyS), formally established in 2003 [9], conducts field data collection through a stratified systematic two-stage cluster sampling design. Sampling units are spaced at intervals of 5 × 5 km, 10 × 10 km, or 20 × 20 km, depending on the forest ecosystem and vegetation type. The Primary Sampling Unit, consistent with international forest resource assessment methodologies, is conceptually a one-hectare circular plot (with a 56.42 m radius). This plot contains four Secondary Sampling Units, or circular subplots of 400 m2, arranged in an inverted “Y” shape relative to the north [17]. SSU 1 is the cluster centre, while SSUs 2, 3, and 4 are peripheral, positioned 45.14 m from the centre at azimuths of 0°, 120°, and 240° [18]. This design closely resembles the spatial geometric configuration used in the Forest Inventory and Analysis (FIA) program in the United States [19,20].
Additionally, the efficiency of data collection is an important factor in plot shape selection. Circular plots can be more efficient to establish and measure, as they require the determination of a single radius, whereas square plots involve the measurement of four sides. However, this efficiency advantage may be offset by the challenges of navigating and accurately delineating the plot boundaries, particularly in dense or uneven forest stands [21]. Different sampling unit shapes, such as circular or rectangular plots, have been shown to affect the precision of biomass estimates and the assessment of forest volume and other critical variables [11,22,23,24,25]. This underscores the importance of carefully considering the sampling design in national forest inventory projects since previous inventories utilised various plot shapes and revealed significant discrepancies in biomass and volume calculations [26,27].
This study aims to investigate the impact of the Secondary Sampling Unit (SSU) shape on the estimation of key forest variables, including aboveground biomass and basal area, within the context of the National Forest and Soil Inventory (INFyS). We used a Multivariate Generalised Linear Mixed Model to assess how different sampling unit shapes influence the accuracy of the estimates. Generalised linear mixed models are a class of statistical models that can accommodate various distributions for the outcome variable, handle non-linear models, and account for correlated data [28]. These models are advantageous in forest inventory data, where the observations are often nested within larger spatial or temporal structures, such as plots, stands, or management units.
Therefore, the objective of this study was to evaluate the effect of circular and square subplots, concentrically overlapped and arranged in an inverted Y cluster structure, over tree density, basal area, and aboveground biomass in a managed temperate forest in central Mexico by using a Multivariate Generalised Linear Mixed Model, which considers the Gamma distribution of the variables and accounts for spatial correlation between Secondary Sampling Units nested within the Primary Sampling Unit.

2. Materials and Methods

2.1. Study Area

The study area is located in a forest areas in western Tlaxcala, Mexico, in the municipality of Nanacamilpa, within the region known as the Sierra Nevada. The Sierra Nevada is located at the confluence of the two biogeographic realms in central Mexico: the Nearctic and the Neotropical, which harbours highly diverse forests stands. The forests of Nanacamilpa and Tlaxcala are part of the Trans-Mexican Volcanic Belt and host a rich biodiversity. A study recorded 129 bird species in the area, with coniferous forests showing the highest species richness and diversity [29]. These forests are considered priority areas for bird conservation, especially for endemic or at-risk species, but face strong anthropogenic pressure. The dominant vegetation is mixed coniferous forests made up mainly of pine species (Pinus teocote, P. montezumae, P. pseudostrobus); between 2900 and 3400 masl, sacred fir (Abies religiosa) predominates, growing primarily in the canyons, while pines, oaks and cedars are dominant on the hillsides. There are relicts of madrone trees (Arbutus xalapensis) and alder (Alnus firmifolia) (Figure 1).

2.2. Forest Inventory Data

The conceptual framework of the research involved establishing 15 Primary Sampling Units (PSUs) across the study area (Figure 2). Within each PSU, four circular Secondary Sampling Units (SSUs) were positioned, following the cluster design of Mexico’s INFyS [18]. Due to the terrain’s topographical conditions, the SSU centres’ geolocation showed slight variation from the theoretical inverted Y design, resulting in a random pattern (Figure 2). Consequently, a strict two-stage cluster sampling design was generated. Within each 400 m2 circular SSU (11.28 m radius), another 400 m2 square SSU (20 m side length) was concentrically superimposed (Figure 2). In total, 120 SSUs were studied (60 circular and 60 square). According to the forest sampling literature, the PSU is also referred to as the “plot” or “cluster”, and the SSUs as “subplots”.
In each circular or square SSU, trees with a diameter at breast height greater than 7.5 cm were recorded, including scientific species names and total height.

2.3. Database Preparation

Diameter at breast height (DBH), tree height (H), and species name were extracted from each sampling plot and used for calculating individual tree aboveground biomass (AGB) for the species found in the study area. We used different allometric models for each species (Table 1). The allometric models use DBH (cm) and height (m) as independent explanatory variables to obtain total biomass per tree (Kg). These individual AGB data (trees) were added on a per-plot basis.

2.4. Statistical Analysis

In addition to conducting data quality control through exploratory statistical analysis, which considered the behaviour of dasometric variables (plausible values) and their relationships, correlations were obtained at the SSU level under the two conditions evaluated for the following attributes: density (DE trees), basal area (BA m2), and aboveground biomass (AGB kg). Using PROC SEVERITY from SAS® Version 9.4 [34], we determined the most appropriate probability distribution for the data. To select the probabilistic models that best represent the values of the variables under study, the following goodness of fit criteria were considered: Akaike Information Criterion (AIC), finite sample corrected Akaike Information Criterion (AICC), Schwarz’s Bayesian Information Criterion (BIC), Kolmogorov–Smirnov (KS), Anderson–Darling (AD), and Cramer–von Mises (CvM) [35] (Table 2). In general, the probabilistic model with the lowest value of the fit criterion was the best. Additionally, sample size, empirical distribution (symmetry), and type of forest variable (DE—discrete, BA—continuous, AGB—continuous) were considered [36,37,38,39,40].
To obtain desirable statistical properties for the residuals, zero mean and normal distribution, following the methodologies presented by [41,42], a Multivariate Generalised Linear Mixed Model (mGLMM) was proposed, allowing for the joint modelling of DE, BA, and AGB through identifying their probabilistic distribution and incorporating the cluster as a random effect to model the correlation between and within the cluster or PSU. The intra-PSU correlation was modelled using a First-Order Autoregressive Covariance Structure [AR(1)], while the correlation between SSUs was achieved through Compound Symmetry [43]; in both cases, parameters were estimated for each response variable of interest. Following [44], the mean of the model considered for the joint statistical was specified as
E Y i j l k b i k = g − 1 μ l k = g − 1 ( b 0 i k + β 1 k X l k )
where
Y i j l k   i s   t h e observation of the k -th the variable of interest recorded in the j -th SSU (secondary sample unit) of the i -th PSU (plot) of the l -th form of the SSU ( k = 1,2 , 3 ; i = 1,2 , … , 15 ; j = 1,2 , … , 4 ; l = 1,2 ) . Furthermore, Y i j l k ~ f μ l k , ϕ l k and n = 15 × 4 × 2 × 3 = 360 .
g is the monotonic differentiable link function and g − 1 (.) is its inverse.
b i k : is the vector of random effects normally distributed with zero mean and covariance matrix G k . The cluster was considered a random effect.
β 1 k is the coefficient that quantifies the fixed effect of the subplot type in interaction with the variable k .
X l k is the categorical variable related to the type of SSU (circular or square) for variable k .
The estimation of fixed and random parameters of the model was carried out using the PROC GLIMMIX procedure of SAS® statistical software Version 9.4 [43]. In PROC GLIMMIX, the random was the PSU. Additionally, through PROC CAPABILITY [45], the model fit quality was verified by jointly analysing Pearson residuals to determine their normal probability distribution and expected value. The criteria for evaluating normality were as follows: (1) statistical tests based on the Empirical Distribution Function and Theoretical Distribution Function (normal): Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling, and (2) graphical representation (histogram of residuals resembling a theoretical normal distribution).

3. Results

In total, 1833 trees were recorded, with diameters at breast height ranging from 7.5 cm to 110 cm and total heights varying from 4 m to 48 m. The main species studied were Pinus montezumae (moctezuma pine), Pinus teocote. (ocote), Abies religiosa (sacred fir), Alnus firmifolia (alder), Arbutus xalapensis (madrone), and Quercus laurina (oak). In general, total height as a function of diameter at breast height exhibited a non-linear, asymptotic pattern characteristic of most forest species (Figure 3).
When analysing all Secondary Sampling Units (SSUs) across different Primary Sampling Units (PSUs), it was observed that the highest densities were recorded in PSUs 10, 1, and 2. In contrast, the lowest densities were found in PSUs 6, 7, and 5. In comparison, the highest basal area values were observed in PSUs 11, 12, and 14, whereas the lowest were found in clusters 1, 7, and 6. Specifically, PSUs 12, 11, and 14 exhibited the highest basal area values, while cluster 1 showed the lowest. This pattern in the variables was consistent across both circular and square subplots. Overall, the differences in density, basal area, and aboveground biomass between the plot shapes were minimal (Figure 4).
These differences were practically imperceptible based on averages, standard deviations, and extreme order statistics, except for aboveground biomass (Table 3).
These differences are explained by the strong positive correlations (greater than 0.98) between the same variables when analysed using the two plot shapes, as indicated by the light horizontal shading in Table 3. In contrast, the relationships between different dasometric variables, whether within the same plot shape (light right oblique shading in Table 4) or across different shapes (light grey shading in Table 4), were generally weak. However, exceptions were noted for basal area and biomass, which showed stronger correlations in either the circular or square plot conditions or both, as highlighted by the light vertical shading in Table 4.
The high positive correlations between the same stand variables—when evaluated under both circular and square plot conditions—are represented by the lightly horizontally shaded cells in Table 4. These correlations are also evident in the corresponding scatter plots, where, in all three cases, the observations closely align along a 45° line passing through the origin (Figure 5).
Given the sample size (120 observations for each variable), the right-skewed empirical distribution, and the type of forest attributes (discrete or continuous), along with the small and similar fit criteria values, the Gamma distribution was identified as the most suitable model for representing the density, basal area, and biomass data when modelling these variables together (Table 5). Additionally, the Gamma distribution is part of the exponential family, making it well suited as the random component in Generalised Linear Models [28]. The Gamma distribution and other distributions derived from it have numerous applications in situations involving continuous random variables [46].
The analysis revealed a notable proximity in the values of the fit criteria, particularly AIC and BIC. It was consistently observed that the predicted values derived from the evaluated probabilistic models for the three variables remained within the 95% confidence limits of the empirical cumulative distribution function, as depicted in Figure 6. Notably, the Gamma distribution demonstrated strong performance across all three scenarios. Indeed, the efficacy of the Gamma distribution as a suitable probabilistic model for modelling forest attributes has been recognised in previous studies, further supported by its commendable results in this analysis [35,47,48].
The joint statistical analysis (based on multivariate response distribution, logarithmic link, residual pseudo-likelihood estimation technique, and dual Quasi-Newton optimisation) highlighted a significant interaction between subplot type and variable at a significance level of 0.05 (F = 683.95, p < 0.0001). This indicates that the effect of the SSU varies across different combinations of subplot types with each variable.
Table 6 details the calculated means for each variable, derived from the estimated model coefficients (both fixed and random), along with their corresponding 95% confidence intervals. Across all three instances, a remarkable similarity is evident in both the point estimates and intervals. However, it is worth noting that for biomass, the circular subplot yielded slightly higher estimates compared to the square subplot.
Notably, the multiple means comparison analysis conducted on the link function scale using Tukey–Kramer revealed, at a significance level of 0.05, that there are no significant differences between the means of the three variables when recorded in both circular and square subplots (p-values > 0.05). This implies that the mean density, basal area, and biomass observed in circular SSUs do not exhibit different statistically significant values from those in square SSU, both structured within a cluster.
Moreover, the confidence intervals for the differences (mean of the same variable recorded in two SSUs of different shapes) in all three scenarios encompass the zero value, as illustrated in Table 7. Conversely, contrasting outcomes were observed when other variables were recorded in distinct or identical SSUs. This discrepancy was anticipated and elucidates why the joint global F-test identified an impact of the subplot shape in conjunction with the variable.
Consequently, across all three scenarios, the confidence intervals of the predicted means (at 95%) for the same variable measured in two SSUs of varying shapes are virtually indistinguishable, as depicted in Figure 7. This suggests that while there is a theoretical preference for population units to be square, in practical terms, forest inventories utilising circular sampling units can prove valuable without introducing statistical implications that would jeopardise the estimation of crucial forest parameters.
The benefit of utilising multivariate statistical analysis lies in its ability to collectively consider all observations of the three variables concurrently, leading to a significant enhancement in the distribution of residuals compared to univariate analysis. Consequently, based on the KS, CvM, and AD tests, it was determined that the Pearson residuals adhered to a normal distribution, with corresponding p-values of 0.150, 0.250, and 0.250. This normal distribution exhibited a mean of zero and a standard deviation of one, as illustrated in Figure 8.
Furthermore, owing to the comprehensive joint statistical analysis conducted, the Pearson residuals’ empirical percentiles and the normal distribution’s modelled percentiles were found to align precisely along a straight line passing through the origin (Figure 9).

4. Discussion

Forest inventory data analysis has traditionally been conducted within the framework of design-based inference, where population values are treated as fixed constants, and the randomisation distribution resulting from the sampling design is the basis of inference [49]. However, recent advancements in statistical modelling have introduced generalised linear mixed models as a powerful tool for analysing forest inventory data [50]. In this study, we aimed to evaluate the impact of sampling unit shape on three conventional forest variables. In Mexico, both the National Forest and Soils Inventory (INFyS) and forest management program predominantly utilise circular sampling units. Thus, this research focused on comparing circular subplots to square ones. We used a Multivariate Generalised Linear Mixed Model, which considers the Gamma distribution of the variables and accounts for the spatial correlation between Secondary Sampling Units nested within the Primary Sampling Unit.
Our study presents a detailed and systematic comparison between circular and square plots for estimating key forest metrics such as tree density, basal area, and aboveground biomass. This evaluation provides valuable insights into whether the shape of Secondary Sampling Units significantly impacts the accuracy of forest inventory estimations, which is crucial for national and regional inventory methods. Our study effectively corroborates its findings with prior studies [11,21] and strengthens the argument that plot shape does not significantly affect the accuracy of estimates if areas are equivalent.
Using a multivariate approach allowed us to simultaneously assess multiple forest variables, capturing their complex relationships and interdependencies. The generalised linear mixed model framework used here accounted for the hierarchical structure of the data, including both fixed effects (e.g., plot shape) and random effects (e.g., PSU for modelling the correlation between SSU), thus providing a more robust and comprehensive analysis. Prior research suggests that bias in forest inventory estimates can stem from the choice of plot design and its placement, particularly in uneven terrains where specific shapes may enhance the accuracy of tree inclusion criteria, thereby affecting overall biomass and volume assessments [8,12,24].
The methodology followed in this study deepens our understanding of how plot shape influences estimations and addresses potential biases that may arise, especially in heterogeneous landscapes where traditional sampling methods may falter. This reinforces the need for an optimised sampling unit design in national forest inventories [24,51,52,53]. To explore these dynamics, we compared the performance of circular versus square plots in accurately representing forest aboveground biomass, basal area, and density, as previous studies have demonstrated that plot shape can significantly influence inventory outcomes, particularly in areas with uneven tree distributions or undulating terrain [24], providing crucial data for sustainable forest management and carbon accounting.
However, the shape of the sampling units can significantly influence the accuracy of estimations, such as forest biomass and volume [5,54], as variations in design can lead to differences in the representation of forest structure and biomass density across diverse ecosystems [26,53]. In this study, we did not find significant differences in the mean values of forest variables such as tree density (trees), basal area (m2), and aboveground biomass (kg). The outcomes of this study are consistent with prior research that has explored the comparison of different sampling unit shapes in the context of forest inventories [11,55].
For instance, aboveground biomass modelled utilising 100,000 repetitions, drawing from field data gathered in Finland, indicated the absence of systematic prediction errors when fitting a model using simple circular plots and subsequently making predictions with square plots, and vice versa. This suggests that the estimation of total or mean biomass remains equally dependable regardless of whether circular or square plots are employed, provided that the sampling unit areas are equivalent—such as the standardised area of 198.53 m2 utilised in this study (with a circular plot radius of 7.95 m and a square plot side length of 14.09 m) [22], which agrees with our findings.
Another similar study [56] compared forest variables such as density, basal area, and timber volume stocks in Nepal using 15 rectangular plots, 15 square plots, and 15 circular plots, each with an area of 500 m2. In this study, the one-way analysis of variance revealed no significant differences in these variables between plot shapes at a 5% significance level. The study also evaluated the efficiency of different methods regarding time required and sampling error. Circular plots proved to be the most time efficient, with the shortest delineation and measurement times, while rectangular plots took the longest. The authors concluded that circular plots offer greater efficiency for forest inventory due to their precision and time-saving advantages in Nepal.
Flores and Flores [57] analysed the effects of 400 m linear transects and circular plots of 400, 300, and 150 m2, using 400 m2 square plots as a reference, on the basal area and density of three mangrove species—Rhizophora mangle (RM), Laguncularia racemosa (LR), and Avicennia germinans (AG)—in California, Mexico. The study found no significant differences in basal area or density across the sampling methods for RM. However, for LR (diameter > 2.5 cm), there were statistical differences in basal area among methods, but post hoc analysis showed no differences between the 400 and 300 m2 circular plots. All methods, however, did show density differences for this species. In the case of AG, significant differences were observed for both variables, although further analysis indicated no differences between 400 m2 circular and square plots.
It is worth noting that this study was carried out under certain conditions valid for temperate forest of the central part of Mexico. This may limit the generalizability of the results to other forest ecosystems with different topographies, climates, or species compositions. Forest inventory dynamics may vary significantly between temperate and tropical environments, leading to potential biases when applying the findings to other regions. Also, although the terrain variations within the sampling plots were considered, their impact was not explicitly modelled beyond descriptive mentions of topographical differences affecting the layout. This could lead to inconsistencies when extrapolating the outcomes to more heterogeneous or complex landscapes where accessibility and topography could heavily influence biomass and basal area estimations and heterogeneity of the forest stand [16].
Aside from the results reported here, the advantages of circular sampling units—such as ease of establishment from a single central point, straightforward verification of trees within the plot, and the inclusion of fewer boundary trees compared to other shapes of the same area—have made them a common choice in forest inventories at both national and property levels [58]. When using circular plots, the tree inclusion zone is a circle centred on the tree itself, with the same radius as the plot [59]. As a result, forest parameter estimates from circular plots are unlikely to be biased, whereas those from square plots may show positive bias [24]. However, in remote sensing-assisted inventories, a mismatch can occur if circular plots are used to fit regression models while predictions are made using square tessellated population elements. This discrepancy arises because square units cover the entire area of interest without gaps, which differs from the characteristics of circular plots.
The findings of this study will contribute to ongoing discussions surrounding the optimal sampling design for national and local forest inventories, providing valuable insights for policymakers, forest managers, and researchers involved in forest resource assessment, management, and monitoring. The implementation of advanced methodologies, such as lidar sampling frameworks, may also offer innovative solutions to improve the accuracy of forest characterisation, particularly in contexts where conventional plot designs may underrepresent variability in tree distribution due to their shape [60,61]. Incorporating lidar-based techniques can enhance the efficiency and precision of forest inventories, enabling the collection of data that better reflects the complexities of forest structure and biomass distribution. This approach addresses the limitations of traditional sampling designs and ensures that forest management practices are grounded in reliable, comprehensive information that captures the full extent of ecosystem variability [24,61].
Ultimately, our study underscores the necessity for forest professionals to carefully consider sampling unit shapes, as these choices have far-reaching implications beyond statistical accuracy, impacting the management decisions that depend on precise forest assessments. Adopting innovative methodologies can lead to more robust and sustainable forest management strategies. In conclusion, the interplay between sampling unit shape and forest parameters estimation is a critical factor that deserves further exploration, as it affects data accuracy and our broader understanding of ecological systems and resource management in forestry. Refining sampling methodologies through multivariate statistical modelling and technological innovation is essential to ensuring the long-term sustainability of our forest ecosystems.
Finally, it is worth noting that integrating forest mapping and inventory data can further enhance the utility of generalised linear mixed models in forest management. Incorporating multisource data into the models, such as remote sensing and GIS information, allows for the exploration of the relationship between forest attributes and environmental or landscape-level covariates—a promising research area for future investigations.

5. Conclusions

From all tree’s species analysed, total height as a function of diameter at breast height exhibited a non-linear pattern, which is an asymptotic pattern characteristic of most forest species.
The joint statistical analysis highlighted a significant interaction between subplot type and variable at a significance level of 0.05 (F = 683.95, p < 0.0001). This indicates that the effect of the SSU varies across different combinations of subplot types with each variable. However, when analysing all Secondary Sampling Units (SSU’s), the difference in density, basal area, and aboveground biomass between the subplot shapes were minimal.
The study results indicate that the choice of Secondary Sampling Unit (SSU), whether circular or square and of equal area, does not significantly influence the mean values of forest variables such as tree density (trees), basal area (m2), and aboveground biomass (kg).
Our findings suggest that the conceptual discrepancy between these two types of units, assuming the same centroid, should not raise concerns regarding differences in the mean values of variables like basal area. Basal area is closely linked to economically significant forest parameters such as timber volume and environmental factors like aerial biomass.
The statistical methodology employed in this study, based on a Multivariate Generalised Linear Mixed Model, is deemed robust. Not only did it yield robust p-values, but the distribution of the residuals was also found to be normal, further underscoring the reliability of the analysis.

Author Contributions

Conceptualisation, E.V.-B., M.E.R.-S. and V.G.-D.L.C.; methodology, E.V.-B., A.G.-H. and M.E.R.-S.; formal analysis, E.V.-B.; investigation, E.V.-B., M.E.R.-S. and A.G.-H.; writing—original draft preparation, E.V.-B. and M.E.R.-S.; writing—review and editing E.V.-B., M.E.R.-S., A.G.-H., V.G.-D.L.C. and R.P.-M. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Institute of Forestry, Agriculture and Livestock Research of Mexico through the project “Explicit spatial estimation of forest aboveground biomass under different remote sensing approaches in the Sierra Nevada, Tlaxcala, Mexico” grant number 12251135086.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

The authors want to acknowledge Nicolas Abad-Zabaleta and Gil Espinoza-Lopez for their technical assistance during the field campaign and Ejido San Jose Nanacamilpa, Nanacamilpa, Tlaxcala, Mexico for their support during the development of this research. The authors are deeply grateful for comments and suggestions during the review process which were helpful in improving the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the study’s design; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

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Figure 1. Localisation of the study area.
Figure 1. Localisation of the study area.
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Figure 2. Localisation of the sampling plots and sampling design.
Figure 2. Localisation of the sampling plots and sampling design.
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Figure 3. Total height and diameter at breast height for all species recorded (blue dots: sacred fir, red dots: moctezuma pine; green dot: oaks, purple dots: ocote pine, brown dots: other species).
Figure 3. Total height and diameter at breast height for all species recorded (blue dots: sacred fir, red dots: moctezuma pine; green dot: oaks, purple dots: ocote pine, brown dots: other species).
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Figure 4. Density (# tress/SSU), basal area (m2/SSU), and aboveground biomass (kg/SSU) in circular and square plots.
Figure 4. Density (# tress/SSU), basal area (m2/SSU), and aboveground biomass (kg/SSU) in circular and square plots.
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Figure 5. Scatter plot for density (r = 0.9961), basal area (r = 0.9875), and aboveground biomass (r = 0.9811) from square and circular subplots (r = 0.9811). The colours identify the SSU for SPU (cluster).
Figure 5. Scatter plot for density (r = 0.9961), basal area (r = 0.9875), and aboveground biomass (r = 0.9811) from square and circular subplots (r = 0.9811). The colours identify the SSU for SPU (cluster).
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Figure 6. Empirical cumulative distribution function (blue) and modelled (Gamma, green; Burr, magenta; Weibull, pink) for density, (Gamma, green; Lognormal, red; Burr, magenta) for basal area and (Gamma, green; Weibull, red; Burr, magenta) for aboveground biomass. FDA: cumulative distribution function.
Figure 6. Empirical cumulative distribution function (blue) and modelled (Gamma, green; Burr, magenta; Weibull, pink) for density, (Gamma, green; Lognormal, red; Burr, magenta) for basal area and (Gamma, green; Weibull, red; Burr, magenta) for aboveground biomass. FDA: cumulative distribution function.
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Figure 7. Mean and confidence intervals for aboveground biomass, density, and basal area resulting from the joint statistical model.
Figure 7. Mean and confidence intervals for aboveground biomass, density, and basal area resulting from the joint statistical model.
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Figure 8. Histogram and normal distribution of residuals of multivariate statistical analysis of density, basal area, and aboveground biomass.
Figure 8. Histogram and normal distribution of residuals of multivariate statistical analysis of density, basal area, and aboveground biomass.
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Figure 9. Observed and normal percentiles of Pearson’s residuals.
Figure 9. Observed and normal percentiles of Pearson’s residuals.
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Table 1. Allometric models used in this study.
Table 1. Allometric models used in this study.
Allometric ModelSpeciesSource
A G B = 0.0713 ∗ D B H 2.5104 Abies religiosa[30]
A G B = 0.0130 ∗ D B H 3.0464 Pinus montezumae[31]
A G B = 0.0345 ∗ D B H 2.9334 Quercus sp.[32]
A G B = 0.3764 ∗ D B H 2 − ( 2.3146 ∗ D B H ) − 1.9106 : Arbutus xalapensis and Alnus firmifolia[33]
Table 2. Description of the statistical criteria for model selection.
Table 2. Description of the statistical criteria for model selection.
CriteriaDefinition
Akaike Information Criterion (AIC)AIC is a measure of the relative quality of a statistical model for a given set of data. AIC helps in comparing multiple models, selecting the one that minimises information loss. A lower AIC value indicates a better model.
Finite Sample Corrected Akaike Information Criterion (AICc)AICc is a modified version of AIC, adjusted for small sample sizes. It corrects the bias that arises when the number of parameters is close to the sample size.
Schwarz’s Bayesian Information Criterion (BIC)BIC is similar to AIC but introduces a stronger penalty for models with more parameters, making it more conservative.
Kolmogorov–Smirnov (KS) TestThe KS test is a non-parametric test used to compare a sample with a reference probability distribution, or to compare two samples. It measures the largest difference between the empirical distribution function of the sample and the cumulative distribution function of the reference.
Anderson–Darling (AD) TestThe AD test is used to check if a sample comes from a specific distribution. It gives more weight to the tails compared to the KS test, making it more sensitive to deviations in these areas.
Cramer–von Mises (CvM) TestThe CyM test determines if a sample comes from a specified distribution. It is based on the integrated squared difference between the empirical and the theoretical cumulative distribution functions.
Table 3. Simple statistics for density (number of trees), basal area (m2), and aboveground biomass (kg) under two study conditions.
Table 3. Simple statistics for density (number of trees), basal area (m2), and aboveground biomass (kg) under two study conditions.
Sampling Unit TypeVariableMeanStandard DeviationMinimumMaximum
CircularDE15.339.47242
CircularBA1.190.660.153.19
CircularAGB99765813434.7227,854
SquareDE15.179.82141
SquareBA1.150.670.153.19
SquareAGB95095798434.7226,772
Table 4. Pearson correlation coefficients between the same variables under the two study conditions and between different variables in the same or different subplot shapes.
Table 4. Pearson correlation coefficients between the same variables under the two study conditions and between different variables in the same or different subplot shapes.
SSU/VariableCircularCircularCircularSquareSquareSquare
DEBAAGBDEBAAGB
CircularDE1.0000.3866 (0.0023)0.2617 (0.0434)0.9961 (<0.0001)0.4246 (0.0007)0.3141 (0.0145)
CircularBA0.3866 (0.0023)1.0000.8584 (<0.0001)0.3904 (0.0020)0.9875 (<0.001)0.8754 (<0.0001)
CircularAGB0.2617 (0.0434)0.8584
(<0.0001)		1.0000.2510 (0.0530)0.8199 (<0.0001)0.9811
(<0.0001)
SquareDE0.9961 (<0.0001)0.3904
(0.0020)		0.2510 (0.0530)1.0000.4383 (0.0005)0.3148 (0.0143)
SquareBA0.4246 (0.0007)0.9875 (<0.0001)0.8199 (<0.0001)0.4383 (0.0005)1.0000.8650 (<0.0001)
SquareAGB0.3141 (0.0145)0.8754
(<0.0001)		0.9811 (<0.0001)0.3148 (0.0143)0.8650 (<0.0001)1.000
Table 5. Fit criteria for evaluating the probabilistic distribution for each variable.
Table 5. Fit criteria for evaluating the probabilistic distribution for each variable.
VariableDistributionAICAICCBICKSADCvM
DensityWeibull860.25 *860.34 *865.82 *1.13 *2.40 *0.24
Gamma861.53861.64867.111.312.440.23 *
Burr862.25862.45870.611.132.400.24
Basal areaGamma211.15 *211.25 *216.73 *0.970.790.15
Burr212.71212.92221.070.720.44 *0.07
Lognormal213.05213.15218.630.64 *0.530.07 *
Aboveground biomassGamma2393 *2393 *2399 *1.27 *1.23 *0.21 *
Burr2396239624021.311.290.23
Weibull2396239624041.421.680.30
* Model selected based on the smallest criterion value.
Table 6. Means and confidence limits (original units, at 95%) of studied variables obtained from the joint analysis.
Table 6. Means and confidence limits (original units, at 95%) of studied variables obtained from the joint analysis.
SSU TypeVariableMeanStandard ErrorLower LimitUpper Limit
CircularDE18.1612.864.5173.17
CircularBA1.110.270.681.81
CircularAGB9250.071355.566932.9712,342.00
SquareDE17.4412.354.3370.28
SquareBA1.050.260.651.72
SquareAGB8689.631273.536512.9211,594.00
Table 7. Estimation of mean differences (link function scale) for the interaction between subplot and variable, adjusted by Tukey–Kramer.
Table 7. Estimation of mean differences (link function scale) for the interaction between subplot and variable, adjusted by Tukey–Kramer.
SSUVariableSSUVariableEstimationStandard Errort-Valuep-Value Lower Limit Upper Limit
CircularBACircularAGB−9.03140.2879−31.37<0.0001−9.8570−8.2058
CircularBACircularDE−2.79820.7504−3.730.0031−4.9500−0.6465
CircularBASquareBA0.04900.05510.890.9489 *−0.10910.2072
CircularBASquareAGB−8.96890.2879−31.15<0.0001−9.7945−8.1433
CircularBASquareDE−2.75790.7504−3.680.0038−4.9097−0.6062
CircularAGBCircularDE6.23320.72338.62<0.00014.15928.3072
CircularAGBSquareBA9.08040.287931.54<0.00018.25489.9060
CircularAGBSquareAGB0.06250.06930.900.9457 *−0.13610.2611
CircularAGBSquareDE6.27350.72338.67<0.00014.19958.3475
CircularDESquareBA2.84730.75043.790.00240.69564.9990
CircularDESquareAGB−6.17070.7233−8.53<0.0001−8.2447−4.0967
CircularDESquareDE0.04030.06580.610.9901 *−0.14830.2289
SquareBASquareAGB−9.01790.2879−31.32<0.0001−9.8435−8.1923
SquareBASquareDE−2.80700.7504−3.740.0030−4.9587−0.6553
SquareAGBSquareDE6.21100.72338.59<0.00014.13698.2850
Degrees of freedom = 312 and alpha = 0.05. * Significant differences with alpha = 0.05.
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Velasco-Bautista, E.; Gonzalez-Hernandez, A.; Romero-Sanchez, M.E.; Guerra-De La Cruz, V.; Perez-Miranda, R. Does It Matter Whether to Use Circular or Square Plots in Forest Inventories? A Multivariate Comparison. Forests 2024, 15, 1847. https://doi.org/10.3390/f15111847

AMA Style

Velasco-Bautista E, Gonzalez-Hernandez A, Romero-Sanchez ME, Guerra-De La Cruz V, Perez-Miranda R. Does It Matter Whether to Use Circular or Square Plots in Forest Inventories? A Multivariate Comparison. Forests. 2024; 15(11):1847. https://doi.org/10.3390/f15111847

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Velasco-Bautista, Efrain, Antonio Gonzalez-Hernandez, Martin Enrique Romero-Sanchez, Vidal Guerra-De La Cruz, and Ramiro Perez-Miranda. 2024. "Does It Matter Whether to Use Circular or Square Plots in Forest Inventories? A Multivariate Comparison" Forests 15, no. 11: 1847. https://doi.org/10.3390/f15111847

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