Efficiency of Data Clustering for Stratification and Sampling in the Two-Phase ALS-Enhanced Forest Stock Inventory

Highlights

What are the main findings?

• Ward’s clustering was advantageous over other data clustering methods.
• Inconsiderable reduction in RMSE observed above around 200 sample plots.
• Complex stands benefit more from increased sample size than homogeneous ones.

What are the implications of the main findings?

• Data clustering methods can aid more optimal forest inventory stratification.
• Structurally guided sampling can be effectively performed with the data clustering.

Abstract

Within the last few decades, ALS-enhanced two-phase forest inventory has emerged as viable alternative to standard inventory designs. As a relatively new and compound method, there still remains significant potential for its optimisation. One key aspect concerns the design of the second-phase sampling. Apart from well-known designs such as random, systematic, or stratified sampling—which often involve some degree of uncertainty regarding their realisations—there are relatively less common, structurally guided sampling designs (SGS), which can facilitate the unambiguous allocation of balanced and well-optimised samples. Unlike traditional stratification, the SGS design does not rely on fixed divisions, which may induce additional errors due to pre-defined and potentially non-representative strata. Instead of geographical (spatial) sample deployment, the SGS uses the multidimensional space of covariates, e.g., ALS metrics, to optimise sample allocation. SGS can be powered by different engines. While some algorithms for SGS, such as the cube method or local pivotal method, have been briefly tested in recent studies, no thorough attention has yet been paid to data clustering algorithms. Therefore, this study compares the performance of several popular data clustering algorithms for structurally guided sampling to train the model for growing stock volume estimation in a two-phase ALS-enhanced forest inventory design. The results showed that hierarchical clustering was competitive with other methods but outperformed them in terms of the highest stability of estimates, even at lower sampling intensity levels. The use of data clustering methods can ensure unambiguous yet more optimal sample distribution, minimising sampling variation or estimation error caused by the randomness of other sampling methods or the inflexibility of pre-defined strata.

1. Introduction

Accurate information about forest resources is becoming increasingly relevant as pressure on forest environments grows to sustainably provide multiple services related to, among others, climate change mitigation, the global carbon cycle, natural environment preservation, renewable wood production, sustainable forest management, biodiversity restoration, landscape management, social services, and tourism [1,2,3,4,5,6,7]. This issue is becoming more pressing as forest areas are generally shrinking and are increasingly isolated into smaller patches [8,9,10]. Another key factor justifying the need for accurate environmental information is the range of policies that use growing stock volume (GSV) or biomass data for carbon budget management [11,12,13,14,15]. In such circumstances, the margin for forest management error should be minimised. Therefore, the demand for precise information about forest resources has become apparent and pronounced by numerous studies.

As single-tree inventories are not yet practically operational at larger scales and satellite remote sensing (RS) solutions may yield some local bias when applied to small areas without in situ calibration data, traditional ground-based sampling remains the standard in most forest censuses at various scales [16,17,18]. Among the many sampling designs, different realisations of systematic sampling (SYS) are by far the most commonly applied in practical forest inventories [16,19,20,21,22,23,24]. The key benefits of this approach are reported as follows: (1) economical and efficient as compared to random sampling, (2) capable of providing uniform coverage of the entire sampling population, (3) relatively simple in terms of sample deployment and operational field logistics, and (4) not prone to plot clumping [25,26,27]. Despite these advantages, systematic sampling may be challenging due to following reasons: (1) substantial bias and/or non-representative samples if the population exhibits strong patterns with spatial autocorrelation, (2) lack of a design-based variance estimator, causing analysts to often resort to simple random sampling (SRS) formulas for uncertainty approximation, which can inflate estimation intervals, (3) less true randomness, meaning not every possible sample configuration is equally likely, (4) inflexibility in the sampling interval, and (5) omission of rare specimens or clusters often individually scattered over the sampling area [22,28,29,30,31,32,33].

Alternative sampling methods are noticeably less common in forest inventory practice. Although considered unbiased and straightforward, simple random sampling (SRS) results in irregular plot distribution and related logistical and ergonomic issues—the aspects especially pertinent in difficult forest terrain. Moreover, this approach requires a substantial sample size to minimise random effects and ensure the stability of results [33,34]. Cluster sampling (CS) addresses some of the aforementioned issues of SRS and is therefore often combined with systematic sampling (SYS), particularly for large and dispersed populations, as in the case of many national forest inventories (NFI), e.g., Poland [35,36], Finland [37], Germany [38,39], USA [40], and Romania [41]. On the other hand, CS still has notable potential for optimisation in forest inventory. Key areas for optimisation include the following: reduction in the clustering effect, i.e., minimisation of the standard error and inherent uncertainty [42,43]; cluster definition and their universality [44,45]; numerical optimisation with spatial information [46]; and improvements in forest attributes estimation by integration with other techniques [47]. Furthermore, CS poses a similar risk of systematic errors as stratified sampling (STRS) when clusters or strata are poorly defined and non-representative of the population, which often occurs when no a priori knowledge is available about the target area [24,33,48,49].

This is where remote sensing (RS)-enhanced methods often come in handy. RS data are known to aid contemporary forest inventories in several ways: enabling continuous spatial coverage of the entire area of interest (AOI), providing initial knowledge about the sampling population [50,51], increasing the precision of stand-level attributes estimation [52], reducing sample size [50,53,54,55,56], and enabling objective monitoring of structurally complex and hardly accessible sites [57,58]. The two-phase sampling design with an RS regression estimator (often referred to as the area-based approach) [59] is of particular interest to forest inventory data analysts, researchers, and an increasing number of practitioners. The suitability of this method for forest inventory purposes, especially in connection with airborne laser scanning (ALS), have been demonstrated by many studies [60,61,62,63]. Already, for over a dozen years, RS-based methods have been under development, and consequently adopted to aid practical forestry [64,65,66,67,68,69,70]. In short, the core aspect of this method relies on the relationship between RS metrics (first-phase plots) and ground (second-phase) plots, on which variables of interest are measured, usually as calibration/control data for some kind of model that serves as an estimator for the whole sampling area [71,72].

As previously mentioned, in practice, ground plots are usually sampled using a systematic rather than a random approach, where general sample size is often determined by formulas in which one term represents the minimal/base sample size (either a fixed number, area, or allowable error-dependent quantity), and the other term reflects some measure of variance in the population [25,42,70,73,74,75,76,77]. Nowadays, thanks to the application of RS data, it is practically feasible to assess stand diversity and variance within the target population in advance, which can help stratify the area of interest before ground measurements and approximate optimal sample size [54,78,79]. As determining the required number of plots based on the desired sampling error and/or population variability seems logical and well justified [80], some research challenges methods which incorporate sampling intensity (e.g., surface area) [81,82], instead seeking to establish sampling thresholds for specific forest regions [55,83,84,85]. Moreover, aside from the previously mentioned caveats regarding sample plot deployment strategies, the grid origin in practical systematic sampling is often a single arbitrary point (usually some AOI corner) [19,70], which when appointed may influence inventory results if not randomly selected or when the resulting sample is non-representative [42,86,87]. This highlights the need to address the common pitfalls of systematic sampling, for instance, by using RS auxiliary data.

Similarly to SYS, CS is often used, especially in NFI-sampling designs. While much research focuses on the spatial (geographical) optimisation of the cluster/plot distribution [44,46,88,89,90,91,92,93,94], less attention is given to sampling in the feature domain consisting of auxiliary variables (e.g., RS/ALS data) [54,95,96,97,98,99]. Perhaps, instead of seeking an optimal spatial pattern for sample plot distribution, which, if not random or representative, may yield biased results [100,101], investigating the multidimensional space of independent variables may offer an alternative perspective on how to view modern forest surveys. Bearing in mind the above caveats, the use of SYS or CS for second-phase sample distribution may appear as a hindrance to revealing the full potential of RS-enhanced forest inventories (EFI). Therefore, the aspects of second-phase sample plot distribution that consider sample size determination and its spread in both spatial and feature domains could still be optimised, especially as some promising results in this area have been reported by Junttila et al. (2013) [21] and more recently by Heikkinen et al. (2025) [102].

To reduce the variance and uncertainty of an estimator, as well as the excessive networking associated with SYS/CS, some researchers propose RS data-driven stratification for sample plot selection. These attempts are collectively referred to as structurally guided sampling (SGS) [103]. In model-based inventory design, this approach can improve the accuracy and precision of estimates [53,104,105,106,107]. One of the most important assumptions in this design is that adequate strata representing the feature space variability of the target population are objectively composed. Dividing the inventory area into homogeneous patches, based on accurate and up-to-date ALS metrics (which can also serve as predictors of forest attributes), can help capture the variability in the target area, while maintaining the sample size at an optimal level, as some portion of variance may already be explained by the strata division [54,98]. Moreover, such stratification can guide representative and more precise sample distribution, entailing all related benefits [104,108,109,110]. Some techniques for stratification and sampling in auxiliary data space have already been tested by Hawbaker et al. (2009) [111], who used a fixed number of strata (30), to delineate forest patches homogeneous in relation to ALS-height metrics, showing substantial improvements over SRS. Similarly, Gobakken et al. (2012) [53] used an arbitrary ALS-assisted stratification (8 strata) for sample plot designation in Scotch pine/Norway spruce-dominated stands, which allowed for RMSE and sample size reduction. Melville et al. (2015) [109] employed balanced sampling (the cube method [112]) as an effective alternative to stratification. A year later, they combined relatively simple nearest centroid (NC) sampling with k-means stratification for both small area estimations and totals, reporting a 50% gain over SRS [109]. Another SGS technique was adopted by Grafström et al. (2014) [79], who introduced the local pivotal method (LPM), finding its performance competitive with SRS. Nevertheless, hitherto, only few studies have covered the application of data clustering algorithms for sampling and/or strata designation [98]. In most cases, only the popular k-means approach was studied [109], though not always explicitly concerning forest stock inventories [113]. More recent insights into other clustering techniques were obtained by Xu et al. (2024) [99], who used Ward’s hierarchical clustering to improve inter-strata heterogeneity in coniferous sites and reduce sample size by about 20%. This promising results and peculiar niche of the subject should encourage further and more comprehensive studies.

Data clustering methods can be distinguished based on the fundamentals behind the data discretization, i.e., partitioning, hierarchical, and density-based approaches, and/or the type of data they handle, i.e., numerical, categorical, ordinal, or mixed [114]. Nonetheless, the principal goal of all these methods is to split the full dataset into groups of observations characterised by similar variance, while maximising the variance between the groups [115]. These procedures are often performed in a multidimensional space consisting of independent or auxiliary variables (covariates), where the number of clusters is often the sole or the most significant hyperparameter to set [98,116]. There are no standardised guidelines regarding the use of the most optimal method [117]. The debate about the suitability of particular methods is lively among data scientists, with various conclusions, greatly depending on the research field, goal, and data specification [98,116]. The key aspects in this regard usually concern the following: appropriate agglomeration algorithm selection, coupled with the appropriate number of clusters and clustering variables [98,116,118]; the type and scale of heterogeneity in the population [45]; branch specification and thus specific data characteristics and complexity [119,120]; the goal of the data analysis [121]; the amount and type of inherent noise and the presence of outliers [122,123]; results consistency [124,125]; and computational efficiency [117,126]. The variety of subjects concerning data clustering algorithms motivated us to draw a summary table (

Table 1. A brief characteristics of selected data clustering methods.

Algorithm	Common Usage	Pros	Cons
K-means (1)	Partitioning data into k clusters with roughly spherical, equally sized clusters. Market segmentation, image compression, etc.	Simple, intuitive, fast for large datasets Easy to implement and scale Deterministic once initialization fixed Cluster centroid interpretable	Must choose k ahead of time Assumes spherical clusters Sensitive to initialization and outliers Hard clustering only
Hierarchical Clustering (2) (Agglomerative/ Divisive)	Exploratory data analysis for hierarchical structures; used in biology, taxonomy, document clustering, etc.	No need to pre-specify number of clusters Provides nested clusters via dendrogram Flexible with linkage criteria (single, complete, average, Ward’s)	Computationally expensive for large datasets Sensitive to linkage and distance metric choice Irreversible decisions Less useful for non-hierarchical clusters
HDBSCAN (3) (Hierarchical Density-Based Spatial Clustering of Applications with Noise)	Clustering with noise, uneven densities, irregular shapes; used in spatial, astrophysics, geospatial data, etc.	Finds clusters of varying density and shape Handles noise well No need for epsilon parameter (unlike DBSCAN) Produces hierarchical structure with flat clustering	More computationally intensive Some points considered noise (not clustered) Parameter tuning needed Less effective in very high dimensions
IDS (Individual Dimension Sampling)	Original authors’ algorithm tested in this study

⁽¹⁾ [127,128,129]; ⁽²⁾ [130,131,132]; ⁽³⁾ [133,134,135].

) that presents a brief comparison of selected data clustering methods evaluated in this study.

As shown in the previous paragraphs, there is still some progress to be made in the context of practical optimisation of forest inventory campaigns. As concluded by Tompalski et al. 2019 [136], the details concerning the required sample size for model calibration still remain to be determined. The range of possible solutions presented, along with their potential and inherent uncertainty, create the ground for the analysis concerning their aptitude in the reciprocal aspects of desired accuracy, precision, available resources, sampling scheme, sample size, and plot distribution. Clearly, the issues in question are complex, thus addressing them all within the scope of a single study may prove challenging. Therefore, the aim of this study was to compare the performance of popular data clustering methods, fed with multiple ALS metrics, in a two-phase forest inventory design, driven by the sample size and the diversity of the population. The authors sought to address the following questions: (1) How can data clustering methods aid optimal stratification for GSV estimation in a two-phase ALS sampling design? (2) What is the influence of sample size in this context? (3) Do the results of particular clustering methods vary with the level of stand complexity? (4) What is the sensitivity or stability of particular methods? (5) How do the scores compare with other, more traditional sampling designs?

2. Methods

2.1. Input Data

The reference data come from 5870 sample plots located in 10 forest districts in Poland: Białowieża, Supraśl, Milicz, Katrynka, Herby, Gorlice, Pieńsk (all measured in 2015), Taczanów, Głogów Małopolski (measured in 2020), and Leżajsk (measured in 2021). Ground surveys were conducted in compliance with Polish standard forest management inventory procedures established in IUL 2012 [137]. Tree species, age, height, and breast height diameters were recorded for every tree exceeding 7 cm at DBH, within 500 m² circular sample plots, systematically distributed in each forest district. Volumes of individual trees were estimated using the official allometric equations applied by Polish State Forests [138] and aggregated at the single plot level. In this study, the GSV was the sole target variable, as it is one of the most important forest attributes, closely related to biomass and carbon stock [139], which in order for its accurate determination typically requires larger samples than other variables [80]. Plot coordinates were recorded using Topcon HiPer V (Topcon Positioning Systems Inc., Livermore, U.S.) survey-grade GNSS receivers, achieving approximately 1 m positioning accuracy, which should be sufficient for the two-phase ALS-aided forest inventory [140,141].

Table 2. Brief characteristics of selected forest districts.

District	Average Age	GSV [m3/ha]	Moran’s I *	Major Tree Species
Białowieża	110	401	0.012	Norway spruce, Oak, Black alder, Scotch pine, Hornbeam
Głogów	57	309	0.066	Scotch pine, Beech, Silver fir, Oak
Gorlice	64	373	0.035	Beech, Silver fir, Scotch pine
Herby	60	325	0.031	Scotch pine, Oak
Katrynka	60	395	0.059	Scotch pine, Norway spruce
Taczanów	77	304	0.042	Scotch pine, Oak
Leżajsk	62	354	0.012	Scotch pine, Beech, Oak, Silver fir
Milicz	60	379	0.013	Scotch pine, Oak, Beech
Pieńsk	54	303	0.005	Scotch pine, Norway spruce, Birch
Supraśl	58	400	0.002	Scotch pine, Norway spruce, Oak

* Spatial autocorrelation index for the target variable: 1—full correlation, 0—no correlation.

contains the main characteristics of selected forest districts, while Figure 1 shows their locations.

Airborne laser scanning campaigns were conducted simultaneously with field inventories. Point cloud tiles covering all forest districts were provided in LAS 1.2 file format. Height normalisation was performed using Digital Terrain Models (DTM) with 0.5 m resolution. Spatial co-registration between ground sample plots and the corresponding 3D space of the point clouds was then carried out. Clipped point clouds were subsequently equalised to a density of 10 pulses per square metre using an original thinning algorithm [141] for better control over the analysed factors. Finally, a set of standard ALS metrics was computed for each individual plot. All data processing in this study was performed employing the lidR library within the R programming environment [142].

In order to account for non-linearity among some explanatory variables, log and power transformations were applied to the original ALS metrics. Categorical variables, such as species information, were converted to numerical values and expressed as the volumetric share of each species within a plot. These procedures resulted in a large number of potential predictors (over 360), thereby a data dimensionality reduction steps were necessary. At first, metrics with a coefficient of variation lower than 10% were eliminated. Next, a random forest (RF) generic model was run on the entire dataset to evaluate importance of each variable. Metrics contributing less than 10% towards the GSV prediction (according to the RF importance index embedded in the randomForest R package (v. 4.7-1.1) [143]) were excluded from further analysis. Finally, an autocorrelation matrix was constructed, and variables were assigned to groups of similar categories. In each group, one or two strong GSV predictors were retained.

Table 3. Grouping and explanatory variables.

Group	Variables	Source/Definition
ALS central tendency	mean height	lidR [142]
ALS dispersion	height sd	lidR [142]
ALS quantiles	height 1st quartile, 95th height percentile	lidR [142]
ALS cumulative histograms	square of the ratio between the number of points above 2nd threshold to all points, square of ratio between the number of points below 7th height threshold to all points	[53,144,145]
Species related	coniferous species share, dominant species share, Shannon diversity index	ground inventory [15]
Age related	average age, natural logarithm of average age	ground inventory, GIS layers, previous inventories [146]

presents the final grouping and explanatory variables used as input data for the main analysis.

2.2. Forest Generation

An underlying assumption had to be fulfilled to make the results of this study comparable, viz. equal-sized populations for each sampling intensity and forest complexity level. Therefore, the core analysis of this study was based on mutated, semi-artificial (synthetic) forests, generated from real inventory data described in the previous sub-chapter. We called them semi-artificial, as they were only merged and replicated based on real ground forest inventory measurements. The procedures behind forest generation were as follows. First, the real data sample plots were ordered according to their reference GSV values. Next, every 10th plot was selected as an independent forest kernel. This step was necessary to reduce the tremendous number of possible forest instances. Subsequently, all explanatory variables were scaled, and the k-Nearest Neighbours (kNN) algorithm was employed to expand the kernel with the most similar plots. The final procedure in forest generation was the replication/mutation of the selected plots. To ensure a sufficiently large sampling population and to keep computational constraints at feasible levels, the forest ceiling was set at 50,000 plots. The concept of mutation of the original plots was based on the addition of small noise/variance to the vectors of variables describing the original plots (up to a few percent, depending on the variable). This approach is known in the literature as synthetic oversampling (bootstrap) with stochastic variance induction [147,148,149].

Thus, generated synthetic forest was evaluated for its structural complexity. For this purpose, a Weighted Coefficient of Variation (CVW) was introduced as the proposed forest complexity indicator. This is essentially the average coefficient of variation from all covariates, weighted by the Pearson coefficients of determination between each predictor and the dependent variable, i.e., the GSV. The entire procedure of forest generation and evaluation was repeated 30 times for each scenario, to account for some randomness inherent in the kNN algorithm, while maintaining a statistically sufficient number of replications [150,151,152,153]. In total, 211,320 uniquely generated forests were produced.

Table 4. Forest generation contingency table.

Factors	Levels
Kernel plot ID	1, 10, 20, 30, …, 580
k-neighbours	100, 250, 500, 1000
Sampling intensity	100, 200, 300
Repetitions	1,2,3, …, 30

presents the combination of all factors and levels used to create unique scenarios for forest generation.

2.3. Clustering/Stratification and Sampling

Several data clustering methods was evaluated in terms of their ability to define optimal stratification groups for drawing a sample for model calibration. The goal of both clustering and stratification is coherent. Essentially, it involves transitioning from a set of variables into discrete, more general groups, by minimising within-group (co)variance and maximising between-group distances [115]. While fundamentals behind most data clustering methods are richly described in the literature (Table 1), we need to explain the principles behind our original, yet relatively simple, IDS algorithm, which stands for Individual Dimensions Sampling. The main difference from other clustering methods is that in IDS, the distribution and variance are evaluated separately for each predictor. In IDS, predictors are ordered according to their importance tag, e.g., the value of GSV correlation. Next, each predictor receives one plot (starting from the strongest to the weakest). The loop is repeated until no more plots are available. The vector of each independent variable is then divided into groups using the natural breaks algorithm [154], with the number of groups equivalent to the number of sample plots assigned to the given variable. As plots selection in IDS was independent for each variable, no covariates scaling was performed before clustering operations, unlike for other clustering methods, where normalisation of covariates was performed using the base R scale function [155].

For most data clustering algorithms, the number of clusters, k, is often the sole and the most important hyperparameter to set. For these algorithms, to adequately capture the full range of variance within the target population and to decimate overly saturated groups (reducing redundant plots), we decided to directly link the number of clusters to the number of sample plots available in a given scenario (N). Here, it is important to distinguish the correct notation, i.e., N refers to the number of plots available in a given scenario, while n refers to the number of plots assigned to a given cluster. In most cases, k = N. This approach allowed us to avoid k tuning, which would otherwise have made the already large number of simulations even greater. On the contrary, the architecture of HDBSCAN requires setting a minimum cluster size instead of the number of clusters. To mimic the behaviour of other clustering algorithms, i.e., to equalise the number of clusters k with the number of plots N, the minimum cluster size for HDBSCAN was set as the quotient of k-neighbours (from the forest generation stage) and the number of plots available in the given scenario.

Intra-group sampling was the next step after forest stratification by clustering algorithms. As in most scenarios, the number of k clusters equalled the number of plots N, two main sampling solutions remained feasible, i.e., random and cluster-centroid sampling. As random in-cluster sampling induces another source of variance, and to better control the experiment of comparing selected clustering methods, the authors focused on centroid-based sampling, which has unambiguous properties in selecting precisely specified plots. Cluster-centroid sampling aimed to identify the plot closest to the centre point of a cluster in its multidimensional space, expressed with scaled covariates. The distance was defined as the average difference between the values of particular covariates characterising a plot and the averaged values of the corresponding covariates from all plots within the cluster. In scenarios where N > k (HDBSCAN), cluster quantile interval sampling was used. For example, if there were two plots in a cluster, those selected were at the 25% and 75% quartiles of the distance distribution; if three plots were available in the cluster, the selected ones were, respectively, at 25%, 50%, and 75% quartiles, and so on. To contrast the performance of cluster-stratified sampling methods, results from simple random sampling were juxtaposed, as well as those from the use of generic models (GM), i.e., models trained on all real sample plots except those chosen to generate the target (artificial) forest.

2.4. GSV Estimation and Validation

Random forest models were trained on the samples selected in each scenario. The RF estimator was implemented as it is relatively universal and does not require a priori knowledge about the data structure. It handles non-linearity among the input data autonomously, and different types of variables can be processed in its engine. The hyperparameters of the RF model as implemented in the R library randomForest [143] were as follows: number of trees—701, number of variables—7, node size—5. To reduce another source of contingency, the same RF settings were used in each simulation. We did not expect significant influence from this factor as, in general, the RF hyperparameters are known to have marginal effects on the predictive performance of RF models, provided that the number of trees is set to the default [156], around 100 [157], at least 128 [158], or a few hundred [159]. Thus, calibrated models were subsequently used to estimate GSV value of the synthetic forest. The criteria for the performance of each method were normalised estimation errors, i.e., nBIAS and nRMSE. Validation of both models and clustering methods was ensured in 30 repetitions. For better understanding of the methodology applied, a graphical concept presenting the workflow of analysis and data processing steps is provided in Figure 2.

3. Results

Figure 3 and Figure 4 present an essential part of the results. Figure 3 traces nRMSE and nBIAS trends as a function of stand complexity expressed by the CVW indicator for each investigated sampling design. In Figure 3, to illustrate central tendencies, the results from 30 repetitions were aggregated as arithmetic means. Figure 4 assesses the stability of each method, expressed as the standard deviation of the error measures. Second-degree polynomial functions were fitted to the raw scatter plots to better highlight subtle differences between some methods and to clear out the jitter caused by the iterations.

Variance among the original sample plots, which mostly comes from managed Polish forests, could derive synthetic stands of small to medium complexity, i.e., CVW values between 0.15 and 0.4. Therefore, the displayed trends should only be considered valid within this range. A strong positive correlation was detected between RMSE and CVW values, indicating less precise estimates for more-complex stands (R² from 0.80 for GM to 0.92 for HDBSCAN). Obtained RMSE values ranged from 10% to 24%. This type of error depended more on stand complexity level than on the particular sampling scheme or sampling intensity (Figure 3—left column). The gain in performance due to increased sampling intensity was most evident in the transition from 100 to 200 samples, whereupon differences gradually became more subtle. Similarly, with random sampling in the two-phase ALS-enhanced forest inventory, there is usually no benefit in increasing sampling intensity above about 200–300 plots, as only negligible performance gains can be attained [55]. The congruous optimal sampling zone was identified by Grafström et al. (2014) [79], who used the local pivotal method (LPM) [160] to draw a spatially balanced sample in a similar study design. Structurally guided sampling and stratification methods analysed in this study also proved competitive with the systematic sampling design presented in Parkitna et al. (2021) [145], who used 900 evenly distributed sample plots to train random forest models, boosted regression trees, and ordinary least squares regression. Regarding systematic error (Figure 3—right column), even 100 samples were sufficient to keep BIAS values within ±1% for all compared methods except HDBSCAN and generic models, the former being visibly more susceptible to sampling intensity levels. The same figure also shows that complex stands benefit more than simple ones (higher RMSE reduction) from an increased number of sample plots. The largest observed change was for HDBSCAN (from 23% to 18%) when transitioning from 100 to 300 plots. This trend was also confirmed by Li [80], who reported that higher heterogeneity in the vertical structure of forest stands entails an increased minimum number of required sample plots.

Relative differences between particular methods were not substantial—up to 3 percentage points (pp), with similar trends along the CVW axis. Nevertheless, hierarchical clustering (H-CLUST) proved to be the most efficient and stable method (Figure 3 and Figure 4). Unsurprisingly, simple random sampling and k-means were the least-biased designs (Figure 3), but required more sample plots to achieve stability comparable to that of hierarchical clustering (Figure 4). The performance of the two remaining methods, that is, IDS and HDBSCAN, was usually intermediate between the most efficient methods (H-CLUST/k-means) and the contrast (GM) method, with HDBSCAN being the most susceptible to the number of sample plots. Generic models yielded the worst results among all methods. The discrepancies compared to other methods became more pronounced as the number of sample plots used to train the RF model in other sampling methods increased.

Some transitions from underestimation to overestimation (and vice versa) around the CVW value of 0.2 may be observed in certain methods. However, the magnitude of this questionable effect is not substantial and is influenced either by random chance from the iterations or, more likely, by the proximity of the scenario sample mean to the total sample mean in this region. Finally, a flattening effect may be observed at around 0.3–0.4 of CVW, indicating no further increase in RMSE errors due to the stand complexity level. However, this behaviour should be interpreted with caution, as it is most likely caused by the smaller amount of available data inputs (kernels) at higher forest complexity levels and increased variance in those regions.

4. Discussion

It is necessary to acknowledge certain limitations concerning our study; however, this can simultaneously cast light on the potential for further research. Specifically, the presented array of 5870 real sample plots, along with their range of complexity, is characteristic of Central European forests; therefore, the results should be considered applicable only to this population. Consequently, there is a well-founded need to apply the proposed methodology to more diverse and complex stands in order to determine whether estimation error measures stabilise above a certain CVW level. This is particularly important, as shown in Figure 3 and by Tompalski et al. (2019) [136], because applying models outside their sampling region (generic models) may result in increased inaccuracies if calibration and estimation forests differ substantially. Nonetheless, the nearly 6000 real data sample plots (both ground measurements and ALS) provide a solid validation for the applicable regions.

Secondly, as model-based inference is a crucial component in the two-phase sampling design with regression estimator [59], this study compared the performance of selected data clustering methods for optimal model calibration sampling. The performance was evaluated using standard measures of predictive capability, namely, BIAS and RMSE. As shown in the previous section, there is a very strong correlation between RMSE and variance in the population. Therefore, with a large array of original ground sample plots, where every single tree was measured, the model estimation errors should closely approximate the true variance in the population. For this reason, we did not evaluate variance estimators as is performed in more traditional design-based inference.

One should also bear in mind other aspects that could slightly influence the observed trends which, however, were not covered within the scope of this study, mainly to avoid excessively convoluted research. First, grouping procedures constitute the initial stage in stratified sampling design. Strata can be delineated in various ways, but their optimisation in terms of required accuracy and available resources for any given area should always be a goal. For the reasons mentioned in the Section 2, in most scenarios, the authors directly linked the number of clusters (strata) to the number of available sample plots, i.e., k = N. Nevertheless, for data clustering algorithms, the number of clusters is often the most important parameter to set, having a significant impact on their performance [98,161,162]. For the same reasons, we only checked a few popular data clustering methods, whereas some research indicates attractive alternatives such as local pivotal methods (LPM) [162] or mixture models [163,164,165]. Moreover, only one unique and explicitly defined set of clustering variables was used in this study. On the other hand, Ref. [98] reported that the type of algorithm and set of clustering variables can have secondary importance. Therefore, these issues still remain to be confirmed.

After strata definition is set, the subsequent procedure involves within-stratum plot selection. This includes both per-stratum sample size allocation and its deployment. Generally, in this regard, random sampling, regular sampling, and centroid sampling can be distinguished, with either proportional or equal sample distribution. Centre-based (centroid) within-stratum plot selection (used in this study) ensures unambiguous sample plot designation, which eliminates part of the error related to random effects, thus increasing the stability (Figure 4). However, not all possible combinations of clustering and in-cluster sampling variants were tested. In this regard, further detail could be explored. Namely, the cluster centre can be defined in various ways. In our study, it was the averaged point in multidimensional space built from scaled covariates with no weights. In reality, however, some predictors have a stronger effect on the target variable than others. Therefore, in future studies, different approaches to identifying cluster centres should also be considered. Perhaps this could help to better optimise sample deployment (favouring only the most important and non-autocorrelated predictors) and thus result in a sample reduced to an even smaller size.

Another point for discussion is that only one type of estimator, namely, random forest, was applied, with a fixed set of values for its hyperparameters in each scenario. Nevertheless, as mentioned in the methodology and demonstrated by other research, RF regression should not differ significantly from other types of estimators for this type of analysis and input data [145]. In fact, in some cases it can even outperform other modelling approaches, especially for GSV estimation [136]. Regarding the hyperparameters, numerous studies have reported their negligible influence on RF predictive power, provided that the number of trees is set to a sufficiently large quantity [156,157,158,159].

Certain caveats can be attributed to the different types of predictors used in this study. In addition to ALS metrics, dominant species and age information were used for both forest stratification and target-variable estimation. These metrics are recognised as relatively significant auxiliary predictors of biomass or GSV [166,167]. The absence of such auxiliary information could potentially yield slightly different results. Nonetheless, species and age do not appear to be very dynamic forest attributes in mostly managed Central European forests, and can now be accurately determined, for instance, using RS data classifications [168,169] or previous inventories. This constitutes another aspect related to data dimensionality reduction, as different types, quality, and quantity of available predictors and auxiliary variables can influence optimal inference and strata designation [170,171,172,173]. After dimensionality reduction in this study, authors did not shuffle the fixed set of 11 input variables between simulations (although this was partially addressed by the use of the RF model, where some degree of randomness in the variables selection is inherently embedded). In contrast, Ref. [98] showed that effective stratification can be achieved with as few as one to three clustering variables, and that the choice of clustering variables could have a secondary impact on clustering performance. Therefore, this issue still warrants further exploration. One should also expect even better performance from each investigated method if outlier-handling analysis was performed, which was not the case in this study, in order to avoid arbitrary judgements. Surprisingly, HDBSCAN, which claims to handle noisy data [133,134,135], performed poorly in this regard (Figure 3 and Figure 4). Nevertheless, in this study, relative differences between particular methods and general trends were more pertinent than their absolute effectiveness, which nonetheless yielded decent GSV estimates, as shown by the comparison to similar studies.

Some aspects related to spatial autocorrelation in forest environments were simplified in this study. The k-NN algorithm for forest replication was not fed with geographical data. This was due to the specific study design and small effect of spatial autocorrelation, as expressed with Moran’s I index (Table 2), in mostly managed forest districts used in this study, where even adjacent stands can drastically differ in their structures. Moreover, as stated in the introduction, the SGS is designed for sampling in the feature space, rather than in geographical sense. Furthermore, remote sensing data (as used in this study), are known to mitigate/explain the influence of the spatial structure of forests [30]. Additionally, SGS fosters an unambiguous sample distribution, meaning that this design is unaffected by a relocation of units within the population (the same units are selected regardless their proximity). This seems like a great advantage of structural guided sampling, especially for populations with trends [162].

Despite the depicted limitations, the results obtained were generally consistent with the literature reports. Stratified sampling usually led to better precision and stability than simple random sampling (Figure 3 and Figure 4), provided that the designed strata adequately captured naturally occurring clusters in the population [42], or when the sampling intensity exceeded 50 plots, with a flattening effect observed at around 200–250 plots [79,174,175]. Further reduction in sampling intensity for model calibration should be possible, as ref. [104] found that even 100 plots (or less) might suffice, provided the sample covers the population variability. An adaptive sampling appears as an apt extension in this regard.

Another advantage of pre-stratification based on auxiliary or RS metrics is that data captured immediately before planned field measurements can provide valuable a priori knowledge about the sampling population, resulting in less biased and more precise estimates and/or less costly inventory campaigns [176]. Therefore, in the presented sampling approaches, adequate and objective strata delineation, as well as better optimised plot allocation (than in post-stratification) [24], could, for instance, be achieved using one of the analysed data clustering algorithms with explicit in-cluster sampling. Their implementation is both unambiguous and flexible, meaning that the procedure behind sample selection is clearly defined, but the exact realisation depends heavily on the given population variance, as indicated by RS or other metrics known in advance.

The choice of a specific sampling method depends primarily on the study goal, available resources, allowable errors, and the characteristics of the target population [33]. Each sampling method has its advantages and disadvantages. Consequently, scientists often combine their features to develop more efficient, universal, and reliable solutions [177]. This can lead to the creation of sophisticated and complex systems, which may, however, face difficulties in gaining acceptance among practitioners [178,179,180]. It is generally preferable to understand the process before implementation. Nevertheless, the high automation capabilities of contemporary digital systems can often relieve users from intricate pipeline details, provided that the “customers” can verify the reliability of the final outcome (estimates) in a manner they trust, such as through control surveys, reference methods, previous assessments, or expertise. This, in turn, can motivate the development and implementation of more comprehensive and auto-optimised methods.

Hierarchical clustering (H-CLUST), using the Ward’s agglomeration algorithm, outperformed the other data clustering methods analysed in this study, as well as the two control methods: random sampling and generic models. K-means produced similar estimates (Figure 3), but was less stable, particularly at lower sampling intensity levels (Figure 4). In contrast, generic models were the most stable approach but returned the least accurate and most biased estimates. Similar findings were reported by [83] who stated that incorporating sample plots from the target populations improved predictions and reduced systematic error. Therefore, it is advisable always to secure a set of in situ sample plots, either for local model development, external model parameter calibration, or at least for control purposes and assessment of estimation errors or variance. Based on the results obtained, the H-CLUST method is preferable, especially when fewer second-phase sample plots are available.

Yet, in this study, the authors did not compare the analysed methods with systematic sampling, which is essentially the most common sampling method applied in practical forestry [16,19,20,21,22,23,24]. The reason for this omission was that it would require a completely different approach, making an already complex methodology even more convoluted. Comparison and evaluation of SYS performance would require a complete reference survey of the entire forest district, which is practically impossible for large-area inventories. On the other hand, contemporary machine learning algorithms, geostatistical interpolation methods, and more recently AI models are capable of generating spatially continuous population (often with RS data) from which one can draw samples [181,182,183,184,185,186,187,188]. It should also be noted that algorithms for digitally generated forests have their own hyperparameters to set and can be sensitive to input data. Although such reconstructions tend to be merely smoother versions of the true population, some research has indicated SGS methods as a good alternative [104], or more recently, even superior to systematic sampling [102]. Our results showed that for model-based inference, SGS sampling can be competitive with SYS [145] in two-phase ALS-enhanced forest inventory, using fewer sample plots for model calibration. Nevertheless, the rapid development of numerical deep learning, neural networks, and generative AI methods, along with an appropriate research approach based on such synthetic data, could further help to explore the following aspects of forest inventory optimisation: (1) Which sampling method is most adequate for the forest inventory of a given area, for a desired performance level and with the available resources? (2) Is it always the same method in the context of forest type and its heterogeneity? (3) What is the extent of the differences in efficiency and reliability between SRS, SYS, STRS, and SGS? These questions seem to highlight the right direction for further research in the field of forest inventory sampling optimisation. They will be addressed in another study.

5. Conclusions

The following conclusions can be drawn from the results of this study. RMSE values were highly correlated with stand complexity. Structurally more-complex stands benefit more from increased sample size than homogeneous stands, in terms of decreased GSV estimation error. Based on the results from synthetic forests, even 100 sample plots may be sufficient to achieve unbiased GSV estimates for the means (±1%), using structurally guided sampling in a two-phase inventory design with the ALS random forest estimator. However, real-world complexity would likely require larger samples. Notwithstanding, a major potential of ALS-enhanced forest inventories lies in their capability for spatially continuous estimation of forest attributes for small areas and individual stands. If this aspect of forest inventory is of interest, the sampling intensity saturation appears to occur at about 200–300 plots. This observation is also generally consistent with some similar study reports discussed in the previous sections. Another conclusion is that the lack of in situ forest inventory data can produce biased and less-accurate GSV estimates. Hierarchical data clustering with the Ward’s agglomeration algorithm outperformed other data clustering methods investigated in this study. The differences were not prominent, but became more pronounced when more sample plots were used for model calibration. Among other clustering methods, the Ward’s criterion was also the most stable in GSV predictions. Further research in the field of structurally guided sampling for forest stock inventory optimisation should, among other things, focus on in-cluster sampling, cluster centre identification, the influence of clustering variables, and comparison with other SGS algorithms, like cube or local pivotal methods. Moreover, a comparison with the most common sampling design in forestry, i.e., systematic, should be made for different areas and at different sampling intensity levels.