Performance of Selected Sampling Methods for Growing Stock Volume Estimation in Model-Based Two-Phase Forest Inventory

Highlights

What are the main findings?

• Structurally guided stratification and sampling suitable for sample size optimization.
• Differences between sampling methods were more visible at lower sampling intensity levels.

What are the implications of the main findings?

• ALS-guided forest stratification and in-data-space plot allocation can potentially reduce sampling intensity at the forest district level.

Abstract

Selection of an adequate sampling method is perhaps one of the most crucial stages of any survey. Researchers and decision-makers often face difficulties in selecting appropriate sampling techniques and sample sizes in relation to the specific research problem and the desired accuracy of estimates. Although general rules of thumb are relatively well established, the optimization of these elements is often based on rough approximations. Direct comparisons of the accuracy of different sampling methods in large-scale, real-world forest inventories are, in most cases, not practically feasible. Consequently, some studies rely on resampling limited observational datasets, whereas others generate artificial populations from which predefined samples can be drawn to evaluate sampling efficiency. Nevertheless, such populations tend to be simplified representations of the often complex structures of real forests. This study proposes an approach that could serve as a bridge between real and artificial data. Specifically, the efficiency of three popular sampling designs—simple random sampling (SRS), systematic sampling (SYS), and stratified sampling (StrS)—was compared for growing stock volume (GSV) estimation in a two-phase model-based forest inventory. The comparisons were performed using semi-synthetic forest populations, in which only the target variable was generated using a machine-learning approach, whereas all other parameters were derived from real forest inventory and remote sensing (RS) data. Stratification and sampling were performed using hierarchical data clustering and the concept of structurally guided sampling (SGS), which, in addition to sampling in geographical space, enables sampling in the multidimensional space of auxiliary variables. The proposed stratification technique allows for potentially more effective delineation of strata than predefined spatial layers, which should be crucial for the overall inventory performance. The results showed that hierarchical stratification (clustering) combined with SGS in the airborne laser scanning (ALS) data space was competitive with SRS and SYS, and could even outperform them at lower sampling intensity levels. The convergence zone in the accuracy and precision of SGS-GSV estimates could be reached at lower sampling intensities than with the classical sampling methods. These results demonstrate a potential for optimizing sampling in ALS model-based forest inventories and highlight opportunities for reducing sample sizes, without practically compromising the accuracy and precision of GSV estimates at the forest district level. Nonetheless, further validation on real-world datasets is needed, as synthetic populations may not fully capture the heterogeneity of actual forest structures, which would most likely require more intensive sampling. Therefore, the results presented in this study should be regarded as indicative of potential trends rather than as definitive guidelines for conducting remote sensing-supported forest inventories.

1. Introduction

Forest inventories have traditionally been a primary source of information concerning wood resources. This information provides an essential basis for long-term sustainable forest management, and in recent decades has become increasingly important due to growing sociopolitical demand for multifunctional forest services [1]. Recent trends in forest research indicate a growing shift towards more sustainable and efficient forest management, with precision forestry emerging as a promising concept to address these challenges [2]. Contemporary forest management increasingly requires optimization and decision-support solutions capable of balancing and prioritizing the diverse needs of society, forest owners, stakeholders, and other interested parties [3]. Despite the recent decline in the global rate of forest loss, with some regional variation, forests continue to face substantial pressures from deforestation, degradation, fragmentation, and land-use change, highlighting the need for more sustainable, efficient, and precise approaches to forest management [4]. This need is further reinforced by the long-term implications of forest-management decisions, as errors or inadequacies may be difficult to detect at an early stage and can lead to severe, often irreversible, consequences over time [5]. Consequently, the availability of accurate, reliable, and appropriately optimized forest inventory data is essential to supporting informed decision-making and ensuring sustainable forest management.

As forest censuses are in most cases impractical at large spatial scales, various sampling methods have been developed over the decades to balance the desired levels of accuracy and precision, on the one hand, with financial costs and survey feasibility, on the other [6]. Sampling methods differ in terms of their reliability, required sampling intensity, and suitability for particular inventory objectives and conditions. There is no single sampling method that is universally optimal for all inventory situations. Even for the same inventory objective, the choice and optimization of a sampling method may vary among populations, depending on their characteristics and the availability of auxiliary data and prior knowledge [7].

Systematic sampling is widely considered one of the most common sampling methods for medium and large-area forest stock inventories [8,9,10,11,12,13]. SYS owes its popularity to the following reasons: (1) under non-random spatial structure (as is the case in many managed forest areas), it is considered as relatively simple in terms of operational field logistics and thus cost-effective [14]; (2) in general, it provides competitive performance to SRS [15,16,17]; (3) it can exploit spatial structure between stands and thereby influence statistical efficiency in the presence of spatial autocorrelation [18]; (4) it ensures uniform spatially balanced coverage without large non-sampled areas thereby preventing plots clumping [19]; and (5) its clear and regular spatial pattern is readily understood and operationally appealing to practitioners and decision-makers [18].

A well-recognized caveat concerning SYS is the lack of a design-unbiased estimator of the variance for the target variable, which often leads analysts to resort to the SRS estimator, which is generally considered more conservative [17,20,21], particularly given less true randomness inherent to SYS [22]. Another important consideration is the potential presence of periodic spatial patterns in managed forest districts, which, in combination with an unfavourable starting point or grid orientation, may introduce substantial statistical risk [18,23].

While SYS can provide excellent spatial coverage and reasonable accuracy, many surveys alter the design into systematic cluster sampling (SCS), in which several plots are located in close proximity to each grid node. Although this approach is often associated with slightly higher estimation errors, it usually reduces sampling costs related to logistical aspects and can be particularly efficient when variability within clusters is substantial [24,25,26]. Moreover, by concentrating several plots around each node, SCS can also reduce the sensitivity of SYS to the random starting point when the surveyed area exhibits strong spatial patterns. Consequently, SCS is widely implemented in national forest inventories (NFI), where statistical efficiency must often be balanced against the operational and economic requirements of repeated inventories [27,28].

Stratified sampling constitutes another widely used survey method in forest inventories. It is designed to improve sampling efficiency in heterogeneous populations by grouping sampling units with similar characteristics. If appropriately designed, stratification can increase the precision of estimates within individual strata and thereby improve overall sampling efficiency [29,30,31,32]. In this design, both the delineation of representative strata and the allocation of the sample among strata remain important considerations [30,33]. In practice, fixed strata are often predefined based on auxiliary variables such as species–age groups [34], stand height classes, site conditions and indices, or combinations of these factors [35,36], sometimes regardless of the importance of the particular covariates [37] and/or using regular breakpoints for continuous variables, such as stand age or height, which are not necessarily linearly correlated with stand growth. Consequently, suboptimal stratification may introduce yet another source of inaccuracies in the estimation of crucial forest attributes [38].

Pre-stratification requires prior knowledge of the sampling population before the actual survey begins. If no suitable auxiliary variables are available, post-stratification (PS) is an alternative. However, the accuracy of PS is highly dependent on the initial selection of sampling plots, especially at low sampling intensities. One possible way to address this limitation could be adaptive sampling (ADA), which expands the total number of observations during sampling by recalculating and tracking changes in assumed sampling efficiency indicators at each subsequent draw [39]. A theoretical challenge inherent to ADA is the selection of an appropriate criterion for determining when and where additional samples should be collected. Model-based adaptive approaches commonly use measures of predictive uncertainty, variance, or prediction error to identify informative locations. Such criteria tend to prioritize areas where the current model performs poorly or exhibits high uncertainty, potentially resulting in a concentration of supplementary samples in heterogeneous or otherwise difficult-to-predict areas. Consequently, the resulting sample may have a spatial distribution that differs substantially from that of the target population, creating a need to balance the local exploitation of high-uncertainty areas with broader exploration of the population [40]. In practical terms, this adaptive process can make it difficult to determine the required sample size in advance. One potential solution could be a combination of adaptive and cluster sampling (CS), whereby additional plots are located in the vicinity of selected initial observations. Such an approach can be particularly effective for spatially sparse and/or clustered populations [41,42]. Nevertheless, uncertainty regarding the ultimate sample size remains a potential limitation of this approach and may complicate financial and logistical planning [43].

In practice, uncertainty related to sampling error is often addressed by applying conservative variance estimators, which usually entail increased sampling efforts that might prove redundant in otherwise well-optimized designs [44,45,46]. Structurally guided sampling (SGS) [47] has been proposed as an alternative approach to improving sampling efficiency and addressing issues related to sampling optimization [48]. This approach not only enables a well-defined spatial distribution of sampling units but can also facilitate balanced sample allocation (SA) across the multidimensional space of auxiliary variables [49,50]. This sampling concept can help accurately represent the full structural variation in forest resources while maintaining sampling intensity at reasonable levels [35,38]. Some of the most common tools used to implement SGS in forestry include the cube method (CM) [51,52], the local pivotal method (LPM) [53], the nearest centroid algorithm [54], and data clustering methods such as k-means [55,56] and Ward’s clustering [46,57,58].

The efficiency of SGS is strongly related to the quality of a priori knowledge about the sampling area. Therefore, it is often combined with various types of remote sensing data, which provide spatially continuous information on forest structure across the entire inventory area [59,60]. The development of remote sensing (RS) methods has made two-phase and multistage sampling designs increasingly attractive to both researchers and practitioners [61]. One of the major benefits of using remote sensing-enhanced forest inventories is the ability to estimate the target variable not only for population totals and means but also for any given non-sampled area [62], usually by means of regression or machine learning models (estimators) [63,64]. Additionally, the complete spatial coverage of the surveyed area provided by airborne or satellite RS missions creates vast opportunities for sampling optimization. RS data can therefore serve as an excellent basis for initial sampling and stratification. Furthermore, certain variables, for instance, those describing the vertical structure of forest stands, can be derived directly from RS data without extensive statistical modelling [65,66].

Although numerous studies have demonstrated the potential of RS-assisted forest inventories for optimizing sample size [67,68], sample plot area [69,70], stand-level estimation accuracy [71,72], point cloud density [69,73], SGS intensity [46,48,58,74,75], economic efficiency [46,71,76], and their combined effects [64,72,77,78,79], relatively few studies have examined the influence of different sampling methods on the efficiency of GSV estimation [80] or closely related biomass estimation [81]. To the best of the authors’ knowledge, no study has yet provided convincing empirical evidence that SYS consistently outperforms other sampling designs in GSV estimation within two-phase, model-based forest inventories [82,83]. This is probably due to the practical difficulties involved in conducting direct empirical comparisons between sampling methods across vast forest areas using real forest inventory data. Consequently, existing studies either limit their scope to small-area estimations [82,83], rely on purely theoretical inference [17,84], resample existing inventory data [49,80,], or generate multivariate artificial populations from which specified samples are drawn and tested through simulations [85,86,87]. Nevertheless, artificially generated study areas often provide simplified representations of forest structure and may fail to reproduce the complexity and spatial patterns encountered in real forests [88,89,90]. Moreover, the use of synthetic data to train biomass prediction models may result in substantial underprediction [91]. Finding an appropriate balance between model-driven generalization and the realistic representation of actual forest structures remains a classical challenge in ecological modelling [92]. Consequently, inferences based on such simulations should be interpreted cautiously and regarded primarily as general approximations of potential trends in sampling optimization [58,93].

To address the aforementioned limitations of analyses based either on limited, small-scale empirical datasets or on artificially generated forests, we sought to bridge these two approaches by generating so-called “semi-synthetic” populations. In this framework, only the response variable, i.e., GSV, was generated using a combination of two machine-learning techniques, residual imputation and stochastic variance induction. All other data—including more than 4000 ground inventory sample plots, spatially continuous ALS point clouds, hyperspectral orthomosaics, and stand-level volume estimates—were obtained from real forest management inventories (FMI) conducted in nine forest districts. The main reason for referring to these populations as “semi-synthetic” was to highlight their differences from the artificial populations presented in other studies. The key distinguishing features were as follows: (1) the response variable was generated based on real stand-level survey data, with independent GSV also included as a predictor; (2) high temporal consistency was maintained, as all data were collected within the same year; (3) the independent variables were derived from real data; and (4) the original spatial dependencies among the auxiliary variables were preserved, with no resampling, replication, or extrapolation beyond individual stand boundaries. We would like to emphasize that the term “semi-synthetic” was introduced by the authors and is not widely recognized in the literature. Consequently, we identified only two studies whose methodologies bore similarities to those used to generate the populations in this study. In [19], the authors also used the same variable as both the response and a predictor, although the predictor was estimated from large-area mean volumes. In [58], the authors used the term “semi-artificial”; however, their approach did not preserve spatial dependencies, nor was the response variable generated using real GSV estimates from another source. Nevertheless, as the populations presented in this study bear certain affinity to entirely artificially generated populations, similar caution should be exercised when drawing inferences from these types of datasets.

Based on these semi-synthetic populations, three commonly used sampling designs were evaluated: simple random sampling, systematic sampling, and stratified sampling. The primary objective was to assess the efficiency of GSV estimation in a model-based, two-phase forest inventory under different sampling designs and combinations of available auxiliary variables. The resulting estimates were subsequently compared across sampling designs. To provide a comprehensive assessment, we examined the effects of sampling intensity (sample size), estimator type, i.e., ordinary linear regression (OLR) versus random forest (RF), and the availability of different types of auxiliary variables, including ALS-derived metrics, species information (SP), stand age, and preliminary GSV estimates obtained from ex situ generic models (GM). For the stratified sampling design, rather than using conventional predefined strata, delineation was based on a structurally guided approach using hierarchical clustering of auxiliary variables. In addition, three within-stratum sampling strategies were evaluated: random selection (RS), nearest-centroid selection (NC), and regular in-data-space sampling (REG).

2. Materials and Methods

2.1. Forest Inventory Data

The real forest inventory dataset used in this study consisted of 4052 ground sample plots and 16,165 stand-level surveys. Field measurements were conducted in 2015 in nine forest districts located in different parts of Poland: Supraśl, Dojlidy, Białowieża, Browsk, Hajnówka, Gorlice, Herby, Milicz, and Pieńsk (Figure 1). Diameters at breast height (DBH) were measured for all trees with a DBH exceeding 7 cm within circular plots of 500 m², distributed systematically across the inventoried areas. At each plot, heights of all trees were measured using rangefinders in all study areas except Białowieża, Browsk, and Hajnówka. In these three districts, owing to their higher biodiversity and greater tree density, heights were measured only for representative trees within each species–age layer. Heights of unmeasured trees were estimated using DBH–height relationship curves, as this simplification is known to have a very limited influence on plot-level aggregated GSV [94]. Because Białowieża, Browsk, and Hajnówka are directly adjacent to one another, these three districts were treated as a single study object, hereafter referred to as BBH. This aggregation enabled comparisons between study objects of different sizes, allowing for an assessment of the potential influence of inventory area on the required sample size. Volumes of individual trees were estimated using allometric equations commonly applied by the Polish State Forests [34,95] and subsequently aggregated at the plot level. The age of individual species–age layers at each plot was extrapolated from previous inventories. In cases where age could not be reliably determined, such as in natural or less intensively managed stands, it was approximated using increment borers or inferred from neighbouring stands by experienced survey crews. The resulting plot-level GSV values were used as the response variable for calibrating the XGBoost model (XGB), which was subsequently used to generate reference GSV values for the semi-synthetic forest populations (see Section 2.4 for more details). Table 1 provides a brief quantitative summary of selected basic characteristics from particular forest districts.

Figure 1. Localization and extent of all sites where reference data were collected. Forest districts for which semi-synthetic populations were generated are in red. BBH stands for: Białowieża, Browsk, Hajnówka, treated as one study area.

Table 1. Basic characteristics of selected forest districts.

Forest District	Area [km2]	GSV [X]	GSV [CV%]	AGE [X]	AGE [CV%]	Dominant Tree Species [In Descending Share Order]
Białowieża	81	357	66	99	51	Norway spruce, Oak sp., Hornbeam, Black alder, Scots pine
Browsk	115	397	53	94	50	Norway spruce, Black alder, Scots pine, Oak sp., Hornbeam
Hajnówka	117	424	55	112	52	Norway spruce, Scots pine, Black alder, Oak sp., Silver birch
Gorlice	159	373	45	64	28	European beech, Silver fir, Scots pine, Acer sp.
Herby	34	317	39	58	32	Scots pine, Oak sp., Silver birch
Dojlidy	158	399	37	60	26	Scots pine, Norway spruce
Milicz	53	371	46	60	38	Scots pine, Oak sp., European beech, Silver birch, Minor sp.
Pieńsk	86	303	36	54	29	Scots pine, Silver birch, Norway spruce
Supraśl	73	401	38	58	27	Scots pine, Norway spruce, Oak sp., Silver birch

In each study object, stand-level forest inventories were conducted between 2014 and 2016 as part of forest management planning. In Poland, forest management plans are prepared every 10 years for each forest district under the jurisdiction of the State Forests. According to the official forest management guidelines [34], an initial GSV estimate was obtained for each stand using the relascope (angle-count) method. These initial stand-level estimates were subsequently refined using the previously described plot-level GSV values. Specifically, a correction coefficient was calculated separately for each species–age layer as the ratio of averaged initial relascope-based estimates to the corresponding plot-level inventory result. These standard procedures were carried out by the data provider. The authors did not have access to the original measurements.

2.2. Remote Sensing Data

Airborne laser scanning data were acquired simultaneously with the forest inventory campaigns. The ALS point clouds provided full spatial coverage of each forest district. The LiDAR surveys were conducted using a RIEGL LiteMapper LMS-Q680i scanner (RIEGL Laser Measurement Systems GmbH., Horn, Austria) operating at pulse repetition frequencies ranging from 300 to 400 kHz. The average flight altitude above mean ground level was approximately 550 m, with a projected overlap of 30–40% between adjacent point-cloud (PC) tiles. The resulting average point-cloud densities varied among study objects within a relatively narrow range of 8–14 pulses per square metre (ppsm). To eliminate potential effects arising from differences in point-cloud density, all PC tiles were standardized to a common average density of 6 ppsm. Nevertheless, at the point densities considered in this study, such differences are not expected to have a substantial practical effect on ALS-based GSV estimates [72], even in the absence of density standardization [73]. In addition to the PC tiles, the RS data provider supplied high-accuracy digital terrain models (DTMs) with a spatial resolution of 0.5 m. The DTMs were used to normalize the heights of the ALS point clouds. The normalized ALS data were subsequently used to derive metrics for forest generation, as well as for stratification, sampling, and GSV estimation in the tested inventory scenarios (see subsequent sections).

At the same time, airborne hyperspectral data were provided for the Milicz, Herby, and BBH forest districts. The imagery was acquired using HySpex VNIR-1800 and HySpex SWIR-384 sensors (Norsk Elektro Optikk AS., Oslo, Norway). All hyperspectral data layers were preprocessed by the data provider, including orthorectification, geometric correction, and atmospheric calibration. The datasets acquired from the two sensors were subsequently merged, resampled, and stacked, resulting in 451 spectral bands with a spatial resolution of 5 m. These RS layers provided spatially continuous information on tree species composition. Species classification followed the procedures described in [96,97], using a support vector machine (SVM) classifier [98] for species identification. The initial classification performance is presented in Table 2. Although airborne hyperspectral imagery classified using SVM has been shown to provide reliable information on tree species composition in forested areas [99,100], additional RGB orthomosaics and stand-level field survey data were used to visually validate and refine the initial SVM classification results in areas where species identification was uncertain. The resulting species classifications were subsequently used for forest generation, stand stratification, and GSV prediction in scenarios assuming the availability of this type of data.

Table 2. Initial species classification accuracies.

Forest District	OA [%]	Kappa
Milicz	91.2	0.81
Herby	93.7	0.83
BBH	72.1	0.63

For all study areas, both ground surveys and ALS acquisitions were conducted within the same year. In this controlled experiment, temporal consistency minimized the influence of unaccounted-for variability associated with stand growth, tree mortality, and harvesting. We consider this close temporal alignment a major advantage of the study, as it enabled us to simulate a potential real-world forest management inventory scenario.

2.3. Auxiliary Variables

Four types of auxiliary variables were used in this study: ALS metrics, tree species information, stand age, and preliminary GSV estimates obtained from a generic model. The influence of auxiliary variables was tested in all possible combinations. These variables were used for forest generation, stratification, sampling, and GSV estimation.

Standard ALS metrics were computed in the R programming environment (v. 4.3.0) [101] using the lidR package (v. 4.1.1) [102]. In addition to the original metrics, their natural logarithms and power-transformed versions were calculated and included in the subsequent analyses. To reduce data dimensionality, given the noticeable autocorrelation among numerous computed metrics, only the three strongest GSV predictors derived from ALS data were retained based on the importance index embedded in the R randomForest function (v. 4.7-1.1) [103]. This pre-selection was performed only once, before the main simulations, using real inventory sample plots from all forest districts. Stand age was obtained from the stand-level inventories described in the previous sections, whereas species information was derived from hyperspectral data classifications. Preliminary GSV estimates were obtained using a generic random forest model trained on all available sample plots except those located within the forest district for which the preliminary estimates were being generated. Thus, for each forest district, the GM was calibrated using data from the remaining districts, thereby preventing information from the target district from being directly incorporated into the preliminary predictions. All types of auxiliary variables were used as predictors in the GM, except for stand-level GSV estimates from the real inventories. The latter were used exclusively as predictors for generating the reference GSV values. Table 3 provides a summary of all auxiliary variables used in this study.

Table 3. Auxiliary variables used in the study.

Type	Variable Name	Info
ALS metrics	Zmean	Mean height value of all returns
	Zq95_p	Power function of the 95th quantile of height values from all returns
	pZbv2_p	Power function from the ratio of the number of returns above the second height interval to the number of all returns [104]
Generic model *	Pre-GSV estimates	Pre-GSV estimates from the RF model trained on ex situ data
Stand-level inventories **	Rel+	GSV estimated for individual stands via relascope plots (see details in Section 2.1).
Species information (SP)	Conifer	Share of coniferous tree species
	Sp_dom	Dominant tree species (if share greater than 0.7, otherwise labelled as “mixed”)
Age	Age_ln	Natural logarithm of the average age of a sampling unit

* This variable was not used for the reference GSV generation. ** This variable was used only at a stage of the reference GSV generation.

2.4. Forest Generation

Semi-synthetic populations were generated for Milicz, Herby, and BBH, as hyperspectral datasets were available only for these study objects. To achieve spatially continuous coverage of the target variable, the inventoried areas were tessellated using hexagonal grids (Figure 2). Each grid cell had an area of 500 m², corresponding exactly to the area of the real inventory sample plots. Subsequently, a complete set of auxiliary variables was assigned to each grid cell.

Figure 2. Forest district tessellation.

The target variable was generated using the following procedure. First, an XGBoost model (v. 3.2.1.1) [105] was fitted using all available auxiliary variables observed at in situ real inventory sample plots. The initial XGB model was then used to predict raw GSV for each grid cell. Only a limited, arbitrarily selected set of XGB hyperparameter values was tested, as preliminary analyses indicated that model performance was relatively insensitive to their values in this study. The final settings were therefore selected heuristically, with reference to the characteristics of the real inventory data. The selected XGB hyperparameter values and the results of the sensitivity tests are provided in Appendix A.

Subsequently, leave-one-out (LOO) validation was performed, yielding one model residual for each real inventory sample plot. The k-nearest neighbours (kNN) algorithm [106], implemented in the FNN package for R (v. 1.1.4.1) [107], was applied to identify the three nearest inventory plots to the ith grid cell in the multidimensional space of auxiliary variables. The mean residual of these three nearest neighbours was added to the initial XGB prediction for the respective grid cell. In selecting the value of k, we aimed to balance the preservation of local variability with robustness while avoiding excessive smoothing resulting from an increasing number of neighbours [108]. Moreover, we drew on experience from similar studies in which kNN imputations were used to estimate forest attributes, with k = 3 being a commonly adopted choice [109,110,111,112,113]. Nevertheless, we acknowledge that imputing residuals differs in certain respects from imputing observed variable values.

Finally, a stochastic variation ranging from −5% to +5% was introduced. For each grid cell, the perturbation was randomly drawn from a discrete uniform distribution with a 0.1 percentage point [pp] precision, and applied to the residual-adjusted GSV predictions. This procedure was intended to introduce additional stochastic variability and reduce the deterministic component of the response-variable generation process. The concept of introducing stochastic variation into model-generated populations was also applied by [114], who incorporated Gaussian noise into synthetic populations, and by [115], who introduced a small jitter component when generating anticipated data (±0.02). Additionally, the authors performed sensitivity analyses to assess the effects of the number of residual neighbours (k) and the magnitude of stochastic variation on the characteristics of the generated populations (Appendix A). With the exception of k = 1, these factors were found to have very limited influence within the investigated ranges.

The resulting synthetic GSV was subsequently treated as the reference (“true”) GSV for each grid cell. The mathematical definition of the synthetic target variable is provided in Equation (1), while Figure 3 compares the GSV distributions of the semi-synthetic population and the real forest inventory plots.

$$V_i=\left[f_{\mathit{XGB}}\left(x_i\right)+{\displaystyle \frac{1}{3}}{\displaystyle {\sum }_{s\in N_3\left(i\right)}}\left(V_s-{\widehat{\mathrm{V}}}_{s,-s}^{\mathit{XGB}}\right)\right](1+u_s)$$

(1)

where

• V_i—synthetic GSV for ith grid cell;
• f_XGB()—basic XGBoost GSV prediction for ith cell;
• x_i—vector of predictor variables for ith cell;
• N₃(i)—3 nearest real inventory plots to ith synthetic plot/grid cell in multidimensional data space;
• V_s—observed GSV of sth real inventory plot;
• ${\widehat{\mathrm{V}}}_{s,-s}^{\mathit{XGB}}$—LOO-XGBoost GSV prediction for real sth plot;
• u_s—random multiplicative perturbation, where u_s ~ U(−0.05, 0.05).

Figure 3. GSV distributions according to real forest inventory data (grey bins) and synthetic response (blue line). Mean and standard deviation were provided for synthetic population and inventory data (in brackets).

Figure 3. GSV distributions according to real forest inventory data (grey bins) and synthetic response (blue line). Mean and standard deviation were provided for synthetic population and inventory data (in brackets).

2.5. Sampling Designs

The performance of three sampling designs was evaluated in this study: simple random sampling, systematic sampling, and stratified sampling, each examined under multiple variants (scenarios). The grid cells described in the previous section were used as the basic sampling units. The efficiency of each sampling design was evaluated based on the mean normalized BIAS and normalized root mean squared error (RMSE) across simulation scenarios that differed in terms of: sample selection method, sampling intensity, estimator type, and availability of auxiliary variables. For sampling designs incorporating an inherent stochastic component, namely SRS and SYS, 100 iterations were performed to assess the variability of the resulting estimators.

Uniform-probability simple random sampling without replacement was tested in its most basic form. For each forest district in each systematic sampling scenario, a rectangular sampling grid was overlaid on the study area 100 times. Sampling intervals ranged from 100 to 2000 m. However, only scenarios resulting in a maximum sampling intensity of 1000 plots or fewer were further considered. To preserve the statistical validity of the systematic sampling design, SYS was implemented using both a random starting point and a random grid orientation [116].

Hierarchical agglomerative clustering using the Ward.D linkage method [57,117], implemented through the hclust function in the fastcluster R package (v. 1.3.0) [118], was used to group sampling units in StrS. A previous study [58] demonstrated the particular suitability of this algorithm for stratifying forested areas based on ALS metrics. Instead of Euclidean distances, Gower’s coefficient [119], implemented in the daisy function of the cluster R package (v. 2.1.4) [120], was used to determine pairwise dissimilarities for clustering, as some scenarios included non-numerical auxiliary variables, such as tree species information. Owing to computational and memory limitations imposed by the software packages used, a random subset of sampling units (grid cells) was selected for clustering and actual sample selection, whereas GSV predictions and error estimation were performed using the full populations. Appendix D compares the main characteristics of the response variable distributions in the full semi-synthetic populations and their reduced counterparts. Given the similarity between these distributions, we expect this reduction to have had, at most, a marginal influence on the final results.

For StrS, three methods of allocating the sample size among strata were evaluated: allocation proportional to stratum size (Size), allocation proportional to stratum variance (Var), and Neyman’s optimal allocation (NOA) [121], which accounts for both factors. Within-stratum variability was evaluated using only the auxiliary variables available in a given scenario. For numerical variables, the coefficient of variation was calculated, whereas for categorical variables, normalized Pielou’s evenness was derived from Shannon entropy [122,123]. For each stratum, a weighted mean of the aforementioned variability measures was calculated, using the RF variable importance index as the weighting factor [103].

For within-stratum sample selection, three methods were evaluated: random selection (RAN), nearest-centroid selection (NC) akin to [54], and regular in-data-space sampling (REG). Before applying NC and REG within each stratum, all variables were standardized using z-scores (mean-centering and standard deviation-based scaling) to prevent biased sample selection resulting from differences in the scales of individual auxiliary variables.

For each stratum under REG, the stratum centroid was calculated as a vector of mean values across all numerical auxiliary variables available in a given scenario. Subsequently, the Euclidean distance between each sampling unit in a given stratum and the centroid was calculated. The range of distances was divided into equally spaced intervals based on the sample size allocated to the respective stratum. The sampling unit closest to each successive interval boundary was selected, with ties resolved by random selection. When only categorical variables were available, REG selection was based on sorting by sampling unit ID. A pseudocode representation of this procedure is provided in Appendix B.

2.6. GSV Estimation and Evaluation

Two GSV estimators, namely random forest regression and ordinary linear regression, were independently fitted using samples selected according to the procedures described in the previous section.

The RF estimator was used as implemented in R following Breiman’s random forest algorithm [103,124]. This implementation involves a stochastic component and may therefore produce slightly different results across runs. To account for this variability, the RF procedure was repeated 10 times for each REG and NC sampling variants of StrS. No RF repetitions were designed for SRS, SYS and StrS-REG, after observing relative insensitivity of scores between RF runs, and assuming that sampling repetitions should already account for this source of limited variability. Details about the exact numbers of RF runs in particular sampling methods, as well as their lack of practical influence on the variability of the results, are demonstrated in Appendix G. The following RF hyperparameters were used: ntree = 500, mtry = 0.6 × the number of available predictors, and node size = 5. A brief hyperparameter tuning was performed (Appendix E); however, previous studies have reported that these parameters generally have a relatively limited influence on model performance beyond certain threshold values [125,126,127].

The OLR estimator included all predictors available in a given scenario. However, to ensure computational feasibility and avoid the excessive number of possible model specifications, interactions between auxiliary variables were not included in the OLR models. Nevertheless, such interactions are inherently accounted for by RF. In scenarios involving categorical variables, separate linear regression models were fitted for each group. When only categorical variables were available, the arithmetic mean was used as the estimator for each respective stratum.

No outlier treatment was applied at this stage. The complete set of selected samples was used for model fitting and subsequent GSV prediction in every scenario. Given that model-based forest populations tend to simplify distributions (Figure 3) [88,89], we assumed that this effect would have a limited influence on our study design. In every scenario, the models resulting from each iteration were used to predict GSV for all hexagonal grid cells representing the units of the semi-synthetic forest population. The performance of each model instance in every scenario was evaluated using mean normalized BIAS (Equation (2)) and normalized RMSE (Equation (3)), calculated across the entire population, with grid cells treated as individual observations. Figure 4 provides a graphical summary of the overall methodological framework.

$$\mathit{BIAS}(\%)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}({\hat{y}}_i-y_i)}{n}}{\displaystyle \frac{100}{\mathsf{\mu }}}$$

(2)

$$\mathit{RMSE}(\%)=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{({\hat{y}}_i-y_i)}^2}{n}}}{\displaystyle \frac{100}{\mathsf{\mu }}}$$

(3)

where

• ${\hat{y}}_i$—estimated GSV value for i-th grid cell;
• y_i—reference GSV value of i-th grid cell;
• n—number of observations (grid cells);
• μ—population GSV mean.

Figure 4. Graphical summary of the methodology.

Figure 4. Graphical summary of the methodology.

3. Results

In this section, each forest inventory method is first examined separately to reduce the multidimensionality of the analysis and identify the most suitable variants. The final subsection compares the most promising variants across all sampling methods.

3.1. Simple Random Sampling

Under this sampling design, ALS metrics alone resulted in the lowest estimation errors in most cases (Table 4 and Table 5). Adding GM pre-estimates slightly reduced RMSE and BIAS in some cases; however, the improvements were marginal, amounting to only 1–2 pp. When used alone, stand age or species information resulted in substantially higher estimation errors, particularly at lower sampling intensities. Combining these variables with ALS metrics or GM pre-estimates resulted in only marginal improvements, again of approximately 1–2 pp. Differences between the RF and OLR estimators in terms of GSV estimation errors were negligible in most SRS configurations. However, at lower sampling intensities, OLR tended to produce slightly lower error values than RF, whereas the opposite pattern was observed at higher sampling intensities. This observation is consistent with the general understanding that machine learning methods typically require larger sample sizes than conventional regression models to produce stable estimates [128,129,130]. Regarding the influence of sampling intensity under SRS, the results suggest that the convergence zone for the most effective configuration, i.e., ALS predictors combined with the RF model, occurs at approximately 150–200 plots (Figure 5 and Figure 6).

Table 4. BIAS scores in particular configurations of SRS (the highest error from 100 iterations).

	BIAS [%]—Estimator: RF (OLR)
Forest District	BBH			Herby			Milicz
Variables\n	100	250	500	100	250	500	100	250	500
GM	6 (5)	4 (3)	2 (3)	7 (7)	4 (5)	3 (2)	4 (4)	3 (3)	2 (2)
ALS	6 (5)	3 (3)	2 (2)	4 (4)	2 (3)	1 (2)	3 (4)	2 (3)	1 (1)
AGE	10 (13)	8 (7)	4 (4)	8 (8)	6 (6)	3 (3)	8 (8)	5 (6)	4 (4)
SP	12 (9)	7 (7)	4 (4)	10 (8)	5 (5)	4 (4)	10 (10)	9 (9)	6 (6)
ALS + GM	4 (5)	3 (3)	2 (2)	4 (4)	3 (3)	1 (2)	3 (4)	2 (3)	1 (1)
AGE + GM	6 (5)	4 (4)	2 (3)	6 (6)	4 (5)	2 (2)	4 (4)	2 (3)	1 (2)
AGE + ALS	5 (5)	3 (3)	2 (2)	4 (4)	3 (3)	1 (1)	2 (4)	2 (3)	1 (1)
SP + GM	6 (5)	4 (4)	2 (3)	7 (6)	4 (4)	2 (2)	4 (3)	2 (2)	1 (2)
SP + ALS	5 (5)	3 (3)	2 (2)	4 (4)	2 (3)	1 (1)	3 (3)	2 (3)	1 (1)
SP + AGE	13 (13)	8 (7)	4 (4)	8 (8)	5 (5)	3 (3)	7 (8)	5 (5)	3 (4)
AGE + ALS + GM	4 (5)	3 (3)	2 (2)	4 (4)	3 (3)	1 (1)	3 (4)	2 (3)	1 (1)
SP + ALS + GM	5 (5)	3 (4)	2 (2)	3 (3)	3 (3)	1 (1)	3 (3)	2 (2)	1 (1)
SP + AGE + GM	5 (5)	3 (3)	2 (3)	6 (6)	3 (4)	2 (3)	3 (4)	2 (3)	1 (2)
SP + AGE + ALS	6 (5)	3 (3)	2 (2)	4 (4)	2 (3)	1 (1)	3 (4)	2 (2)	1 (1)
SP + AGE + ALS + GM	5 (5)	3 (3)	2 (3)	4 (3)	3 (3)	1 (1)	3 (3)	2 (2)	1 (1)

RF—Random Forest, OLR—Ordinary Linear Regression.

Table 5. RMSE scores in particular configurations of SRS (the highest error from 100 iterations).

	RMSE [%]—Estimator: RF (OLR)
Forest District	BBH			Herby			Milicz
Variables\n	100	250	500	100	250	500	100	250	500
GM	24 (21)	24 (20)	23 (20)	30 (26)	29 (26)	28 (25)	18 (15)	16 (14)	16 (14)
ALS	21 (20)	19 (20)	18 (19)	14 (14)	13 (14)	12 (13)	15 (16)	14 (15)	12 (15)
AGE	47 (42)	46 (39)	44 (39)	35 (33)	32 (32)	31 (32)	42 (37)	37 (36)	36 (36)
SP	45 (46)	43 (42)	42 (41)	40 (38)	37 (36)	37 (36)	47 (46)	44 (43)	43 (42)
ALS + GM	20 (20)	18 (19)	18 (19)	13 (14)	12 (13)	11 (12)	13 (15)	12 (14)	11 (13)
AGE + GM	24 (21)	21 (20)	21 (20)	26 (25)	25 (25)	24 (24)	18 (15)	15 (14)	14 (14)
AGE + ALS	20 (20)	18 (20)	18 (19)	14 (14)	12 (14)	12 (13)	14 (16)	13 (15)	11 (15)
SP + GM	23 (22)	22 (21)	21 (20)	23 (22)	21 (21)	20 (20)	16 (15)	14 (14)	13 (14)
SP + ALS	20 (21)	18 (20)	18 (19)	13 (13)	11 (13)	10 (12)	13 (15)	12 (13)	10 (13)
SP + AGE	43 (42)	42 (40)	41 (39)	36 (34)	33 (33)	32 (32)	37 (40)	34 (36)	32 (35)
AGE + ALS + GM	20 (20)	18 (19)	18 (19)	13 (14)	11 (13)	11 (12)	13 (15)	12 (14)	11 (13)
SP + ALS + GM	20 (22)	18 (21)	17 (19)	13 (14)	12 (13)	11 (12)	13 (14)	12 (13)	10 (12)
SP + AGE + GM	22 (23)	21 (21)	20 (20)	23 (22)	20 (20)	20 (20)	16 (15)	14 (14)	13 (14)
SP + AGE + ALS	21 (21)	18 (20)	17 (19)	13 (14)	11 (13)	10 (12)	14 (15)	12 (14)	10 (13)
SP + AGE + ALS + GM	20 (22)	18 (20)	17 (19)	13 (14)	11 (13)	10 (12)	13 (14)	12 (13)	10 (12)

Figure 5. BIAS of selected SRS configuration: ALS predictors, RF estimator (the highest error).

Figure 6. RMSE of selected SRS configuration: ALS predictors, RF estimator (the highest error).

3.2. Systematic Sampling

Table 6 clearly indicates that the choice of estimator in the two-phase systematic sampling design was of secondary importance for GSV estimation in SYS. At all sampling intensity levels, the differences in BIAS between the two estimator types did not exceed 0.2 percentage points (values in brackets). In contrast, a slight improvement in RMSE, of up to 2.2 percentage points, was observed when RF was used as the GSV estimator, particularly at higher sampling intensities. These results provide further evidence that RF models can benefit from larger sample sizes [131]. Given the slight advantage of RF as a GSV estimator, the subsequent analysis focuses primarily on this estimator.

Table 6. Most efficient types of estimators in systematic sampling.

BIAS %	Object			RMSE %	Object
N	Milicz	Herby	BBH	N	Milicz	Herby	BBH
N ~ 500	0.5 (0.1)	0.7 (0.1)	0.7 (0.1)	N ~ 500	16 (2.2)	18 (1.2)	22 (0.6)
N ~ 250	0.8 (0.1)	1.0 (0.1)	1.1 (0.1)	N ~ 250	17 (0.8)	18 (0.7)	23 (0.2)
N ~ 100	1.5 (0.2)	1.4 (0.2)	1.9 (0.0)	N ~ 100	18 (0.6)	19 (0.0)	24 (0.1)

Winning estimator type: RF (green cells), no difference greater than 1 pp (light-blue cells). Values outside brackets are mean absolute percentage scores from all configurations. Values within brackets are differences (percentage points) between score of the winning estimator and score of the inferior estimator.

Table 7 presents the performance of the RF estimator under systematic sampling for different combinations of auxiliary variables used for GSV prediction. The main purpose of this analysis was to narrow down the number of possible variable combinations and identify the most promising ones. In most cases, ALS predictors resulted in the lowest RMSE values, whereas BIAS remained relatively stable across the different configurations, possibly due to the averaging of estimates. Adding preliminary GSV estimates from the generic model slightly reduced mean RMSE values by up to 1–2 percentage points (pp) in the Herby and Milicz forest districts, which exhibited lower structural complexity than BBH. The addition of other auxiliary variables did not substantially improve estimation accuracy. Using only stand age or species information resulted in considerably higher RMSE values and slightly higher BIAS, with differences of up to 3 pp observed mainly at lower sampling intensities.

Table 7. RF in SYS efficiency, BIAS% (RMSE%), across all variable config. (mean absolute values over 100 iterations).

N	~500			~250			~100
Variables\Object	Herby	BBH	Milicz	Herby	BBH	Milicz	Herby	BBH	Milicz
ALS	0 (11)	1 (16)	0 (11)	1 (12)	1 (17)	1 (12)	1 (13)	2 (18)	1 (13)
GM	1 (27)	1 (22)	0 (15)	1 (27)	1 (22)	1 (15)	2 (28)	2 (22)	1 (15)
AGE	1 (31)	2 (43)	1 (34)	2 (31)	2 (43)	2 (35)	2 (32)	4 (44)	2 (36)
SP	2 (36)	1 (43)	1 (42)	3 (37)	2 (44)	2 (42)	5 (38)	4 (44)	4 (45)
ALS + GM	0 (10)	1 (16)	0 (11)	1 (10)	1 (17)	1 (11)	1 (11)	2 (18)	1 (12)
AGE + GM	1 (23)	1 (20)	1 (14)	1 (24)	1 (21)	1 (14)	2 (24)	2 (21)	1 (16)
AGE + ALS	0 (11)	1 (16)	0 (11)	1 (11)	1 (17)	1 (11)	1 (12)	2 (18)	1 (12)
SP + GM	1 (19)	1 (20)	0 (12)	1 (20)	1 (20)	1 (13)	2 (21)	2 (21)	1 (14)
SP + ALS	0 (10)	1 (16)	0 (10)	1 (10)	1 (17)	1 (10)	1 (12)	2 (18)	1 (12)
SP + AGE	1 (31)	1 (39)	1 (31)	2 (32)	2 (40)	1 (32)	3 (33)	3 (41)	2 (34)
AGE + ALS + GM	0 (10)	1 (16)	0 (11)	1 (10)	1 (17)	1 (11)	1 (11)	1 (18)	1 (12)
SP + ALS + GM	0 (10)	1 (16)	0 (10)	1 (10)	1 (17)	1 (10)	1 (11)	2 (18)	1 (12)
SP + AGE + GM	1 (19)	1 (20)	0 (12)	1 (19)	1 (20)	1 (13)	2 (21)	2 (21)	1 (14)
SP + AGE + ALS	0 (10)	1 (16)	0 (10)	1 (10)	1 (16)	1 (10)	1 (11)	2 (18)	1 (12)
SP + AGE + ALS + GM	0 (10)	1 (16)	0 (10)	1 (10)	1 (17)	1 (10)	1 (11)	2 (18)	1 (12)

The lines in Figure 7 and Figure 8 illustrate the worst-case performance of the RF estimator under SYS for GSV prediction in a two-phase forest inventory using only ALS metrics. They represent the worst results observed across 100 iterations at each sampling intensity level. The highest estimation errors were observed in BBH, whereas the lowest were recorded in Herby and Milicz, which exhibited similar performance. This pattern corresponds well with the structural complexity of the respective study areas. BIAS values stabilized at approximately 400 sample plots (Figure 7), whereas RMSE values reached a plateau somewhat earlier, at approximately 300 plots, across all study areas (Figure 8).

Figure 7. BIAS of selected config. of SYS: ALS variables, RF estimator (the highest errors).

Figure 8. RMSE of selected config. of SYS: ALS variables, RF estimator (the highest errors).

3.3. Stratified Sampling

Table 8 and Table 9 show the distributions of absolute BIAS and RMSE values for selected configurations of the stratified sampling design. These results are based exclusively on ALS variables, as their suitability for GSV prediction was confirmed in Table 10. Again, the RF estimator showed a slight advantage over OLR in some configurations. Among the variants considered, one configuration (highlighted in aquamarine) provided consistently low and relatively stable BIAS values, regardless of the study area or sampling intensity. This configuration combined Neyman’s optimal allocation (NOA) of the total sample size among strata, regular in-data-space sampling (REG), and RF as the GSV estimator. Similar performance was achieved using sample allocation proportional to stratum size. However, at lower sampling intensities, this approach was less stable than the selected configuration, resulting in BIAS values of up to 4% in the Milicz forest district, compared with consistently observed values of 1–2% across all study areas for the selected configuration.

Table 8. BIAS distribution [%] for selected configurations of StrS (worst score from all iterations).

		Estimator: RF									Estimator: OLR
SA	ICS	n = 100			n = 250			n = 500			n = 100			n = 250			n = 500			Obj
		k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15
NOA	NC	3	2	0	4	1	2	1	3	1	7	2	1	6	2	2	7	3	1	BBH
	RAN	5	6	4	3	3	3	2	2	2	5	5	4	3	3	3	2	2	2
	REG	1	1	2	1	3	1	1	0	0	3	2	2	2	2	1	0	1	0
Size	NC	3	2	1	3	1	0	3	3	1	6	2	1	6	3	1	6	3	1
	RAN	4	5	4	3	2	3	2	2	2	5	5	4	3	3	3	2	2	2
	REG	1	2	2	2	0	3	1	0	1	0	5	2	1	1	3	2	1	1
Var	NC	3	2	1	2	2	0	3	1	1	6	1	1	6	2	2	5	2	1
	RAN	4	7	6	3	3	4	2	3	3	4	5	6	3	4	4	2	3	2
	REG	2	2	4	1	2	1	1	0	1	3	2	3	3	2	1	2	1	1
NOA	NC	1	1	2	6	1	1	5	2	1	2	1	1	3	1	0	3	1	1	Herby
	RAN	4	3	4	2	3	2	1	1	1	3	3	4	2	3	2	1	2	2
	REG	1	1	1	2	1	2	1	1	1	2	1	0	4	3	1	4	2	2
Size	NC	1	1	1	8	0	0	9	3	2	2	2	1	3	1	1	3	1	1
	RAN	3	2	3	2	2	1	1	1	1	3	2	3	4	2	2	2	1	1
	REG	1	1	0	1	0	1	2	1	0	3	0	1	5	1	1	3	2	1
Var	NC	1	2	2	5	2	0	5	1	0	3	2	1	3	1	1	3	1	1
	RAN	3	6	5	2	4	3	1	2	2	4	6	6	2	4	4	2	2	2
	REG	2	1	4	0	0	1	1	0	1	2	0	4	3	0	1	3	1	1
NOA	NC	1	2	3	1	0	1	1	1	0	1	1	4	1	2	3	2	1	2	Milicz
	RAN	4	2	4	3	2	2	2	1	1	5	4	4	2	2	2	2	1	1
	REG	1	4	0	1	0	1	1	2	0	5	5	1	3	2	3	4	4	3
Size	NC	3	1	3	0	1	1	0	2	0	2	2	5	2	2	3	2	1	2
	RAN	3	3	2	2	2	2	1	1	2	3	4	3	3	2	2	1	1	1
	REG	4	1	2	2	2	2	0	0	1	5	1	0	5	4	4	5	3	2
Var	NC	2	3	3	1	1	1	3	2	1	2	2	4	3	2	3	2	1	2
	RAN	5	4	3	3	2	2	2	2	1	6	3	3	2	3	2	3	2	2
	REG	3	2	1	1	2	1	1	1	0	5	2	1	5	3	2	6	4	3

Selected config. highlighted in aquamarine; SA—sample allocation; Var—sample allocation by strata variance; Size—sample allocation by strata size; NOA—Neyman optimal allocation; ICS—in cluster/stratum sampling.

Table 8. BIAS distribution [%] for selected configurations of StrS (worst score from all iterations).

		Estimator: RF									Estimator: OLR
SA	ICS	n = 100			n = 250			n = 500			n = 100			n = 250			n = 500			Obj
		k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15
NOA	NC	3	2	0	4	1	2	1	3	1	7	2	1	6	2	2	7	3	1	BBH
	RAN	5	6	4	3	3	3	2	2	2	5	5	4	3	3	3	2	2	2
	REG	1	1	2	1	3	1	1	0	0	3	2	2	2	2	1	0	1	0
Size	NC	3	2	1	3	1	0	3	3	1	6	2	1	6	3	1	6	3	1
	RAN	4	5	4	3	2	3	2	2	2	5	5	4	3	3	3	2	2	2
	REG	1	2	2	2	0	3	1	0	1	0	5	2	1	1	3	2	1	1
Var	NC	3	2	1	2	2	0	3	1	1	6	1	1	6	2	2	5	2	1
	RAN	4	7	6	3	3	4	2	3	3	4	5	6	3	4	4	2	3	2
	REG	2	2	4	1	2	1	1	0	1	3	2	3	3	2	1	2	1	1
NOA	NC	1	1	2	6	1	1	5	2	1	2	1	1	3	1	0	3	1	1	Herby
	RAN	4	3	4	2	3	2	1	1	1	3	3	4	2	3	2	1	2	2
	REG	1	1	1	2	1	2	1	1	1	2	1	0	4	3	1	4	2	2
Size	NC	1	1	1	8	0	0	9	3	2	2	2	1	3	1	1	3	1	1
	RAN	3	2	3	2	2	1	1	1	1	3	2	3	4	2	2	2	1	1
	REG	1	1	0	1	0	1	2	1	0	3	0	1	5	1	1	3	2	1
Var	NC	1	2	2	5	2	0	5	1	0	3	2	1	3	1	1	3	1	1
	RAN	3	6	5	2	4	3	1	2	2	4	6	6	2	4	4	2	2	2
	REG	2	1	4	0	0	1	1	0	1	2	0	4	3	0	1	3	1	1
NOA	NC	1	2	3	1	0	1	1	1	0	1	1	4	1	2	3	2	1	2	Milicz
	RAN	4	2	4	3	2	2	2	1	1	5	4	4	2	2	2	2	1	1
	REG	1	4	0	1	0	1	1	2	0	5	5	1	3	2	3	4	4	3
Size	NC	3	1	3	0	1	1	0	2	0	2	2	5	2	2	3	2	1	2
	RAN	3	3	2	2	2	2	1	1	2	3	4	3	3	2	2	1	1	1
	REG	4	1	2	2	2	2	0	0	1	5	1	0	5	4	4	5	3	2
Var	NC	2	3	3	1	1	1	3	2	1	2	2	4	3	2	3	2	1	2
	RAN	5	4	3	3	2	2	2	2	1	6	3	3	2	3	2	3	2	2
	REG	3	2	1	1	2	1	1	1	0	5	2	1	5	3	2	6	4	3

Selected config. highlighted in aquamarine; SA—sample allocation; Var—sample allocation by strata variance; Size—sample allocation by strata size; NOA—Neyman optimal allocation; ICS—in cluster/stratum sampling.

Table 9. RMSE distribution [%] for selected configurations of StrS (worst score from all iterations).

		Estimator: RF									Estimator: OLR
SA	ICS	n = 100			n = 250			n = 500			n = 100			n = 250			n = 500			Obj
		k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15
NOA	NC	28	23	20	26	23	20	24	22	20	27	30	25	22	25	20	20	25	21	BBH
	RAN	20	20	19	18	17	18	17	17	17	21	20	22	19	19	19	19	18	18
	REG	18	17	18	17	17	17	16	16	17	21	20	19	19	19	18	19	18	18
Size	NC	30	23	20	30	23	20	25	22	19	26	33	21	25	26	21	21	24	20
	RAN	22	19	20	17	18	17	17	17	17	21	21	21	19	19	18	19	18	18
	REG	17	17	17	17	17	17	16	16	16	19	19	19	19	19	18	19	18	18
Var	NC	28	24	20	30	23	19	24	23	19	25	30	22	24	23	22	20	24	20
	RAN	21	22	22	17	18	18	16	17	17	20	21	23	20	20	20	19	19	18
	REG	17	18	18	17	18	17	17	16	17	20	21	20	19	20	18	19	19	18
NOA	NC	21	15	14	21	14	14	20	14	14	24	18	16	16	16	15	14	15	15	Herby
	RAN	14	14	14	12	13	13	12	12	12	14	15	15	14	13	14	14	13	13
	REG	14	13	13	13	12	13	12	12	12	16	14	15	15	14	14	15	14	13
Size	NC	19	15	15	24	14	14	24	14	13	21	19	15	14	14	16	16	13	13
	RAN	13	14	14	12	12	12	12	12	12	15	14	14	14	13	13	13	13	12
	REG	12	13	13	12	12	12	12	12	12	14	13	13	15	12	13	14	13	13
Var	NC	18	14	14	21	14	14	20	15	14	15	18	17	15	18	16	15	17	16
	RAN	15	16	16	13	13	13	12	12	13	15	17	18	14	15	15	14	14	14
	REG	13	14	14	13	12	13	12	12	13	15	16	16	15	14	15	15	14	14
NOA	NC	21	16	15	20	14	14	19	15	13	49	43	19	25	15	19	15	17	15	Milicz
	RAN	15	16	14	13	13	13	12	12	12	17	17	15	15	14	14	15	14	13
	REG	13	14	12	12	12	12	12	12	11	16	14	13	16	13	14	16	14	13
Size	NC	22	15	15	21	14	13	20	14	13	18	19	42	20	21	27	14	17	18
	RAN	14	15	14	13	13	13	12	12	12	14	16	14	14	14	13	14	13	12
	REG	13	12	13	13	12	12	12	12	12	16	14	14	15	14	13	15	13	13
Var	NC	23	14	15	20	14	14	21	14	13	55	25	27	19	14	17	27	14	13
	RAN	15	15	14	14	13	12	13	12	12	20	16	14	18	14	13	18	14	13
	REG	14	13	12	12	12	12	11	12	11	17	14	13	17	14	13	18	14	13

Selected configuration was highlighted in aquamarine; description identical to that of Table 8.

Table 9. RMSE distribution [%] for selected configurations of StrS (worst score from all iterations).

		Estimator: RF									Estimator: OLR
SA	ICS	n = 100			n = 250			n = 500			n = 100			n = 250			n = 500			Obj
		k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15	k3	k9	k15
NOA	NC	28	23	20	26	23	20	24	22	20	27	30	25	22	25	20	20	25	21	BBH
	RAN	20	20	19	18	17	18	17	17	17	21	20	22	19	19	19	19	18	18
	REG	18	17	18	17	17	17	16	16	17	21	20	19	19	19	18	19	18	18
Size	NC	30	23	20	30	23	20	25	22	19	26	33	21	25	26	21	21	24	20
	RAN	22	19	20	17	18	17	17	17	17	21	21	21	19	19	18	19	18	18
	REG	17	17	17	17	17	17	16	16	16	19	19	19	19	19	18	19	18	18
Var	NC	28	24	20	30	23	19	24	23	19	25	30	22	24	23	22	20	24	20
	RAN	21	22	22	17	18	18	16	17	17	20	21	23	20	20	20	19	19	18
	REG	17	18	18	17	18	17	17	16	17	20	21	20	19	20	18	19	19	18
NOA	NC	21	15	14	21	14	14	20	14	14	24	18	16	16	16	15	14	15	15	Herby
	RAN	14	14	14	12	13	13	12	12	12	14	15	15	14	13	14	14	13	13
	REG	14	13	13	13	12	13	12	12	12	16	14	15	15	14	14	15	14	13
Size	NC	19	15	15	24	14	14	24	14	13	21	19	15	14	14	16	16	13	13
	RAN	13	14	14	12	12	12	12	12	12	15	14	14	14	13	13	13	13	12
	REG	12	13	13	12	12	12	12	12	12	14	13	13	15	12	13	14	13	13
Var	NC	18	14	14	21	14	14	20	15	14	15	18	17	15	18	16	15	17	16
	RAN	15	16	16	13	13	13	12	12	13	15	17	18	14	15	15	14	14	14
	REG	13	14	14	13	12	13	12	12	13	15	16	16	15	14	15	15	14	14
NOA	NC	21	16	15	20	14	14	19	15	13	49	43	19	25	15	19	15	17	15	Milicz
	RAN	15	16	14	13	13	13	12	12	12	17	17	15	15	14	14	15	14	13
	REG	13	14	12	12	12	12	12	12	11	16	14	13	16	13	14	16	14	13
Size	NC	22	15	15	21	14	13	20	14	13	18	19	42	20	21	27	14	17	18
	RAN	14	15	14	13	13	13	12	12	12	14	16	14	14	14	13	14	13	12
	REG	13	12	13	13	12	12	12	12	12	16	14	14	15	14	13	15	13	13
Var	NC	23	14	15	20	14	14	21	14	13	55	25	27	19	14	17	27	14	13
	RAN	15	15	14	14	13	12	13	12	12	20	16	14	18	14	13	18	14	13
	REG	14	13	12	12	12	12	11	12	11	17	14	13	17	14	13	18	14	13

Selected configuration was highlighted in aquamarine; description identical to that of Table 8.

Table 10. Influence of variable types for GSV estimation in selected configuration of StrS.

	BIAS (RMSE) [%]
Forest District	BBH			Herby			Milicz
Variables\n	100	250	500	100	250	500	100	250	500
GM	11 (25)	8 (24)	7 (23)	6 (28)	4 (27)	4 (26)	6 (21)	6 (19)	7 (18)
ALS	1 (18)	1 (17)	1 (16)	1 (14)	1 (13)	1 (12)	1 (13)	1 (12)	1 (12)
AGE	6 (42)	6 (40)	4 (38)	8 (34)	2 (31)	2 (31)	3 (35)	1 (34)	3 (34)
SP	6 (44)	4 (43)	7 (47)	7 (39)	5 (37)	3 (36)	14 (47)	12 (53)	4 (44)
ALS + GM	2 (18)	1 (18)	1 (17)	1 (12)	1 (11)	1 (11)	2 (13)	1 (11)	1 (11)
AGE + GM	2 (20)	1 (21)	2 (21)	3 (25)	2 (24)	2 (24)	2 (15)	1 (15)	1 (14)
AGE + ALS	3 (17)	1 (17)	1 (16)	1 (12)	1 (12)	1 (12)	2 (12)	1 (12)	1 (11)
SP + GM	7 (23)	1 (20)	1 (20)	2 (22)	2 (21)	1 (20)	5 (16)	3 (15)	2 (14)
SP + ALS	1 (17)	1 (18)	1 (16)	1 (12)	1 (10)	1 (10)	2 (12)	2 (10)	1 (10)
SP + AGE	4 (39)	4 (40)	4 (39)	11 (32)	10 (34)	6 (34)	17 (41)	11 (42)	1 (36)
AGE + ALS + GM	3 (17)	1 (17)	1 (16)	1 (12)	2 (11)	1 (10)	2 (13)	1 (12)	1 (11)
SP + ALS + GM	1 (19)	2 (18)	1 (17)	4 (11)	1 (10)	1 (10)	1 (12)	1 (11)	1 (10)
SP + AGE + GM	4 (21)	4 (20)	1 (20)	2 (24)	2 (22)	2 (21)	2 (16)	3 (14)	3 (13)
SP + AGE + ALS	4 (18)	1 (17)	2 (16)	1 (12)	1 (11)	1 (10)	3 (12)	1 (11)	1 (10)
SP + AGE + ALS + GM	2 (19)	1 (17)	1 (17)	2 (12)	1 (10)	1 (10)	1 (11)	1 (10)	1 (10)

Selected configuration: k = 3, SA: NOA, ICS: REG.

Table 10. Influence of variable types for GSV estimation in selected configuration of StrS.

	BIAS (RMSE) [%]
Forest District	BBH			Herby			Milicz
Variables\n	100	250	500	100	250	500	100	250	500
GM	11 (25)	8 (24)	7 (23)	6 (28)	4 (27)	4 (26)	6 (21)	6 (19)	7 (18)
ALS	1 (18)	1 (17)	1 (16)	1 (14)	1 (13)	1 (12)	1 (13)	1 (12)	1 (12)
AGE	6 (42)	6 (40)	4 (38)	8 (34)	2 (31)	2 (31)	3 (35)	1 (34)	3 (34)
SP	6 (44)	4 (43)	7 (47)	7 (39)	5 (37)	3 (36)	14 (47)	12 (53)	4 (44)
ALS + GM	2 (18)	1 (18)	1 (17)	1 (12)	1 (11)	1 (11)	2 (13)	1 (11)	1 (11)
AGE + GM	2 (20)	1 (21)	2 (21)	3 (25)	2 (24)	2 (24)	2 (15)	1 (15)	1 (14)
AGE + ALS	3 (17)	1 (17)	1 (16)	1 (12)	1 (12)	1 (12)	2 (12)	1 (12)	1 (11)
SP + GM	7 (23)	1 (20)	1 (20)	2 (22)	2 (21)	1 (20)	5 (16)	3 (15)	2 (14)
SP + ALS	1 (17)	1 (18)	1 (16)	1 (12)	1 (10)	1 (10)	2 (12)	2 (10)	1 (10)
SP + AGE	4 (39)	4 (40)	4 (39)	11 (32)	10 (34)	6 (34)	17 (41)	11 (42)	1 (36)
AGE + ALS + GM	3 (17)	1 (17)	1 (16)	1 (12)	2 (11)	1 (10)	2 (13)	1 (12)	1 (11)
SP + ALS + GM	1 (19)	2 (18)	1 (17)	4 (11)	1 (10)	1 (10)	1 (12)	1 (11)	1 (10)
SP + AGE + GM	4 (21)	4 (20)	1 (20)	2 (24)	2 (22)	2 (21)	2 (16)	3 (14)	3 (13)
SP + AGE + ALS	4 (18)	1 (17)	2 (16)	1 (12)	1 (11)	1 (10)	3 (12)	1 (11)	1 (10)
SP + AGE + ALS + GM	2 (19)	1 (17)	1 (17)	2 (12)	1 (10)	1 (10)	1 (11)	1 (10)	1 (10)

Selected configuration: k = 3, SA: NOA, ICS: REG.

Based on the visual inspection of Table 9, the RMSE scores were less susceptible to the sample allocation method than to in-cluster sampling (ICS). The NC usually provided the worst scores, whereas RAN and REG were competitive, with a slight advantage for REG.

Table 10 presents observed upper limits of BIAS and RMSE in StrS, using RF as the GSV estimator, considering different sets of variables used for stratification, NOA/REG sample selection and GSV prediction. The results indicate that ALS variables alone provided the most robust and stable performance across sampling intensity levels and forest districts. Their use maintained BIAS at a consistent level of approximately 1% and yielded relatively low RMSE values, ranging from 13% in Milicz to 18% in BBH. Adding other types of auxiliary variables, such as preliminary GSV estimates from the generic model or stand age and species information, resulted in further reductions in RMSE of up to 2 percentage points (pp) in some scenarios. However, these additions could also introduce additional noise and consequently increase BIAS by up to 2 pp in some cases. Table 10 further indicates that including more than two types of auxiliary variables generally provided no additional benefit and could even be detrimental, potentially due to increased data dimensionality, collinearity, and additional noise associated with the inclusion of further variables and their quality. Most importantly, all configurations that excluded ALS metrics resulted in noticeably higher estimation errors.

Figure 9 illustrates the influence of the number of strata on GSV estimation performance under selected stratified sampling configurations at different sampling intensities. The trend lines represent power functions fitted to the scores obtained across all iterations within each scenario. Figure 9 suggests a convergence zone at approximately 150–200 sample plots. Above this range, no practically meaningful differences in performance were observed, whereas below this threshold, the results appeared to be more strongly influenced by random variation and other unexplained effects, making it difficult to draw consistent conclusions. For example, at lower sampling intensity levels, the BIAS results showed distinct patterns across individual forest districts, depending on the number of strata.

Figure 9. Influence of the number of clusters on performance of selected StrS configuration: 3—green; 9—orange; 15—blue.

As the results demonstrate, several StrS configurations produced competitive results, with slight variations across study areas. To reduce the dimensionality of the analysis and facilitate a clear and concise comparison of the tested sampling designs, we ultimately selected one preferred StrS configuration. Based on a visual inspection of Table 8, Neyman’s optimal allocation combined with regular in-data-space sampling was selected. The results presented in Table 10 indicated that ALS variables alone were the most suitable predictors for GSV estimation. A slight advantage of the RF estimator over OLR was also evident in Table 8 and Table 9. Finally, three strata were selected, as their number had only a limited influence on estimation performance when the sample size exceeded approximately 150–200 plots (Figure 9).

3.4. Direct Comparison

Figure 10 provides a direct comparison of the analyzed sampling methods using the selected configurations. The results represent the least favourable iteration in each scenario, i.e., the largest estimation error observed across the finite set of simulation runs. At medium and higher sampling intensities (approximately 250–300 plots and above), the choice of sampling design was of secondary importance for GSV estimation in the two-phase, model-based, ALS-assisted forest inventory. In particular, the results indicated that increasing the sample size beyond approximately 300 plots provided little practical benefit. However, at lower sampling intensities, particularly around 100 plots, the advantage of the proposed StrS configuration (green line) became more pronounced, as reflected by its relatively stable BIAS and RMSE values. Notably, this advantage was more evident in structurally complex forest districts, such as BBH and, to a lesser extent, Milicz, than in areas consisting predominantly of relatively simple stands, such as Herby. Apart from StrS, SRS and SYS exhibited relatively similar performance across the entire range of tested sampling intensities.

Figure 10. Direct comparison of GSV estimation performance across selected variants of SRS, SYS, and StrS. The highest estimation errors observed across all iterations. SRS—blue line; SYS—orange line; StrS—green line.

Figure 11 and Table 11 and Table 12 present the assessment of uncertainty associated with GSV estimation under the different sampling designs. For SRS and SYS, standard deviations (Equation (4)) and standard errors (Equation (5)) were calculated from BIAS and RMSE values obtained across 100 independent sampling iterations. In the selected StrS configuration, the only source of limited variability was associated with the stochastic component between 10 RF runs. Details about the exact numbers of RF runs in particular sampling methods, as well as their marginal influence on the variability of the results, are demonstrated in Appendix G.

Figure 11. Standard deviation (SD) of BIAS and RMSE for selected configurations of sampling methods: SRS—blue line; SYS—orange line; StrS—green line.

Table 11. Standard error (SE) of BIAS [pp] for selected config. of sampling methods.

	BBH			Herby			Milicz
n	SRS	SYS	StrS	SRS	SYS	StrS	SRS	SYS	StrS
100	0.17	0.19	0.07	0.14	0.12	0.03	0.12	0.13	0.06
200	0.12	0.11	0.03	0.09	0.09	0.02	0.09	0.08	0.04
300	0.10	0.10	0.02	0.07	0.07	0.02	0.06	0.06	0.02
400	0.08	0.08	0.03	0.06	0.05	0.03	0.06	0.05	0.03
500	0.08	0.07	0.02	0.04	0.05	0.02	0.05	0.04	0.02

Table 12. Standard error (SE) of RMSE [pp] for selected config. of sampling methods.

	BBH			Herby			Milicz
n	SRS	SYS	StrS	SRS	SYS	StrS	SRS	SYS	StrS
100	0.07	0.06	0.02	0.04	0.04	0.03	0.06	0.06	0.02
200	0.04	0.03	0.02	0.02	0.02	0.01	0.04	0.03	0.02
300	0.02	0.03	0.02	0.02	0.02	0.01	0.03	0.02	0.02
400	0.02	0.02	0.02	0.01	0.01	0.02	0.02	0.02	0.02
500	0.02	0.02	0.02	0.01	0.01	0.01	0.02	0.02	0.02

Names, definitions and default values of all hyperparameters can be found in the randomForest library documentation.

The observed trends were similar for SRS and SYS across all forest districts. Except at very low sampling intensities, SD values remained relatively stable (within 1–2 percentage points) across all sampling designs. However, the variability of BIAS under StrS was slightly lower and stabilized at a sample size of approximately 200 plots (Table 11). At this sampling intensity, the variability of RMSE was similar across all investigated sampling designs (Figure 11, Table 12). Overall, the selected StrS configuration provided more stable estimation errors than SRS or SYS at lower sampling intensities, as the stochastic component of this configuration was minimized. The differences between sampling designs were more pronounced at lower sampling intensities and were more evident for BIAS than for RMSE.

$$\mathit{SD}=\sqrt{{\displaystyle \frac{\sum {(x_i-\overline{x})}^2}{(n-1)}}}$$

(4)

$$\mathit{SE}={\displaystyle \frac{\mathit{SD}}{\sqrt{n}}}$$

(5)

where

• SD—standard deviation of BIAS/RMSE scores from particular iterations;
• SE—standard error of BIAS/RMSE scores from particular iterations;
• x_i—single BIAS/RMSE score;
• $\overline{x}$—arithmetic mean of BIAS/RMSE scores from particular iterations;
• n—number of iterations in the given sampling method:
• SRS = 100, SYS = 100, StrS RAN = 50, StrS NC = 10, StrS REG = 10.

4. Discussion

4.1. General Trends

This study demonstrated the potential of ALS-based, structurally guided sampling for low-sampling-intensity forest stock inventories. In StrS, the convergence zone appeared at smaller sample sizes than in SRS or SYS (Figure 10). This effect was even more pronounced in the structurally more complex study area. For instance, in a relatively simple forest district such as Herby, the differences in error values between sampling methods were up to about 2 pp. However, in a more complex study area, such as BBH, the differences at lower sampling intensity levels, i.e., approximately 100 plots and below, could be as high as ~4 pp, representing a notable improvement relative to the average range of observed errors. On the other hand, the range of RMSE values was slightly higher in BBH, and any improvements are usually more visible at higher error levels. Therefore, validation in at least one or two even more complex forest districts would be desirable to strengthen this hypothetical effect. Further increases in sample size resulted in limited and non-proportional gains in GSV prediction accuracy (up to 2 pp). This trend was characteristic of all analyzed methods. In fact, beyond approximately 200–300 plots, the choice of sampling method appeared to become of secondary importance for GSV prediction performance (Figure 10) [80].

4.2. Limitations

The study design was oriented towards a single-response survey, with GSV as the sole target variable. As noted by [132], unequal-probability sampling approaches, such as SGS, can be efficient for single-response surveys but may not necessarily be optimal for multipurpose forest inventories. Therefore, drawing separate SGS samples for individual response variables and comparing their efficiency with that of a single multipurpose sampling design would certainly merit further investigation. On the other hand, GSV, or closely related biomass, is among the most important forest attributes to be determined in forest management and (NFIs) [133,134,135]. Moreover, other forest attributes, such as DBH or tree height, are often treated as auxiliary, substitute, or intermediate variables [136] and may require smaller sample sizes than GSV when considered as target attributes [68]. Finally, with the widespread use of RS techniques, attributes such as tree or stand height can often be estimated with substantially lower errors than GSV [68] or even measured directly from ALS data [66], in some cases with greater accuracy than conventional ground measurements [65].

The results of this study were presented at the forest-district level. The authors did not plan comparisons of the performance of the selected sampling methods within predefined stratification layers, as is the case in many practical forest inventories [34,36,37,137,138,139]. Such an investigation would require a significant extension of the already broad study. It is therefore possible that the results could vary among post-stratified subpopulations and that optimal sampling could consequently be applied separately within each user-defined stratum. On the other hand, the architecture of the SGS method should inherently account for naturally occurring subpopulations and thus provide a well-balanced sample distribution, irrespective of the complexity of the surveyed area [38]. Therefore, the results were presented for three distinct forest districts. Nevertheless, if population totals and/or means are the sole quantities of interest, for example, in less structurally complex areas, monocultures, or plantations, this issue may be of lesser importance. However, from a broader scientific perspective, this aspect should remain within the scope of further analysis, particularly given that recent European policies increasingly emphasize the conservation, restoration, and enhancement of forest biodiversity [140,141,142].

Although the overall BIAS and RMSE levels were relatively small, it should be remembered that the results were evaluated against a semi-synthetic response variable, and model-generated outputs often tend to provide smoother and less complex representations of real populations [88,89]. As shown in Figure 3 and Table 9 (RMSE), the present study was no exception in this regard. Therefore, the results presented here should be treated as indicative of general trends rather than as definitive guidelines. Nevertheless, the conditions were identical for all tested sampling methods and, in accordance with the objectives of the study, the relative differences in performance between the investigated methods were more important than the absolute error values. In the presented simulations, the authors tried to imitate real inventory cases as closely as possible. However, in practice, sampling is performed only once. Therefore, it is desirable to know the approximate range of possible error characteristics associated with the considered sampling methods. For practical reasons, in our opinion, the limited influence of the stochastic element specific to the presented StrS/SGS method is a major benefit compared with SRS and SYS, where their randomness must often be balanced by increasing the sample size. Therefore, in addition to the standard deviations of RMSE/BIAS scores (Figure 11) and standard errors (Table 11 and Table 12), the final results presenting direct comparisons between the investigated sampling methods also include the largest error observed within a finite set of repetitions in a given scenario (Figure 10). On this occasion, the authors would like to highlight that GSV estimates from real stand-level inventories were used as auxiliary predictors for the generation of the reference GSV values. This constitutes an additional reason for treating the generated forests as semi-synthetic populations.

4.3. Complex Sites and Additional Variables

Although the selected study areas were characterized by relatively diverse species composition and structural complexity (with regard to Polish lowland environmental conditions), no mountainous regions were included in the analysis. This was mainly due to the lack of hyperspectral data and, consequently, sufficiently detailed species information for mountain regions, which was necessary to create reliable semi-synthetic forest populations. However, based on the results presented in Figure 10, perhaps complex sites (akin to BBH) could benefit more from SGS than homogeneous sites, although this broad premise definitely requires additional testing across more sites with varying characteristics.

Under the discussed conditions, SGS could potentially utilize additional variables, such as site characteristics, climatic drivers, slope, aspect, elevation, and XY coordinates, to achieve more effective sampling than SRS or SYS. Nonetheless, a substantial impact of these auxiliary variables should not necessarily be expected, as even predictors more strongly correlated with GSV and used in this study, such as stand age and species composition, generally did not improve the results substantially once ALS metrics were available (Table 10). This result suggests that the exclusive use of ALS metrics can already explain most of the variance in GSV. Height metrics, in particular, were usually the most strongly correlated with tree and stand volume (Appendix F). Perhaps these limited improvements could also be partially explained by the authors of [143], who demonstrated a method incorporating species information into ALS-based GSV prediction models. That method reduced RMSE by 2 to 9 pp. Nonetheless, in their research, the additional species information was also derived from ALS data. Similar observations were reported by [37], who clearly stated the need to replace species information with other auxiliary variables for stratum delineation, having found a low correlation between species information and GSV (0.12).

Similarly, information on stand age could not visibly improve the overall BIAS and RMSE scores. If present, gains of up to 1–2 pp were more often observed at medium and higher sampling intensity levels. However, the addition of species data was usually more informative. This result can be partially explained by the naturally higher autocorrelation between ALS height metrics and stand age than between ALS height metrics and species data. Moreover, increasing the dimensionality of the auxiliary data space at lower sampling rates could be another source of uncertainty and instability in RF-derived GSV estimates. For instance, ALS or ALS + SP/AGE could be more informative configurations of StrS than ALS + SP + AGE when using 100 plots for model calibration (Table 10). Another explanation for the noisy/unstable behaviour following the addition of age information could be that stand age was assigned based on the dominant group of trees across the entire stand. Theoretically, individual stands should be delineated in a way that ensures the most structurally homogeneous conditions [34]. However, in practice, such distinction is usually challenging, especially when accounting for the vertical structure of multilayered stands, as individual stands can be very heterogeneous in terms of, for instance, age information and species composition across different layers. We would also like to note that the accuracy of age estimation was not reported by the data provider.

Furthermore, ref. [144] found little influence of climatic drivers and site conditions on tree-level volume estimates when these variables were combined with ALS metrics. On the other hand, potential latent effects and interactions should always be considered. For instance, ref. [127] demonstrated that slope had no meaningful influence on GSV estimates, either for the entire sample or for particular groups, with the exception of deciduous stands. This effect was most likely related to reduced DTM accuracy resulting from the weaker penetration of LiDAR signals through the denser crowns of broadleaved tree species.

4.4. Ex Situ Data

Preliminary GSV estimates obtained from generic models trained exclusively on ex situ data were not as attractive as initially presumed. One possible explanation is that the performance of ex situ models could have been improved if spatial autocorrelation had been explicitly taken into account. On the other hand, ref. [58] demonstrated little evidence of spatial autocorrelation among mostly managed forest districts in Poland. Moreover, for the relatively small forest areas considered in this study, the effect in question could be reduced even further [145]. Regarding the spatial separation between sites, ref. [146] found ex situ models to be competitive with local forest inventories within a 100 km range; however, the distances between some of the forest districts in our study could have been slightly greater. In contrast, Table 5, Table 7 and Table 10 suggest that, wherever possible, ALS in situ models should be preferred, as they naturally provided better results than models based exclusively on generic ex situ information. This finding is in agreement with the results of [147], who concluded that at least some number of local plots should always be secured for model calibration, or at least for controlling the magnitude of possible estimation errors. Quite surprisingly, ref. [146] found that only 20 in situ plots, selected using SRS, might be sufficient to calibrate external models in such a way as to provide precision and accuracy levels comparable with those of a new local inventory. Nevertheless, the authors of that study used relatively small datasets, resulting in unusually high sampling intensity rates. To mitigate that effect, the authors used replacements for sample plot selection. Regarding the spatial scale, ref. [148] obtained the best performance when training plots were limited to a 20 km radius, although the results were based on Landsat TM data. Based on the results presented in our study, the provision of preliminary GSV estimates as auxiliary predictors for in situ calibrated models could improve the results by only approximately 1–2 pp across all sampling methods (Table 5, Table 7 and Table 10). Such an approach bears some affinity to stacking ensemble architectures, which seek to leverage the complementary advantages of predictions derived from separate models or data sources [149,150,151,152]. Nevertheless, in most cases, ALS + GM scenarios produced BIAS and RMSE values similar to those achieved by models utilizing only in situ ALS metrics. Moreover, additional caution may be warranted when using ex situ information, as in some cases, GM metrics combined with other predictors produced slightly inferior results compared with scenarios in which different metric types were used separately. Possibly, the introduction of additional variables may have induced some extra noise related to multidimensionality and/or the quality of certain auxiliary variables. For instance, the inclusion of either stand age or species information could visibly improve GM performance in StrS. This behaviour could be partially explained by the fact that SGS can utilize additional data to a greater extent than SRS or SYS. However, quite interestingly, when age and species information were combined with GM, the aforementioned gain was observed to a lesser extent (Table 10). Obviously, the conducted analysis cannot exclude other possible causes of the observed patterns. The inclusion of ex situ data from generic models constituted only a secondary aspect of this study, rather than its main objective. Therefore, based on the current analysis, it is challenging to draw firm conclusions about these contingencies. The plausible causes of this behaviour remain to be investigated in further analyses.

4.5. Practical Implications

In general, the obtained results followed rather logical expectations. Flexible stratification based on population parameters can facilitate the minimization of within-group variance and differentiation between groups, which, in turn, can result in a more optimal and well-balanced sample distribution. In SRS, as well as in SYS, the stochastic element plays a crucial role in practical realizations, whereas in the proposed SGS, the random component is kept to a minimum, with the RF iterations being its main source. This aspect can be considered a meaningful practical benefit of StrS/SGS. It was also one of the reasons for preferring hierarchical clustering over k-means partitioning, as the former, when combined with Ward linkage, has been shown to be more stable than the latter [58]. Nevertheless, the observed randomness could potentially be reduced even further by using OLR instead of RF as the estimator. In fact, linear regression performed slightly better than RF in some scenarios, particularly at lower sampling intensity levels. This observation is consistent with the general tendency of machine learning techniques to require larger sample sizes than OLR to achieve stable predictions [128,129,130]. Nonetheless, the predictive performance of GSV estimation was similar for both estimators, with differences of up to approximately 1–2 pp and a slight advantage of RF in the majority of scenarios.

Based on the observed results, some practical recommendations can be formulated regarding the choice between OLR and RF for GSV estimation. Namely, given a very small sample size, OLR appears to be the more stable choice. At larger sample sizes, RF seems to be preferable, as it can provide a slight improvement in predictive performance. Compared with linear regression, RF is generally less sensitive to the inclusion of irrelevant or noisy predictors. Moreover, it inherently accommodates non-linear relationships and interactions between predictors, thus reducing the need for rigorous a priori variable selection and the formulation of an explicit model equation [124,153,154]. In the case of OLR, both the model formula and potential interactions must be explicitly specified, which can drastically increase the number of possible model configurations.

We would also like to stress that, in this research, the same model was used for each stratification group. The application of stratum-specific models could potentially contribute to a further reduction in error rates. Although RF and OLR produced comparable results, only two types of estimators were tested in this study. The literature remains ambiguous regarding the selection of the most appropriate estimator. Similarly to the present study, ref. [155] found little influence of the prediction method (up to 3 pp), whereas other studies reported quite contrary results. For instance, ref. [80] showed a notable influence of the prediction method on the required sample size, favouring RF over other methods when the target precision level was set to a 5% standard error. Ref. [131] also supported the use of RF for biomass estimation over other prediction methods, such as kNN, SVM, OLR, or Gaussian Processes. This aspect certainly warrants further investigation, as positive results could indicate that, under certain circumstances, the use of ALS metrics might become redundant. As a further consideration, if only approximate population totals or means were of interest, regardless of the local spatial variability of the response variable, data from previous inventories could be reused in managed forest districts. For instance, according to Table 4 and Table 10, when only stand age information was used for GSV prediction based on 500 observations, some BIAS values fell within a reasonable range of approximately 2–4% at the forest district level.

While the proposed StrS approach outperformed SRS and SYS at lower sampling intensity levels, it is generally more complex and may therefore require additional preparation. Several issues should be considered before implementing StrS: (1) the required level of accuracy or available sample size, (2) the selection of appropriate auxiliary variables for stratification and/or sampling, (3) the selection of adequate stratification rules, (4) the allocation of sample sizes among strata, (5) the within-stratum sampling design, and (6) preferences regarding the type of estimator. Moreover, the potential interactions among these factors and the consequences of their particular combinations should always be taken into account. The above considerations were the main motivations for undertaking the present study. The authors sought to approximate the expected inventory error rates associated with decisions regarding the selection of sampling methods and their respective variants.

The results showed that structurally guided stratification of ALS metrics can help distinguish meaningful subpopulations within heterogeneous forest districts. Although only one “stratification engine” was employed, namely hierarchical agglomerative clustering with Ward.D linkage, previous research has identified this clustering approach, as well as k-means or k-medoids, as potentially suitable for GSV-oriented forest inventories [36,58,156]. Nevertheless, SGS can also be driven by other “engines”, including the Cube Method [51], the Local Pivotal Method (LPM) [53], and Latin Hypercube Sampling (LHS) [157]. Their suitability for FMI warrants further investigation, as previous studies have indicated the considerable potential of these SGS algorithms for sampling optimization. For instance, ref. [19] reported that LPM can be more efficient than SYS for certain variables in NFI, while [48] demonstrated that LPM and LHS could substantially improve sampling efficiency compared with SRS. However, GSV was not among the forest attributes examined in the latter study.

4.6. Selected Variants

Regarding the StrS configuration, as might be expected, the Neyman allocation method showed considerable potential for achieving a well-balanced sample distribution, as it combines the advantages of the two other sample allocation methods evaluated in this study, namely allocation proportional to stratum size and allocation proportional to stratum variance. Although differences in error scores due to the allocation methods were relatively subtle, Neyman allocation combined with regular in-data-space sampling and the RF estimator consistently maintained relatively low BIAS values, even at limited sample sizes and across all study areas (Table 8). Random in-stratum sampling yielded similar RMSE results but slightly higher BIAS values and was, as expected, somewhat less stable than the regular sampling approach. Nearest-centroid selection produced the highest error values in most scenarios with low sampling intensity, most likely because it failed to adequately represent the full range of variability within the subpopulations.

Interestingly, the choice of k, i.e., the number of clusters/strata, had no major impact on the results at medium and higher sampling intensity levels (Figure 9, Table 8 and Table 9). Although relatively small, differences associated with the number of clusters became apparent only at low (~100) and very low (~50) sampling intensity levels, reaching approximately 2–3 percentage points (pp.). At larger sample sizes, the influence of this factor was negligible. With regard to BIAS, however, different numbers of stratification groups resulted in varying error levels across the study areas, with differences reaching up to 4 pp at lower sampling intensities. No clear pattern could be identified that would allow the optimal number of clusters to be determined. As demonstrated by [156], this may be a highly complex issue and was therefore considered beyond the scope of the present study. Nevertheless, future research aimed at exploring further reductions in sampling intensity should include a thorough investigation of this behaviour. Meanwhile, the present results provide possible justification for avoiding very low sampling intensity levels.

Even though many variants of the selected sampling designs were tested in this study, by no means were all available sampling methods covered. In order to maintain a clear and well-organized research framework, the authors focused on what are perhaps the most common sampling designs [158], supplemented by one relatively new approach which, based on previous research [35,38,46,57,80,137], had shown promising results and therefore appeared worthy of further testing and development. First, SCS was not tested, although it is a very popular design in many national forest inventories [27,28]. This sampling technique is considered particularly useful for NFIs, as it provides a compromise between statistical efficiency and the economic considerations associated with repeated inventories over large areas, such as at the national level. For this reason, it may be less common in forest district inventories. Moreover, under specific circumstances, SYS can be perceived as a special case of systematic cluster sampling [43,159,160], and therefore, drastically different results should not necessarily be expected. Second, ADA initially appeared to be a promising option worthy of analysis in terms of its suitability for forest inventory. However, preliminary tests showed poor performance, particularly at lower sampling intensities, and the approach was therefore omitted from the final analysis. This poor performance could likely be attributed to the fact that, in the briefly tested ADA implementation, the newly selected sample plots at each recursion tended to fall on so-called “hard points”, i.e., areas characterized by higher variance that were generally not representative of the entire population. Perhaps combining ADA with SYS and CS could mitigate the weaknesses of the basic ADA approach and contribute to further optimization of forest inventory designs [43].

In this study, the authors focused exclusively on “clean” forms of the analyzed sampling designs. Nevertheless, in practical forest management inventories, the final sampling approach is often a combination of two or more basic sampling designs [133]. For instance, in Poland, one of the most popular and officially recommended approaches to forest management inventory is, in fact, a combination of stratified and systematic random sampling [34]. Additionally, the potential benefits of applying post-stratification to already completed forest inventories should not be overlooked. PS can be particularly useful for less flexible sampling designs and/or inventories based on repeated measurements of fixed sample plots, as it may improve precision without requiring additional observations [161,162]. It may also be suitable for small areas represented by a limited number of sample plots [163]. An important consideration in PS is that the quality of stratum delineation can substantially affect its efficiency [161]. Moreover, because strata are defined after sampling has been completed, PS cannot fully exploit the advantages of optimal sample allocation among strata. Another important issue concerns endogenous post-stratification (EPS), in which strata are constructed using predictions from a model calibrated with in situ sample data. Such an approach may lead to substantial underestimation of variance if the dependence between the estimated strata and the sample data is not properly accounted for [164,165]. Finally, EPS-based re-estimation has traditionally been applied to data from large-scale forest inventories, such as NFIs. Nonetheless, ref. [162] recently demonstrated the successful application of EPS in a forest management inventory as well. More generally, combining different sampling techniques at different stages of an inventory may help overcome some of the limitations of their individual “clean” forms. However, there is unlikely to be a universal approach suitable for all inventory situations, and the efficiency of combined sampling designs should therefore also be evaluated under different circumstances.

Finally, one should always bear in mind that the simplicity of a sampling design and the associated statistical inference can often be strong arguments for choosing a particular inventory method. The regular pattern and uniform spatial coverage provided by SYS are particularly appealing to practitioners, making this method a very popular choice [18]. Nevertheless, SGS, as well as combinations of different sample selection methods with increasingly available RS/ALS data, offer vast opportunities for further research, methodological development and optimization, as well as potential practical applications.

5. Conclusions

This study demonstrated the potential of hierarchical clustering for stratification in a two-phase model-based forest management inventory. The proposed stratification approach, combining Neyman optimal allocation with regular sampling in ALS data space, allowed for a noticeable reduction in sample size compared with simple random sampling and systematic sampling, while maintaining relatively low GSV estimation errors at the district level. The preferred configuration of the stratified sampling design produced relatively stable results across all study areas at sampling intensities of approximately 200 plots or more. However, multipurpose inventories and real-world forest conditions would likely require a greater number of field observations. At the investigated sampling intensities, the number of stratification groups had only a limited effect on GSV estimation accuracy, whereas at lower sampling intensities, its influence was more variable.

ALS metrics were the strongest GSV predictors among the variable types tested in this study. Moreover, ALS data constituted the most appropriate basis for the structural stratification of the forests considered. The addition of pre-GSV estimates from generic ex situ models could slightly improve the accuracy and precision of the final GSV estimates. However, at lower sampling intensities, their inclusion could also introduce additional noise, probably due to increased multidimensionality and certain dissimilarities among the inventoried objects. Similarly, the inclusion of species and/or age information did not notably improve the results when combined with ALS variables. When used alone, these variables generally resulted in higher estimation errors, particularly RMSE. Further exploration of different types of variables and methods may bring additional benefits to forest inventories.

Differences between the two estimators were relatively small, with a slight overall advantage of random forest over ordinary linear regression, particularly at higher sampling intensities. At smaller sample sizes, however, ordinary linear regression often produced more stable results. The performance of simple random sampling and systematic sampling was similar across all sampling intensities, indicating that, under the conditions examined in this study, the differences between these two designs were considerably smaller than those observed between either of them and the proposed stratified approach.

Finally, it is important to remember that the results of this study were based on partially simulated data. Such populations tend to simplify the heterogeneity of real forest structures. Therefore, all findings should be treated as approximations of general trends rather than as definitive guidelines. Accordingly, a direct comparison of the proposed stratified inventory design with simple random sampling, systematic sampling, and potentially other sampling methods not tested in this study, using entirely real forest data, would constitute an important direction for future research and provide valuable empirical validation of the findings presented here. In particular, further tests across more diverse study areas, including mountain regions, are needed to better assess the potential of the investigated forest inventory methods. The performance of SGS should therefore be evaluated also at the subpopulation level and in multipurpose forest inventories.