1 Introduction

Forest surveys, particularly large-scale surveys such as National Forest Inventories (NFIs), are crucial for conducting international and national forest monitoring and assessment programs. Forest data are essential for various purposes, including reporting on forest resources (FAO-FRA 2020), monitoring biodiversity (Corona et al. 2011; Forest Europe 2020), tracking deforestation and forest degradation (Corona et al. 2023), as well as improving decision-making processes, silvicultural practices, harvesting, and conservation activities (Chirici et al. 2020).

Traditionally, forest surveys are based on probabilistic sampling and aim to provide estimates of averages and totals of forest attributes such as forest extents, growing stock volumes and their increments at national and regional level (Tomppo et al. 2010). Commonly, the estimation of these quantities is approached from a design-based perspective in which the values of the attributes of interest are considered as fixed, thus avoiding any model assumption on them, and the properties of the adopted estimators derive only from the sampling schemes actually performed in the survey (Mandallaz 2008). In contrast, in model-based approaches those values are random variables that are assumed to be generated by a super-population probability model, while sampling is of purposive or ignorable in nature (Rubin 1976), so that the properties of the adopted predictors derive from the assumed model (Chambers and Clark 2012).

Here, we do not intend to discuss the advantages and disadvantages of the design-based versus model-based approaches, as these are well delineated in the statistical literature (Smith 1994, 2001; Gregoire 1998; Thompson 2002, chapter 10; Little 2004). Our aim is simply to highlight that design-based inference is the traditional approach used in most forest surveys, including NFIs, and that: i) this approach ensures design-unbiased estimators of averages and totals of forest attributes, with analytical expressions of their sampling variances, which can be estimated using standard survey sampling methods (Gregoire and Valentine 2008); ii) the design-based properties of these estimators can be considered objective because they derive directly from the sampling scheme actually used in the survey, rather than being derived from an assumed model (Särndal et al. 1992, page 21).

Furthermore, the design-based approach enables the use of large datasets obtained from remote sensing (e.g., satellite multispectral data), which are available either for free or at low cost for the entire survey area, thanks to the increasing capabilities of remote sensing technologies (Barrett et al. 2016; Di Biase et al. 2018). In particular, remotely sensed data can be utilized at the design level, for example, in the initial stratification of sampling units based on major land uses (McRoberts and Tomppo 2007), or at the estimation level, within the framework of model-assisted inference. In this approach, models that incorporate remotely sensed data as covariates are used to build improved estimators, while still retaining the properties derived from the sampling scheme used in the field. Specifically, model-assisted inference ensures approximate design-unbiasedness for the resulting estimators of averages and totals, as well as approximate design-based variance expressions, with subsequent variance estimators (Opsomer et al. 2007; Breidt and Opsomer 2017).

However, even if design-based estimation of averages and totals of forest attributes provides reassuring statistical soundness, the resulting estimates can support forest monitoring and management at large scale only, without providing explicit spatial details. On the other hand, the construction of wall-to-wall maps and the provision of small-area estimates or predictions of forest attributes are becoming essential in most large-scale forest surveys. In particular, wall-to-wall forest mapping is now considered as unavoidable in modern NFIs, usually referred to as Enhanced Forest Inventories (Stinson and White 2018).

The purpose of this discussion paper is to argue how forest attribute mapping should be conducted from a statistically sound perspective, while ensuring consistency with the design-based inference used for estimating averages and totals. Section 2 focuses on the dominant role of kNN and Random Forest (RF) methods in forest resource mapping. Section 3 outlines the methodological shortcomings of the approaches applied over the past thirty years in forest literature regarding these mapping techniques. Section 4 highlights the necessary steps to achieve methodological soundness. Finally, concluding remarks are presented in Section 5.

2 Forest attributes mapping

For a long time, mapping has typically relied on geostatistical methods. Among these, kriging predictors are probably the most widely used and well-established techniques. In the most general case, known as kriging with external drift or universal kriging, it is assumed that the values to be mapped follow a deterministic trend. This trend is expressed as a function of a set of covariates available for the entire survey region, plus error terms that stem from a centered, intrinsically stationary, and Gaussian random process, characterized by a spatial autocovariance function.

The theoretical appealing of kriging and its large use are due to the fact that, under the assumed model, kriging ensures the best linear unbiased predictors with an analytical variance expression that can be readily estimated from sample observations (Cressie 1993). Alternatively, a theoretically founded, well-experienced mapping criterion is the geographically weighted regression, based on the assumption that the relationship between the interest variable and covariates varies across the survey region and tend to be similar for nearby locations. Therefore, the values to be mapped are predicted by a regression model in which the covariate values at sample locations are weighted in relation to their distances to the location under estimation (Fotheringham et al. 2002). Surprisingly, despite their solid theoretical basis, both the techniques has been scarcely applied for mapping forest resources. Sales et al. (2007) and Wang et al. (2021) constitutes rare, even if not unique, examples.

In contrast, towards the end of the last century, mapping techniques based on the nearest neighbor method became increasingly popular in forest studies. This mapping approach was first introduced in forest research by Tomppo (1990, 1991) for use in the Finnish NFI. Among these techniques, the kNN method has been the most widely applied (Corona et al. 2014). Its popularity is likely due to its simplicity and flexibility. Specifically, kNN predictions are easily generated by linearly combining the k sample observations that are “nearest” to the location where the prediction is being made, according to some distance criterion and in some space, thus avoiding the computational challenges of more complex geostatistical and weighted regression methods. Furthermore, the number of neighbors (k) involved in the predictions determines the smoothness of the resulting map, and various distance metrics (e.g., Euclidean distance, Mahalanobis distance, etc.) can be chosen, offering great flexibility in its implementation. Another advantage of kNN is its ability to efficiently incorporate auxiliary information by determining the k nearest neighbors in the space of the auxiliary variables, rather than in geographical space. After the two seminal papers by Tomppo, hundreds of applications of kNN stemmed from forest studies. Chirici et al. (2016) provide a detailed review of these applications and, among the listed plethora of articles, Landsat imagery is the most frequent source of auxiliary information followed by airborne laser scanning metrics, digital aerial imagery, SPOT imagery and combinations of these sources. At least in this review article, there is no mention of kNN applications directly performed in the geographical space or including spatial coordinates as auxiliary information.

In recent years, the use of kNN applications has decreased in favor of machine learning techniques, with RF imputation becoming increasingly popular (e.g., Baccini et al. 2008; Cartus et al. 2014; Chirici et al. 2020). However, Lin and Jeon (2006) demonstrated that RF can be viewed as an adaptively weighted kNN method, suggesting that it is likely to share similar properties. The similarity between the two techniques is evident from the results of a large simulation study by Fattorini et al. (2024), in which the performances of kNN and RF are found to be virtually identical. Therefore, from this point onward, the two mapping methods will be considered jointly whenever possible.

3 What was done

As already stated, one of the main reasons for the popularity of kNN and RF in forest resource mapping is the computational simplicity of the first with respect to the more challenging computational tasks involved in geostatistical and weighted regression methods, and the wide disposal of software for the automated applications of the second (e.g., package randomForest, R Core Team 2021). Unfortunately, this utilitarian choice has not considered the subsequent difficulties in performing a rigorous statistical inference.

In fact, although the properties of kNN and RF predictors have a long tradition in the statistical literature, starting from the seminal works by Stone (1977) and Devroye et al. (1994) to the more recent papers by Breiman (2001), Hall et al. (2008), Samworth (2012) and Gadat et al. (2016), the results are derived by considering the sample data as realizations of independent and identically distributed random variables. This restriction prevents the extension of these results to the mapping of forest attributes in which sample data traditionally arise from a probabilistic sampling of locations, usually performed without replacement (thus precluding independence) and possibly with different inclusion probabilities of locations (thus precluding identical distributions).

Specific studies would have been necessary for determining the properties of kNN and RF predictors in forest mapping. In contrast, while at least to our knowledge no theoretical study regarding RF mapping is present in forest literature, the review by Chirici et al. (2016) found that most papers regarding kNN are of applicative nature (about 80%) while only about 7% are of methodological nature, with proposals invariably supported by practical applications or simulations, with no theoretical investigation able to justify their use. The theoretical weakness of kNN studies in forest literature has been evidenced by Eskelson et al. (2009), who are the unique authors, among the myriads who have dealt with kNN, pointing out that “In general, estimation techniques are chosen based on statistical properties such as unbiasedness, consistency and efficiency. These properties are not well understood for NN imputation”.

The main methodological issues faced – with little success – in forest literature regarding kNN mapping is bias mitigation and estimation of precision. The bias issue has been well recognized by Magnussen et al. (2010b) and Baffetta et al. (2012). Both papers evidence that the largest values of the interest variable tend to be underestimated and the smallest ones to be overestimated and propose empirical machineries to reduce bias. Nevertheless, the performance of the proposed methods is evaluated only by case studies, in the first case, and by a simulation study, in the second case, without any theoretical demonstration of their effectiveness.

Regarding the estimation of precision, Baffetta et al. (2012, section 4) pessimistically argue that any rigorous inference at the pixel level is precluded in kNN mapping. Other authors, however, have been more optimistic. Model-based inference is the approach typically used to estimate the mean squared errors of kNN predictors, reflecting foresters’ skepticism toward the possibility of adopting design-based methods. For instance, Kim and Tomppo (2006, equation 3) ignore the presence of bias and propose a variance estimator based on a variogram model defined in the space of auxiliary variables, estimated using the Matern class of variograms. Three years later, Tomppo and co-authors (Magnussen et al. 2009, equation 4 and appendices A.1–A.5) develop a mean squared error estimator, again model-based, because in their view, “distributional assumptions are needed for estimating the uncertainty of kNN predictions”. This estimator is derived under the assumption that population values are random variables with expectations and variances depending on a vector of auxiliary variables. Surprisingly, one year later, the same authors (Magnussen et al. 2010a, equation 4), neglecting bias, propose another estimator for the variance of kNN predictions based on modified balanced repeated replications. Once again, the authors’ skepticism toward design-based approaches is evident in their statement that “the nonparametric nature of kNN estimators precludes an analytical design-based kNN variance estimator.” As we will discuss in the next section, this view turns out to be unjustified in light of more recent studies.

The statistical weaknesses of these proposals are evident from the fact that they have been all tested on real or artificial data without any theoretical investigation into their properties, and without even being subjected to simulations. Furthermore, at least to our knowledge, none of these proposals have applied in the subsequent literature. In practice, they have been only used in the papers in which they were introduced.

In conclusion, drawing on the extensive forest literature on kNN and RF mapping, it is clear that assessments of map precision are, in most cases, based on highly empirical cross-validation or leave-one-out techniques. While these methods may be useful as global indices of fitting performance, they do not reflect the actual precision of kNN and RF predictions at any specific location within the survey area. In other words, the forest literature has yet to provide reliable tools for estimating the true precision of kNN and RF maps.

4 What needs to be done

As inference on averages and totals of forest attributes, from their estimation to the estimation of their precision, is traditionally performed in a design-based framework, the estimation of the precision of forest maps should be congruent and should be performed in the same design-based framework without resorting to models. As we shall see in this section, the track toward a design-based estimation of the precision of forest maps, even if essential to ensure inferential congruence, is much more complex than the fast tracks that can be travelled by supposing and exploiting more or less realistic model assumptions, in which the resulting estimates of precision crucially depend on the validity of these assumptions.

In this context, it is important to note that the kNN mapping criterion, as well as similar – albeit much less commonly used – criteria such as the inverse distance weighting (IDW) and nearest neighbour (NN) mapping, are invariably based on the naïve assumption that locations that are close (in some sense) tend to be more similar than those that are farther apart. This is typically referred to as Tobler’s Law (Tobler 1970). Based on this assumption, estimation at unsampled locations is naturally performed by linear combinations of the sample values observed at locations that are neighbour (in some sense) to the locations under estimation. Regarding the NN mapping, note that it can be viewed as a variation of kNN when a single neighbour (the nearest) is exploited instead of k, or a variation of IDW mapping when the smoothing parameter adopted to weight distances approaches infinity (Fattorini et al. 2022).

Obviously, when applying these criteria to estimate the value at a single location an error invariably occurs, but nothing ensures that the error has a design-based distribution centred and clumped at the true value. In practice, for finite samples, the design-based finite sample properties of these mapping estimators are not clearly delineated. Little is known about their design-based bias and mean squared errors and how these are related to the sampling effort. Accordingly, the sole way to render statistically sound these mapping techniques is to determine conditions ensuring the design-based consistency of the corresponding estimators. In addition, if the variables to be mapped are bounded, as always happens in practical applications, consistency also involves asymptotic design-based unbiasedness.

Surprisingly, consistency has been invariably neglected in forest literature, probably for the fact that it holds as the number of sampled locations approaches infinity, so that consistency is viewed as a pure theoretical property never attainable in practice, as in practice the number of sample locations is always finite. On the other hand, as pointed out by Sarndal et al. (1992, section 5.3), design-based consistency has a very practical importance, in the sense that, if consistency holds, then we can trust in the resulting estimates because the sampling distribution of the estimators of the population values can be considered tightly concentrated around the true values when the number of sample locations is large enough. Moreover, owing to the asymptotic unbiasedness, these sampling distributions can be considered centred around the trues values.

Fattorini et al. (2018a,b, 2022) demonstrate the design-based consistency of IDW and NN mapping in geographical space under mild continuity conditions on the surfaces to be mapped, highlighting the crucial role of the sampling schemes used in the surveys. In particular, they prove that consistency holds under random selection of locations, as well as under tessellated schemes widely applied in forest surveys and NFIs, such as tessellation-stratified sampling and systematic grid sampling for continuous populations, and their counterparts for finite populations of areas (pixels).

Furthermore, for the first time in over thirty years of kNN applications, Fattorini et al. (2024) prove the consistency of kNN mapping (and presumably RF mapping) under the same mild conditions on the surfaces to be mapped, and the same random and tessellated sampling schemes. These results may be surprising to forest scientists who have traditionally applied kNN and RF in the space of auxiliary variables, often neglecting geographical coordinates, as in such cases, the two mapping criteria may still lack consistency. These findings emphasize the critical need to incorporate geographic coordinates into the distance calculations, alongside the auxiliary variables.

These findings are especially relevant because design-based consistency is also crucial for the design-based estimation of map uncertainty. The core idea, first introduced by Fattorini et al. (2022) in NN mapping and later applied to IDW mapping with the smoothing parameter selected from the sample data using leave-one-out procedures (Fattorini et al. 2023), should be easily extendable to kNN, where the smoothing parameter is the number k of neighbors used in the interpolation. The approach involves treating the estimated map as a pseudo-population (Quatemberg 2016), from which bootstrap samples are drawn using the same spatial scheme adopted for the original sample. Then, all the steps performed on the original sample to estimate the map are repeated in each bootstrap sample. The key intuition behind this proposal is that, under consistency, as the estimated maps converge to the true maps, the bootstrap distributions arising from these maps should also converge to the true distributions of the estimators, thus providing consistent estimators of their mean squared errors. Franceschi et al. (2022) empirically investigate the effectiveness of NN maps as pseudo-populations, showing that they are likely to be good representations of the true maps.

In this framework, the pseudo-population bootstrap constitutes a very effective tool that allows to consider the design-based uncertainty entailed not only by the sampling scheme, but also by any estimation step performed on the sample data, from the choice of the “best” smoothing parameter to the choice of the “best” auxiliary variables to be exploited, irrespective of the criterion adopted to define what is “best”. That has been experienced in a model-assisted approach in which the auxiliary variables are exploited as covariates in a linear prediction model and the estimated values are obtained from the predictions plus an error component achieved from the IDW mapping of the sample errors in the geographic space (Di Biase et al. 2022). The strategy has been subsequently automated in the R programming language by adopting Sentinel-2 remote sensing data as auxiliary information downloaded from the Google Earth Engine cloud computing platform (Francini et al. 2024).

A further issue that may sound awkward in the reporting phase is that the estimates of averages and totals of forest attributes achieved by the familiar design-based estimators for the whole study area and for domains partitioning the area (e.g., the administrative regions of a country) will invariably differ from the total estimates achieved from the resulting map as the sum of the estimated values. To obtain non-discrepant results, Marcelli et al. (2022) suggest a harmonization procedure in which the values estimated by means of the IDW technique are rescaled, so that their totals coincide with those achieved by traditional estimation. The capacity of the harmonization procedure to maintain consistency of the rescaled maps is proven providing that the traditional estimators are themselves consistent, a feature that has been proved to hold under the above mentioned random and tessellated sampling schemes (Fattorini et al. 2020). Therefore, under IDW mapping and tessellated sampling schemes, the uncertainty involved by harmonization can be included in the pseudo-population bootstrap, viz. by resampling from the harmonized maps and including the harmonization as a further step to be performed in each bootstrap sample.

Obviously, as the number of domains increases and their size decreases, the number of sample locations available for traditional estimation within each domain becomes smaller. This inevitably leads to an increase in the variance of traditional estimators, making the resulting estimates unreliable. Therefore, harmonization with excessively small domains should be avoided. However, the sum of interpolated values within these domains can still be used as appropriate design-based small-area estimates, thus bypassing the complex modeling typically employed in small-area estimation (Rao and Molina 2015). The design-based consistency of these small-area estimators has been proven under IDW mapping and tessellated sampling schemes (Fattorini et al. 2018a, appendix B), while their mean squared errors can once again be estimated within the framework of pseudo-population bootstrap.

5 Conclusions

The integration of sample survey inventory data and mapping information has become a key issue in the development and implementation of surveys designed to monitor and assess forests and forest ecosystems. The ever-growing availability of remote sensing data provides valuable ancillary information for spatially estimating forest attributes, i.e., for forest resource mapping, such as wall-to-wall mapping, which is now considered essential in modern NFIs.

From this perspective, this paper has explored the opportunity to use a design-based approach to ensure inferential congruency between mapping and the estimation of averages and totals of forest attributes, while also addressing the methodological challenges identified over more than thirty years of forest literature in attempts to estimate map precision.

The results achieved for IDW and NN mapping in geographical spaces, as presented and discussed in this paper, can be easily extended, with minimal adjustments, to kNN and RF mapping in the space of auxiliary variables, as long as geographical coordinates are included to ensure consistency. This approach would ultimately enable statistically sound, design-based estimation of map precision, at least within the tessellated sampling schemes commonly used in forest surveys, particularly large-scale ones such as NFIs.

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