The interest in forest observation and monitoring has surged in recent years, particularly due to the essential role occupied by forests in the energy and ecological transition our societies are undergoing e.g.,. Namely, as important carbon sinks and renewable energy sources, forests represent indispensable, but also very vulnerable, levers in mitigating the climate crisis, along with all the biodiversity they harbor . As these are challengingly distributed targets, being relatively inaccessible for in situ measurements in some parts of the globe, their observation and monitoring have received quite a bit of attention from the remote sensing community .

As in other domains of remote sensing applications , the forest remote sensing research community has witnessed a rapid increase in the utilization of machine learning and deep learning techniques, also referred to throughout the article as artificial intelligence (AI), in recent years. Notably, this has led to numerous developments of remote-sensing- and AI-based models for high-resolution mapping of the forest canopy height . Many of these approaches involve utilizing spatial or airborne lidar measurements, such as the Global Ecosystem Dynamics Investigation (GEDI) or airborne laser scanning (ALS) data. These are often complemented by imaging multi-spectral (Sentinel-2, Landsat, Planet, Spot) and sometimes radar (Sentinel-1, ALOS) data in order to achieve a wider and/or denser coverage .

Lidar measurements provide three-dimensional scattering information, generally allowing us either to reconstruct, to a degree, the forest stand structure or, at least, to estimate its average shape over a certain footprint. They have an outstanding potential for inferring numerous forest attributes (canopy height, wood volume, aboveground biomass, etc.), even in cases of relatively complex forest environments . However, often, they either do not have a recurrent acquisition character (e.g., the development of diachronic acquisitions at regional to national scales is at its early ages) or, as is the case with the GEDI mission, do not provide a continuous spatial coverage .

Multi-spectral and radar imagers typically offer wide, recurrent, and spatially continuous coverage of forests. Yet, except for particular acquisition setups, such as photogrammetry or polarimetric synthetic aperture radar interferometry (PolInSAR) , they generally do not provide vertically resolved information about the forest stand. Aside from that, they are also prone to a series of non-negligible issues, such as optical signal saturation or the multiplicity of forest structure properties simultaneously influencing the radar backscattering signal, causing its apparent saturation .

There have been numerous attempts within the forest observation community to reconcile the benefits of lidar and imaging measurements while also mitigating their respective limitations. AI methods have played a significant role in achieving this by constructing links between the lidar-derived forest attributes, such as canopy height, and broad-coverage images. Nevertheless, these remain models and are therefore obviously far from being faultless. Firstly, electromagnetic interactions in remote sensing data cannot theoretically explain all forest attribute variabilities. Even if they could, the data would still be prone to imperfections from lidar or imaging sources and from modeling choices and parameterizations. Therefore, it makes sense that all of these factors combined cause models to struggle to spatiotemporally reproduce a substantial part of the variability of the forest attributes. Similar effects were also observed in other kinds of spatial modeling when it comes to either resources or ecological modeling .

One way to attenuate these effects would be to combine different models in a way which might optimize their joint performances while enabling a comparative evaluation . This can be done in various ways, depending, first and foremost, on the availability of validation and/or reference data. If there are no reference data, the most intuitive way to proceed is the simple average or median of models (e.g., simple model averaging), aiming, respectively, to smooth predictions among models or to remove dissident predictions. If reference data are available, one could think of a more sophisticated way to construct an average, such as weighted model averaging. This could be based on analyzing the model variable (input) dynamics, i.e., how well it matches with the observed one , or, as is far more frequently the case, on evaluating model predictions (output). In the latter case, at least in the environmental sciences, we often rely on Bayesian model averaging (BMA) . The BMA can be perceived as a weighted mean of various predictions, with weights reflecting the performances of different models. Alternatively, it can be viewed as a finite-mixture model, estimating the probability that each observation from an independent validation dataset has been generated by one of the models belonging to an ensemble .

In this article, we apply the BMA with the aim of analyzing five selected AI-based models aimed at spatializing the GEDI- or ALS-estimated canopy height across metropolitan France using optical multi-spectral and contingent radar data . In order to do so, we use in situ measurements and estimations from the French National Forest Inventory (NFI) plots as an independent validation dataset. The approximately 5500 plots enable us to estimate both the overall and local weights of selected models based on four different variants of height measurements or estimations: dominant height, mean height, maximum height, and Lorey's height . By involving auxiliary data related to the topography, the dominant tree species, the forest stand vertical structure, and the type of the forest ownership, we also investigate factors influencing the models spread, i.e., where the models have similar weights but contrasting predictions. Finally, we contrast the performance of individual models against the fitted mixtures at the reference measurement sites, allowing us to highlight the advantages and the limitations (some of which have been previously noted in other fields, ) while also identifying potential perspectives of the proposed approach.

The article is organized as follows: in Sect. , we present the five employed models, while Sect.  introduces the NFI reference datasets. Section  contains the detailed description of the used BMA approach. In Sect. , we present the results, followed by the corresponding discussion in Sect. . Finally, Sect.  provides the concluding remarks of the article.

The selected remote-sensing- and AI-based models differ in terms of the remote sensing data used but also in the way these data are processed and in terms of the AI method employed. While not the sole commonality among them, a particularly relevant aspect for this study is that they all encompass metropolitan France, where we have access to the reference NFI data. Therefore, in this section, we briefly present their principal characteristics, which are partly illustrated in Fig.  and will be recalled in Sects.  and  while interpreting their mutual differences as highlighted by the BMA.

National Forest Inventory (NFI) programs are surveys of the forest resources over a certain territory . The French NFI is based on a spatially systematic stratified sampling design, which takes place in two phases: photo-interpretation of around 100000 points per year for the assessment of forest area and field observations and measurements at up to 7000 of these points for assessing forest resource variables .

For each visited point, i.e., a plot with a radius of 25 m, numerous attributes are accessible, including various measurements or estimations of forest canopy height. In this study, we focus on four particular variants among these options:

Hdom, which the average tree height of the seven largest trees per plot;

Given that the tree height is measured at the plot only for a sample of trees (one measurement per diameter class and species), complementary values were imputed using a random forest MissForest approach . The method was applied per species and sylvo-ecological region using the diameter at breast height, the height, and the plot-level variables (the stem density, the basal area, and the wood volume). Validation was done using data acquired during 2005–2009, the period during which all height measurements were performed. To do so, the current protocol was simulated, and imputations were compared with measurements, leading to a mean bias estimate (MBE) of -0.1 m, a mean absolute error (MAE) of 1.98 m (14.8 %, normalized relative to the target mean value), and a root mean squared error (RMSE) of 2.66 m (16.6 %, normalized relative to the target mean value).

This allows us to establish a sample that is big enough to obtain a potentially better estimate of the height (notably, the mean and dominant height), which is, in this case, annotated throughout the article with the superscript i, i.e., Hdomi, Hmeani, Hmaxi, and Hlori.

In this study, we utilize canopy height estimates from the four variants, sourced from 5475 NFI plots dispersed throughout metropolitan France for the year 2020. All analyses presented in Sect.  are based on the supplemented version (estimation – including imputations), except for the overall weight analysis in Sect. , which encompasses both the original version (measurements – without imputations) and the supplemented version.

Now that we have introduced the five different models and the reference dataset, the obvious question would be as follows: which one should we select? The one which compares the best with the reference data? We know that every model, as asserted in the Introduction, has its own intrinsic uncertainty. By relying on only one selected model when many are available, we somehow potentially misjudge the total uncertainty, given that other models could have different predictions with different uncertainties. By applying the BMA method, as introduced in this section, to all available models, we aim to mitigate the above-mentioned issue to some extent , assuming that all models have respectable performance and possess the potential to complement one another. The persistent limitation is that all these uncertainties are assessed at (almost) randomly selected points, namely the NFI plots, leaving the behavior of uncertainty between these points somewhat unpredictable. As will be further elaborated upon in the article, this limitation also emerges as an important yet motivating challenge. It affects our ability to apply the BMA approach as effectively for purposes of synthesizing new spatialized higher-order mixture models, as is the case for the analysis presented in this study, when relying on sparsely distributed reference datasets.

Here, the BMA assumes the combination of model outputs without affecting their internal structure called “the BMA of deterministic models” by. Alternatively, one could also consider employing the BMA to optimize some of the model parameters simultaneously called “the BMA of statistical models” by. While this approach could be relevant for the type of models used in this study, it would first require significant computational resources, as well as a revision of AI-based models to allow certain parameters to remain tunable beyond the training and validation phases.

Therefore, let H be the forest canopy height, predicted using the input data x obtained by one of the K=5 considered models, introduced in Sect. , denoted as M1,…,MK. Similarly, let H denote the reference dataset, introduced in Sect. , containing N=5475 NFI estimates (referred to along with the observations across the article). According to the law of total probability, we can decompose the posterior distribution of the forest canopy height as follows: 1f(H|H)=∑k=1Kf(H|Mk,H)⋅PrMk|H, with f(H|Mk,H) being the posterior distribution of the canopy height under model Mk and PrMk|H being the posterior probability of model Mk. These sum up to 1 and can therefore somehow be interpreted as “importance” weights (PrMk|H≡wk), implying that the posterior distribution of the forest canopy height f(H|H) represents a weighted average of the distributions under participating individual models.

A reasonable assumption when dealing with the canopy height is that its conditional distribution given model Mk can be approximated by a Gaussian distribution centered at the model output mk: 2H(x)|Mk,H∼N(mk(x),σk2), with σk2 being the variance of the kth model, describing its uncertainty with respect to the H NFI observation data. Equation () thus takes the following form: 3f(H(x)|H)=∑k=1Kwk⋅ϕH;mk(x),σk, where ϕ(⋅) denotes the Gaussian probability density function. The conditional mathematical expectation of the canopy height can thus be expressed as follows: 4EH(x)|H=∑k=1Kwk⋅mk(x), representing, essentially, the weighted sum of the canopy height predictions of individual models. This comes at the cost of increased complexity as the mixture involves both the complexity of individual models and the addition of new weights.

Perhaps even more interesting than the mathematical expectation is the variance estimation : 5VarH(x)|H=∑k=1Kwk⋅mk(x)-∑l=1Kwl⋅ml(x)2+∑k=1Kwk⋅σk2, which is decomposed into the between-equation variance (first term of Eq. ) and the within-equation variance (second term of Eq. ). The former quantifies the model spread; i.e., it indicates when models have similar weights but contrasting predictions. The latter denotes the weighted average of the individual model uncertainties, reflecting the uncertainty of the ensemble of models, or the total uncertainty outlined at the outset of this section. This implies that the overall uncertainty could be mis-estimated if only a single model is chosen, even if it performs best in comparison with the reference data.

Once applied to the models and the NFI data from Sects.  and , the method introduced in the previous section gives the overall weights of each model across the entire territory of interest (wk), as well as the local weights at every observation site i, corresponding to the converged “final” value of the missing data (zki). The former are analyzed in Sect. , while the latter are addressed in Sect. . The variance estimations (VarH(x)|H) can also be expressed in the overall or local fashion, with the latter being the subject of Sect.  and . The local estimation of VarH(x)|H at the point i is obtained by substituting the weights wk and wl in Eq. () with the respective missing data (zki and zli).

Since not all models share the same spatial resolution (M1, M3, and M5 at 10 m and M2 and M4 at 30 m), instead of upscaling M1, M3, and M5 to 30 m, we opted to downscale M2 and M4 to 10 m. This was achieved by subdividing a 30 m pixel into nine identical pixels.

The findings outlined in Sect.  indicate that the various models inadvertently tend to predominantly predict different types of forest canopy height. This could, indeed, be a significant finding for the community as describing forests with high spatial precision in terms of four canopy height definitions instead of just one could have positive implications for allometric estimations of wood volume or aboveground biomass. Namely, instead of relying on only one allometric relation, one could simultaneously rely on four of them, differing not only in terms of parameterizations but also in terms of input variables (a type of height) .

This very first portion of the results demonstrated the relatively strong impact of complementing the NFI height measurements or estimations with the MissForest imputations, which may prompt consideration of the potential benefits of incorporating this approach into the NFI sampling design prior to stratified inference.

The findings outlined in Sect.  imply that the fusion of remote-sensing-based observational models may need to be scale dependent, indicating that the contributions of different models vary depending on the focal spatial scale. This aligns closely with a similar message conveyed by  regarding predictive forest species distribution models.

The results in Sect.  pointed out that the models considered in this study diverge, i.e., have similar weights but contrasting predictions in mountainous terrain. Even though the between variance remained below the level of the within variance, with the mixture therefore being reliable, even in the mountains, this implies worse performances of the models in mountain environments, which are more challenging . The hypothesis we formulate is that this comes mostly from the quality of remote sensing data, which is lower in mountainous regions, be it from lidar sensors, multi-spectral imagers, or radar sensors. The list of reasons for the latter is long, with the most prominent effects being the following:

Mountainous terrain introduces distortions into remote sensing images, particularly due to shadows and slope effects, a phenomenon accentuated in radar data by shadows, layovers, and foreshortenings . The GEDI data, in particular, are significantly impacted by steep areas as the distortions of backscattered waveforms within a 25 m footprint introduce additional uncertainty into derived RH profiles .

Thus, we pinpoint (high) mountainous regions as an important challenge for ongoing model advancements, particularly since studies like that of  demonstrate that a combination of Sentinel-1 and Sentinel-2, along with digital terrain models (DTMs), can enhance performance in mountainous areas, at least concerning tree type classification (broad-leaved vs. coniferous). Many well-preserved forest areas worldwide are located in mountainous terrain due to accessibility issues, which makes this particularly important.

The findings exposed in Sect.  suggest that any information about the three out of four considered categorical variables (dominant tree species, vertical structure of forest stand, type of forest ownership) could potentially be a useful modality for the remote-sensing-based models for the high-resolution mapping of the forest canopy height.

The tree species and the topography impacts (Sect. ) are undoubtedly mixed due to the altitudinal zonation, suggesting that tree species can potentially be perceived as a predictor of the variation caused by topographic effects or vice versa.

An interesting influence of the forest stand vertical structure on the estimation of the canopy height can be explained by the remote sensing signal sensitivity over the entirety of the forest stand scanned. This is obviously the case for the lidar but also appears to be the case for the radar , as well as for the optical sensors which can be used to detect and are therefore sensitive to the understory presence and/or composition . Discrepancies among models appear to be greater for regular low forests, other low stands, or irregular structures compared to for regular high stands (Fig. ). This is not surprising as low structures tend to be more heterogeneous. As a result, they are poorly captured by models that target a specific height proxy. This effect could potentially be reinforced by the fact that all models exhibit some degree of saturation (Fig. ). Therefore, in contrast to lower heights, the models tend to converge in the case of higher stands, which cannot exactly be corrected by fitting a higher-order mixture model.

While there are no pristine forests in metropolitan France, not all forests are managed to the same extent, and this often depends on ownership type, making it a strong proxy for forest stand complexity . The impact of ownership type is likely to be linked to the previously discussed influence of forest stand vertical structure. For instance, lightly managed private forests, where we observed the highest between variance, show greater structural variability (26 % of low or irregular stands compared to 17 % of public forests or only 30 % of regular high stands without understory). This is a result of public policies and the tendency of large private properties (which are managed) to prioritize large-dimension timber production. This contrasts with currently lightly managed or unmanaged private properties which have been managed in the past under some form of the coppice system . Moreover, the between variance increases as unmanaged private forests account for between 55 % and 60 % of the forested area in metropolitan France, naturally covering the widest observation gradients.

Beyond demonstrating that, for each of the four forest canopy height definitions, the BMA successfully fits a mixture that surpasses any individual model and the SMA in terms of R2 and NRMSE, the results shown in Sect.  indicate that the mean bias estimates (MBEs) for the mixtures are not lower than those of any of the individual models included. This aligns with the conclusions reported by  and , suggesting that, while the BMA effectively addresses variance, it may need to be supplemented with bias correction methods to ensure that the finite mixture exhibits not only significantly reduced variance compared to the participating models but also lower bias. One alternative approach that might yield less biased mixtures and that is worth exploring would be to adapt the well-established E–M algorithm employed by incorporating restricted maximum likelihood estimation (REML) instead of maximum likelihood estimation (MLE) .

The 5-fold cross-validation confirmed the stability of the estimated local weights, demonstrating a certain competence in predicting the appropriate weights despite changes in the composition of the reference data. An intriguing auxiliary observation worth mentioning in the discussion is that Hdomi and Hlori seem to exhibit the most sensitivity in terms of weight variability (Fig. ). This could be because these forest height types heavily depend on diameter distribution, order statistics, or weighted mean and therefore represent local population properties, unlike Hmaxi and Hmeani.

Equally important is that this exercise highlighted the challenge of predicting weights between sparse reference data points, i.e., evaluating model uncertainty in a spatially continuous manner . While not demonstrated in this article, we believe that the BMA offers a promising approach to tackle this issue, potentially facilitating a major shift from the “analysis” presented here to “synthesis” – combining different models spatially continuously as their estimated uncertainties evolve. One of the avenues we have explored, which certainly merits further attention, is how to stratify the local weights obtained from the “training plots” (80 % of the data in the 5-fold cross-validation) for application to the “test plots” (remaining 20 % of the data). Namely, as elaborated by , under the stated Gaussian assumption, the weighted sum presented in Eq. (), i.e., without any stratification, does not necessarily provide a better fit to the reference data than any individual model.

We have explored several possible avenues for addressing this issue, such as stratifying based on the proximity of estimated height distributions among models or considering the forest ownership criterion. Additionally, we have examined potential modifications to the employed method that could enhance predictive skills. However, the marginal improvements obtained were insufficient for inclusion in this work, which is currently focused on analysis rather than prediction (i.e., interpolation–extrapolation). Instead, the avenues discussed serve as a foundation for future research, which could tackle the broader issue of spatial uncertainty evaluation. Another perspective for future work, and a more straightforward solution to this issue, would be to utilize a spatially continuous reference, such as the canopy height map derived from high-density (HD) lidar data once it becomes available for the entire area of interest. This approach would enable the automatic synthesis of spatialized mixtures. However, it is crucial to acknowledge that such reference height maps will not be entirely flawless due to various sources of heterogeneity, such as differences in sensors and acquisition seasons. As a result, they will need to be thoroughly processed before being considered to be as reliable as the NFI field measurements used in this study.

As for the methodological decision mentioned in Sect. , i.e., the decision not to include the updated weights in the convergence, as suggested by , we also tested the opposite approach, which is more common in the literature. However, this did not improve the goodness of fit shown in Fig.  but only increased the contrast between the local weights of individual models, favoring the overall dominant model. Therefore, we chose to present an analysis that yields at least equally effective higher-order mixtures while emphasizing the potential contributions of models that are not overall dominant.

It is important to note that limiting this study to France may lead to an overly optimistic evaluation of the models and their combinations. This is likely due to the availability of higher-quality training data in Europe compared to, e.g., most tropical regions (e.g., fewer clouds, superior ALS data, and clearer topographic visibility in less dense forests) and boreal regions where no GEDI data are available. Additionally, the range of forest types, while extensive by European standards, is narrower compared to that of tropical forests, which are also characterized by denser stands and greater biomass. Consequently, remote-sensing-based forest attribute mapping faces significant challenges in tropical forests, which are not adequately addressed in this study focused on temperate European forests.

In this study, we jointly interpreted the performances of five different remote-sensing- and AI-based models for the high-resolution mapping of the forest canopy height by combining them using the Bayesian model averaging framework and NFI in situ measurements. We observed that the participation of the different models varies depending on the height reference employed – maximum, mean, dominant or Lorey's – which can be directly linked to the different remote sensing input data. We also observed significant spatial variations in terms of the local weights, indicating that any attempt at model fusion should be tailored to the scale. A much more pronounced spread (comparable weights but contrasting predictions) of the analyzed models was observed in the regions with a pronounced topography, clearly indicating that the real challenge, at least in the temperate regions, is to do better in the mountains by means of a better remote sensing data correction, as well as better modeling. The observed spread is also significantly impacted by the dominant tree species observed, the forest stand vertical structure, and the forest ownership type, with the last-mentioned factor being a very good proxy for the forest stand complexity. The latter suggests that including these as modalities when possible could potentially improve the performances of the analyzed models. Nevertheless, the spread observed when using the BMA remained inferior to the within variance, which was not the case when relying on the simple model average. All this suggests that the response to the paraphrased George Box aphorism posed in the title – “some models are useful, but might they provide us with even more insights when combined?” – could indeed be affirmative in our context, particularly when employing the BMA. However, it is important to note that these models should be complemented by a bias correction method, which is not addressed by the BMA methodology employed. Accordingly, it also failed to correct for the saturation of predicted height observed in individual models. It is also important to note that the BMA method we applied must be complemented by an appropriate stratification strategy before enabling the prediction, i.e., the fitting of spatially continuous higher-order mixture models, when using sparse reference data. In other words, such a strategy is necessary for estimating uncertainty between the in situ measurement points.

From an analytical standpoint, the most apparent direction for the presented work would be to move towards establishing a well-defined framework for evaluating models, possibly incorporating a denser network of references using GEDI measurements. The advantage of using the GEDI measurements would be the possibility of more easily relating the evaluated differences to the employed AI method or the choice of the complementary non-lidar measurements. Another perspective, particularly applicable when relying on GEDI measurements, would involve adapting the method for evaluating tree cover maps. This adaptation would necessitate a change in the probability distribution assumption (Eq. ) from Gaussian to Bernoulli.

From a synthesis standpoint, the most compelling perspective of the presented work would be the ability to use the obtained local weights to effectively produce four finite spatialized mixtures, corresponding to dominant height, mean height, maximum height, and Lorey's mean height. This approach should facilitate the creation of an ensemble model for allometric wood volume or aboveground biomass (AGB) estimation. However, it necessitates either the availability of spatially continuous reference data (lidar HD data) or further methodological research at least partially focused on the stratification of sparsely distributed local weights.