Thanks for these comments. There are a copy of the comments by Reviewer 1, so please see our response to Reviewer 1 at https://doi.org/10.5194/egusphere-2024-2302-AC2.

Methodology

Question 1. Because it partitions the prediction error between bias, plot variability and coefficient variability, we believe that the method we propose is particularly relevant to study how biomass allometry varies when scaling up across different forest types. If we fit an allometric equation to a local forest type and then applies it to another forest with a different structure and species composition, we expect the bias component of MSS to be important. However, when dealing with the heterogeneity of tropical forests, the central question remains to know whether it is worth to address it by introducing in allometric equations additional covariates that explain this heterogeneity, or whether it is enough to leave it as a random noise that will cancel off if the plot is large enough. This question is a question of compromise between bias and variance. Our dataset provides an example of such a compromise linked to species composition. When looking at the tree-level, the model with the best predictive performance is model (21) that fits a different allometry for each species genus. It confirms that different tree genera have different biomass allometries. However, at the scale of the forest where the species composition is not exactly the same as in the calibration dataset, model (21) is also the one that results in the highest bias and the weakest overall predictive performance. The conclusion for this example is that, even there are differences in allometry between tree genera, if our objective is to predict the biomass of large plots, it is statistically more efficient to leave the heterogeneity in species composition as a random noise. We cannot say that this conclusion is general because it is based on our particular dataset. Nevertheless, the methods we provide will allow other people to address such questions in other contexts.

Question 2. We can address the question of the bias arising from the non-representativeness of the sample of trees by using a different dataset. To be more precise, we can fit the allometric equations using a dataset and assess their predictive performance using another dataset. This computation corresponds to the validation case that we discussed in Section 4.2 of the manuscript.

To exemplify this bias, we can fit the allometric equation using the dataset of Fayolle et al. (2018) and assess its predictive performance using the subset of the dataset of Chave et al. (2014) corresponding to the Amazon (with trees coming from Brazil, Colombia, French Guiana and Peru). We may expect indeed central African forests not be representative of Amazonian forests. The following partition of MSS is then obtained for model (20):

This partition of MSS is to be compared to model (20) in Figure 3. The coefficient variability and the plot variability are of the same order for Amazonian forests as for central African forests. However, the bias component is about 30 times bigger for Amazonian forests than for central African forests. This is not a surprise since the allometric equations were fitted using central African trees. Therefore, assessing the predictive performance of the allometric equations on a dataset that is not representative of the forest where the equations were fitted will inflate the role of bias in the overall performance.

We will add these elements in Section 4.2 of the manuscript.

Results, broader implications, and limitations

Questions 1 to 3. These three questions are indeed very important, but we are afraid these questions make sense at geographical scales beyond the forest (landscape, region, country, continent…). Our study deals with the scales between the tree and the forest. The overall objective of our study is to improve the strategy in developing allometric equations, by changing its target. The target of this strategy used to be the minimization of the prediction error at the tree level. Our proposal is to set it as the minimization of the prediction at the plot level. We expect our results to contribute to a better efficiency in the estimation of biomass stocks, but we are not providing actual estimates of biomass stocks. This latter objective would require forest inventory data. Such forest inventory data do exist, but it is out of the scope of our study. We prefer to make it clear, so that there is no misunderstanding between the results we are providing in this manuscript and the results the Reviewer might want to see.

For instance, there is no temporal dimension in our study, for the mere reason that biomass allometry is assumed to be a timeless characteristic of trees, deriving from their biological characteristics only. Changes in forest biomass stocks over time basically result from (1) changes in forest areas, and (2) changes in biomass density. Based on the FAO definition of forests that mainly refer to land use, the former changes (the so-called “activity data” in IPCC’s terminology) are driven by human factors. The latter changes may result from changes in forest structure and composition over time. To address this component of change, we would need forest inventory data at different dates.

This being said, we acknowledge that the objective of our study was not clearly stated in the introduction. We will modify the introduction to better explain it.

Question 4. Our study shows that minimizing the prediction error at the tree level may no not be the most efficient strategy to develop allometric equations when the objective is to assess forest biomass stocks at a large scale (at the forest scale and beyond). As a consequence, introducing additional covariates in the models, with additional measurements costs incurred, may not be needed. Our results may thus contribute to save efforts in measuring tree biomass for the future development of allometric equations. However, the development of allometric equations may be motivated by other objectives than the estimation of forest biomass stocks at large scales. For instance, there may be a theoretical interest in understanding the biological basis of biomass allometry. In that case, the focus will remain on the tree level.

We realize that our manuscript focused too much on the technical aspects of the question and did not explain the more general implication of our results. We will add a sentence in the introduction (line 32) to clarify that the development of new tree biomass allometric equations is still mobilizing a great deal of scientific efforts around the world. We will also add a few sentences in the conclusion to clarify that our results may contribute to be more efficient in these efforts.

Question 5. The question of research gaps on changes in carbon stocks in tropical forests is a very broad question. We here restrict it to the sub-question of the development of allometric equations. The determinants of tree biomass allometry are still not well understood. There is considerable effort in developing new allometric equations around the world (for more species, for more locations). However, with a few exceptions (like the metabolic scaling theory), there is no general theory underlying tree biomass allometry. As a consequence, the various efforts are difficult to aggregate under a common framework.

We will add some sentences in the discussion (after line 285) to clarify that this is a research gap that undermines our understanding of forest biomass stocks.

Question 6. Tropical forests are an important carbon pool at the global level. There are still uncertainties on the quantification of this pool. For instance, there are still major divergent estimates among large-scale biomass maps obtained through remote sensing and field data (Rodda et al., 2024). There are many factors contributing to these uncertainties. At the plot level, the uncertainty on the choice of the allometric equation used to convert inventory data into biomass estimates is one of the greatest sources of error (Picard et al., 2016). Our study contributes to making more informed choices among models.

Question 7. Additional sources of errors could be included in the MSS partition, such as measurement errors. However, previous studies have shown that measurement errors have a minor contribution to the overall biomass prediction error at the plot level (Picard et al., 2015). Therefore, we also expect measurement errors to have a minor influence on our results.

Another limitation of our method is the use of a simulated forest to generate plot data instead of actual forest inventory data. We are not aware of any field data giving the measured biomass of every tree in a forest. Such a dataset probably does not exist given the tremendous cost it would incur, even if LiDAR may change the game in a near future. Therefore, some minimal assumptions are needed to generate large-scale plot biomass data (where biomass is measured, not estimated by a model). At least, a starting point to generate such data could be to use forest inventory data. We could imagine for instance that every tree in the forest inventory data would be assigned the observed biomass of the individual that is closest in size to this tree. Even if tree biomass would not be actual measurements, this process would preserve the shifts in forest structure and species composition that are observed in real forests.

In our study, the bias contribution to MSS is almost independent of plot size (see Figure 2c). We expect this result to change if forest inventory data was used in place of the simulated forest. We would then expect the bias to increase with plot size. Therefore, the relative importance of bias in model selection for large plots would be even greater than it is at present. We thus expect our results to be conservative with respect to the role of bias in model selection.

Question 8. As already said, we are not aware of any dataset giving the measured biomass of every tree in a large plot, not to say a forest. Terrestrial LiDAR may change the game in a near future by providing non-destructive measurement of tree volume for all trees in a plot. We circumvented this issue by resampling measured trees. The bias contribution to MSS may thus have been underestimated in large plots.

Question 9. The method we propose to assess the predictive performance of allometric equations is general and can be applied to any forest. At explained in lines 256-257, the specific ranking of allometric equations is then dependent on the specific forest under study. We could investigate the influence of the forest structure on the model ranking by varying the parameter μ of the simulated forest.

The calibration dataset X has a greater proportion of large trees than the simulated forests F. Therefore, by comparing the bias statistics b_X and the bias statistics b_F given in Table 1, we can already predict how the model ranking would change if the forest had many large trees. Because the model performance for large plots is driven by the bias component, we expect the model ranking to shift from the ranking of b_F  to the ranking of b_X as the proportion of large trees in the forest increases.

Main scientific contributions

Question 1. The main methodological innovation lies in the strategy to rank and select allometric equations. When the objective is to estimate plot-level biomass, allometric equations should be selected on their bias performance at the forest level rather than on their residual error at the tree level.

Question 2. Our study could contribute to reduce the uncertainty on the model choice that represents an important part of the uncertainty of biomass estimates in tropical forests. The drivers of spatial variation of biomass in tropical forests could then be more easily identified and understood.

Question 3. There is no restriction to apply the method we propose to other tropical regions. On the contrary, we encourage this method to be replicated in other tropical areas and other biomes. We expect the relative importance of bias, plot variability and coefficient variability in model selection to change from one place to another. However, the decreasing importance of plot variability with plot size should be a constant feature.

Additional references

• Picard, F. Boyemba Bosela, and V. Rossi, “Reducing the error in biomass estimates strongly depends on model selection,” Annals of Forest Science, vol. 72, no. 6, pp. 811–823, 2015, doi: 10.1007/s13595-014-0434-9.
• Picard et al., “Error in the estimation of emission factors for forest degradation in central Africa,” Journal of Forest Researc, vol. 21, no. 1, pp. 23–30, 2016, doi: 10.1007/s10310-015-0510-5.
• R. Rodda et al., “LiDAR-based reference aboveground biomass maps for tropical forests of South Asia and Central Africa,” Scientific Data, vol. 11, no. 1, p. 334, 2024, doi: 10.1038/s41597-024-03162-x.

Issue 1. We acknowledge that we overlooked BEF methods and other plot-level models in the introduction. We acknowledge too that we overlooked biomass studies from Asia. BEF and other plot-level models necessarily rely on plot-level biomass data for their calibration. Not all studies on BEF and plot-level models clearly indicate how they obtained these biomasses at plot-level. In many cases, plot-level biomass was actually obtained from tree-level measurements and tree-level allometric equations to convert these measurements into biomass. Pan et al. (2004), cited by the Reviewer, proceeded in this way. So did Brown et al. (1989), which is also an often-cited reference on BEF methods. Using allometric equations to derive plot-level biomass is precisely the point we are addressing in our manuscript. If the tree-level allometric equation is biased, so will be the plot-level biomass data obtained from it, and so will be the BEF and plot-level model fitted to these data.

In other words, BEF and plot-level models are not really an alternative to tree-level allometric equation. They are rather an intermediary step to scale up biomass estimates from the tree level to higher levels (plot, forest, region, country, continent…).

We will change the introduction to clarify that in the chain of measurements that starts from the tree and end with remote sensing techniques, there is an additional intermediary step that consists in BEF and plot-level methods. Lines 29 & sqq. will be changed as follows: “Plot-level biomass can be used to fit plot-level models that predict plot biomass from plot volume and other plot characteristics, using biomass expansion factors or related approaches (Pan et al., 2004; Fang et al., 2007; Guo et al., 2010). These plot-level models can then be used to estimate forest biomass at the country (Fang et al., 2007) and continental (Fang et al., 2014) scales. Plot-level biomass can be used too to calibrate remote sensing indices to predict the biomass of pixels in satellite images […]”.

Issue 2. For smoother reading, we will compile the mathematical expressions of the predictive performance statistics in a table that will be merged with Table 1. The new Table 1 will have three columns: statistics; mathematical expression; level (tree, plot or forest). All the equations related to the decomposition of the sum of squared errors (viz. equations (5) to (19)) will be collated and put into supplementary material.

Issue 3. We will streamline the introduction by moving some technical parts (lines 46-48, lines 54-57, and lines 68-72) to the methods section.

As regards the model form, the models we propose are all rooted in the concept of allometry as defined by Huxley & Teissier (1936). It assumes that the relative growth rates of two parts of an individual correlate (Gould, 1966). Models (20) to (23) correspond to simple allometry, where the ratio between relative growth rates is fixed. As discussed by White & Gould (1965), the biologically meaningful parameters are the coefficients associated to covariates. Model (24) correspond to complex allometry, where the relative growth rate of biomass is an increasing function of the relative growth rate of diameter. After back-transformation from the log-transform, model (24) also correspond to a log-normal model. Its parameters correspond to maximal biomass, the diameter where biomass reaches its maximum, and a shape parameter. These parameters also have a biological meaning. As a tree grow, it accumulates biomass as its diameter increases, until it reaches senescence. When senescent, it may lose biomass (because of dead branches, holes in the trunk, etc.) while its diameter still increases.

Apart from allometry, we could have indeed included other families of models that predict tree-level biomass. Geometric models are rooted in the tree taper concept. They predict biomass as wood density times volume, where volume is integrated from a taper equation (Manso et al., 2024). Another family of models emerges from the carbon allocation strategy of trees (Wolf et al., 2011; Yang et al., 2023).

While we agree that biomass models should aim at providing a biological interpretation and relate to theories, we disagree that model fitting should be restricted to those models with a theoretical basis. Theories need to be tested against observations and data, not the other way around.

We will add wording in the Methods section (§2.4) to clarify the biological concepts underlying the model studied. We will also add a paragraph in the discussion (after line 285) to clarify that other families of models could be used, thus contributing to the debate on the biological basis of biomass allometry.

As regards model performance, one way to assess the prediction error when covariates extend beyond the calibration range is to partition the dataset in two subsets depending on tree size, calibrate the allometric equation using one subset, and assess the prediction error using the other dataset. This approach is very similar to what we did in Section 4.2 based on tree height. For instance, we may partition the dataset into the subset X₁ of trees with diameter < 48.9 cm and the subset X₂ of trees with diameter ≥ 48.9 cm, where 48.9 cm is the median diameter of the dataset. If we use X₂ to calibrate the allometric equations and predict the biomass of trees in X₁, the following partition of MSS is obtained:

The other way around, if we use X₁ to calibrate the allometric equations and predict the biomass of trees in X₂, the following partition of MSS is obtained:

Therefore, the error of predicting the biomass of large trees with an allometric equation fitted to small trees is much greater than the error of predicting the biomass of small trees with an allometric equation fitted to large trees. Moreover, the bias is comparatively greater in the former case than in the latter case. There is nothing really new in these results. Due to heteroscedasticity, there is much more variability in tree biomass in large trees than in small trees. Including large trees in biomass datasets is a recommendation that is known for long (Chave et al., 2005). We may include these additional elements in the discussion (Section 4.2) even if, in our view, there is limited interest in doing so.

References (in addition to those cited by the Reviewer)

• Brown, A. J. R. Gillespie, and A. E. Lugo, “Biomass estimation methods for tropical forests with applications to forest inventory data,” Forest Science, vol. 35, no. 4, pp. 881–902, 1989, doi: 10.1093/forestscience/35.4.881.
• Chave et al., “Tree allometry and improved estimation of carbon stocks and balance in tropical forests,” Oecologia, vol. 145, no. 1, pp. 87–99, 2005, doi: 10.1007/s00442-005-0100-x.
• J. Gould, “Allometry and size in ontogeny and phylogeny,” Biological Reviews, vol. 41, no. 4, pp. 587–638, 1966, doi: 10.1111/j.1469-185X.1966.tb01624.x.
• S. Huxley and G. Teissier, “Terminology of relative growth,” Nature, vol. 137, pp. 780–781, 1936, doi: 10.1038/137780b0.
• Manso, A. Price, A. Ash, E. Macdonald, “Volume prediction of young improved Sitka spruce trees in Great Britain through Bayesian model averaging,” Forestry: An International Journal of Forest Research, cpae010, doi: 10.1093/forestry/cpae010
• F. White and S. J. Gould, “Interpretation of the coefficient in the allometric equation,” American Naturalist, vol. 99, no. 904, pp. 5–18, 1965, http://www.jstor.org/stable/2459251
• Wolf, C. B. Field, and J. A. Berry, “Allometric growth and allocation in forests: a perspective from FLUXNET,” Ecological Applications, vol. 21, no. 5, pp. 1546–1556, 2011, doi: 10.1890/10-1201.1.
• Yang et al., “Dynamic carbon allocation trade-off: A robust approach to model tree biomass allometry,” Methods in Ecology and Evolution, vol. 15, no. 5, pp. 886–899, 2024, doi: 10.1111/2041-210X.14315.