Biomass Estimation and Uncertainty Quantification From Tree Height

We propose a tree-level biomass estimation model approximating allometric equations by LiDAR data. Since tree crown diameter estimation is challenging from spaceborne LiDAR measurements, we develop a model to correlate tree height with biomass on the individual-tree levels employing a Gaussian process regressor. In order to validate the proposed model, a set of 8342 samples on tree height, trunk diameter, and biomass has been assembled. It covers seven biomes globally present. We reference our model to four other models based on both, the Jucker data and our own dataset. Although our approach deviates from standard biomass–height–diameter models, we demonstrate the Gaussian process regression model as a viable alternative. In addition, we decompose the uncertainty of tree biomass estimates into the model- and fitting-based contributions. We verify the Gaussian process regressor has the capacity to reduce the fitting uncertainty down to below 5%. Exploiting airborne LiDAR measurements and a field inventory survey on the ground, a stand-level (or plot-level) study confirms a low relative error of below 1% for our model. The data used in this study are available at https://github.com/zhu-xlab/BiomassUQ.

of accumulated organic material in an ecosystem.It has been applied widely as an index of forest volume.Moreover, it is key to monitoring ecosystems and modeling climate change [1], [3].In addition, biomass provides a tool to evaluate carbon sequestration.Accurate biomass estimates help assess loss caused by wildfire [4].
Forest AGB evaluation is categorized into three levels: fine, middle, and coarse-grained, cf.Fig. 1.In situ measurements of biomass include tree harvest and desiccation to weigh the wood on a scale [5].Despite errors in the harvesting process and scale uncertainties, this method most accurately evaluates biomass on the tree level.However, such a destructive approach is costly in labor and time.As an alternative, allometric equations provide estimates of biomass on individual tree level [6]- [9].Tree biomass is considered a function of tree height, trunk diameter at breast height (also known as trunk diameter or simply diameter), wood density, crown diameter, etc. Luo et al. review state-of-the-art allometric equations applied to tree species in China [10].However, the method presented is not easily transferred to a global scale.
To generate large-scale (regional to global scale) biomass maps, spaceborne hyperspectral and synthetic aperture radar (SAR) sensors with wide swath and global or near-to-global coverage, have been applied in the literature [11]- [14].Based on a regression model trained with ground reference and remote sensing data pairs, the total amount of biomass for each pixel in the remote sensing imagery can get estimated.However, limited by the coarse-grained resolution of remote sensing imagery 2 , ground truth reference data is costly to collect.According to [18] approximately 234 trees per ha grow in the Black Forest, Germany.Also, since the values of pixels in remote sensing imagery are not related to trees' biomass one-to-one, the relative errors might raise up to 37% [1].
A trade-off compared with in-situ biomass measurements is the estimation of tree-level parameters (such as height, crown diameter, etc.) from high-resolution remote sensing data as input to allometric equations [19]- [22].Although highly correlated with biomass, parameters such as wood density and diameter cannot reliably get estimated by aerial imagery.Jucker et al. confirmed that the height and crown diameter of trees are sufficient to estimate the trunk diameter by a single equation.Crown diameter and height are easily derived from airborne laser scanning (ALS) data [19], [20], [23]- [25].However, the crown diameter estimation is a source of significant model error.Deciduous tree's crown changes with season, and extraction of the crown profile for individual trees in dense forests is a major challenge.As demonstrated by Figure 2, the crown profile of isolated trees may get reliably estimated, cf.label A. However, in densely populated areas such as labeled B, a reliable separation is close to impossible.The analysis of high-resolution aerial imagery in [26] revealed the accuracy of estimated crown diameter significantly varied with plots (0.63 and 0.85) when compared with height estimations.According to [27], the relative error of tree crown diameter derived from airborne/UAV-borne LiDAR data is significantly larger than that of derived tree height (19.22% and 20.7% for crown diameter, 11.70% and 10.97% for tree height estimation respectively).In fact, the crowns of individual trees in e.g.rainforests may significantly intersect.
A few studies investigated stand-level height-biomass allometry [28]- [30].Due to the lack of tree density information, these methods either focused on regional biomass estimation [28], or additional metrics such as percentile heights [31] and horizontal structure index [30] are integrated in the allometric relationships.Alternatively, tree-level heightbiomass allometry may be beneficial as it directly includes tree density information.But as indicated in [32], heightdiameter relationship varies even within a small scale given the compositional diversity.Therefore, to build a general treelevel height-biomass allometric relationship for fast biomass estimation, it is necessary: 1) the relationship is derived based on a dataset that collected over large areas; 2) the regression model should be less biased, so that aggregation of individual tree biomass within homogeneous forests would reduce biomass errors.In this paper, we evaluate such an approach: We assemble a ground truth dataset including 8,342 measurements of tree height, trunk diameter, and biomass drawn from global sampling.We proposed a Gaussian process regressor (GPR) which is a noise-aware model to capture the nonlinear relationship of biomass and tree height and reduce the estimation bias.
In addition, uncertainty quantification is crucial for evaluating the confidence level of derived models.Uncertainty related on plot-level biomass estimation based on tree al-lometry mainly comes from two sources [5]: independent tree variables derivation uncertainty due to inaccurate and/or insufficient measurements and estimation uncertainty led by residual model noise and imperfect reference data.In this paper, we focus on the estimation uncertainty.Most existing publications reported their model's residual noise by metrics such as root mean square root (RMSE) [1], [21].On the other hand, Monte Carlo simulation approach was used to quantify the parameter uncertainty by selecting different samples as training data [33].Usually, they assume that the reference data are noise-free.However, due to the imperfect sampling strategy, the impact of reference data on the uncertainty quantification is non-negligible [5].In this paper, we propose an algorithm to decompose the estimation uncertainty into model uncertainty and fitting uncertainty.This way, we analyze the model uncertainties of four allometric equations in contrast to the fitting uncertainties of five candidate models.Stand-level uncertainties of the proposed GPR and two other models is evaluated based on LiDAR measurements and field surveys.

II. DATASETS
Our experiments employ a dataset collected from the opensource allometry databases [8], [36], [37].Dataset provided by [8] consists of 4004 tree measurements at 58 sites in tropical forests over the globe.The Biomass And Allometry Database (BAAD) [36] includes 258,526 measurements over the globe collected from 175 studies.Each measurement records the tree height, tree components' biomass, etc.The biomass data collected in [37] includes 6,604 records of trees in Eurasian forests (mainly in Russia).Then part of the measurements in the datasets were removed so that the left ones obey the following characteristics: • the total height and trunk diameter of the tree are recorded; • the geographic location of the tree is recorded; • the tree is harvested to measure its biomass; • the tree's diameter exceeds 5 centimeters (cm); • the tree's biomass passes the threshold of 2 kg.[34]).And (b) depicts corresponding aerial imagery over Prospect Park in Brooklyn, New York, USA in 2017.A and B mark areas of an isolated tree vs. a cluster of trees, respectively.LiDAR statistics and NAIP imagery got harmonized by the Big GeoData platform PAIRS.[35] Information on ecoregions defining geospatial boundaries of biome types (TEOW) has been downloaded from the world wide life fund (WWF) [38].According to geographic location, each measurement is allocated by one of the seven biome types: tropical and subtropical forests, temperate mixed forests, temperate coniferous forests, boreal forests, grasslands and shrublands, tundra, savannas, woodlands, and mediterranean forests, or deserts and xeric shrublands.Based on previous research, the parameters of the allometric equations depend on species, climate, and environmental conditions.Instead of training multiple models for each species and ecoregion, we explored the potential of using a single model to capture the variation caused by ecological factors.
Here we utilize a total number of 8,342 pairs of tree heightdiameter-biomass measurements.In addition, we employ the Jucker data [21] as a reference for the proposed model.In addition to height, diameter, and biomass, the Jucker dataset [21] also records crown diameter.Although the Jucker data includes trunk diameter information, it contains 2,395 samples, only.Figure 3 (a) visualizes the geographic distribution and the number of records for various sites.We set the diameter of the blue circles in proportion to the number of measurements.Distinct biome regions are colored differently.The dataset has global coverage containing all four forest types defined by the Food and Agriculture Organization of the United Nations (FAO) report 3 .Figure 3 (b)-(d) present violin and box plots including median values (blue circles) and outliers (grey circles) for each: height, diameter, and biomass of every biome type, respectively.Figures in plot Figure 3 (b) indicate the number of measurements in the biome regions.All three tree parameters cover a wide range of measured values: biomass may get as little as two kilograms (kg), and it may exceed 300 tons; tree height varies from 1.2 to 138 meters; tree diameters span a range from 5 cm up to more than two meters.Figure 4 plots the distribution of tree diameter D vs. tree height H in double-logarithmic scale.Point colors indicate the 3 https://fra-data.fao.org/log-scaled amount of biomass.Obviously, biomass increases with tree height and trunk diameter.We observe: besides a small number of outliers, tree height and trunk diameter are highly correlated.

A. Allometric Equation
Biomass refers to the total amount of dry weight of organic material in a unit area, i.e. the unit of biomass has dimension of e.g.kilograms per square meter m 2 or tons (t) per hectare (ha).A tree's biomass accumulates from the biomass of stump, trunk, branches, twigs, and leaves [39].To accurately measure the biomass of trees, trees are felled, and dried at 105 • C for scaling.Large trees is impossible to gauge.Instead, wood densities ρ i and volumes V i of all tree components labeled i get recorded to estimate the biomass as i ρ i V i .In [40]- [42], the authors elaborated on the process of dry biomass measurements.
The study [43] demonstrates trunk biomass constitutes about 83% of the total biomass of the tree.In addition, based on measurements with total above-ground biomass larger than 2 kg in BAAD [36], leaf mass accounts for about 10% of a tree's biomass.Consequently, trunk biomass estimation needs the most attention.Assuming trunk biomass modeled by a cone, the biomass B = ρV = 1  12 πρD 2 H, where B, D, and H denote the tree's biomass, diameter, and height, respectively.ρ is the average wood density of dried trees that remote sensing is unable to capture.A central assumption of our work reads: tree height is able to predict tree diameter such that biomass is predominantly determined by tree height.Our experiments in Section IV indicate complex relationships beyond a loglog linear model.Hence, we establish a non-linear model B ∼ GP (H), where GP corresponds to a Gaussian process regressor detailed in Section III-B.The mapping of height H to biomass B is sensitive to average wood density and the biome-dependent relationship of tree diameter vs. tree height.

B. Gaussian Process Regressor
In order to model noisy biomass B = B(H) depending on tree height measurements H based on a set of samples (H 1 , B 1 ), (H 2 , B 2 ), . . ., (H n , B n ), we employ Gaussian process regression.
Gaussian processes implement distributions over sequences of variables y = (y 1 , y 2 , . . ., y n , y n+1 , . . . ) fully parameterized by mean values and the (symmetric) two-point correlation function where we introduced the statistical averaging operator over the Gaussian distribution N .Higher order (centralized) moments y i y j y k , y i y j y k y l , . . .
can get expressed as products of two-point correlation functions [44].Thus, for the below samples sequences y from a multivariate Gaussian distribution with zero mean and covariance matrix K defined by matrix elements K ij .
It is observed that in most physical systems (spatial) correlations exponentially decay proportional to the length scale l, ∝ e −l .In fact, algebraic decays, i.e. ∝ l −α , indicate strongly correlated systems close to phase transitions.It is therefore reasonable to model the kernel where the x i and x j is associated with either height measurements H or model inputs Ĥ. Figure 5 depicts the two-point correlation for two measurements y i = B i and y j = B j far apart (x i = H i H j = x j ), close (H i ≈ H j ), and identical (H i = H j ).
In order to predict N − n values ŷ given a set of n observation pairs (x 1 , y 1 ), (x 2 , y 2 ), . . ., (x n , y n ) we cast both into an N -variate Gaussian distribution over variables y = (y, ŷ) and corresponding (given) parameters x = (x, x) to sample from the resulting conditional Gaussian probability distribution 4ŷ|y ∼ N (κK −1 y, where we decomposed the covariance matrix K according to y = (y, ŷ): with κ T the transposed matrix of κ.Note that K depends on x, only.Similarly, K takes prediction inputs x, only.In contrast, κ entails a mix of x and x.By design K ii = 0 such that we may flexibly add uncorrelated Gaussian measurement noise to x through where σ ∈ R quantifies the variance of the uncorrelated measurement noise.The Kronecker-delta δ ij turns zero for all indices except for i = j where it assumes the value 1.In addition, we may want to explicitly model a mean/expected (biomass) function m(x ) = b(H ) that translates into In summary: Given • the one-dimensional radial basis function kernel at length scale l where 1 denotes the unit matrix, b 0 is a constant hyperparameter mean biomass, i.e., getting optimized alongside with l, and 2) associated covariance matrix Σ Note that the various elements of K contain both, data tree heights H i of sample biomass terms i.e. when taking the logarithm, it is minimized the scalar (loss) function where we exploited log det = Tr log [46], and defined the ldependent eigenvalues k i of the symmetric matrix K + 1σ.β i denotes the b 0 -dependent components of vector B − b 0 in the eigenbasis of K.When using the Gaussian process for tree biomass estimation, the mean biomass offset can be predicted as a linear combination of observed biomass offsets weighted by the covariance matrix (closer samples have higher weights), as formulated in Eq. 12.In Eq. 16, the prediction uncertainty (covariance matrix) consists of two parts, the inherent noise level, and the height distances between the observed data and the prediction inputs.
Notice: Gaussian process regressors can get casted into the framework of non-parametric Bayesian models.An approach that has proven efficient in many non-linear regression tasks [47]- [49].

C. Evaluation Methods
We compare the Gaussian process biomass-height model with a random forest (RF) model and three allometric equations, specifically: biomass-height-crown diameter (LR), biomass-height (LR2), and biomass-height-diameter (LR3).Random forest is a data-driven non-linear regressor, which has been widely applied to biomass estimation [1].The form of the three allometric equations read: where a, b, c are the coefficients and bias terms determined by the training data; CD refers to the crown diameter; and is model residuals.Since no crown diameter measurements in curated data in section II, we utilize an alternative biomassdiameter model: 1) Tree-level Results Evaluation: To evaluate model accuracy, three indices get derived: R-squared (R 2 ), root mean square error (RMSE), and model bias.R-square refers to the coefficient of determination, and is defined according to where y i and ŷi are the i-th ground truth and predicted values.ȳ amounts for the average mean of ground truth.ESS, TSS, and RSS abbreviate the definitions of explained sum of squares, total sum of squares, and residual sum of squares in line with According to these definitions, the R-squared score may receive impact by a single, strongly biased estimation.Thus, calculating R 2 , we exclude outliers when the corresponding absolute error exceeds the mean absolute error by at least three times, cf.red circles in Figure 7. RMSE is calculated as follows Bias relates to relative systematic error.It is defined as The negative or positive value of the bias indicates biomass under-or overestimation.
In the following, we use a binning method to visualize prediction errors for input dimensions.That is, the residuals between observed and predicted biomass y i − ŷi are calculated.According to the percentile input values such as height get assigned on a logarithmic scale, and residuals are split into separate groups (bins).The interval spanning the mean plusminus half a standard deviation for each bin is presented alongside the fitted curve.The result visually depicts the mean and standard deviation of the prediction errors.In addition, it illustrates the level of over-or underestimation.
2) Plot-level Results Evaluation: We quantify uncertainty on stand level by relative error (RE) and relative root mean square error (relative RMSE, denoted as %RM SE).Since insitu data for biomass is unavailable, the biomass obtained by the LR3 model trained on our curated dataset using the filed inventoried tree heights and diameters serves as ground truth.
By aggregating individual tree information, the relative error denotes the ratio of the sum over residuals and the sum over predicted biomass values by LR3: Here H i , D i represent tree height and diameter of the i-th tree in the plot; f indicates one of the candidate models, and x i signals model input parameter(s).
The relative RMSE refers to the ratio of RMSE and the mean of biomass predicted by LR3: where i indexes the i-th plot.
3) Uncertainty Evaluation: In the following, we elaborate on the uncertainty evaluation algorithm in use.We concern with two sources of uncertainties, namely: model uncertainty and fitting uncertainty, cf. Figure 6.Model uncertainty indicates variance rooted in model selection such as the choice of input parameters for allometric equations, etc.In practice, tree biomass depends on many factors such as annual rainfall, species, and average annual temperature.The model at hand might bear the limited capacity to capture such dependencies.As a result, the mapping from input to biomass remains noisy with the model unable to capture such residuals.We define model uncertainty by the variability of measured biomass.Concerning the wide range and heavy tail of the tree biomass distribution, we work with log-scaled quantities.Specifically, the model uncertainty index is calculated as follows: 1) measurements get sorted by the input parameter such as tree height, crown diameter, etc., and are grouped into n buckets; 2) for each of the n groups, the ratio of standard deviation to mean of the biomass is calculated on a logarithmic scale; 3) the overall model uncertainty is calculated as the averaged ratios.In some allometric models, biomass is correlated with multiple parameters such as height and diameter.Subsequently, the measurements are sorted by one of the parameters.
In general, increasing the number of input parameters has the potential to decrease model uncertainty at the price of additional effort to collect data.An alternative provides training separate models for each biome, species, and age.On the downside, this approach requires vast amounts of in-situ measurements harvesting trees.Also, there exists an option to reduce model uncertainty for stand-level products where spatial aggregation of tree biomass may cancel over-and underestimation [1].
When employing various forms of regression models, the fitting precision of the regressors varies.We refer to the discrepancy in average biomass and predicted biomass as fitting uncertainty.Computation of the fitting uncertainty is similar to model uncertainty calculation: 1) the measurements are sorted by input parameter and assigned into n evenly spaced pockets; 2) the absolute error of predicted biomass subtracted by the mean observed biomass is determined for each measurement of every group; 3) for each of the n groups, the ratio of mean absolute error (MAE) to mean observed biomass is computed on a logarithmic scale; 4) the overall fitting uncertainty is computed as an averaged ratio.
We are going to demonstrate that Gaussian process regressors reduce the fitting uncertainty with regard to the random forest and linear regression models.

A. Jucker Data
We adopt the Jucker data in order to benchmark the Gaussian process regressor in reference to the other models introduced.The dataset includes 2,395 measurements including records on crown diameter.In order to compare with and validate trained models such as LR proposed by Jucker et al., we filter training data to exclude diameters smaller than 5 cm.We apply a random split into training and test sets in proportion to 9:1.
Table I summarizes the performance of the five regressors.LR3, the biomass-height-diameter in Section III-C performs best in terms of all three indicators picked.R 2 , RM SE, and Bias yield values 0.95, 424.68 kg, and 0.08, respectively.As detailed in Section III-A, a linear model on a log scale is insufficient to fit tree height-biomass data, and the LR2 model performance is reflected by an increased RMSE (1.47 Mg), the most prominent bias (0.29), and an R 2 score equating to 0.53.Non-linear models-such as random forest and Gaussian process regressor-reduce the RMSE to 1.15 Mg and 1.12 Mg, respectively.
Our experiment indicates a low RMSE for the LR, RF, and GPR models, namely: 1.11 Mg, 1.15 Mg, and 1.12 Mg.Compared with LR 20 (the most widely used model), RF and GPR yield higher R-square scores by margins of 21% and 27%, respectively.Consistently, the bias drops by 19% and 15%.Based on the above findings, our tree heightonly Gaussian process regressor provides a serious option for biomass modeling when compared with state-of-the-art biomass-height-crown diameter models.
The left column in Figure 7 lists fitted curves (blue lines) and corresponding errors distributions (blue areas) for the five models we did investigate.The background resembles density maps of biomass-input parameter pairs.We observe the LR3 model fits best with the data, it yields the lowest uncertainty, cf. it exhibits the most narrow range of green-dashed, vertical lines in the plots of the right most column of Figure 7.In Figure 7 (a), although the fitted line doesn't align perfectly with the data, the actual biomass is linearly correlated with the product of tree height and crown diameter, which implies that a linear log-log model can describe the relationship between them.
In terms of single-parameter models, LR2 overestimates biomass predictions for medium range tree height values, and it strongly underestimates the biomass for small and large heights-a linear model does not properly capture the nonlinear biomass-height relationship.Both, the random forest and Gaussian process regressor render well with the data.However, the fitted curve of the RF model is less regular compared with GPR bearing risk of less robustness with respect to outliers.
The scatter plots of Figure 7 (center column) contrast modeled biomass with observed ground truth.Red circles label outliers.All plots exhibit strong correlation between predicted and observed biomass.Results in Figure 7 (h) is best aligned with the diagonal y = y(x) = x suggesting the biomassheight-diameter model as preferred.Unfortunately, in many remote sensing scenarios estimating tree diameter is out of reach.The data in Figure 7 (e) document the LR2 model tend to overestimate the biomass when observed biomass is around 40 kg, while underestimating above about 200 kg.In terms of height-only models, random forest and Gaussian process regressor, Figure 7 (k) and (n), is ruled by comparable performance with lower bias for the full range of input data when referenced to their linear counterparts.
The right column in Figure 7 evaluates the density distributions of residuals B i − Bi .Dashed lines are the 20th (left) and 90th (right) percentiles of the errors.Obviously, LR and LR2 models are significantly biased with positive errors dominating.Besides, RF and GPR distribute errors alike.

B. Collected Data
The five candidate models are then trained and tested using the collected data in Section II.Since the dataset does not record crown diameter, as an alternative to LR in Equation 20, we exploit the biomass-diameter model of Equation (23).Table II and Figure 8 present corresponding results.The plots in Figure 8 is arranged in line with Figure 7.
The LR and LR3 models yield significantly less bias-0.14 and 0.11, respectively-when compared with the other models exceeding 0.34.R 2 scores for LR and LR3 read 0.73 and 0.78, respectively.Our findings suggest that • tree diameter is relevant in biomass estimation; • tree height information improves model accuracy.It seems a linear regressor is sufficient to render the biomasstree diameter relationship.The dominant root mean square errors (8.2 Mg) stem from outliers (red circles) in Figure 8 (h).
A linear biomass-height model results in most poor performance, with R 2 as low as 0.25, and a bias of 0.50.The plot in Figure 8 (e) illustrates a significant underestimation of model predictions vs. ground truth when the observed biomass exceeds 1 Mg.We conclude the log-log linear model misses to represent the above-ground biomass-tree height relationship.In fact, the nonlinear models outperform the linear model in terms of all three indicators.Moreover, the residual errors in Figure 8 (l)(o) better centers on zero compared with the results of Figure 8 (f); an indication of the nonlinear models more closely agree with the test data.Compared with the random forest model the Gaussian process regressor is less biased.However, it ships with larger RM SE of 5.0 Mg and lower R 2 score equal to 0.66.The exceptionally high R 2 score roots in top generalization ability for B > 2 Mg.In Section V below we demonstrate that the GPR model outperforms RF.

C. Uncertainty Evaluation
We consult the Jucker data [21] to quantify model uncertainties.Figure 9 contrasts the model uncertainties of the four models: biomass-height, biomass-diameter, biomass-crown diameter, and biomass-height-crown diameter, respectively.It suggests that diameter is closely related to biomass.The biomass-diameter model exhibits the lowest model uncertainty of about 14%.The biomass-height model reaches medium performance at overall model uncertainty of 18.25%.The overall model uncertainty of the single-parameter biomass-crown  10 aggregates fitting uncertainties of our five candidate biomass models learned from the Jucker data.In general, fitting uncertainties stay below model uncertainties.The overall fitting uncertainties of LR, LR2, LR3, random forest, and the proposed Gaussian process regressor read 8.80%, 11.45%, 6.13%, 6.90%, and 4.50% respectively.We conclude estimation errors is dominated by model uncertainty: All five models exhibit higher fitting uncertainty when the observed biomass is less than 2.5 log kg) with the GPR model (marked by star) performing best.Because of the non-linear biomassheight relationship, the LR2 model (marked by diamond) scores highest with respect to fitting uncertainty, LR indicates medium performance, while LR3 and RF unveil performance scores on equal level.The GPR model demonstrates the lowest overall fitting uncertainty.Moreover, it constantly performs in all the groups suggesting the Gaussian process regressor over the other models in terms of low fitting uncertainty.
V. VALIDATION BY LIDAR DATA Finally, we study the uncertainty of trained models on stand level.We utilize a dataset that get assembled from forests in Baden-Württemberg, Germany in the years 2019 and 2020 [27].It embraces 12 separate plots, each covering a spatial area of about one hectare.For each plot, point clouds of individual trees get segmented from terrestrial, UAV-borne and airborne LiDAR devices.Field inventory measurements are available for a fraction of trees, too.We exclude from the validation process three out of the 12 plots with less than 20 trees available.
For each tree, its height, diameter, and crown diameter are derived either by field measurements and LiDAR point cloud data, and discard from the analysis trees without field measurements.For a single tree, there may exist multiple LiDAR measurements and the number of measurements fluctuates from plot to plot.Therefore, we average all measurements.Note that these measurements are incomplete, for example, tree heights were not inventoried (or measured by LiDAR data).In those cases, LiDAR-measured (or inventoried) variables were used instead.In Figure 11 (a), we present box plots of the tree biomass grouped by stands where it is indicated the     in seven out of nine plots, thus there is a significant bias for errors to accumulate.In Table III we compute the relative error of the three models given all reference data.We notice the LR model rendering more biased compared with the GPR.The scatter plot in Figure 11 (c) supports that LR tends to overand underestimate tree biomass when assuming values less and larger than 500 kg, respectively.GPR-predicted biomass estimates well correlate with the predicted biomass of the LR3 model reference.
As listed in Table III, the relative RMSE of models LR, RF, and GPR assume values 16.93%, 46.08%, and 24.46%, respectively.Although GPR is less accurate compared with the LR model, the relative RMSE of GPR is acceptable when referenced to the state-of-the-art biomass estimation errors on a national and global scale, cf.[1] quoting %RMSE values in 37% to 67%.

VI. CONCLUSION
We proposed a Gaussian process regressor model to estimate biomass on individual tree level taking tree height as input, only.It enables rapid regional-to-national above-ground biomass evaluation from high-resolution LiDAR data.As a single input parameter model, a series of existing allometry databases contribute to model training.We benchmarked GPR against four established biomass models training on Jucker data and a dataset curated by this work.Results confirm GPR performs best when compared with two biomass-height models, and it achieves reasonable results in reference to a biomass-height-crown diameter model.GPR generates a low fitting uncertainty of 4.50%.Stand-level uncertainty analysis of GPR yielded an averaged relative root mean square error of 24.46%.Moreover, GPR renders less biased at a mean relative error of 0.0021.Future work may explore a stratified approach where biome-specific models [21] have the potential to decrease model uncertainty.

Fig. 2 .
Fig.2.Sample of rasterized statistics of LiDAR return count (a) (for details in methodology cf.[34]).And (b) depicts corresponding aerial imagery over Prospect Park in Brooklyn, New York, USA in 2017.A and B mark areas of an isolated tree vs. a cluster of trees, respectively.LiDAR statistics and NAIP imagery got harmonized by the Big GeoData platform PAIRS.[35]

Fig. 3 .Fig. 4 .
Fig.3.Summary of data collected[8],[36],[37]: Geospatial distribution of the measurements plotted on top of the biome classification map.Circle diameters represent the number of records at each geo-location (a); violin plots of the distributions of tree heights in meters (b), tree diameters in centimeters (c), and above-ground biomass in kilograms (d) for various biomes.The number of records for each biome is shown as a number to the right.
B i , and values Ĥi for biomass values Bi to predict.Also, the constant offset b 0 could get replaced by a more generic (known) functional dependence, e.g. a linear model b(H ) = b 1 H + b 0 , etc.The scalar hyperparameters l and b 0 get optimized by maximization of the predictor variables B-marginalized likelihood

Fig. 6 .
Fig.6.Illustration of model uncertainty vs. fitting uncertainty.Dots and the solid lines refer to the sample measurements and the averaged biomass, respectively.Two sources of uncertainties we focus on: the model uncertainty that corresponds to the standard deviation of the sample measurements, and the fitting uncertainty-that is: the deviation of averaged biomass and the regressor-predicted biomass.

Fig. 7 .
Fig. 7. Plots of the fitted curves with corresponding prediction errors (left column), scatters of predicted and observed biomass shown in middle column, and the distributions of errors depicted by the contents of the right column.The evaluation is based on the Jucker data.Each row corresponds to one of the five models-from top to down: LR (a)-(c), LR2 (d)-(f), LR3 (g)-(i), RF (j)-(l), GPR (m)-(o).

Fig. 9 .
Fig. 9. Study of model uncertainties when working with tree height H, tree diameter D, crown diameter CD, and the product H × CD as input parameter of the allometric equation.The overall model uncertainties read: 18.25%, 14.13%, 29.81%, and 20.57% respectively.

TABLE I SUMMARY
OF R-SQUARE SCORES, RMSE AND BIAS OF A SERIES OF REGRESSION MODELS FOR BIOMASS ESTIMATION BENCHMARKED ON THE

TABLE II SUMMARY
OF R-SQUARED, RMSE, AND BIAS FOR FIVE REGRESSION MODELS ESTIMATING BIOMASS FROM THE DATASET CURATED.

TABLE III COMPARISON
OF MODEL PERFORMANCE IN TERMS OF RELATIVE ERROR AND RELATIVE RMSE FOR CANDIDATE MODELS: LR, RF, AND GPR