Allometric Scaling and Resource Limitations Model of Tree Heights : Part 3 . Model Optimization and Testing over Continental China . Remote Sensing , 2014 , 6 , 3533-3553

1 State Key Laboratory of Remote Sensing Science, Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100101, China; E-Mails: nixl@irsa.ac.cn (X.N.); cao413@irsa.ac.cn (C.C.) 2 Department of Earth and Environment, Boston University, 675 Commonwealth Avenue, Boston, MA 02215, USA; Emails: schoi@bu.edu (S.C.); ylshi.nuist@gmail.com (Y.S.); ranga.myneni@gmail.com (R.B.M.) 3 School of Remote Sensing, Nanjing University of Information Science and Technology, Nanjing 210044, China 4 Survey Planning and Design Institute, State Forest Administration of China, Beijing 100714, China; E-Mail: wangxuejun320@126.com 5 Center for Ecological Analysis of Lidar, Natural Resource Ecology Laboratory, Colorado State University, Fort Collins, CO 80523, USA; E-Mail: lefsky@cnr.colostate.edu 6 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA; E-Mail: marc.simard@jpl.nasa.gov


Introduction
Forest structural attributes are crucial indicators for demonstrating forest ecosystem services [1].In particular, the spatio-temporal patterns of forest height (potentially biomass) provide regional or global contexts of terrestrial carbon fluxes and storages [2][3][4].Due to the broad geographical extent, large spatial variability and the limited accessibility for field inventories, mapping forest heights at a larger scale generally relies on three-dimensional (3D) remote sensing techniques such as light or radio detection and ranging (lidar or radar) systems [5][6][7][8].For instance, the Geoscience Laser Altimeter System (GLAS) is the first spaceborne lidar instrument providing reasonable vertical profiles of forests with global coverage [9].However, because of its intrinsic discrete sampling scheme, GLAS needs additional extrapolation procedures to produce wall-to-wall forest height maps.Recent studies have used several statistical models based on valid relationships between forest heights and geospatial predictors, which have produced spatially continuous forest height maps depicting the spatial variation in vertical forest structures [5][6][7][8]10,11].Nevertheless, their extrapolation approaches are likely to depend on input data quality and quantity, and often neglect the physical/physiological mechanisms governing tree growth.
For better mechanistic understanding in mapping forest heights, Kempes et al. [12] presented the Allometric Scaling Resource Limitation (ASRL) model composed of two conceptual schemes: (a) allometric scaling laws may provide a valid tree geometry [13,14] and (b) potential tree growth is constrained by local resources (e.g., water, temperature, sunlight, and wind) [15].However, the ASRL model with universal allometric coefficients and scaling exponents shows less predictive power when applied to diverse eco-climatic regimes and forest types of varying age classes [16,17].Thus, a synergistic combination of physical/physiological mechanism (i.e., ASRL approach) and 3D remote sensing techniques (i.e., GLAS) has recently been explored using parametric optimizations [16,17].
The first [16] in a multi-article series has provided a thorough description of the optimized ASRL model.The GLAS altimetry is implemented to iteratively adjust three selected ASRL parameters (area of single leaf, α, exponent for canopy radius, η, and root absorption efficiency, γ).This parametric optimization minimizes disparities between the reference and modeled tree heights.The optimized ASRL model results in a contiguous map of the maximum forest heights over the continental USA (CONUS) at the 1 km spatial resolution, and shows a moderate predictive power with a root mean square error (RMSE) of 3.31 m and a coefficient of determination (R 2 ) of 0.59 (from a two-fold cross validation).This showed the feasibility of the optimized ASRL model for large-scale mapping.The second study [17] in this series examined the above approach at a finer spatial scale, i.e., FLUXNET sites, where more accurate input information is available.This site-scale test shows a satisfactory performance of the optimized ASRL model through the two-fold cross validation (R 2 = 0.85; RMSE = 1.81 m) and bootstrapping evaluation (R 2 = 0.66; RMSE = 2.60 m).
Based on the experiments over the CONUS in the previous two articles, our scientific question aroused here is if the optimized ASRL model is applicable to other forest ecosystems, e.g., over the continental China (CONNA).The geographical extents and the diversity in climate, topography, forest types of the CONNA are comparable to those of the CONUS.Therefore, this third study aims to test the optimized ASRL model over the CONNA at both site and continental scales.In this study, we produced the modeled heights for Chinese forests using the same manner used in previous articles [16,17].The evaluations were made against field-measured and GLAS-derived tree heights.In addition, the spatially continuous field of maximum tree heights (1 km resolution) over the CONNA was compared against the existing forest height maps from Lefsky [7] and Simard et al. [8].This study briefly introduced the optimized ASRL model along with input variables and key allometric scaling parameters.Since we are examining the applicability of the model across different forest ecosystems, our discussions focus more on the final outcomes over the CONNA rather than the initial predictions or the optimization processes whose details are given in previous articles [16,17].

Climate Data
Input climatic variables for the ASRL model include precipitation, temperature, relative humidity, wind speed, and incoming solar radiation, which determine the available and evaporative flow rates.The ASRL model quantifies resource limitations on tree's growth using these flow rates [12].This study obtained two types (weather station-based and gridded data) of climate datasets over the CONNA (Table 1).
First, weather station-based climatic variables were used for the site-scale ASRL simulations.The China Meteorological Data Sharing Service System (CMDSSS) provides the annual meteorological records observed from 754 weather stations [18] (Figure S1 of Supplementary Information).This study implemented such observations spanning over 50 years , comprised of annual total precipitation, average temperature, relative humidity, wind speed, and solar radiation time.We further converted the observed solar radiation time into the incoming solar radiation rate, as needed for the model predictions, using a statistical method suggested by Liao et al. [19].
Second, gridded climate datasets (1 km resolution) were exploited in the continental-scale ASRL model simulations.The WorldClim [20] produces long-term  climate maps based on the thin-plate smoothing spline algorithm [21].As annual total precipitation and average temperature were only available input climate data from the WorldClim, we interpolated other climatic variables from the CMDSSS using ordinary kriging [22] to generate the gridded maps of annual average relative humidity, wind speed, and solar radiation.Uncertainties related to this interpolation process are discussed in Section 4.3.

Ancillary Data
Simulation and optimization procedures of the ASRL model also require other ancillary datasets including digital elevation model (DEM), leaf area index (LAI), land cover (LC), and percent tree cover (Table 1).The DEM and LAI were input ASRL variables to calculate the potential evaporative flow rate of trees, while LC and percent tree cover were used to identify forested lands over the CONNA.In this study, we used the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) Global DEM (GDEM V2, 30 m grid) [23].The approximate growing season (June to September) averaged LAI data from 2003 to 2006 were obtained from a refined version of the Moderate Resolution Imaging Spectroradiometer (MODIS) LAI product (1 km grids) [24].The MODIS global LC (MCD12Q1, 500 m grid) [25] and Vegetation Continuous Field (VCF; MOD44B, 250 m grid) [26] for the year 2005 accounted for the rest of ancillary datasets.To standardize the spatial resolutions, finer gridded ancillary data were resampled into the 1 km resolution using the majority principle technique for categorical values and the cubic convolution method for numerical values [27].We took into account three forest types (evergreen, deciduous, and mixed forests) aggregated from the five International Geosphere-Biosphere Programme (IGBP) forest classes, and forested lands were defined by a percent tree cover (≥50%) based on the VCF product (Figure 1).The above-described ancillary datasets were used in the ASRL model simulations at both site and continental scales.

Eco-Climatic Zone
For the computationally practical model optimizations (see Section 3.1), we separately defined the eco-climatic zones over the CONNA, accounting for three forest types, annual total precipitation (30 mm intervals) and annual average temperature (2 °C intervals).This classification resulted in 537 zones at the site scale and 5805 zones at the continental scale.This zonal information was used as the basic element (unit or level) in the model optimization and evaluation process.

GLAS-Derived Tree Heights
The GLAS-derived tree heights over the CONNA were used to optimize the ASRL parameters and to assess the model's performance.In this study, we obtained the latest release of GLAS laser altimetry data (Release-33) from the National Snow and Ice Data Center.Amongst various altimetry products of the Release-33, the level-2 data (i.e., GLA14) were selected for the maximum tree height retrieval.The GLA14 provides metrics for land surface elevation along with the geo-location of laser footprints and the waveform parameters, such as signal beginning and echo energy peaks [28].We exploited the GLAS data recorded from May to October (2003)(2004)(2005)(2006) because this seasonal period represents the best approximate of forests' leaf-on condition.The defoliation of deciduous forests may generate underestimations in the GLAS-derived maximum tree heights.As it is impractical to estimate the ellipsoidal shapes and sizes for each footprint [29], this study assumed that all the GLAS footprints are likely to have a circular shape with a diameter of 70 m [30].* Annual incoming solar radiation was computed from the observed solar radiation time [19].
The GLAS waveforms might be degraded by the atmospheric forward scattering, signal saturation, background noise (low cloud), and slope gradient effects.In addition, GLAS shots over non-forested lands were not of interest.This study, thus, adopted the five screening steps of previous studies [16,17] to remove the invalid GLAS shots based on the GLA14 internal quality flags and the ancillary datasets (LC, VCF, and GDEM: see Table S1).From the full-waveform profile of each valid GLAS shot, we computed the distance between signal beginning (W SB ) and ground peak (W GP ) points using the Gaussian decomposition approach [31].This distance theoretically represents the maximum tree height within circular GLAS footprint since the W SB and W GP correspond to the highest point of tree and the ground surface elevation, respectively.Here, we additionally employed a simple physically-based slope correction approach of Lee et al. [32], taking into account the topographic condition (slope, θ) and GLAS footprint size (d GLAS ).The final metric of GLAS-derived maximum tree heights (H GLAS ) over the CONNA was calculated using Equation (1).Distribution of final GLAS height at the 1 km resolution is shown in Figure S2a.products.Three aggregated forest types (evergreen, deciduous, and mixed forests) with percent tree cover (≥50%) were considered in this study.(1) where, H GLAS refers to the GLAS-derived maximum tree heights, respective W SB and W GP represent the signal beginning and the ground peak of GLAS full-waveform, d GLAS is the GLAS footprint size, and θ corresponds to the slope.

Field-Measured Tree Heights
The State Forestry Administration (SFA) of China has systematically investigated national forest resources and updated its database through the repetitive National Forest Inventory (NFI) projects since 1973.In this study, we used tree height information from 5,069,905 plots surveyed for the 7th NFI period (2004-2008) [33].Most of the NFI sampling plots are squares (25.8 m × 25.8 m, approximately 0.0667 ha), although some plots have rectangular, round or diamond shapes.Density of plots has been determined by the expected precision level, and distance between plots varies from 2 × 2 to 8 × 8 km 2 [33].The NFI reports 23 surveyed attributes for each plot (e.g., geo-location, forest type, individual diameter at breast height (DBH), and mean height).To derive maximum tree height (H NFI ) for each eco-climate zone, we first developed a power function based allometric equation (Equation ( 2)) between mean DBH (i.e., averaged individual DBHs) and height information from primary inventory dataset [34][35][36].In this process, we extracted and used only 157 effective climate zones with ≥20 field surveyed plots (continental scale study).Maximum tree height was then calculated from the largest individual DBH within each effective climate zone under the assumption that the tallest individual tree has the largest DBH.Distribution of in situ maximum tree heights is shown in Figure S2b and their locations are limited over the northeastern regions of CONNA.This H NFI will be exploited for model validation only at the continental scale. (2)

Method
We examined the performance of the optimized ASRL model over the CONNA.Our analysis consisted of two main parts, predicting maximum tree heights at the (a) site and (b) continental scales (Figure 2).This section briefly describes the ASRL model and its optimization process.The model evaluations at both site and continental scales are presented in the following sub-sections.

Figure 2.
Overall scheme of research.The performances of the optimized ASRL model was examined over the CONNA.This study was divided into tests at the site and continental scales.

Overview of the Optimized ASRL Model
The ASRL model can predict the potential maximum tree heights based on a combination of the allometric scaling laws and local resource availability.In this model, hydraulic [13,14] and mechanical [15] constraints determine the potential for tree growth.A key premise of the ASRL is that a tree maintains its evaporative flow by collecting sufficient resources (water and light) while satisfying its basal metabolic needs, in turn, limiting maximum growth.This mechanism can be quantified using a simple inequality (Q p ≥ Q e ≥ Q 0 ) where Q 0 refers to the required metabolic flow rate, Q e represents the potential evaporative flow rate, and Q p corresponds to the available flow rate.These three flows are associated with tree geometry (e.g., canopy radius and leaf area), light (e.g., soil reflectance and leaf absorptivity), and water (e.g., root absorption coefficient and depth of a stomata).The Q o increases as a tree grows.Simultaneously, the Q e and Q p also change but their alterations may occur in different degrees and directions.The ASRL approach, basically, finds a point violating the inequality between given three flow rates.Thus, the intersection either between Q p and Q e or between Q e and Q 0 represents a limitation of tree growth, which is the potential maximum height (see [12,16] for more detailed information of the ASRL model).
Although the unoptimized ASRL model might grant a mechanistic understanding in the prediction of potential tree heights [12,16], it still has less predictive power when applied to diverse eco-climatic zones and forest types with varying age classes [16,17].A plausible reason for this drawback is that the model implements universal allometric coefficients and scaling exponents (e.g., α = 13 cm 2 , η = 1.14, and γ = 0.33 as in [12]).Obviously, we expect inconsistent allometric relationships across different environmental conditions.Shi et al. [16] recommend a parametric optimization of the ASRL model: such that (a) iteratively adjusts three ASRL parameters (i.e., α, η, and γ) for each eco-climatic zone; and (b) minimizes disparities between the reference and modeled heights, in turn, improving the overall prediction accuracy.Amongst a number of ASRL coefficients and exponents, they have tested those three parameters governing the canopy geometry, energy absorption and transmission, and water absorption capacity of trees [12,16,17].In this study, we applied the same processes at both site and continental scales.Here, the Powell's optimization approach [37,38] was used for each eco-climatic zone (Section 2.1.3)to enable computationally practical model simulations (see Section S1 and [16] for detail of parametric optimization process).
Powell's optimization algorithm is a straightforward and intuitive optimization process to find the minima of a multidimensional function.This algorithm starts with initial parameter values (e.g., area of single leaf, α, exponent for canopy radius, η, and root absorption efficiency, γ), and the search directions are evaluated sequentially.The best possible improvement in modeled results is explored in a search direction through line minimization methods.Here, it iteratively initializes the parameters before proceeding to the next search direction.In this study, Powell's algorithm minimizes the difference between the reference and modeled tree heights based on a merit function (Press et al. [38]; Equation ( 3)) feasible to optimize three variables.
where, p k is the ASRL model parameter (k = 1, 2, and 3 represent α, η, and γ, respectively), p k0 refers to the initial values of each parameter, p kb is the boundary limit of each parameter defined by Equation (3.1), and Δ pk is the standard deviation of each parameter value with respect to its initial values.In this process, we set variable ranges (lower and upper boundaries as in the TRY database [39]) of each parametric tree trait (1 cm 2 ≤ α < 100 cm 2 , 0.8 ≤ η < 1.5, and 0.1 ≤ γ < 0.8) and initial value of each parameter (α = 13 cm 2 , η =1.14, and γ = 0.33 according to Kempes et al. [12]).w k is a scalar weight set by the condition in Equation (3.2), N is the total number of comparison sets (i) for GLAS height (H GLAS ) and the model prediction (H ASRL ) with given parameter values for each climatic zone and Δ H is the standard deviation associated with H GLAS and H ASRL .From this iterative process, the minimum of the function F(p 1 , p 2 , p 3 ) of each climatic zone was derived to predict maximum tree height.

Site Scale Model Simulation
Among 754 available meteorological stations (537 eco-climatic zones), this study selected 14 sites (14 eco-climatic zones) for the site-specific model simulations over the CONNA (Table 2; Figure S1).The selected study sites accounted for a sufficient number of valid GLAS shots (n ≥ 20) whose geo-locations are concurrently registered within 10 km radius circle from each station.These GLAS data are necessary for the iterative optimization (training) and validation (test) purposes.The initial ASRL predictions were made using the weather station-based input climate and ancillary data for each study site.We then used the Powell's optimization process to calculate the least differences between the GLAS-derived and modeled heights, resulting in 14 sets of the three adjusted ASRL parameters.
In this study, the model evaluations at the site scale maintained independence between the training and test sets of the GLAS data using (a) a two-fold cross validation and (b) a bootstrapping technique.First, the two-fold cross validation assessed the model performance by randomly partitioning the GLAS samples into two equal batches (i.e., training and test sets).Respective training and test GLAS tree heights were used for the ASRL model optimizations and evaluations for each study site.Second, the bootstrapping approach provided more iterations (n = 250) in the random splitting of the GLAS data.Each iteration optimized and evaluated the ASRL model in the same way, as did the two-fold cross validation.When interpreting the evaluation results, this study implemented three statistical metrics: RMSE, R 2 , and relative errors (RE = (H obs − H pred )/H obs ; where H obs refers to the observed height and H pred represents the predicted height).RE was an important statistic for determining ASRL model performance due to the large variation in tree heights across study sites.

Continental Scale Model Simulation
In addition to the site scale test, we also examined the optimized ASRL model at the continental scale (1 km resolution) over the forested lands of the CONNA (Figure 1).Candidate forest pixels for this test lie in the effective eco-climatic zones (409 out of 5805 zones over the CONNA; Section 2.1.3)where ≥20 valid GLAS shots were geo-registered.The initial ASRL predictions were carried out over the individual candidate pixels by implementing the gridded climate and ancillary data.Unlike the site-scale test, we formulated the model optimizations at the eco-climatic zone level: assigning identical ASRL parameters if the forest pixels were associated with the same eco-climatic zone.The optimization for every pixel is not practical since there are over a million forest pixels.For each eco-climatic zone, the ASRL parameters were iteratively adjusted to minimize the overall differences between tree heights derived from the valid GLAS data and the ASRL model.Table 2. 14 selected study sites where ≥20 valid GLAS shots were geo-registered within a 10 km radius.The site information includes geo-location (latitude (Lat.) and longitude (Lon.)),forest types, and climatic variables (temperature (Tmp.), precipitation (Pre.), incoming solar radiation (Rad.), wind speed (Wnd.) and relative humidity (Reh.)).The optimized parameter sets (area of single leaf (Opt.α), exponent for canopy radius (Opt.η), and root absorption efficiency (Opt.γ)) were also listed.The model evaluations at the continental scale were three-fold: (a) a two-fold cross validation using the valid GLAS altimetry, (b) a comparison to the in-situ tree height estimates (H NFI ; Section 2.2.2), and (c) inter-comparison with the existing forest height maps developed by Simard et al. [8] (H Simard ) and Lefsky [7] (H Lefsky ).Just as in the site-scale tests, the two-fold cross validation randomly divided the valid GLAS shots into two equal sets (training and test) for each eco-climatic zone.Additionally, overall means of the modeled tree heights for each eco-climatic zone were assessed by the H NFI .The inter-comparison with H Simard and H Lefskey required pre-processes of the forest height maps, in terms of spatial resolution, projection, and tree height units, to be comparable with our results [16].

Predicted Tree Heights at the Site Scale
Following the initial model predictions of the potential maximum tree heights over the 14 selected study sites, three ASRL parameters (α, η, and γ) were iteratively optimized using the independent training GLAS estimates.Results from the two model assessments (two-fold cross validation and bootstrapping) are shown in Figure 3.The optimized ASRL presented a significant agreement between the modeled and test GLAS maximum tree heights (n = 14; RMSE = 1.72 m; R 2 = 0.97).The performance of the model was satisfactorily stable regardless of the bootstrapping iterations (n = 14 × 250 = 3500; RMSE = 4.39 m; R 2 = 0.81).The relative error terms (RE; Figure S3) showed a fairly tight distribution: standard deviation (SD) of RE = 0.09 for the two-fold validation; and SD of RE = 0.17 for the bootstrapping.The optimized ASRL model produced slight overestimations by 2% from the reference tree heights (mean of REs (MREs) = −0.02 in both evaluation cases).These site scale tests over the CONNA are comparable to those over the CONUS documented in Choi et al. [17] (n = 12; RMSE = 1.81 m; R 2 = 0.85; MRE = 0.02 for the two-fold validation, and n = 500; RMSE = 2.60 m; R 2 = 0.66; MRE = −0.04 for the bootstrapping).The performance of the model was generally stable across different forest ecosystems at the site scale, as supported by this and previous studies resulting in relatively high agreement between the modeled and reference tree heights.The optimized parameter values significantly varied depending on the different eco-climatic conditions (Table 2).Looking at the adjusted parameters resulting from the optimization, the single leaf area (initial α = 13.0 cm 2 ) showed conspicuous change from the initial value (mean = 8.34 cm 2 ; SD = 2.63) while the root absorption efficiency (initial γ = 0.33) narrowly ranged from 0.33 to 0.56 (mean = 0.42; SD = 0.07).The exponent for canopy radius (initial η = 1.14) was expected to remain stable, as it was not likely to depend on the eco-climatic conditions (mean = 1.1397;SD = 0.0003).In the sensitivity analysis of single ASRL parameters, Kempes et al. [12] also reported stability in η.The morphological characteristics of leaves for each forest type were well represented by α (mean α values for deciduous, evergreen, and mixed forests = 11.84,7.74, and 8.29 cm 2 , respectively).Interestingly, the γ for evergreen forests (mean γ = 0.45) showed relatively higher optimized γ values when compared to deciduous forests (mean γ = 0.34).Similar results for γ have been observed in previous field (e.g., [40,41]), and modeling studies (e.g., [16]).This parametric optimization allows the ASRL model to adjust estimates of canopy geometry, energy absorption/transmission, and hydrological absorption capacity for each study site.The optimized ASRL model successfully represents the balances in evaporative and available flow rates along with the required metabolic flow.This provides a more mechanistic understanding of the relationship between maximum tree heights and concurrent local resource limitations.

Predicted Tree Heights at the Continental Scale
A contiguous map of the maximum tree heights over the CONNA (1 km resolution) was generated from the optimized ASRL model using the gridded climate and ancillary data (Figure 4a).This map closely represents the spatial distribution of tree heights in Chinese forests (mean = 36.44m; SD = 23.85),showing a high correspondence to the spatial pattern of GLAS tree heights (mean = 30.00m, SD = 6.55; Figure S2a).The optimized ASRL model simulated relatively short tree heights over the northeastern CONNA (Heilongjiang and Inner Mongolia), whereas relatively tall trees were mostly distributed around the central and southern regions (Sichuan, Yunnan, and Chongqing).Such distinct spatial patterns of tree heights were strongly governed by local resource limitations based on water availability and energy balances.Because of sufficiently warm and humid conditions along with high solar energy availability, trees over the southern CONNA likely have higher photosynthetic activity than those under resource-limited conditions (northeastern regions) [42,43].Some regions including Jilin and Liaoning provinces close to the border of North Korea were likely to show taller tree heights than forests near the border of Russia (Heilongjiang and Inner Mongolia) due to the higher temperature and adequate precipitation in Jilin and Liaoning.
Variations across eco-climatic conditions of optimized ASRL parameters at the continental scale were similar to those at the site scale (Figure 4b-d and Table 2).The α values of each eco-climatic zone were noticeably adjusted to demonstrate the characteristics of tree growth given local resource conditions (Figure 4b).The forest types were satisfactorily reflected in the optimized α (mean α of deciduous, evergreen, and mixed forests = 21.31,13.96 and 17.51 cm 2 , respectively).Both η (mean = 1.28;SD = 0.19; Figure 4c) and γ (mean = 0.37; SD = 0.15; Figure 4d) did not significantly deviate from the initial values (initial η and γ = 1.14 and 0.33, respectively), while η and γ still produced distinct spatial distributions across different eco-climatic conditions (described in Section 4.1).We believe that this spatial pattern of the modeled tree heights represent a degree of predictive power over the CONNA, and the distributions of the optimized parameters support a mechanistic understanding for relationships between the predicted maximum tree height and local resource limitations.

Evaluations Using GLAS and Field Measured Tree Heights
The modeled tree heights at the continental scale were evaluated using (a) the two-fold cross validation with the GLAS altimetry and (b) the one-to-one comparisons with the field-measured tree heights.The two-fold validation showed a moderate agreement between the modeled and independent GLAS data (n = 409; RMSE = 6.63 m; R 2 = 0.63; Figure 5a).The optimized ASRL model predicted 8% taller tree heights than GLAS estimates (MRE = −0.08, Figure S4a).This overestimation tended to be distributed over the wet environment (southern CONNA; Figure S5a) where there are likely other competitive factors limiting tree height [12].Additionally, the available water flow rate (Q p ) may lead to an overestimation of tree heights for the wet forests since the ASRL model neglects soil water behavior and water holding capacity of given soil type and condition.Large underestimations were found in complex terrain regions (high sloped) including Sichuan and Chongqing, and in arid regions near the Gobi desert.Two plausible reasons for the discrepancy over the complex terrain are that (a) topographic effects on the GLAS altimetry might not be completely rectified by our correction scheme [21,44,45] and (b) input climatic variables are highly sensitive to complex terrain, which might produce large uncertainties [46,47].The discrepancy over arid regions is likely due to trees that have specialized traits (e.g., drought tolerance) that deviate from the specific parameters used here [12].As shown in Figure 5b, the second evaluation (one-to-one comparison) revealed that the optimized ASRL model could explain 52% of the variance of field-measured tree heights (n = 157; RMSE = 6.70 m; R 2 = 0.52).Here, we noted that the model predicted 6% shorter tree heights when compared to the field-measured tree heights (MRE = 0.06, Figure S4b).Most underestimates were situated over the densely forested northeastern region of the CONNA (Figure S5b).This was likely caused by errors propagated from the training GLAS dataset.The dense forest canopy (e.g., LAI > 4) or the conical crown shaped trees might result in relatively shorter GLAS-derived tree heights [48][49][50].In addition, the ASRL model neglects the forest age, which may be associated with the underestimates because growth rates of the northeastern mature forests (>50 years) might contrast to those of younger forests (<40 years) over the central and southern region of the CONNA [51].

Inter-Comparisons with Other Existing Forest Height Maps
We additionally conducted inter-comparisons with two independent forest height maps.A moderate degree of agreement (RMSE = 7.69 m; R 2 = 0.43) was obtained from the comparison between H ASRL and H Simard (Figure 5c), but systematic discrepancies existed between the two tree heights.Simard et al. [11] used a different definition of forested lands, which included mosaic crops, open forest, and saline flooded forests.Also, their regression model of tree heights did not allow the prediction of forest heights > 40 m.Thus, the predicted tall trees (>40 m) over the CONNA produced larger errors when compared to H Simard .Inter-comparison with H Lefsky (Figure 5d) implemented Lefsky's input GLAS-based tree heights, rather than the Lorey's height as in [7].As depicted in Figure 5d, the optimized ASRL model explained approximately 40% of the variance (R 2 = 0.38) of H Lefsky .However, this comparison showed larger differences between the modeled tree heights and H Lefsky (RMSE = 8.85 m).These relatively large errors might be caused by the difference in GLAS height retrieval processes between this study (standard Gaussian decomposition) and Lefsky's study (statistical approach).

Limitations and Further Directions
The evaluation procedures at both scales showed that the optimized ASRL model has the potential to be applied to different forest ecosystems.We successfully demonstrated the spatio-temporal variability of tree heights over the CONNA across various eco-climatic zones and forest types.Nevertheless, the optimized ASRL model still produced larger uncertainties (Figures 3 and 5 Climate variables (e.g., precipitation and wind speed) are generally highly sensitive to topography and land cover conditions due to radiative and hydrological properties [52,53].Thus, the quality of climate variables varies depending on the interpolation methods [54,55].Since climatic variables largely determine resource conditions errors may be induced when resources, quantified by climate variables limit height growth rates.In addition, the topographic correction of GLAS data may still produce errors due to the simplified footprint size/shape and the simple slope correction scheme [17,44].It is well known that due the radiative characteristics of lidar waveform, the tree height retrievals are strongly influenced by the asymmetric shapes and sizes of an individual footprint, off-nadir angles, sensor zenith/azimuth angle, and slope orientation conditions [48,56].The uncertainty induced from the GLAS height retrieval might propagate and influence the final ASRL height product.Input data quality is an ongoing challenge, and these issues should be resolved through further related studies [48,[54][55][56].
In the case of the sampling scheme, this study assumed that the number of valid GLAS shots (≥20 footprints) and field plots (≥20 plots) were sufficient to reflect the variability of local tree heights for each climate zone.In practice, however, our assumption suffered from the sequential discrete GLAS footprints along with ascending or descending orbits and roughly gridded field measurements.In other words, sampling scheme limitations and aforementioned uncertainty of GLAS retrieval [21,44,45] collectively caused disagreement between tree height distributions derived from GLAS altimetry, field observation, and the optimized ASRL model (Figure S5).This problem seems ineludible in this study, yet we expect solutions from newly planned field and 3D Earth observing campaigns (e.g., iss-jem Lidar for Observation of Vegetation Environment [57]).
Forest age and soil condition are important drivers in the physical/physiological mechanisms for tree growth.Topographic characteristics and soil texture primarily determine the water availability, which is associated with tree metabolism [58].Thus, neglecting soil condition may degrade predictive power across diverse eco-climatic regions, because the modeled internal plant flow rate may not fully explain the actual water flow rate.We assumed that GLAS heights bring indirect information about forest age into the ASRL model, however more explicit forest age-based scaling parameters may be needed to improve model performance.To alleviate these influences, subsequent studies will incorporate soil water capacity [58] and stand age [59] maps.
In this study, we optimized three selected ASRL parameters (α, η, and γ) that are strongly associated with the canopy radiation energy budget and precipitation uptake rate of trees.Other parameters (e.g., stomatal density) were optimized by Kempes et al. [12], which also showed better prediction than that of the unoptimized ASRL model.As seen in this and previous studies, parametric optimization clearly improves the predictive power of the model.Thus, further study will explore the applicability of optimizing additional other ASRL model parameters.

Concluding Remarks
Our multi-article series ultimately aims to produce spatially continuous maps of forest heights and biomass using the Allometric Scaling and Resource Limitation (ASRL) approach.This third study tested the feasibility of the optimized ASRL model over the Continental China (CONNA) at both site (14 study sites) and continental scales.The Geoscience Laser Altimeter System (GLAS) data were used for model optimization.Three selected ASRL parameters (area of single leaf, α; exponent for canopy radius, η; and root absorption efficiency, γ) were adjusted to minimize the differences between the reference (GLAS and in situ data) and modeled tree heights.
In the site-scale tests, the optimized ASRL model was assessed using the two-fold cross validation and bootstrapping evaluation.Both evaluations maintained the independence between training and test GLAS data.Results of the two-fold cross validation at the 14 study sites showed that predicted tree heights could explain 97% of the variability in reference tree heights (here, GLAS altimetry).On average, estimated height errors were 1.72 m.The model performance was sufficiently stable regardless of bootstrapping iterations (n = 250 random splitting of training and test data).This exercise also indicated that the optimized ASRL model produced satisfactory predictive power (RMSE = 4.39 m; R 2 = 0.81) at the site scale.
With regard to the continental-scale tests, the GLAS and in-situ tree heights were exploited for evaluation purposes.The existing forest height maps (H Lefsky [7] and H Simard [8]) were also used for inter-comparison.The model was capable of showing reasonable correspondence with GLAS and in-situ datasets (RMSE = 6.63 m; R 2 = 0.63 against the GLAS, RMSE = 6.70 m; R 2 = 0.52 against in-situ).The predicted heights tended to be overestimated (8%) and underestimated (6%) when comparing to the GLAS and in-situ tree heights, respectively.The inter-comparison indicated that the optimized ASRL model generated a moderate degree of agreement to H Simard (RMSE = 7.69 m; R 2 = 0.43) and to H Lefsky (RMSE = 8.85 m; R 2 = 0.38).The modeled tree heights were overestimated when compared to H Simard while being underestimated when compared to H Lefsky -these discrepancies might be influenced by the different definitions of the tree height and forested land among these studies.
We have assessed the feasibility of the model to different forest ecosystems (Chinese forests).The optimized ASRL model satisfactorily performed across scales (both site and continental scales), and has also provided a mechanistic understanding of relationships between tree heights and local resource limitations.Unlike previous works, this study made attempts to evaluate the model using in situ maximum tree heights.In this multi-article series, subsequent studies for tree height prediction will also test the optimized ASRL model over different study sites (e.g., Amazonian Basin) in addition to the CONNA.After addressing the discussed limitations, we are interested in the woody biomass prediction using the same methods as in this series.To examine the feasibility of using the model for biomass prediction, further study will begin by assessing performance of the model at the site and continental scales of the continental US.

Figure 1 .
Figure 1.Forested area defined by the MODIS LC (MCD12Q1) and VCF (MOD44B)products.Three aggregated forest types (evergreen, deciduous, and mixed forests) with percent tree cover (≥50%) were considered in this study.

Figure 3 .
Figure 3. Comparisons between the predicted and test GLAS tree heights over the 14 selected study sites: (a) Two-fold cross validation and (b) bootstrapping approach.

Figure 4 .
Figure 4. (a) Spatial distribution of tree heights predicted by the optimized ASRL model over the CONNA (1 km resolution).Spatial patterns of the optimized (b) single leaf area; (c) exponent for canopy radius; and (d) root absorption efficiency.

Figure 5 .
Figure 5. Comparisons between the predicted tree height and test GLAS altimetry dataset over the CONNA.(a) Two-fold cross validation between the GLAS height and ASRL model prediction (409 effective climatic zones); (b) comparison between the field-measured height and ASRL model prediction (157 effective climatic zones); (c,d) inter-comparisons between the ASRL predictions and Simard's and Lefsky's height estimations, respectively.
).These uncertainties might be explained by considering limitations from external (a,b) and internal (c,d) factors: (a) quality of input climate and GLAS data; (b) consistency of sampling scheme; (c) consideration of forest age and soil condition; and (d) selection of optimized scaling parameters.

Table 1 .
Climatic and ancillary datasets prepared for the Allometric Scaling and Resource Limitation (ASRL) model.