The key input climatic variables include annual total precipitation, annual average temperature, annual incoming solar radiation, annual average wind speed and annual average relative humidity. Additionally, two categories of ancillary input data are needed: (a) Digital Elevation (DEM) and Leaf Area Index (LAI) for initializing the ASRL model and (b) land cover and tree cover for delineating forested lands.

Table 2 lists input data (climate data and ancillary inputs) required for the ASRL model. Finer gridded data (e.g., 30 m or 250 m) were resampled to 1 km resolution in this study using the majority principle for categorical values and cubic convolution for numerical values [26]. Wind speed data at 32 km resolution were spatially interpolated to 1 km resolution using an Inversed Distance Weighting (IDW) method [27].

The GLAS laser altimetry data provide information related to land elevation and vegetation height at a spatial resolution of ∼70 m (ellipsoidal footprints) and at ∼170 m spaced intervals [35,36]. The latest release (Release-33) of the standard GLAS product corresponding to the GLAS Level-2 Land Surface Altimetry (GLA14; L2 Land Surface Altimetry) for the period 2003 to 2006 was obtained from the National Snow and Ice Data Center (NSIDC) for this study. The GLA14 product was used to estimate forest canopy heights within each footprint (e.g., [1,3]) using geolocation information and waveform parameters, such as signal beginning and echo energy peaks [37]. There are notable heterogeneities in the dimension and shape of the individual GLAS footprints. The GLAS instrument was designed to have a fixed footprint size, but the dimensions of footprints are significantly changed depending on the laser periods (e.g., orbits and spans of campaigns) [38]. To simplify, we assumed that all GLAS footprints have a circular diameter of 70 m [39]. Data from May to October of each year were used, as these come from the growing season and correspond to the MODIS LAI product.

GLAS waveform data are affected by three degrading factors: (a) atmospheric forward scattering and signal saturation, (b) background noise (low cloud) and (c) slope gradient effects. Additionally, GLAS footprints over non-forest and/or bare ground must be filtered from analysis. Five screening steps were applied to remove invalid GLAS waveform data prior to retrieval of tree heights (Table 3). Note that we removed any remaining outliers using two standard deviations from the mean of GLAS tree heights (5 m < H_GLAS ≤ 100 m) in this analysis. Final valid GLAS footprints were intersected with the pixels (=1 km) over forested lands. We averaged tree heights derived from the GLAS footprints falling in a pixel. This generates a raster distribution of GLAS heights (Figure S1).

There are two approaches of retrieving tree heights from GLAS waveform data [40]: (a) the “statistical analysis for examining full GLAS waveform extents” [41–44] and (b) the “decomposition of GLAS waveforms into multiple Gaussian distribution curves” [1,3,6,40,45–47]. We considered only the second approach in this study.

We used two standard altimetry variables of GLA14 product (signal begin range increment, SigEndOff; centroid range increment for the last Gaussian Peak, gpCntRngOff 1). Theoretically, gpCntRngOff 1 is assumed to represent the ground level elevation within a GLAS field-of-view, while SigBegOff refers to the highest point of a surface. Amongst five possible GLAS height metrics representative of tree heights [25], we used the best metric that closely resembled field-measured and Laser Vegetation Imaging Sensor tree heights (R² = 0.70; RMSE = 4.42 m; [25]): that is, “SigBegOff – gpCntRngOff 1” with correction of the potential bias [40] (Equation (1)). (1) H GLAS = ( SigBegOf f − gpCntRngOff ) − d × tan θ 2Here, SigBegOff is the signal begin range increment and gpCntRngOff 1 refers to the last peak of Gaussian of GLAS waveform, d is the footprint diameter of GLAS data (∼70 m) and θ is the slope value nearest to the geolocation of GLAS footprint center.

The forested area in the CONUS was categorized into 841 climatic zones based on three forest types, annual total precipitation (30 mm intervals) and annual average temperature (2 °C intervals). An empirical orthogonal panel [48,49] was used to identify the pattern of these two climatic variables (horizontal axis—annual total precipitation and vertical axis—annual average temperature), and to associate forested grids to climatic zones (Figure S2; Table 4).

The reasons for defining climatic zones are twofold: First, a direct comparison of GLAS heights, which represent actual tree heights, with potential tree heights predicted by the ASRL model is not valid. Second, optimization of the ASRL model for every forested pixel (over 1.3 million pixels) is not computationally practical. Thus, the optimization was performed at the climatic zone level for each of the three forest types.

ASRL model simulations were performed over forested areas (Section 3.1.2). The ASRL model predicts potential tree heights at 1 km spatial resolution using input climatic and ancillary variables (Section 3.1). There are notable disparities between model predicted tree heights and actual observations—the reason being that the ASRL model includes constant scaling exponents and parameters across different climatic regimes and forest types.

Optimization of the ASRL model was designed to simultaneously adjust multiple scaling parameters. This optimization was aimed to minimize the difference between actual tree heights derived from GLAS data and tree heights predicted by the ASRL model (Figure 2). The underlying theoretical framework is based on Powell’s optimization methodology [50] that results in finding the minima of a multidimensional function. This algorithm is efficient for generating the convergence of a function due to (a) its bi-directional search algorithm over the vector of multi-variables and (b) nonessential calculation of derivatives for each variable [50]. A function involving three variables (merit function as in Equation (2)) was formulated and implemented based on Press et al.[51] and Kuusk and Nilson [52].

Three parameters of the ASRL model—area of single leaf (α), exponent for canopy radius (η) and root absorption efficiency (γ)—were selected for optimization. Initial values of these three parameters (α, η, and γ) were set to 13 cm², 1.14, and 0.33, respectively. These values are comparable to the representative values (averages) from the TRY database [53] and also based on Kempes et al.[14]. Although there are other physiological traits available from the TRY database [54], this study was limited to optimization of these three parameters.

The collection of solar radiation for plant growth is associated with the coefficient for canopy transmissions. Here, α produces the total leaf area based on the branching generation theory [55]. In the ASRL model, the canopy-level budget is collected from the energy budget in a single leaf based on the allometric geometry of canopy [14]. The value η controls the scaling of canopy radius with tree height, which is related to the rate of absorbed solar radiation. Lastly, γ determines the available flow rate given the incoming rate of precipitation within the root capture area. The tallest tree takes γ (=1/3) on average and local γ varies across different soil type and hydrology [14].

Kempes et al.[14] have performed the sensitivity tests of several allometric scaling exponents, and η showed the second least sensitivity. The value γ was tested in the optimization procedures of Kempes et al.[14], generating clear improvement of the model predictions. In this study, α was additionally selected for optimization due to (a) its strong relationship to net absorbed radiation, sensible and latent heat fluxes (e.g., [56]) and (b) considerable variability of α across different climatic zones and forest types [57,58].

The optimization process stops the iterative adjustment of the three parameters when it finds the maximum likelihood estimates of each parameter that result in minimizing the merit function. To reach an optimal solution, we implemented variable ranges (lower and upper boundaries as in the TRY database) for each of the input parameters such that 1 cm² ≤ α < 100 cm², 0.8 ≤ η < 1.5, and 0.1 ≤ γ < 0.8. Equation (2) shows the merit function, (2) J ( p 1 , p 2 , p 3 ) = ∑ k = 1 3 [ ( p k − p kb ) 4 w k 2 + ( p k − p k 0 ) 2 Δ p k 2 ] + ∑ i = 1 N ( H ASRL , i − H GLAS , i ) 2 Δ H 2Here p_k is the selected ASRL model parameter (k; 1 = area of single leaf, 2 = exponent for canopy radius, and 3 = root absorption efficiency), p_k0 refers to the initial values of each parameter in the original ASRL model, p_kb refers to the boundary limits ((lower limit + upper limit)/2) for each parameter, Δ_pk is the standard deviation associated with each parameter with respect to initial values, w_k is a scalar weight (w_k = 0 when p_k ∈ [p_{k_lower-limit}, p_{k_upper-limit}], otherwise w_k = 10), N is the total number of comparison sets (i) for GLAS heights (H_GLAS) and ASRL model predictions (H_ASRL) with given parameter values for each climatic zone and Δ_H is the standard deviation associated with H_GLAS and H_ASRL. The iterative adjustment process continues until it finds the minimum of the function J (p₁, p₂, p₃) for each of the climatic zones.

A noteworthy limitation of this optimization exercise is that forest stand ages are not directly involved in the optimization process. Tree heights and growth rates vary depending on forest types and sites due to different growing conditions. Those are clearly related to forest stand ages [59–61]. When a tree ages, its height increases along with decline in the rate of its vertical growth over time—young forest stands (∼10 years) grow in the southeastern region, while old forest stands (∼900 years) inhabit the western coasts in CONUS [62]. However, it does not necessarily mean that our methodology neglects forest stand ages in tree height estimations. GLAS waveform data indirectly brings age information of forests into the ASRL model to find appropriate scaling parameters, as actual heights are associated with forest stand ages.

Performance of the ASRL model was tested by comparing GLAS tree heights and model predicted heights (with and without optimization). The goal was to show the efficacy of the optimization process. We calculated R² and root-mean-square-error (RMSE) from relationships between GLAS tree heights and model predicted heights in each climatic zone (Equation (3)). (3) RMSE ASRL = ∑ i = 1 n ( H ¯ ASRL , i − H ¯ GLAS , i ) 2 nHere H̄_ASRL is the mean of tree heights from ASRL model predictions (with and without optimization) in each climatic zone, H̄_GLAS is the mean of tree heights derived from GLAS waveform data in each climatic zone, and i refers to the number of climatic zones (n = 841).

The training datasets of GLAS used in model optimization are identical to the test datasets of GLAS used for evaluation [63]—this was first done to assess whether the optimization scheme was correctly implemented or not. It was not meant to establish validity of the optimized ASRL model. The actual evaluation of the optimized ASRL model was performed as detailed below.

The prediction of the optimized ASRL model was evaluated in two parts: (a) two-fold cross validation approach and (b) two inter-comparisons of optimized ASRL model prediction (H_{opt ASRL}) with forest canopy heights produced by Simard et al.[1] (H_Simard) and Lefsky [2] (H_Lefsky). Each evaluation performs inter-comparisons at the climatic zone level and at the pixel level.

A continuous map of potential tree heights (H_{potential ASRL}) was generated with the unoptimized ASRL model at 1 km resolution (Figure 3(a)). Maximum potential tree heights were greater than 50 m in both the Northeastern Appalachian and Pacific Northwestern forest corridors. The model predicted lower values of potential tree heights (≤35 m) in the Southeast.

We noted discrepancies between model predictions and GLAS tree heights (H_GLAS; actual tree height; Figure S1). A low correlation was observed (Figure 3(b)) in each climatic zone (R² = 0.06; RMSE = 22.8 m). In addition, there was significant skewness in the histograms of actual (mean = 31.3 m; standard deviation = 11.5) and potential (mean = 45.5 m; std. = 23.6) tree heights at the pixel level (Figure 3(c)). Tree heights were overestimated especially in the northeastern forests as compared to H_GLAS (Figure S3(a)). A plausible reason could be that the ASRL model does not accurately reflect the spatial/temporal dynamics in the estimation of internal flow balances (metabolic flow requirement, available flow, and evaporative flow) across different eco-climatic regimes and forest types [14].

The optimized model was then used to generate a spatially continuous map of tree heights (H_{opt ASRL}; Figure 4(a)). We noted a significant improvement in predictions of tree heights both at the climatic zone level (Figure 4(b)) and individual pixel level (Figure 4(c); Figure S3(b)): (a) the RMSE decreased from 22.8 m (without optimization) to 3.1 m (after optimization) with an increase in R² from 0.06 to 0.8 (P < 0.01); (b) the histograms show a better agreement between distributions of GLAS tree heights (mean = 31.3 m; std. = 11.5) and the optimized model predictions (mean = 30.4 m; std. = 8.5); and (c) relatively smaller model prediction errors over the Northeastern Appalachian and Pacific Northwestern forest corridors as compared to the unoptimized ASRL model predictions.

Figure S4 shows that the ASRL model prediction errors at the individual pixel level decreased from 15.10 m (H_GLAS − H_{potential ASRL}) to −0.80 m (H_GLAS − H_{opt ASRL}). However, the optimized ASRL model poorly predicted tree heights over complex terrains (e.g., ∼20 m underestimation for the redwood stands in the Pacific Northwestern mountains of California and Oregon; Figure S3b). Other GLAS-based models also reported relatively large prediction errors in the estimation of tree heights [1] and biomass [3] in these forests. Interpolation of annual precipitation (e.g., [67]) and temperature (e.g., [68]) may have produced large uncertainties in climatic variables that are sensitive to topographic features. Note that these are critical inputs to the ASRL model. Other plausible reasons for this discrepancy may be: (a) GLAS undersampling for some of the climatic zones (Figure S5) that resulted in fewer comparison sets in the merit function (Equation (2)) and (b) topographic influence on GLAS waveform data which could not perfectly be rectified by our slope gradient filter (Table 3).

The optimized parameters are shown in Figure 5. There are notable changes in the optimal area of single leaf (initial value α: 13.0 cm²) that ranged from 1.5 cm² to 90.0 cm². The root absorption efficiency (initial value γ: 0.33) converged to a relatively narrower range of values (from 0.05 to 0.65), while ∼80% of the optimized exponent for canopy radius (η) fell within the range of ±10% of its initial value (1.14). Kempes et al.[14] have also reported a stable median relative error against the percent change of a single scaling parameter (i.e., η).

The area of single leaf of deciduous forests (mean α = 19.3 cm²) was higher than that of evergreen forests (mean α = 9.1 cm²). The original ASRL model precludes inclusion of forest types. The optimization process allows combining allometric scaling laws with features that are representative of specific forest types. Optimized α values are well correlated with the variability in forest types, annual total precipitation and annual average temperature in each climatic zone. Warm (annual average temperature = ∼15 °C) and wet (annual total precipitation ≥ ∼1,500 mm) regions displayed a larger value of α for both deciduous and evergreen forests. In cold regions (annual average temperature = ∼5 °C), the optimized value of α for evergreen forests increased with annual total precipitation. These results are supported by other studies that examined relationships between leaf traits and environmental conditions [69–71].

Similar trends in the optimized γ values were observed in warm and wet regions. However, evergreen forests generally showed higher optimized γ values compared to deciduous forests in relatively dry regions. Water availability is spatially heterogeneous for an individual species within a location [72]. For example, evergreen and deciduous plants in dry regions have different root systems and water use efficiencies (evergreen > deciduous as in [73,74]). Kempes et al.[14] have demonstrated an improvement of the ASRL model based on optimization of γ that generated a lower variance in the model error.