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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

A methodology to generate spatially continuous fields of tree heights with an optimized Allometric Scaling and Resource Limitations (ASRL) model is reported in this first of a multi-part series of articles. Model optimization is performed with the Geoscience Laser Altimeter System (GLAS) waveform data. This methodology is demonstrated by mapping tree heights over forested lands in the continental USA (CONUS) at 1 km spatial resolution. The study area is divided into 841 eco-climatic zones based on three forest types, annual total precipitation classes (30 mm intervals) and annual average temperature classes (2 °C intervals). Three model parameters (area of single leaf, α, exponent for canopy radius, η, and root absorption efficiency, γ) were selected for optimization, that is, to minimize the difference between actual and potential tree heights in each of the eco-climatic zones over the CONUS. Tree heights predicted by the optimized model were evaluated against GLAS heights using a two-fold cross validation approach (R^{2} = 0.59; RMSE = 3.31 m). Comparison at the pixel level between GLAS heights (mean = 30.6 m; standard deviation = 10.7) and model predictions (mean = 30.8 m; std. = 8.4) were also performed. Further, the model predictions were compared to existing satellite-based forest height maps. The optimized ASRL model satisfactorily reproduced the pattern of tree heights over the CONUS. Subsequent articles in this series will document further improvements with the ultimate goal of mapping tree heights and forest biomass globally.

Several recent articles have reported generating spatially continuous maps of forest canopy heights and/or biomass using a combination of remote sensing data,

Physical/physiological models for mapping tree heights or biomass rely on mechanisms governing plant growth. The Allometric Scaling and Resource Limitations (ASRL) model [

Generating continuous fields of tree heights and biomass is the larger objective of this multi-part series of articles. In this first article, we focus on how the ASRL model can be used with GLAS data to map actual tree heights over the continental USA (CONUS) at 1 km spatial resolution. The ASRL model is briefly explained in Section 2 together with key equations and parameters. Section 3 includes descriptions of input data for ASRL model and GLAS data preprocessing. Information of the model optimization and evaluation is provided in Section 4 followed by results and discussion (Section 5) and concluding remarks (Section 6). The second paper of this series [

The ASRL model [

In the fundamental premises of the ASRL model, a tree obtains sufficient resources (water and nutrients) to meet its needs for the growth and the availability of local resources limits the maximum potential growth. This is expressed by an inequality equation of basal metabolic rates (_{p}_{e}_{0}_{p}_{e}_{0}_{0}_{p}_{e}_{p}_{e}

The maximum tree growth varies depending on many factors (e.g., climatic and soil condition, forest types and stand ages), but the ASRL model implements consistent allometric scaling parameters and exponents across different eco-climatic regimes and forest types of varying age classes [

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.

The ASRL model was mostly driven by climatic variables derived from the DAYMET model [

The first set of ancillary data (DEM and LAI) is used for the initial ASRL predictions of potential tree heights. The growing season (June to September) average LAI data were calculated from a refined version of the standard Moderate Resolution Imaging Spectroradiometer (MODIS) LAI products (1 km grids) for the time period from 2003 to 2006 [

The second set of ancillary data is required during the ASRL model prediction and the parametric optimization to identify forested lands. The model simulations were conducted on spatial regions categorized into three forest classes—deciduous, evergreen, and mixed forests—each with percent tree cover ≥50 percent based on the MODIS Vegetation Continuous Field (VCF) product (

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 [

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 (_{GLAS}

There are two approaches of retrieving tree heights from GLAS waveform data [

We used two standard altimetry variables of GLA14 product (signal begin range increment, ^{2} = 0.70; RMSE = 4.42 m; [

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 [

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 (

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 cm^{2}, 1.14, and 0.33, respectively. These values are comparable to the representative values (averages) from the TRY database [

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 [

Kempes

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 cm^{2} ≤ α < 100 cm^{2}, 0.8 ≤ η < 1.5, and 0.1 ≤ γ < 0.8. _{k}_{k0}_{kb}_{pk}_{k}_{k}_{k}_{k_lower-limit}_{k_upper-limit}_{k}_{GLAS}_{ASRL}_{H}_{GLAS}_{ASRL}_{1}_{2}_{3}

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 [

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 R^{2} and root-mean-square-error (RMSE) from relationships between GLAS tree heights and model predicted heights in each climatic zone (_{ASRL}_{GLAS}

The training datasets of GLAS used in model optimization are identical to the test datasets of GLAS used for evaluation [

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 (_{opt ASRL}_{Simard}_{Lefsky}

A two-fold cross validation approach was performed: That is, we randomly divided the original sample input data into two sets of training and test data. The first half of the GLAS tree heights was used as a training data to optimize the ASRL model in each climatic zone. The test data was generated by averaging the remaining half of the GLAS tree heights in each climatic zone and used for model evaluation purposes (_{opt ASRL}_{training} − _{GLAS}_{test}). We selected spatially corresponding tree height values (the nearest pixels) in pixel level comparisons.
_{opt ASRL}_{training} is the predicted height by the optimized ASRL model using the first set of GLAS training data for each climatic zone, _{GLAS}_{test} is the mean of tree heights computed from the second set of GLAS test data in each climatic zone, and

The optimized model evaluations were additionally performed by comparing model predicted heights with _{Simard}_{Lefsky}

To facilitate inter-comparison with _{Simard}

On the other hand, a direct comparison between Lefsky [

There are certain limitations of our analysis in the ASRL model predictions and evaluations: (a) up- and down-scaling approaches of resampling may cause potential errors due to the aggregation of heterogeneity in finer grids and the neglect of discontinuity in coarser datasets [

A continuous map of potential tree heights (_{potential ASRL}

We noted discrepancies between model predictions and GLAS tree heights (_{GLAS}^{2} = 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 (_{GLAS}

The optimized model was then used to generate a spatially continuous map of tree heights (_{opt ASRL}^{2} 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.

_{GLAS}_{potential ASRL}_{GLAS}_{opt ASRL}

The optimized parameters are shown in ^{2}) that ranged from 1.5 cm^{2} to 90.0 cm^{2}. 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

The area of single leaf of deciduous forests (mean α = 19.3 cm^{2}) was higher than that of evergreen forests (mean α = 9.1 cm^{2}). 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 [

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 [

^{2} = 0.59; RMSE = 3.31 m; P < 0.01). Histograms comparing the test GLAS heights (mean = 30.8 m; std. = 10.7) and tree heights predicted by the optimized model (mean = 30.6 m; std. = 8.4) show considerable similarity (

Forest height maps from Simard

_{opt ASRL}_{Simard}^{2} = 0.45; RMSE = 8.01 m; P < 0.01). Average values of _{opt ASRL}

We also compared our forest height map with Lefsky’s [_{opt ASRL}_{Lefsky}^{2} = 0.41; RMSE = 6.72 m; P < 0.01) between _{opt ASRL}_{Lefsky}_{opt ASRL}_{opt ASRL}_{Lefsky}

An optimization of the Allometric Scaling and Resource Limitations (ASRL) model with Geoscience Laser Altimeter System (GLAS) waveform data was performed to generate a spatially continuous map of tree heights over the continental USA (CONUS) at 1 km resolution. The optimization is designed to minimize differences between actual heights (based on GLAS waveforms) and potential tree heights predicted by the ASRL model. This study covered all forested lands with over 50% tree cover. These were categorized into 841 climatic zones based on forest types (deciduous, evergreen, and mixed forests), fixed intervals of annual total precipitation (30 mm) and annual average temperature (2 °C). The optimization procedure simultaneously adjusted three model parameters (area of single leaf, α; exponent for canopy radius, η; and root absorption efficiency, γ) in each of the climatic zones.

After testing for correctly implementing the optimization technique, tree heights predicted by the optimized model were first evaluated using a two-fold cross validation approach. Regression analysis was used to assess the correlation between predictions of tree heights by the optimized model (_{opt ASRL}_{GLAS}_{test}) in all climatic zones. Mean values of _{opt ASRL}_{GLAS}_{test} mean estimates in each of the climatic zones and, on average, showed an estimation error of 3.31 units of height. A similar evaluation of the optimized ASRL model was performed at FLUXNET sites—this is detailed in the second of this multi-article series [_{opt ASRL}_{GLAS}

Second, tree height predictions by the optimized ASRL model were compared with available forest height products derived independently but from the GLAS data—Simard _{Simard}_{Lefsky}^{2} = 0.45 and RMSE = 8.01 m for _{Simard}^{2} = 0.41 and RMSE = 6.72 m for _{Lefsky}_{opt ASRL}_{Simard}_{Lefsky}

Predictions of tree heights by the ASRL model were clearly improved by the optimization technique reported in this article. The optimization successfully compensated for certain limitations of the original ASRL model, which did not account for effects related to spatio-temporal variability in climatic-regimes and forest types. The results demonstrate the potential for a more generic applicability of the ASRL model for estimation of tree heights. Nevertheless, the optimized ASRL model still yields ambiguous results over complex terrains, possibly due to uncertainties in input climatic data and topographic effects in the GLAS waveform data. The optimization methodology reported in this article has certain limitations: e.g., (a) a limited number of scaling parameters (α, η, and γ) were explored in the model optimization, (b) stand age was not directly considered in the optimization, (c) soil conditions were neglected in the optimization and (d) we assumed that allometric scaling laws at individual tree level were applicable at larger scales. Also, our analysis could not take into account the uncertainties derived from resampling and reprojection of maps and data at different scales and projections. Alleviation of these limitations should be addressed in future articles in this series.

This study was partially funded by the National Natural Science Foundation of China (Grants No. 40801139 and 41175077), China Scholarship Council and the Fulbright Foundation.

Forested lands (1 km spatial resolution) over the continental USA (CONUS) based on the National Land Cover Database (NLCD) 2006 land cover. Three forest types—deciduous, evergreen, and mixed forests—with percent tree cover ≥50% were considered in this study.

Diagram showing the optimization of the ASRL model. The model predicts potential tree heights (initial prediction) based on input climatic and ancillary variables. Three allometric scaling parameters (area of single leaf, α, exponent for canopy radius, η and root absorption efficiency, γ) are adjusted in the optimization process to minimize the difference between GLAS tree heights and ASRL modeled tree heights. This optimization process is done separately for each of the climatic zones.

(

(

Spatial distribution of the three parameters selected for model optimization. (

A two-fold cross validation approach showing comparisons between test GLAS tree heights and the optimized ASRL model predictions using training GLAS tree heights. We randomly divided the GLAS height data into two equal sets of training and test data: (_{opt ASRL}_{training} − _{GLAS}_{test}) from pixel level comparison. Number of bins of histograms is 50. Frequencies have been normalized by total grids (frequency %).

Inter-comparison of tree heights predicted by the optimized ASRL model with forest canopy heights from Simard _{opt ASRL}_{Simard}_{opt ASRL}_{Simard}

Inter-comparison of tree heights predicted by the optimized ASRL model with tree heights from Lefsky [_{opt ASRL}_{Lefsky}_{opt ASRL}_{Lefsky}

Key equations of the Allometric Scaling and Resource Limitations (ASRL) model [

Equations of Basal Metabolic Rates | Available Flow Rate | _{p} |
_{p}_{root}^{2}_{inc} |
_{root}_{inc} |

Evaporative Flow Rate | _{e} |
_{e}_{f} E_{can} μ_{w} ρ_{w}^{−1} |
_{f}_{can}_{w}^{−2} kg·mol^{−1});_{w}^{3} kg·m^{−3}) | |

Required Flow Rate | _{0} |
_{0}_{2}^{η}^{2} |
_{2} = Proportionality Constant for Metabolism^{−7} L·day^{−1} cm·^{−}^{η}^{2});_{2} = Exponent for Metabolism (≈2.7) | |

Sub-equations of Evaporative Flow Rate | Effective Area over the Latent Heat Flux Loss | _{f} |
_{f}_{L} δ_{s} a_{s} |
_{L}_{s}^{−2});_{s}^{2}) |

Total One-sided Area of All Leaves on a Tree | _{L} |
_{L}^{N} |
||

Number of Branching Generations | _{0}_{N} |
_{0}_{N} | ||

Evaporative Flux of Canopy | _{can} |
Refers to [ |
Equation uses Rate of Absorbed Solar Radiation (_{abs}_{can} | |

Sub-allometric Scaling Equations | Radial Extent of Root System | _{root} |
_{root}_{3}^{1/4} |
_{3} = Root to Stem Mass Proportionality (≈0.423) |

Maximum Stem Radius | _{0} |
_{0}_{2}/_{1})^{1/}^{η}^{1}^{η}^{2/η1} |
_{1} = Proportionality Constant for Metabolism (=0.257 L·day^{−1} cm^{−}^{η}^{1});_{1} = Exponent for Metabolism (=1.8) | |

Canopy Radius | _{can} |
_{can}_{5}^{η} |
_{5} = Proportionality Constant for Canopy Radius (=35.24 cm·m^{−η}); |

Climatic and other ancillary variables used for ASRL model simulations.

Climatic Variables | Annual Total Precipitation | mm | 1980–1997 | 1 km | DAYMET model [ |

Annual Average Temperature | °C | 1980–1997 | 1 km | ||

Annual Incoming Solar Radiation | W/m^{2} |
1980–1997 | 1 km | ||

Annual Average Vapor Pressure | hPa | 1980–1997 | 1 km | ||

Annual Average Wind Speed | m/s | 2000–2008 | 32 km | North American Regional Reanalysis (NARR) data [ | |

Ancillary Variables I | Digital Elevation (DEM) | m | 2009 | 30 m | National Elevation Dataset (NED) [ |

Growing Season Average Leaf Area Index (LAI) | N/A | 2003–2006 Jun–Sep | 1 km | Post-processed Moderate Resolution Imaging Spectroradiometer (MODIS) LAI products [ | |

Ancillary Variables II | Land cover | N/A | 2006 | 30 m | National Land Cover Database (NLCD) [ |

Percentage of Tree Cover | % | 2005 | 250 m | MODIS Vegetation Continuous Fields (VCF) Collection 5 [ |

Five screening steps to remove invalid Geoscience Laser Altimeter System (GLAS) footprints over the CONUS. Final valid GLAS footprints are 126693 in this study.

1. Atmospheric Forward Scattering and Signal Saturation Filter |
- Cloud-free and saturation-free GLAS waveform data; - Internal flag of GLAS data “FRir_qaFlag = 15” and “satNdx = 0” |
1,822,739 | [ |

2. NLCD and VCF Filters |
- GLAS footprints over forested lands; - Geolocation of NLCD and VCF pixels (pixels nearest to the center of a GLAS footprint); - Deciduous, evergreen, and mixed forests with greater than 50% of the tree cover |
1,659,061 | - |

3. Background Noise Level (Low Cloud) Correction Filter |
- No background noise level in GLAS waveform data; - Absolute difference (≤ 50 m) between the NED DEM and the internal elevation (“i_elev”) of GLAS waveform data |
161,533 | [ |

4. Slope Gradient Correction Filter |
- GLAS footprint over non- high topographic condition; - Slope value < 20 ° of the nearest pixel from GLAS data; - Additionally correction of the potential bias (= footprint size × tan (slope)) |
129,705 | [ |

5. Removal of Remaining Outliers |
- Using two standard deviations from the mean of GLAS tree heights (5 m < |
- |

Definition of climatic zones for grouping pixels within a forested area. Three forest types with fixed ranges of annual total precipitation (30 mm intervals) and annual average temperature (2 °C intervals) yield a total of 5805 segments (3 × 129 × 15). Of these, only 841 climatic zones have at least one GLAS observation.

3 | 300 | 4,170 | 30 | −5 | 25 | 2 | 841 |