# Generalized Hierarchical Model-Based Estimation for Aboveground Biomass Assessment Using GEDI and Landsat Data

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Overview

#### 2.2. Generalized Hierarchical Model-Based Estimation (GHMB)

- The first dataset, denoted S, contains a sample of field data for which sampled RS data are also available. The dataset is used for estimating the model parameters $\mathit{\beta}$. Each estimator based on this set is given the subscript S; the data S comprise n field observations.
- The second dataset, denoted $Sa$, contains the enlarged sample of RS data and the corresponding RS data from the wall-to-wall dataset. It is used for estimating the model parameters $\mathit{\alpha}$, and any estimator based on this set has the subscript $Sa$; the data $Sa$ comprise M sampled RS observations.
- The third dataset contains the wall-to-wall RS data for the entire target population, U. The target population U comprises N population elements.

- with clustered data structure, $\mathsf{\Omega}$ and $\mathbf{\Sigma}$ are block-diagonal matrices, where the blocks correspond to the clusters;
- under spatial autocorrelation, the matrices’ off-diagonal elements, corresponding to covariances between errors, are non-zero;

#### 2.3. Reference Methods for Comparison

#### 2.3.1. Generalized Two-Stage Model-Based Estimation (GTSMB)

`HMB`” by Saarela et al. [30] was used to obtain estimates based on (17) and (18).

#### 2.3.2. Hybrid Estimation

#### 2.3.3. Conventional Model-Based Inference (MB)

## 3. Material

#### 3.1. Reference Data

#### 3.1.1. Simulated GEDI Data

#### 3.1.2. Landsat 7 ETM+ Data

#### 3.1.3. Field Data

#### 3.2. Simulated Populations

#### 3.3. Regression Modeling

#### 3.4. Evaluation Criteria

## 4. Results

#### Sources of Uncertainty for the GHMB and Hybrid Estimation Methods

## 5. Discussion

- One important issue is that all the case study data were simulated, based on sparse samples of reference data. The simulations are simplified generalizations of the real world, that leave out many important issues that must be handled in practical surveys, such as delineating forests from non-forest land [42]. In practical applications, land-use maps need to be applied to delineate forests before the GHMB method is employed.
- Another important restriction of the present study is that the results are based on estimates from a small number of iterations. A future expanded study should be based on Monte Carlo simulations of both the populations and the sampling from these populations, as a basis for empirical evaluation of the proposed estimators. Another method for simulating the multivariate variable $\mathbf{a}$ in Equation (23) might be applied to demonstrate explicitly abilities of the proposed estimators.
- Further, future studies on this subject should also deal with the details of how to estimate the correlation matrices of model errors that are required for the GLS regression; in this article these details are only briefly addressed. One of the solutions to this problem could be iterative re-weighted least squares regression methods, such methods are often applied in geostatistical approaches.
- The current GHMB estimator is derived under the assumption that the target population is large, i.e., so that the population mean is at least approximately equal to the superpopulation mean. Modifications of the GHMB estimators for small-area estimation should be addressed in a potential future study.
- In the current study we assumed that the regression models involved in the AGB assessment by means of GEDI and Landsat data are linear. However, in reality the relationship between AGB (or growing stock volume) and height-like measures tends to be nonlinear e.g., [23]. A further elaboration of the GHMB method for nonlinear models would be needed to handle such cases.
- Lastly, the performance of GHMB method in comparison to other methods should be analyzed as a basis for making recommendations on what method is appropriate under different conditions.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AGB | AboveGround Biomass |

GEDI | Global Ecosystem Dynamics Investigation |

GHMB | Generalized Hierarchical Model-Based |

GLS | Generalized Lest Squares |

GTSMB | Generalized Two-Stage Model-Based |

Landsat 7 ETM+ | Landsat 7 Enhanced Thematic Mapper Plus |

LiDAR | Light Detection And Ranging |

MB | Model-Based |

NFI | National Forest Inventory |

PALS | Portable Airborne Laser System |

RS | Remote Sensing |

UNFCCC | United Nations’ Framework Convention on Climate Change |

## Appendix A. Generalized Hierarchical Model-Based Estimators

## Appendix B. A Comparison between Expectations and Variance Estimators of the Two Methods: GHMB and GTSMB

## Appendix C. Field and LiDAR Data Collection Methods

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**Figure 3.**(

**a**) Simulated spatial pattern of GEDI footprints and the 1-km grid delineation; (

**b**) Case A: data for a single 1-km grid-cell were used; (

**c**) Case B: data from neighboring grid-cells were used as well, for the GHMB and GTSMB estimators.

Study Site | Forest Type | Number of Field Plots | AGB, [Mg/ha] | |||
---|---|---|---|---|---|---|

min | mean | max | sd | |||

OR—Oregon | Douglas Fir (24%), Ponderosa Pine (33%), Fir Spruce/Mountain Hemlock (22%), Lodgepole Pine (20%) | 49 | 4.7 | 227.5 | 775.1 | 205.7 |

ME—Maine | Spruce/Fir (49%), Maple/Beech/Birch (50%) | 48 | 0.2 | 64.1 | 264.8 | 69.7 |

PANJ—Pennsylvanian/New Jersey border | Loblolly/Shortleaf Pine (25%), Oak/Hickory (62%), Maple/Beech/Birch (6%) | 49 | 0.0 | 89.1 | 378.8 | 106.7 |

SC—South Carolina | Loblolly/Shortleaf Pine (66%), Oak/Gum/Cypress (28%) | 50 | 0.0 | 72.7 | 373.3 | 80.1 |

CO—Colorado | Pinyon/Juniper (17%), Fir/Spruce/Mountain Hemlock (27%), Aspen/Birch (36%), Western Oak (12%) | 46 | 0.0 | 133.8 | 353.5 | 96.2 |

MN—Minnesota | Spruce/Fir (28%), Aspen/Birch (68%) | 47 | 0.0 | 48.1 | 202.0 | 48.5 |

Study Site | AGB | GEDI rh60 | GEDI rh90 | Landsat B3 | Landsat B4 | |
---|---|---|---|---|---|---|

OR | 1 | 0.79 | 0.85 | −0.49 | −0.19 | AGB |

– | 1 | 0.78 | −0.55 | −0.03 | GEDI rh60 | |

– | – | 1 | −0.43 | −0.12 | GEDI rh90 | |

– | – | – | 1 | −0.17 | Landsat B3 | |

– | – | – | – | 1 | Landsat B4 | |

ME | 1 | 0.90 | 0.87 | −0.26 | −0.44 | AGB |

– | 1 | 0.83 | −0.41 | −0.35 | GEDI rh60 | |

– | – | 1 | −0.28 | −0.41 | GEDI rh90 | |

– | – | – | 1 | 0.14 | Landsat B3 | |

– | – | – | – | 1 | Landsat B4 | |

PANJ | 1 | 0.82 | 0.70 | −0.54 | 0.06 | AGB |

– | 1 | 0.87 | −0.62 | 0.24 | GEDI rh60 | |

– | – | 1 | −0.63 | 0.45 | GEDI rh90 | |

– | – | – | 1 | −0.37 | Landsat B3 | |

– | – | – | – | 1 | Landsat B4 | |

SC | 1 | 0.89 | 0.83 | −0.36 | 0.07 | AGB |

– | 1 | 0.85 | −0.37 | 0.03 | GEDI rh60 | |

– | – | 1 | −0.34 | 0.05 | GEDI rh90 | |

– | – | – | 1 | −0.26 | Landsat B3 | |

– | – | – | – | 1 | Landsat B4 | |

CO | 1 | 0.82 | 0.78 | −0.38 | −0.36 | AGB |

– | 1 | 0.82 | −0.42 | −0.22 | GEDI rh60 | |

– | – | 1 | −0.46 | −0.35 | GEDI rh90 | |

– | – | – | 1 | 0.02 | Landsat B3 | |

– | – | – | – | 1 | Landsat B4 | |

MN | 1 | 0.83 | 0.89 | −0.24 | 0.04 | AGB |

– | 1 | 0.80 | −0.33 | 0.06 | GEDI rh60 | |

– | – | 1 | −0.27 | 0.12 | GEDI rh90 | |

– | – | – | 1 | −0.14 | Landsat B3 | |

– | – | – | – | 1 | Landsat B4 |

Study Site | AGB | GEDI rh60 | GEDI rh90 | Landsat B3 | Landsat B4 | |
---|---|---|---|---|---|---|

OR | 227.5 | 10.3 | 22.1 | 378.3 | 1990.3 | $\mu $ |

205.7 | 11.1 | 13.7 | 205.5 | 540.7 | sd | |

ME | 64.1 | 5.2 | 10.2 | 278.2 | 3074.5 | $\mu $ |

69.7 | 4.6 | 5.4 | 94.5 | 562.7 | sd | |

PANJ | 89.1 | 8.2 | 14.1 | 279.6 | 3505.6 | $\mu $ |

106.7 | 9.2 | 9.7 | 89.4 | 93.6 | sd | |

SC | 72.7 | 7.5 | 13.2 | 289.3 | 2825.7 | $\mu $ |

80.1 | 7.9 | 8.3 | 128.8 | 417.4 | sd | |

CO | 133.8 | 5.9 | 12.8 | 388.6 | 2369.6 | $\mu $ |

96.2 | 4.9 | 6.4 | 155.0 | 834.4 | sd | |

MN | 48.1 | 4.9 | 10.2 | 280.0 | 3243.0 | $\mu $ |

48.5 | 4.7 | 6.4 | 89.0 | 761.3 | sd |

**Table 4.**Simulated spatial autocorrelation between two square plots center-points at 20 m distance for AGB variable for each study site.

Study Site | ${\mathit{\rho}}_{\mathit{i},\mathit{j}}$ | a | Study Site | ${\mathit{\rho}}_{\mathit{i},\mathit{j}}$ | a | Study Site | ${\mathit{\rho}}_{\mathit{i},\mathit{j}}$ | a |
---|---|---|---|---|---|---|---|---|

OR | 0.72 | 1.64 × 10${}^{-2}$ | PANL | 0.82 | 0.99 × 10${}^{-2}$ | CO | 0.44 | 4.10 × 10${}^{-2}$ |

ME | 0.59 | 2.64 × 10${}^{-2}$ | SC | 0.59 | 1.64 × 10${}^{-2}$ | MN | 0.77 | 1.31 × 10${}^{-2}$ |

Notation | Description | Application | Dataset |
---|---|---|---|

AGB-GEDI | A model is linking AGB as a response variable with GEDI variables as predictor variables | GHMB (first level of modeling hierarchy), GTSMB and Hybrid | S |

AGB-Landsat (Sa) | A model linking predicted AGB as a response variable with Landsat predictor variables | GHMB (second level of modeling hierarchy) | $Sa$ |

GEDIrh60-Landsat GEDIrh90-Landsat | Models linking GEDI response variables with Landsat predictor variables | GTSMB | $Sa$ |

AGB-Landsat (S) | A model linking AGB as a response variable with Landsat predictor variables | MB | S |

Case | Model | df |
---|---|---|

A, B | AGB-GEDI and AGB-Landsat (S) | 47 |

A | AGB-Landsat (Sa), GEDI(rh60)-Landsat and GEDI(rh90)-Landsat | 91 |

B | AGB-Landsat (Sa), GEDI(rh60)-Landsat and GEDI(rh90)-Landsat | 724 |

Study Site | $\mathit{\mu}$ | ${\overline{\mathit{y}}}_{\mathit{U}}$ | Two Sources of RS Data | One Source of RS Data | ||||
---|---|---|---|---|---|---|---|---|

Case A | Case B | Hybrid | MB | |||||

GHMB | GTSMB | GHMB | GTSMB | |||||

OR | 227.5 | 226.8 | 227.4 | 227.4 | 226.5 | 226.5 | 226.3 | 225.6 |

ME | 64.1 | 64.4 | 64.6 | 64.6 | 64.0 | 64.0 | 64.5 | 64.5 |

PANJ | 89.1 | 88.1 | 88.0 | 88.0 | 88.3 | 88.3 | 88.0 | 88.2 |

SC | 72.7 | 72.6 | 73.4 | 73.4 | 72.7 | 72.7 | 73.3 | 73.6 |

CO | 133.8 | 133.1 | 133.7 | 133.7 | 133.6 | 133.6 | 133.7 | 133.2 |

MN | 48.1 | 48.9 | 49.0 | 49.0 | 48.7 | 48.7 | 48.9 | 49.2 |

Study Site | Two Sources of RS Data | One Source of RS Data | ||||
---|---|---|---|---|---|---|

Case A | Case B | Hybrid | MB | |||

GHMB | GTSMB | GHMB | GTSMB | |||

OR | 14.8 | 14.8 | 9.4 | 9.4 | 13.1 | 13.3 |

ME | 14.0 | 14.0 | 8.0 | 8.0 | 13.8 | 15.2 |

PANJ | 25.4 | 25.4 | 17.5 | 17.5 | 20.3 | 21.2 |

SC | 15.2 | 15.2 | 9.0 | 9.0 | 14.2 | 16.3 |

CO | 8.6 | 8.6 | 6.3 | 6.3 | 8.8 | 9.1 |

MN | 19.5 | 19.5 | 10.8 | 10.8 | 15.7 | 18.4 |

**Table 9.**Proportion (percentage) of the variance due to different sources for the GHMB and hybrid estimation methods, [%].

Study Site | GHMB | Hybrid | ||||
---|---|---|---|---|---|---|

Due to AGB-GEDI | Due to AGB-Landsat (Sa) | Due to Modeling | Due to Sampling | |||

Case A | Case B | Case A | Case B | |||

OR | 30.4 | 76.1 | 69.6 | 23.9 | 39.3 | 60.7 |

ME | 22.7 | 69.0 | 77.3 | 31.0 | 23.5 | 76.5 |

PANJ | 37.9 | 80.3 | 62.1 | 19.7 | 60.5 | 39.5 |

SC | 26.2 | 73.1 | 73.8 | 26.9 | 30.2 | 69.8 |

CO | 46.5 | 87.0 | 53.5 | 13.0 | 44.8 | 55.2 |

MN | 19.1 | 62.6 | 80.9 | 37.4 | 29.7 | 70.3 |

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## Share and Cite

**MDPI and ACS Style**

Saarela, S.; Holm, S.; Healey, S.P.; Andersen, H.-E.; Petersson, H.; Prentius, W.; Patterson, P.L.; Næsset, E.; Gregoire, T.G.; Ståhl, G. Generalized Hierarchical Model-Based Estimation for Aboveground Biomass Assessment Using GEDI and Landsat Data. *Remote Sens.* **2018**, *10*, 1832.
https://doi.org/10.3390/rs10111832

**AMA Style**

Saarela S, Holm S, Healey SP, Andersen H-E, Petersson H, Prentius W, Patterson PL, Næsset E, Gregoire TG, Ståhl G. Generalized Hierarchical Model-Based Estimation for Aboveground Biomass Assessment Using GEDI and Landsat Data. *Remote Sensing*. 2018; 10(11):1832.
https://doi.org/10.3390/rs10111832

**Chicago/Turabian Style**

Saarela, Svetlana, Sören Holm, Sean P. Healey, Hans-Erik Andersen, Hans Petersson, Wilmer Prentius, Paul L. Patterson, Erik Næsset, Timothy G. Gregoire, and Göran Ståhl. 2018. "Generalized Hierarchical Model-Based Estimation for Aboveground Biomass Assessment Using GEDI and Landsat Data" *Remote Sensing* 10, no. 11: 1832.
https://doi.org/10.3390/rs10111832