# AM–GM Algorithm for Evaluating, Analyzing, and Correcting the Spatial Scaling Bias of the Leaf Area Index

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## Abstract

**:**

## 1. Introduction

## 2. Materials

#### 2.1. Simulated Data

#### 2.2. Site Data

## 3. Methods

#### 3.1. Mathematical Theory of the AM–GM Algorithm

#### 3.1.1. Holder’s Defect and AM–GM Inequality

#### 3.1.2. Calculation of LAI Scaling Bias

#### 3.2. Calculate Factor μ

#### 3.3. Simplify the AM–GM Algorithm

## 4. Results

#### 4.1. Validation Results of the AM–GM Algorithm

#### 4.2. Analysis of LAI Scaling Bias

- Spatial resolution.

- Model nonlinearity.

- Surface heterogeneity.

#### 4.3. Scaling Bias Calculated by the Simplified Algorithm

## 5. Discussion

#### 5.1. Algorithm Comparison

#### 5.2. The Influence of NDVI Aggregation on the AM–GM Algorithm

#### 5.3. Limitation of the AM–GM Algorithm

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Standard false-color images of two forest scenes simulated using the LESS model: (

**a**) Genhe scene; (

**b**) virtual scene.

**Figure 2.**Standard false-color images of the VALERI sites: (

**a**) Plan-de-Dieu; (

**b**) Puéchabon; (

**c**) Sud-Ouest; (

**d**) Les Alpilles; (

**e**) Barrax; (

**f**) Demmin; (

**g**) Haouz.

**Figure 3.**The flowchart for calculating, analyzing, and correcting the scaling bias of LAI using the AM–GM algorithm.

**Figure 4.**Schematic diagram of the scaling bias of LAI at coarse resolution in quantitative remote sensing. The LAI retrieval model $f$ is built at fine resolution. $DAT{A}_{n}$ represents the fine-resolution remote sensing data, and their corresponding LAI values at fine resolution ($LA{I}_{n}$ ) are retrieved using $f$. $LA{I}_{exa}$ represents the exact LAI at coarse resolution; this is acquired by averaging $LA{I}_{n}$. $DAT{A}_{m}$ represents the coarse-resolution remote sensing data, which is obtained by averaging $DAT{A}_{n}$. $LA{I}_{app}$ represents the approximate LAI at coarse resolution, which is obtained by applying model f to data $DAT{A}_{m}$. $bia{s}_{real}$ represents the LAI scaling bias of a coarse resolution pixel, indicated by the difference between $LA{I}_{app}$ and $LA{I}_{exa}$.

**Figure 5.**Numerical and spatial correction effectiveness of the AM–GM algorithm over two simulated scenes. The exact LAI ($LA{I}_{exa}$, m

^{2}/m

^{2}) at 9 m spatial resolution is obtained from directional gap probability at 1 m spatial resolution and then aggregated to 9 m spatial resolution; the approximated LAI at 9 m spatial resolution ($LA{I}_{app}$, m

^{2}/m

^{2}) is evaluated from the aggregated directional gap probability at 9 m spatial resolution. The corrected $LA{I}_{app}$ ($LA{I}_{cor}$, m

^{2}/m

^{2}) are achieved using the AM–GM algorithm at 9 m spatial resolution. The errors between $LA{I}_{app}$ and $LA{I}_{exa}$ over (

**a**) the Genhe scene and (

**c**) the virtual scene; the errors between $LA{I}_{cor}$ and $LA{I}_{exa}$ over (

**b**) the Genhe scene and (

**d**) the virtual scene. (

**a1**–

**d1**) spatial distributions of $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{exa}$ and $LA{I}_{cor}$ over the Genhe scene; (

**a2**–

**d2**) spatial distributions of $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{exa}$ and $LA{I}_{cor}$ over the virtual scene.

**Figure 6.**Numerical and spatial correction effectiveness of the AM–GM algorithm over three VALERI sites. The exact LAI ($LA{I}_{exa}$, m

^{2}/m

^{2}) at 500 m spatial resolution is obtained from reflectance at 20 m spatial resolution and then aggregated to 500 m spatial resolution. The approximate LAI ($LA{I}_{app}$, m

^{2}/m

^{2}) at 500 m spatial resolution is estimated from the aggregated reflectance at 500 m spatial resolution. The corrected $LA{I}_{app}$ ($LA{I}_{cor}$, m

^{2}/m

^{2}) is achieved using the AM–GM algorithm at 500 m spatial resolution. The errors between $LA{I}_{app}$ and $LA{I}_{exa}$ over (

**a**) the Plan-de-Dieu site, (

**c**) the Puéchabon site, and (

**e**) the Sud-Ouest site; the errors between $LA{I}_{cor}$ and $LA{I}_{exa}$ over (

**b**) the Plan-de-Dieu site, (

**d**) the Puéchabon site, and (

**f**) the Sud-Ouest site. (

**a1**–

**d1**) spatial distributions of $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{exa}$ and $LA{I}_{cor}$ over the Plan-de-Dieu site; (

**a2**–

**d2**) spatial distributions of $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{exa}$ and $LA{I}_{cor}$ over the Puéchabon site; (

**a3**–

**d3**) spatial distributions of $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{exa}$ and $LA{I}_{cor}$ over the Sud-Ouest site.

**Figure 7.**Average of LAI scaling bias calculated using the AM–GM algorithm at multiple scales for each scene and site: (

**a**) based on the Beer–Lambert Law at two LESS simulated scenes, and (

**b**) based on the NDVI-LAI semi-empirical transfer function at three VALERI sites.

**Figure 8.**Spatial distribution maps of the LAI scaling bias and two factors ($Var\left(X\right)$ and ${\mu}_{\rho \_AM-GM}$ ) computed by the AM–GM algorithm based on the NDVI-LAI semi-empirical transfer function at 500 m resolution over three VALERI sites. (

**a1**–

**a3**) the local variance of directional gap probability at the Plan-de-Dieu site, the Puéchabon site, and the Sud-Ouest site, respectively. (

**b1**–

**b3**) the second derivative ${\mu}_{\rho \_AM-GM}$ absorbs the impact of all higher-order moments at the Plan-de-Dieu site, the Puéchabon site, and the Sud-Ouest site, respectively. (

**c1**–

**c3**) the scaling bias of LAI at the Plan-de-Dieu site, the Puéchabon site, and the Sud-Ouest site, respectively.

**Figure 9.**The statistical relationships between $\mathrm{ln}\left(G\left[p\left({\rho}_{NIR}^{},{\rho}_{R}^{}\right)\right]\right)$ and $\mathrm{ln}\left(p\left(A\left[{\rho}_{NIR}^{}\right],A\left[{\rho}_{R}^{}\right]\right)\right)$ investigated at multiple resolutions, (

**a**) 200 m, (

**b**) 500 m, (

**c**) 1000 m, and (

**d**) 1500 m, over Les Alpilles, Barrax, Demmin, and Haouz cropland sites.

**Figure 10.**Correction of LAI scaling bias using the simplified AM–GM algorithm at multiple resolutions (200 m, 500 m, 1000 m, and 1500 m) over the Sud-Ouest site. Panels (

**a**–

**d**) show the errors between $LA{I}_{app}$ and $LA{I}_{exa}$ before correction, while panels (

**e**–

**h**) show the errors after correction.

**Figure 11.**Spatial distributions of the exact LAI ($LA{I}_{exa}$), approximate LAI ($LA{I}_{app}$ ), corrected LAI ($LA{I}_{cor}$ ), and the differences between $LA{I}_{cor}$ and $LA{I}_{exa}$ at multiple resolutions (200 m, 500 m, 1000 m, and 1500 m) over the Sud-Ouest site. (

**a1**–

**d1**) represent $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{cor}$ and $LA{I}_{exa}$ at 200 m spatial resolution, respectively; (

**a2**–

**d2**) represent $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{cor}$ and $LA{I}_{exa}$ at 500 m spatial resolution, respectively; (

**a3**–

**d3**) represent $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and the difference between $LA{I}_{cor}$ and $LA{I}_{exa}$ at 1000 m spatial resolution, respectively; (

**a4**–

**d4**) represent $LA{I}_{exa}$, $LA{I}_{app}$, $LA{I}_{cor}$, and difference between $LA{I}_{cor}$ and $LA{I}_{exa}$ at 1500 m spatial resolution, respectively.

Sensor | Parameter Value | Optical Database | Parameter Value |
---|---|---|---|

Type | Orthographic | Larch Branch | Ground Measurements |

Width (pixels) | 45 | Brown Loam | Ground Measurements |

Height (pixels) | 45 | Larch Leaf | Ground Measurements |

Samples (/pixel) | 64 | Terrain | Parameter Value |

Spectral Bands | 482:60, 561.5:57, 654.5:37, 865:28 | Type | Plane |

Image Format | Spectrum | XSize (m) | 45 |

NoData Value | −1 | YSize (m) | 45 |

Width Extent (m) | 45 | BRDF Type | Lambertian |

Height Extent (m) | 45 | Optical Property | Brown Loam |

Four Components Product | Tick | ||

Observation | Parameter Value | Objects | Parameter Value |

View Zenith (°) | 0 | Single-Tree Models of Larch | Constructed by Xu et al. [36] |

View Azimuth (°) | 180 | ||

Sensor Height (m) | 30 | ||

Illumination and Atmosphere | Parameter Value | Advanced | Parameter Value |

Sun Zenith (°) | 30 | Minimum Iterations | 5 |

Sun Azimuth (°) | 90 | Number of Cores | 20 |

Sky-Type | SKY_TO_TOTAL | ||

Sky-Percentage | 0, 0 |

Site Name | Land Cover | Day of Year | Image Year | Spatial Resolution | Site Size | Location |
---|---|---|---|---|---|---|

Plan-de-Dieu | Crops | 181 | 2004 | 20 m | 3 km × 3 km | 44°11′N, 4°56′E |

Puéchabon | Mediterranean Forests | 163 | 2001 | 20 m | 3 km × 3 km | 43°43′N, 3°38′E |

Sud-Ouest | Nine Crops | 201 | 2002 | 20 m | 3 km × 3 km | 43°30′N, 1°14′E |

Les Alpilles | Crops | 204 | 2002 | 20 m | 3 km × 3 km | 43°48′N, 4°42′W |

Barrax | Crops | 195 | 2003 | 20 m | 5 km × 3 km | 39°40′N, 2°60′W |

Demmin | Crops | 164 | 2004 | 20 m | 5 km × 3 km | 53°53′N, 13°12′E |

Haouz | Crops | 73 | 2003 | 20 m | 3 km × 3 km | 31°39′N, 7°36′W |

Site Name | Before Correction | After Correction | ||
---|---|---|---|---|

RMSE | Bias | RMSE | Bias | |

Genhe | 0.71 | −0.67 | 0.00 | 0.00 |

Virtual | 0.74 | −0.62 | 0.00 | 0.00 |

Plan-de-Dieu | 0.05 | −0.03 | 0.00 | 0.00 |

Puéchabon | 0.19 | −0.14 | 0.00 | 0.00 |

Sud-Ouest | 0.25 | −0.22 | 0.00 | 0.00 |

Spatial Resolution | 200 m | 500 m | 1000 m | 1500 m | |
---|---|---|---|---|---|

Parameters | |||||

$a$ | 0.052 | 0.089 | 0.056 | 0.043 | |

$b$ | 0.011 | 0.022 | 0.063 | 0.081 |

**Table 5.**Model nonlinearity factor ($\mu $) and input variable for the AM–GM and TSEM algorithms at 500 m resolution.

Site Name | $\mathit{\mu}$ | Input Variable | ||
---|---|---|---|---|

RMSE | Bias | RMSE | Bias | |

Plan-de-Dieu | 5.56 | −3.99 | 0.01 | −0.00 |

Puéchabon | 5.48 | −4.58 | 0.02 | −0.01 |

Sud-Ouest | 13.43 | 7.16 | 0.02 | 0.01 |

Site Name | Considering the Scaling Bias of NDVI | Ignoring the Scaling Bias of NDVI | ||
---|---|---|---|---|

RMSE | Bias | RMSE | Bias | |

Plan-de-Dieu | 0.00 | 0.00 | 0.18 | 0.01 |

Puéchabon | 0.00 | 0.00 | 0.49 | 0.04 |

Sud-Ouest | 0.00 | 0.00 | 0.61 | −0.08 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, J.; Sun, R.; Xiao, Z.; Zhao, L.; Xie, D.
AM–GM Algorithm for Evaluating, Analyzing, and Correcting the Spatial Scaling Bias of the Leaf Area Index. *Remote Sens.* **2023**, *15*, 3068.
https://doi.org/10.3390/rs15123068

**AMA Style**

Zhang J, Sun R, Xiao Z, Zhao L, Xie D.
AM–GM Algorithm for Evaluating, Analyzing, and Correcting the Spatial Scaling Bias of the Leaf Area Index. *Remote Sensing*. 2023; 15(12):3068.
https://doi.org/10.3390/rs15123068

**Chicago/Turabian Style**

Zhang, Jingyu, Rui Sun, Zhiqiang Xiao, Liang Zhao, and Donghui Xie.
2023. "AM–GM Algorithm for Evaluating, Analyzing, and Correcting the Spatial Scaling Bias of the Leaf Area Index" *Remote Sensing* 15, no. 12: 3068.
https://doi.org/10.3390/rs15123068