# Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}= 0.76 and 0.78, RMSE = 0.0016 g/cm

^{2}and 0.0017 g/cm

^{2}, respectively for the pooled datasets), and their results were superior to the corresponding single Vis, 2D matrices, and two machine learning methods established with the same VI combinations.

## 1. Introduction

_{w}) [19,20,21]. This can lead to serious ill-posed problems in physics-based LMA inversion [22,23,24]. To improve its LMA estimation, Féret et al. [25] determined the range of 1700 to 2400 nm for LMA retrieval through the PROSPECT model. Qiu et al. [4] developed a new version of PROSPECT-g to consider the anisotropic scattering within leaves. The physical method has a rigorous mathematical foundation, yet its inversion can be complicated and time-consuming because of the numerous amounts of parameters [26]. Among the empirical methods, fitting relationships to vegetation indices (VIs) are one of the most popular. Several VIs have been developed for LMA estimation [27,28,29]. They require observations of only a few spectral bands and are very convenient to use. Constructing regression models between vegetation indices (VIs) and leaf physiological traits with machine learning (ML) algorithms have also achieved success [30,31]. Partial least squares regression (PLSR) was used by Serbin et al. [32] to establish a widely applicable LMA prediction model. However, a substantial amount of prior information is needed to train the regression equations, and their quality is found to be restricted by the training data, especially when the model is to be applied on a dataset independent from the training set [25].

## 2. Materials and Methods

#### 2.1. Data Description

#### 2.1.1. PROSPECT Model

_{ab}), carotenoid content (C

_{ar}), anthocyanin content (C

_{ant}), brown pigment content (C

_{brown}), leaf water content (C

_{w}), and leaf mass per area (LMA). The forward running of the model can generate accurate simulations of leaf hemispherical reflectance and transmittance from 400 nm to 2500 nm at a resolution of 1 nm. Leaf chemical traits can be inverted using PROSPECT-D by assigning the component content of the most similar spectrum to the measured electromagnetic spectrum.

#### 2.1.2. Description of the Synthetic Datasets

_{anth}and C

_{brown}were not considered here as they are insensitive to leaf reflectance in the SWIR domain based on sensitivity analysis and therefore set at 0 μg/cm

^{2}[36].

#### 2.1.3. Description of the Experimental Datasets

#### 2.2. 3D Matrix Approach for Estimating LMA

#### 2.2.1. VIs for Building the 3D Matrices

#### 2.2.2. Establishment of the 3D Matrices

_{1}-VI

_{2}-VI

_{3}space into several small cells of the same size (Figure 1). Each cell was linked to an LMA value, which corresponded to a small range of the three VI values. Thus, the smaller the size of the cells in the 3D matrix, the more simulations are needed for building up the whole 3D matrix. In the present study, the 3-VI space was partitioned into 100 × 100 × 100 combining both accuracy and efficiency. In this way, two LMA 3D matrices were generated by using the two VI combinations in Section 2.2.1.

#### 2.2.3. Estimation of the LMA

#### 2.3. Estimation of LMA through ML Algorithms

#### 2.4. Accuracy Evaluation

^{2}), root mean square error (RMSE), and normalized root mean square error (NRMSE) [45,46]:

## 3. Results

#### 3.1. 3D Matrices of VIs for LMA

^{2}) was approximately three times that of MSR-ND-R2300 (0.001 g/cm

^{2}). These results showed that MSR-ND-R2300 had a lower uncertainty than R1800-MND-D for LMA estimation.

#### 3.2. Evaluation of the LMA Matrices

#### 3.2.1. Evaluation against Synthetic Data

^{2}= 0.99, RMSE = 0.0005 g/cm

^{2}, NRMSE = 1.7%). The NRMSE of LMA retrieved by either of the two 3D matrices was less than 2%, indicating that the 3D matrices significantly optimized LMA estimation, and the effects of involving more than three VIs may not be obvious.

#### 3.2.2. Evaluation against the Experimental Datasets

^{2}= 0.78, RMSE = 0.0017 g/cm

^{2}, NRMSE = 10.5%). Thus, the matrix of MSR-ND-R2300 was considered a more robust and high-precision method than the relevant 2D matrices, single VIs, and ML algorithms.

^{2}= 0.76, RMSE = 0.0016 g/cm

^{2}, NRMSE = 9.9%). For both experimental datasets, the retrieval of LMA using matrix of R1800-MND-D yielded the highest accuracy (RMSE = 0.0016 g/cm

^{2}for MA, RMSE = 0.0015 g/cm

^{2}for LOPEX). To conclude, the matrix of R1800-MND-D is also a superior method for estimating LMA compared with the corresponding 2D matrices, single VIs, and the two ML models.

^{2}(0.85 for MA, 0.74 for LOPEX, and 0.78 for pooled dataset, Figure 6a,c,e in all experimental datasets, whereas R1800-MND-D achieved lower RMSE (0.0016 g/cm

^{2}for MA, 0.0015 g/cm

^{2}for LOPEX, and 0.0016 g/cm

^{2}for pooled dataset, Figure 6b,d,f. Moreover, the estimation bias of R1800-MND-D for a few samples with LMA around 0.005 g/cm

^{2}was relatively large, which may lead to the decrease of R

^{2}. Nevertheless, the RMSE of LMA estimation using the matrix of R1800-MND-D was still lower than that using MSR-ND-R2300. Consequently, the 3D matrix of R1800-MND-D was considered more promising than MSR-ND-R2300 for estimating LMA.

## 4. Discussion

#### 4.1. Improvements over Traditional VIs

_{w}in its sensitive bands, the detection of variations in LMA by VIs was limited [39]. The 3D matrix proposed in this study can effectively strengthen the relationship between VIs and LMA, and thus facilitate the retrieval of LMA. The range of LMA in the 3D matrices established in this study was between 0.001 and 0.036 g/cm

^{2}(Figure 2 and Figure 3), which covered the range of most vegetation species and growing stages according to the TRY database [47] and previous researches [32,48]. In addition, these 3D matrices had achieved good accuracy when estimating LMA on all the validation datasets (Table 4, Table 5, Table 6 and Table 7). The especially high accuracy obtained based on synthetic dataset was partly due to that the same Gaussian distribution used in generating synthetic datasets G and T, and that random and systematic noises were not considered. Yet it proved that the three dimensions were adequate for LMA estimation. Using the 3D matrix improved LMA retrieval compared with any single corresponding VI because inversion is based on similarity of the values of multiple VIs rather than an inaccurate regression relationship susceptible to C

_{w}absorption interference.

^{2}using the LOPEX dataset. Wu et al. [49] adopted the spectral invariants theory at the leaf level. The inversion accuracy of LMA was 0.0018 g/cm

^{2}(RMSE) using the same dataset. Validation using LOPEX showed that our method had a slightly higher inversion accuracy (RMSE = 0.0015 g/cm

^{2}).

#### 4.2. Impact of VI Selection in 3D Matrix Construction

_{ab}and LAI is hard to differentiate on the canopy scale, Xu et al. [50] proposed the use of two VIs sensitive to LAI and C

_{ab}in generating a matrix to decrease the influence of LAI and were then able to improve the accuracy of the model predictions. In the case of LMA, although C

_{w}affects the detection of the spectral signals of LMA in the SWIR domain, combining VIs related to LMA and C

_{w}would not result in the same degree of improvement in LMA estimation according to our prior experiments (not shown here). This is due to that the correlation between C

_{ab}and LAI is strong, while that between LMA and C

_{w}is much poorer [27]. As a result, a C

_{w}-sensitive VI, while acting as an axis of the matrix, does not assist retrieval of LMA because it has little contribution to LMA differentiation. The results in this study showed that a 3D matrix based on VIs related to LMA yields a high accuracy for LMA estimation, indirectly making up the absorption of LMA. This 3D matrix-based method can potentially be applied to invert other leaf parameters, particularly those with weak light absorption features, such as leaf carotenoid content.

#### 4.3. Sources of Error and Further Development of the 3D Matrix Approach

^{2}often accompanied by smaller errors, thus the average STD of MSR-ND-R2300 was relatively lower. The utilization of synthetic data set may introduce bias to the 3D matrix when applied in experimental datasets. Therefore, the R

^{2}of the two 3D matrices from experimental datasets was therefore consistent with the STD matrices, while the RMSE was not.

## 5. Conclusions

_{w}in the SWIR region. Thus, this study developed a novel approach for optimizing LMA estimation using a 3D VI matrix based on PROSPECT-D simulations. The results showed that compared with the corresponding single VIs and 2D matrices, the 3D matrix improved the estimation of LMA effectively. Compared with the machine learning models constructed using the same VI combinations, the 3D matrices estimate LMA more accurately. Between the two 3D matrices, R1800-MND-D achieved slightly better comprehensive performance than MSR-ND-R2300, with a lower RMSE for both experimental datasets. Given the physical principle used in constructing the 3D matrix, and that little prior information was involved in calibrating the models, the 3D matrix shows high potential for a stronger generalization capability. The proposed method can provide valuable guidance for reflecting vegetation growth, development stages, as well as physiological response to environmental stresses.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**3D matrix composed of the modified simple ratio (MSR), the normalized difference-type index (ND), and leaf reflectance at 2300 nm (R2300) using dataset T of 200,000 simulations (the converging point locates around MSR of 0.4353, ND of 0.0321, and R2300 of 0.0201).

**Figure 3.**3D matrix composed of leaf reflectance at 1800 nm (R1800), the modified normalized difference-type index (MND), and the difference-type index (D) using dataset T of 200,000 simulations (the converging point locates around R1800 of 0.0896, MND of −1, and D of 0.0078).

**Figure 6.**Estimated versus measured LMA based on matrix MSR-ND-R2300 (

**a**,

**c**,

**e**), and R1800-MND-D (

**b**,

**d**,

**f**) using experimental datasets LOPEX, MA, and POOLED.

**Table 1.**Distribution characteristics of the leaf parameters used in the PROSPECT-D simulations for generating the synthetic datasets (STD: standard deviation).

Title 1 | Min | Max | Mean | STD |
---|---|---|---|---|

N | 1 | 3.5 | 1.6 | 0.3 |

C_{ab} (μg/cm^{2}) | 0 | 110 | 32.81 | 18.87 |

C_{ar} (μg/cm^{2}) | 0 | 30 | 8.51 | 3.92 |

EWT (g/cm^{2}) | 0 | 0.07 | 0.0115 | 0.007 |

LMA (g/cm^{2}) | 0.001 | 0.04 | 0.01 | 0.07 |

Index | Index ID | Formula | Reference |
---|---|---|---|

Modified simple ratio-type index | MSR | $\frac{\left(\mathrm{R}2265-\mathrm{R}2400\right)}{\left(\mathrm{R}1620-\mathrm{R}2400\right)}$ | [28,40] |

Normalized difference-type index | ND | $\frac{\left(\mathrm{R}1368-\mathrm{R}1722\right)}{\left(\mathrm{R}1368+\mathrm{R}1722\right)}$ | [27,39] |

Single reflectivity-type index | R2300 | $\mathrm{R}2300$ | [28] |

Index | Index ID | Formula | Reference |
---|---|---|---|

Difference-type index | D | $\mathrm{R}2395-\mathrm{R}2295$ | [28] |

Modified normalized difference-type index | MND | $\frac{(\mathrm{R}2285-\mathrm{R}1335)}{(\mathrm{R}2285+\mathrm{R}1335-\mathrm{R}2400\times 2)}$ | [28,41] |

Single reflectivity-type index | R1800 | $\mathrm{R}1800$ | [32] |

**Table 4.**Retrieval performances of LMA using MSR-ND-R2300, corresponding 2D matrices and single VIs (the optimal results are indicated in bold).

VI or Matrix | Synthetic Dataset E | ||
---|---|---|---|

R^{2} | RMSE (g/cm^{2}) | NRMSE (%) | |

MSR | 0.71 | 0.0040 | 13.2 |

ND | 0.85 | 0.0022 | 7.3 |

R2300 | 0.39 | 0.0044 | 14.6 |

MSR-ND | 0.91 | 0.0017 | 5.6 |

MSR-R2300 | 0.93 | 0.0015 | 5.0 |

ND-R2300 | 0.98 | 0.0008 | 2.6 |

MSR-ND-R2300 | 0.99 | 0.0005 | 1.7 |

**Table 5.**Retrieval performances of LMA using R1800-MND-D, corresponding 2D matrices, and single VIs (the optimal results are indicated in bold).

VI or Matrix | Synthetic Dataset E | ||
---|---|---|---|

R^{2} | RMSE (g/cm^{2}) | NRMSE (%) | |

R1800 | 0.20 | 0.0051 | 16.9 |

MND | 0.69 | 0.0032 | 10.6 |

D | 0.57 | 0.0037 | 12.3 |

R1800-MND | 0.91 | 0.0017 | 5.6 |

R1800-D | 0.82 | 0.0024 | 7.9 |

MND-D | 0.95 | 0.0013 | 4.3 |

R1800-MND-D | 0.99 | 0.0006 | 2.0 |

**Table 6.**Corresponding to the MSR-ND-R2300 matrix, the coefficient of determination (R

^{2}), root mean square error (RMSE, in g/cm

^{2}), and normalized RMSE (NRMSE, %) of the estimated LMA from experimental datasets using the 3D matrix, corresponding 2D matrices, single VIs, support vector machine (SVM), and partial least-squares regression (PLSR) (the optimal results are indicated in bold).

VI or Matrix or ML | MA | LOPEX | POOLED | ||||||
---|---|---|---|---|---|---|---|---|---|

R^{2} | RMSE | NRMSE | R^{2} | RMSE | NRMSE | R^{2} | RMSE | NRMSE | |

MSR | 0.69 | 0.0030 | 20.1 | 0.45 | 0.0025 | 17.9 | 0.59 | 0.0029 | 17.9 |

ND | 0.79 | 0.0019 | 12.8 | 0.56 | 0.0025 | 17.9 | 0.68 | 0.0020 | 12.3 |

R2300 | 0.36 | 0.0024 | 16.1 | 0.14 | 0.0040 | 28.6 | 0.26 | 0.0028 | 17.3 |

MSR-ND | 0.49 | 0.0030 | 20.1 | 0.41 | 0.0024 | 17.1 | 0.45 | 0.0029 | 17.9 |

MSR-R2300 | 0.70 | 0.0021 | 14.1 | 0.48 | 0.0019 | 13.6 | 0.62 | 0.0021 | 13.0 |

ND-R2300 | 0.84 | 0.0023 | 15.4 | 0.73 | 0.0026 | 18.6 | 0.76 | 0.0024 | 14.8 |

MSR-ND-R2300 | 0.85 | 0.0018 | 12.1 | 0.74 | 0.0016 | 11.4 | 0.78 | 0.0017 | 10.5 |

SVM | 0.81 | 0.0022 | 14.8 | 0.69 | 0.0023 | 16.4 | 0.73 | 0.0022 | 13.6 |

PLSR | 0.82 | 0.0020 | 13.5 | 0.69 | 0.0022 | 15.7 | 0.74 | 0.0021 | 13.0 |

**Table 7.**Corresponding to R1800-MND-D matrix, the coefficient of determination (R

^{2}), root mean square error (RMSE, in g/cm

^{2}), and normalized RMSE (NRMSE, %) of the estimated LMA from experimental datasets using 3D matrix, corresponding 2D matrices, single VIs, support vector machine (SVM), and partial least-squares regression (PLSR) (the optimal results are indicated in bold).

VI or Matrix or ML | MA | LOPEX | POOLED | ||||||
---|---|---|---|---|---|---|---|---|---|

R^{2} | RMSE | NRMSE | R^{2} | RMSE | NRMSE | R^{2} | RMSE | NRMSE | |

R1800 | 0.09 | 0.0041 | 27.5 | 0.03 | 0.0054 | 38.6 | 0.07 | 0.0044 | 27.2 |

MND | 0.72 | 0.0021 | 14.1 | 0.47 | 0.0022 | 15.7 | 0.62 | 0.0021 | 13.0 |

D | 0.40 | 0.0019 | 12.8 | 0.28 | 0.0024 | 17.1 | 0.36 | 0.0020 | 12.3 |

R1800-MND | 0.81 | 0.0019 | 12.8 | 0.61 | 0.0017 | 12.1 | 0.73 | 0.0018 | 11.1 |

R1800-D | 0.51 | 0.0017 | 11.4 | 0.41 | 0.0020 | 14.3 | 0.48 | 0.0018 | 11.1 |

MND-D | 0.65 | 0.0022 | 14.8 | 0.46 | 0.0022 | 15.7 | 0.57 | 0.0022 | 13.6 |

R1800-MND-D | 0.83 | 0.0016 | 10.7 | 0.67 | 0.0015 | 10.7 | 0.76 | 0.0016 | 9.9 |

SVM | 0.68 | 0.0036 | 24.1 | 0.39 | 0.0038 | 27.1 | 0.57 | 0.0036 | 22.3 |

PLSR | 0.66 | 0.0031 | 20.7 | 0.37 | 0.0036 | 25.7 | 0.55 | 0.0032 | 19.8 |

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**MDPI and ACS Style**

Chen, Y.; Sun, J.; Wang, L.; Shi, S.; Gong, W.; Wang, S.; Tagesson, T.
Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices. *Remote Sens.* **2021**, *13*, 3761.
https://doi.org/10.3390/rs13183761

**AMA Style**

Chen Y, Sun J, Wang L, Shi S, Gong W, Wang S, Tagesson T.
Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices. *Remote Sensing*. 2021; 13(18):3761.
https://doi.org/10.3390/rs13183761

**Chicago/Turabian Style**

Chen, Yuwen, Jia Sun, Lunche Wang, Shuo Shi, Wei Gong, Shaoqiang Wang, and Torbern Tagesson.
2021. "Optimized Estimation of Leaf Mass per Area with a 3D Matrix of Vegetation Indices" *Remote Sensing* 13, no. 18: 3761.
https://doi.org/10.3390/rs13183761