# Exploring the Impact of Noise on Hybrid Inversion of PROSAIL RTM on Sentinel-2 Data

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. General Methodology

#### 2.2. PROSAIL

#### 2.3. Machine Learning Algorithms

#### 2.3.1. Random Forest

#### 2.3.2. Artificial Neural Networks

#### 2.3.3. Gaussian Processes

#### 2.4. Step 1: Sensitivity Analysis

#### 2.5. Step 2: Hyperparameter Tuning

#### 2.6. Step 3: RTM Inversion

#### 2.7. Impacts of Noise

## 3. Results

#### 3.1. Sensitivity Analysis

#### 3.2. Hyperparameter Tuning

#### 3.3. RTM Inversion

#### 3.4. Impacts of Noise

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Code Availability

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Spectral Variability in Response to Trait Variability

**Figure A1.**Neural networks training and validation error in function of epochs. Error bars represent the standard deviation of a 5-fold cross-validation procedure.

#### Appendix A.2. Artificial Neural Networks Mean Absolute Percentage Error in Function of Training Epochs

**Figure A2.**With this figure it is possible to visualize the variation in PROSAIL simulated S2-like data reflectance spectra when stabilizing all traits except for one. This shows that variations in Car are likely not measurable by Sentinel-2.

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**Figure 1.**The first step consists of identifying which biophysical variables can be detected by S2 while the second step uses Bayesian optimization to find the best parameters for the various models. Lastly, the third and fourth steps test the performance of the various algorithms for inverting simulated S2-like data using Radiative Transfer Models in pure and noisy conditions.

**Figure 2.**Flowchart representing the training and validating procedure of the experiments. All algorithms were trained with the same data to exclude any random effect. The noise is only added in the second part of the experiment when testing its effect on the error.

**Figure 3.**Scatterplot between each trait and band. To improve visualization, only a fraction of 100 samples were used in this plot.

**Figure 5.**The sensitivity index as a function of each trait. The first-order sensitivity is the sensitivity associated with each trait while the total order is the sum of the first order with the interactions.

**Figure 6.**Model performance on pure PROSAIL RTM simulation. While each model has its own color, the darker colors represent the training error. The data used for these plots is accessible in the Github repository provided at the end of this manuscript.

**Figure 7.**Detail of Figure 6 between 0 and 5%.

**Figure 8.**Effect of noise on model performance. While each model has its own color, the darker colors represent the training error. The data used for these plots is accessible in the Github repository provided at the end of this manuscript.

**Figure 9.**Detail of Figure 8 between 0 and 10% added noise.

**Table 1.**Overview of PROSAIL input parameters used. Trait combinations were sampled representatively using a Latin hypercube sampling approach across the range identified. For the ML sections, $Car$ was fixed at 10.

Model | Domain | Description | Parameter | Units | Range | |
---|---|---|---|---|---|---|

PROSPECT | Leaf | Leaf structure index | N | - | 1 | |

Chlorophyll a + b content | ${C}_{ab}$ | ug/cm${}^{2}$ | 0 | 120 | ||

Total carotenoid content | $Car$ | ug/cm${}^{2}$ | 0 | 60 | ||

Equivalent water thickness | ${C}_{w}$ | cm | 0.001 | 0.008 | ||

Dry matter content | ${C}_{m}$ | g/cm${}^{2}$ | 0.001 | 0.008 | ||

Brown pigments | ${C}_{brown}$ | - | 0 | |||

Total anthocyanin content | ${C}_{anth}$ | ug/cm${}^{2}$ | 0 | |||

SAIL | Canopy | Leaf area index | $LAI$ | - | 0 | 10 |

Average leaf slope | LIDF_{a} | ° | −0.35 | |||

Leaf distribution bimodality | LIDF_{b} | ° | −0.15 | |||

Hot spot parameter | hspot | - | 0.01 | |||

Soil | Soil reflectance | ${\rho}_{soil}$ | - | 0.5 | ||

Soil brightness factor | ${\alpha}_{soi}$ | - | 0.1 | |||

Positional | Solar zenith angle | tts | ° | 45 | ||

Sensor zenith angle | tto | ° | 45 | |||

Relative azimuth angle | phi | ° | 0 |

**Table 2.**Overview of the hyperparameters tested during the optimization. The final set of parameters used are given in Tables 4 and 5.

Model | Parameter | Data Type | Range |
---|---|---|---|

Random Forests | Number of trees | Integer | {50, 100, 150, …, 1000} |

Minimum samples node split | Continuous | [0; 0.5] | |

Minimum samples leaf node | Continuous | [0; 0.5] | |

Gaussian Processes | Number of optimizer restarts | Integer | {10, 20, 30, 40, 50, 60, 70, 80, 90, 100} |

Kernel functions | Categorical | radial basis function, rational quadratic, matérn, dot product | |

Artificial Neural Network | Number of hidden layers | Integer | 1 to 3 |

Number of neurons | Integer | {5, 10, 15, 20} | |

Activation functions | Categorical | Linear, Sigmoid, Tanh, Exponential, Softplus, ReLU, Softsign | |

Optimizer | Categorical | Adam, RMSprop, Adadelta | |

Multi-task Neural Network | Number of shared layers | Categorical | {1, 2} |

Number of single task layers | Integer | {1, 2} | |

Number of neurons (shared) | Integer | {5, 10, 15, 20} | |

Number of neurons (single) | Integer | {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} | |

Activation functions | Categorical | Linear, Sigmoid, Tanh, Exponential, Softplus, ReLU, Softsign | |

Optimizer | Categorical | Adam, RMSprop, Adadelta |

**Table 3.**Summary of Spearman’s $\rho $ correlation coefficients for different biophysical variables and S2 bands. The significance levels are represented as: $p\le 0.001$ (***); $p\le 0.01$ (**); $p\le 0.05$ (*); $p\le 0.1$ (.); $p>0.1$: no symbol.

Band | ${\mathit{C}}_{\mathbf{ab}}$ | ${\mathit{C}}_{\mathit{w}}$ | ${\mathit{C}}_{\mathit{m}}$ | $\mathit{LAI}$ | $\mathit{Car}$ |
---|---|---|---|---|---|

B2 | −0.10 (***) | 0.00 | −0.04 (.) | 0.2 (***) | −0.46 (***) |

B3 | −0.95 (***) | 0.03 | −0.02 | −0.04 (.) | −0.10 (***) |

B4 | −0.56 (***) | 0.02 | −0.02 | 0.01 | −0.02 |

B5 | −0.97 (***) | 0.03 | −0.02 | −0.04 | −0.01 |

B6 | −0.61 (***) | 0.00 | −0.19 (***) | 0.49 (***) | 0.01 |

B7 | −0.01 | −0.01 | −0.47 (***) | 0.82 (***) | 0.01 |

B8A | 0.01 | −0.02 | −0.47 (***) | 0.82 (***) | 0.01 |

B11 | 0.04 | −0.65 (***) | −0.49 (***) | 0.46 (***) | −0.01 |

B12 | 0.02 | −0.67 (***) | −0.69 (***) | 0.03 | −0.02 |

**Table 4.**Summary of selected hyperparameters. * on the multi-task network are related to the shared layers; task-specific parameters are given in Table 5.

Model | Parameter | Selected |
---|---|---|

Random Forests | Number of trees | 850 |

Minimum samples node split | 0.00053 | |

Minimum samples leaf node | 0.00286 | |

Gaussian Processes | Number of optimizer restarts | 80 |

Kernel functions | Rational Quadratic | |

Artificial Neural Network | Number of hidden layers | 3 |

Number of neurons | (20, 20, 15) | |

Activation functions | Softsign, Softsign, ReLU | |

Optimizer | Adam | |

Multi-task Neural network | Number of shared layers | 2 |

Trait-specific layers | 2 | |

Number of neurons * | (15, 20) | |

Activation functions * | Tanh, Exponential | |

Optimizer | Adam |

Model | Trait | Parameter | Selected |
---|---|---|---|

Multi-task Neural network | Cab | Number of neurons | (3, 6) |

Activation functions | Softplus, Sigmoid | ||

Cw | Number of neurons | (3, 4) | |

Activation functions | Softplus, Tanh | ||

Cm | Number of neurons | (8, 6) | |

Activation functions | ReLU, Sigmoid | ||

LAI | Number of neurons | (6,10) | |

Activation functions | Softsign, Softsign |

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**MDPI and ACS Style**

de Sá, N.C.; Baratchi, M.; Hauser, L.T.; van Bodegom, P. Exploring the Impact of Noise on Hybrid Inversion of PROSAIL RTM on Sentinel-2 Data. *Remote Sens.* **2021**, *13*, 648.
https://doi.org/10.3390/rs13040648

**AMA Style**

de Sá NC, Baratchi M, Hauser LT, van Bodegom P. Exploring the Impact of Noise on Hybrid Inversion of PROSAIL RTM on Sentinel-2 Data. *Remote Sensing*. 2021; 13(4):648.
https://doi.org/10.3390/rs13040648

**Chicago/Turabian Style**

de Sá, Nuno César, Mitra Baratchi, Leon T. Hauser, and Peter van Bodegom. 2021. "Exploring the Impact of Noise on Hybrid Inversion of PROSAIL RTM on Sentinel-2 Data" *Remote Sensing* 13, no. 4: 648.
https://doi.org/10.3390/rs13040648