# Retrieval of Maize Leaf Area Index Using Hyperspectral and Multispectral Data

^{1}

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## Abstract

**:**

_{d: 725; 715; 565}) for the hyperspectral dataset and the modified simple ratio (mSR

_{c: 740; 705; 865}) for the multispectral dataset of field spectra and the three band spectral index (TBSI

_{b: 665; 865; 783}) for the Sentinel-2 dataset. The relevant vector machine was the selected MLRA for the two datasets of field spectra (multispectral and hyperspectral) while the support vector machine was selected for the Sentinel-2 data. When using the LUT inversion technique, the minimum contrast estimation and the Bhattacharyya divergence cost functions were the best performing. The vegetation indices outperformed the other two approaches, with the TBSI

_{b}as the most accurate index (RMSE = 0.35). At the field scale, spectral data from Sentinel-2 can accurately retrieve the maize leaf area index in the study area.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area Description

#### 2.2. Characteristics of the Sampled Fields

^{2}parcels were established for monitoring and data collection. Table 1 summarizes the cropping practices in each sampled field.

#### 2.3. Data Collection

#### 2.3.1. Field LAI Data Collection

^{2}= 0.90; n = 30; p < 0.000) [1]. Additionally, an independent validation of the model was performed using data from the crop variety, ‘PAN 67’, grown in 2018 (R

^{2}= 0.914, n = 16, p < 0.000; data not published).

#### 2.3.2. Field Equivalent Water Thickness and Dry Mater Content

^{2}), the first full developed leaf, defined from the top to the bottom of each sampled plant, was cut and enclosed in a sealed plastic bag and brought to the laboratory inside a cooler. In the laboratory, a 2 cm diameter disc was cut in each leaf and the wet weight taken before the discs were dried at 65 °C in an oven until constant weight (dry weight). The Cw (cm) was then calculated according to the following equation:

_{w}(g) and D

_{w}(g) are, respectively, the fresh and dry weights of the leaf discs; A (cm

_{2}) is the area of the disc; and dw (1 g cm

^{−3}) is the water density. For the estimation of Cm, we used the following equation:

#### 2.3.3. Field Spectral Data Collection

#### 2.3.4. Spectral Data from Sentinel-2

#### 2.4. Data Analysis

#### 2.4.1. Field Spectral Data Pre-Processing

#### 2.4.2. Statistical Modelling of LAI Based on Vegetation Indices

#### 2.4.3. Statistical Modelling of LAI Based on Machine Learning Regression Algorithms

#### 2.4.4. Retrieval of LAI Based on Radiative Transfer Models

#### Simulation of the Look-Up-Table (LUT)

#### LUT Inversion for LAI Retrieval

#### 2.5. LAI Model Calibration, Cross Validation, and External Validation

#### 2.6. Model Assumptions, Accuracy Assessment, and Selection

^{2}). These measures were compared within the different retrieval approaches (VI, MLRA, and LUT inversion) and between them to decide which approach performed better in estimating LAI in the study area.

## 3. Results

#### 3.1. LAI Ground Measurements and Derived from Sentinel-2

#### 3.2. Vegetation Index Based LAI Models and Validation

#### 3.2.1. Field Spectral Data Resampled to 10 nm (FSP_10)

_{b}and mDI

_{d}fitted to a second degree polynomial function outperformed the other VI, showing consistent results for the LAI estimations when applied to the calibration, cross validation, and external validation datasets. For the external validation dataset, the mSR

_{b}presents R

^{2}= 0.80, comparatively higher than the mDI

_{d}(R

^{2}= 0.62), while the RMSE = 0.80 of mSR

_{b}shows lower accuracy than the mDI

_{d}with RMSE = 0.58. The three bands of VI, mSR

_{b,}and mDI

_{d}include bands within the same spectral regions of the green (565 nm) and red edge (715 nm, 725 nm, and 735 nm), suggesting reliability of the band optimization process for LAI estimation.

_{b}and the mDI

_{d}for the cross validation and external validation datasets. For the cross-validation dataset (Figure 6 left panel), the two indices exhibit good matching in the whole range of LAI values, but there is an underestimation for values greater than three. For the independent validation dataset (Figure 6 right panel), the two VIs estimate well the lower values of LAI, but for values above one, the estimation is poorer with an underestimation for the mDI

_{d}and an overestimation for the mSR

_{b}.

#### 3.2.2. Field Spectral Data Resampled to Sentinel-2 (FSP_S2)

_{c}, based on the polynomial model with bands located at the red edge (705 nm, 740 nm) and near infrared (865 nm) region of the spectra, presents better LAI estimates with R

^{2}≥ 0.82 and RMSE of 0.40, 0.43, and 0.62, respectively, for the calibration, cross validation, and external validation datasets. The mDI

_{c}presents very similar statistics for the calibration and cross validation datasets, but relatively lower R

^{2}(0.71) and higher RMSE (0.77) for the independent dataset. It is interesting to note that all the validated VIs derived from FSP_S2 were based on bands within the red edge and near infrared region of the spectra, which may suggest consistency of the band optimization and selection process.

#### 3.2.3. Sentinel-2 Spectral Data (SP_S2)

_{b}, mDI

_{c}, TBSI

_{c}) present very similar statistics and all were constructed with the same spectral bands, centred at red (665 nm), the red edge (783 nm), and near infrared (865 nm). However, the TBSI

_{b}fitted to a second degree polynomial function outperforms the other VIs with an RMSE = 0.32, R

^{2}= 0.79 and RMSE = 0.35, R

^{2}= 0.74 for calibration and cross validation, respectively.

_{b}, SR

_{,}TBSI

_{a}), with the TBSI

_{b}the outperforming one with an RMSE = 0.14, R

^{2}= 0.83 and RMSE = 0.18, R

^{2}= 0.76, respectively, for calibration and cross validation. This was expected because the SP_S2 was used as input data to derive the LAI_S2 products. However, careful interpretation is recommended given the lower correlation between the Field_LAI and the LAI_S2 (r = 0.69). The TBSI

_{b}was constructed with green (560 nm), red edge (705 nm), and near infrared (865 nm) bands.

#### 3.3. Machine Learning Regression Based LAI Models and Validation

#### 3.3.1. Field Spectral Data Resampled to 10 nm (FSP_10)

^{2}(0.77) was obtained with three band models while the minimum average R

^{2}(0.70) was acquired with the total number of bands that comprises the hyperspectral dataset (70 bands) (Figure A1, Appendix A). The three selected bands are centered at 565 nm, 675 nm, and 775 nm, with a 10 nm spectral resolution, corresponding to green, red, and red edge regions of the spectra, respectively.

#### 3.3.2. Field Spectral Data Resampled to Sentinel-2 Bands (FSP_S2)

^{2}= 0.51, when only one band is involved, to a maximum average of R

^{2}= 0.77 when three bands are involved in the modelling (Figure A2, Appendix A). There was no accuracy improvement with additional band inclusion in the models. The selected bands match to the red and red edge Sentinel-2 bands, centered at 665 nm, 705 nm, and 783 nm.

^{2}= 0.73, which is slightly poorer than the SVM values of RMSE = 0.48 and R

^{2}= 0.78. However, in the independent dataset, the RVM presents higher accuracy with RMSE = 0.53, compared to the SVM values of RMSE = 0.90. The bagging trees (BaT) algorithm is equally consistent, but with relatively higher RMSE (0.63 and 0.64) for the cross validation and external validation datasets, respectively.

#### 3.3.3. Sentinel-2 Spectral Data (SP_S2)

^{2}= 0.52), RTF (RMSE = 0.52, R

^{2}= 0.51), and GPR (RMSE = 0.50, R

^{2}= 0.49). However, as was expected, the combination, SP_S2 vs. LAI_S2, yielded very good statistics: RFF (RMSE = 0.22, R

^{2}= 0.64), RVM (RMSE = 0.23, R

^{2}= 0.61), and SVM (RMSE = 0.23, R

^{2}= 0.60).

#### 3.4. LUT Inversion Based LAI Estimation and Validation

#### LUT Inversion

^{2}, the K(x) = x(log(x)) – x, and the Bhattacharyya divergence, evidenced good LAI estimation. Nevertheless, the highest LAI estimation accuracy was achieved applying the Bhattacharyya divergence cost function to the SP_S2 vs. LAI_S2 dataset, resulting in an RMSE = 0.20 and R

^{2}= 0.70. Furthermore, this dataset combination showed better performance than the others with all the evaluated cost function. However, the SP_S2 vs. LAI_S2 dataset produced relatively lower association measure values, 0.65 ≤ R

^{2}≤ 0.72, compared to the range of 0.82 ≤ R

^{2}≤ 0.86 for FSP_S2 vs. Field LAI.

## 4. Discussion

^{2}), when inverting through the field spectral data (FSP_S2), and the Bhattacharyya divergence cost function, when using the SP_S2 dataset for the inversion. Similar findings were reported by [68] for LAI.

## 5. Conclusions

_{b}spectral index. This VI is built with three bands centered in the red, red edge, and near infrared regions of the electromagnetic spectrum with widely known biophysical significance for LAI estimation. The hyperspectral data (aggregated to 10 nm) failed to improve the LAI estimation accuracy comparatively to Sentinel-2 multispectral data. This finding is of particular relevance for the operational application of spectral data in crop monitoring, though Sentinel-2 data is freely available and presents good spatial and spectral resolutions. However, future research should consider using field LAI data acquired with high precision equipment, including other crop types and extensive sampling, in order to increase the ground truth data and, as a result, improve the accuracy of LAI retrieval.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Variation of the models’ R

^{2}as a function of the number of bands involved in machine learning modelling using the field spectral data resampled to 10 nm (FSP_10). The highest R

^{2}was achieved using three bands centered at 565 nm, 675 nm, and 775 nm.

**Figure A2.**Variation of the models’ R

^{2}as a function of the number of bands involved in machine learning modelling using the field spectral data resampled to Sentinel-2 bands (FSP_S2). The highest R

^{2}was achieved using three bands: Centered at 665 nm, 705 nm, and 783 nm.

**Figure A3.**Variation of the models’ R

^{2}as a function of the number of bands involved in machine learning modelling using the spectral data from Sentinel-2 images (SP_S2). The highest R

^{2}was achieved using two bands: Centered at 490 nm and 665 nm.

Authors | [56] | [67] | [84] | [85] | [2] | [86] | [14] |
---|---|---|---|---|---|---|---|

Model: PROSPECT 5 | |||||||

Ca+b: Clorofila a+b (µg/cm²) | 30–70 | 20–80 | 15–55 | 5–70 | 10–70 | 10–70 | 05–75 |

Equivalent water thickness (g/cm^{2}) | 0.01–0.06 | 0.01–0.04 | 0.01–0.02 | - | - | 0.01–0.03 | 0.002–0.05 |

N: Leaf Structural Parameter | 1–3 | 1 | 1.5–1.9 | 1.5 | 1.3–1.7 | 1–1.6 | 1.3–2.5 |

Car: Carotenoids (µg/cm²) | - | 1 | - | - | - | - | - |

Cbrown: Brown pigments (g/cm²) | - | 0.05 | - | - | - | 0–2 | - |

Cm: Dry matter content (g/cm²) | 0.008–0.025 | 0.0046 | 0.005–0.01 | 0.009 | 0.004–0.007 | 0.005–0.021 | 0.001–0.03 |

Model: 4SAIL | |||||||

LAI: Leaf area Index | 1–7 | 0.1–6 | 0.3–7.5 | 0–8 | 0–6 | 0–7 | 0.1–7 |

ALA: Leaf angle distribution (°) | 20–60 | 70, 57, 45 | 40–70 | 35 | 40–70 | 40–70 | 40–70 |

skyl: Diffuse/Direct light | - | 0.1 | - | - | - | 10 | 0.05 |

psoil: Soil Coefficient | - | 0.1 | 0.5–1.5 | - | 0.7–1.3 | 0–1 | 0–1 |

hspot: Hot spot | 0.5/LAI | 0.78, 0.40, 0.32 | 0.05–0.1 | 0.01 | 0.05–1 | 1–1.6 | 0.05–0.5 |

tts: Solar Zenit Angle (°) | −20–+80 | 51, 45, 33 | - | 30 | 35 | 20–50 | 22.3 |

tto: Observer zenit Angle (°) | 0–55 | 0 | - | 10 | 0 | 0 | 20.19 |

psi: Azimut Angle (°) | −120–+120 | 0 | - | - | 0 | 0 | 0 |

Crop/vegetation type | Wheat | Wheat | Rangelands | Maize, vegetables, sunflower, alfafa and vine | Maize and sugar beet | Maize, vegetables and alfafa | Maize, vegetables, alfafa, sunflower, vines |

Dataset | Modelling Approach | CV | EV | |||
---|---|---|---|---|---|---|

BP | JB | BP | JB | |||

FSP_10 | VI | mSRb | 0.95 | 13.4 | 0.12 | 6.18 |

mDId | 0.2 | 3.15 | 0.28 | 6.98 | ||

MLRA | RVM | 0.17 | 12.42 | 0.07 | 7.29 | |

VHGPR | 0.54 | 12.42 | 0.3 | 7.29 | ||

FSP_S2 | VI | mDIc | 0.08 | 5.9 | 0.17 | 4.9 |

mSRc | 0.52 | 13.59 | 0.81 | 2.89 | ||

MLRA | BaT | 0.02 | 12.42 | 0.41 | 7.29 | |

RVM | 0.13 | 12.42 | 0.2 | 7.29 | ||

SP_S2 vs. Field_LAI | VI | mNDc | 0.004 | 0.79 | ||

TBSIb | 0.65 | 6.76 | --- | |||

TBSIc | 0.0003 | 1.89 | ||||

MLRA | GPR | 0.41 | 2.99 | |||

RFF | 0.47 | 2.99 | --- | |||

SVM | 0.18 | 2.99 | ||||

SP_S2 vs. LAI_S2 | VI | SR | 0.77 | 1.22 | ||

TBSIa | 0.88 | 5.68 | --- | |||

TBSIb | 0.88 | 1.83 | ||||

MLRA | RFF | 0.65 | 10.5 | |||

RVM | 0.51 | 10.5 | --- | |||

SMV | 0.39 | 10.5 |

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**Figure 1.**Location of the sampling fields overlaid to a true color composite (Red, Green and Blue) of Sentinel-2 image (

**right**), and the boundaries of the country of Mozambique and the district of Vilankulo (

**left**).

**Figure 2.**Average monthly precipitation (Prec.) and temperature (Temp.) for the three years of data collection (2015, 2016, and 2018) and comparison with a reference period (1991–2015) [23].

**Figure 3.**Statistical modelling process with the leaf are index (LAI) and spectral data used in the calibration and validation. In the calibration section, the full line arrows indicate the process for the vegetation indices approach while the dashed arrows indicate the process for the machine learning approach.

**Figure 4.**Look-up table (LUT) generation through PROSAIL and LUT inversion process for LAI retrieval.

**Figure 5.**Boxplot of the field LAI for the three years: 2015 (n = 40), 2016 (n = 137), 2018 (n = 23), and derived from sentinel products, 2016/18_LAI_S2 (n = 22).

**Figure 6.**Comparisons between measured and estimated LAI with the two best performing spectral vegetation indices using the FSP_10 dataset for both cross validation (CV) and external validation (EV) datasets. mSR—modified simple ratio; mDI—modified difference index. The full line depicts the linear regression and the dashed line the linear regression through origin.

**Figure 7.**Comparisons of the measured and estimated LAI with the two best performing spectral vegetation indices using the FSP_S2 dataset for both cross validation (CV) and external validation (EV) datasets. mSR—modified simple ratio; mDI—modified difference index. The full line depicts the linear regression and the dashed line the linear regression through the origin.

**Figure 8.**Comparisons of the measured and estimated LAI with the best performing vegetation indices using the Sentinel-2 spectral data (SP_S2) with the field leaf area index (Field_LAI) and with Sentinel-2 leaf area index product (LAI_S2). mNDc—modified difference index; TBSI—three band spectral index; SR—simple ratio. The blue line depicts the linear regression and the black dashed line the linear regression through origin.

**Figure 9.**Comparisons of the LAI measured and estimated with the two best performing machine learning regression algorithms (MLRA) using the field spectra resampled to 10 nm (FSP_10). CV—cross validation, EV—external validation. RVM—relevance vector machine; VH-GPR—VH. Gaussian processes regression. The full line depicts the linear regression and the black dashed line the linear regression through origin.

**Figure 10.**Comparisons of the LAI measured and estimated with the two best performing machine learning regression algorithms (MLRA) using the field spectra resampled to Sentinel-2 bands (FSP_S2). CV—cross validation, EV—external validation. RVM—relevance vector machine; BaT—bagging tree. The blue line depicts the linear regression and the black dashed line the linear regression through origin.

**Figure 11.**Comparisons of the LAI measured and estimated with the best performing machine learning regression algorithms (MLRA) using the Sentinel-2 spectral data (SP_S2) with the field leaf area index (Field_LAI) and the Sentinel-2 leaf area index product (LAI_S2). GPR—Gaussian process regression; SVM—support vector regression; RFF—random forest Fitensemble; RVM—relevance vector machine. The full line depicts the linear regression and the dashed line the linear regression through the origin.

**Figure 12.**Comparisons of the measured and estimated LAI of the best LUT inversion cost functions using the field spectral data resampled to Sentinel-2 bands with the field leaf area index (FSP_S2 vs. F_LAI—first two plots), Sentinel-2 spectral data with field leaf area index data (SP_S2 vs. F_LAI—third and fourth plots), and Sentinel-2 spectral data with Sentinel-2 leaf area index product data (SP_S2 vs. LAI_S2—last two plots). The full line depicts the linear regression and the dashed line the linear regression through origin.

Item | Field 1 (2015) | Field 2 (2016) | Field 3 (2016) | Field 4 (2018) |
---|---|---|---|---|

Latitude | 21°56′24.19″ | 21°59′02.53″ | 21°56′20.88″ | 21°59′24.62″ |

Longitude | 35°07′24.19″ | 35°09′30.95″ | 35°07′18.12″ | 35°09′55.53″ |

Surface | 3 ha | 1 ha | 1 ha | 3 ha |

Soils | Sandy-loam soils | Sandy-loam soils | Sandy-loam soils | Sandy-loam soils |

Irrigation system and scheduling | Sprinkler: irrigation schedule conditioned by the water pumping availability | Drip irrigation: 3 days interval from V3–V8 and 6 days interval in the following stages | Drip irrigation: 3 days interval from V3–V8 and 6 days interval in the following stages | Central pivot: 5 days irrigation interval (with punctual constraints due power fluctuations) |

Variety | PAN 53, a medium maturity variety (125–140 days to harvest) | PAN 53 | PAN 53 | PAN 67, a medium maturity variety (120–130 days to harvest) |

Planting geometry | 0.5 × 0.25 cm; | 0.75 × 0.2 cm; | 0.75 × 0.2 cm; | 0.9 × 0.15 cm; |

Sowing and harvest dates | 9 June/30 October | 2 June/2 November | 4 July/4 December | 10 December 2017/25 April 2018 |

Crop yield | 2.5 Ton/ha | 5 Ton/ha | 5 Ton/ha | 4 Ton/ha |

Agricultural practices | Surface fertilization with Urea; manual weed removal; insect control of insect Sesamia monogriodes with cipermetrine at stages V8, VT and R | Deep fertilization with Guano (1200 kg/ha) before sowing; 3 applications of Mono-Ammonium Phosphate (MAP) (200 kg/ha) and Ammonium Sulphate (100 kg/ha) throughout the season | Deep fertilization with Guano (1200 kg/ha) before sowing; 3 applications of Mono-Ammonium Phosphate (MAP) (200 kg/ha) and Ammonium Sulphate (100 kg/ha) throughout the season | Deep fertilization with Guano (1200 kg/ha) early before sowing; 3 applications of Mono Amonium-Phosphate (MAP) (200 kg/ha) and Ammonium Sulphate (100 kg/ha) throughout the season; application of insecticides to control the Spodoptera frugiperda |

Crop Stage | Dates of Data Collection | |||
---|---|---|---|---|

Field 1 (2015) | Field 2 (2016) | Field 3 (2016) | Field 4 (2018) | |

V3 | 2 July | 28 Jun | 30 July | 9 January |

V6 | 20 July | 19 July | 12 August | 25/January |

V8 | 31 July | 30 July | 24 August | 5 February |

VT | 28 August | 17 August | 13 August | 4 March |

R1 | 3 September | 28 August | 27 September | |

RT | 16 September | 25 October |

Type of Index | Formulation | Original Index and Source |
---|---|---|

2 bands index | ||

ND | (ρ a − ρ b)/(ρ a + ρ b) | NDVI; [31] |

mND_{a} | [(ρ a − ρ b)/(ρ a + ρ b + 0.5)] * 1.5 | SAVI; [32] |

SR | ρ a/ρ b | SR; [33] |

mSR_{a} | ρ a/ρ b − 1 | CI Green; [34] |

DI | ρ a − ρ b | DI; [35] |

mDI_{a} | (1/ρ a) − (1/ρ b) | ARI; [36] |

3 bands index | ||

mDI_{b} | (ρ a − ρ b) − 0.2 * (ρ a − ρ c) | CARI; [37] |

mND_{b} | 2.5 * [(ρ a − ρ b)/(ρ a + 6 * ρ b + 7.5 * ρ c + 1)] | EVI; [38] |

mND_{c} | (ρ a − ρ b)/(ρ a + ρ b − ρ c) | VARI; [39] |

mSR_{b} | (ρ a − ρ b)/(ρ a − ρ c) | SIPI; [40] |

mSR_{c} | (ρ a − ρ b)/ρ c | PSRI; [41] |

mDI_{c} | [(ρ a − ρ b) − 0.2 * (ρ a − ρ c)] * (ρ a/ρ b) | mCARI; [42] |

mDI_{d} | [(1/ρ a) − (1/ρ b)] * ρ c | mARI; [36] |

mDI_{e} | [(ρ a + ρ b)/2] − ρ c | RVSI; [43] |

TBSI_{a} | (ρ a − ρ c)/(ρ b + ρ a) | [14] |

TBSI_{b} | (ρ a − ρ b + 2 ρ c)/(ρ a + ρ b + ρ c) | [44] |

TBSI_{c} | (ρ a − ρ b − ρ c)/(ρ a + ρ b + ρ c) | [45] |

_{Green}—green chlorophyll index; ARI—anthocyanin reflectance index; mARI—modified anthocyanin reflectance index; SIPI—structure insensitive pigment index; PSRI—plant senescence reflectance index; RVSI—reflectance vegetation stress index. Figure 3 presents the flowchart of the two statistical modelling approaches.

Algorithm | Source |
---|---|

Regression tree (RT) | [48] |

Random Forest (TreeBagger) (RFTB) | [49] |

Bagging trees (BaT) | [50] |

Relevance vector machine (RVM) | [51] |

Kernel ridge regression (KRR) | [52] |

Gaussian process regression (GPR) | [53] |

Variation Heteroscedastic Gaussian process regression (VH-GPR) | [54] |

Support Vector Regression (SVM) | [55] |

Random Forest (Fitensemble) (RFF) | [49] |

Model/Parameter | Abbreviation | Unit | Range of Values | Fixed Values |
---|---|---|---|---|

Prospect 5 model | ||||

Equivalent water thickness | Cw | cm | 0.001–0.030 | - |

Leaf chlorophyll content | Cab | µg/cm² | 5–40 | - |

Leaf structure coefficient | N | No dimension | 1–1.4 | - |

Dry matter content | Cm | g/cm^{2} | 0.001–0.008 | - |

Carotenoids content | Car | µg/cm² | - | 10 |

Brown pigments content | Cbrown | g/cm² | - | 5 |

4SAIL model | ||||

Leaf area index | LAI | m^{2}/m^{2} | 0.01–3.5 | - |

Average leaf angle | ALA | Degree | 20–60 | - |

Hot-spot size parameter | Hspot | m/m | 0.25–1 | - |

Diffuse/Direct light | Skyl | No dimension | - | 10 |

Soil Coefficient | Psoil | No dimension | - | 0.6 |

Solar Zenith Angle | Tts | Degree | - | 10 |

Observer zenith Angle | Tto | Degree | - | 5 |

Azimuth Angle | Psi | Degree | - | 0 |

**Table 6.**Goodness-of fit statistics of the best performing vegetation indices (VI), using different combinations of spectral and leaf area index data.

VI | Bands | CA | CV | EV | Equation Parameters | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

FSP_10 vs. Field_LAI | |||||||||||

Observations | 137 | 137 | 63 | ||||||||

mSR_{b} | 735;565;715 | 0.39 | 0.85 | 13.4 | 0.41 | 0.83 | 16.2 | 0.8 | 0.8 | 20.8 | a_{2} = 0.07; a_{1} = −1.056; a_{0} = 4.65 |

mDI_{d} | 725;715;565 | 0.4 | 0.84 | 14.7 | 0.42 | 0.82 | 15.3 | 0.58 | 0.62 | 21.6 | a_{2} = −319.73; a_{1} = −64.78; a_{0} = −0.59 |

mSR_{a} | 705;755 | 0.47 | 0.78 | 15.8 | 0.49 | 0.76 | 17.1 | 1.74 | 0.58 | 19.4 | m = −6.05; b = −0.17 |

FSP_S2 vs. Field_LAI | |||||||||||

Observations | 137 | 137 | 63 | ||||||||

mDI_{c} | 705;740;865 | 0.41 | 0.83 | 12.6 | 0.42 | 0.83 | 13.4 | 0.77 | 0.71 | 22.4 | a_{2} = −587.33; a_{1} = −75.51; a_{0} = 0.37 |

mSR_{c} | 740;705;865 | 0.4 | 0.84 | 12.4 | 0.43 | 0.82 | 14.7 | 0.62 | 0.82 | 23.6 | a_{2} = −21.63; a_{1} = 17.03; a_{0} = −0.63 |

mSR_{b} | 842;783;705 | 0.43 | 0.82 | 14.9 | 0.44 | 0.8 | 15.3 | 0.97 | 0.75 | 21.8 | m = −6.87; b = 3.33 |

SP_S2 vs. Field_LAI | |||||||||||

Observations | 22 | 22 | |||||||||

TBSI_{b} | 665;865;783 | 0.32 | 0.79 | 19.2 | 0.35 | 0.74 | 16.1 | - | - | a_{2} = 166.6; a_{1} = 199.5; a_{0} = 59.9 | |

mDI_{c} | 865;665;705 | 0.36 | 0.74 | 20.3 | 0.38 | 0.71 | 17.4 | - | - | m = 3.1; b = −0.76 | |

TBSI_{c} | 865;665;783 | 0.36 | 0.73 | 20.5 | 0.38 | 0.71 | 19.8 | - | - | m = 9.6; b = 2.6 | |

SP_S2 vs. LAI_S2 | |||||||||||

Observations | 22 | 22 | |||||||||

TBSI_{b} | 705;842;560 | 0.14 | 0.83 | 11.2 | 0.18 | 0.76 | 11.7 | - | - | k = −2.9; n = 0.83 | |

SR | 665;783 | 0.18 | 0.73 | 12.4 | 0.19 | 0.72 | 11.5 | - | - | m = 0.20; b = 0.9 | |

TBSI_{a} | 560;705;842 | 0.17 | 0.75 | 12.2 | 0.19 | 0.73 | 10.3 | - | - | m = −1.07; b = −0.16 |

_{0}, a

_{1}, a

_{2}—coefficients of the second degree polynomial function (y = a

_{2}X

^{2}+ a

_{1}X + a

_{0}); m, b—coefficients of the linear function (y = mX + b); k, n—coefficients of the exponential equation (y = ne

^{kx}); m, b—coefficients of the linear function (y = mX + b).

**Table 7.**Goodness-of-fit statistics for the LAI models of the best performing machine learning regression algorithms (MLRA) applied to different combinations of spectral and leaf area index.

MLRA/Type of Data | CV | EV | ||||
---|---|---|---|---|---|---|

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

FSP_10 vs. Field_LAI | ||||||

Observations | 137 | 63 | ||||

Support Vector Regression (SVM) | 0.48 | 0.77 | 14.9 | 0.67 | 0.65 | 17.5 |

Variation Heteroscedastic Gaussian Processes Regression (VHGPR) | 0.53 | 0.73 | 16.5 | 0.63 | 0.83 | 16.3 |

Relevance vector Machine (RVM) | 0.54 | 0.72 | 16.7 | 0.5 | 0.67 | 13.9 |

FSP_S2 vs. Field_LAI | ||||||

Observations | 137 | 63 | ||||

Support Vector Regression (SVM) | 0.48 | 0.78 | 14.9 | 0.9 | 0.72 | 24.9 |

Relevance vector Machine (RVM) | 0.52 | 0.73 | 26.3 | 0.53 | 0.62 | 15.9 |

Bagging trees (BaT) | 0.63 | 0.63 | 19.5 | 0.64 | 0.72 | 19.9 |

SP_S2 vs. Field_LAI (n = 22) | ||||||

Observations | 22 | - | ||||

Support Vector Regression (SVM) | 0.51 | 0.52 | 27.6 | - | - | |

Random Forest (Fitensemble) | 0.52 | 0.51 | 28.4 | - | - | |

Gaussian Processes Regression (GPR) | 0.5 | 0.49 | 27.4 | - | - | |

SP_S2 vs. LAI_S2 (n = 22) | ||||||

Observations | 22 | - | ||||

Random Forest (Fitensemble) | 0.22 | 0.64 | 14.1 | - | - | |

Relevance vector Marchine (RVM) | 0.23 | 0.61 | 14.8 | - | - | |

Support Vector Regression (SVM) | 0.23 | 0.6 | 14.9 | - | - |

**Table 8.**Goodness-of-fit statistics of look-up-table (LUT) inversion using the field spectral data aggregated according to Sentinel-2 bands and the field LAI (FSP_S2 vs. Field LAI); the spectral data of Sentinel-2 images and the LAI product derived from Sentinel-2 (SP_S2 vs. LAI_S2); the spectral data of Sentinel-2 images and the field LAI (SP_S2 vs. Field LAI), while considering different cost function (CF) algorithms.

CF Algorithm | FSP_S2 vs. Field LAI (n = 63) | SP_S2 vs. LAI_S2 (n = 22) | SP_S2 vs. Field LAI (n = 22) | ||||||
---|---|---|---|---|---|---|---|---|---|

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

K(x) = (log(x))^{2} | 0.43 | 0.82 | 18.2 | 0.28 | 0.71 | 17.9 | 0.53 | 0.6 | 28.9 |

K(x) = x(log(x)) − x | 0.49 | 0.82 | 20.7 | 0.24 | 0.7 | 15.4 | 0.56 | 0.6 | 30.6 |

Bhattacharyya divergence | 0.61 | 0.83 | 25.9 | 0.2 | 0.7 | 12.9 | 0.62 | 0.7 | 33.9 |

RMSE | 0.85 | 0.86 | 36.1 | 0.26 | 0.65 | 16.9 | 0.77 | 0.7 | 41.8 |

K(x) = −log(x) + x | 1.09 | 0.85 | 46.2 | 0.41 | 0.72 | 26.6 | 0.92 | 0.6 | 49.7 |

**Table 9.**Multi-comparison of LAI measured and estimated based on different combinations of modelling approaches and remote sensing validation data-set.

Spectral Data | Modelling Approach | Field LAI | LAI_S2 | ||
---|---|---|---|---|---|

RMSE | b * | RMSE | b * | ||

FSP_10 | VI | 0.42 | 0.99 | - | - |

FSP_S2 | VI | 0.43 | 1.0 | - | - |

FSP_10 | MLRA | 0.54 | 0.95 | - | - |

FSP_S2 | MLRA | 0.52 | 0.99 | - | - |

FSP_S2 | LUT | 0.43 | 1.11 | - | - |

SP_S2 | VI | 0.35 | 0.82 | 0.18 | 0.80 |

SP_S2 | MLRA | 0.51 | 0.78 | 0.22 | 0.62 |

SP_S2 | LUT | 0.53 | 1.20 | 0.20 | 0.88 |

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**MDPI and ACS Style**

Mananze, S.; Pôças, I.; Cunha, M.
Retrieval of Maize Leaf Area Index Using Hyperspectral and Multispectral Data. *Remote Sens.* **2018**, *10*, 1942.
https://doi.org/10.3390/rs10121942

**AMA Style**

Mananze S, Pôças I, Cunha M.
Retrieval of Maize Leaf Area Index Using Hyperspectral and Multispectral Data. *Remote Sensing*. 2018; 10(12):1942.
https://doi.org/10.3390/rs10121942

**Chicago/Turabian Style**

Mananze, Sosdito, Isabel Pôças, and Mario Cunha.
2018. "Retrieval of Maize Leaf Area Index Using Hyperspectral and Multispectral Data" *Remote Sensing* 10, no. 12: 1942.
https://doi.org/10.3390/rs10121942