# Chlorophyll Concentration Retrieval by Training Convolutional Neural Network for Stochastic Model of Leaf Optical Properties (SLOP) Inversion

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Stochastic Model of Leaf Optical Properties

- For each up (similarly down) state, the up (down), scattering and absorption probabilities are$$\begin{array}{cc}\hfill {P}_{\mathrm{absorption}}\left(\lambda \right)& =\frac{a\left(\lambda \right)}{a\left(\lambda \right)+s}\xb7(1-{e}^{-\left(a\right(\lambda )+s)\xb7L}),\hfill \end{array}$$$$\begin{array}{cc}\hfill {P}_{\mathrm{scattering}}\left(\lambda \right)& =\frac{s}{a\left(\lambda \right)+s}\xb7(1-{e}^{-\left(a\right(\lambda )+s)\xb7L}),\hfill \end{array}$$$$\begin{array}{cc}\hfill {P}_{\mathrm{up}\left(\mathrm{down}\right)}\left(\lambda \right)& =1-{P}_{\mathrm{absorption}}-{P}_{\mathrm{scattering}}.\hfill \end{array}$$
- For each scattered state the scattering and absorption probabilities are same as for the up and down states on the same layer. The up and down probabilities are$${P}_{\mathrm{down}}\left(\lambda \right)={P}_{\mathrm{up}}\left(\lambda \right)=\frac{1-{P}_{\mathrm{absorption}}-{P}_{\mathrm{scattering}}}{2}.$$
- The probability of direct reflection is given as a parameter and the probability of entering the first layer is $1-{P}_{\mathrm{direct}\mathrm{reflection}}.$
- The probability of going from absorbed state, reflected state, or emitted state to itself is 1.
- All other transition probabilities are 0.

- $a\left(\lambda \right)$ is the absorption coefficient,$$a\left(\lambda \right)=\frac{\pi}{4}\xb7(1+\frac{2{e}^{-\rho \left(\lambda \right)}}{\rho \left(\lambda \right)}+\frac{2({e}^{-\rho \left(\lambda \right)}-1)}{\rho {\left(\lambda \right)}^{2}})\xb7{d}_{\mathrm{chloroplast}}^{2}\xb7{c}_{\mathrm{chloroplast}}+{a}_{{H}_{2}O}\left(\lambda \right)\xb7{w}_{{H}_{2}O},$$$$\rho \left(\lambda \right)=\frac{6}{\pi}\xb7\frac{1}{{d}_{\mathrm{chloroplast}}^{2}\xb7{c}_{\mathrm{chloroplast}}}\sum _{\mathrm{pigments}}{a}_{i}\xb7{c}_{i},$$
- s is the scattering coefficient,
- L is the length of the light path, which is assumed to be the same as the thickness of the layer,
- ${c}_{i}$ are their concentrations,
- ${d}_{\mathrm{chloroplast}}$ is the diameter of chloroplast ($\mathrm{c}\mathrm{m}$) and ${c}_{\mathrm{chloroplast}}$ is its concentration (1/$\mathrm{c}{\mathrm{m}}^{3}$).

- Initialize the state vector: The initial state of the network of states, for example “all photons coming from above, none inside” corresponds to the following state vector:$${\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{k}\end{array}\right]}_{0}=\left[\begin{array}{c}1\\ 0\\ 0\\ \vdots \\ 0\end{array}\right]$$
- Matrix multiplication: The new state vector is the transition probability matrix multiplied with the old state vector:$${\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{k}\end{array}\right]}_{n+1}=\left[\begin{array}{cccc}{P}_{0,0}& {P}_{0,1}& \cdots & {P}_{0,k}\\ \vdots & \vdots & \cdots & \vdots \\ {P}_{k,0}& {P}_{k,1}& \cdots & {P}_{k,k}\end{array}\right]{\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{k}\end{array}\right]}_{n}$$
- Check the end condition: If the new and old state vectors are close enough to each other, end; otherwise, repeat from step 2. If$$\left(\right)open="|"\; close="|">{\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{k}\end{array}\right]}_{n+1}-{\left[\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ \vdots \\ {x}_{k}\end{array}\right]}_{n}$$

#### 2.2. Convolutional Neural Network

#### 2.3. Empirical Dataset

#### 2.4. Validation

- Calculation of chlorophyll a and b maps and comparison with the $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ index (TCARI = transformed chlorophyll absorption reflectance index, OSAVI = optimized soil adjusted vegetation index), a spectral index that is known for having a strong negative correlation with chlorophyll concentration [42],
- Comparison of simulated and measured $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ indices,
- Chlorophyll a/b ratio comparison to literature.

**Figure 3.**Median spectrum from each plot 1–7 described in Figure 5.

#### 2.4.1. Simulated Validation Dataset

- ${r}^{2}$ score between predicted and original values,
- Correlation coefficient between original and predicted values,
- MSE of their difference,
- Average difference,
- Standard deviation for the difference, and
- 95% confidence interval for the difference.

#### 2.4.2. Empirical Validation Dataset

#### 2.4.3. Comparison of Simulated and Measured $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ Indices

#### 2.4.4. Chlorophyll a/b

## 3. Results and Discussion

#### 3.1. Simulated Validation Dataset

#### 3.2. Empirical Validation Dataset

#### 3.3. Comparison of Simulated and Measured $\frac{\mathit{TCARI}}{\mathit{OSAVI}}$ Indices

#### 3.4. Chlorophyll a/b

#### 3.5. Other Observations

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1DCNN | One Dimensional Convolutional Neural Network |

CNN | Convolutional Neural Network |

FGI | Finnish Geospatial Research Institute |

FPI | Fabry–Pérot interferometer |

FWHM | Full width of the half maximum |

GPU | Graphics Processing Unit |

GSD | Ground Sampling Distance |

LAI | Leaf Area Index |

MSE | Mean Square Error |

OSAVI | Optimized Soil Adjusted Vegetation Index |

RTM | Radiative Transfer Model |

SLOP | Stochastic model of Leaf Optical Properties |

TCARI | Transformed Chlorophyll Absorption Reflectance Index |

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**Figure 1.**Diagram representation of stochastic model of leaf optical properties (SLOP). A tree leaf is assumed to have two major layers: a palisade layer and a spongy layer. In both layers, a photon can go straight through, be absorbed or scatter until it is absorbed or it moves to the previous or next layer. Adapted from [22].

**Figure 2.**Two spectra produced with SLOP. The graph on the left is produced with minimum values from Table 1 while the graph on the right is produced with maximum values.

**Figure 5.**Plots selected for in-depth analysis plotted over a false color image of the research forest. The wavelength bands used in making the figure are approximately 800 $\mathrm{n}$$\mathrm{m}$, 700 $\mathrm{n}$$\mathrm{m}$, and 500 $\mathrm{n}$$\mathrm{m}$. The edges show a rainbow artefact produced by some of the bands being empty in the plot. Plots 1, 5, and 6 are in a spruce-dominated plot, plots 2, 3, and 5 are in a birch-dominated plot and plot 4 is on a forest road. Plot 8 is a larger plot that consists mainly of birch forest, while having a significant amount of spruce on the border plots.

**Figure 6.**Convolutional neural network (CNN) training and testing results for chlorophyll a (

**a**–

**d**) and b (

**e**–

**h**). Figures (

**a**,

**e**) contain training and testing ${r}^{2}$ scores and figures (

**b**,

**f**) contain training and testing mean square error (MSE) scores. In figures (

**c**,

**g**) the estimated values are compared to the values in the validation dataset and in figures (

**d**,

**h**) their difference is computed and the matching normal distribution is calculated. Red and green lines represent the 95% confidence interval.

**Figure 7.**The chlorophyll a (top), b (middle), and $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ index (bottom) maps. The chlorophyll maps are calculated by feeding the hyperspectral data to the inverse SLOP model.

**Figure 8.**Chlorophyll a, b and $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ maps (left column) and correlation between the chlorophylls and the $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ index (right column) in plots 1–7. Figures (

**a**,

**b**) are related to plot 1, (

**c**,

**d**) to plot 2, (

**e**,

**f**) to plot 3, (

**g**,

**h**) to plot 4, (

**i**,

**j**) to plot 5, (

**k**,

**l**) to plot 6 and (

**m**,

**n**) to plot 7.

**Figure 9.**Correlation between chlorophyll and the $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ index in larger plot 8.

**Figure 10.**Comparison between simulated and empirical $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ indexes. The simulated index is calculated from data simulated using SLOP. (

**a**) corresponds to the research plot 1, (

**b**) to plot 2, (

**c**) to 3, (

**d**) to 4, (

**e**) to 5, (

**f**) to 6 and (

**g**) to 7.

**Figure 11.**Chlorophyll a values divided by chlorophyll b values for the whole forest (

**a**), for a birch (

**b**) and for a spruce (

**c**).

**Table 1.**Constants and variables used in making training, testing, and validation data with SLOP. The training, testing, and validation data consist of 500,000 spectra made with SLOP. Each spectrum is produced by taking a random value from each interval in the table and calculating SLOP for each specified wavelength. The spectra are divided into training, testing, and validation data randomly, with constant sizes.

Leaf Layer | |||
---|---|---|---|

Palisade | Spongy | ||

Variables | Chlorophyll a concentration ($\mathrm{m}$$\mathrm{g}$/$\mathrm{c}{\mathrm{m}}^{3}$) | [1, 10] | [0, 4] |

Chlorophyll b concentration ($\mathrm{m}$$\mathrm{g}$/$\mathrm{c}{\mathrm{m}}^{3}$) | [0.5, 5.5] | [0, 3] | |

$\beta $-carotene concentration ($\mathrm{m}$$\mathrm{g}$/$\mathrm{c}{\mathrm{m}}^{3}$) | [0, 1] | [0, 0.5] | |

Lutein concentration ($\mathrm{m}$$\mathrm{g}$/$\mathrm{c}{\mathrm{m}}^{3}$) | [0, 1] | [0, 0.5] | |

Violaxanthin concentration ($\mathrm{m}$$\mathrm{g}$/$\mathrm{c}{\mathrm{m}}^{3}$) | [0, 0.5] | [0, 0.25] | |

Neoxanthin concentration ($\mathrm{m}$$\mathrm{g}$/$\mathrm{c}{\mathrm{m}}^{3}$) | [0, 0.5] | [0, 0.25] | |

Water content ($\mathrm{c}{\mathrm{m}}^{3}$/$\mathrm{c}{\mathrm{m}}^{3}$) | [0.8, 1] | [0.1, 0.5] | |

scattering coefficient (1/$\mathrm{c}$$\mathrm{m}$) | [3.5, 5.5] | [1000, 1100] | |

Probability of direct reflection | [0.4, 0.06] | ||

Constants | Chloroplast diameter ($\mathrm{c}\mathrm{m}$) | 0.0005 | |

Chloroplast concentration (1/$\mathrm{c}$${\mathrm{m}}^{3}$) | $5\times {10}^{-9}$ | $6.7\times {10}^{-8}$ | |

Thickness ($\mathrm{c}\mathrm{m}$) | 0.0069 | 0.0069 |

**Table 2.**Wavelength and full width of the half maximum (FWHM) values of the measured hyperspectral data [41].

Wavelength (nm): | 507.60, 509.50, 514.50, 520.80, 529.00, 537.40, 545.80, 554.40, 562.70, 574.20, 583.60, 590.40, 598.80, 605.70, 617.50, 630.70, 644.20, 657.20, 670.10, 677.80, 691.10, 698.40, 705.30, 711.10, 717.90, 731.30, 738.50, 751.50, 763.70, 778.50, 794.00, 806.30, 819.70 |

FWHM (nm): | 11.2, 13.6, 19.4, 21.8, 22.6, 20.7, 22.0, 22.2, 22.1, 21.6, 18.0, 19.8, 22.7, 27.8, 29.3, 29.9, 26.9, 30.3, 28.5, 27.8, 30.7, 28.3, 25.4, 26.6, 27.5, 28.2, 27.4, 27.5, 30.5, 29.5, 25.9, 27.3, 29.9 |

Layer | Kernel/Pool Size and Activation | Filters/Units |
---|---|---|

Batch Normalization | ||

Conv1D | 3 ReLU | 64 |

Batch Normalization | ||

MaxPooling1D | 3 | |

Conv1D | 3 ReLU | 128 |

Batch Normalization | ||

Dropout (0.15) | ||

Flatten | ||

Dense | ReLU | 100 |

Dense | ReLU | 1 |

Optimiser: | Adam | |

Loss: | Mean square error | |

Accuracy: | ${r}^{2}$-score |

Chlorophyll a | Chlorophyll b | |
---|---|---|

${r}^{2}$ score between original and predicted values | 0.97 | 0.95 |

Correlation coefficient between original and predicted values | 0.98 | 0.98 |

MSE | 0.07 | 0.03 |

Average difference | 0.05 | 0.02 |

Standard deviation | 0.26 | 0.18 |

95% Confidence interval of the difference between predicted and original values | $[-0.46,0.56]$ | $[-0.34,0.37]$ |

**Table 5.**Correlation coefficients between predicted chlorophyll values and the $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ index in the eight researched plots.

Correlation Coefficient for Chlorophyll a and Index | Correlation Coefficient for Chlorophyll b and Index | Plot Description | |
---|---|---|---|

Plot 1 | −0.91 | −0.81 | Spruce |

Plot 2 | −0.90 | −0.83 | Birch |

Plot 3 | −0.84 | −0.61 | Birch |

Plot 4 | −0.81 | −0.61 | Forest road |

Plot 5 | −0.93 | −0.88 | Birch |

Plot 6 | −0.91 | −0.74 | Spruce |

Plot 7 | −0.88 | −0.55 | Spruce |

Plots 1, 6 and 7 | −0.90 | −0.70 | Spruce |

Plots 2, 3 and 5 | −0.89 | −0.77 | Birch |

Plots 1–7 | −0.87 | −0.68 | Spruce and Birch |

Plot 8 | −0.82 | −0.76 | Larger plot. Mainly birch, spruce on the border areas |

**Table 6.**Correlation coefficients between simulated and empirical $\frac{\mathrm{TCARI}}{\mathrm{OSAVI}}$ indexes in the seven investigated plots.

Correlation Coefficient between Simulated and Empirical $\frac{\mathbf{TCARI}}{\mathbf{OSAVI}}$ Indexes | Plot Description | |
---|---|---|

Plot 1 | 0.89 | Spruce |

Plot 2 | 0.83 | Birch |

Plot 3 | 0.75 | Birch |

Plot 4 | 0.68 | Forest road |

Plot 5 | 0.88 | Birch |

Plot 6 | 0.85 | Spruce |

Plot 7 | 0.81 | Spruce |

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**MDPI and ACS Style**

Annala, L.; Honkavaara, E.; Tuominen, S.; Pölönen, I.
Chlorophyll Concentration Retrieval by Training Convolutional Neural Network for Stochastic Model of Leaf Optical Properties (SLOP) Inversion. *Remote Sens.* **2020**, *12*, 283.
https://doi.org/10.3390/rs12020283

**AMA Style**

Annala L, Honkavaara E, Tuominen S, Pölönen I.
Chlorophyll Concentration Retrieval by Training Convolutional Neural Network for Stochastic Model of Leaf Optical Properties (SLOP) Inversion. *Remote Sensing*. 2020; 12(2):283.
https://doi.org/10.3390/rs12020283

**Chicago/Turabian Style**

Annala, Leevi, Eija Honkavaara, Sakari Tuominen, and Ilkka Pölönen.
2020. "Chlorophyll Concentration Retrieval by Training Convolutional Neural Network for Stochastic Model of Leaf Optical Properties (SLOP) Inversion" *Remote Sensing* 12, no. 2: 283.
https://doi.org/10.3390/rs12020283