# A Spectral Fitting Algorithm to Retrieve the Fluorescence Spectrum from Canopy Radiance

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## Abstract

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_{2}bands and spectrally-integrated values). Further, the algorithm is employed to process experimental field spectroscopy measurements collected over different crops during a long-lasting field campaign. The reliability of the retrieval algorithm on experimental measurements is evaluated by cross-comparison with F values computed by an independent retrieval method (i.e., SFM at O

_{2}bands). For the first time, the evolution of the F spectrum along the entire growing season for a forage crop is analyzed and three diverse F spectra are identified at different growing stages. The results show that red F is larger for young canopy; while red and far-red F have similar intensity in an intermediate stage; finally, far-red F is significantly larger for the rest of the season.

## 1. Introduction

_{2}bands are considered to be more suitable to detect F because these spectral features are broader and deeper (darker), but a complex atmospheric correction is required to retrieve F. This is challenging for satellite and airborne remote sensing data, but it does not represent an issue for field spectroscopy measurements.

## 2. F Spectrum Retrieval Approach

_{0}) and the sensor view direction (Ω). Additionally, the Equation (1) considers isotropic R and F and neglects a possible adjacency contribution from surrounding, which is a reasonable assumption for TOC observations. The main idea underlying the proposed algorithm consists in a simplified modeling of the F spectrum (two parameters), leveraging on the spectral information included in the canopy R as a proxy of the F reabsorption and scattering. Figure 1 summarizes the main steps involved in the retrieval algorithm that basically consists in: (i) modeling the reflectance; (ii) modeling the fluorescence; (iii) estimating actual R and F spectra by means of an iterative optimization technique.

_{2}bands, are excluded in this first stage, since the spline is in the end targeted at modeling reflectance (free of F). Thus, a smooth function (R*fit) is initially obtained, considering R* but avoiding O

_{2}bands and, from this, the reflectance is finally estimated iteratively in the optimization algorithm (consistently with F).

_{RED}) and far-red (F

_{FAR-RED}) fluorescence emission (Equations (3)–(6) respectively). Each Lorentzian peak is characterized by one single state parameter only (${x}_{1}$, ${x}_{2}$) which modulates the overall peak intensity. The other two parameters required to describe the spectral distribution of each peak (u

_{RED}and u

_{FAR-RED}) are left constant to their characteristic value (i.e., wavelength of maximum emission and spectral width). The F spectrum obtained in a first step by combining F

_{RED}and F

_{FAR-RED}is further adjusted by a spectrally-variable correction factor (1-R) (Figure 3) which is intended as a proxy of canopy reabsorption (red wavelengths) and scattering (far-red wavelengths). The [1-(1-R)] term could be simply reduced to R, but the extended notation is maintained in Equation (2) because the element-wise product (Hadamard product) between the sum of F

_{RED}and F

_{FAR-RED}and the (1-R) term explicitly provides the F radiance reabsorbed by the canopy. The correction factor is mainly introduced to simplify the overall modeling of the red and far-red peaks, including the valley in between. This spectral region is particularly critical and difficult to model because: (i) there is a substantial contribution of both red and far-red peaks; and (ii) it corresponds to reflectance red-edge wavelengths characterized by a sharp transition from strong absorption (red) to scattering (far-red). Similarly, it is reasonable to assume that F reabsorption has a progressive decrease from red toward far-red wavelengths, introducing a spectrally-variable effect that has a complex behavior in this spectral region.

_{1}and x

_{2}coefficients are properly adjusted in the iterative optimization to disentangle F from the reflected radiance.

_{2}bands and the spectrally-integrated value. The Matlab source code of the present algorithm is available for download through the git repository https://gitlab.com/ltda/flox-specfit.

## 3. Theoretical and Experimental Data

#### 3.1. RT Simulations

^{−2}) are considered, while all the possible combinations between the two variables levels are simulated. The other model parameters are left to their default values as suggested in the input_data_default.xlsx file available in the SCOPE model software package. This permits us to obtain a wide range of different F spectra, characterized by low to high F values and different spectral behaviors. Typical atmospheric conditions for clear-sky days during summer at mid-latitudes are considered to run MODTRAN5, specifically the following values are used: aerosol optical thickness of 0.1, a mid-latitude summer atmospheric model, rural aerosol model, a water content of 2.0 (g/cm

^{2}), and a spectral resolution of 0.1 cm

^{−1}. The line-of-sight parameters are settled with a solar zenith angle of 30° and a nadir viewing angle, since most of the ground-based measurements, considered later on in this study, are collected with this nadiral configuration. A single atmospheric condition is simulated since atmospheric influence is mainly removed by the direct measurement of downwelling radiance when considering field spectroscopy measurements.

#### 3.2. Field Spectroscopy

## 4. Evaluation Metrics

_{2}bands and the spectrally-integrated value.

_{2}bands by the much more consolidated and widely used SFM approach. The SFM is intrinsically less prone to errors since the approach exploits only narrow spectral windows around the O

_{2}bands: (i) the mathematical functions used to fit F and R are simpler than those required by the full spectrum algorithm; (ii) the F contribution to the total signal is larger and therefore less prone to uncertainties.

## 5. Results

#### 5.1. Accuracy Quantification

_{2}-A and O

_{2}-B bands. In particular, simulated cases with intermediate and high values of Cab (40–80 µg cm

^{−2}) and LAI (3–7) show the better results at all wavelengths (cases 17–49). Conversely, few cases are instead characterized by a slight spectral mismatch in the valley in between the two F emission peaks, corresponding to the spectral region among 690–730 nm. This issue can be observed for cases n° 1–14 characterized by LAI values of 1–2. This behavior can be caused by a slight overcorrection introduced in the F spectrum by the retrieval algorithm, probably caused from a larger contribution of soil onto the canopy reflectance signature. The cases characterized by lower chlorophyll content (Cab of 20, 30 µg cm

^{−2}) also show an imperfect match of the red peak and a subsequent slight overestimation of the minimum of the valley.

^{2}ranges between 0.99 for F

_{FAR}-

_{RED}, F

_{760}and F

_{INT}to 0.94 in case of F

_{RED}. The slopes estimated by ordinary least square linear regression model for the different metrics are almost close to 1:1 line (range 0.87 to 1.10) and intercepts are close to zero for all the different metrics. The F

_{RED}is estimated with RMSE = 0.027 mW m

^{−2}sr

^{−1}nm

^{−1}(RRMSE = 2.3%), the F

_{FAR}-

_{RED}with RMSE = 0.061 mW m

^{−2}sr

^{−1}nm

^{−1}(RRMSE = 2.3%) and the spectrally-integrated value F

_{INT}with RMSE = 3.308 mW m

^{−2}sr

^{−1}nm

^{−1}(RRMSE = 1.8%). Similarly, the F values at the O

_{2}bands are estimated with RMSE = 0.023 mW m

^{−2}sr

^{−1}nm

^{−1}(RRMSE = 1.9%) at 687 nm; and RMSE = 0.011 mW m

^{−2}sr

^{−1}nm

^{−1}(RRMSE = 0.5%) at 760 nm.

_{FAR}-

_{RED}, F

_{760}, and F

_{INT}are retrieved more accurately compared to F

_{RED}and F

_{687}and this behavior is retained for the diverse SNRs considered (Table 2). In particular, the retrieval accuracy for all the F spectrum derived metrics resulted almost unaffected when considering “noise-free” and SNR = 1000 scenarios. The F

_{FAR}-

_{RED}, F

_{760}and F

_{INT}are estimated with similar accuracy also from spectra with SNR = 200 (RRMSE of 2.3%, 0.5%, 1.9% respectively), while retrieval error slightly increases when considering data with SNR = 50 (RRMSE of 2.7%, 1.3%, and 2.9%). In contrast, the retrieval accuracy for F

_{RED}and F

_{687}appear more affected by noise, in fact a first negligible decrease of accuracy is found for SNR = 200 (RRMSE = 2.6%, 2.3%) and a larger significative drop is observed when spectra with SNR = 50 are analyzed (RRMSE = 8.5%, 8.7%).

#### 5.2. F Spectrum from Experimental Field Spectroscopy

_{RED}and F

_{FAR-RED}at the O

_{2}bands.

_{2}bands estimated by the traditional SFM. The dots represent hourly average measurements (and standard deviation), between morning 10 am until afternoon 5 pm local time. The data refer to clear-sky days (50 days in total) extracted from the entire time series, from February to August, for different crops investigated (see Table 1) which result in a total of 303 points. The F values at the O

_{2}bands extracted from the F spectrum show a very close agreement with the corresponding values estimated at the O

_{2}bands by the SFM. The linear regression model is estimated minimizing χ

^{2}function calculated using both uncertainties on x and y axes (Deming fit), the uncertainty of the fitted parameters is computed using Monte Carlo assuming errors are Gaussian and centered. Specifically, F

_{760}shows a coefficient of determination close to one (R

^{2}= 0.997), the slope of the linear regression model of 0.96 and intercept −0.018, with RMSE = 0.102 mW m

^{−2}sr

^{−1}nm

^{−1}(RRMSE = 16.5%). Analogously, F

_{687}values also show a close agreement between the two retrieval methods (R

^{2}= 0.934), slope and intercept (1.104 and −0.021 respectively) are close to 1:1 line and the overall RMSE = 0.099 (RRMSE = 12.5%). Finally, an even better relationship is found for reflectance R

_{760}with RMSE = 0.002 (RRMSE = 0.4%) and for R

_{687}RMSE = 0.001 (RRMSE = 1.8%).

_{RED}, F

_{FAR-RED}, F

_{INT}values are selected to show the temporal evolution of fluorescence spectrum behavior in comparison with incoming irradiance (${L}^{\downarrow})$ and the MERIS Terrestrial Chlorophyll Index (MTCI) [59].

_{RED}at the photosystem level typically has a larger amplitude compared to F

_{FAR-RED}[47]. Furthermore, in these first days, the crop has a lower chlorophyll content, which on one hand implicates a general lower fluorescence emission, but on the other hand also implicates lower reabsorption. After a few days, the F spectrum develops towards an intermediate stage characterized by F

_{RED}and F

_{FAR-RED}with comparable values (DOYs = 68 − 80). In a more developed stage, later in the growing season, the F spectrum assumes a more common spectral behavior characterized by F

_{FAR-RED}significantly higher than F

_{RED}(DOYs = 80 − 145). In this condition, the larger chlorophyll content entails an overall larger amount of F emitted at photosystem level, but in contrast it also causes larger reabsorption at red wavelengths. It results that F

_{RED}develops weakly across the growing season, from 0.29 in the early days to 0.89 mW m

^{−2}sr

^{−1}nm

^{−1}when the canopy reaches its full development. Analogously F

_{FAR-RED}is also characterized by very low values about of 0.14 mW m

^{−2}sr

^{−1}nm

^{−1}in the early days, but conversely it quickly rises until reaching values larger than 4.0 mW m

^{−2}sr

^{−1}nm

^{−1}when the canopy is completely developed. The F

_{INT}ranges between an average value of 14 mW m

^{−2}sr

^{−1}up to more than 250 mW m

^{−2}sr

^{−1}nm

^{−1}when the canopy is mature.

## 6. Discussion

#### 6.1. Retrieval Algorithm Assumptions and Limitations

_{2}bands are retrieved better than the values corresponding to the F emission peaks maximum. The higher accuracy observed at the O

_{2}-A band compared to the O

_{2}-B (and F

_{FAR-RED}compared to F

_{RED}) is due to the simpler spectral behavior of F and R in this spectral region, together with the fact that the atmospheric absorption feature at 760 nm is broader and therefore more prone to detect fluorescence.

_{FAR-RED}, F

_{760}and F

_{INT}are faintly affected by noise and retrieval accuracy only slightly reduces from “noise-free” to lower SNR levels, whereas F

_{RED}and F

_{687}resulted more sensitive. The retrieval accuracy reported in this work (TOC level) cannot be evaluated against the threshold accuracy defined for the FLEX mission. However, we might consider that the 10% error requested for FLEX is defined to enable the further exploitation of F retrievals for plant stress detection and gross primary productivity modeling studies. Therefore, on the basis of this threshold, we might argue that the F spectra retrieved by the presented algorithm (uncertainty largely lower than 10%) can be used for successive physiological analysis.

#### 6.2. Observations on Experimental Field Data

_{2}absorption bands for the large data set considered that includes several crops observed along the season at different growing stages (young/mature canopies).

_{RED}larger than F

_{FAR-RED}, (ii) F

_{RED}comparable to F

_{FAR-RED}; (iii) F

_{FAR-RED}larger than F

_{RED}. These results are expected from a theoretical point of view considering the typical temporal development of agricultural canopies and the absorption, scattering, emission and reabsorption processes which determinate R and F spectra. The F

_{RED}is strongly reabsorbed by the chlorophyll within the leaf and the canopy, therefore the amount of F

_{RED}escaping the canopy is typically low, especially when the canopy is fully developed. The results show F

_{RED}with a lower seasonal trend because as soon as the chlorophyll content increases, the red F tends to saturate due to reabsorption. However, it must be considered that F

_{RED}is mainly released from photosystem II and thus it is directly linked to electron transport rate which is one of the key processes driving photosynthesis. Therefore, further studies need to be developed to better understand how to fully exploit F

_{RED}signal and two potential approaches can be delignated: (i) correcting F spectrum for reabsorption; (ii) exploiting canopy RTMs in which all these radiative processes are properly modeled. Conversely, F

_{FAR-RED}shows a more pronounced seasonal dynamic that mainly follows the progressive increase of canopy green biomass which in turn determinates a larger amount of absorbed photosynthetic active radiation from the canopy and thus higher F emission. However, F

_{FAR-RED}is composed by fluorescence emitted from both photosystems (photosystems II and I) and consequently its expected relationship with actual photosynthetic rate is weaker compared to F

_{RED}(corrected for reabsorption). All these considerations must be taken into account to understand the behavior of the spectrally-integrated fluorescence. In fact, the temporal dynamic of F

_{INT}results from the relative contributions of F

_{RED}and F

_{FAR-RED}peaks and how they are differently modulated across the season: red peak mostly contributes early, while far-red plays a major role for the remaining part of the growing season.

#### 6.3. Implications for Remote Sensing

## 7. Conclusions

_{2}bands estimated by an independent method. The F spectrum of canopies characterized by intermediate and large Cab and LAI values were estimated with better accuracy, while the soil effect introduces a slight disturbance when retrieving the F spectrum for sparse canopies. Further studies are suggested to improve this specific topic, while also permitting more accurate retrievals for heterogeneous scenarios. The proposed algorithm resulted robust also considering different levels of noise introduced in the RT simulations, suggesting the possibility to employ the algorithm on measurements collected by different instruments (different SNRs) or when spectral measurements are collected without optimal illumination conditions.

_{RED}and F

_{FAR-RED}have a different relative intensity. This behavior implies a diverse contribution of red and far-red onto F

_{INT}across the growing season, therefore we expect that F

_{INT}might provide diverse information according to the different canopy development stages. Specifically, the F

_{RED}contributes to the overall F

_{INT}more at the beginning of the season when the canopy is young, whereas F

_{FARE-RED}dominates the successive phenological phases. This behavior is mainly caused by the strong reabsorption of the F

_{RED}, and thus reabsorption correction methods shall be developed to enable further analyses which involve F

_{RED}and consequently F

_{INT}. In general, future studies are suggested to better investigate these observations, and a synergic analysis between spectral and eco-physiological measurements (e.g., CO

_{2}flux measurements) will help on this scope. These interesting observations provide the basis for more detailed studies about the dynamic of the F spectrum in relation to canopy biophysical variables and plant functioning. The possibility to observe the F spectrum at global scale is one of the challenges for the ESA’s upcoming FLEX mission and retrieval algorithms like the one described in this work can be used for the purpose of FLEX.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flow chart of the fluorescence spectrum retrieval algorithm based on the Spectral Fitting technique for top-of-canopy observations (field spectroscopy).

**Figure 2.**Apparent reflectance (

**R***,

**dark green line**), spline fit of apparent reflectance (

**R*fit**,

**blue line**) and true reflectance (

**R**,

**light green**). Inner boxes show details at O

_{2}bands.

**Figure 3.**Red and far-red F emission peaks initially modeled by the retrieval algorithm (

**gray peaks**); 1-R correction function (

**dashed green line**); resulting F spectrum (

**red line**).

**Figure 5.**Surface reflectance (

**left**), sun-induced fluorescence (

**middle**), upwelling radiance (

**right**) simulated for the different RT simulations.

**Figure 6.**Field measurement set-up over the different targets investigated: (

**A**) forage; (

**B**) alfalfa; (

**C**) corn; (

**D**) chickpea.

**Figure 7.**Retrieval of full F spectrum in the 650–780 nm spectral window from RT simulations: (

**left**) apparent reflectance (

**dark green**), reference reflectance from SCOPE (

**dashed blue**) and retrieved reflectance (

**light green**); (

**middle**) reference fluorescence from SCOPE (

**blue**) and retrieved fluorescence (

**red**); (

**right**) reference and modeled upwelling radiance ((

**dark and light blue**) respectively).

**Figure 8.**Comparison between F spectrum retrieved (

**red**) and reference (

**blue**) for all the simulated cases.

**Figure 9.**Scatterplots between reference and retrieved values at selected wavelengths: (

**A**) maximum of the red peak; (

**B**) maximum of far-red peak; (

**C**) spectrally integrated F; (

**D**) F at 687 nm (O

_{2}-B band); (

**E**) F at 760 nm (O

_{2}-A band). The red line is the linear least square fit whereas the dashed blue line represents the 1:1. Data refers to SNR = 1000.

**Figure 10.**Retrieval of full F spectrum from FLOX: (

**left**) apparent reflectance (

**dark green**) and retrieved reflectance (

**light green**); (

**middle**) full F spectrum; (

**right**) measured (

**dark blue**) and modeled (

**light blue**) canopy upwelling radiance and downwelling radiance (

**gray**).

**Figure 11.**Scatterplot between fluorescence (

**top**) and reflectance (

**bottom**) at 760 nm (

**left**) and 687 nm (

**right**) estimated by SFM and the novel F spectrum algorithm. The dots represent hourly average values (standard deviation) for the entire time series acquired on different crops during the entire season. The red line is the OLS linear regression model; dashed blue line the 1:1.

**Figure 12.**Temporal evolution of fluorescence and reflectance along growing season of forage crop between February to May: (

**top-left**) reflectance spectrum; (

**bottom-left**) fluorescence spectrum; (

**top-right**) MTCI and downwelling radiance; (

**bottom-right**) F

_{RED}, F

_{FAR-RED}and F

_{INT}values.

**Table 1.**Field spectroscopy measurements collected on different crops. The crop species, FLOX serial number, start/end dates of measurements for different targets and latitude/longitude of the different point measured are indicated.

Target | Instrument Id | Measurements Date | Location (Lat/Lon) | |
---|---|---|---|---|

Start | End | |||

Forage | JB-009-ESA | 21 February | 25 May | 42.828 N; 11.069 E |

Alfalfa | JB-009-ESA | 26 May | 12 July | 42.828 N; 11.076 E |

Corn | JB-009-ESA | 13 July | 31 August | 42.825 N; 11.068 E |

Chickpea | JB-013-ESA | 10 June | 42.818 N; 11.078 E |

**Table 2.**Statistical metrics in terms of slope, intercept, coefficient of determination (R

^{2}), Root Mean Square Error (RMSE), Relative Root Mean Square Error (RRMSE) of the linear regression model between reference and retrieved values considering different noise levels.

Noise | Statistics | F_{RED} | F_{FAR-RED} | F_{INT} | F_{687} | F_{760} |
---|---|---|---|---|---|---|

No noise | slope | 0.87 | 0.94 | 1.10 | 0.89 | 0.99 |

intercept | 0.15 | 0.21 | −19.20 | 0.11 | 0.02 | |

R^{2} | 0.940 | 0.995 | 0.995 | 0.985 | 0.999 | |

RMSE | 0.027 | 0.061 | 3.307 | 0.023 | 0.011 | |

RRMSE(%) | 2.3 | 2.3 | 1.9 | 1.9 | 0.5 | |

SNR = 1000 | slope | 0.87 | 0.94 | 1.10 | 0.89 | 0.99 |

intercept | 0.15 | 0.21 | −19.11 | 0.11 | 0.02 | |

R^{2} | 0.940 | 0.995 | 0.995 | 0.985 | 0.999 | |

RMSE | 0.027 | 0.061 | 3.308 | 0.024 | 0.011 | |

RRMSE(%) | 2.3 | 2.3 | 1.9 | 1.9 | 0.5 | |

SNR = 200 | slope | 0.88 | 0.95 | 1.11 | 0.90 | 0.99 |

intercept | 0.13 | 0.21 | −19.80 | 0.10 | 0.02 | |

R^{2} | 0.930 | 0.995 | 0.996 | 0.981 | 0.999 | |

RMSE | 0.031 | 0.061 | 3.267 | 0.027 | 0.012 | |

RRMSE(%) | 2.6 | 2.3 | 1.9 | 2.3 | 0.5 | |

SNR = 50 | slope | 0.69 | 0.95 | 1.08 | 0.71 | 0.99 |

intercept | 0.28 | 0.17 | −18.39 | 0.24 | −0.01 | |

R^{2} | 0.770 | 0.995 | 0.995 | 0.857 | 0.999 | |

RMSE | 0.105 | 0.048 | 4.762 | 0.106 | 0.025 | |

RRMSE(%) | 8.5 | 2.7 | 2.9 | 8.7 | 1.3 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cogliati, S.; Celesti, M.; Cesana, I.; Miglietta, F.; Genesio, L.; Julitta, T.; Schuettemeyer, D.; Drusch, M.; Rascher, U.; Jurado, P.;
et al. A Spectral Fitting Algorithm to Retrieve the Fluorescence Spectrum from Canopy Radiance. *Remote Sens.* **2019**, *11*, 1840.
https://doi.org/10.3390/rs11161840

**AMA Style**

Cogliati S, Celesti M, Cesana I, Miglietta F, Genesio L, Julitta T, Schuettemeyer D, Drusch M, Rascher U, Jurado P,
et al. A Spectral Fitting Algorithm to Retrieve the Fluorescence Spectrum from Canopy Radiance. *Remote Sensing*. 2019; 11(16):1840.
https://doi.org/10.3390/rs11161840

**Chicago/Turabian Style**

Cogliati, Sergio, Marco Celesti, Ilaria Cesana, Franco Miglietta, Lorenzo Genesio, Tommaso Julitta, Dirk Schuettemeyer, Matthias Drusch, Uwe Rascher, Pedro Jurado,
and et al. 2019. "A Spectral Fitting Algorithm to Retrieve the Fluorescence Spectrum from Canopy Radiance" *Remote Sensing* 11, no. 16: 1840.
https://doi.org/10.3390/rs11161840