# Can Vegetation Indices Serve as Proxies for Potential Sun-Induced Fluorescence (SIF)? A Fuzzy Simulation Approach on Airborne Imaging Spectroscopy Data

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## Abstract

**:**

_{fuzzy}and SIF

_{fuzzy-APAR}. The SIF emitted from the core of the photosynthesis and observed at the top-of-canopy is regulated by three major controlling factors: (1) light interception and absorption by canopy plant cover; (2) escape fraction of SIF photons (fesc); (3) light use efficiency and non-photochemical quenching (NPQ) processes. In our study, we proposed and validated a fuzzy logic modelling approach that uses different combinations of spectral vegetation indices (SVIs) reflecting such controlling factors to approximate the potential SIF signals at 760 nm and 687 nm. The HyPlant derived and field validated SVIs (i.e., SR, NDVI, EVI, NDVIre, PRI) have been processed through the membership transformation in the first stage, and in the next stage the membership transformed maps have been processed through the Fuzzy Gamma simulation to calculate the SIF

_{fuzzy}. To test whether the inclusion of absorbed photosynthetic active radiation (APAR) increases the accuracy of the model, the SIF

_{fuzzy}was multiplied by APAR (SIF

_{fuzzy-APAR}). The agreement between the modelled SIF

_{fuzzy}and actual SIF airborne retrievals expressed by R

^{2}ranged from 0.38 to 0.69 for SIF

_{760}and from 0.85 to 0.92 for SIF

_{687}. The inclusion of APAR improved the R

^{2}value between SIF

_{fuzzy-APAR}and actual SIF. This study showed, for the first time, that a diverse set of SVIs considered as proxies of different vegetation traits, such as biochemical, structural, and functional, can be successfully combined to work as a first-order proxy of SIF. The previous studies mainly included the far-red SIF whereas, in this study, we have also focused on red SIF along with far-red SIF. The analysis carried out at 1 m spatial resolution permits to better infer SIF behaviour at an ecosystem-relevant scale.

## 1. Introduction

^{2}, Japanese space agency’s (JAXA) in-orbit Greenhouse gases Observing Satellite (GOSAT) has a circular footprint of 10.5 km in diameter, ESA’s Tropospheric Monitoring Instrument (TROPOMI) onboard prosecutor of Sentinel-5P has the spatial resolution of 3 × 7 km, while National Aeronautics and Space Administration (NASA)’s Orbiting Carbon Observatory 2 (OCO-2) has a footprint size of 1.3 × 2.3 km. Apart from that: 1) spatial inconsistency in homogeneous and heterogeneous landscapes (e.g., boreal evergreen forests, the US Midwest cropland, the Indo-Gangetic wheat belt, etc.) [51] cause inaccuracy in the SIF–GPP relationship, 2) low signal-to-noise ratio (SNR) [52] and 3) low temporal revisit time (except GOSAT, 3 days) from 16 days (OCO-2, TROPOMI, Tan-Sat) to 29 days (GOME-2) fails to capture the short-term temporal diversity of structural, phenological and functional variability of plants for regional and local level studies.

_{760}). FCVI was defined as the difference between near-infrared (NIR) and broad-band (400–700 nm) reflectance acquired under stable Sun-canopy-observer geometrical conditions. Similarly, the NIRv index, which is the product of total scene NIR reflectance (NIR

_{T}), and the normalized difference vegetation index (NDVI) also worked as the proxy of SIF and were proved to have good agreement with large pixel GOME-2 SIF products [54]. Joiner et al. [48] also proposed a new methodology based on principal component analysis (PCA) to retrieve SIF in around 740 nm (SIF

_{740}) based on GOME-2 data with a spatial resolution of 0.5°. Gentine and Alemohammad [7] developed neural network (NN) architecture to reconstruct SIF (RSIF) from 500 meters Moderate Resolution Imaging Spectroradiometer (MODIS) Aqua reflectance products and compared it with GOME-2 SIF

_{740}data developed by Joiner et al. [48]. Following a similar path, Zhang et al. [50] trained the NN model over surface reflectance products of the MODIS data to develop contiguous SIF (CSIF) datasets and compared it with OCO-2 SIF products. Based on a good agreement between the SIF–GPP relationship, many studies have simulated SIF at moderate spatial and temporal resolutions using MODIS products [35,55,56]. Raychaudhuri [57], as well as Irteza and Nichol [58], simulated SIF from Hyperion data. However, due to the lack of adequate atmospheric corrections, the outcomes were not satisfactory.

_{760}), but it can also approximate the narrower red band SIF at 687 nm (SIF

_{687}). For the first time, we have quantitatively approximated SIF signals at both oxygen absorption bands through different combinations of SVIs. This study is also the first experimental evidence that SIF signals for both SIF

_{760}and SIF

_{687}can be approximated by using high spatial resolution (1 m) airborne imaging spectroscopic data that can better describe the natural variability of the investigated area.

_{760}(O

_{2}A/far-red) and SIF

_{687}(O

_{2}B/red) positions.

_{760}and SIF

_{687}retrieved from airborne data. To achieve the best replica of SIF from the integrated fuzzy model and test whether the inclusion of APAR makes the model more accurate, we adopted two different approaches: 1) direct assimilation of SVIs into the fuzzy logic model termed as SIF

_{fuzzy}, 2) injection of APAR into the SVIs-based fuzzy logic modelled outputs termed as SIF

_{fuzzy-APAR}. Furthermore, SIF

_{fuzzy}and SIF

_{fuzzy-APAR}were validated based on the HyPlant-derived actual SIF data at both oxygen absorption bands. We hypothesized that the modelled proxies of SIF will capture the structural and functional traits of diverse vegetation groups and are in strong agreement with actual SIF signals. Further, we also hypothesized that the inclusion of APAR into SIF

_{fuzzy-APAR}model will increase the model accuracy. We believe that the proposed method might be applicable for ground, UAV, airborne as well as spaceborne-based observations for both homogenous and heterogeneous ecosystems.

## 2. Material and Methods

#### 2.1. Site Description

#### 2.2. Airborne Data Acquisition

#### 2.3. Computation of SVIs, SIF, and APAR

_{687}/SIF O

_{2}B) and 760 nm (SIF

_{760}/SIF O

_{2}A) respectively were retrieved from the HyPlant FLUO module spectra based on the Spectral Fitting Method (SFM) introduced by Meroni and Colombo [67] and Meroni et al. [68] and optimized by Cogliati et al. [69,70]. The retrieval of SIF maps at 687 nm and 760 nm using the SFM method and data processing chain are broadly described in Bandopadhyay et al. [1].

_{687}and SIF

_{760}maps, as well as SVIs maps of SR, NDVI, EVI, NDVIre, and PRI, were validated based on the in-situ TOC reflectance and SIF measurements conducted on the same day of airborne campaign discussed broadly in Bandopadhyay et al. [1]. The validation of the HyPlant derived SVIs and SIF maps at both the 687 nm and 760 nm bands shows a good agreement with in-situ ground measured SVIs and SIFs (see Bandopadhyay et al. [1]). This valid agreement showed the authenticity of HyPlant derived SVIs and SIF maps which were further used in this study for fuzzy simulation.

^{−2}s

^{−1}was recorded between 13:00 and 14:00.

#### 2.4. Identification and Selection of Experimental Vegetation Groups

#### 2.5. Fuzzy Logic Modelling of SIF Proxy from Reflectance-Based Vegetation Indices

_{fuzzy}) were multiplied by APAR. Further, the final modelled outputs without (termed as SIF

_{fuzzy}) and with the injection of APAR (termed as SIF

_{fuzzy-APAR}) were compared and validated with the actual SIF at 687 and 760 nm maps based on validated airborne SIF data. The associated standard error (SE) and uncertainties (expressed by the root mean square error, RMSE) were also computed. The detailed methodology of the modelling approach is presented in Figure 3.

#### 2.5.1. Fuzzy Membership Transformation

#### 2.5.2. Fuzzy Overlay Operation

_{fuzzy}and the final APAR-Fuzzy Gamma modelled outputs as SIF

_{fuzzy-APAR}. The validation and agreement of both modelled outputs in reference to actual SIF signals at 760 nm and 687 nm have been conducted at the vegetation group scale along with associated error and uncertainty estimations.

#### 2.5.3. Experiment on Different Fuzzy Combinations

_{fuzzy}and SIF

_{fuzzy-APAR}.

#### 2.6. Validation of the Model, Error and Uncertainty Estimation

^{2}) and statistical significance (p-value) of the relationships between modelled and actual data. The analysed relationships were considered to be significant if the p-value obtained from the test was lower than 0.05 (with 95% confidence interval). In order to estimate the error and associated uncertainty between modelled and actual data, standard error (SE) and root mean square error (RMSE) have been estimated accordingly.

_{i}and y

_{i′}are the observed and predicted values, respectively.

## 3. Results

#### 3.1. Outcome of the Membership Maps

#### 3.2. Performance of SIF_{fuzzy}

_{fuzzy}simulation based on the integration of membership SVIs and fuzzy modelling under six different combinations (C1–C6) showed a wide diversity of signals over the experimental landscape (Figure 6). The C6 SIF

_{fuzzy}map (Figure 6F) that incorporates all the MF_SVIs shows a good consistency of signals over pine forests and meadows. The modelled signal over the peatland region is quite complex, whereas non-vegetated zones like forest clearings and post-agricultural lands are characterized with very low signals. The C6 SIF

_{fuzzy}map ranges between 0 and 0.86. Characterized with the same consistency of signals C5 SIF

_{fuzzy}map (Figure 6E) ranges between 0 and 0.91, whereas the C1 SIF

_{fuzzy}(Figure 6A) and C3 SIF

_{fuzzy}(Figure 6C) maps range between 0 and 0.92 in both scenarios. The simulated signal for C2 SIF

_{fuzzy}(Figure 6B) map that incorporates MF_SR and MF_EVI ranges between 0 and 0.95 and C4 SIF

_{fuzzy}(Figure 6D) map that incorporates MF_SR and MF_PRI ranges between 0 and 0.96. The distribution of modelled signals for C2 SIF

_{fuzzy}and C4 SIF

_{fuzzy}were very similar to previous combinations where pine forests and meadows were characterized with high signals and non-vegetated zones were characterized with low signals. The complexity of signals has been observed within peatland for all six combinations due to the wide heterogeneity of the vegetation groups. We have also observed clear differences in the absolute values of all the modelled maps (C1–C6) where C1 and C2 with the exclusion of PRI and inclusion of EVI showed lower pixel values, clearly visible mainly for forests. Whereas, intensities and contrasts are much higher for C3, C4, and C5 models with the inclusion of PRI, indicating the highest pixel values for forest ecosystems.

_{fuzzy}and actual SIF bands at SIF

_{760}and SIF

_{687}for all six combinations. However, the degree of agreement differs for different combinations (Table 4, Figure 7). The SIF

_{fuzzy}model under C4 (f(SR+PRI)) was identified as the best performing proxy combination for SIF

_{760}recorded with the highest R

^{2}of 0.69 and RMSE of 0.235 mW·m

^{−2}·sr

^{−1}nm

^{−1}(Table 4). The second-best performing proxy combination for SIF

_{760}was the SIF

_{fuzzy}model under C6 (f(NDVI+EVI+NDVIre+SR+PRI)) recorded with the R

^{2}of 0.62 and RMSE of 0.268 mW·m

^{−2}·sr

^{−1}nm

^{−1}. However, the simulated values of SIF

_{fuzzy}for these models (Figure 7A,E), as well as for C1 and C2 (Figure S2), were underestimated in reference to actual SIF signals at 760 nm. SIF

_{fuzzy}models for SIF

_{760}under C3 (f(NDVI+PRI)) and C5 (f(NDVI+EVI+PRI)) were recorded with the lowest RMSE of 0.184 and 0.193 mW·m

^{−2}·sr

^{−1}nm

^{−1}, respectively.

_{fuzzy}model under C6 (f(NDVI+EVI+NDVIre+SR+PRI)) has been identified as the best performing proxy combination for SIF

_{687}recorded with the highest R

^{2}of 0.92 and RMSE of 0.082 mW·m

^{−2}·sr

^{−1}nm

^{−1}(Table 4). The second-best performing proxy combination for SIF

_{687}recorded with the RMSE of 0.09 mW·m

^{−2}·sr

^{−1}nm

^{−1}was the SIF

_{fuzzy}model under C1 (f(NDVI+EVI)). The SIF

_{fuzzy}simulations under C3 (f(NDVI+PRI)), C4 (f(SR+PRI)) and C5 (f(NDVI+EVI+PRI)) models, although correlated very well with the measured SIF

_{687}(R

^{2}= 0.90), were recorded with a higher RMSE of 0.154, 0.109, 0.143 mW·m

^{−2}·sr

^{−1}nm

^{−1}, respectively, and tend to overestimate the simulated SIF

_{fuzzy}(Figure 7D,F and Figure S2). The model outputs of C2 (f(SR+EVI)) recorded with RMSE of 0.114 mW·m

^{−2}·sr

^{−1}nm

^{−1}tend to slightly underestimate the simulated SIF

_{fuzzy}(Figure S2), but the rate of underestimation and difference from the best C6 model outputs is very low.

_{760}(Figure 7C) and SIF

_{687}(Figure 7B), respectively. The agreements between modelled and actual data demonstrated that SIF

_{fuzzy}worked very well and can approximate the actual SIF signals at 760 nm and 687 nm with an error smaller than 10% for ecosystems where the maximum SIF values are close to 1.0 mW·m

^{−2}·sr

^{−1}nm

^{−1}. The diversity of SIF

_{fuzzy}signals from different vegetation groups have been observed from these agreements too (Figure 8 and Figure S3) and they correspond well to the diversity of SIF values shown in Bandopadhyay et al. [1].

_{fuzzy}under C3 and C6 combinations. The WDPS and RF gained moderate signals from the modelled data, whereas, due to no vegetation cover, HV is characterized with poor modelled signals. MMP was the only vegetation group within the grassland ecosystem that received the highest signals, whereas CM, PG, PVS3 received low signals from the modelled SIF

_{fuzzy}data. The signals obtained from the peatland ecosystem were quite complex. The AF and LBB were the two vegetation groups within peatland that received the highest modelled SIF

_{fuzzy}signals and RV, RVAF, SV, TM received moderate signals. The weakest modelled signal under both combinations received CF within the peatland vegetation group. The estimation of modelled SIF

_{fuzzy}signals for other combinations from 19 vegetation groups is provided in Supplementary Figure S3.

#### 3.3. Performance of SIF_{Fuzzy-APAR}

_{fuzzy-APAR}modelled data developed through the injection of APAR into the SIF

_{fuzzy}under six different combinations (C1–C6) showed a very prominent wide diversity of signals over different vegetation groups as well as over different ecosystems. The detailed outcome of the SIF

**including the SIF**

_{fuzzy-APAR}_{fuzzy-APAR}maps and the agreement between modelled and actual data at both SIF bands have been provided in the Supplementary Materials SM2.

## 4. Discussion

_{760}and SIF

_{687}with individual SVIs at vegetation group scale under different ecosystems using HyPlant data. Although there were quite good agreements between some of the SVIs and SIF, the relationships were not optimum. However, an integrated fuzzy logic modelling technique incorporating different airborne SVIs allowed us for the first time to develop a reliable proxy of SIF

_{760}and SIF

_{687}evident by strong agreement, statistical significances, and low uncertainties. The SIF is influenced by different factors such as light absorption and interception, canopy structure influencing the fluorescence escape, and light use efficiency partially related to NPQ [80,81]. NDVI, SR, EVI, and NDVIre are greenness indices that are related to light absorption and interception, whereas EVI and NDVIre better present the canopy structure, thus they can be more related to fesc, while PRI is the component related to the light use efficiency and NPQ processes [13,17]. Considering the above, we combined different SVIs to approximate the potential SIF signals by including indices reflecting different factors controlling SIF emission. We proceeded by combining the greenness indices such as NDVI and EVI (C1), as well as SR and EVI (C2) which covered SIF influencing factors related to photon interception and canopy structure, whereas addition of PRI to greenness indices NDVI and SR (C3, C4) covered also the light use efficiency and NPQ processes influencing the SIF. We also combined NDVI, EVI, and PRI (C5) and NDVI, SR, EVI, NDVIre, and PRI (C6) to cover all three SIF influencing factors to some extent. Through C1 and C2 combinations, the modelled SIF

_{fuzzy}explained between 38% and 55% of the variability in SIF

_{760}and from 85% to 89% for SIF

_{687}. By the C3 and C4 combinations, we approximated SIF

_{fuzzy}which explained from 61% to 69% and 90% of the variability in SIF

_{760}and SIF

_{687}, respectively. Through C5 and C6 combinations, our modelled SIF

_{fuzzy}explained from 51% to 62% and from 90% to 92% of SIF

_{760}and SIF

_{687}variability, respectively. The SIF

_{fuzzy}showed the lowest RMSE with a high value of R

^{2}in the C3 f(NDVI + PRI) model for SIF

_{760}, and C6 f(NDVI+EVI+NDVIre+SR+PRI) model for SIF

_{687}. SIF

_{760}is known to be influenced by canopy chlorophyll concentration and is having less influence on canopy structure in comparison to SIF

_{687}where fesc plays a key role in SIF emission [65,82]. This is probably why the addition of EVI did not improve model accuracy for SIF

_{760}. Whereas, the best result with the combination of five SVIs for SIF

_{687}showed the sensitivity and complexity of SIF

_{687}. It showed that the combination of different SVIs reflecting variable factors controlling SIF emission like photon interception, fesc, light-use efficiency, and NPQ process improved the approximation of SIF in our models. The remaining residuals (from 62% at C1 to 38% at C6 for SIF

_{760}and from 15% at C1 to 8% at C6 for SIF

_{687,}see Table 3) decreased with the increasing number of SVIs considered in the study. This may indicate that through the inclusion of different SVIs we were able to reflect the functional contribution of different factors controlling SIF at the top-of-canopy level. At the same time, relatively large residuals indicated that 100% prediction by this method is not possible as the fuzzy logic modelling process cannot mimic the functional contribution of all the different physiological processes influencing SIF emission (such as e.g., fluorescence quantum efficiency). This may be due to the different proportions and accuracy of SIF controlling factors covered through SVIs. We would assume that the contribution of such residuals may increase under the influence of stress factors over different vegetation types and hence the applicability of such a fuzzy logic modelling approach in SIF approximation under this condition might be limited. However, it has been evident that the ensemble of SVIs can approximate the SIF signals in more efficient and promising ways rather than individual SVIs representing different SIF emission factors and different vegetation traits.

_{fuzzy-APAR}was compared with SIF

_{687}and SIF

_{760}and it was observed to be best fitted to the outputs of the C1 model including NDVI and EVI (Table S5 in SM2 Supplementary Material). The inclusion of APAR slightly overestimated the SIF but also improved the R

^{2}in the case of SIF

_{760.}SIF and APAR both are energy flux, therefore an improvement of SIF

_{fuzzy}to SIF

_{760}signal by the injecting of APAR was expected and supported by, e.g., Yang et al. [13], who expressed SIF as a product of APAR and effective light use efficiency that directly connects to the fluorescence yield of the canopy. Moreover, this study includes a particular time image of the APAR and injects it with SIF

_{fuzzy}to boost the model. We assume that after some training the use of different APAR images composed with different irradiance values can be considered as a prime variable to scale the SIF

_{fuzzy}at a different time of the day. The overestimation of SIF

_{fuzzy-APAR}has been particularly observed for dense forest canopies (i.e., DF, SFPS, SFAG, SeFPS) in reference to SIF

_{760}which was in agreement with Jung et al. [85] and Forrester et al. [86]. For SIF

_{687}, the modelled overestimations of SIF

_{fuzzy-APAR}have been observed in overall vegetation groups. Despite this, the overestimation of SIF was minor and our model outputs showed that an integrated fuzzy logic model can replicate the actual SIF values at both oxygen absorption bands and the derived model output values were also reasonable in reference to actual ranges. The existing studies that simulate SIF signals from reflectance spectra mostly focused on the far-red SIF range (ranging from around 740 nm to 760 nm) [35,50]. While, our model is proved to approximate SIF signals not only in the far-red region but also in a narrower SIF

_{687}.

_{fuzzy-APAR}produced a higher correlation (observed by improved R

^{2}values) with actual SIF signals in comparison to SIF

_{fuzzy}under all C1–C6 combinations. The SIF

_{fuzzy-APAR}or SIF

_{fuzzy}not only have an agreement with the actual SIF signals at 760 nm and 687 nm but also well represent the diversity of SIF signals within different homogenous and heterogenous vegetation groups.

_{fuzzy}and SIF

_{fuzzy-APAR}for different vegetation groups. Thus, it means that the assimilation of plant physiological and biophysical information through fuzzy logic modelling was capable not only to approximate SIF but can also provide vital information on vegetation diversity [88]. The vegetation canopy structure influences the absorption, scattering, transmission of plants that directly affects the overall emission of fluorescence photons at the top of the canopy [89,90]. This is why, due to lower photosynthetic activities and no green cover, the deforested area, mowed meadows, and post-agricultural lands have been characterized with no values in our modelled data. Inside the peatland, alder forest and low birch bush were characterized with higher values of SIF

_{fuzzy}and SIF

_{fuzzy-APAR}, whereas transition mires were characterized with low values of SIF

_{fuzzy}.

## 5. Conclusions

_{fuzzy}and SIF

_{fuzzy-APAR}was the first experimental evidence for quantitative demonstration of the fuzzy logic approach showing that the combination of SIF influencing factors represented by different SVIs can approximate the potential SIF signals at both oxygen absorption bands at 760 nm and 687 nm. It also demonstrated the efficiency of the integrated fuzzy logic model towards the step-by-step approximation of SIF signals through a process-based approach. This is also the first study that considers the red fluorescence into the prediction process which is extremely new and not covered in other published works.

_{fuzzy}and SIF

_{fuzzy-APAR}worked quite accurately to approximate the SIF signals, where SIF

_{fuzzy}were closer to SIF

_{687}values, whereas SIF

_{fuzzy-APAR}were better correlated with SIF

_{760}as expressed by higher R

^{2}and lower RMSE.

_{fuzzy}under models C3 and C4 for SIF

_{760}and C6 for SIF

_{687}or SIF

_{fuzzy-APAR}under C6 for both SIF

_{760}and SIF

_{687}would be the optimum solution to develop the proxy of potential SIF signals at 760 nm and 687 nm with high accuracy. The study also showed that EVI and NDVI, which can be available from spaceborne products, can be used to approximate the SIF signals to some extent. Although this study employed one-day airborne campaign data, the outcome of this study demonstrated a promising method to develop a proxy of potential SIF, where and when SIF data are not easily available or under data constraint situations. Though the proposed method does not have a significant impact on the change of observation day, slight changes in the agreements may occur during seasonal changes and atmospheric anomalies.

_{fuzzy}, and SIF

_{fuzzy-APAR}models will enrich our understanding of SIF science and global carbon cycles. We have not performed the SIF

_{fuzzy}and SIF

_{fuzzy-APAR}on the satellite dataset, but we believe that the model will work for it (however, it still needs to be validated). Thus, the proposed model is possible to be applied through satellite-derived SVIs from Sentinel 2, Landsat, MODIS, etc. to develop a SIF proxy and can be related to SIF products retrieved from spaceborne OCO-2 or GOME-2 satellites. We proved that our modelled SIF

_{fuzzy}and SIF

_{fuzzy-APAR}are capable to reflect the diversity of potential plant photosynthetic activity from multiple ecosystems. Therefore, such studies may further develop our knowledge of the local, regional and global photosynthetic activity and the carbon cycle in natural ecosystems.

## Supplementary Materials

_{fuzzy-APAR}, Figure S1: Correlation between NDVIre and SIF at 760 nm and 687 nm, Figure S2: Scatterplots of fuzzy model outputs (SIF

_{fuzzy}) and original SIFs, Figure S3: Bar diagrams represent the SIF

_{fuzzy}values obtained from 19 ROIs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of the Rzecin peatland, Wielkopolska region, Poland. An RGB composite map was obtained by combining reflectance bands for the red, green, and blue bands of the HyPlant DUAL module during the Spectrometry of a Wetland and Modelling of Photosynthesis (SWAMP) campaign on 11 July 2015. (Adopted from Bandopadhyay et al. [1]).

**Figure 2.**Location and boundaries of the 158 ROIs identified in the HyPlant RGB image and categorized into 19 unique vegetation groups. Adopted after Bandopadhyay et al. [1].

**Figure 3.**Scheme of the research methodology and different steps of data processing from HyPlant data acquisition to model building and validation of the outputs.

**Figure 4.**Schema of the fuzzy logic modelling system adopted in this study showing the progress from the input variable to the membership transformation in order to overlay operation to the final output.

**Figure 5.**Membership maps of the different SVIs: (

**A**) MF_NDVI; (

**B**) MF_SR; (

**C**) MF_NDVIre; (

**D**) MF_EVI and (

**E**) MF_PRI, derived from the fuzzy membership transformation functions. The membership maps ranging from 0 to 1 represent no membership to high membership, respectively.

**Figure 6.**Simulated SIF

_{fuzzy}maps developed through the integration of membership maps and Fuzzy Gamma approach for C1–C6 combinations: (

**A**) C1 SIF

_{fuzzy}; (

**B**) C2 SIF

_{fuzzy}; (

**C**) C3 SIF

_{fuzzy}; (

**D**) C4 SIF

_{fuzzy}; (

**E**) C5 SIF

_{fuzzy}; and (

**F**) C6 SIF

_{fuzzy}. The colour stretch in the left represents the range of C1–C6 SIF

_{fuzzy}maps.

**Figure 7.**Scatterplots of the best performing fuzzy logic model outputs (SIF

_{fuzzy}) and actual SIFs (SIF

_{760}and SIF

_{687}) were determined based on HyPlant airborne data. (

**A**,

**B**) SIF

_{fuzzy}expressed by f(NDVI+EVI+NDVIre+SR+PRI) under model C6; (

**C**,

**D**) SIF

_{fuzzy}expressed by f(NDVI+PRI) under model C3; (

**E**,

**F**) SIF

_{fuzzy}expressed by f(SR+PRI) under model C4. Standard deviations are represented in error bars. The letter abbreviations correspond to the codes of vegetation groups presented in Figure 2.

**Figure 8.**Example of bar diagrams representing the modelled values of SIF

_{fuzzy}obtained from 19 ROIs; (

**A**) SIF

_{fuzzy}as expressed by f(NDVI+PRI) under C3; (

**B**) SIF

_{fuzzy}as expressed by f(NDVI+EVI+NDVIre+SR+PRI) under C6. Error bars represent the standard deviations.

**Table 1.**Vegetation indices derived from HyPlant DUAL module reflectance data. R in the formulas represents the reflectance while the numbers refer to wavelengths in nm. Adopted after Bandopadhyay et al. [1].

Vegetation Indices | Equations | References |
---|---|---|

Simple Ratio (SR) | $\mathrm{SR}=\frac{{\mathrm{R}}_{\u2329795-810\u232a}}{{\mathrm{R}}_{\u2329665-680\u232a}}$ | [71] |

Normalized Difference Vegetation Index (NDVI) | $\mathrm{NDVI}=\frac{{\mathrm{R}}_{\u2329795-810\u232a}-{\mathrm{R}}_{\u2329665-680\u232a}}{{\mathrm{R}}_{\u2329795-810\u232a}+{\mathrm{R}}_{\u2329665-680\u232a}}$ | [72] |

Enhanced Vegetation Index (EVI) | $\mathrm{EVI}=2.5\left[\frac{{\mathrm{R}}_{\u2329795-810\u232a}-{\mathrm{R}}_{\u2329665-680\u232a}}{{\mathrm{R}}_{\u2329795-810\u232a}+6\xb7{\mathrm{R}}_{\u2329665-680\u232a}-7.5\xb7{\mathrm{R}}_{\u2329475-490\u232a}+1}\right]$ | [39] |

Red-edge Normalized Difference Vegetation Index (NDVIre) | ${\mathrm{NDVI}}_{\mathrm{re}}=\frac{{\mathrm{R}}_{\u2329735-750\u232a}-{\mathrm{R}}_{\u2329695-710\u232a}}{{\mathrm{R}}_{\u2329735-750\u232a}+{\mathrm{R}}_{\u2329695-710\u232a}}$ | [73] |

Photochemical Reflectance Index (PRI) | $\mathrm{PRI}=\frac{{\mathrm{R}}_{\u2329570\pm 2.5\u232a}-{\mathrm{R}}_{\u2329531\pm 2.5\u232a}}{{\mathrm{R}}_{\u2329570\pm 2.5\u232a}+{\mathrm{R}}_{\u2329531\pm 2.5\u232a}}$ | [74] |

**Table 2.**The mathematical equations of membership functions and justifications for considering a membership transformation function for individual HyPlant derived SVIs.

HyPlant SVIs | Membership Functions | Equations | Justifications | References |
---|---|---|---|---|

SR | Fuzzy MS Large | $\mathsf{\mu}\left(\mathrm{x}\right)=1-\frac{\mathrm{bs}}{\mathrm{x}-\mathrm{am}+\mathrm{bs}}\text{}\mathrm{if}$ $\mathrm{x}>\mathrm{am}\text{}\mathrm{otherwise}\text{}\mathsf{\mu}\left(\mathrm{x}\right)=0$ | Positive strong correlation with SIF | [1] |

NDVI | Positive strong correlation with SIF | [1] | ||

NDVIre | Positive strong correlation with SIF | (Supplementary Materials Figure S1) | ||

EVI | Fuzzy Linear | $\mathsf{\mu}\left(\mathrm{x}\right)=\left\{\frac{\mathrm{x}-\mathrm{min}}{\mathrm{max}-\mathrm{min}}\right\}$ | Positive poor correlation with SIF | [1] |

PRI | Fuzzy MS Small | x > am otherwise μ(x) = 1 $\mathsf{\mu}\left(\mathrm{x}\right)=\frac{\mathrm{bs}}{\mathrm{x}-\mathrm{am}+\mathrm{bs}}\text{}\mathrm{if}$ $\mathrm{x}>\mathrm{am}\text{}\mathrm{otherwise}\text{}\mathsf{\mu}\left(\mathrm{x}\right)=1$ | Negative correlation with SIF | [1] |

Combinations | Objectives | Equations | Code |
---|---|---|---|

Combination 1 | approximate SIF based on greenness and biomass related SVIs (without/with the inclusion of APAR) | ${\mathrm{SIF}}_{\mathrm{fuzzy}}=f\left(\mathrm{NDVI}+\mathrm{EVI}\right)$ ${\mathrm{SIF}}_{\mathrm{fuzzy}-\mathrm{APAR}}=f\left(\mathrm{NDVI}+\mathrm{EVI}\right)\ast \mathrm{APAR}$ | C1 |

Combination 2 | ${\mathrm{SIF}}_{\mathrm{fuzzy}}=f\left(\mathrm{SR}+\mathrm{EVI}\right)$ ${\mathrm{SIF}}_{\mathrm{fuzzy}-\mathrm{APAR}}=f\left(\mathrm{SR}+\mathrm{EVI}\right)\ast \mathrm{APAR}$ | C2 | |

Combination 3 | approximate SIF based on greenness and xanthophyll cycle-related SVIs (without/with the inclusion of APAR) | ${\mathrm{SIF}}_{\mathrm{fuzzy}}=f\left(\mathrm{NDVI}+\mathrm{PRI}\right)$ ${\mathrm{SIF}}_{\mathrm{fuzzy}-\mathrm{APAR}}=f\left(\mathrm{NDVI}+\mathrm{PRI}\right)\ast \mathrm{APAR}$ | C3 |

Combination 4 | ${\mathrm{SIF}}_{\mathrm{fuzzy}}=f\left(\mathrm{SR}+\mathrm{PRI}\right)$ ${\mathrm{SIF}}_{\mathrm{fuzzy}-\mathrm{APAR}}=f\left(\mathrm{SR}+\mathrm{PRI}\right)\ast \mathrm{APAR}$ | C4 | |

Combination 5 | approximate SIF based on greenness, biomass, and xanthophyll cycle-related SVIs (without/with the inclusion of APAR) | ${\mathrm{SIF}}_{\mathrm{fuzzy}}=f\left(\mathrm{NDVI}+\mathrm{EVI}+\mathrm{PRI}\right)$ ${\mathrm{SIF}}_{\mathrm{fuzzy}-\mathrm{APAR}}=f\left(\mathrm{NDVI}+\mathrm{EVI}+\mathrm{PRI}\right)\ast \mathrm{APAR}$ | C5 |

Combination 6 | approximate SIF based on greenness, biomass, xanthophyll cycle, and red-edge position related SVIs (without/with the inclusion of APAR) | ${\mathrm{SIF}}_{\mathrm{fuzzy}}=f\left(\mathrm{SR}+\mathrm{NDVI}+\mathrm{EVI}+\mathrm{NDVIre}+\mathrm{PRI}\right)$ ${\mathrm{SIF}}_{\mathrm{fuzzy}-\mathrm{APAR}}=f\left(\mathrm{SR}+\mathrm{NDVI}+\mathrm{EVI}+\mathrm{NDVIre}+\mathrm{PRI}\right)\ast \mathrm{APAR}$ | C6 |

**Table 4.**Summary of the statistics (R

^{2}—coefficient of determination, p-value, SE—standard error, R—correlation coefficient and RMSE—root mean square error) of linear regressions between SIF

_{fuzzy}vs. SIF

_{760}and SIF

_{fuzzy}vs. SIF

_{687}. The statistical operational outputs were derived based on 19 ROIs representing vegetation groups of the forest, grassland and peatland.

Combinations | SIF_{fuzzy} Functions | R^{2} | p-Value | SE | Pearson’s r | RMSE mW·m ^{−2}·sr^{−1} nm^{−1} |
---|---|---|---|---|---|---|

SIF_{fuzzy} vs. SIF_{760} | ||||||

C1 | SIF_{fuzzy} (NDVI+EVI) | 0.38 | <0.05 | 0.172 | 0.61 | 0.259 |

C2 | SIF_{fuzzy} (SR+EVI) | 0.55 | <0.001 | 0.167 | 0.74 | 0.300 |

C3 | SIF_{fuzzy} (NDVI+PRI) | 0.61 | <0.001 | 0.185 | 0.78 | 0.184 |

C4 | SIF_{fuzzy} (SR+PRI) | 0.69 | <0.001 | 0.176 | 0.83 | 0.235 |

C5 | SIF_{fuzzy} (NDVI+EVI+PRI) | 0.51 | <0.01 | 0.195 | 0.71 | 0.193 |

C6 | SIF_{fuzzy} (NDVI+EVI+NDVIre+SR+PRI) | 0.62 | <0.001 | 0.159 | 0.78 | 0.268 |

SIF_{fuzzy} vs. SIF_{687} | ||||||

C1 | SIF_{fuzzy} (NDVI+EVI) | 0.85 | <0.001 | 0.083 | 0.92 | 0.090 |

C2 | SIF_{fuzzy} (SR+EVI) | 0.89 | <0.001 | 0.083 | 0.94 | 0.114 |

C3 | SIF_{fuzzy} (NDVI+PRI) | 0.90 | <0.001 | 0.092 | 0.95 | 0.154 |

C4 | SIF_{fuzzy} (SR+PRI) | 0.90 | <0.001 | 0.098 | 0.95 | 0.109 |

C5 | SIF_{fuzzy} (NDVI+EVI+PRI) | 0.90 | <0.001 | 0.086 | 0.95 | 0.143 |

C6 | SIF_{fuzzy} (NDVI+EVI+NDVIre+SR+PRI) | 0.92 | <0.001 | 0.069 | 0.96 | 0.082 |

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**MDPI and ACS Style**

Bandopadhyay, S.; Rastogi, A.; Cogliati, S.; Rascher, U.; Gąbka, M.; Juszczak, R.
Can Vegetation Indices Serve as Proxies for Potential Sun-Induced Fluorescence (SIF)? A Fuzzy Simulation Approach on Airborne Imaging Spectroscopy Data. *Remote Sens.* **2021**, *13*, 2545.
https://doi.org/10.3390/rs13132545

**AMA Style**

Bandopadhyay S, Rastogi A, Cogliati S, Rascher U, Gąbka M, Juszczak R.
Can Vegetation Indices Serve as Proxies for Potential Sun-Induced Fluorescence (SIF)? A Fuzzy Simulation Approach on Airborne Imaging Spectroscopy Data. *Remote Sensing*. 2021; 13(13):2545.
https://doi.org/10.3390/rs13132545

**Chicago/Turabian Style**

Bandopadhyay, Subhajit, Anshu Rastogi, Sergio Cogliati, Uwe Rascher, Maciej Gąbka, and Radosław Juszczak.
2021. "Can Vegetation Indices Serve as Proxies for Potential Sun-Induced Fluorescence (SIF)? A Fuzzy Simulation Approach on Airborne Imaging Spectroscopy Data" *Remote Sensing* 13, no. 13: 2545.
https://doi.org/10.3390/rs13132545