Introduction

Monitoring the biochemical constituents of plant leaves using remote sensing techniques can improve our understanding of the dynamics of vegetation physiological and ecological functions. In the 400–800 nm region, leaf photosynthetic pigments are the main absorbers of harvesting light in plants1, 2 and these pigments mainly consist of total chlorophylls (Chls) (chlorophyll a (Chla), chlorophyll b (Chlb)) and carotenoids (Cars)3,4,5. Chlorophyll a can act as a light harvesting pigment and the reaction centre for leaf photosynthesis6. Chlorophyll b can act as an accessory light-harvesting pigments and helps Chlorophyll a to perform leaf photosynthesis7. Carotenoids also act as accessory light-harvesting pigments and perform an essential photo-protective role on non-photochemical quenching of excess light energy8. As the physiological functions of these photosynthetic pigments in plant are different, their proportions vary according to leaf age and physiological stress. especially chlorophyll a and b. For instance, a decrease in chlorophyll a/b ratios in rice seedlings is associated with leaf senescence9, whereas an increase of the chlorophyll a/b ratios is associated with nitrogen limitation and high light during acclimation of tropical woody seedlings10. Therefore, an improved discrimination between the key pigments is important for physiological and ecological applications of remote sensing11. However, to date, efforts to simultaneously retrieve the concentrations of individual photosynthetic pigments from remotely-sensed data have been restricted due to the overlap of absorption spectra of different pigments which can mask the contribution of individual pigments to the reflectance and/or transmittance spectrum.

PROSPECT models are a type of leaf radiative transfer (RT) models, which have been widely used in the remote sensing community4, 12. The earlier versions of PROSPECT models only considered the effects of total chlorophylls pigments. Until PROSPECT-5 version, this model separated photosynthetic pigments into total chlorophylls and carotenoids. Within this version (PROSPECT-5), the separation of Chls and Cars absorption coefficients was band-by-band derived based on a minimum distance fitting method using modeled and measured in vivo spectra13. However, with this method, the specific absorption coefficients of Chla and Chlb could not be separated in PROSPECT-511. This is because band-by-band fitting method is a purely mathematical fitting and absent of physical significance. Thus, uncertainties in indexing absorption peaks to their respective pigments may occur during the determination of these pigment absorption coefficients.

Therefore, a new approach with physical significance that can limit the masking phenomenon between multiple pigment spectra is needed in PROSPECT model. The Gauss-Lorentz (G-L) function fitting method can be used to define individual material absorption spectra and provide a chance to explicitly deal with the issue of overlapping absorption features14,15,16. G-L function fitting can be applied in the spectral absorption peak separation of mixed constituents by fitting the parameters (absorption peak height, full width at half maximum (FWHM) and Gauss ratio), given the absorption peak number and position14. These parameters holds explicit physical significance of material absorption spectra. As the given absorption peak positions can help the G-L function to index and characterize its own absorption spectra, the separation of absorption spectra of mixed materials is not susceptible to the problem of band overlapping and masking phenomena between absorption features17.

In view of the above therefore, the present study develops an algorithm for the separation of multiple photosynthetic pigment absorption coefficients (Chla, Chlb and Cars) by using a modified G-L function, and then proposes an extended version of the PROSPECT model in the 400–800 nm range, herein referred to as PROSPECT-Multiple Pigment (PROSPECT-MP). PROSPECT-MP is capable of incorporating the in vivo absorption coefficients of Chla, Chla and Cars pigments and describes leaf optical properties from 400 to 800 nm in order to facilitate the simultaneous retrieval of these multiple individual photosynthetic pigment concentrations by model inversion.

Results and Discussion

Parameter Calibration

Characteristics of the PROSPECT-MP parameter calibration

In order to describe the absorption features present within these spectra, we used the full width at half maximum (FWHM, \({{\boldsymbol{K}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\boldsymbol{w}}}\)) and position of each absorption peak (\({{\boldsymbol{K}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\boldsymbol{p}}}\)) to derive the Range of Absorption Feature (\({\boldsymbol{RAF}}\)) using the following:

$$RA{F}_{i,j,lb}={K}_{i,j,p}-{K}_{i,j,w}/2;\,RA{F}_{i,j,ub}={K}_{i,j,p}+{K}_{i,j,w}/2.$$
(1a)

Where

$${K}_{i,j,p}={A}_{i,j,p}+{K}_{i,j,{\rm{\Delta }}\lambda }$$
(1b)

where i is the determinable pigment type (Chla, Chlb or Cars); \({j}\) is the peak number within the pigment-specific absorption coefficient;\({A}_{{\rm{i}},{\rm{j}},{\rm{p}}}\) is the peak position of the \({\boldsymbol{j}}{\rm{t}}{\rm{h}}\) absorption peak for the \({\boldsymbol{i}}{\rm{t}}{\rm{h}}\) pigment type in organic solution; \({{\boldsymbol{K}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\rm{\Delta }}{\boldsymbol{\lambda }}}\) is the spectral displ.cement of the \({\boldsymbol{j}}{\rm{t}}{\rm{h}}\) absorption peak for the \({\boldsymbol{i}}{\rm{t}}{\rm{h}}\). pigment type in vivo; \({\boldsymbol{RA}}{{\boldsymbol{F}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\boldsymbol{lb}}}\) and \({\boldsymbol{RA}}{{\boldsymbol{F}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\boldsymbol{ub}}}\) stand for the ler and upper boundary of the \({\boldsymbol{j}}{\rm{th}}\) absorption peak for the ith pigment in vivo leaf. The main spectral characteristics of the pigment absorption coefficients are expressed by the metrics (\({\boldsymbol{RA}}{{\boldsymbol{F}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\boldsymbol{lb}}}\) and \({\boldsymbol{RA}}{{\boldsymbol{F}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\boldsymbol{ub}}}\)) in Table 1.

Table 1 Absorption peak characteristics determined from pigment absorption coefficients within PMP.
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The spectra in Fig. 1 and data within Table 1 together allow us to describe the key characteristics of the pigment-specific and baseline absorption coefficients from PROSPECT-MP. For this parameter calibration, the characteristics of \({{\boldsymbol{K}}}_{{\boldsymbol{Chla}}}\) include: 1) two prominent absorption peaks (1st and 4th) positioned at 417 and 680 nm; 2) spectral displacements of those peak positions relative to the corresponding absorption peaks in the organic solution (acetonitrile/methanol/dichloromethane; 60:20:20 v/v/v) are −15 and 16 nm, respectively; 3) the two main regions of absorption as defined by the RAFs are located at 400–464 and 668–692 nm, which are regions that have been used previously in empirical spectral indices for Chla1, 3.

Figure 1
figure 1

Spectral characteristics of the determined PROSPECT-5 (P5, solid line) and PROSPECT-MP (PMP, dotted line) parameters in vivo leaf. (a) show for Chla absorption coefficient (\({{K}}_{{C}{h}{l}{a}}\)); (b) for Chlb absorption coefficient (\({{K}}_{{C}{h}{l}{b}}\)); (c) for \({{K}}_{{C}{a}{r}{s}}\); (d) for leaf baseline absorption coefficient (\({{K}}_{0}\)); and (e) for leaf average refractive index (\({\bar{{m}}}_{{l}{a}}\)).

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The characteristics of \({{\boldsymbol{K}}}_{{\boldsymbol{Chlb}}}\) are: 1) two prominent absorption peaks (1st and 3rd) at 482 and 665 nm with spectral displacements of 18 and 15 nm; 2) two RAFs located at 455–509 and 640–690 nm.

For \({{\boldsymbol{K}}}_{{\boldsymbol{Cars}}}\): 1) there is only one absorption peak which is located at 520 nm with a spectral displacement of 50 nm; 2) its absorption spectrum has a Gaussian shape and the RAF is 497–543 nm, which regions have been used for the previous spectral indices for Cars18. Attempts to separate specific absorption coefficients of individual pigments in the Cars group (Lu, An, Ze, Vi, Ne and β-Car) using the same method unfortunately failed. This could be due to the small distances between absorption peaks of individual pigments, which might mean that directional hemispherical reflectance (DHR) and transmittance (DHT) with higher spectral resolution are required for separating these pigments.

The \({{\boldsymbol{K}}}_{0}\) spectrum shows a decreasing trend with wavelength (Fig. 2(d)), which is consistent with Jacquemoud and Baret11. Leaf average refractive index (\({\bar{{\boldsymbol{m}}}}_{{\boldsymbol{la}}}\)) shows similar general variations with wavelength change for both versions (Fig. 2(e)).

Figure 2
figure 2

Leaf directional hemispherical reflectance (DHR) spectra which reveal the in vivo pigment absorption characteristics (Mutant1 = 11.864 μg/cm2; Mutant2 = 15.452 μg/cm2; Mutant3 = 20.521 μg/cm2 and Mutant4 = 27.944 μg/cm2).

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Figure 1(a and b) shows that the separated \({{\boldsymbol{K}}}_{{\boldsymbol{Chla}}}\) derived from PMP is reasonably consistent with the combined \({{\boldsymbol{K}}}_{{\boldsymbol{Chls}}}\) derived from P5, but this is not the case for \({{\boldsymbol{K}}}_{{\boldsymbol{Chlb}}}\). This could be the reason that Chla concentration is generally higher than Chlb in the LOPEX 93 dataset (Chla: Chlb ratio is around 3:1)11. Figure 2(c) shows that the form of the \({{\boldsymbol{K}}}_{{\boldsymbol{Cars}}}\) coefficient derived from PMP is consistent with absorption principles and differs substantially from the \({{\boldsymbol{K}}}_{{\boldsymbol{Cars}}}\) coefficient in P5.

As PROSPECT-MP uses the modified G-L function fitting, the determined pigment-specific absorption coefficients: 1) are all in accordance with the physical principles underpinning pigment absorption spectra19; 2) can directly account for peak position variations in environmental polarity between the organic solution and a leaf in vivo by using the spectral displacement parameter; 3) can also quantify the absorption characteristics of the corresponding pigment in vivo using the RAF parameter derived from FWHM.

Analysis of peak displacement within the absorption coefficients for PROSPECT-MP

In order to evaluate the effectiveness of the spectral displacement parameter in accounting for shifts in the absorption peak positions in vivo compared with the organic solution, we compared the PROSPECT-MP pigment absorption coefficients (as described above) with reflectance spectra of leaves with particular biochemical compositions that revealed the absorption features of specific pigments in vivo. A set of leaves from a Chlb-deficient rice mutant (IG20) were provided by the College of Life Sciences, Zhejiang University, China20. In the 580–700 nm region, leaf reflectance is dominated by chlorophyll absorption, and in Fig. 3b, the reflectance spectra of the Chlb deficient leaves reveal the in vivo absorption characteristics of Chla. The absorption peaks for Chla are found at 590, 628 and 680 nm. In the 450–500 nm region, leaf absorption is dominated by Cars and Chlb. Thus, as Fig. 2 shows, the Chlb-deficient mutant leaves reveal the in vivo absorption peak for Cars (480 nm).

Figure 3
figure 3

Comparison of measured (black) and simulated (grey) reflectance and transmittance for (a) low (Chla = 15.11 μg/cm2; Chlb = 4.25 μg/cm2; Cars = 5.94 μg/cm2), (b) medium (Chla = 47.86 μg/cm2; Chlb = 13.35 μg/cm2; Cars = 9.92 μg/cm2) and (c) high (Chla = 90.52 μg/cm2; Chlb = 29.34 μg/cm2; Cars = 27.41 μg/cm2) pigment concentration levels from P5; and (d) low, (e) medium and (f) high levels from PMP.

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As noted earlier (Table 1) the absorption peaks of K chla (λ) (2nd, 3rd and 4th peaks) and Cars from PMP were observed at 590, 626, 680 and 520 nm, respectively, in which Chla and Chlb correspond closely with the in vivo absorption peaks (Fig. 2), but Cars performance is poor. This indicates that the process used to calibrate the Chla and Chlb absorption coefficients in PROSPECT-MP was effective. However, this analysis was not able to reveal all the in vivo absorption peaks for pigments (i.e. neither the Chlb peaks nor the 1st Chla peak were determined). Nevertheless, this analysis did provide some evidence with which to evaluate the PROSPECT-MP absorption coefficients.

Performance Evaluation

Spectral modelling performance

Figure 3 illustrates the performance of P5 and PMP in simulating directional hemispherical reflectance (DHR) and transmittance (DHT) for leaves at low, medium and high pigment levels. The performances for DHR and DHT are very similar at medium and high pigment levels. For the low pigment concentration, the simulations from P5 were worse than those at the other two concentration levels, with a tendency to underestimate reflectance and transmittance, which is consistent with the results from Feret et al.11. The main disparities are located in the 500–650 nm regions. Furthermore, for the low pigment concentration, PMP performs better than P5.

Spectral modelling evaluation

Global performance evaluation of simulated leaf spectra. The global performance assessment used averaged values across the whole spectrum (400–800 nm) for each evaluation metric (Root Mean Square Error, RMSE; BIAS; and Standard Error Corrected, SEC) from the measured and modeled spectrum of the 32 leaf samples (those not used for model calibration). For all model implementations, the results for DHR and DHT are encouraging in that the global RMSE values are less than 0.03, BIAS values are lower than ±0.01 and SEC values are similar to the global RMSE results (Table 2). These findings indicate that PROSPECT-MP can accurately compute leaf DHR and DHT using input data on Chla, Chlb and Cars concentrations, and its performance is similar to that of P5.

Table 2 Global performance evaluation of simulated leaf spectra from P5 and PMP (n = 32).
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Local performance evaluation of simulated leaf spectra. In contrast to the global evaluation, the local assessment is able to quantify errors in the leaf DHR and DHT simulations band-by-band. Figure 4 illustrates the results of the evaluation metrics from the two version implementations. RMSE values were variable across the whole spectrum. In the 460–510 nm regions where the absorption bands of Cars are located, PMP generated lower RMSE values for both leaf DHR and DHT modelling compared with P5. These results indicate that the PMP can better simulate leaf spectra by incorporating the contribution from Cars absorption. There are generally larger RMSE values in the 510–580 nm regions, where the anthocyanin absorption bands are located. Burger and Edwards21 reported that green leaves contain some determinable anthocyanin information, but the pigment was not considered in the LOPEX93 dataset. Thus, the incorporation of anthocyanin in the PROSPECT model will be required in subsequent studies. In the 600–700 nm region for Chls absorption, the RMSEs for DHR are lowest for PMP, then P5, which indicates that PROSPECT-MP can better simulate leaf DHR. In this region, the RMSEs for DHT are similar to DHR across the two version implementations. However, in the 730–800 nm region, the RMSE values for PMP are larger than P5 implementations. For the BIAS metric, the two version implementations perform similarly, except for PMP in the 510–580 nm region. For the SEC metric, the results for the two inversion implementations are similar to those for RMSE. These results demonstrate that PROSPECT-MP can successfully simulate leaf spectra in a similar fashion to PROSPECT-5, but PROSPECT-MP appears to perform better in the Cars absorption bands.

Figure 4
figure 4

Simulated DHR (left column) and DHT (right column) spectra from P5 (grey line) and PMP (black line) (n = 32). (a) and (b) show RMSE values; (c) and (d) show BIAS; (e) and (f) show SEC.

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Pigment concentration retrieval performance

The capacity of the PROSPECT model to retrieve the concentrations of pigments from spectra of fresh leaves depends on the types of pigment absorption coefficients used in the model. Hence, the P5 and PMP implementations tested here differ in their capacity to retrieve different pigments: P5 can only directly retrieve Chls and Cars; while PMP can retrieve Chla, Chlb and Cars. As Chls in a leaf is usually the sum of Chla and Chlb22, we can derive Chls from PMP, to facilitate the comparison.

Figure 5 Illustrates the relationships between measured pigment concentrations and the corresponding values retrieved from the measured spectrum of the 32 leaf samples (those not used for model calibration) by model inversion. Vacant plots denote that P5 implementation is incapable of retrieving the specific pigment. The results show that the performance of PMP enables the retrieval of Chls, Chla, Chlb and Cars concentration from leaf spectra, but P5 only does for Chls and Cars.

Figure 5
figure 5

Comparison between measured and retrieved pigment concentrations (µg/cm2, n = 32). (a) and (c) are for Chls concentration; (d) is Chla; (e) is Chlb; and (b) and (f) are Cars; Retrievals (a) and (b) are from inversions of P5; (c), (d), (e) and (f) from PMP.

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Pigment concentration retrieval evaluation

The evaluation metrics (RMSE, BIAS, SEC and Coefficient Variability (CV)) for the retrieved concentrations are shown in Table 3. For Chls retrieval, the performance of the two version implementations is similar, but with PMP performing better than P5 in terms of RMSE and BIAS. For Cars retrieval, PMP model implementations substantially outperformed P5. Thus, the results demonstrate that for Chls concentration retrievals, PROSPECT-MP and PROSPECT-5 have a similar performance, while PROSPECT-MP produces a reliable estimate of Chla and Chlb concentrations and a more accurate estimate of Cars than PROSPECT-5.

Table 3 The validation of pigment concentration retrievals from in vivo leaf spectra by P5 and PMP (n = 32).
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Although PROSPECT-MP can provide a reliable retrieval capability of Chls and Cars concentration comparing to PROSPECT-5, and can also retrieve Chla and Chlb concentrations, these performances of pigment retrieval from PMP are still weak based on the R2 values and regression equations in Fig. 5, especially Chlb and Cars. The reasons will be given as follows. When the G-L function fitting method is used to separate the mixed absorption spectra with band overlapping features, two requirements should be met: 1) All material responded on spectra must be considered14. Plant leaves commonly contain the determinable Ants information21, but in the LOPEX93 dataset, Anthocyanins (Ants) was not considered. Due to band overlapping features in the absorption spectra of Ants and Chlb and Cars3, 4, the case of unconsidering Ants in PMP could produce the distorted calibration of pigment absorption coefficients, leading to weak performance of pigment retrievals, especially Chlb and Cars. 2) The measurements of all material should be accurate enough. In the LOPEX93 dataset23, the spectrophotometric method for Cars measurement might forget its component pigments (lutein, neoxanthin, etc.), which could lead to the measurements of Cars not accurate enough24. The inaccurate measurements of Cars could firstly produce the over-fitting of pigment absorption coefficients using a minimum distance fitting method25, and then lead to the distorted absorption coefficients of Chla, Chlb and Cars, ultimately resulting in weak performances of these pigment retrievals.

Based on the analysis above, there are two probable causes for the poor performances of pigment retrievals in PMP: 1) Anthocyanins (Ants) was not considered in the LOPEX93 dataset. 2) The measurement of Cars was not accurate enough. Thus, a consideration of Ants information and an improvement of the measurement of individual photosynthetic pigment concentrations are needed in the future studies for the more accurate retrievals of plant pigments from leaf spectra.

Conclusions

This paper demonstrates that the extended version of PROSPECT (PROSPECT-MP) proposed in this study can reliably simulate leaf hemispherical reflectance and transmittance in the 400–800 nm region, and it can retrieve accurately, multiple photosynthetic pigment concentrations comparing to PROSPECT-5 from spectra of fresh leaves by model inversion. The modified G-L function is employed to create a new function for the overall leaf absorption coefficient, which can limit the masking phenomenon between pigments. Consequently PROSPECT-MP is parameterized using pigment absorption coefficients (K chla (λ), K chlb (λ), and K cars (λ)). The determined pigment absorption coefficients possess three key features: 1) they are consistent with the physical principles of pigment absorption spectra; 2) they account for the spectral displacement of absorption peaks within media of different polarities; 3) they quantify the main absorption characteristics of each pigment with the RAF parameter.

In order to test the effectiveness of these developments in the treatment of leaf optics within PROSPECT-MP, the LOPEX93 dataset was used to evaluate the ability of PROSPECT-MP to simulate leaf DHR and DHT spectra and retrieve pigment concentrations from measured fresh leaf spectra by model inversion. To provide some context, the performance of PROSPECT-MP was compared with that of PROSPECT-5. The results were encouraging in that: 1) PROSPECT-MP was able to simulate accurately, in vivo leaf DHR and DHT spectra; 2) PROSPECT-MP can be used to retrieve leaf Chls and Cars with similar accuracies to PROSPECT-5; and, 3) PROSPECT-MP provides an additional capability for retrieving individual Chla and Chlb concentrations.

Our ongoing work is now focusing on improving the description within PROSPECT-MP of the optical properties of leaf photo-protective pigment (anthocyanins). Future developments of PROSPECT-MP would improve the robustness and transferability in the capabilities for retrieval of multiple pigment concentrations, and a synthesis of PROSPECT-MP with a canopy RT model will offer new opportunities for improving the estimates of pigment variations across larger spatial scales. PROSPECT-MP will provide a framework for future developments in the modelling of leaf optical properties and for the hyperspectral remote sensing of vegetation biochemistry.

Data and Method

Data

The LOPEX93 dataset23 and the absorption spectrum of pure pigments in an organic solution were used for calibrating and evaluating the performance of PROSPECT-MP. The LOPEX93 dataset has been used previously for the calibration and validation of leaf and canopy RT models26,27,28 and incorporates a wide range of biochemical concentrations and plant species. Therefore, in the present study, PROSPECT-MP was calibrated and compared with the PROSPECT-5 model using this dataset. To achieve this, we selected all 64 single-fresh leaf samples from the LOPEX93 dataset for which DHR and DHT were available along with measures of Chla, Chlb, Cars content11. In order to determine the numbers of absorption peaks and their positions for individual pigments, absorption spectra were obtained for pure pigments (Chla, Chlb, β-Car, Vi, An, Ze, Ne, and Lu) in a mixed organic solution using a Shimadzu UV-VIS detector29. The pigments and the organic solution were of chromatographic purity and purchased from the Sigma-Aldrich Co. LLC in January 2013.

Developing PROSPECT-MP method

Principles of PROSPECT-MP

Description of leaf absorption coefficient function. In fresh leaves, the presence of Chla, Chlb and Cars can influence the observed reflectance and transmittance spectra3, 4. However, leaf absorption coefficient (\({\boldsymbol{k}}\)) of PROSPECT-5 incorporates Chls and Cars without separating Chla and Chlb. Therefore, PROSPECT-5 cannot simulate the influence of Chla and Chlb on leaf spectra and cannot retrieve the Chla and Chlb from spectra by model inversion. To overcome the problem, we extended the pigment absorption function to incorporate Chla, Chlb and Cars leaf absorption coefficients, and these were described based on previous reports11, 12, 30.

$$k(\lambda )=\frac{{K}_{Chla}(\lambda ){C}_{Chla}+{K}_{Chlb}(\lambda ){C}_{Chlb}+{K}_{Cars}(\lambda ){C}_{Cars}}{N}+{K}_{0}(\lambda )$$
(2)

where \({\boldsymbol{N}}\) stand leaf structure index.

Characterization of pigment absorption coefficients in vivo leaf using modified Gauss-Lorentz function. Although Chla, Chlb and Cars have selective absorption characteristics, there is considerable overlap in their absorption spectra3, 31 and this can make the masking phenomenon to block the simultaneous separation of these individual pigment absorption coefficients in equation 2. In order to limit masking in the separation of these pigment absorption features, our approach was to employ a modified G-L function fitting method which uniformly describes the peaks of multiple individual pigment absorption coefficients in vivo leaf, as given in the expression below.

$${K}_{i,j}(\lambda )={K}_{i,j,v}\cdot {K}_{i,j,h}\cdot {e}^{-4ln2\cdot {(\frac{{A}_{i,j,p}+{K}_{i,j,{\rm{\Delta }}\lambda }-\lambda }{{K}_{i,j,w}})}^{2}}+(1-{K}_{i,j,v})\frac{{K}_{i,j,h}}{1+4{({A}_{i,j,p}+{K}_{i,j,{\rm{\Delta }}\lambda }-\lambda )}^{2}{{K}_{i,j,w}}^{-2}}$$
(3)

where\(\,{{\boldsymbol{K}}}_{{\boldsymbol{i}},{\boldsymbol{j}}}\) represents the \({\boldsymbol{j}}{\rm{t}}{\rm{h}}\) peak function within the absorption coefficient for the \({\boldsymbol{i}}{\rm{t}}{\rm{h}}\) pigment type; The factor \({{\boldsymbol{K}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\rm{\Delta }}{\boldsymbol{\lambda }}}\) is introduced to account for the shifts in absorption peak positions of specific pigments in vivo compared with the organic solution extract4, 32, and they when linked to the corresponding \({{\boldsymbol{A}}}_{{\boldsymbol{i}},{\boldsymbol{j}},{\boldsymbol{p}}}\), can help the modified G-L function to index and characterize the corresponding potential absorption region of \({{\boldsymbol{K}}}_{{\boldsymbol{i}},{\boldsymbol{j}}}\). This treatment can limit the masking phenomenon in the separation of multiple pigment absorption coefficients.

As the pigment absorption feature comes from the sum of absorption features of each peak, the pigment-specific absorption coefficients can be expressed as:

$${K}_{i}(\lambda )=\sum _{j=1}^{j}{K}_{i,j}(\lambda ).\,$$
(4)

These pigment-specific absorption coefficients then allow us to solve for K in equation 2.

For the G-L function, the determination of the pigment absorption number and positions of peaks is the key issue for the parameterization of absorption spectra, which will be given as follows.

Determination of pigment absorption peak number and position for the modified G-L function. The observed number of absorption peaks for photosensitive material in a weaker polarity medium can be regarded as the potential maximum number in a stronger polarity medium33. For fresh leaves, the water content is commonly over 50%11. Therefore, leaves can be regarded as essentially a water-based medium. Cotton et al.34 reported the polarity order of different media as: water > acetonitrile > methanol > dichloromethane. Hence, the polarity of leaf pigments in vivo is greater than that in the mixed organic solution (acetonitrile/methanol/dichloromethane; 60:20:20 v/v/v). As a result, the observed number of absorption peaks for pure pigments in the mixed organic solution can be regarded as the potential number of absorption peaks of the corresponding pigments in vivo.

The absorption spectra of pure pigments in mixed organic solution are shown in Fig. 6. It can be seen that the positions of the three absorption peaks are adjacent within Cars group (Fig. 6(a) and (b)). Table 4 demonstrates that the largest distance between absorption peak positions within the Cars group did not exceed 8 nm. Hence, we merged all of the individual peak positions from Lu, An, Ze, Vi, Ne and β-Car into a single set of peak positions to represent the Cars and the averaged value of these peaks are for the peak positions for the Cars. Therefore, the determinable leaf pigment groups are Chla, Chlb and Cars. The number and positions of absorption peaks were obtained by calculating first and second derivatives of the absorption spectra and the results are shown in Table 4. Given the above argument about differences in environmental polarity35, Table 4 shows the potential maximum number of peaks within A Chla , A ChlB , and A Cars in vivo.

Figure 6
figure 6

Absorption spectra of pure pigments in acetonitrile/methanol/dichloromethane (60:20:20 v/v/v). The concentrations of Lu (Lutein), An (Antheraxanthin), Ze (Zeaxanthin) in (a) and Ne (Neoxanthin), Vi (Violaxanthin), β-Car (β-carotene) in (b) were all 0.2 mg/ml; Chla and Chlb in (c) were 0.01 mg/ml.

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Table 4 The number of absorption peaks and their wavelength positions in the 400–800 nm region, for pure pigments in a mixed organic solution.
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Extension of PROSPECT to PROSPECT-MP for multiple pigment retrieval. In the PROSPECT model, leaf DHR (\({{\boldsymbol{R}}}_{{\boldsymbol{N}}}\)) and DHT (\({{\boldsymbol{T}}}_{{\boldsymbol{N}}}\)) are given11 by:

$${R}_{N}={R}_{1{\rm{st}}}+\frac{{T}_{1{\rm{st}}}{R}_{N-1}T}{1-{R}_{N-1}R};\,{T}_{N}=\frac{{T}_{1{\rm{st}}}{T}_{N-1}}{1-{R}_{N-1}R}$$
(5)

where \({{\boldsymbol{R}}}_{1{\bf{s}}{\bf{t}}}\) and \({{\boldsymbol{T}}}_{1{\bf{s}}{\bf{t}}}\), \({\boldsymbol{R}}\) and \({\boldsymbol{T}}\), and \({{\boldsymbol{R}}}_{{\boldsymbol{N}}-1}\) and \(\,{{\boldsymbol{T}}}_{{\boldsymbol{N}}-1}\) are the DHR and DHT of the first layer, the internal elementary layer and the internal N-1elementaayers. They can be parameterized by parameters N and \({\bar{{\boldsymbol{m}}}}_{{\boldsymbol{la}}}\) and \({\boldsymbol{\tau }}\). (leaf transmission coefficient). The paramter \({\boldsymbol{\tau }}\). is related to\({\boldsymbol{k}}({\boldsymbol{\lambda }})\) through the following equation36:

$$\tau =(1-k(\lambda )){e}^{-k(\lambda )}+k{(\lambda )}^{2}{\int }_{\,k(\lambda )}^{\infty }{x}^{-1}{e}^{-x}dx$$
(6)

With \({\boldsymbol{k}}\) derived as in equation 2, incorporating the absorption coefficients of multiple individual photosynthetic pigments in vivo, we use the improved \(k\) in equation 2 to replace \(k\) of PROSPECT model in equation 6 to extend the model at 400–800 nm to a leaf optical model (PROSPECT-MP) for the simultaneous retrieval of multiple individual photosynthetic pigments (Chla, Chlb and Cars).

In addition, the methods of calibration and evaluation for PROSPECT-MP and PROSPECT-5 in the LOPEX93 dataset were shown in the Note S1 and S2 of ‘Supplementary Information’.