# Quantifying leaf optical properties with spectral invariants theory

Shengbiao Wu, Yelu Zeng, Dalei Hao, Qinhuo Liu, Jing Li, Xiuzhi Chen, Ghassem Asrar, Gaofei Yin, Jianguang Wen, Bin Yang, et al.

► To cite this version:

Shengbiao Wu, Yelu Zeng, Dalei Hao, Qinhuo Liu, Jing Li, et al. Quantifying leaf optical properties with spectral invariants theory. Remote Sensing of Environment, 2021, 253, pp.112131. 10.1016/j.rse.2020.112131. hal-04228584

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https://hal.science/hal-04228584v1

Submitted on 3 Aug 2025

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<figure><img src="image_1.png" /></figure>
1 Quantifying leaf optical properties with spectral invariants theory

2 Shengbiao Wu a,1, Yelu Zeng b,c,*,1, Dalei Hao b, Qinhuo Liu d, Jing Li d, Xiuzhi Chen e, Ghassem

3 R. Asrar f, Gaofei Yin g, Jianguang Wen d, Bin Yang h, Peng Zhu a, Min Chen b,*

4 a Laboratoire des Sciences du Climat et de l'Environnement, Gif sur Yvette, France

5 b Joint Global Change Research Institute, Pacific Northwest National Laboratory, College Park,

6 MD 20740, USA

7 c Department of Global Ecology, Carnegie Institution for Science, Stanford, CA 94305, USA

8 d State Key Laboratory of Remote Sensing Science, Aerospace Information Research Institute,

9 Chinese Academy of Sciences, Beijing 100101, China

10 e Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, School

11 of Atmospheric Sciences, Sun Yat-sen University, Guangzhou 510275, China

12 f Universities Space Research Association, Columbia, MD 21046, USA

13 g Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University,

14 Chengdu 610031, China

15 h College of Electrical and Information Engineering, Hunan University, Changsha 410082, China

17 * Corresponding author e-mail address: yelu.zeng@pnnl.gov (Y.Z.); min.chen@pnnl.gov (M.C.)

18 ¹ These authors contributed equally.

19

20
21 Abstract

22 Leaf optical spectra reflect the combination of leaf biochemical, morphological and
23 physiological properties, and play an important role in many ecological and Earth system
24 processes. Radiative transfer models are widely used to simulate leaf spectra by quantifying
25 photon transfer processes of reflection, transmission and absorption within a plant leaf. Recent
26 advances in spectral invariants theory offer a unique and efficient approach for modeling the
27 canopy-scale radiative transfer processes, but remain underexplored for applications at the leaf
28 scale. In this study, we developed a leaf-scale optical property model based on the spectrally
29 invariant properties (leaf-SIP) of plant leaves. Similar to the canopy-scale model, the leaf-SIP
30 model decouples leaf-scale radiative transfer process into two parts: wavelength-dependent
31 contribution from leaf chemical components and wavelength-independent contribution from leaf
32 structures, described by two spectrally invariant parameters (i.e., a photon recollision probability
33 *p* and a scattering asymmetry parameter *q*). We implemented the leaf-SIP model by
34 parameterizing *p* and *q* with a measurable leaf morphological trait, the leaf mass per area (LMA).
35 We evaluated the performance of the leaf-SIP model with two *in situ* datasets (i.e., LOPEX and
36 ANGERS) and the widely used PROSPECT leaf optical model. The results show that the leaf
37 spectra simulated by the leaf-SIP model agreed well with *in situ* datasets and the simulations of
38 the PROSPECT model, with a small root mean squared error (RMSE), bias, and high coefficients
39 of determination (*R*²) of 0.026, 0.035, 0.95 and 0.037, 0.049, 0.91 for leaf reflectance and leaf
40 transmittance, respectively. Our results also show that the leaf-SIP model can be used with
41 measured leaf spectra to accurately estimate several key leaf functional traits, such as the leaf
42 chlorophyll content, equivalent water thickness and LMA. The leaf-SIP model provides an
43 efficient and physical way of accurately simulating leaf spectra and retrieving key leaf functional
44 traits from hyperspectral measurements.
45
**Keywords:** Leaf optical properties; leaf reflectance/transmittance; leaf functional traits; spectral invariants theory; photon recollision probability

# 1. Introduction

Leaf spectra have been recognized as a key indicator of many biochemical, biophysical and physiological properties of plant leaves. They are strongly correlated to multiple important leaf functional traits, such as leaf maximum carboxylation rate (Ainsworth et al., 2014; Serbin et al., 2011), instantaneous water use efficiency (Yendrek et al., 2017; Yonemura et al., 2019), chlorophyll and nitrogen content (Asner and Martin, 2008; Yendrek et al., 2017), and leaf dry matter content (Asner et al., 2011; Serbin et al., 2014). Increasing evidences show that the spectroscopy technique offers a simple and powerful tool to track the dynamics of leaf functional traits across biomes and light environments (Meacham-Hensold et al., 2019; Schneider et al., 2017; Yang et al., 2016). Since plant leaf is a fundamental element for the vegetation-light interaction, leaf spectra essentially affect the radiative transfer processes at canopy and ecosystem levels (Monson and Baldocchi, 2014; Shiklomanov et al., 2019), and are commonly used as a *priori* information for retrieving canopy structural (e.g., leaf area index (LAI)) and leaf biochemical properties (e.g., chlorophyll and nitrogen) from hyperspectral measurements (Yi et al., 2008). Therefore, accurate characterization of leaf spectra is critically important to facilitate understanding and quantification of key ecological processes within the plant canopy.

It is well understood that leaf spectra are influenced by leaf internal structures of tissues (Féret et al., 2017; Jacquemoud and Baret, 1990) and leaf biochemical composition (e.g., water, chlorophylls, carotenoids, cellulose, lignin and proteins). Physical models have been developed for quantifying leaf spectra using a variety of methods/models, such as the Kubelka-Munk theory (Kubelka and Munk, 1931), plate model (Allen et al., 1970; Féret et al., 2017; Jacquemoud and Baret, 1990), ray-tracing approach (Allen et al., 1973; Govaerts et al., 1996) and stochastic method (Maier et al., 1999; Tucker and Garatt, 1977). Particularly, the PROSPECT plate model
(Jacquemoud and Baret, 1990) serves as an accurate and probably the most extensively-used leaf optical model, and pioneers the simulations of leaf spectra for various plant species, the sensitivity analysis of canopy radiative transfer simulations, and the retrieval of leaf functional traits using physically-based inversions (Féret et al., 2019; Jacquemoud and Baret, 1990; Pedrós et al., 2010; Stuckens et al., 2009).

Recent advances in the spectral invariants theory (Stenberg et al., 2016) open a new opportunity to rethink the approach for vegetation radiative transfer modeling. The concept of the spectral invariants theory has been initiated and evolved from the 3-D radiative transfer processes because the interaction probabilities for photons in vegetation media rely on vegetation structure rather than photon frequency or vegetation optical properties (Stenberg et al., 2016). It demonstrates how to eliminate vegetation spectra dependencies on wavelength using a set of simple algebraic combinations along with one spectrally invariant parameter, i.e., photon recollision probability (*p*). This parameter describes the probability that a photon scatters due to interactions with phytoelements (leaf or needle) and collides again with vegetation before escaping the vegetation, which satisfies the law of energy conservation (Knyazikhin et al., 2013). For its simple and accurate parameterization, the spectral invariants theory has been widely employed for a variety of canopy-scale applications, including the simulations of canopy bidirectional reflectance factor (BRF) (Möttus and Stenberg, 2008; Zeng et al., 2018) and canopy albedo and absorption (Stenberg et al., 2013), quantification of atmospheric radiative transfer processes (Marshak et al., 2011; Marshak and Knyazikhin, 2017), retrieval of LAI and fraction of absorbed photosynthetically active radiation (fPAR) from satellite measurements (Majasalmi et al., 2014; Yan et al., 2016a, 2016b), and estimation of surface fluxes in land surface models (Majasalmi and Bright, 2019). The spectral invariants theory also provides an unprecedented simple approach for quantifying canopy-scale solar-induced chlorophyll fluorescence (SIF) (Liu et al., 2019; Köhler et al., 2018; Yang et al., 2018; Zeng et al., 2019, 2020), which is a new and powerful proxy for estimating gross primary productivity of terrestrial ecosystems. These studies
collectively suggest that the spectral invariants theory offers an important tool for characterizing canopy-scale radiative transfer process.

However, the application of the spectral invariants theory for the leaf-scale radiative transfer remains to be underexplored. A preliminary modeling effort by Lewis and Disney (2007) used the recollision probability *p* within the leaf to parameterize the leaf single scattering albedo and expressed it as a function of wavelength-dependent refractive index (see Eq. 15 of Lewis and Disney, 2007). This work did not fully follow the spectral invariants theory because *p* is expected to be wavelength-independent (Mõttus and Stenberg, 2008; Smolander and Stenberg, 2005; Stenberg, 2007). In addition, leaf reflectance and leaf transmittance, which are more widely used than leaf single scattering albedo in various applications (Sun et al., 2018a), have not yet been fully quantified in a way following the spectral invariants theory. Addressing these deficiencies demands fundamental researches that explore the feasibility of applying spectral invariants theory for simulating leaf-level radiative transfer and quantifying leaf-scale optical properties.

This study aims to develop a new leaf-scale radiative transfer model based on the spectrally invariant properties (hereafter referred to the leaf-SIP model) of plant leaves for quantifying leaf spectra (i.e., leaf single scattering albedo, leaf reflectance, and leaf transmittance). Specifically, we first modeled leaf single scattering albedo with a leaf-scale *p* in a similar way for modeling the canopy albedo (Stenberg, 2007; Stenberg et al., 2013, 2016). We then partitioned the leaf single scattering albedo into leaf reflectance and leaf transmittance using a spectral-invariant scattering asymmetry parameter (hereafter referred to *q*; Mõttus and Stenberg, 2008). In the following sections, we first described the methodology for model development and parameterization, followed by the model evaluation against two observational datasets and the well-validated PROSPECT leaf optical model. We then presented the results and conclusions of this study in the subsequent sections.
## 2. Model development

### 2.1 The leaf-SIP model

The leaf-SIP model assumes that radiative transfer within plant leaves is a stochastic process. In principle, a photon will be absorbed or scattered when it interacts with a 'basic' leaf internal scattering element (the green ellipses in Fig.1). This 'basic' leaf internal scattering element is a conceptual term in our model, and it can be interpreted as an element that is composed of the chloroplasts, nucleus, vacuole etc. and represents the average optical properties in the successive collision and scattering processes. Particularly, the probability that a photon collides with leaf internal scattering element again (recollision) only depends on the leaf structure rather than its wavelength (Stenberg et al., 2016). Similar to the canopy-scale single scattering albedo, the leaf single scattering albedo (ω) in the leaf-SIP model is calculated as the accumulated geometric series of scatterings from a smaller structural organization with a basic reference scattering albedo of ω₀ and a recollision probability p (Lewis and Disney, 2007):

$$ 
\begin{aligned}
\omega &= \omega_0 \times (1-p) + \omega_0 \times p \times \omega_0 \times (1-p) + \omega_0 \times (p \times \omega_0)^2 \times (1-p) + \dots \\
&= \omega_0 \frac{1-p}{1-p\omega_0}
\end{aligned}
\qquad (1) $$

where the reference scattering albedo ω₀ is directly related to the absorption spectrum Aλ of the leaf biochemical constituents (Lewis and Disney, 2007; Marshak and Knyazikhin, 2017), and ω₀ is defined as:

$$ \omega_0 = \exp(-A_{\lambda}) = \exp(-\sum_{i=1}^{m} \frac{C_i \times k_i(\lambda)}{\beta \times LMA}) \qquad (2) $$

where m is the number of leaf biochemical constituents; Cᵢ and kᵢ (λ) are the content per unit leaf area and specific absorption coefficient of ith leaf biochemical constituent, respectively, which usually includes chlorophyll a+b (Cₐb), carotenoids (Cₓc), anthocyanin (Cₐnth), brown pigments (Cₐbrown), equivalent water thickness (Cₐw), and leaf mass per area (LMA). Note that Cᵢ in µg/cm² is the accumulated constituent per unit leaf area, thus we used β×LMA in the denominator (Eq. 2) to
scale it to a density metric (µg/cm³), because LMA is a measurable key leaf structural trait that
can represent the leaf thickness (Choong et al., 1992). β is a constant for the scaling to the
optimal size of the leaf internal scattering element (see Section 3.2).

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Schematic representation of the leaf-SIP model. *R* and *T* are the leaf reflectance and leaf transmittance, respectively; *ω*₀ is the reference albedo of the leaf internal scattering element; *p* is the recollision probability; *q* is the asymmetry parameter determining the fractions of photon scattering in the upward and downward directions. All variables are wavelength-dependent except *p* and *q*. The green ellipse represents the leaf internal scattering element, which is composed of the chloroplasts, nucleus, vacuole, etc.</figcaption></figure>

When a photon is scattered by the leaf internal scattering element, it can escape from leaf
in either upward or downward direction. The upward escape probability ρ_up,i of a scattered
photon depends on the scattering order i (Mõttus and Stenberg, 2008), and we thus used an
asymmetric parameter q to quantify such directional dependence, i.e., ρ_up,i = ρ/2 × (1+q|q|⁻¹), where
ρ is the photon escape probability (i.e., ρ=1-p). Here -1<q<1, where q>0 refers to the case of
dominant upward scattering (i.e., leaf reflectance > leaf transmittance) and q<0 refers to the case
of dominant downward scattering (i.e., leaf reflectance < leaf transmittance). We used |q| to
ensure that the direction of each order of photon scattering is consistent. With that, similar to the
derivation of canopy reflectance with spectrally invariant parameters, the leaf reflectance R can
be calculated using the following geometric series (Mõttus and Stenberg, 2008):
$$
\begin{align*}
R &= \omega_0 \rho_{up,1} + \omega_0 p \omega_0 \rho_{up,2} + \omega_0 (p \omega_0)^2 \rho_{up,3} + \dots \\
&= \omega_0 \frac{\rho}{2} (1+q) + \omega_0^2 p \frac{\rho}{2} (1+q|q|) + \omega_0^3 p^2 \frac{\rho}{2} (1+q|q|^2) + \dots \\
&= \frac{1-p}{2p} \sum_{i=1}^{\infty} \omega_0^i p^i + \frac{1-p}{2p} \sum_{i=1}^{\infty} \omega_0^i p^i q|q|^{i-1} \\
&= \frac{1-p}{2p} \frac{\omega_0 p}{1-\omega_0 p} + \frac{1-p}{2} \frac{\omega_0 q}{1-\omega_0 p|q|} \\
&= \frac{1}{2} \frac{\omega_0 (1-p)}{1-\omega_0 p} + \frac{1-p}{2p} \frac{\omega_0 (1-p)}{1-\omega_0 p} \frac{1-\omega_0 p}{\omega_0 (1-p)} \frac{\omega_0 p q}{1-\omega_0 p|q|} \\
&= \frac{\omega}{2} + \frac{\omega q (1-\omega_0 p)}{2 (1-\omega_0 p|q|)} \\
&= \omega \left( \frac{1}{2} + \frac{q (1-\omega_0 p)}{2 (1-\omega_0 p|q|)} \right) \tag{3}
\end{align*}
$$

and the leaf transmittance $T$ is calculated as the difference between leaf single scattering albedo and leaf reflectance (Mõttus and Stenberg, 2008):

$$ T = \omega - R = \omega \left( \frac{1}{2} - \frac{q (1 - \omega_0 p)}{2 (1 - \omega_0 p |q|)} \right) \quad (4) $$

## 2.2 Modeling spectrally invariant properties p and q

The $p$ and $q$, as described in Section 2.1, are two critical parameters that determine the leaf spectra and by definition are only related to leaf structure as they are wavelength-independent (spectrally invariant) properties. Here we used LMA to parameterize $p$ and $q$. Similar to the canopy-scale SIP model (Stenberg, 2007; Zeng et al., 2018), $p$ can be modeled as:

$$ p = 1 - \frac{1 - \exp(-f(LMA))}{f(LMA)} \quad (5) $$

where $f(LMA)$ is a monotonically increasing function of LMA (see Section 3.2), and in concept it is comparable to LAI in the canopy-scale SIP model (see Eq. A2 in Appendix A).

When the plant leaf is infinitely thin (i.e., $LMA \to 0$), we have $p \to 0$ and $R \to 0$. With $p=0$ in Eq. 3, leaf reflectance $R$ can be simplified as:

$$ R = \omega \left( \frac{1}{2} + \frac{q (1 - \omega_0 \cdot 0)}{2 (1 - \omega_0 \cdot 0 \cdot |q|)} \right) = \omega \left( \frac{1}{2} + \frac{q}{2} \right) = 0 \quad (6) $$

We thus have $q=-1$. Similarly, when the leaf is infinitely thick (i.e., $LMA \to \infty$), we have
$q>0$ (i.e., dominant leaf reflectance) and $T→0$, and then Eq. 4 can be derived as:

$$T = \omega \cdot \frac{1}{2} \cdot \frac{(1 - \omega_0 pq) - q(1 - \omega_0 p)}{1 - \omega_0 pq} = \omega \cdot \frac{1}{2} \cdot \frac{1 - q}{1 - \omega_0 pq} = 0 \quad (7)$$

We thus have $q=1$. We further modeled $q$ with an exponential function of LMA that meets the above two conditions (i.e., $q=-1$ when $LMA→0$ and $q=1$ when $LMA→∞$), following the Beer's law which is commonly used in understanding the light attenuation, and resulting in:

$$q = 1 - 2 \times \exp(-g(LMA)) \quad (8)$$

where $g(LMA)$ is a monotonically increasing function of LMA and calibrated in Section 3.2.

# 3. Materials and methods

## 3.1 Datasets

We used two independent *in situ* datasets (i.e., LOPEX and ANGERS, Féret et al., 2008) for calibrating and evaluating the leaf-SIP model. Both the LOPEX and ANGERS datasets cover a wide range of plant species (88 needle leaf and broadleaf species in total). They provide a total of 606 sets (with exclusion of 10 sets due to the missing of leaf functional traits) of leaf reflectance and leaf transmittance spectra from 400 nm to 2400 nm (with 1-nm spectral sampling interval), and the associated leaf functional traits, including the leaf structure parameter $N$ inverted by the PROSPECT model, chlorophyll a+b content $C_{ab}$, carotenoid content $C_{xc}$, equivalent water thickness $C_w$, and LMA (Féret et al., 2008). The two datasets are by far the most complete datasets suitable for the development and evaluation of leaf-scale optical model. In this study, we used the whole leaf spectra measurements of the two datasets across all species as the calibration and validation datasets (see Section 3.2 below). Table 1 and Table 2 summarize the leaf structural and biochemical parameters of the LOPEX and ANGERS datasets, respectively.

<table><caption>**Table 1.** Leaf structural and biochemical parameters of the LOPEX dataset. This dataset is available at http://opticleaf.ipgp.fr/index.php?page=database</caption><thead><tr><td>Parameter</td><td>Symbol</td><td>Unit</td><td>Min</td><td>Max</td><td>Mean</td><td>Std.</td></tr></thead><tbody><tr><td>Leaf structure parameter</td><td>N</td><td>—</td><td>1.093</td><td>3.000</td><td>1.668</td><td>0.326</td></tr></tbody></table>
<table><caption>**Table 2.** Leaf structural and biochemical parameters of the ANGERS dataset. This dataset is available at http://opticleaf.ipgp.fr/index.php?page=database</caption><tr><td>Chlorophyll content</td><td>C<sub>ab</sub></td><td>µg/cm<sup>2</sup></td><td>1.362</td><td>98.802</td><td>47.280</td><td>17.312</td></tr><tr><td>Carotenoid content</td><td>C<sub>xc</sub></td><td>µg/cm<sup>2</sup></td><td>3.450</td><td>28.350</td><td>10.306</td><td>4.219</td></tr><tr><td>Water content</td><td>C<sub>w</sub></td><td>g/cm<sup>2</sup></td><td>0.002</td><td>0.052</td><td>0.011</td><td>0.007</td></tr><tr><td>Leaf mass per area</td><td>LMA</td><td>g/cm<sup>2</sup></td><td>0.002</td><td>0.016</td><td>0.005</td><td>0.002</td></tr></table>

<table><thead><tr><th>Parameter</th><th>Symbol</th><th>Unit</th><th>Min</th><th>Max</th><th>Mean</th><th>Std.</th></tr></thead><tbody><tr><td>Leaf structure parameter</td><td>N</td><td>—</td><td>1.046</td><td>2.709</td><td>1.525</td><td>0.256</td></tr><tr><td>Chlorophyll content</td><td>C<sub>ab</sub></td><td>µg/cm<sup>2</sup></td><td>0.780</td><td>106.718</td><td>33.988</td><td>21.722</td></tr><tr><td>Carotenoid content</td><td>C<sub>xc</sub></td><td>µg/cm<sup>2</sup></td><td>0.000</td><td>25.280</td><td>8.687</td><td>5.076</td></tr><tr><td>Water content</td><td>C<sub>w</sub></td><td>g/cm<sup>2</sup></td><td>0.004</td><td>0.034</td><td>0.012</td><td>0.005</td></tr><tr><td>Leaf mass per area</td><td>LMA</td><td>g/cm<sup>2</sup></td><td>0.002</td><td>0.033</td><td>0.005</td><td>0.004</td></tr></tbody></table>

## 3.2 Calibrating *p* and *q*

We used the LOPEX and ANGERS datasets to calibrate *p* and *q* following the next steps. First, we obtained an optimal *β* for determining the optimal size of the leaf internal scattering element by minimizing the difference between the leaf-SIP simulations and *in situ* datasets using a least-square approach. Specifically, we sampled *β* within the range of 50 to 1000 at a step of 50, and *p* within the range of 0 to 1 at a step of 0.001. Then we calculated *ω* using Eqs. 1 and 2 together with the leaf structural and biochemical parameters as shown in Tables 1 and 2, and finally compared the leaf-SIP model simulated *ω* with the *in situ* reference (i.e., leaf reflectance plus leaf transmittance). We found the RMSE of simulated *ω* decreases with *β* and becomes stable when *β* is larger than 600 (Fig. S1). We thus selected 600 as the optimal value of *β*. Second, we randomly separated the full dataset into calibration (80%) and independent validation (20%) subsets and repeated this separation for 100 times to generate the bootstrapped calibration and validation datasets. Third, for each *LMA* within the calibration datasets, we estimated *p* by solving Eq. 1 with the known *ω*₀ from the optimal *β* and leaf functional traits, and generated the
pairs of LMA and p. Then we generated the pairs of LMA and q by solving q from Eqs. 3 and 4 with the known R, T, p, and ω.

As a result, we derived the p~LMA (Fig. 2a) and q~LMA (Fig. 2b) relationships by fitting the terms of f(LMA) and g(LMA) in Eqs. 5 and 10 using a least-square approach, where these two terms are modeled as a linear function of LMA:

$$f(LMA) = k_1 \cdot LMA \qquad (9)$$

and

$$g(LMA) = k_2 \cdot LMA \qquad (10)$$

resulting in the optimal values of 2765.0 and 102.8 for k₁ and k₂, respectively.

The p~LMA and q~LMA relationships were tested/evaluated using the calibration/validation data subsets (Figs. S2-S4) and showed good agreements with the in situ reference.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Relationships between two spectrally invariant parameters (i.e., p and q) of the leaf-SIP model and LMA, including (a) p~LMA and (b) q~LMA relationships.</figcaption></figure>

### 3.3 Evaluating the leaf-SIP model

The leaf-SIP model was evaluated in two ways. First, we used the leaf chemical constituents provided by the LOPEX and ANGERS datasets to drive the leaf-SIP model, and compared the simulated leaf spectra with the corresponding measured leaf spectra reported in the
two datasets. We also compared the performance of the leaf-SIP model with the widely-used PROSPECT leaf optical model (Jacquemoud and Baret, 1990). Since the leaf anthocyanin $C_{anth}$ and brown pigments $C_{brown}$ contents were not available in these two datasets, the default values of these two parameters were used (i.e., $C_{anth}=0$ and $C_{brown}=0$, Vilfan et al., 2016) in the leaf-SIP and PROSPECT models to simulate leaf reflectance and leaf transmittance.

Second, for testing the leaf-SIP model's capability of retrieving leaf traits via spectrum, we compared three key leaf structural and biochemical traits (i.e., $C_{ab}$, $C_{w}$, and LMA) that derived from the leaf-SIP model with *in situ* measurements. Specifically, we searched for the optimal values of leaf function traits that minimize the difference between the simulated leaf single scattering albedo and *in situ* observation. The values of leaf functional traits (i.e., minimum, maximum, and initial values) were constrained in the reasonable ranges (Table S1) as suggested by Vilfan et al. (2016). Particularly, we used the weighted merit function with field measurements to retrieve LMA according to Colombo et al. (2008) and Féret et al. (2019). The optimization process was performed with a nonlinear least squares problem solver, i.e., the 'lsqnonlin' function in Matlab R2018a (MathWorks, Natick, Massachusetts, USA).

## 4. Results

### 4.1 Performance of leaf-SIP model for leaf spectra simulations

We compared the means and standard deviations of leaf single scattering albedo, leaf reflectance and leaf transmittance from both the LOPEX and ANGERS datasets, with the corresponding simulations of the leaf-SIP and PROSPECT models. Our results show that the leaf-SIP and PROSPECT models well captured the shapes of leaf spectra, and overall their simulations were in good agreements with the *in situ* measurements (Fig. 3). Compared with the PROSPECT model, the leaf-SIP model showed higher accuracy in simulating leaf optical properties for the wavelength region of 1400-2000 nm, lower accuracy for the wavelength regions below 500 nm and beyond 2000 nm, and comparable accuracy in other wavelength
258 regions.

<figure><img src="image_4.png" /></figure>

259

260 **Fig. 3.** Comparisons between the *in situ* measurements and the leaf-SIP model simulations for
261 (*a*) means of leaf single scattering albedo, (*b*) means of leaf reflectance, and (*c*) means of leaf
262 transmittance. The simulation results from the PROSPECT model (blue lines) are also presented
263 for comparison. Shade regions of slight gray, slight pink, slight blue indicate the one standard
264 deviation of the *in situ* measurements, leaf-SIP simulations, and PROSPECT simulations,
265 respectively.

266 The leaf spectra simulated by the leaf-SIP model were highly consistent with *in situ*
267 measurements of LOPEX and ANGERS, with a bias of 0.030, 0.026, and 0.036, a RMSE of
268 0.041, 0.035, and 0.048, a R² of 0.98, 0.95, and 0.91 for leaf single scattering albedo, leaf
269 reflectance and leaf transmittance, respectively (Fig. 4). The performance of the leaf-SIP model
270 was comparable with the PROSPECT model (Fig. S5). Similar conclusions can be drawn from
271 the comparisons between the simulated leaf spectra and data from LOPEX or ANGERS (Fig. S6
272 for the leaf-SIP model and Fig. S7 for the PROSPECT model).
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Density scatter plots of *in situ* measurements and leaf-SIP simulations across the entire solar reflective domain of 400-2400 nm, including (**a**, **d**) measured vs. leaf-SIP simulated leaf single scattering albedo, (**b**, **e**) measured vs. leaf-SIP simulated leaf reflectance, (**c**, **f**) measured vs. leaf-SIP simulated leaf transmittance. Top panels show the calibration results using the training data from 80% of *in situ* leaf spectra (i.e., 477 sets of leaf spectra), with 100 replicates. The bottom panels show validation results using the dataset set aside 20% of *in situ* leaf spectra (i.e., 119 sets of leaf spectra) with 100 replicates.</figcaption></figure>

## 4.2 Performance of leaf-SIP on leaf structural and biochemical trait retrievals

We also compared the measured leaf structural and biochemical traits of the LOPEX and ANGERS datasets with those obtained by inversing the leaf-SIP model. Our results show that both leaf structural and biochemical traits were accurately retrieved by the leaf-SIP model with small errors compared to the ANGERS dataset (Fig. 5a-c): $C_{ab}$ (RMSE=6.94 µg/cm²), $C_w$ (RMSE=0.0026 g/cm²), and *LMA* (RMSE=0.0022 g/cm²). For the LOPEX dataset (Fig. 5d-f), the leaf structural and biochemical traits were also accurately retrieved: $C_w$ (RMSE=0.0019 g/cm²), and *LMA* (RMSE=0.0018 g/cm²). We found less accuracy in $C_{ab}$ estimations (RMSE=13.81 µg/cm²), primarily shown as the “vertical lines” for the leaf-SIP retrieved results
as shown in Fig. 5d. These errors are due to that the reference LOPEX dataset assigns the mean leaf $C_{ab}$ of five samples to each of them, while the actual leaf $C_{ab}$ vary for individual samples (Sun et al., 2018a). Similar comparable results in retrieving leaf structural and biochemical traits were shown for the PROSPECT model (Fig. S8).

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Evaluation of leaf structural and biochemical traits retrieved by the leaf-SIP model using *in situ* measurements of leaf chlorophyll $C_{ab}$ (**a**, **d**), equivalent water thickness $C_w$ (**b**, **e**), and dry matter content LMA (**c**, **f**). Top panels show the retrieval results based on the ANGERS dataset, and bottom panels show the retrieval results based on the LOPEX dataset.</figcaption></figure>

## 5. Discussion

Leaf spectra provide a unique source of information for characterizing plant physiological and biophysical properties in a noninvasive manner. It is a powerful indicator of the spatio-temporal variations of many plant functional traits (Wu et al., 2019) and governs a number of ecological and biogeochemical processes. In this study, we demonstrated that the spectral invariants theory is suitable for quantifying leaf-scale radiative transfer process by decoupling leaf spectra into wavelength-dependent contribution by leaf chemical components,
and wavelength-independent contribution by leaf structures, through only two parameters of *p* and *q*. With *LMA* as an indicator of leaf structure, the proposed leaf-SIP model shows promising performance when compared with the well-established and widely-used PROSPECT model, and two independent *in situ* datasets. Our work thus represents a major step toward using the spectral invariants theory to characterize the leaf-canopy radiative transfer process. As the spectrally invariant properties are directly linked with the leaf internal structures, the leaf-SIP model also has a good potential to effectively account for the complex 3-D leaf structure effects on leaf spectra, which requires further exploration.

The spectral invariants theory emerges as a new approach for quantifying vegetation radiative transfer, including both forward modeling and backward inversion, which offers great potential for deriving leaf and plant canopy attributes from remotely sensed hyperspectral measurements that have been intensively conducted by *in situ*, airborne, and satellite sensors. On the model development side, this theory complements the classic approaches. The leaf-SIP model developed in this study, for the first time to our knowledge, adapted the spectral invariants theory to the leaf scale for simulating leaf reflectance and leaf transmittance. Our results indicate that this model has a similar level of accuracy as compared with the PROSPECT model, suggesting the leaf-SIP model may provide an alternative approach to simulate leaf spectra. Moreover, the spectral invariants theory has also been used to quantify the radiative transfer process for vegetation canopy (Zeng et al., 2018) and atmosphere layer (Marshak et al., 2011). Therefore, there is a potential to integrate the soil-leaf-canopy and biosphere-atmosphere radiative transfer processes into a unified leaf-SIP model framework. Such integration will closely link the ground hyperspectral measurements with the top-of-atmosphere (TOA) radiance observations, and enable the realistic simulation of TOA radiances that are directly comparable to the remote sensing satellite observations (Verhoef and Bach, 2007). Such capability will also help in the definition, design, and development of the next generation of space-based spectroradiometers, as well as being used in the analysis and interpretation of observations from such instruments.
A recent evaluation of leaf-level optical properties that used in the land surface model also suggested the feasibility of using spectral invariants theory for facilitating the estimation of the global land surface energy budget (Majasalmi and Bright, 2019). Due to the simplicity and high accuracy of the leaf-SIP model, it can be possibly used in future land surface models to improve the representation of land surface heterogeneity from the perspective of trait-based theory (Funk et al., 2017; Reich 2014; Yuan et al., 2014). In addition, spectra-based plant functional trait has received increasing attention from multiple ecology fields (e.g., plant ecophysiology, functional trait, ecology and evolution) because of the close connections between key leaf traits and vegetation spectra (Osnas et al., 2013; Wu et al., 2019). Our results suggest that the leaf-SIP model can be employed as an efficient physical approach to estimate the key leaf structural and functional traits, such as chlorophyll content, water content, LMA, and potentially other traits.

The spectral invariants theory recently has also been successfully extended for advancing the radiative transfer process relating to solar-induced chlorophyll fluorescence (SIF) (Yang et al., 2018; Zeng et al., 2019), which represents a major breakthrough for space-borne observation of global vegetation photosynthesis that controls global carbon and major component of water cycle. Such advances include estimating the escape probability of SIF (Zeng et al., 2019), downscaling SIF from canopy scale to photosystem scale (Liu et al., 2019), and quantifying total SIF from measurements of observed directional SIF (Zeng et al., 2020) in a simple and analytical way. Despite such advances, with very few studies on leaf-scale SIF, we expect a leaf-scale SIP model for SIF can be derived from the optical SIP model because they share the same model physics except for the emission process for the SIF. Combining the SIP-based SIF models across the leaf and canopy scales (Zeng et al., 2020), will enable downscaling SIF from canopy level to photosystem level, which is more closely related to plant photosynthetic activity.

Despite the encouraging results from our study, there remain a few limitations for the leaf-SIP model to be resolved for further advance. First, the ‘basic’ leaf internal scattering
element, on which the spectrally invariant properties *p* and *q* were built, is still conceptual and not directly explainable with real leaf structural constituents, and does not consider the contributions of some real leaf constituents (e.g., cell walls and wax cuticle) to the leaf optical properties (Jacquemoud and Baret, 1990). These deficiencies might explain some of the errors in the simulated leaf optical properties when compared with the *in situ* measurements (Fig. 3).

Second, the leaf-SIP model assumes *LMA* can be an indicator of the leaf thickness (Choong et al., 1992) in the estimation of the reference scattering albedo $\omega_0$, *p* and *q* (Eqs. 2, 5 and 10). This assumption might lead to some uncertainties in estimating spectrally invariant parameters, such as *q* (Fig. 2b), although it has limited impact on the partition of leaf reflectance and leaf transmittance from leaf single scattering albedo (Fig. 3). Finally, the leaf-SIP model neglects the bidirectional reflectance distribution function (BRDF) effects due to variations in the solar-sensor geometry, which could have an considerable influence on leaf optical properties because of the undulating shapes of the upper side of top leaf surface layer (Bousquet et al., 2005; Comar et al., 2014). Several efforts have been conducted for the characterization of leaf-scale BRDF effects by quantifying leaf surface roughness as a function of roughness parameter and incident angle $\alpha$ to account for the leaf specular reflection components (Bousquet et al., 2005; Sun et al., 2018b; Xie et al., 2016; Yang et al., 2018). Such BRDF effects are needed to be examined and considered in the leaf-SIP model in future, which however requires more high-quality leaf-scale BRDF measurements.

## 6. Conclusions

In this study, we adopted the concept of spectral invariants theory to quantify leaf optical properties. The spectrally invariant properties used in this study were designed to account for the effects of leaf structural properties (e.g., leaf intercellular structures) and the contents of light-absorbing compounds within leaves (e.g., pigment contents, equivalent water thickness, and dry matter content). Two *in situ* leaf spectrum datasets were used for developing and evaluating
the performance of the leaf-SIP model through the forward model simulations, and backward inversions of leaf functional trait.

The forward simulation of the leaf-SIP model matches very well with the *in situ* leaf spectra measurements obtained from the LOPEX and ANGERS datasets. The statistical analysis of accuracy metrics are promising: for leaf single scattering albedo, RMSE=0.041, Bias=0.030, R²=0.98; for leaf reflectance, RMSE=0.035, Bias=0.026, R²=0.95; and for leaf transmittance, RMSE=0.048, Bias=0.036, R²=0.91, respectively. For the retrieval of leaf functional traits, the leaf-SIP model was used for estimating chlorophyll content $C_{ab}$ with RMSEs of 6.94-13.81 µg/cm²; equivalent water thickness $C_w$, with RMSE of 0.0019-0.0026 g/cm²; and LMA with the RMSE of 0.0018-0.0022 g/cm². The results collectively suggest that the leaf-SIP model provides a physical and efficient approach for modeling leaf optical properties, and offers a good potential for retrieving leaf functional traits, as well as the other broader applications, such as improving trait-based ecosystem models and advancing the interpretation of remote sensed SIF measurements.

## Acknowledgments

This work was supported by the Pacific Northwest National Laboratory (PNNL) Directed Research and Development funds. The authors thank Drs. Dong Huang, Yuri Knyazikhin, and Ranga B. Myneni for the discussions on the spectral invariants theory and radiative transfer modeling.

## Appendix A. Derivation of *p*-LAI relationship

The canopy-scale *p* can be expressed as a function of the canopy directional light interceptance $i_0$ and leaf area index (LAI) (see Eq. 10 in Stenberg, 2007)

$$p = 1 - \frac{i_0}{LAI} = 1 - \frac{1 - \exp\left(-\frac{G(\theta_0, \theta_l) \cdot LAI}{\cos\theta_0}\right)}{LAI} \quad (A1)$$

where $\theta_0$ is the solar zenith angle; $\theta_l$ is the leaf inclination angle; G is the leaf projection function.
For the reflectance/transmittance of the leaf layer in the vertical direction, the situation is similar to the canopy with horizontal leaves (i.e., $\theta_l=0^\circ$) and when the sun is at nadir (i.e., $\theta_0=0^\circ$). This leads to $G(\theta_0, \theta_l)=1$ and $\cos\theta_0=1$, and then

$$p = 1 - \frac{1 - \exp(-LAI)}{LAI} \qquad (A2)$$

## Appendix B. How to get access to the leaf-SIP model

We have released the source code of the leaf-SIP model at https://github.com/chenminbnu/SIP-RTM-Leaf. Please contact the authors by email: yelu.zeng@pnnl.gov (Y.Z.) and min.chen@pnnl.gov (M.C.) in case of any questions.
415 **Reference**

416 Ainsworth, E.A., Serbin, S.P., Skoneczka, J.A., Townsend, P.A. 2014. Using leaf optical
417 properties to detect ozone effects on foliar biochemistry. *Photosynthesis Research*, 119, 65-76.
418 https://doi.org/10.1007/s11120-013-9837-y

419 Allen, W.A. 1973. Transmission of isotropic light across a dielectric surface in two and three
420 dimensions. *Journal of the Optical Society of America*, 63, 664-666.
421 https://doi.org/10.1364/JOSA.63.000664

422 Allen, W.A., Gausman, H.W., Richardson, A.J. 1970. Mean effective optical constants of
423 cotton leaves. *Journal of the Optical Society of America*, 60, 542-547

424 Asner, G.P., Martin, R.E. 2008. Spectral and chemical analysis of tropical forests: Scaling
425 from leaf to canopy levels. *Remote Sensing of Environment*, 112, 3958-3970.
426 https://doi.org/10.1016/j.rse.2008.07.003

427 Asner, G.P., Martin, R.E., Tupayachi, R., Emerson, R., Martinez, P., Sinca, F., Powell, G.V.,
428 Wright, S.J., Lugo, A.E. 2011. Taxonomy and remote sensing of leaf mass per area (LMA) in
429 humid tropical forests. *Ecological Applications*, 21, 85-98. https://doi.org/10.1890/09-1999.1

430 Bousquet, L., Lachérade, S., Jacquemoud, S., Moya, I. 2005. Leaf BRDF measurements and
431 model for specular and diffuse components differentiation. *Remote Sensing of Environment*, 98,
432 201-211. https://doi.org/10.1016/j.rse.2005.07.005

433 Choong, M., Lucas, P., Ong, J., Pereira, B., Tan, H., Turner, I. 1992. Leaf fracture toughness
434 and sclerophylly: their correlations and ecological implications. *New Phytologist*, 121, 597-610.
435 https://doi.org/10.1111/j.1469-8137.1992.tb01131.x

436 Colombo, R., Meroni, M., Marchesi, A., Busetto, L., Rossini, M., Giardino, C., Panigada, C.
437 2008. Estimation of leaf and canopy water content in poplar plantations by means of
438 hyperspectral indices and inverse modeling. *Remote Sensing of Environment*, 112, 1820-1834.
439 https://doi.org/10.1016/j.rse.2007.09.005

440 Comar, A., Baret, F., Obein, G., Simonot, L., Meneveaux, D., Viénot, F., De Solan, B. 2014.
441 ACT: A leaf BRDF model taking into account the azimuthal anisotropy of monocotyledonous
442 leaf surface. *Remote Sensing of Environment*, 143, 112-121.
443 https://doi.org/10.1016/j.rse.2013.12.006

444 Féret, J.-B., François, C., Asner, G.P., Gitelson, A.A., Martin, R.E., Bidel, L.P., Ustin, S.L.,
445 Le Maire, G., Jacquemoud, S. 2008. PROSPECT-4 and 5: Advances in the leaf optical properties
446 model separating photosynthetic pigments. *Remote Sensing of Environment*, 112, 3030-3043.
447 https://doi.org/10.1016/j.rse.2008.02.012

448 Féret, J.-B., Gitelson, A., Noble, S., Jacquemoud, S. 2017. PROSPECT-D: towards
449 modeling leaf optical properties through a complete lifecycle. *Remote Sensing of Environment*,
450 193, 204-215. https://doi.org/10.1016/j.rse.2017.03.004

451 Féret, J.-B., Le Maire, G., Jay, S., Berveiller, D., Bendoula, R., Hmimina, G., Cheraiet, A.,
452 Oliveira, J., Ponzoni, F., Solanki, T. 2019. Estimating leaf mass per area and equivalent water
thickness based on leaf optical properties: Potential and limitations of physical modeling and machine learning. *Remote Sensing of Environment*, 231, 110959.
https://doi.org/10.1016/j.rse.2018.11.002

Funk, J.L., Larson, J.E., Ames, G.M., Butterfield, B.J., Cavender-Bares, J., Firn, J., Laughlin, D.C., Sutton-Grier, A.E., Williams, L., Wright, J. 2017. Revisiting the Holy G rail: using plant functional traits to understand ecological processes. *Biological Reviews*, 92, 1156-1173. https://doi.org/10.1111/brv.12275

Govaerts, Y.M., Jacquemoud, S., Verstraete, M.M., Ustin, S.L. 1996. Three-dimensional radiation transfer modeling in a dicotyledon leaf. *Applied Optics*, 35, 6585-6598.
https://doi.org/10.1364/AO.35.006585

Jacquemoud, S., Baret, F. 1990. PROSPECT: A model of leaf optical properties spectra. *Remote Sensing of Environment*, 34, 75-91. https://doi.org/10.1016/0034-4257(90)90100-Z

Knyazikhin, Y., Schull, M.A., Stenberg, P., Mõttus, M., Rautiainen, M., Yang, Y., Marshak, A., Carmona, P.L., Kaufmann, R.K., Lewis, P. 2013. Hyperspectral remote sensing of foliar nitrogen content. *Proceedings of the National Academy of Sciences*, 110, E185-E192.
https://doi.org/10.1073/pnas.1210196109

Köhler, P., Guanter, L., Kobayashi, H., Walther, S., Yang, W. 2018. Assessing the potential of sun-induced fluorescence and the canopy scattering coefficient to track large-scale vegetation dynamics in Amazon forests. *Remote Sensing of Environment*, 204, 769-785.
https://doi.org/10.1016/j.rse.2017.09.025

Kubelka, P., Munk, F. 1931. An article on optics of paint layers. *Z. Tech. Phys*, 12

Lewis, P., Disney, M. 2007. Spectral invariants and scattering across multiple scales from within-leaf to canopy. *Remote Sensing of Environment*, 109, 196-206.
https://doi.org/10.1016/j.rse.2006.12.015

Liu, X., Guanter, L., Liu, L., Damm, A., Malenovský, Z., Rascher, U., Peng, D., Du, S., Gastellu-Etchegorry, J.-P. 2019. Downscaling of solar-induced chlorophyll fluorescence from canopy level to photosystem level using a random forest model. *Remote Sensing of Environment*, 231, 110772. https://doi.org/10.1016/j.rse.2018.05.035

Maier, S., Lüdeker, W., Günther, K. 1999. SLOP: A revised version of the stochastic model for leaf optical properties. *Remote Sensing of Environment*, 68, 273-280.
https://doi.org/10.1016/S0034-4257(98)00118-7

Majasalmi, T., Bright, R.M. 2019. Evaluation of leaf-level optical properties employed in land surface models. *Geoscientific Model Development*, 12, 3923-3938.
http://hdl.handle.net/11250/2635910

Majasalmi, T., Rautiainen, M., Stenberg, P. 2014. Modeled and measured fPAR in a boreal forest: validation and application of a new model. *Agricultural and Forest Meteorology*, 189, 118-124. https://doi.org/10.1016/j.agrformet.2014.01.015

Marshak, A., Knyazikhin, Y. 2017. The spectral invariant approximation within canopy radiative transfer to support the use of the EPIC/DSCOVR oxygen B-band for monitoring
vegetation. *Journal of Quantitative Spectroscopy and Radiative Transfer*, 191, 7-12.
https://doi.org/10.1016/j.jqsrt.2017.01.015

Marshak, A., Knyazikhin, Y., Chiu, J.C., Wiscombe, W. 2011. Spectrally invariant approximation within atmospheric radiative transfer. *Journal of the Atmospheric Sciences*, 68, 3094-3111. https://doi.org/10.1175/JAS-D-11-060.1

Meacham-Hensold, K., Montes, C.M., Wu, J., Guan, K., Fu, P., Ainsworth, E.A., Pederson, T., Moore, C.E., Brown, K.L., Raines, C. 2019. High-throughput field phenotyping using hyperspectral reflectance and partial least squares regression (PLSR) reveals genetic modifications to photosynthetic capacity. *Remote Sensing of Environment*, 231, 111176. https://doi.org/10.1016/j.rse.2019.04.029

Monson, R., Baldocchi, D. (2014). *Terrestrial biosphere-atmosphere fluxes*. Cambridge University Press

Mõttus, M., Stenberg, P. 2008. A simple parameterization of canopy reflectance using photon recollision probability. *Remote Sensing of Environment*, 112, 1545-1551. https://doi.org/10.1016/j.rse.2007.08.002

Osnas, J.L., Lichstein, J.W., Reich, P.B., Pacala, S.W. 2013. Global leaf trait relationships: mass, area, and the leaf economics spectrum. *Science*, 340, 741-744. 10.1126/science.1231574

Pedrós, R., Goulas, Y., Jacquemoud, S., Louis, J., Moya, I. 2010. FluorMODleaf: A new leaf fluorescence emission model based on the PROSPECT model. *Remote Sensing of Environment*, 114, 155-167. https://doi.org/10.1016/j.rse.2009.08.019

Reich, P. B. 2014. The world-wide 'fast-slow' plant economics spectrum: a traits manifesto. *Journal of Ecology*, 102(2), 275-301.

Schneider, F.D., Morsdorf, F., Schmid, B., Petchey, O.L., Hueni, A., Schimel, D.S., Schaepman, M.E. 2017. Mapping functional diversity from remotely sensed morphological and physiological forest traits. *Nat Communication*, 8, 1-12. https://doi.org/10.1038/s41467-017-01530-3

Serbin, S.P., Dillaway, D.N., Kruger, E.L., Townsend, P.A. 2011. Leaf optical properties reflect variation in photosynthetic metabolism and its sensitivity to temperature. *Journal of Experimental Botany*, 63, 489-502. https://doi.org/10.1093/jxb/err294

Serbin, S.P., Singh, A., McNeil, B.E., Kingdon, C.C., Townsend, P.A. 2014. Spectroscopic determination of leaf morphological and biochemical traits for northern temperate and boreal tree species. *Ecological Applications*, 24, 1651-1669. https://doi.org/10.1890/13-2110.1

Shiklomanov, A.N., Bradley, B.A., Dahlin, K.M., M Fox, A., Gough, C.M., Hoffman, F.M., M Middleton, E., Serbin, S.P., Smallman, L., Smith, W.K. 2019. Enhancing global change experiments through integration of remote-sensing techniques. *Frontiers in Ecology and the Environment*, 17, 215-224. https://doi.org/10.1002/fee.2031

Smolander, S., Stenberg, P. 2005. Simple parameterizations of the radiation budget of uniform broadleaved and coniferous canopies. *Remote Sensing of Environment*, 94, 355-363.
530 https://doi.org/10.1016/j.rse.2004.10.010

531 Stenberg, P., 2007. Simple analytical formula for calculating average photon recollision
532 probability in vegetation canopies. *Remote Sensing of Environment*, 109, 221–224.
533 https://doi.org/10.1016/j.rse.2006.12.014

534 Stenberg, P., Lukeš, P., Rautiainen, M., Manninen, T. 2013. A new approach for simulating
535 forest albedo based on spectral invariants. *Remote Sensing of Environment*, 137, 12-16.
536 https://doi.org/10.1016/j.rse.2013.05.030

537 Stenberg, P., Mõttus, M., Rautiainen, M. 2016. Photon recollision probability in modelling
538 the radiation regime of canopies—A review. *Remote Sensing of Environment*, 183, 98-108.
539 https://doi.org/10.1016/j.rse.2016.05.013

540 Stuckens, J., Verstraeten, W.W., Delalieux, S., Swennen, R., Coppin, P. 2009. A
541 dorsiventral leaf radiative transfer model: Development, validation and improved model
542 inversion techniques. *Remote Sensing of Environment*, 113, 2560-2573.
543 https://doi.org/10.1016/j.rse.2004.10.010

544 Sun, J., Shi, S., Yang, J., Du, L., Gong, W., Chen, B., Song, S. 2018a. Analyzing the
545 performance of PROSPECT model inversion based on different spectral information for leaf
546 biochemical properties retrieval. *ISPRS Journal of Photogrammetry and Remote Sensing*, 135,
547 74-83. https://doi.org/10.1016/j.isprsjprs.2017.11.010

548 Sun, Z., Wu, D., Zhao, Y. 2018b. Improved kernel-driven semi-empirical bidirectional
549 reflectance factor models for characterizing the reflection of vegetation covers: considering a
550 specular kernel. *Agricultural and Forest Meteorology*, 260, 95-108.
551 https://doi.org/10.1016/j.agrformet.2018.06.005

552 Tucker, C.J., Garratt, M.W. 1977. Leaf optical system modeled as a stochastic process.
553 *Applied Optics*, 16, 635-642. https://doi.org/10.1364/AO.16.000635

554 Verhoef, W., Bach, H. 2007. Coupled soil-leaf-canopy and atmosphere radiative transfer
555 modeling to simulate hyperspectral multi-angular surface reflectance and TOA radiance data.
556 *Remote Sensing of Environment*, 109, 166-182. https://doi.org/10.1016/j.rse.2006.12.013

557 Vilfan, N., Van der Tol, C., Muller, O., Rascher, U., Verhoef, W. 2016. Fluspect-B: A
558 model for leaf fluorescence, reflectance and transmittance spectra. *Remote Sensing of
559 Environment*, 186, 596-615. https://doi.org/10.1016/j.rse.2016.09.017

560 Wu, J., Rogers, A., Albert, L.P., Ely, K., Prohaska, N., Wolfe, B.T., Oliveira Jr, R.C.,
561 Saleska, S.R., Serbin, S.P. 2019. Leaf reflectance spectroscopy captures variation in
562 carboxylation capacity across species, canopy environment and leaf age in lowland moist
563 tropical forests. *New Phytologist*, 224, 663-674. https://doi.org/10.1111/nph.16029

564 Xie, D., Qin, W., Wang, P., Shuai, Y., Zhou, Y., Zhu, Q. 2016. Influences of leaf-specular
565 reflection on canopy BRF characteristics: A case study of real maize canopies with a 3-d scene
566 brdf model. *IEEE Transactions on Geoscience and Remote Sensing*, 55, 619-631.
567 https://doi.org/10.1109/TGRS.2016.2598442

568 Yan, K., Park, T., Yan, G., Chen, C., Yang, B., Liu, Z., Nemani, R.R., Knyazikhin, Y.,
569 Myneni, R.B. 2016a. Evaluation of MODIS LAI/FPAR product collection 6. Part 1: Consistency and improvements. *Remote Sensing*, 8, 359. https://doi.org/10.3390/rs8050359

571 Yan, K., Park, T., Yan, G., Liu, Z., Yang, B., Chen, C., Nemani, R.R., Knyazikhin, Y., 572 Myneni, R.B. 2016b. Evaluation of MODIS LAI/FPAR product collection 6. Part 2: Validation and intercomparison. *Remote Sensing*, 8, 460. https://doi.org/10.3390/rs8060460

574 Yang, B., Knyazikhin, Y., Zhao, H., Ma, Y. 2018. Contribution of leaf specular reflection to 575 canopy reflectance under black soil case using stochastic radiative transfer model. *Agricultural and Forest Meteorology*, 263, 477-482. https://doi.org/10.1016/j.agrformet.2018.08.024

577 Yang, P., van der Tol, C. 2018. Linking canopy scattering of far-red sun-induced chlorophyll 578 fluorescence with reflectance. *Remote Sensing of Environment*, 209, 456-467. https://doi.org/10.1016/j.rse.2018.02.029

580 Yang, X., Tang, J., Mustard, J.F., Wu, J., Zhao, K., Serbin, S., Lee, J.-E. 2016. Seasonal 581 variability of multiple leaf traits captured by leaf spectroscopy at two temperate deciduous 582 forests. *Remote Sensing of Environment*, 179, 1-12. https://doi.org/10.1016/j.rse.2016.03.026

583 Yendrek, C.R., Erice, G., Montes, C.M., Tomaz, T., Sorgini, C.A., Brown, P.J., McIntyre, L.M., Leakey, A.D., Ainsworth, E.A. 2017. Elevated ozone reduces photosynthetic carbon gain 585 by accelerating leaf senescence of inbred and hybrid maize in a genotype- specific manner. *Plant, Cell & Environment*, 40, 3088-3100. https://doi.org/10.1111/pce.13075

587 Yi, Y., Yang, D., Huang, J., Chen, D. 2008. Evaluation of MODIS surface reflectance 588 products for wheat leaf area index (LAI) retrieval. *ISPRS Journal of Photogrammetry and Remote Sensing*, 63, 661-677. https://doi.org/10.1016/j.isprsjprs.2008.04.004

590 Yonemura, S., Kodama, N., Taniguchi, Y., Ikawa, H., Adachi, S., Hanba, Y.T. 2019. A 591 high-performance system of multiple gas-exchange chambers with a laser spectrometer to 592 estimate leaf photosynthesis, stomatal conductance, and mesophyll conductance. *Journal of 593 Plant Research*, 1-14. https://doi.org/10.1007/s10265-019-01127-5

594 Yuan, H., Dickinson, R.E., Dai, Y., Shaikh, M.J., Zhou, L., Shangguan, W., Ji, D. 2014. A 595 3D canopy radiative transfer model for global climate modeling: Description, validation, and 596 application. *Journal of Climate*, 27, 1168-1192. https://doi.org/10.1175/JCLI-D-13-00155.1

597 Zeng, Y., Badgley, G., Chen, M., Li, J., Anderegg, L.D., Kornfeld, A., Liu, Q., Xu, B., Yang, 598 B., Yan, K. 2020. A radiative transfer model for solar induced fluorescence using spectral 599 invariants theory. *Remote Sensing of Environment*, 240, 111678. https://doi.org/10.1016/j.rse.2020.111678

600 Zeng, Y., Badgley, G., Dechant, B., Ryu, Y., Chen, M., Berry, J.A. 2019. A practical 601 approach for estimating the escape ratio of near-infrared solar-induced chlorophyll fluorescence. 602 *Remote Sensing of Environment*, 111209. https://doi.org/10.1016/j.rse.2019.05.028

604 Zeng, Y., Xu, B., Yin, G., Wu, S., Hu, G., Yan, K., Yang, B., Song, W., Li, J. 2018. Spectral 605 invariant provides a practical modeling approach for future biophysical variable estimations. 606 *Remote Sensing*, 10, 1508. https://doi.org/10.3390/rs10101508