# MARMIT-2: An improved version of the MARMIT model to predict soil reflectance as a function of surface water content in the solar domain

Alice Dupiau, Stéphane Jacquemoud, Xavier Briottet, Sophie Fabre, Françoise Viallefont-Robinet, William Philpot, Claudia Di Biagio, Mathieu Hébert, Paola Formenti

► To cite this version:

Alice Dupiau, Stéphane Jacquemoud, Xavier Briottet, Sophie Fabre, Françoise Viallefont-Robinet, et al.. MARMIT-2: An improved version of the MARMIT model to predict soil reflectance as a function of surface water content in the solar domain. Remote Sensing of Environment, 2022, 272, pp.112951. 10.1016/j.rse.2022.112951. hal-03620272

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Distributed under a Creative Commons Attribution 4.0 International License
1 **Date:** January 28, 2022

2 **Title:** MARMIT-2: an improved version of the MARMIT model to predict soil reflectance as a function of surface water content in the solar domain

4 **Authors:** A. Dupiau<sup>a,b</sup>, S. Jacquemoud<sup>a</sup>, X. Briottet<sup>b</sup>, S. Fabre<sup>b</sup>, F. Viallefont-Robinet<sup>b</sup>, W. Philpot<sup>c</sup>, C. Di Biagio<sup>d</sup>, M. Hébert<sup>e</sup>, P. Formenti<sup>d</sup>

6 <sup>a</sup> Université de Paris, Institut de physique du globe de Paris, CNRS, F-75005 Paris, France (dupiau@ipgp.fr, jacquemoud@ipgp.fr)

8 <sup>b</sup> ONERA/DOTA, Université Fédérale de Toulouse, 2 avenue Edouard Belin, BP 74025, 31055 Toulouse Cedex 4, France (xavier.briottet@onera.fr, sophie.fabre@onera.fr, francoise.viallefont@onera.fr)

10 <sup>c</sup> School of Civil & Environmental Engineering, Cornell University, 220 Hollister Hall, Ithaca, NY 14850, USA (philpot@cornell.edu)

12 <sup>d</sup> Université de Paris and Université Paris Est Créteil, CNRS, LISA, F-75013 Paris, France (claudia.dibiagio@lisa.ipsl.fr, paola.formenti@lisa.ipsl.fr)

14 <sup>e</sup> Univ Lyon, UJM-Saint-Etienne, CNRS, Institut d'Optique Graduate School, Laboratoire Hubert Curien 42023, Saint-Etienne, France (mathieu.hebert@institutoptique.fr)

16 **Keywords:** Hyperspectral remote sensing; Soil moisture content; MARMIT model; Mineral soil particles; Effective refractive index; Dot gain effect

18 **Highlights:**

19 • MARMIT-2, an improved version of the MARMIT model, is presented

20 • The spectral reflectance of wet soils is predicted with better accuracy

21 • The optical properties of a water layer containing soil particles are calculated

22 • A mixing function is used to calculate an effective complex refractive index
* A database of 225 soil samples measured in the laboratory is used for validation

**Journal:** *Remote Sensing of Environment*

**Abstract**

This paper presents MARMIT-2, a radiative transfer model that predicts the spectral reflectance of soils in the solar domain (0.4-2.5 µm) as a function of their surface moisture.

This is an improved version of MARMIT (multilayer radiative transfer model of soil reflectance) that represents a wet soil as a dry soil topped with a thin layer of liquid water.

The changes brought in this article concern the mixing of the spectral reflectance of the dry and wet soil areas, the transmission of diffuse light in the water layer, and the inclusion of soil particles in the water layer. Wet soil reflectance is now expressed in terms of dry soil reflectance and three free parameters: the thickness of the water layer, the surface fraction of the wet soil, and a new parameter, the volume fraction of soil particles in the water layer.

With more accurate physical modeling, MARMIT-2 simulates soil spectral reflectance with better accuracy than MARMIT. In particular, the fit of the soil reflectance spectra is much better for high water contents, both in the visible range (0.4-0.7 µm) and in the water absorption bands around 1.45 µm and 1.95 µm. The average root mean square error between measured and predicted reflectance obtained on a set of 225 soil samples is about 0.8% with MARMIT-2 versus 1.8% with MARMIT.

**1. Introduction**

Wet soils are darker than dry soils, even though water absorbs very little visible light (Lekner and Dorf, 1988). This commonly observed optical phenomenon is due to the fact that the
refractive index of soil particles is closer to that of water than to that of air. Light scattering decreases as the refractive index of soil particles approaches that of the surrounding medium (Bohren 1983; Twomey et al., 1986). If the soil particles and the surrounding medium had the same refractive index, the incident beam would not be deflected when encountering a particle, and would continue its path deeper into the medium until it is totally absorbed. The effect of water on soil reflectance is actually noticeable beyond the visible range (0.4-0.7 µm), in the near-infrared (0.7-1.4 µm) and shortwave infrared (1.4-2.5 µm) domains, where absorption by water is appreciable, with particularly strong absorption bands around 1.45 µm and 1.95 µm. Measuring soil reflectance between 0.4 µm and 2.5 µm can therefore be useful to estimate soil moisture content (SMC).

Soil moisture content is a key variable at the Earth's surface that controls many processes including erosion, evapotranspiration, infiltration, and energy budget (Berg and Sheffield, 2018; Green et al., 2019; Jaeger and Seneviratne, 2011; Kirkby, 2016). Because SMC is highly variable in time and space, field measurements are not well suited (Gardner et al., 2000) for mapping over large areas. Since the 1970s, remote sensing has been used to estimate continuous SMC measurements in the solar (0.4-2.5 µm), thermal infrared (3-12 µm), and microwave (0.5-100 cm) domains for applications in agriculture, hydrology, planetary science, defense, etc. (e.g., Babaeian et al., 2019; Mohanty et al., 2017; Ochsner et al., 2013; Petropoulos et al., 2018; Wang and Qu, 2009; Zhang and Zhou, 2016). The depth at which water content is estimated varies with wavelength. Microwaves give access to soil moisture at depths ranging from a few meters to a few decimeters and thermal infrared at a depth of a few millimeters. In the solar spectrum, light penetrates the soil to a maximum depth of about one millimeter. To retrieve the SMC from the fraction of solar radiation reflected by the soil, semi-empirical methods such as spectral indices are widely used (e.g.,
69 Bryant et al., 2003; Haubrock et al., 2008a, 2008b; Khanna et al., 2007; Zhang et al., 2019). These methods are often soil-specific and require a calibration phase to quantify their moisture. There are very few physical models capable of directly estimating SMC and validated on a wide range of soils.
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73 The MARMIT model (Bablet et al., 2018) is a radiative transfer model predicting the spectral reflectance of wet soils in the solar domain. It belongs to the family of two-layer models that represent a wet soil as a dry soil covered with a thin layer of liquid water. The concept was introduced by Angstrom (1925) and improved by Lekner and Dorf (1988) over 60 years later. Bach and Mauser (1994) extended it to the near infrared and shortwave infrared, taking into account the absorption of light by water. Their specific absorption coefficient of water is parameterized to fit a set of 10 soil reflectance spectra. The MARMIT model is based on this approach and has been validated on 217 soil samples covering a wide range of colors and textures, whose spectral reflectance was measured in the laboratory at different moisture levels. It has also been tested on hyperspectral images acquired by unmanned aerial systems (Eon and Bachmann, 2021). Although MARMIT estimates SMC with good accuracy (Bablet et al., 2018), a mismatch between measured and modeled reflectance spectra remains, especially for high SMC values. In particular, the reflectance is overestimated in the visible range and the match between measured and simulated spectra is poor in the water absorption bands at 1.45 µm and 1.95 µm, with a RMSE (root mean square error) > 3% around 1.95 µm. Indeed, the bilayer model describes a wet soil too simplistically and does not accurately portray a complex, granular, porous medium. Part of the problem is likely related to modeling the surface water layer using the spectral optical properties of pure water. A potentially significant correction would be to add mineral particles and dissolved organic matter, whose chemical composition is generally unknown, to the water layer.
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This article presents several modifications to MARMIT that make it more physically realistic: the modeling of light scattering at the air/water interface and within the water layer is improved, and soil particles are included within the water layer. Section 2 details the modifications made to the model to better account for the identified phenomena; Section 3 describes the datasets used to validate the model; Section 4 focuses on the parameterization of the model and Section 5 on the performance analysis; Section 6 is dedicated to the discussion. Section 7 is the conclusion.

## 2. Model

In this section, the basis of the MARMIT model is recalled and then the improvements are presented. The first improvement concerns the consideration of diffuse light in the water layer, the second the inclusion of soil particles in the water layer, and the third the scattering of light at the soil surface.

### 2.1. The MARMIT model

The MARMIT model represents a wet soil as a dry soil covered with a thin layer of liquid water (Bablet et al., 2018). The dry soil can be fully or partially covered by water, with a coverage fraction equal to $\epsilon$ ($0 \le \epsilon \le 1$). The modeled reflectance of the wet soil, $R_{mod}$, is then expressed as follows (wavelengths are not shown in the equations for ease of reading):

$$R_{mod} = \epsilon R_w + (1 - \epsilon) R_d \quad (1)$$

with $R_d$ the reflectance of a completely dry soil, assumed known, and $R_w$ the reflectance of a completely wet soil expressed as follows:

$$R_w = \frac{t_{12}t_{21}R_dT_w^2}{1 - r_{21}R_dT_w^2} \quad (2)$$
where $t_{12}$ and $t_{21}$ are the transmittance of the air/water interface for light entering and exiting the water layer. $t_{12}$ is calculated according to the Fresnel equations for unpolarized collimated light: it depends on the refractive index of water and on the angle of incidence of the light beam $\theta_i$. The transmitted rays then undergo multiple reflections in the water layer between the water/air interface of reflectance $r_{21}$ and the soil background of reflectance $R_d$. Once light has entered the water layer, it can thus be treated as completely diffuse. We assume, as Lekner and Dorf (1988) and Bablet et al. (2018), that the reflectance ($r_{21}$) and transmittance ($t_{21}$) at the water/air interface are the Fresnel angular reflectance and transmittance integrated over a hemisphere (Duntley, 1942; Judd, 1942; Stern, 1964, see Eqs 2 and 3 in Bablet et al., 2018). The coefficients $r_{21}$ and $t_{21}$ depend only on the refractive index of water and are linked through:

$$t_{21} = 1 - r_{21}. \qquad (3)$$

The imaginary part of the refractive index of water is ignored in the calculation of Fresnel coefficients at the air/water interface because it is too small to generate a significant variation in their value.

The transmittance of the water layer, $T_w$, is calculated by a Beer-Lambert-Bouguer law as a function of $\alpha_w$ ($m^{-1}$), the specific absorption coefficient of pure liquid water, and $L$ ($m$), the thickness of the water layer.

## 2.2. Diffuse light in the water layer

In MARMIT-2, we assume that the incident light is diffuse and that the air/water interface is not flat but follows the microtopography of the grain surface, as shown in Fig. 3 of Sadeghi et al. (2015). Thus, $t_{12}$ does not depend on $\theta_i$ and is calculated for Lambertian lighting by
integration of the Fresnel angular reflectance over a hemisphere as in Bablet et al. (2018) for $r_{21}$ and $t_{21}$. Note that the model is now independent of $\theta_i$ since all interface coefficients are integrated.

To account for multiple scattering of light in the water layer, we extend the Beer-Lambert-Bouguer law used for the transmittance of the water layer $T_w$ to diffuse light as in Allen et al. (1969) and Jacquemoud and Baret (1990) who modeled light transmission through a plant leaf as:

$$T_w = (1 - \alpha_w L)e^{-\alpha_w L} + \alpha_w L \int_{\alpha_w L}^{\infty} \frac{e^{-x}}{x} dx \quad (4)$$

where $\alpha_w$, thereby $T_w$, depend on the wavelength. Equation 4 is the result of the integration of the Beer-Lambert-Bouguer law over one hemisphere.

## 2.3. Inclusion of soil particles in the water layer

The representation of wet soil in MARMIT is rather simplistic and far from reality. It leads to a poor prediction of its spectral reflectance, especially in the visible range and in the water absorption bands. In the previous section, we assumed that the light was scattered in the water layer due to soil roughness and multiple scattering between the interfaces. In addition to this, we now make the assumption that the water layer above the soil is not pure liquid water but contains mineral particles much smaller than the wavelength of light. This does not change the model itself but only the complex refractive index of the water layer, which is an input variable; indeed, the real part is needed to calculate the reflectance and transmittance at the air/water interface ($t_{12}, t_{21}, r_{21}$) and the imaginary part is needed to calculate the transmittance of the particle-infused water layer ($T_w$). We propose to estimate the effective complex refractive index of this turbid medium. We assume that the soil particles are
spherical, homogeneously distributed in the water layer, and occupy a volume fraction $\delta$ between 0 and 0.25 so that they are sufficiently spread out not to form aggregates (Fig. 1). The effective complex refractive index of the medium is calculated from the refractive index (or permittivity) of the soil particles and water and $\delta$ by a mixing function.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic representation of the inclusion of soil particles in the water layer and key parameters of MARMIT-2. $n_w^*$ and $n_p^*$ refer to the complex refractive index of water and soil particles.</figcaption></figure>

The complex refractive index of an absorbing material, $n^*$, is:

$$n^* = n + ik. \qquad (5)$$

The relative permittivity, commonly referred to as the dielectric constant and noted $\epsilon^*$, is a dimensionless, complex number expressed relative to the permittivity of vacuum ($\epsilon_0 = 8.854.10^{-12} \text{ F.m}^{-1}$):

$$\epsilon^* = \epsilon' + i\epsilon'' = n^{*2} \qquad (6)$$

Let $\epsilon_{eff}^*$ and $n_{eff}^*$ be the effective complex permittivity and the effective complex refractive index of the water + soil particles layer. Table 1 summarizes some mixing functions from the literature to calculate $\epsilon_{eff}^*$ or directly $n_{eff}^*$. The real ($n_{eff}$) and imaginary ($k_{eff}$) parts of the effective refractive index are expressed in terms of $\epsilon_{eff}^*$ as follows:
$$n_{eff} = Re \left( \sqrt{\epsilon_{eff}^*} \right) \qquad (7)$$

$$k_{eff} = Im \left( \sqrt{\epsilon_{eff}^*} \right) \qquad (8)$$

$\epsilon_p^*, n_p^*, \epsilon_w^*$ and $n_w^*$ are the relative complex permittivities and the refractive indices of particles and water, respectively, and $\delta$ is the volume fraction of soil particles in the water layer.

<table><caption>**Table 1.** Some mixing functions extracted from the literature and used to calculate the effective permittivity or effective refractive index of a mixture of water and soil particles. The superscript * indicates complex quantities. The subscripts *eff*, *p* and *w* denote the effective medium, soil particles and water respectively.</caption><thead><tr><th></th><th>Name</th><th>Formula</th><th>Reference</th></tr></thead><tbody><tr><td></td><td>Dielectric average</td><td>ε<sub>eff</sub><sup>*</sup> = δε<sub>p</sub><sup>*</sup> + (1 - δ)ε<sub>w</sub><sup>*</sup></td><td>Park et al. (2017)</td></tr><tr><td></td><td>Lichtenecker</td><td>ε<sub>eff</sub><sup>*</sup> = ε<sub>p</sub><sup>*</sup>δε<sub>w</sub><sup>*</sup><sup>1-δ</sup></td><td>Lichtenecker (1926); Simpkin (2010)</td></tr><tr><td rowspan="2">Permittivity mixing</td><td>α-model</td><td>ε<sub>eff</sub><sup>*</sup> = (δε<sub>p</sub><sup>*</sup>α + (1 - δ)ε<sub>w</sub><sup>*</sup>α)<sup>1/α</sup><br/>-1 ≤ α ≤ 1</td><td>Brovelli and Cassiani (2008)<br/>α = 1/2 (Birchak et al., 1974)<br/>α = 1/3 (Looyenga, 1965)</td></tr><tr><td>Maxwell-Garnett</td><td>ε<sub>eff</sub><sup>*</sup> = ε<sub>w</sub><sup>*</sup> (ε<sub>p</sub><sup>*</sup> + 2ε<sub>w</sub><sup>*</sup> + 2δ(ε<sub>p</sub><sup>*</sup> - ε<sub>w</sub><sup>*</sup>)) / (ε<sub>p</sub><sup>*</sup> + 2ε<sub>w</sub><sup>*</sup> - δ(ε<sub>p</sub><sup>*</sup> - ε<sub>w</sub><sup>*</sup>))</td><td>Cosenza et al. (2009); Riziq et al. (2007)</td></tr><tr><td>Refractive index mixing</td><td>Refractive average</td><td>n<sub>eff</sub><sup>*</sup> = δn<sub>p</sub><sup>*</sup> + (1 - δ)n<sub>w</sub><sup>*</sup></td><td>Park et al. (2017); Riziq et al. (2007)</td></tr></tbody></table>

The main assumption underlying the mixing functions is that the particle is much smaller than the wavelength of light. Therefore, they have been used in the microwave domain (0.5-100
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cm) for calibration of time domain reflectometry (TDR) probes (Zakri et al., 1998). Here we
assume that particles suspended in water are small enough to satisfy this assumption in the
solar domain (0.4 – 2.5 µm).

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The *dielectric average* is defined as the arithmetic average of the permittivities of water and soil particles, weighted by the volume fractions of the two components. If the volume fraction of the soil particles is $\delta$, the volume fraction of the water is $1 - \delta$. The *refractive average* is similarly defined from the refractive indices. Park et al. (2017) state that the permittivities follow the superposition principle. Hence, from a theoretical perspective, it is better to use the *dielectric average* rather than the *refractive average*.

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Simpkin (2010) showed that the Lichtenecker equation can be directly derived from Maxwell's equations and the charge conservation principle. The α-model is also often referred to as the Lichtenecker model. Brovelli and Cassiani (2008) suggest using $\alpha = 1/2$ for water-saturated porous media while Looyenga (1965) uses $\alpha = 1/3$.

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The Maxwell-Garnett mixing formula considers a matrix (water) containing small inclusions (soil particles). Cosenza et al. (2009) applied it to soils in the microwave domain, and Riziq et al. (2007) calculated the effective index of very small aerosol particles at 0.532 µm in this manner. Simpkin (2010) showed that the Maxwell-Garnett mixing formula was an approximation to the Lichtenecker formula. The mixing functions listed here will be applied and compared to each other in Section 4.

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## 2.4. Light scattering on soil surface

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**Equation 1** assumes that the spectral reflectance of a wet soil is a linear combination of the spectral reflectance of a completely dry soil and a completely wet soil, weighted by their

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fraction of surface coverage. This model is approximate, in part because light entering the soil in a dry area can be scattered and exit in a wet zone, and vice versa. This phenomenon is known as the optical dot gain, or as the Yule-Nielsen effect, in printing where light is also scattered in a scattering medium, i.e. paper, around ink dot areas. It has been studied for the prediction of halftone print reflectance (Arney and Yamaguchi, 1999; Hébert, 2014; Yule and Nielsen, 1951). Equation 9 accounts for this nonlinear effect:

$$R_{mod} = (\epsilon R_w^{\frac{1}{\nu}} + (1 - \epsilon)R_d^{\frac{1}{\nu}})^{\nu} \quad (9)$$

where $\nu$ is an empirical parameter for which a physical interpretation has been proposed by Hébert et al. (2015). When applied to halftone prints, it is related to the scattering power of the printed medium. In most cases, $\nu$ is a real number greater than 1 but can be negative (Lewandowski et al., 2006; Viggiano, 2006). In Section 4, we show the performance of the model with various values of $\nu \ge 1$.

Table 2 summarizes the model inputs of MARMIT and MARMIT-2. The reflectance of a wet soil is simulated with MARMIT-2 from the reflectance of a dry soil, three free parameters that must be estimated (the thickness of the water layer $L$, the surface coverage of wet soil $\epsilon$, and the volume fraction of soil particles $\delta$) and two parameters that are assumed to be known (the Yule-Nielsen factor $\nu$ and the complex refractive index of soil particles $n_p^*$). The effective refractive index of the soil particle-water mixture is calculated from $n_p^*$ and $\delta$ with a mixing function that will be chosen hereafter among several options (Section 2.3). Although the effects of $\nu$ and $n_p^*$ may be physically interdependent, we consider them separately (Section 4). We quantify the model error by first varying the real and imaginary parts of the soil refractive index with $\nu = 1$, and then varying $\nu$ with $n_p^* = 1.53 + 0.001i$.
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<table><caption>**Table 2.** Model inputs of MARMIT and MARMIT-2: free and fixed parameters.</caption><thead><tr><th></th><th>MARMIT</th><th>MARMIT-2</th></tr></thead><tbody><tr><td>Surface coverage fraction of water (ε)</td><td>Free</td><td>Free</td></tr><tr><td>Thickness of water layer (L)</td><td>Free</td><td>Free</td></tr><tr><td>Volume fraction of soil particles (δ)</td><td>-</td><td>Free</td></tr><tr><td>Yule-Nielsen factor (ν)</td><td>-</td><td>Fixed</td></tr><tr><td>Complex refractive index of water (n<sub>w</sub><sup>*</sup>(λ))</td><td>Fixed</td><td>Fixed</td></tr><tr><td>Complex refractive index of soil particles (n<sub>p</sub><sup>*</sup>(λ))</td><td>-</td><td>Fixed</td></tr><tr><td>Light incidence angle (θ<sub>i</sub>)</td><td>Fixed</td><td>-</td></tr></tbody></table>

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## 3. Data

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This section presents the database used to validate MARMIT-2 which contains reflectance spectra of various soils measured in the laboratory for several moisture contents. We then provide an overview of the refractive index values of mineral particles in soils and aerosols published in the literature.

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### 3.1. Soil reflectance databases

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The MARMIT-2 model was validated on seven datasets (Liu02, Lob02, Whit04, Les08, Mar12, Phil14, and Bab16) described in Bablet et al. (2018) plus a new dataset, Dup20, acquired as part of this study. This represents a total of 225 soil samples and 1984 reflectance spectra measured in the 0.4-2.5 µm range at different moisture levels. Most of these datasets can be downloaded at https://pss-gitlab.math.univ-paris-diderot.fr/marmit/marmit. Dup20 was acquired at ONERA (Toulouse, France) in 2020 following the same protocol as for Bab16 and Les08. It contains the reflectance spectra of 8 soil samples measured at different moisture
245 contents in the 0.4-2.5 µm spectral range using an ASD FieldSpec 3HR spectrophotometer.

246 **Table 3** describes the soil samples in Dup20.

247 **Table 3.** Dup20 dataset information. The complex refractive index of the aerosol phase of samples #2, #3, #4, and #7 has been published in the literature. *n* is the real part averaged over the wavelength range 0.37-0.95 µm; the imaginary part *k* was measured at seven wavelengths: 0.37, 0.47, 0.52, 0.59, 0.66, 0.88, and 0.95 µm.

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<table><thead><tr><th rowspan="2">ID</th><th rowspan="2">Origin</th><th colspan="3">Texture</th><th colspan="2">Refractive index</th><th rowspan="2">Reference</th></tr><tr><th>Clay (%)</th><th>Silt (%)</th><th>Sand (%)</th><th>n</th><th>k</th></tr></thead><tbody><tr><td>#1</td><td>Tunisia</td><td>73.8</td><td>19.4</td><td>6.8</td><td>-</td><td>-</td><td>Bab16 (Bablet et al., 2018)</td></tr><tr><td>#2</td><td>Chili (Atacama)</td><td>-</td><td>-</td><td>-</td><td>1.54</td><td>0.0006 - 0.0016</td><td>Di Biagio et al. (2019)</td></tr><tr><td>#3</td><td>Tunisia (Maouna)</td><td>-</td><td>-</td><td>-</td><td>1.51</td><td>0.0012 - 0.0045</td><td>Di Biagio et al. (2019)</td></tr><tr><td>#4</td><td>USA (Sonora desert, Arizona)</td><td>-</td><td>-</td><td>-</td><td>1.51</td><td>0.0004 - 0.0011</td><td>Di Biagio et al. (2019)</td></tr><tr><td>#5</td><td>France (Saint-Julien, Tarn)</td><td>13.5</td><td>49.0</td><td>37.5</td><td>-</td><td>-</td><td>Les08 (Lesaignoux et al., 2013)</td></tr><tr><td>#6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>Bab16 (Bablet et al., 2018)</td></tr><tr><td>#7</td><td>China (Ulan Buh)</td><td>-</td><td>-</td><td>-</td><td>1.48</td><td>0.0011 - 0.0041</td><td>Di Biagio et al. (2019)</td></tr><tr><td>#8</td><td>Tunisia</td><td>73.8</td><td>19.4</td><td>6.8</td><td>-</td><td>-</td><td>Bab16 (Bablet et al., 2018)</td></tr></tbody></table>

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251 The refractive index of the aerosol phase of samples #2, #3, #4, and #7 was measured in the visible range by Di Biagio et al. (2019). The spectral reflectance of samples #1, #5, #6, and #8 was also measured at different moisture contents in Les08 or Bab1. This redundancy allows to verify the consistency of the measurements. Due to the appearance of cracks on the surface of
sample #1 during the drying process, we made a replica of this sample, sample #8, to check the measurement.

The soils were placed in 5.7 cm diameter, 1.3 cm high Petri dishes for samples #1 to #7, and in a 5 cm diameter, 1.6 cm high Petri dish for sample #8. The mass of the dry soils ranged from 50 to 80 g. Spectral reflectance was acquired at nadir for different moisture levels. The soil was fully illuminated by a halogen lamp placed at an angle of 15° to the normal of the sample and the measurement area was a disk of about 3.5 cm diameter. The ratio of the radiance measured on the sample to the radiance measured on a Spectralon panel assumed to be Lambertian, multiplied by the reflectance of the panel provided by the manufacturer, is calculated, and hereafter referred to as the reflectance.

We first measured the reflectance of a dry sample, then moistened it to saturation using a drip system and measured its reflectance and SMC; the sample was then oven-dried at 60°C and its reflectance and SMC were measured regularly until it was completely dry. In total, one reflectance spectrum was measured on the dry soil and nine reflectance spectra were measured on the wet soils. The dry soil provided slightly different results before and after the experiment due to the appearance of shrinkage cracks during drying, deposition of dissolved matter on the surface, or clay swelling during humidification. In Dup20, the dry soil spectrum refers to the measurement made after oven drying because it corresponds to the limit value of the other spectra measured at different moisture contents. The SMC (%) is determined by the gravimetric method (Bablet et al., 2018):

$$SMC = 100 \cdot \frac{m_{ws} - m_{ds}}{m_{ds}} = 100 \cdot \frac{m_{H_2O}}{m_{ds}} \qquad (10)$$
with $m_{ws}$ the mass of the wet sample, $m_{ds}$ the mass of the dry sample, and $m_{H_2O}$ the mass of water. SMC = 0% means that the soil is dry and SMC = 100% means that the sample contains as much water as soil.

Different protocols were applied depending on the dataset: some soils were humidified and others dried; some measurements were performed at an incidence angle of 15°, others at higher angles of up to 30°. In addition, soil texture is not available for all samples and information on their mineralogical composition is even scarcer. Therefore, it is difficult to draw general conclusions for specific soil classes and comparison between datasets must be done with caution.

## 3.2. Refractive index database

Because of the difficulty of experimentally determining the refractive index of a soil, data are scarce and, in some cases, the result of old studies. Many were measured only in the visible range or with coarse spectral sampling (Egan and Hilgeman, 1979). Moreover, different studies may provide different values for the same mineral (Zhang et al., 2015). Figure 2 shows the spectral variation of the real and imaginary parts of the refractive index of major soil minerals in the solar domain.
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Real (a) and imaginary (b) parts of the refractive index of soil minerals. Aluminum oxide and magnetite are from Querry (1985) and are distributed by Polyanskiy (refractiveindex.info); calcite is from Gosh (1999) and is distributed by Polyanskiy (refractiveindex.info); illite, kaolinite, feldspar, and montmorillonite are from Egan and Hilgeman (1979); gypsum is from Roush et al. (2007); hematite is from Longtin et al. (1988) and is distributed by Grainger et al. (ARIA); olivine is from Fabian et al. (2001) and is distributed by Grainger et al. (ARIA); quartz is from Drummond (1936), Gray (1963), Longtin et al. (1988), Philipp (1985) and Spitzer and Kleinman (1961), and is distributed by Jacquinet-Husson et al. (GEISA database); wüstite is from Henning et al. (1995) and is distributed by Grainger et al. (ARIA). The imaginary part of the refractive index of quartz and calcite is almost zero, so it is not shown in Fig. 2b. If the measurements for calcite, hematite, and quartz were made for both ordinary and extraordinary polarizations, only the average of the two polarizations is shown.</figcaption></figure>

The refractive index of minerals has been the subject of several studies. Smith and Grainger (2013) compiled the real and imaginary parts of the refractive index of the main minerals composing desert dust aerosols in the range 0.5-1.9 µm (Fig. 2). They concluded that their optical properties were controlled more by the imaginary part of the refractive index than by its real part and proposed 1.53 + 0.001i as a reasonable value for dust. Zhang et al. (2015)
compared published complex refractive index values for iron oxides. They pointed out the
scarcity of continuous measurements between 0.2 µm and 50 µm and their inconsistency.
Moreover, Ishida et al. (1991) and Ishida and Price (1996) studied 37 soil samples from
northern Japan and developed a model to derive their effective complex refractive index
between 0.4 µm and 0.9 µm. The real part varied between 1.42 and 1.73, but the authors
pointed out strong variations in the imaginary part with wavelength, particle size, and soil
composition. More recent experiments have measured the refractive index of aerosols.
Kandler et al. (2007) calculated the refractive index of a Saharan aerosol and obtained 1.59 +
0.007i at 0.7 µm. Kandler et al. (2011) also derived the refractive index of 242 aerosol
samples collected in Cape Verde from their chemical composition for 12 wavelengths
between 0.355 µm and 2 µm. They showed that the imaginary part of the aerosol indices
varied with particle size, reflecting their complex chemical composition. Wagner et al. (2012)
measured the complex refractive index of resuspended aerosols of Saharan dust samples in
the UV-A (ultraviolet) and visible ranges: they found that the real part was rather constant
over this spectral range and that 1.53 was a realistic value to consider. Finally, Di Biagio et al.
(2019) measured the refractive indices of 19 desert dust aerosols from various locations for 7
wavelengths between 0.37 µm and 0.95 µm. They noted little spectral variation in the real
part.

In summary, the real part of the refractive index of soil particles is often spectrally constant
while the imaginary part is more variable and is more important for modeling the optical
properties of soil. Two studies agree to take 1.53 for the real part. In the next section, the
choice of the refractive index of soil particles in MARMIT-2 is discussed.
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# 4. Evaluating the model parameters

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We first chose the optimal mixing function used to calculate the effective refractive index of the soil particle-water mixture. Figure 3 shows the real and imaginary parts of this effective index obtained with different mixing functions, for several volume fractions of particle $\delta$. The first two columns correspond to a mixture of water and kaolinite, one of the most common minerals in clays, the last column corresponds to a mixture of water and quartz. The absorption of quartz being negligible, only the real part of the effective refractive index is plotted. The choice of the mixing function has little influence on the effective refractive index, especially for the imaginary part. Thus, it does not have a significant impact on the prediction of soil reflectance. In MARMIT-2, we chose the dielectric average which is the simplest rule.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Real (n) and imaginary (k) parts of the effective refractive index of a kaolinite particle - water mixture (first two columns) and real part of the effective refractive index of a quartz particle - water mixture (third column). The black solid curve represents pure liquid water, the black dashed curve represents mineral particles, and the colored lines (all nearly overlapping) represent the mixtures.</figcaption></figure>

After selecting the mixing function, the question arises as to the value of the refractive index of the soil particles. The model must be general and validated over a wide range of soils, so it is preferable that the refractive index of the soil particles remains the same from soil to soil.
To determine the range of refractive index values tolerated by the model, we calculated the RMSE between measured and modeled reflectance spectra at 0.550 µm and 0.860 µm (negligible water absorption), 1.447 µm and 1.928 µm (center of the water absorption bands), 1.700 µm (in-between bands), and 2.200 µm (hydroxyl and clay absorption band) for different values of the real and imaginary parts of the indices. The RMSE is averaged for the wet soil spectra of all our datasets (except Whit04 because the values for 0.860 µm are missing). To cover most of the values found in the literature, the real part, $n_p$, varied between 1.35 and 1.65, and the imaginary part, $k_p$, between $10^{-6}$ and $10^{-1}$ (Fig. 4). The Yule-Nielsen parameter was set here to $\nu = 1$. The RMSE is calculated as follows:

$$RMSE(\lambda) = \sqrt{\frac{\sum_{s=1}^{n_{\text{spectra}}} (R_{\text{meas},s}(\lambda) - R_{\text{mod},s}(\lambda, L, \epsilon, \delta))^2}{n_{\text{spectra}}} \quad (11)$$

with $\lambda$ the wavelength, $n_{\text{spectra}}$ the number of wet soil spectra ($n_{\text{spectra}} = 1274$ for all datasets except Whit04), $R_{\text{meas}}$ the measured soil reflectance, and $R_{\text{mod}}$ the modeled soil reflectance expressed with Eqs. 2 and 9. $L$, $\epsilon$, and $\delta$ are estimated by minimizing the cost function:

$$\chi^2(L, \epsilon, \delta) = \sqrt{\frac{\sum_{\lambda_1}^{\lambda_2} (R_{\text{meas}}(\lambda) - R_{\text{mod}}(\lambda, L, \epsilon, \delta, n_p, k_p))^2}{n_\lambda}} \quad (12)$$

with $n_\lambda$ the total number of wavelengths. The lower and upper bounds of the model parameters are 0 and 1 for $\epsilon$, 0 and 0.2 cm for $L$, and 0 and 0.25 for $\delta$.

The RMSE ranges from 0.5% to 3.1% for the six wavelengths. At 0.550 µm and 0.860 µm, it is highest for low values of $n_p$ and $k_p$, and decreases for increasing values of $k_p$. For values
of $k_p$ greater than $10^{-4}$, it seems to be independent of variations in $n_p$ and $k_p$. At 1.700 µm, 368 369 1.928 µm and 2.200 µm, we observe the same behavior but the RMSE increases again for 370 high values of $k_p$. At 1.447 µm, the RMSE is lowest for high values of $n_p$ and low values of 371 $k_p$. For these six wavelengths, the variation of $n_p$ seems to have little impact on the RMSE 372 provided that the imaginary part ranges between $10^{-4}$ and $10^{-2}$ and in this case the RMSE is 373 relatively low. Therefore, we chose to use the constant value of $1.53 + 0.001i$ as suggested 374 by Smith and Grainger (2013).
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** RMSE between measured and modeled reflectance spectra for all databases except Whit04 at six wavelengths for varying values of the real and imaginary parts of the refractive index of soil particles. The white cross is placed at $n_p^* = 1.53 + 0.001i$.</figcaption></figure>

To quantify the impact of the Yule-Nielsen parameter $v$, we plotted the spectral average of the RMSE between measured and modeled reflectance spectra for all databases except Whit04 as a function of $v$ (Fig. 5) using $n_p^* = 1.53 + 0.001i$. The minimum value is obtained for $v =$
2.27 (Fig. 5). Calculating the RMSE values in Fig. 4 with $v = 2.27$, we again observe that it is lowest for values of $k_p$ between $10^{-4}$ and $10^{-2}$ and in this range of $k_p$, relatively independent of $n_p$. Therefore, MARMIT-2 is parameterized with $n_p^* = 1.53 + 0.001i$ and $v = 2.27$.

<figure><img src="image_6.png" /><figcaption>Fig. 5. RMSE between measured and modeled reflectance as function of the Yule-Nielsen parameter $v$ for all databases except Whit04, averaged at all wavelengths.</figcaption></figure>

# 5. Model performance

To assess the effect of including soil particles in the water layer, we first simulate the reflectance of wet soils for various volume fractions of soil particles using Eqs. 2, 6 and 9. The reflectance spectra in Fig. 6 are simulated for different values of $\delta$, with $L$ and $\epsilon$ fixed. These simulations show a decrease in reflectance in the visible range as more soil particles are added to the water layer. This also deepens the absorption band at 1.45 $\mu$m and decreases the
rightward slopes of the absorption bands at 1.45 µm and 1.95 µm. These changes should
improve the fit of the laboratory spectra by MARMIT, which overestimates the reflectance in
the visible range and is not able to correctly reproduce the reflectance in the water absorption
bands.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Measured reflectance spectra of a dry soil and reflectance spectra of wet soils simulated with MARMIT-2 for $L = 0.005$ cm, $\epsilon = 0.5$, and variable $\delta$. (a) Sandy soil (Colluviosol 26d1, Bab16), (b) silty soil (Luvisol 208TE, Bab16), and (c) clayey soil (30Pujaut, Les08).</figcaption></figure>

The model inversion consists in estimating the parameter set ($L, \epsilon, \delta$) by minimizing the cost function of Eq. 12. Figure 7 shows the measured and modeled reflectance spectra of a white
sand (Bab16 database) at different moisture levels using MARMIT and MARMIT-2 for different values of refractive index of soil particles. The agreement between measurements and simulations is better with MARMIT-2, especially in the visible range and around the water absorption band at 1.4 µm. The refractive index used in Fig. 7b is that of quartz (assuming that its imaginary part is negligible), the main constituent of this soil (Fig. 2). In Fig. 7c, the refractive index of the soil particles is equal to $n_p^* = 1.53 + 0.001i$. Although this makes physical sense, using the refractive index of quartz instead of an arbitrary constant value does not improve the fit. The agreement between the measured and modeled reflectance spectra appears to be better when a constant refractive index is used (Fig. 7c).

<figure><img src="image_8.png" /></figure>
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**Fig. 7.** Measured (solid line) and modeled (dashed line) reflectance spectra of a white sand (Bab16 database) at different SMCs. (a) MARMIT, (b) MARMIT-2 using the refractive index of quartz, and (c) MARMIT-2 using 1.53+0.001i for the refractive index of soil particles. The estimated model parameters are indicated in the plot legends as (a) (L(cm); ε) and (b, c) (L(cm); ε; δ).

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**Figure 8** shows the measured and modeled reflectance spectra at different SMCs for a desert soil collected in China (Dup20 database). The value of $n_p^* = 1.48 + 0.002i$ in **Fig. 8b** is the average value measured by Di Biagio et al. (2019) in 7 wavelengths of the visible range. The real part was rather constant while the imaginary part varied spectrally. In **Fig. 8c**, the refractive index used was $n_p^* = 1.53 + 0.001i$. The agreement between predictions and measurements is better when soil particles are included in the water layer, and it appears that using $1.53 + 0.001i$ produces similar performance to using the measured value of the refractive index. This is in agreement with **Fig. 4**: the RMSE between the measured and modeled reflectance spectra is little affected by the value of the real part of the refractive index of the soil particles if chosen between 1.35 and 1.65.
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Measured (solid lines) and modeled (dashed lines) reflectance spectra of China sand (Dup20 database) at different SMCs. (a) MARMIT, (b) MARMIT-2 using 1.48+0.002i for the refractive index of soil particles (Di Biagio et al., 2019), and (c) MARMIT-2 using 1.53+0.001i for the refractive index of soil particles. The estimated model parameters are indicated in the plot legends as (a) ($L(cm); \epsilon$) and (b, c) ($L(cm); \epsilon; \delta$).</figcaption></figure>

MARMIT-2 predicts soil spectral characteristics with better accuracy over the entire solar domain. For low moisture content, the fit of the reflectance spectra was satisfactory with MARMIT so the changes brought by MARMIT-2 have little effect. For high moisture contents however, the inclusion of soil particles in the water layer significantly improves the
fit. The reflectance decreases in the visible range and the depth of the water absorption band around 1.95 µm is reduced.

To quantify the improvements made by MARMIT-2, the RMSE between measured and modeled reflectance spectra is calculated for the different versions of the model and all soils available in our database (Fig. 9.a):

$$RMSE(\lambda) = \sqrt{\frac{\sum_{k=1}^{n_{spectra}} (R_{meas,k}(\lambda) - R_{mod,k}(\lambda))^2}{n_{spectra}}} \quad (13)$$

with $R_{meas}$ and $R_{mod}$ the measured and modeled reflectance spectra and $n_{spectra}$ the total number of spectra (1984 in our datasets). It can be seen that the modifications described in Sections 2.2, 2.3, and 2.4 contribute significantly to reducing the RMSE and that the main improvement is due to the inclusion of soil particles in the water layer. The RMSE spectrum is not significantly changed, with the highest errors observed in the strong water absorption band around 1.95 µm. This band is also located in a region of low atmospheric transmission, so this is not an obstacle to validating the model on airborne or satellite images. The difference between the RMSE calculated with MARMIT and MARMIT-2 is 1.0% on average and up to 1.9% at 1.902 µm. The average RMSE calculated with MARMIT-2 is 0.8%, with a minimum of 0.4% and a maximum of 1.8%.

Figures 9.b and 9.c show that for both MARMIT and MARMIT-2, the RMSE is higher for high SMCs but the error difference between high and low SMCs is reduced with MARMIT-2.

Figures 9.d and 9.e display the RMSE as a function of soil texture. Clay soils are defined by particle size less than 2 µm, silty soils by particle size between 2 µm and 50 µm, and sandy soils by particle size between 50 µm and 2000 µm. Of the 225 samples in our database, texture information is available for 139 of them. For most wavelengths, the RMSE is higher
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for silts than for sands with MARMIT, while it is higher for clays with MARMIT-2. This

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should be considered carefully because there are about half as many clay soils as sandy soils.

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In addition, the performance of the model may depend more on the chemical composition of
the soils than on their texture.

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<figure><img src="image_10.png" /><figcaption>**Fig. 9.** Spectral root mean squared error (RMSE) for all soils in our database (1984 spectra, 225 soil samples) as a function of (a) different model versions: MARMIT, MARMIT with diffuse light in the water layer, MARMIT with diffuse light and soil particles in the water layer, MARMIT with diffuse light and soil particles in the water layer and the Yule-Nielsen parameter (MARMIT-2); (b-c) moisture level (995 spectra measured at SMC > 20% and 989 at SMC ≤ 20%)</figcaption></figure>

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for MARMIT (left) and MARMIT-2 (right); (d-e) texture class (23 clays, 68 silts, and 48 sands) for MARMIT (left) and MARMIT-2 (right). The grey areas represent the spectral bands where atmospheric transmittance is ≤ 0.6. The gap in the spectra between 0.75 µm and 0.92 µm is due to missing data in the Whit04 database.

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# 6. Discussion

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The main improvement to the MARMIT model is the inclusion of soil particles in the water layer that changes its optical properties. Bach (1995) and Philpot (2010) adapted the specific absorption coefficient of the water layer to account for the presence of dissolved organic matter or suspended mineral particles. Their approach was empirical and involved fitting the absorption coefficient for a specific set of soil reflectance spectra. In this work, we provide a physical argument by calculating the effective complex refractive index of the mixture of water and soil particles with a mixing rule. We ensure that the model is versatile with respect to the soil type by using a single constant value for the refractive index of the soil particles.

The low sensitivity of the model to the value of the refracting index of soil particles allows us to do so, provided that the imaginary part is chosen between $10^{-4}$ and $10^{-2}$ which covers a wide range of soils. However, the absorption coefficient of the water layer is different for each SMC because the volume fraction of soil particles in the water layer is adjusted for each reflectance. It makes physical sense that the absorption coefficient of the water layer is not unique for all samples. Indeed, depending on the moisture level, the fraction of dissolved organic matter or suspended mineral particles, thereby the overall optical properties of the water layer, vary.

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Figure 10 shows the water layer absorption coefficients used in Bach (1995) and Philpot (2010), and those calculated in MARMIT-2 for two volume fractions of soil particles. The Bach coefficient is calculated as follows:
$$492 \qquad \alpha_{Bach} = 0.031 + 1.17\alpha_{w} - 0.0088\alpha_{w}^{2} + 0.000025\alpha_{w}^{3} \qquad (14)$$

It is obtained by fitting a polynomial of degree three for 10 soil samples. The Philpot coefficient is:

$$495 \qquad \alpha_{Philpot} = \alpha_{w} + c_{dom} \cdot e^{-0.006(\lambda-440)} + c_{a} \cdot a_{a} + c_{b} \cdot a_{b} + c_{c} \cdot a_{c} \qquad (15)$$

where $a_a$, $a_b$, and $a_c$ are Gaussians centered at 1.45 µm, 1.9 µm, and 2.8 µm. $c_{dom}$, $c_a$, $c_b$, and $c_c$ are fitted for each spectrum. The decreasing exponential function models the spectral absorbance of dissolved organic matter.

<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Pure liquid water and water layer specific absorption coefficients of several soil reflectance models (Bach, 1995; Philpot, 2010; Bablet et al., 2018, and MARMIT-2 for two values of $\delta$).</figcaption></figure>
The specific absorption coefficient of the water layer in MARMIT-2, $\alpha_{MARMIT-2}(\delta)$, shows similarities with $\alpha_{Bach}$ and $\alpha_{Philpot}$. The coefficients $\alpha_{MARMIT-2}$ and $\alpha_{Philpot}$ are higher than $\alpha_w$ between 0.4 µm and 1.3 µm, and tend to decrease in this spectral range, although the decrease of $\alpha_{MARMIT-2}$ seems to follow an inverse function rather than an exponential function. The spectral variation of the coefficients differs in the water absorption band centered at 1.45 µm, where $\alpha_{MARMIT-2}$ and $\alpha_{Philpot}$ are both greater than $\alpha_w$. In the band around 1.95 µm, $\alpha_{MARMIT-2}$ and $\alpha_{Bach}$ are both less than $\alpha_w$. Between these two bands, $\alpha_{MARMIT-2}$ is the only one that is greater than $\alpha_w$.

The specific absorption coefficient $\alpha_{MARMIT-2}$ is expressed as a function of $\delta$, the volume fraction of soil particles, which varies from spectrum to spectrum. The estimated values of $\delta$ do not seem to depend on soil texture but on SMC: they are globally higher for low SMC values. This could be explained by the fact that when a small amount of water is added to the soil, the soil is saturated with suspended particles and dissolved organic matter. When more water is added, the number of particles remains the same but the volume of water increases so the volume fraction of particles is lower.

Despite this slight trend, there is no obvious link between $\delta$ and SMC. Therefore, model calibration between the retrieved parameters and SMC is done as in MARMIT by fitting a sigmoid function between SMC and the product $L \cdot \epsilon$. Since MARMIT-2 has three input parameters instead of two for MARMIT, the retrieved values of $L$ and $\epsilon$ are slightly different between the two versions of the model; therefore, the sigmoid must be recalculated for each soil.
523 MARMIT-2 performs best on soils with low water content. Since the reflectance of a dry soil
524 is used as a reference to predict the reflectance of a wet soil, the wetter the soil, the more
525 difficult it is to predict its spectral reflectance.

526 The RMSE is higher for clay soils than for silty and sandy soils, especially in the visible and
527 near infrared; furthermore, we observe that the shape of the spectral RMSE for each soil class
528 is very different. This is surprising because one would expect more suspended particles in
529 clay soils than in sandy soils, and therefore the addition of $\delta$ would be more useful for the
530 former. This should be viewed with caution as texture information is only available for 139 of
531 the 225 soil samples and only 23 samples are clay soils (17% of soils with known texture).

## 7. Conclusion

532 Modifications to the MARMIT model have overcome several limitations: overestimation of
533 the reflectance spectra in the visible range, and poor prediction of reflectance around the
534 water absorption bands. Although wet soils are still represented in a simple way in MARMIT-
535 2, the model is now much more efficient in terms of fitting the observed reflectance spectra. It
536 better accounts for surface scattering, replacing the flat air/water interface with a rough
537 interface. The optical effect of soil particles in the water layer was introduced, which adds a
538 third input parameter, the volume fraction of soil particles $\delta$, and a new variable, the complex
539 refractive index of soil particles. The dependence of the model on the refractive index of the
540 soil was an issue for the development of an operational model valid for all types of soils. We
541 have shown that, if the value of this index is chosen within a reasonable range, it has a limited
542 impact on the model performance. We therefore decided to use the constant value 1.53 +
543 0.001i for all soils.
544
We calculated the RMSE for 225 soil samples whose reflectance was measured in the laboratory at different SMCs. It is on average equal to 0.8% with MARMIT-2 versus 1.8% with MARMIT. This average value for all databases was satisfactory with MARMIT, but when comparing the measured and simulated reflectance spectra, we observed a significant mismatch at certain wavelengths, particularly for high moisture contents. These poor fits showed that a physical phenomenon was not well taken into account, which justified the need to improve the model. We analyzed the performance of MARMIT-2 as a function of moisture and texture: we found better results for low SMC values and for sands than for clays. We expected a different result: a more complete database with more clay soils could show the benefit of including soil particles in the water layer because clay particles are smaller than sand particles. Due to the lack of information on the mineralogical composition of our soil samples, we were not able to analyze this effect.

The next step will be to validate the model at metric and decametric spatial resolutions using airborne and satellite images. To do this, it may be necessary to link MARMIT-2 to a BRDF (Bidirectional Reflectance Distribution Function) model of bare soils. In the longer term, MARMIT-2 could be coupled to another radiative transfer model to simulate soil reflectance as function of other properties such as vegetation or organic carbon content.

## Data availability

Some of the datasets used for the validation of MARMIT-2 and a Python code simulating wet soil spectra are freely available at https://pss-gitlab.math.univ-paris-diderot.fr/marmit/marmit.
565 Acknowledgements

566 The PhD thesis of Alice Dupiau is funded by ONERA and IPGP. This work was supported by
567 CNES in the frame of the MOLIERE (MOdeLing soIl rEflectance as a function of wateR
568 contEnt) project and by the Labex UnivEarth (ANR-10-LABX-0023 and ANR-18-IDEX-
569 0001) in the frame of the SOILSPECT project.

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782 List of Figure Captions

783 **Fig. 1.** Schematic representation of the inclusion of soil particles in the water layer and key parameters of
784 MARMIT-2. $n_w^*$ and $n_p^*$ refer to the complex refractive index of water and soil particles.

785 **Fig. 2.** Real (a) and imaginary (b) parts of the refractive index of soil minerals. Aluminum oxide and magnetite are from
786 Querry (1985) and are distributed by Polyanskiy (refractiveindex.info); calcite is from Gosh (1999) and is distributed by
787 Polyanskiy (refractiveindex.info); illite, kaolinite, feldspar, and montmorillonite are from Egan and Hilgeman (1979);
788 gypsum is from Roush et al. (2007); hematite is from Longtin et al. (1988) and is distributed by Grainger et al. (ARIA);
789 olivine is from Fabian et al. (2001) and is distributed by Grainger et al. (ARIA); quartz is from Drummond (1936), Gray
790 (1963), Longtin et al. (1988), Philipp (1985) and Spitzer and Kleinman (1961), and is distributed by Jacquinet-Husson et al.
791 (GEISA database); wüstite is from Henning et al. (1995) and is distributed by Grainger et al. (ARIA). The imaginary part of
792 the refractive index of quartz and calcite is almost zero, so it is not shown in Fig. 2b. If the measurements for calcite,
793 hematite, and quartz were made for both ordinary and extraordinary polarizations, only the average of the two polarizations is
794 shown.

795 **Fig. 3.** Real (n) and imaginary (k) parts of the effective refractive index of a kaolinite particle - water mixture (first
796 two columns) and real part of the effective refractive index of a quartz particle - water mixture (third column). The
797 black solid curve represents pure liquid water, the black dashed curve represents mineral particles, and the colored
798 lines (all nearly overlapping) represent the mixtures.
**Fig. 4.** RMSE between measured and modeled reflectance spectra for all databases except Whit04 at six wavelengths for varying values of the real and imaginary parts of the refractive index of soil particles. The white cross is placed at $n_p^* = 1.53 + 0.001i$.

**Fig. 5.** RMSE between measured and modeled reflectance as function of the Yule-Nielsen parameter $\mathcal{V}$ for all databases except Whit04, averaged at all wavelengths.

**Fig. 6.** Measured reflectance spectra of a dry soil and reflectance spectra of wet soils simulated with MARMIT-2 for $L = 0.005$ cm, $\epsilon = 0.5$, and variable $\delta$. (a) Sandy soil (Colluviosol 26d1, Bab16), (b) silty soil (Luvisol 208TE, Bab16), and (c) clayey soil (30Pujaut, Les08).

**Fig. 7.** Measured (solid line) and modeled (dashed line) reflectance spectra of a white sand (Bab16 database) at different SMCs. (a) MARMIT, (b) MARMIT-2 using the refractive index of quartz, and (c) MARMIT-2 using $1.53+0.001i$ for the refractive index of soil particles. The estimated model parameters are indicated in the plot legends as (a) $(L(cm); \epsilon)$ and (b, c) $(L(cm); \epsilon; \delta)$.

**Fig. 8.** Measured (solid lines) and modeled (dashed lines) reflectance spectra of China sand (Dup20 database) at different SMCs. (a) MARMIT, (b) MARMIT-2 using $1.48+0.002i$ for the refractive index of soil particles (Di Biagio et al., 2019), and (c) MARMIT-2 using $1.53+0.001i$ for the refractive index of soil particles. The estimated model parameters are indicated in the plot legends as (a) $(L(cm); \epsilon)$ and (b, c) $(L(cm); \epsilon; \delta)$.

**Fig. 9.** Spectral root mean squared error (RMSE) for all soils in our database (1984 spectra, 225 soil samples) as a function of (a) different model versions: MARMIT, MARMIT with diffuse light in the water layer, MARMIT with diffuse light and soil particles in the water layer, MARMIT with diffuse light and soil particles in the water layer and the Yule-Nielsen parameter (MARMIT-2); (b-c) moisture level (995 spectra measured at SMC > 20% and 989 at SMC ≤ 20%) for MARMIT (left) and MARMIT-2 (right); (d-e) texture class (23 clays, 68 silts, and 48 sands) for MARMIT (left) and MARMIT-2 (right). The grey areas represent the spectral bands where atmospheric transmittance is ≤ 0.6. The gap in the spectra between 0.75 µm and 0.92 µm is due to missing data in the Whit04 database.

**Fig. 10.** Pure liquid water and water layer specific absorption coefficients of several soil reflectance models (Bach, 1995; Philpot, 2010; Bablet et al., 2018, and MARMIT-2 for two values of $\delta$).