An analytical specro-photometric model for layered complex planetary surfaces: limitations of the Hapke model and new formulations

Introduction

Visible and near infrared (VNIR) spectroscopy is a widely used and powerfull tool to investigate planetary surfaces. Most minerals or volatile elements making up surfaces have characteristic absorption bands in the range 0.5 μm < λ < 5 μm. In this spectral range, the ices present in the solar system all have in common a high reflectivity in the visible, and strong absorption bands in the near infrared.  Usually, planetary ices form a layer covering an underlying substrate. This results in both the signals from ice and the surface underneath being present in the VNIR spectroscopic measures at most wavelenghts.  Moreover, ice is volatile and interacts with planetary surfaces and atmospheres, and is therefore often intimately mixed with granular mineral materials (regolith, atmospheric aerosols, etc...). It is important to note that other type of stratified planetary surface can be studied with VNIR spectroscopy, such as a thin space-weathered ice layer covering a well preserved one such as expected on icy moons, or liquid films such as hydrocarbures on Titan. The purpose of this work is to develop a numerically efficient two-layer radiative transfer model that can encompass all these possibilities and allow for massive inversions, across entire hyperspectral cubes, such as CRISM [1], NIMS [2] or SIMBIO-SYS [3].

 

Model :

Figure 1 describes the problem. We consider the two-stream approximation, under collimated radiation inside layered media. The surface is constituted of two layers, medium 1 of optical thickness τ₁ on top, and a semi infinite medium 2 under it. The reflection coefficients at the interfaces are S_e1u, S_i1u, S_i1b and S_i2u. The surface is illuminated by a collimated radiation  of intensity J₀ and incidence θ₀. J₁ is the part of this radiation transmitted to the medium 1, with an incidence θ₁. All the other radiation inside the medium 1 is either a part of the upward stream I₁ or the downward stream  I₂ . The same process happen at the second interface, and medium 2 is under collimated radiation J₂, of incidence θ₂; and all the other radiation inside the medium 2 is either a part of the upward stream I₃  or the downward stream I₄ . We define μ₀  = cosθ₀, μ₁  = cosθ₁   and  μ₂  = cosθ₂   to simplify notations.

Medium 1 has a single scattering albedo ω₁ and medium 2 has a single scattering albedo ω₂.  Radiative transfer is not considered inside medium 0 that can be considered as air for instance. The main goal of the work is to compute the intensity going out upward from medium 1 to medium 0. In this problem, J₀, J₁ and J₂ are considered collimated radiation, and there are reflective interfaces between the different media. Then, the Snell's law of refraction applies and gives the inclinations ofJ₁  from θ₀ and the the inclinations of J₂  from θ₁.

 

Figure 1: Scheme of the fluxes and quantities involved inside the media.

The resolution of the radiative transfer is conducted the way as Bruce Hapke did [4], using isotropic scatterers, giving four differential equations to be solved, as two per surface strata:

Reflectance:

Integrating the differential equations considering continuity at the interfaces and finite energy for large τ gives the following expression for the reflectance, with A1, A2, B, C1 and C2 being the integrations constant, Rspec the specular reflectance at the top interface and γ_1,2²=1-ω_1,2². This expression is then modified to take anisotropy into account, using similarity relations as detailed in section 8.6.4 of Hapke 2012 [4].

Singularities:

This expression of the reflectance contains 3 singularities that must be treated: one in 1/(1/μ-2γ₁) is visible in the previous equation, and two others of the same form hidden in constants C1 and C2. These singularities have no physical basis and should be possible to simplify. They already existed in the granular monolayer version of the two-stream resolution from B. Hapke and the original expression could be reformulated into one without said singularities.

Figure 2 illustrates the singularity of the reflectance for the monolayer case. The blue plot represents the equivalent of the Eq. 2 implemented naively. The orange plot represents the same equation, after simplification. Without simplification, the errors are large in almost all the physical domain ω>0.3 which makes the naïve expression useless.

Figure 2: Illustration of the effect of a singularity in two mathematically identical formulations of the monolayer Hapke model [4]

Challenge:

Contrarily to the monolayer case, the two-layer case faces an analytical expression that is much more complex. For the moment, we did not find a reformulation that eliminates these singularities. The numerical treatment seems challenging but a solution may exist. We will discuss the strategies deployed attempting to solve this problem.

In the literature, numerous studies especially for photometry use a two granular layers resolution provided by Hapke that contains theses singularities. This is a problem because they can introduce significant numerical errors, such as illustrated in fig.2. It is crucial to note that, even if the two formulations are identical on the mathematical point of view, the singularity has a huge impact on the numerical implementation of it and the overall reflectance estimate.

Conclusion

We have developed a new spectrophotometric radiative transfer model under the two-stream approximation for layered planetary surfaces. This model addresses the limitations of existing models, such as the Hapke model, by considering both compact and granular layers with an unlimited number of components in intimate mix. It handles collimated radiation and reflective interfaces between media.

However, the validation process revealed significant challenges, particularly the presence of singularities in the reflectance expression. These singularities, which have no physical basis, can introduce substantial numerical errors and affect the accuracy of reflectance estimates. While we have not yet found a reformulation to eliminate these singularities, addressing them is crucial for improving the model's reliability and applicability in studying complex planetary surfaces. 

References:

[1] Murchie et al., JGR: PLanets, 2007

[2] Brown et al., Space Science Reviews, 2004

[3] Cremonese et al., Space Science Reviews, 2020

[4] Habke, 2012, Cambridge University Press