Radiative transfer and travel time to characterize the surface

Introduction

The next generation of lidar instruments will encompass the travel time measurement for each photon packet that would be available to characterize the planetary surface medium, either spaceborne, in-situ or in the lab. For instance, the BepiColombo laser altimeter (BELA) (Thomas et al., 2021) includes this capability to explore the hermian surface. We propose here a tool to simulate in-silico the time transfer inside the planetary medium.

Methods

We propose a Monte Carlo approach based on the work from Farrell et al., 1992 and Boas et al., 2002. Our model computes the position of the ray during its travel through the medium. The main parameters are incidence angle, the optical thickness, the single scattering albedo, the Henyey-Greenstein or isotropic phase function.

Validation

The validation of the model is a mandatory step in order to compute a coherent and physically correct simulation of light time transfer within a planetary surface medium. In the model, planetary surface mediums can be either described as a semi-infinite granular layer or a homogeneous slab above a bedrock with a certain albedo. The purpose here is to be able to replicate the same simulations in the same conditions from previous works such as  Gao et al., 2013 , Gautheron et al., 2024 and Patterson et al., 1989. We first computed the reflectance in the same conditions as seen in Gao et al., 2013 in the case of a semi-infinite granular layer (Figure 1) which are an incidence angle of 50 degree, an optical thickness of 1000 and an isotropic phase function. The DISORT algorithm (Stamnes et al., 1998) gives a numerical solution for this case that is also computed and displayed for a double check. By comparing both DISORT solutions and Gao et al., 2013 results, our simulation shows a very good agreement, confirming the validity of our model in terms of angular distribution.  

Figure 1 : Reflectance for all azimuth computed in the conditions of semi-infinite granular layer for an incidence of 50 degrees,single scattering albedo omega = 1. Scattered rays follow isotropic diffusion law as expected ; black dots represents the DISORT solution ; black line represents the mean reflectance value for all azimuth ; color lines represents reflectance following the emergence for different azimuth

We propose then a method to compute the trajectory and the time travel of the rays in the medium, that would be simulating the response of a full waveform lidar. Knowing the speed of light in the medium and assuming its constancy according to the medium’s optical properties, we then evaluate the time travel in comparison with the analytical solution from Patterson et al., 1989. Results in Figure 2 shows a very good agreement that validates our algorithm.

Figure 2 : Histogram of time travel for an instance of semi-infinite granular layer compared with the analytical solution from Patterson et al., 1989. Here the source of the rays is isotropic at a depth z0 and observed at a distance 10 mm. The absorption coefficient is 0.005 mm-1, the scattering coefficient is 1 mm-1 and the speed of light is 299792458 m/s.

Result

As an illustration of the capabilities of our tool, we computed the travel time in the following conditions : an input incidence of 0°, a sufficient optical thickness to reach semi-infinite layer conditions and a Henyey-Greenstein phase function ; Figure 3 illustrates this situation. We can observe that most of the photons reach the surface in a short period of time meaning their travel length is rather small. Between 0.4 ns and 1 ns, fewer photons reach the surface implying a longer travel length. Thus, we are now able to discuss how the media properties will affect the response of a full waveform lidar.

Figure 3: Histogram of travel time from an input of 1 million rays, incidence = 0° with omega = 0.99 and a Henyey-Greenstein phase function with a scattering anisotropy of g = 0.8 , tau = 60 (semi-infinite medium). The absorption coefficient is 0.005 mm-1, the scattering coefficient is 1 mm-1 and the speed of light is 299792458 m/s.

Conclusion and perspectives

We propose a new approach to efficiently simulate the travel time of photons inside a planetary surface. We conducted several tests to validate the approach and one simulation in realistic condition. In the future, we will adapt this tool to peculiar planetary science cases, such as the mercury regolith for BELA, or the icy surface for GALA.

References

Boas, D. A., J. P. Culver, J. J. Stott, and A. K. Dunn, "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head," Opt. Express 10, 159-170 (2002)

Farrell, T.J., Patterson, M.S. and Wilson, B. (1992), A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo. Med. Phys., 19: 879-888. https://doi.org/10.1118/1.596777

Gao, M., X. Huang, P. Yang, and G. W. Kattawar, "Angular distribution of diffuse reflectance from incoherent multiple scattering in turbid media," Appl. Opt. 52, 5869-5879 (2013)

Gautheron Arthur, Raphaël Clerc, Vincent Duveiller, Lionel Simonot, Bruno Montcel, and Mathieu Hébert, "On the validity of two-flux and four-flux models for light scattering in translucent layers: angular distribution of internally reflected light at the interfaces," Opt. Express 32, 9042-9060 (2024)

Patterson, M.  S., B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989)

Stamnes, K.; Tsay, S.-C.; Jayaweera, K. & Wiscombe, W. Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media Appl. Opt., 1988, 27, 2502-2509, http://dx.doi.org/10.1364/AO.27.002502

Thomas, Nicolas & Hussmann, Hauke & Spohn, Tilman & Lara, L. & Christensen, Ulrich & Affolter, Michael & Bandy, T. & Beck, T. & Chakraborty, S. & Geissbuehler, U. & Gerber, Michael & Ghose, K. & Gouman, J. & Hosseiniarani, Alireza & Kuske, K. & Peteut, A. & Piazza, Daniele & Rieder, M. & Servonet, A. & Metz, Bodo. (2021). The BepiColombo Laser Altimeter. Space Science Reviews. 217. 10.1007/s11214-021-00794-y.