Pre-conditioned backward Monte Carlo solutions to radiative transport in planetary atmospheres

Abstract

Context. The interpretation of polarised radiation emerging from a planetary atmosphere must rely on solutions to the vector radiative transport equation (VRTE). Monte Carlo integration of the VRTE is a valuable approach for its flexible treatment of complex viewing and/or illumination geometries, and it can intuitively incorporate elaborate physics. Aims. We present a novel pre-conditioned backward Monte Carlo (PBMC) algorithm for solving the VRTE and apply it to planetary atmospheres irradiated from above. As classical BMC methods, our PBMC algorithm builds the solution by simulating the photon trajectories from the detector towards the radiation source, i.e. in the reverse order of the actual photon displacements. Methods. We show that the neglect of polarisation in the sampling of photon propagation directions in classical BMC algorithms leads to unstable and biased solutions for conservative, optically-thick, strongly polarising media such as Rayleigh atmospheres. The numerical difficulty is avoided by pre-conditioning the scattering matrix with information from the scattering matrices of prior (in the BMC integration order) photon collisions. Pre-conditioning introduces a sense of history in the photon polarisation states through the simulated trajectories. Results. The PBMC algorithm is robust, and its accuracy is extensively demonstrated via comparisons with examples drawn from the literature for scattering in diverse media. Since the convergence rate for MC integration is independent of the integral's dimension, the scheme is a valuable option for estimating the disk-integrated signal of stellar radiation reflected from planets. Such a tool is relevant in the prospective investigation of exoplanetary phase curves. We lay out two frameworks for disk integration and, as an application, explore the impact of atmospheric stratification on planetary phase curves for large star-planet-observer phase angles. By construction, backward integration provides a better control than forward integration over the planet region contributing to the solution, and this presents a clear advantage when estimating the disk-integrated signal at moderate and large phase angles.

Towards a comprehensive model of Earth's disk-integrated Stokes vector

“…In the implementation of the PBMC algorithm, the irradiance F at the observer's location is evaluated through a summation: where I ( u i , v i ) is the outgoing radiance Stokes vector from the i -th one-photon experiment, the uniform random variables u, v ∈ [0, 1] sample the planet visible disk and α is the star–planet-observer phase angle (García Muñoz & Mills 2014). Equation (1) is effectively an arithmetic average and its convergence properties depend on the dispersion in the outcome of the one-photon experiments.…”

“…The fundamentals of the PBMC algorithm have been described (García Muñoz & Mills 2014), and the model has been validated for horizontally uniform atmospheres (García Muñoz & Mills 2014; García Muñoz et al 2014). The validation exercise included a few thousand solutions from the literature for plane-parallel media, disk-integrated model solutions for Rayleigh-scattering atmospheres and measurements of the disk-integrated brightness and polarization of Venus.…”

“…in the same order as the actual photon displacements. In the MC approach, the total computational cost is proportional to n ph . Recent work has introduced an alternative approach based on Pre-conditioned Backward Monte Carlo (PBMC) integration of the vector Radiative Transport Equation (García Muñoz & Mills 2014). Past applications of the PBMC model to the problem of scattering from a spatially unresolved planet focused on atmospheres that might be stratified in the vertical but exhibited otherwise uniform optical properties in the horizontal direction.…”

Exoplanet phase curves at large phase angles. Diagnostics for extended hazy atmospheres

“…The radiative transfer calculations presented in this work were done with a Backward Monte Carlo algorithm that has been described elsewhere (García Muñoz & Mills 2015), and used to investigate the phase curves of exoplanets, Venus and Titan (García Muñoz & Isaak 2015;García Muñoz et al 2014. These previous studies omitted the consideration of the finite angular size of the star, which can modify the effective scattering geometry at the particle level.…”

“…Mathematically (Eqs. (9)-(10) in García Muñoz & Mills (2015), and assuming that all stellar rays are attenuated by the same amount), this involves an integral over the solid angle Ω subtended by the star at the collision location: ∫ ∂Ω p a (s · s )dΩ(s ) which can be pre-calculated and re-written as:…”