Radiative-transfer-equation (RTE) inversion of coda-wave energy envelopes was recently demonstrated by Liu and Hu (2025; published 2026 with Jian) on discrete-element (DEM) simulations of granular media: fitting the analytical RTE solution to simulated coda envelopes recovered the scattering mean free path with errors below 3% under absorbing boundaries, versus a 5.71% error under a rigid reflecting boundary. That demonstration rests on idealizations the authors themselves flag: scattering treated as isotropic and effectively scalar, neglecting the vector nature of elastic waves and P–S mode conversion, and a two-dimensional setting. The open problem is to extend grain-scale RTE inversion to the conditions of real granular and planetary-regolith media: incorporate the full vector elastic wavefield with P–S conversion and anisotropic scattering into the forward and inverse theory; validate the inversion beyond two dimensions, in three-dimensional packings; and integrate finite and layered domains — including layers whose thickness is comparable to the scattering mean free path — into the inversion or a correction scheme, quantifying how inversion accuracy and boundary sensitivity behave in each regime.

Background

Seismic coda waves are the long, incoherent tails that follow the direct arrivals in a seismogram, produced by multiple scattering of wave energy off small-scale heterogeneities. Since Aki (1969) interpreted local-earthquake codas as scattered waves, the temporal decay and shape of coda envelopes have been used to measure the statistical scattering properties of the Earth's interior. The radiative transfer equation (RTE) is the standard mesoscopic theory for this purpose: it describes the transport of incoherent wave energy through a scattering medium without resolving the full wavefield, and its key material parameter is the scattering mean free path lsl_{s}, the average distance a wave packet travels between scattering events. Conventional RTE treatments of seismic coda idealize the medium as infinite and homogeneous, treat scattering as isotropic, and often track a single scalar energy density, which neglects the vector character of elastic waves — in particular the conversion between compressional (P) and shear (S) wave modes at heterogeneities (Muller and Shapiro 2001). Layered and bounded geometries were incorporated into RTE coda theory early for the crust (Margerin, Campillo and Tiggelen 1998), and vector-elastic radiative transfer including mode conversion was formulated and validated against full-waveform finite-difference simulations in two-dimensional continuous random media by Przybilla, Korn and Wegler (2006), who then estimated crustal scattering parameters from local-earthquake data with elastic radiative transfer theory (Przybilla, Wegler and Korn 2009). The current frontier of that program checks the consistency of fully three-dimensional elastic spectral-element coda simulations against Monte Carlo solutions of the radiative transfer equation, including the partitioning of energy between P and S modes (Bracale, Margerin, Brossier and Campillo 2025). All of this machinery targets continuous random media with volume-averaged scattering parameters. Unconsolidated granular media are a different and less explored setting. Dense granular media were shown experimentally to exhibit codalike multiple scattering of elastic waves (Jia 2004), and loose granular materials — notably the regolith covering the Moon and Mars — combine strong heterogeneity with finite layer thicknesses that can be comparable to lsl_{s}, so the infinite-medium assumption of standard RTE fails. The discrete element method (DEM) simulates such media by representing every grain explicitly, with contact forces (stiffness, damping, friction) resolved at each contact. Liu and Hu (2025; published 2026 with Jian) combined DEM wave simulations with analytical RTE inversion: in two-dimensional packings of circular grains, they fitted the analytical RTE energy-envelope solution to the mean-square envelope of the simulated coda and recovered the scattering mean free path with errors below 3% when the domain boundary was absorbing — implemented as a perfectly matched layer (PML), an artificial absorbing boundary that emulates an infinite medium — versus a 5.71% error under a rigid reflecting boundary, showing that boundary reflections systematically distort grain-scale inversion. Their inversion, however, rests on the idealizations they explicitly acknowledge: scattering treated as isotropic and effectively scalar, entailing, in their words, "the neglect of the whole vector nature of elastic wave propagation, particularly the conversions between compressional (P) and shear (S) waves", and a two-dimensional setting — while the motivating applications (planetary regolith seismology, laboratory probing of granular packs) involve vector wavefields, three-dimensional packings, and finite or layered domains. The unresolved target is the extension of this grain-scale inversion framework to exactly those conditions, together with a quantitative account of how inversion accuracy and boundary sensitivity behave there.

Problem Statement

Extend radiative-transfer-equation (RTE) based grain-scale inversion for elastic-wave scattering in granular media from its validated idealized setting — scalar, isotropic scattering in two-dimensional DEM packings with absorbing or rigid boundaries — to the physical conditions of real granular media and planetary regoliths, as one coherent objective: determine whether — and under what conditions — RTE-based grain-scale inversion can recover grain-scale scattering and microstructural parameters when (i) the full vector elastic wavefield is accounted for, including conversion between compressional (P) and shear (S) waves and anisotropic scattering; (ii) the medium is a three-dimensional packing; and (iii) the domain is finite or layered — including layers whose thickness is comparable to the scattering mean free path lsl_{s} — with boundary conditions incorporated into the forward RTE theory or corrected for in the inversion. A complete answer must quantify, against known ground truth (synthetic or experimental) that is generated by physics independent of the inversion's own forward model — for example coda envelopes from a DEM or full-waveform simulation with known parameters, or a laboratory measurement; envelopes synthesized from the inversion's own RTE forward model do not count as validation ground truth — the inversion accuracy and boundary sensitivity achieved in each of these regimes, thereby establishing the domain of reliability of RTE-based grain-scale inversion for laboratory and planetary-regolith applications. Deliverables (i)–(iii) are jointly necessary parts of this single objective — the reliability of grain-scale inversion under realistic granular conditions — rather than independent problems. The required deliverable is a single inversion in which all three conditions hold simultaneously, with the accuracy and boundary sensitivity reported for that combined regime and for its vector-elastic, three-dimensional, and finite/layered variants; a quantitatively demonstrated negative or limitative outcome — a narrow or empty domain of reliability in any regime — counts as a complete answer.

References

  1. Liu, Ning; Hu, Wen-Tao (2025). Numerical Investigation of Wave Scattering in Granular Media: Grain-Scale Inversion and the Role of Boundary Effects. arXiv:2507.07455. DOI 10.48550/arXiv.2507.07455. https://arxiv.org/abs/2507.07455
  2. Liu, Ning; Hu, Wen-Tao; Jian, Shi-Kai (2026). Numerical Investigation of Wave Scattering in Granular Media: Grain-Scale Inversion and the Role of Boundary Effects. Proceedings of the International Field Exploration and Development Conference 2025, Springer Series in Geomechanics and Geoengineering, pp. 23-34. DOI 10.1007/978-981-95-8269-3_3. https://doi.org/10.1007/978-981-95-8269-3_3
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