AI-driven Bayesian inference of statistical microstructure descriptors from finite-frequency waves
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Most geological processes are tied to heterogeneous microstructures, e.g. contaminated groundwater flow through porous media, production-induced compaction and/or microfracturing of rocks under stress leading to large-scale earthquakes. Still, the ability to reliably image materials at the microscale from long-wavelength wave data remains a major challenge to the geophysical, engineering and medical fields. At present, when it comes to describing microstructures, we generally rely on tomographic images of specific in-situ-acquired samples; a technique that is reliable and can capture realistic structures. This digital sample approach, however, makes the generalisation of microstructure complexity from single samples difficult to abstract and generalise. Here, we present a novel framework to constrain microstructure geometry and properties from long-scale waves, which are known to be sensitive to microheterogeneities. To quantify microstructures while maintaining generality we use two-point statistics, from which we can analytically compute scale-dependent effective wave properties - such as wavespeeds and attenuation - building on recent strong-contrast expansion (SCE) theory for (visco)elastic wavefields. A key aspect of this SCE approach is that it yields wave-equation-based effective properties at finite propagation frequencies, overcoming limitations of static-elasticity approaches in capturing scale-dependent wave behaviour. We apply the theory to acoustic and elastic waves in the long-wavelength regime and additionally to higher-frequency fields at scales close to the coherent scattering regime. By evaluating both analytical two-point correlation functions and that of image samples we observe that both effective wavespeeds and attenuation of longscale waves predominantly depend on volume fraction and phase properties, and that especially attenuation at small scales is highly sensitive to the geometry of microstructure heterogeneity (e.g. geometric hyperuniformity) due to incoherent inference of sub-wavelenght multiple scattering. Our main goal is not only to discuss effective wave behaviour, but primarily to investigate inferring microstructure properties from observed effective wave parameters. To both solve this highly nonlinear inversion problem and take uncertainty into account, we use the supervised machine learning method of Random Forests (RF) to construct a Bayesian inference approach. Assuming fixed medium contrasts, we can accurately resolve two-point correlation functions sampled from various microstructural configurations, including: a bead pack, Berea sandstone and Ketton limestone samples. Rather importantly, we show that inversion of smallscale-induced effective elastic wave data yields the best results, particularly compared to single-wave-mode (e.g., acoustic only) information. Additionally, we show that the retrieval of microscale medium contrasts is more difficult - as it is highly ill-posed - and can only be achieved with specific a priori knowledge. Our results, being the first fully nonlinear inference models of complex micro-descriptors from finite-frequency elasticwave properties, are promising for many imaging applications, such as earthquake hazard monitoring, non-destructive testing, imaging fluid flow properties in porous media, quantifying sub-wavelength tissue properties in medical ultrasound, or designing materials with tailor-made wave properties.