  The ability to image materials at the microscale from long-wavelength wave
data is a major challenge to the geophysical, engineering and medical fields.
Here, we present a framework to constrain microstructure geometry and
properties from long-scale waves. To realistically quantify microstructures we
use two-point statistics, from which we derive scale-dependent effective wave
properties - wavespeed and attenuation - using strong-contrast expansions (SCE)
for (visco)elastic wavefields. By evaluating various two-point correlation
functions we observe that both effective wavespeeds and attenuation of
long-scale 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-wavelength multiple scattering. Our goal is to
infer microstructure properties from observed effective wave parameters. To
this end, we use the supervised machine learning method of Random Forests (RF)
to construct a Bayesian inference approach. We can accurately resolve two-point
correlation functions sampled from various microstructural configurations,
including: a bead pack, Berea sandstone and Ketton limestone samples.
Importantly, we show that inversion of small scale-induced effective elastic
waves 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 are
promising for many applications, such as earthquake hazard
monitoring,non-destructive testing, imaging fluid flow in porous media,
quantifying tissue properties in medical ultrasound, or designing materials
with tailor-made wave properties.
