  To what extent can particulate random media be characterised using direct
wave backscattering from a single receiver/source? Here, in a two dimensional
setting, we show using a machine learning approach that both the particle
radius and concentration can be accurately measured when the boundary condition
on the particles is of Dirichlet type. Although the methods we introduce could
be applied to any particle type. In general backscattering is challenging to
interpret for a wide range of particle concentrations, because multiple
scattering cannot be ignored, except in the very dilute range. Across the
concentration range from 1% to 20% we find that the mean backscattered wave
field is sufficient to accurately determine the concentration of particles.
However, to accurately determine the particle radius, the second moment, or
average intensity, of the backscattering is necessary. We are also able to
determine what is the ideal frequency range to measure a broad range of
particles sizes. To get rigorous results with supervised machine learning
requires a large, highly precise, dataset of backscattered waves from an
infinite half-space filled with particles. We are able to create this dataset
by introducing a numerical approach which accurately approximates the
backscattering from an infinite half-space.
