  On-chip technology based on acoustic waves is a strong asset in modern
telecommunication with the prospects of becoming a cornerstone of
next-generation devices. In this context, mapping and manipulating acoustic
waves through coherent scattering is pivotal for a non-trivial control of the
flow of acoustic energy, which could consequently enable advanced information
manipulation. To this end, here a technique for mapping acoustic fields is
introduced and used for characterizing $\mu$m-sized phononic crystals defined
in GaAs slabs and excited by GHz surface acoustic waves (SAWs) based on atomic
force microscopy. It is shown that incoherent scattering excites a wide
distribution of modes, which enables the mapping of the dispersion relation of
the two-dimensional structures, while the phononic crystal symmetry directly
correlates with coherent scattering effects. Enabling the use of acoustic
atomic force microscopy and understanding the role of scattering are of
paramount importance for the versatile use of GHz acoustic waves in
technological applications, setting the baseline for advanced operations like
hyperspectral filtering, beam steering or spatial-division multiplexing.
