This disclosure provides a system and method for producing ultrasound images based on Full Waveform Inversion (FWI). The system captures acoustic/(an)elastic waves transmitted through and reflected and/or diffracted from a medium. The system performs an FWI process in a time domain in conjunction with an accurate wave propagation solver. The system produces 3D maps of physical parameters that control wave propagation, such as shear and compressional wavespeeds, mass density, attenuation, Poisson's ratio, bulk and shear moduli, impedance, and even the fourth-order elastic tensor containing up to 21 independent parameters, which are of significant diagnostic value, e.g., for medical imaging and non-destructive testing.