Pseudo-Differential Sweeping Method for Ultrasound Waves with Fractional Attenuation

We derive a pseudo-differential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via a two-way (transmission and reflection) sweeping scheme tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudo-differential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Pad\'e approximations for the pseudo-differential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Pad\'e approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically.