Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian

TLDR

Efficient simulation of wave propagation in lossy media with power‑law absorption is important for biomedical ultrasonics, but existing time‑domain fractional operator equations require storing the full pressure field, limiting their use in many 3‑D problems. The authors derive a wave equation that employs two lossy derivative operators based on the fractional Laplacian. These operators separately model power‑law absorption and dispersion and can be incorporated into Fourier‑based pseudospectral and k‑space methods without the memory overhead of time‑domain fractional counterparts. They present a framework using three coupled first‑order constitutive equations to encode the wave equation and validate it through one‑, two‑, and three‑dimensional simulations.

Abstract

The efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics. Previous wave equations which use time-domain fractional operators require the storage of the complete pressure field at previous time steps (such operators are convolution based). This makes them unsuitable for many three-dimensional problems of interest. Here, a wave equation that utilizes two lossy derivative operators based on the fractional Laplacian is derived. These operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into Fourier based pseudospectral and k-space methods without the increase in memory required by their time-domain fractional counterparts. A framework for encoding the developed wave equation using three coupled first-order constitutive equations is discussed, and the model is demonstrated through several one-, two-, and three-dimensional simulations.