A static memory method for modelling time-fractional power law absorption

The attenuation of ultrasound propagation through tissue is known to follow a frequency power-law in the time domain. This can be modelled as a loss operator in the equation of state within the Euler equations that takes the form of a fraction time derivative operator. This can be treated through a second order accurate transfer between the fractional time derivative and a fractional Laplacian spacial operator in order to avoid storage of the full time history. This is used for example by the k-Wave toolbox. As an alternate finite history methods have been suggested. Building on recent work, here, we re-write the time fractional derivative as a finite sum using a recursion relation. This allows the time fractional derivative to be directly computed with a static memory requirement. For homogeneous media, the advantage of this over the fractional Laplacian methods may not initially be obvious with a base higher memory requirement and computational cost with only a small increase in accuracy. However, upon introducing a heterogeneous medium with regions of different power law attenuation; the increased computational cost of the time fractional method is contained exclusively within the pre-computation, while the increased cost for the fractional Laplacian is applied to each time step.