Reconstruction of a small acoustic inclusion via time-dependent polarization tensors

Abstract

Abstract This paper aims at introducing the concept of time-dependent polarization tensors (TDPTs) for the wave equation associated to a diametrically small acoustic inclusion, with constitutive parameters different from those of the background and size smaller than the operating wavelength. Firstly, the solution to the Helmholtz equation is considered, and a rigourous systematic derivation of a complete asymptotic expansion of the scattered field due to the presence of the inclusion is presented. Then, by applying the Fourier transform, the corresponding time-domain expansion is readily obtained after truncating the high frequencies. The new concept of TDPTs is shown to be promising for performing imaging. In particular, the optimization approach proposed by Ammari et al. (Ammari, H., Kang, H., Kim, & E. Lee, J.-Y. (2012)The generalized polarization tensors for resolved imaging. Part II: shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements. Math. Comp., 81, 839–860.) is extended to TDPTs. Numerical simulations are presented, showing that the TDPTs reconstructed from noisy measurements allow to image fine shape details of the inclusion.