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This paper is concerned with the inverse acoustic scattering problem withphaseless total-field data at a fixed frequency. An approximate factorizationmethod is developed to numerically reconstruct both the location and shape ofthe unknown scatterer from the phaseless total-field data generated by incidentplane waves at a fixed frequency and measured on the circle $\partial B_R$ witha sufficiently large radius $R$. The theoretical analysis of our method isbased on the asymptotic property in the operator norm from$H^{1/2}(\mathbb{S}^1)$ to $H^{-1/2}(\mathbb{S}^1)$ of the phaselesstotal-field operator defined in terms of the phaseless total-field datameasured on $\partial B_R$ with large enough $R$, where $H^s(\mathbb{S}^1)$ isa Sobolev space on the unit circle $\mathbb{S}^1$ for real number $s$, togetherwith the factorization of a modified far-field operator. The asymptoticproperty of the phaseless total-field operator is also established in thispaper with the theory of oscillatory integrals. The unknown scatterer can beeither an impenetrable obstacle of sound-soft, sound-hard or impedance type oran inhomogeneous medium with a compact support, and the proposed inversionalgorithm does not need to know the boundary condition of the unknown obstaclein advance. Numerical examples are also carried out to demonstrate theeffectiveness of our inversion method. To the best of our knowledge, it is thefirst attempt to develop a factorization type method for inverse scatteringproblems with phaseless data.

An approximate factorization method for inverse acoustic scattering with phaseless total-field data