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In this paper we consider the inverse problem of vibro-acoustography, atechnique for enhancing ultrasound imaging by making use of nonlinear effects.It amounts to determining two spatially variable coefficients in a system ofPDEs describing propagation of two directed sound beams and the wave resultingfrom their nonlinear interaction. To justify the use of Newton's method forsolving this inverse problem, on one hand we verify well-definedeness anddifferentiability of the forward operator corresponding to two versions of thePDE model; on the other hand we consider an all-at-once formulation of theinverse problem and prove convergence of Newton's method for its solution.

Simultaneous reconstruction of sound speed and nonlinearity parameter in a paraxial model of vibro-acoustography in frequency domain