An acoustic inverse scattering problem which aims to reconstruct the acoustical parameters of the spherical coating material surrounding a sound-soft spherical object is investigated. The compressibility of the coating material is assumed to have one-dimensional variation along the radial direction. The coupled integral equations, the object and data equations, are reduced to one-dimensional form using the spherical symmetry. The inverse problem is then solved iteratively via Newton method. Numerical simulations demonstrate that the method is highly effective for profiles with smooth variation and provides a useful approximation for layered profiles. However, the success of the method depends on choosing an appropriate initial guess as expected for an iterative solution based on Newton algorithm.