Seismic inversion can be formulated by considering a linearized integral relation which is deduced from the wave equation. This Born inversion approach is equivalent to linear least-squares inversion for a particular parameterization of the medium. The least-squares solution is a member of a family of generalized LP norm solutions which are deduced from a maximum-likelihood formulation. This formulation allows design of various statistical inversion solutions. We present two iterative solutions to the one-dimensional (1-D) seismic inverse problem: the iterative least-squares (ILS) and the iterative reweighted least-squares (IRLS) methods. The ILS method involves solving a distorted background velocity problem after the initial least-squares solution is obtained. The IRLS method is used as a robust regression technique which is better suited for dealing with certain types of noise and is computationally faster than ILS. Several numerical examples demonstrate that the IRLS method accurately estimates impedance profiles despite the presence of large-amplitude noise spikes in the seismic traces. Numerical experiments suggest that the IRLS inversion can also be insensitive to noise bursts which are of a lower frequency band than noise spikes.