  A simple inverse problem for the wave equation requires determination of both
the wave velocity in a homogenous acoustic material and the transient waveform
of an isotropic point radiator, given the time history of the wavefield at a
remote point in space. The duration (support) of the source waveform and the
source-to-receiver distance are assumed known. A least squares formulation of
this problem exhibits the "cycle-skipping" behaviour observed in field scale
problems of this type, with many local minima differing greatly from the global
minimizer. An extended formulation, dropping the support constraint on the
source waveform in favor of a weighted quadratic penalty, eliminates this
misbehaviour. With proper choice of the weight operator, the velocity component
at any local minimizer of this extended objective function differs from the
global minimizer of the least-squares formulation by less than a linear
combination of the source waveform support radius and data noise-to-signal
ratio.
