  In this paper, a two-step regularization method is used to solve an ill-posed
spherical pseudo-differential equation in the presence of noisy data. For the
first step of regularization we approximate the data by means of a spherical
polynomial that minimizes a functional with a penalty term consisting of the
squared norm in a Sobolev space. The second step is a regularized collocation
method. An error bound is obtained in the uniform norm, which is potentially
smaller than that for either the noise reduction alone or the regularized
collocation alone. We discuss an a posteriori parameter choice, and present
some numerical experiments, which support the claimed superiority of the
two-step method.
