  We study Tikhonov regularization for solving ill--posed operator equations
where the solutions are functions defined on surfaces. One contribution of this
paper is an error analysis of Tikhonov regularization which takes into account
perturbations of the surfaces, in particular when the surfaces are approximated
by spline surfaces. Another contribution is that we highlight the analysis of
regularization for functions with range in vector bundles over surfaces. We
also present some practical applications, such as an inverse problem of
gravimetry and an imaging problem for denoising vector fields on surfaces, and
show the numerical verification.
