  We consider an inverse problem associated with some 2-dimensional non-compact
surfaces with conical singularities, cusps and regular ends. Our motivating
example is a Riemann surface $\mathcal M = \Gamma\backslash{\bf H}^2$
associated with a Fuchsian group of the 1st kind $\Gamma$ containing parabolic
elements. $\mathcal M$ is then non-compact, and has a finite number of cusps
and elliptic singular points, which is regarded as a hyperbolic orbifold. We
introduce a class of Riemannian surfaces with conical singularities on its
finite part, having cusps and regular ends at infinity, whose metric is
asymptotically hyperbolic. By observing solutions of the Helmholtz equation at
the cusp, we define a generalized S-matrix. We then show that this generalized
S-matrix determines the Riemannian metric and the structure of conical
singularities.
