In this study we examine the recovery of vertically varying anisotropic crust and upper mantle structure using converted teleseismic waves recorded on three-component seismograms from individual broadband stations. Our analysis is cast in terms of inverse scattering theory for a one-dimensional medium and begins with a derivation of the plane wave Green's function for a homogeneous medium exhibiting arbitrary elastic anisotropy. This Green's function is employed within the single-scattering approximation to derive formulae that relate the amplitude of the scattered wave in time to perturbations in material properties at corresponding depths. Inclusion of seismograms from multiple events representing a range of azimuths and incident angles leads to the construction of a linear system of equations that is readily solved using singular value decomposition. A useful by-product of this amplitude-versus-slowness approach is the identification of simple and compact expressions for linearized reflection and conversion coefficients in anisotropic media that are accurate for small contrasts at near-vertical angles of incidence. We demonstrate the accuracy of the linearized P-to-S transmission coefficient for an idealized upper mantle model. In a companion paper we will examine the application of this approach to field data recorded at stations of the Canadian National Seismograph Network.