  We explicitly extract the structure of higher-derivative curvature-squared
terms at genus 0 and 1 in the d=4 heterotic string effective action
compactified on symmetric orbifolds by computing on-shell S-matrix superstring
amplitudes. In particular, this is done within the context of calculating the
graviton 4-point amplitude. We also discuss the moduli-dependent gravitational
threshold corrections to the coupling associated with the CP even quadratic
curvature terms.
