  In this work, we explore experimentally and theoretically the spectrum of
magnetic excitations of the Fe$^{3+}$ and Yb$^{3+}$ ions in ytterbium iron
garnet (Yb$_3$Fe$_5$O$_{12}$). We present a complete description of the
crystal-field splitting of the $4f$ states of Yb$^{3+}$, including the effect
of the exchange field generated by the magnetically ordered Fe subsystem. We
also consider a further effect of the Fe-Yb exchange interaction, which is to
hybridise the Yb crystal field excitations with the Fe spin-wave modes at
positions in the Brillouin zone where the two types of excitations cross. We
present detailed measurements of these hybridised excitations, and propose a
framework which can be used in the quantitative analysis of the coupled spectra
in terms of the anisotropic $4f-3d$ exchange interaction.
