This paper presents the asymptotic analysis of the sound propagation in an ocean-surface bilinear duct.In the previous study, we have derived alternative sound representations involving rays, diffracted waves, and normal modes, and various combinations of these. These representations have been derived from the integral representation of the sound fields by applying the various evaluation techniques. However, when the duct has a velocity profile that varies more generally in depth direction, the sound fields cannot be expressed in the integral form. In the present study, we directly derive geometric acoustical rays, diffracted waves, and adiabatic modes by applying the ray tracing technique, the geometrical theory of diffraction, and the narrow band parabolic equation method, respectively. It is confirmed, by comparison with the reference solution calculated from the mode theory and with the experimental data reported by Pedersen and Gordon, that the asymptotic representations developed here provide an effective means of predicting the sound fields in the duct. We also discuss how to extend the analytical results to more general environments with inhomogeneous velocity profile which varies vertically as well as laterally with respect to the guiding direction.