Diagram illustrating the basic relationship of the rest-frame frequencies of spectral lines (ν_CMF) to observer’s frame frequency (ν_obs) for one particular (non-core) p-ray in the spherically symmetric geometry (p,z geometry). Shown are two spectral lines of different opacity χ which get shifted across the observer’s frame frequency by the velocity field in the wind. The dots represent the stepping points of the adaptive microgrid used in solving the transfer equation in the radiative line transfer. The method employed is an integral formulation of the transfer equation using an adaptive stepping technique on every p-ray in which the radiation transfer in each micro-interval is solved as a weighted sum on the microgrid (cf. Pauldrach et al. 2001): where I is the specific intensity, S is the source function and τ is the optical depth (increasing from τ₀ on the right to τ_n on the left in the figure). To accurately account for the variation of the line opacities and emissivities due to the Doppler shift, all line profile functions are evaluated correctly for the current microgrid-(z,p)-coordinate on the ray, thus effectively resolving individual line profiles. Based on that, application of the Sobolev technique gives for the radiative line acceleration (cf. Pauldrach et al. 2012): where is the Sobolev optical depth, and ν₀ is the frequency at the center of each line – thus, the effects of line-overlap and multiple scattering are naturally included (χ_line(r) is the local line absorption coefficient, μ is the cosine of the angle between the ray direction and the outward normal on the spherical surface element, and c is the speed of light)⁴.