Schematic illustration of radiative transfer with resonant Compton scattering (RCS) and the computation of the observed polarization flux. The surface element of the NS is ﹩d{{\boldsymbol{A}}}_{s}={\widehat{\mathit{\boldsymbol{\Omega }}}}_{s}d{A}_{s}﹩. The scattering surface element (located at ﹩{{\boldsymbol{r}}}_{{\rm{sc}}\,}={r}_{\,{\rm{sc}}}{\widehat{\mathit{\boldsymbol{\Omega }}}}_{\,{\rm{sc}}\,}﹩) is denoted by dA. The scattered radiation propagates along the direction ﹩\widehat{\mathit{\boldsymbol{\Omega }}}\simeq {\widehat{\mathit{\boldsymbol{\Omega }}}}_{\,{\rm{obs}}\,}﹩, and the incident radiation (before scattering) propagates along ﹩{\widehat{\mathit{\boldsymbol{\Omega }}}}^{{\prime} }=({r}_{\,{\rm{sc}}\,}{\widehat{\mathit{\boldsymbol{\Omega }}}}_{0}-R{\widehat{\mathit{\boldsymbol{\Omega }}}}_{s})/| {r}_{\,{\rm{sc}}\,}{\widehat{\mathit{\boldsymbol{\Omega }}}}_{0}-R{\widehat{\mathit{\boldsymbol{\Omega }}}}_{s}| ﹩. The magnetic axis of the magnetar is along the z-axis, and ﹩{\widehat{\mathit{\boldsymbol{\Omega }}}}_{\,{\rm{obs}}\,}﹩ is in the y–z plane, with the angle between ﹩{\widehat{\mathit{\boldsymbol{\Omega }}}}_{\,{\rm{obs}}\,}﹩ and the z-axis being θ.