$I^e_\nu$ is the local specific intensity (as emitted at the surface patch of the NS); $\textbf{k}$ is the direction of photon propagation at the surface; $\alpha\equiv\cos^{-1}(\textbf{k}\cdot\hat\textbf{r})$ is the angle between $\textbf{k}$ and the local radial vector; and $\mu$ is the cosine of the polar angle (with respect to the observation axis) of the patch. Due to gravitational light bending, $\cos\alpha$ is a non-trivial function of $\mu$. Following Beloborodov (2002), we make an approximation to this function, and calculate the observed specific flux as $$F^o_\nu\simeq\int {g^3\over{D^2}}I^e_\nu({\bf k})\left[1-g^2\left(1-\mu\right)\right]dA,
\label{eq:dFonu3}$$ where the integration is over the entire observable area (including light bending) of the NS surface. Thus, given the knowledge of local radiative intensity \[$I^e_\nu(\textbf{k})$\] of all surface patches, combined with the photon trajectory at different rotational phase (time) \[$\mu(t) \Leftrightarrow \alpha(t)$\], the resultant observed specific flux can be calculated.

Radiative Transfer in Magnetic Neutron Star Atmospheres
-------------------------------------------------------

The full computation of $I^e_\nu(\textbf{k})$ requires construction of atmospheric models in radiative equilibrium for various surface magnetic fields $\bB$ (magnitude and direction), effective temperatures $T_{\mathrm{eff}}$, and compositions (e.g., Shibanov et al. 1992; Pavlov et al. 1995; Zane et al. 2001; Ho & Lai 2001, 2003, 2004; Potekhin et al. 2004; van Adelsberg & Lai 2006; 