 the estimation of thickness is that the effect of magnetic field may influence the disk thickness.
\citet{2006ApJ...640..901H} pointed out that the location of photosphere of the accretion disk is modified by the magnetic
field significantly. However, in our case, the shadowing effect happens at the inner region, where the radiation pressures
dominates. Thus the thickness and the self-shadowing might not be significantly affected by the magnetic effect.
The magnetic field might also change the inner boundary condition significantly \citep[e.g.][and references therein]{2010arXiv1003.0966P,2010ApJ...711..959N}.
To investigate the effects of the magnetic field, we should do the general relativistic MHD simulations
\citep{2003ApJ...589..444G,2003ApJ...599.1238D}, which is beyond the
scope of this work.
}
\subsection{Ray-Tracing Method}
After obtaining the global structure of the relativistic disk, we
use the ray-tracing technique to calculate the emergent spectra with
inclusion of all the relativistic effects. The ray-tracing method is
based on the analytical integration of the null-geodesics
\citep{1983mtbh.book.....C,1994ApJ...421...46R,1998NewA....3..647C},
and the subroutines are the same as those used in
\citet{2009ApJ...699..722Y}.

One major improvement compared to earlier work \citep[e.g.
][]{2005ApJS..157..335L} is that we include the disk thickness in
the ray-tracing calculation. To achieve this, we start from the infinity, and
then search along the each photon trajectory for the intersection point.

Once the intersection points are found, we can calculate the local
temperature of the emitting points in our disk model and
calculate the redshift factor $g$. The emergent spectra are then
obtained by integrating over the photometric plane according to
\begin{equation}
 F_{\nu_o}=\int g^{3} I_{\nu_{e}} \frac{{d} \alpha {d} \beta}{D^2}\;,
\end{equation}
where $\alpha$ and $\beta$ are two impact parameters, and the redshift factor is
\begin{equation}
 g=\frac{\nu_{\rm o}}{\nu_{\rm e}}\;,
\end{equation}
here $D$ is the distance of the object expressed in geometrical units and $I_{\nu}$ takes the diluted blackbody form \citep[e.g.,][]{1995ApJ...445..780S}.

The redshift factors are calculated as follow:
\begin{equation}
 g=\frac{\nu_{\rm obs}}{\nu_{\rm em}}=\frac{\left.p_{\mu} u^{\mu}_{\rm Observer}\right|_{r=\infty,\theta=\theta_{\rm \rm obs}}}{\left. p_{\mu}u^{\mu}_{\rm Fluid}\right |_{r=r_{\rm em}, \theta=\theta_{\rm em}} }\;,
\end{equation}
where $p_{\mu}$ is the four-momentum of the photon, given by \citet{1968PhRv..174.1559C}
\begin{equation}
p_{\mu}=(-1,\pm \frac{\sqrt R}{\Delta},\pm \Theta^{1/2},\lambda)\;.
\end{equation}
The four-velocity of the fluid is given by \citet{2009ApJ...699..722Y}：
\begin{equation}
u^{\mu}_{\rm Fluid}=(\gamma_{r} \gamma_{\phi} e^{-\nu},\gamma_{r}\beta_{\rm r} e^{-\mu_1},\  0\  ,\gamma_{r}\gamma_{\phi}(\omega e^{-\nu}+\beta_{\phi} e^{-\psi}))\;,
\end{equation}
{
where $e^{ \nu}=(\Sigma \Delta / A)^{1/2}$, $e^{ \psi}=(A \sin^2 \theta/\Sigma)^{1/2}$.}
Thus we have the  expression for redshift factor:
 \begin{equation}
g=\frac{\Sigma^{1/2}\Delta^{1/2}}{A^{1/2}}\frac{1}{\gamma_r \gamma_{\phi}}\frac{1}{1\mp \frac{\beta_r}{\gamma_{\phi}}\frac{R^{1/2}}{A^{1/2}}-\lambda \omega -\lambda \beta_{\phi} \frac{r^2 \Delta^{1/2}}{A}}
 \end{equation}
where ``em'' means the emitting points, and the ``$+$'' sign for the trajectories of the 
photons without $r$-turning point and ``$-$'' sign for
those with $r$-turning point. In Equation(20),
\begin{equation}
R=r^4+(a^2-\lambda^2-Q) r^2+2(Q+(\lambda-a)^2 )r -a^2 Q \;.
\end{equation}
here the definition of $\lambda$ and $Q$ are the same as that of \citet{1973ApJ...183..237C}.

In the ray-tracing calculations, the outer boundary condition of the integration is set at $10^5 r_g$. In the cases when the emitting radius is so large that
the disk solutions do not cover, we use the results from SSD instead, because the differences between the transonic disk model and SSD are indeed insignificant at the large radii.

{
One limitation in our calculation is that the returning irradiation \citep{1995xrbi.nasa...58V,2005ApJS..157..335L} is neglected.
In the inner regions of the accretion disk, due to the strong gravity, the radiation emitted at one part of the accretion disk
may reach other parts, which may result in a modification to the disk structure and the emergent spectra. Certain disk geometry
may enhance such kind of effect \citep{1995xrbi.nasa...58V}. In the case of the thin accretion disk, this effect has moderate
influence (about several percents) on the emergent spectra \citep{2005ApJS..157..335L}. In our case, due to the thick geometry, since the disk is shadowed by itself, the self-irradiation effect may be stronger.
As the inclusion of that effect requires significant amount of calculation, for simplicity,
we neglect it in our current work.

Another limitation in our calculation is that the effect of limb-darkening has not been considered. As \citet{2005ApJS..157..335L}
have pointed out, the limb darkening has effect on the emergent spectra, especially when the inclination angle is large,
and the deviation induced by this effect is about $10\%$ at low energy band when the disk is viewed at $85^{\circ}$ \citep{2005ApJS..157..335L}.
In this work, this effect has not been included.
According to the limb-darkening law $I_{\theta} \sim 1/2+3/4 \cos \theta$, our results are still robust at relatively small inclination angles.
}
\begin{figure*}[h!]
\includegraphics[width=0.7 \textwidth]{vrsigma.eps}
\caption{\label{vrsigma}Radial velocity and surface density as a function of radii. From thin to thick, the lines stand for the solutions for the
accretion rates $1/16 \dot M_{\rm edd}$, $1/4\dot M_{\rm edd}$, $1/2 \dot M_{\rm edd}$, $1\dot M_{\rm edd}$ and $2 \dot M_{\rm edd}$ respectively. The left panels correspond to Schwarzschild black hole, and the right panels correspond to Kerr black hole with $a=0.98$. The black hole mass is taken to be $10 M_{\odot}$.}
\end{figure*}


\begin{figure*}[h!]
\includegraphics[width=0.7 \textwidth]{vphivphi.eps}
\caption{\label{vphivphi}Rotation velocity and the ratio of rotation velocity to Keplerian rotation velocity $v_{\phi}/v_{\phi}^{\rm Kep}$ vs. radii. The symbols of the lines are the same as in Figure \ref{vrsigma}. The left panels correspond to Schwarzschild black hole, and the right panels correspond to Kerr black hole with $a=0.98$.}
\end{figure*}

\section{Structure of the Relativistic Disk}

We present our results on the structure of the accretion disk. Throughout this paper, the critical
accretion rate is taken as
\begin{equation}
 \dot M_{\rm edd}=16\frac{L_{\rm edd}}{c^2}\;,
\end{equation}
where the numerical factor comes from the energy generation efficiency of the Schwarzschild black hole. For the sake of comparison,
we adopt the same critical accretion rate for both the Schwarzschild
and Kerr black holes. This is different from the convention used in \citet{2006ApJ...652..518M}. Similar results have already been obtained by \cite{2009ApJS..183..171S}, and one major difference between our work and theirs is that our work is based on