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\begin{sf}
SCATTERING

sample a new incident direction
  $(\theta_{inc},\phi_{inc})$ according to 
\[
g(\theta_{inc},\phi_{inc})=\frac{Z_{11}(\theta_{scat},\phi_{scat},
\theta_{inc},\phi_{inc})\sin(\theta_{inc})}{K_{11}(\theta_{scat},\phi_{scat})
  - K_{a1}(\theta_{scat},\phi_{scat})}
\]
Calculate the matrix $\mathbf{Q_k}=\mathbf{Q_{k-1}q_k}$
, where
\[
\mathbf{q_k}=\frac{\sin(\theta_{inc})_k
  \mathbf{O(s_k,s_{k-1})}\mathbf{Z(n_{k-1},n_k)}}
  {g\left(\Delta s\right)g(\theta_{inc},\phi_{inc}) \tilde{\omega}} ,
\]
and $\mathbf{Q_0}=\mathbbm{1}$.
\end{sf}
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