The first spatial importance sampling scheme we have investigated is directly tied to the zoning. For a problem with high opacity at line center, the small zones near the surface are very important to getting the transport and emission spectrum right, and therefore must be treated with more particles per unit volume than the thicker zones in the center of the problem. We have compared three schemes. The first is straightforward emission of particles with a constant density per unit volume (unbiased). The next scheme involves emission of an equal number of particles in each zone, which translates to a geometric weighting (based on the zoning scheme), in favor of the smaller zones near the surface. The final scheme involves a further geometric biasing within each zone so particles are born closer to interfaces where they have a greater chance of crossing between zones.
Favoring of photons born near the surface of an opaque problem is key to obtaining an emission spectrum with low statistical error. For photons at line center, only those born near the surface have a good chance of streaming out of the problem and making a contribution to the output spectrum. Importance sampling schemes improve the statistics of escaping photons that have a significant weight in the line center frequencies.
A final step in importance sampling is required to efficiently treat the large central zones in problems with high line center opacity. If particles are emitted uniformly within the zone, one emits line center photons with high probability. These photons travel only a short distance within the zone before their weight shrinks to the point of being insignificant, and their particle history is terminated. A lot of computer time is expended computing a deterministic equilibrium solution for the central zone, while getting very noisy results for the transport across zone boundaries, the quantity of interest. We found that just as the geometric zoning scheme with an equal number of photons in each zone improves the emission spectrum from a surface, a similar geometric subzone scheme to importance sample the regions near the surface of the interior zones reduces the noise in the transport between thick interior zones, resulting in an improved solution throughout the problem.
The subzone biasing scheme is very similar to the zone biasing described above. One creates a subzone grid for each real zone in the problem, starting with a thin subzone at each surface of an interior zone. Successive subzone sizes are then obtained in a geometric progression by increasing the subzone size by a factor of 2, working inwards to the interior of the zone from both sides. The process is stopped when the remainder of the zone is just larger than the subzone on each side of it. In Fig. 1, we show this subzone configuration using the dashed lines. Just as was the case for zonal biasing, an equal number of photons is emitted in each subzone of a given zone. The weights of the photons emitted in each subzone are adjusted so that emitted weight is distributed uniformly across the zone and the correct total weight is emitted within the zone.
We would like to note that one could employ directional biasing for photons born near a zone interface, favoring those heading in the direction of the interface; as that would contribute to transport between zones. We do not explore this in this paper but would expect an advantage to be generated for SIMC where there is no effective scattering. For IMC, the effective scattering term would produce angular mixing that would remove the advantage for this type of biasing.