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rtm-sgt-ocr-v1

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In our implementation, each grid point is moved slightly away from its nearest neighbor for a preset number of iterations.	In our implementation, each grid point is moved slightly away from its nearest neighbor for a preset number of iterations.
The effect is illustrated in Fig.	The effect is illustrated in Fig.
3 for a random 2D sampling of a Gaussian density profile.	\ref{smooth} for a random 2D sampling of a Gaussian density profile.
The top panels shows the initial unsmoothed distribution while the lower panels showed the triangulated point distribution after the smoothing algorithm has been applied.	The top panels shows the initial unsmoothed distribution while the lower panels showed the triangulated point distribution after the smoothing algorithm has been applied.
The plots in the right column show the neighbor distances as a function of radius.	The plots in the right column show the neighbor distances as a function of radius.
In the smoothed grid, the distances are much less scattered and follows the Gaussian profile (shown as the light colored full curve) much more accurately than in the top panel.	In the smoothed grid, the distances are much less scattered and follows the Gaussian profile (shown as the light colored full curve) much more accurately than in the top panel.
The smoothing strategy should not be exaggerated, i.e., moving the points too slowly and iterating trough too many stpdf, because the algorithm will then act as an annealing process and it will result in a perfectly regular grid where all variations in the point distribution due to the underlying density field is smeared out.	The smoothing strategy should not be exaggerated, i.e., moving the points too slowly and iterating trough too many stpdf, because the algorithm will then act as an annealing process and it will result in a perfectly regular grid where all variations in the point distribution due to the underlying density field is smeared out.
By doing it right, however, a smooth grid can be obtained while the underlying density structure is still preserved in the grid.	By doing it right, however, a smooth grid can be obtained while the underlying density structure is still preserved in the grid.
We have found empirically that by using 25 iterations and moving the closest neighbors about of the distance away from each other results in a sufficiently smooth grid that preserves the underlying physical structure well.	We have found empirically that by using 25 iterations and moving the closest neighbors about of the distance away from each other results in a sufficiently smooth grid that preserves the underlying physical structure well.
During gridding of our source model, we also distribute a number of points randomly on the surface of a sphere surrounding our model.	During gridding of our source model, we also distribute a number of points randomly on the surface of a sphere surrounding our model.
These points are also Delaunay triangulated and connected to the model grid points, but they do not represent anything except the surface of our computational domain.	These points are also Delaunay triangulated and connected to the model grid points, but they do not represent anything except the surface of our computational domain.
Whenever a photon reaches one of these sink points, it is considered to have escaped the model.	Whenever a photon reaches one of these sink points, it is considered to have escaped the model.
The photon transport itself goes along Delaunay lines only, from one point to another, which makes integration of Eq.	The photon transport itself goes along Delaunay lines only, from one point to another, which makes integration of Eq.
1 particularly simple and very fast.	\ref{radtran} particularly simple and very fast.
In the three-dimensional Delaunay triangulation, the expectation value for the number of lines attached to a grid point is approximately 16 (?) and the spatial sampling of J, is thus limited to this number of directions.	In the three-dimensional Delaunay triangulation, the expectation value for the number of lines attached to a grid point is approximately 16 \citep{ritzerveld2006} and the spatial sampling of $J_\nu$ is thus limited to this number of directions.
However, we still need to trace a number of photons along each Delaunay line, not only in order to sample the frequency band properly, but also because we cannot conserve momentum stringently with a single photon on this grid.	However, we still need to trace a number of photons along each Delaunay line, not only in order to sample the frequency band properly, but also because we cannot conserve momentum stringently with a single photon on this grid.
In principle, a photon passing a grid point froma certain direction should continue to travel in the exact same direction.	In principle, a photon passing a grid point from a certain direction should continue to travel in the exact same direction.
This is in general not possible due to the random orientation of the Delaunay lines, so instead we choose one of the two outgoing Delaunay lines (£1 and £2) that make the smallest angle with the original direction of the photon.	This is in general not possible due to the random orientation of the Delaunay lines, so instead we choose one of the two outgoing Delaunay lines $\ell1$ and $\ell2$ ) that make the smallest angle with the original direction of the photon.
The outgoing line is picked at random, but weighted by the ratio of the two angles, where Z;«4Z».	The outgoing line is picked at random, but weighted by the ratio of the two angles, where $\angle_1 < \angle_2$.
The same procedure is used at all subsequent grid points (using the original momentum vector to determine the outgoing direction) until the photons escape the model.	The same procedure is used at all subsequent grid points (using the original momentum vector to determine the outgoing direction) until the photons escape the model.
By sending a number of photons along each initial Delaunay line, we thus probe, not a single line of sight, but rather a cone, while still conserving momentum on average.	By sending a number of photons along each initial Delaunay line, we thus probe, not a single line of sight, but rather a cone, while still conserving momentum on average.
An example of the photon propagation is shown in Fig.	An example of the photon propagation is shown in Fig.
4 for a single point and a single direction.	\ref{photprop} for a single point and a single direction.
Because of the relative low number of photons needed to probe the spatial directions, we can allow ourself to increase the number of photons used to sample different frequencies, while we still maintain a low (initial) number of photons per grid point.	Because of the relative low number of photons needed to probe the spatial directions, we can allow ourself to increase the number of photons used to sample different frequencies, while we still maintain a low (initial) number of photons per grid point.
The inset in Fig.	The inset in Fig.
4 shows the distribution of the location where the photons reach the surface of the grid.	\ref{photprop} shows the distribution of the location where the photons reach the surface of the grid.
This distribution is reasonably well described by a Gaussian distribution around the intersection of the original momentum vector and the surface.	This distribution is reasonably well described by a Gaussian distribution around the intersection of the original momentum vector and the surface.
The number of initial photons is a user-defined setting, but as a default value, we use five times the number of neighbor points, so that each neighbor is initially probed by five photons.	The number of initial photons is a user-defined setting, but as a default value, we use five times the number of neighbor points, so that each neighbor is initially probed by five photons.
These photons are distributed evenly across a frequency range of «2σ with respect to the line center so that the median photon coincides with the local rest frequency.	These photons are distributed evenly across a frequency range of $\pm 3 \sigma$ with respect to the line center so that the median photon coincides with the local rest frequency.
o is determined by the local turbulent velocity dispersion through the user-defined Doppler b-parameter.	$\sigma$ is determined by the local turbulent velocity dispersion through the user-defined Doppler b-parameter.
Any given grid point will see more Delaunay connections coming from high density regions than from low density regions, simply because the grid point density is higher in high density regions.	Any given grid point will see more Delaunay connections coming from high density regions than from low density regions, simply because the grid point density is higher in high density regions.
Because of this inhomogeneity in the angular distribution of Delaunay connection, care must be taken when averaging the radiation field using Eq. 6..	Because of this inhomogeneity in the angular distribution of Delaunay connection, care must be taken when averaging the radiation field using Eq. \ref{jbar}.
In our implementation, this equation reduces to a discrete sum where N is the number ofDelaunay neighbors.	In our implementation, this equation reduces to a discrete sum where $N$ is the number ofDelaunay neighbors.
ω; is a weight that is proportional to the solid angle represented by the i'th Delaunay line.	$\omega_i$ is a weight that is proportional to the solid angle represented by the $i$ 'th Delaunay line.
This angle corresponds strictly to a surface area on a unit sphere, but we use the area of the Voronoi facet that corresponds to the Delaunay line as a good approximation (within 10%)).	This angle corresponds strictly to a surface area on a unit sphere, but we use the area of the Voronoi facet that corresponds to the Delaunay line as a good approximation (within ).
Figure 5 shows a comparison of J, between andRATRAN.	Figure \ref{flatdisk} shows a comparison of $J_\nu$ between and.
. The input model is a thin flat disk with a density profile oc 77!.	The input model is a thin flat disk with a density profile $\propto r^{-1}$ .
The radius is 500 AU and the height is 50 AU.	The radius is 500 AU and the height is 50 AU.
noise in the observed. data. and necessitates the use of smoothing to the observed. data.	noise in the observed data, and necessitates the use of smoothing to the observed data.
Mazeh Goldberg (1992) introduce an iterative algoritlim whose solution depends on the initial guess.	Mazeh Goldberg (1992) introduce an iterative algorithm whose solution depends on the initial guess.