## SHDOM Convergence Examples

These examples illustrate how to choose the angular resolution, "base"
grid spatial resolution, and the adaptive grid cell splitting parameter
to achieve the desired radiometric accuracy with the minimum CPU time.

### LES Example

Solar radiative transfer computed with SHDOM is performed for a 2D slice
(X-Z) from the LES simulation described below. The liquid water content and effective radius
image shows that the cloud layer is primarily contained within heights
from 500 to 800 meters in the 3.5 km (X) by 1.0 km (Z) domain. A
wavelength of 1.65 micron and a sun angle of 45 degrees is specified.
The extinction field from Mie scattering
has a peak of 100 /km at one point near cloud top.
The accuracy is determined by comparing the radiance and flux output
from SHDOM runs with a reference run. The reference run has Nmu=32
zenith angles by Nphi=64 azimuth angles and Nx=512 by Nz=161 grid points
(6 to 7 meter spatial resolution). The SHDOM runs are done for
Nmu=8,Nphi=16 and Nmu=16,Nphi=32 angular resolutions. Comparisons are
shown for even spaced grids and adaptive grid starting from the Nx=64 by
Nz=21 base grid (55 m by 50 m resolution). Three types of output
radiometric quantities are compared: the upwelling radiance at 64
locations for 7 angles in the X-Z plane (0,+/-15,+/-30,+/-45 degrees),
the upwelling hemispheric flux at every base grid point, and the mean
radiance at every base grid point. The comparisons to the reference are
made by plotting the rms absolute difference divided by the mean as a
function of the number of grid points.

The results show that increasing the spatial resolution (number of grid
points) improves results only up to the point where the error is
dominated by the limited angular resolution. For example, the limiting
accuracy of the upwelling radiance is about 3% for Nmu=8 Nphi=16
resolution, but 1.5% for Nmu=16 Nphi=32. Another important point
illustrated is that the accuracy does not keep improving as the adaptive
grid procedure generates more grid points. This is because the base
grid is needed to represent the radiance field everywhere, while the
adaptive grid concentrates on areas where the source function is
changing rapidly. The plots show that adaptive grid substantially
improves the accuracy from the base grid and with considerably fewer
grid points (i.e. less CPU time) than the evenly spaced grid. A rule
of thumb is that the splitting accuracy parameter should be chosen so
that the number of new grid points is not much larger than the number of
base grid cells. Here the best choice is SPLITACC=0.03 (for a solar
flux of 1) which reduces the radiance error to 1.5% with 2863 total grid
points (from a base grid of 1365 points). The resulting adaptive grid cell plot shows that
new grid points were created in the optically thicker regions,
especially those directly illuminated by sunlight.

### Landsat Example

This example is also for solar radiative transfer, but for a cloud field
derived from Landsat data. The visible channel is used to derive the
optical depth with the independent pixel assumption and the thermal
infrared channel is used to obtain the cloud top height. The vertical
profile of extinction is derived from an adiabatic liquid water
assumption. The extinction and adaptive grid
cells show that this field contains two separate cumulus clouds. The
base grid has 100 meter spacing for 5 km in X and 3 km in Z. A phase
function 10 micron effective radius water droplets at 0.63 micron
wavelength is assumed. A sun angle of 45 degrees is used again. The
surface albedo is 0.4 for the center 2 km and 0.2 for the rest of the
surface.
The same radiometric quantities are compared between even and
adaptive grid cases and a reference case with Nx=400 Nz=241 Nmu=32
Nphi=64.

Each of these results show the adaptive grid to be extremely powerful in
substantially improving accuracy with the addition of a small number of
grid points. Again the accuracy achieved depends on both the angular
resolution and the spatial resolution. Paradoxically the adaptive grid
performs better for the Nmu=8 case, probably because the large base grid
cells serve to smooth the discontinuities in the radiance field caused
by the few number of discrete ordinates. With adequate angular accuracy
the even grid approach has better accuracy for large numbers of grid
points because it can represent the radiance field in clear sky more
faithfully.
Comparing the LES convergence test with the Landsat test illustrates
that for some fields the adaptive grid method performs much better than
for others. In nearly all cases where the base grid gives poor accuracy,
the adaptive grid will improve the accuracy more rapidly than an evenly
spaced grid.

Last modified: June 20, 1997
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