A possible error in the new spherical
code related to polarization?
A few models run with the new spherical program Rob supplied in June suggest discrepancies in
calculations run with 3 x 3 or 4 x 4 phase matrices. Here I present the evidence for your perusal.
All of the calculations below use similar vertical structures.
First, I compared results from the new spherical, VLIDORT (plane parallel) and the Monte Carlo spherical
code of Feng Xu.
Optical depth: 1, all below 80 km
Rayleigh scattering only
Observer at 80 km (for spherical cases)
Solar zenith angle: 37°
Ground albedo: 0.25
Azimuth: 0
Tau = 1, Az = 0
0.5
Spherical
0.48
Plane parallel
0.46
Monte Carlo
0.44
Spherical scalar
Plane parallel scalar
I/F
0.42
Monte Carlo scalar
0.4
0.38
0.36
0.34
0.32
0.3
0
30
60
Nadir Angle
1
90
Note:
1. The scalar calculations (dashed lines) are all in close agreement (except horizontally)
2. The plane parallel and Monte Carlo calculations are in close agreement
3. The spherical vector calculation (solid blue) seems out of family
For the same case, consider the Q Stokes parameter.
0.00
-0.02
0
30
60
-0.04
-0.06
Q
-0.08
-0.10
-0.12
-0.14
-0.16
Monte Carlo
Spherical RT
-0.18
-0.20
Nadir Angle
Q is significantly different for the spherical and Monte Carlo spherical cases.
Next change the azimuth to 30°. Do vector calculations.
2
90
0.50
0.45
0.40
0.35
I
0.30
0.25
0.20
0.15
Monte Carlo
0.10
Spherical RT
0.05
0.00
0
30
60
90
60
90
Nadir Angle
0.00
0
30
-0.05
Q
-0.10
-0.15
Monte Carlo
-0.20
-0.25
Spherical RT
Nadir Angle
3
0.040
0.020
0.000
0
30
60
90
U
-0.020
-0.040
-0.060
Monte Carlo
-0.080
Spherical RT
-0.100
-0.120
Nadir Angle
Large differences are present for Q and U. These persist at larger azimuth angles.
Next we compress the atmosphere to be below 0.1 km (still optical depth 1). We change the solar
zenith angle to be a value in the Coulson Tables (36.8699°). The reason for compressing the atmosphere
is to produce a case where the sphericity is negligible; the thickness of the atmosphere is tiny compared
to the radius of the planet. The azimuth is 0. We no longer compare against Monte Carlo results.
0.5
0.45
Plane parallel
I/F
0.4
0.35
Spherical
0.3
Coulson
0.25
0.2
0.15
0.1
0.05
0
0
30
Nadir Angle
60
4
90
0.18
Plane parallel
0.16
Spherical
0.14
Coulson
0.12
Q
0.1
0.08
0.06
0.04
0.02
0
0
30
Nadir Angle
60
90
Note:
1. The Coulson and plane parallel (VLIDORT) results agree perfectly.
2. The spherical results are different, especially for Q.
Next change the azimuth to 90°.
0.45
Plane parallel
0.4
Spherical
0.35
Coulson
I/F
0.3
0.25
0.2
0.15
0.1
0.05
0
0
30
Nadir Angle
60
5
90
0.06
Plane parallel
0.04
Spherical
Coulson
Q
0.02
0.00
0
30
60
90
60
90
-0.02
-0.04
-0.06
Nadir Angle
0.30
Plane parallel
0.25
Spherical
Coulson
U
0.20
0.15
0.10
0.05
0.00
0
30
Nadir Angle
Again, the Coulson and plane-parallel results agree, and the spherical result is different. The
differences for Q and U are large.
Finally we repeat this last case, but do scalar calculations.
6
0.45
Plane parallel
0.4
Spherical
0.35
Coulson
I/F
0.3
0.25
0.2
0.15
0.1
0.05
0
0
30
Nadir Angle
60
90
The VLIDORT (plane parallel) scalar case differs from the VLIDORT vector case, as expected. The
VLIDORT scalar and spherical scalar results closely agree.
Conclusions
The good agreement among the scalar calculations indicates that they are probably correct. However,
when polarization is included in the calculation, the agreement disappears. This is especially true for the
Q and U Stokes parameters. This is found for comparisons with Monte Carlo results and in cases where
the spherical correction is negligible.
The suggestion is that the polarization is not handled correctly for the spherical case. I recall that
including polarization requires some coordinate transformations between the scattering plane system
and a local normal system, and I wonder if that might be the problem.
7