=Paper=
{{Paper
|id=Vol-2744/short25
|storemode=property
|title=The Cluster Low-Streams Regression Method for Fast Computations of Top-of-the-Atmosphere Radiances in Absorption Bands (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short25.pdf
|volume=Vol-2744
|authors=Ana del Águila,Dmitry Efremenko
}}
==The Cluster Low-Streams Regression Method for Fast Computations of Top-of-the-Atmosphere Radiances in Absorption Bands (short paper)==
https://ceur-ws.org/Vol-2744/short25.pdf
The Cluster Low-Streams Regression Method for Fast
Computations of Top-of-the-Atmosphere Radiances in
Absorption Bands ?
Ana del Águila1[0000−0001−9006−9631] , Dmitry Efremenko1[0000−0002−7449−5072]
1
Remote Sensing Technology Institute, German Aerospace Center (DLR), 82234
Oberpfaffenhofen, Germany
ana.delaguilaperez@dlr.de, dmitry.efremenko@dlr.de
Abstract. Atmospheric composition sensors provide a huge amount of data.
A key component of trace gas retrieval algorithms are radiative transfer mod-
els (RTMs), which are used to simulate the spectral radiances in the absorption
bands. Accurate RTMs based on line-by-line techniques are time-consuming. In
this paper we analyze the efficiency of the cluster low-streams regression (CLSR)
technique to accelerate computations in the absorption bands. The idea of the
CLRS method is to use the fast two-stream RTM model in conjunction with the
line-by-line model and then to refine the results by constructing the regression
model between two- and multi-stream RTMs. The CLSR method is applied to
the Hartley-Huggins, O2 A-, water vapour and CO2 bands for the clear sky and
several aerosol types. The median error of the CLSR method is below 0.001 %,
the interquartile range (IQR) is below 0.1 %, while the performance enhancement
is two orders of magnitude.
Keywords: Radiative transfer model, Regression model, Line-by-line model
1 Introduction
The information about the atmospheric trace gases can be retrieved from the spectral
radiances measured at the top of the atmosphere. The key component of atmospheric
retrieval algorithms are the radiative transfer models (RTMs). Accurate simulations in
the absorption bands are based on the so called line-by-line (LBL) model [1], which
requires thousands of monochromatic RTM computations per absorption band due to
strong spectral variability of the absorption coefficient. Alternatives to computationally
expensive LBL models are the k-distribution method [2, 3] and the principal component
analysis (PCA)-based RTMs [4–9], in which the redundancies in hyper-spectral data are
eliminated and the spectrum can be computed by using a small number of RTM calls.
These methods are reviewed in [10].
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0).
?
Supported by the German Aerospace Center (DLR) and the German Academic Exchange Ser-
vice (DAAD) through the programme DLR/DAAD Research Fellowship (57424731).
2 A. del Águila and D. Efremenko
In our recent work [11], the Cluster Low-Streams Regression (CLSR) method was
developed to accelerate hyper-spectral computations. The idea of the CLSR method
is to perform LBL computations by using a fast two-stream RTM and then to refine
results by using the correlation model for the two-stream and reference multi-stream
RTMs. This approach was applied to the O2 A-band and the weak CO2 band for different
atmospheric scenarios. The results were compared with the PCA-based RTMs showing
an improvement over the last in terms of accuracy. Note that the idea of improving
accuracy of two-stream models was exploited in numerous theoretical studies (see e.g.
[12, 13] and references therein).
In this study, the CLSR method is extended to ozone Hartley-Huggins band and
the water vapour band in the ultra-violet and near infrared spectral ranges, respectively.
Additionally, the CLSR method is applied to several atmospheric models containing
different aerosol types.
2 Methodology
2.1 Data overview
We consider the computations of the reflected spectral radiances at the top of the at-
mosphere (TOA) in the Hartley-Huggins, O2 A-, water vapour and CO2 bands. Table 1
summarizes the spectral bands examined with their corresponding spectral range, spec-
tral resolution and number of spectral points to be simulated. As a reference RTM,
we use the discrete ordinates with matrix exponential (DOME) method [14, 15]. The
number of discrete ordinates (streams) in the polar hemisphere Ndo regulates the com-
putational performance and accuracy. In the following, the model is called multi-stream
(MS) when Ndo ≥ 2 and low-stream (LS) otherwise. Following previous analysis in
[16], the multi-stream RTM with Ndo = 32 discrete ordinates is used as a reference
RTM.
The gaseous absorption coefficients for the O2 A-, water vapour and CO2 bands
are computed with the LBL model Py4CAtS [17], while the ozone absorption cross-
sections in the Hartley-Huggins band are taken from the HITRAN 2016 database [18].
Rayleigh scattering is modeled as proposed in [19].
Table 1. Spectral ranges, spectral resolution and number of spectral points for the absorption
bands used in this study.
Band Spectral range Spectral resolution Number of
(nm) (nm) spectral points
Hartley-Huggins 280 — 335 0.0188 2932
O2 A 755 — 775 0.0010 20000
Water vapour 770 — 1000 0.0058 40000
CO2 1590 — 1620 0.0015 20000
The atmosphere is discretized into 35 layers with a step of 1 km between 0 and
25 km, and a step of 2.5 km between 25 km and 50 km. For all the simulations, we
The Cluster Low-Streams Regression Method... 3
assume a Lambertian surface with an albedo of 0.3. The solar zenith angle, the viewing
zenith angle and the relative azimuth angle are 45◦ , 35◦ and 90◦ , respectively.
The atmosphere can contain one of the aerosols from the OPAC database [20], op-
tical properties of which are summarized in Table 2 and Table 3 .
Table 2. Aerosol optical thickness (AOD) in the middle of the spectral range for the spectral
bands and aerosol types considered.
Aerosol type Hartley-Huggins O2 A Water vapour CO2
(315 nm) (760 nm) (885 nm) (1610 nm)
Tropospheric 0.133 0.018 0.015 0.010
Continental clean 0.42 0.20 0.17 0.08
Urban 4.28 0.46 0.35 0.16
Desert 0.71 0.20 0.20 0.19
Continental polluted 2.4 1.2 0.9 0.4
Table 3. Single scattering albedo (SSA) for each spectral band and aerosol type.
Aerosol type Hartley-Huggins O2 A Water vapour CO2
Tropospheric 0.950 0.947 0.942 0.936
Continental clean 0.959 0.962 0.959 0.958
Urban 0.940 0.935 0.929 0.913
Desert 0.932 0.953 0.950 0.945
Continental polluted 0.951 0.960 0.957 0.950
Figure 1 shows the radiance spectra computed by using the multi-stream and the
low-stream (i.e. two-stream) RTMs for the different absorption bands. The computa-
tions are perfomed using the LBL-framework. Note that both spectra have a similar
spectral behavior. Hence, it is possible to establish a regression model between the low-
and multi-stream radiances, which is a subject to the CLSR method considered further.
2.2 Cluster Low-Streams Regression (CLSR) method
The Cluster Low-Streams Regression (CLSR) method is described in detail in [11] and
can be summarized as follows:
First, let us consider a LBL spectrum {ILS (λi )}N i=1 computed at N spectral points
{λi }N
i=1 by means of a low-stream RTM. We sort the radiances in ascending order
and consider C clusters in the sorted radiance set (with NC = N/C radiance points).
Secondly, we select n equidistant radiance points in each cluster and we compute the
c
multi-stream radiances for the corresponding wavelengths {I MS,q }nq=1 . Assuming that
in each cluster c there is a linear relationship between low- and multi-stream radiances,
we get
4 A. del Águila and D. Efremenko
Hartley-Huggins band O2 A-band
Multi-stream 0.07
Two-stream
0.08
0.06
TOA radiance (rel.u.)
0.05
0.06
0.04
0.04 0.03
0.02
0.02
0.01
0.00 0.00
280 290 300 310 320 330 755.0 757.5 760.0 762.5 765.0 767.5 770.0 772.5 775.0
Water vapour band CO2 band
0.07
0.06
0.06
0.05
TOA radiance (rel.u.)
0.05
0.04
0.04
0.03
0.03
0.02 0.02
0.01 0.01
0.00 0.00
800 850 900 950 1000 1590 1595 1600 1605 1610 1615 1620
Wavelength (nm) Wavelength (nm)
Fig. 1. Top of the atmosphere radiances computed for the absorption bands: Hartley-Huggins, O2
A-, water vapour and CO2 bands for one aerosol case. Black lines correspond to the multi-stream
RTM while purple lines correspond to the two-stream RTM.
IˆMS,i
c
= αc T̂ ci + β c IˆLS,i
c
+ γ c , i = 1, ..., NC , (1)
where αc , β c and γ c are the regression coefficients of the c-th cluster and T̂ is the
corresponding direct transmittance. Equation (1) can be also written as follows:
Y =A·X (2)
h i h i
with Y = IˆMS,i
c
, A = [αc , β c , γ c ] and X = T̂ ci , IˆLS,i
c
, 1 . Finally, we find the
regression coefficients as a solution to the following least square problem:
n h i2
X c
A = arg min I MS,q − Y . (3)
A
q=1
In this way, we can restore the spectra of the multi-stream radiances {I˜MS,i }N
i=1 . Here,
the “hat” notation Iˆ refers to the sorted radiances, the “bar” notation I refers to the
equidistant radiances entering the regression model and the “tilde” notation I˜ refers to
the predicted radiances. The total number of regression points, and thus the number of
calls to the multi-stream RTM, is nC. Note that unlike the k-distribution method, the
CLSR method provides a spectrum at the same spectral resolution as the LBL approach.
The Cluster Low-Streams Regression Method... 5
3 Simulations
3.1 Simulations of the absorption bands by using the CLSR method for the clear
sky atmosphere
In this section we apply the CLSR method to simulate absorption bands. In addition to
O2 A- and CO2 bands analyzed in [11], we consider Hartley-Huggins and water vapour
bands.
To estimate the accuracy of the CLSR method, we compute the residuals, the median
and interquartile range (IQR). The residual for the radiance is computed at each spectral
point λi as follows:
I˜MS,i − IMS,i
∆Ires,i = cont · 100, (4)
IMS
where I˜MS,i is the radiance calculated with the CLSR method (cf. Eq. (1)), while IMScont
is the radiance without absorption (i.e. the continuum radiance, which is used to avoid
radiance values close to zero in the denominator of Eq. (4), when strong gas absorption
is present [11, 7]).
Figure 2 shows the residuals of the CLSR method for different number of regression
points per cluster for the four bands considered.
Hartley-Huggins band O2-A band
0.0100 2.31e-01 -5.59e-07 -1.48e-07 -1.02e-07 5.05e-08 -5.75e-08 1.23e-08 1.01e-07
0.0100 1.33e-01 -2.86e-03 1.18e-05 2.14e-06 1.59e-06 4.12e-06 1.73e-06 -5.88e-07
2.81e+00 5.05e-03 1.39e-03 1.10e-03 5.94e-04 7.21e-04 6.74e-04 6.54e-04 7.48e-01 2.85e-02 1.05e-03 9.52e-04 1.25e-03 1.05e-03 8.02e-04 1.02e-03
0.0075 0.0075
0.0050 0.0050
0.0025 0.0025
ΔIresΔ(%)
0.0000 0.0000
−0.0025 −0.0025
−0.0050 −0.0050
Median
−0.0075 −0.0075 IQR
−0.0100 −0.0100
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
Wate% )apou% band CO2 band
0.100 1.26e-01 3.84e-03 -2.59e-04 6.69e-04 3.06e-04 -2.39e-04 4.24e-04 1.56e-04
0.0100 7.42e-02 -6.78e-05 3.01e-05 2.53e-06 4.91e-06 1.46e-05 1.52e-05 2.06e-06
1.39e+00 4.76e-02 8.98e-03 8.90e-03 5.71e-03 4.05e-03 4.31e-03 3.97e-03 3.16e-01 7.02e-04 4.27e-04 1.86e-04 8.52e-05 1.89e-04 1.31e-04 1.11e-04
0.075 0.0075
0.050 0.0050
0.025 0.0025
ΔIresΔ(%)
0.000 0.0000
−0.025 −0.0025
−0.050 −0.0050
−0.075 −0.0075
−0.100 −0.0100
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
Number of regression points Numbe% of %eg%ession points
Fig. 2. Box plots of the residuals for the CLSR method for the clear sky scenario in the following
absorption bands: Hartley-Huggins, O2 A-, water vapour and CO2 bands. The orange and black
values on top of each box indicate the median values and the IQR values, respectively.
The residuals gradually decrease with the number of spectral points. In fact, they
are significantly reduced when switching from 1––2 to 3 regression points. Therefore,
the median values remain almost constant from 3 regression points. This trend is iden-
tical to the one found in [11] for different atmospheric scenarios. Note that the scale
of residuals for the water vapour band is one order of magnitude higher than for the
Hartley-Huggins, O2 A- and CO2 bands.
6 A. del Águila and D. Efremenko
3.2 Application of the CLSR method in the case of aerosols: accuracy results
In our previous work, we applied the CLSR method to the O2 A- and CO2 bands for sev-
eral atmospheric scenarios like aerosols and clouds at different heights and thicknesses.
In this paper we examine the application of the CLSR method for several aerosol types
and we extend the analysis to the Hartley-Huggins and water vapour bands. The com-
putations are performed for the aerosol types outlined in Section 2.1.
Figure 3 shows the residuals for four bands and 5 aerosol models.
Median
IQR
0.0100
Hartley-Huggins band 0.0100
O2-A band
-1.03e-07 -1.02e-07 -2.35e-07 -1.10e-07 -1.41e-07 4.24e-06 3.25e-05 -6.82e-06 1.29e-05 2.95e-05
1.09e-03 1.14e-03 1.02e-03 1.06e-03 1.05e-03 6.89e-04 4.25e-03 3.34e-03 2.41e-03 6.57e-03
0.0075 0.0075
0.0050 0.0050
0.0025 0.0025
ΔIresΔ(%)
0.0000 0.0000
−0.0025 −0.0025
−0.0050 −0.0050
−0.0075 −0.0075
−0.0100 −0.0100
Wate% )apou% band CO2 band
0.0100
1.61e-04 -7.67e-05 9.43e-04 1.01e-03 -3.43e-05 1.82e-05 1.23e-04 4.27e-06 4.48e-05 6.65e-05
0.04 8.55e-03 8.01e-03 2.16e-02 1.58e-02 1.76e-02 5.95e-04 2.49e-03 1.20e-03 2.68e-03 4.20e-03
0.0075
0.0050
0.02
0.0025
ΔIresΔ(%)
0.00 0.0000
−0.0025
−0.02
−0.0050
−0.04 −0.0075
−0.0100
Tropospheric Clean Urban Desert Polluted T%oposphe% c Clean U%ban Dese%t Polluted
Fig. 3. Residuals for the CLSR method for the following absorption bands: Hartley-Huggins, O2
A-, water vapour and CO2 bands. The color of the boxes represents the type of aerosol: (grey)
tropospheric; (blue) continental clean; (red): urban; (green): desert; (yellow): polluted. Note the
different scale for the water vapour band compared with the other absorption bands.
Note that the residuals for the Hartley-Huggins, O2 A- and CO2 bands are substan-
tially smaller than those for the water vapour band. However, the median residuals are
below 0.001 % and the results of the CLSR model are not biased.
In general, we conclude that the efficiency of the CLSR method is comparable to
that of alternative methods like PCA-based RTMs (e.g. [7]) and our previous studies
([11]).
3.3 Assessment of the CLSR computational efficiency
In this section we analyse the computational performance of the CLSR method. Table
4 shows the number of calls to two- and multi-stream RTMs, the computational time
The Cluster Low-Streams Regression Method... 7
and the corresponding speedup factor with respect to the multi-stream LBL simulations
for the O2 A-band. The computational time for monochromatic computations for the
TS RTM is tTS =1.6e-4 s, while for the MS RTM with Ndo = 32 discrete ordinates
per hemisphere is tMS =0.12 s, i.e., around three order of magnitude larger. However,
most of the computational burden is still due to the MS RTM, and the computations of
the approximate spectrum with a high spectral resolution by using the TS RTM is not
a performance bottleneck in the whole CLSR processing chain. The results show that
using the matrix of coefficients X with 5 clusters and 4 regression points for the CLSR
method is 420 faster than using the LBL model. The speedup factor is of the same order
of magnitude as in [21].
Table 4. Summary of number of calls, computational time and speedup factors for the O2 A-band
with LBL and CLSR methods.
LBL CLSR
Number of calls to MS RTM 20000 20
Number of calls to TS RTM 0 20000
Time for MS RTM (s) 2320 2.32
Time for TS RTM (s) 0 3.20
Total computational time (s) 2320 5.52
Speedup factor — 420
4 Conclusions
In this study, we have analysed the efficiency of the Cluster Low-Streams Regression
(CLSR) method to accelerate spectral computations in several absorption bands. The
CLSR method exploits the linear relationship between the low-stream and multi-stream
models, where the corresponding regression coefficients are found by using the least-
squares method. In our simulations several OPAC aerosol models have been considered.
We reproduced the spectra with a median error below 0.001 % as compared to the
reference multi-stream line-by-line model and IQR values below 0.1%. Thus, the errors
present low variation and stability.
The number of calls to the multi-stream model was reduced by 3 orders of magni-
tude (e.g. from 20000 to 20 calls in the case of O2 A-band). The resulting performance
enhancement is about 400 times. Note, that since the CLSR method is two orders of
magnitude faster than the LBL model, it can be used for computations of the aerosol
spectra in near-real-time applications.
In our future work, we plan to extend the CLSR method by using the asymptotic
radiative transfer theory [22] and the diffuse approximation [23] instead of the two-
stream RTM. Also it is of high interest to apply the CLSR method for modelling of the
Stokes parameters.
8 A. del Águila and D. Efremenko
References
1. Clough, S.A., Rinsland, C.P., Brown, P.D.: Retrieval of tropospheric ozone from simula-
tions of nadir spectral radiances as observed from space. Journal of Geophysical Research
100(D8), 16579 (1995). https://doi.org/10.1029/95jd01388
2. Fomin, B.A.: A k-distribution technique for radiative transfer simulation in inhomogeneous
atmosphere: 2. FKDM, fast k-distribution model for the shortwave. Journal of Geophysical
Research 110(D2) (2005). https://doi.org/10.1029/2004jd005163
3. Fu, Q., Liou, K.: On the correlated k-distribution method for radiative transfer in nonho-
mogeneous atmospheres. Journal of the Atmospheric Sciences 49(22), 2139–2156 (1992).
https://doi.org/10.1175/1520-0469(1992)049¡2139:OTCDMF¿2.0.CO;2
4. del Águila, A., Efremenko, D.S., Molina Garcı́a, V., Xu, J.: Analysis of two dimensionality
reduction techniques for fast simulation of the spectral radiances in the hartley-huggins band.
Atmosphere 10(3), 142 (3 2019). https://doi.org/10.3390/atmos10030142
5. Efremenko, D.S., Loyola, D.G., Doicu, A., Spurr, R.J.D.: Multi-core-CPU
and GPU-accelerated radiative transfer models based on the discrete ordi-
nate method. Computer Physics Communications 185(12), 3079–3089 (2014).
https://doi.org/10.1016/j.cpc.2014.07.018
6. Efremenko, D., Doicu, A., Loyola, D., Trautmann, T.: Optical property dimensionality reduc-
tion techniques for accelerated radiative transfer performance: Application to remote sensing
total ozone retrievals. Journal of Quantitative Spectroscopy and Radiative Transfer 133, 128–
135 (2014). https://doi.org/10.1016/j.jqsrt.2013.07.023
7. Kopparla, P., Natraj, V., Limpasuvan, D., Spurr, R., Crisp, D., Shia, R.L., Somkuti, P., Yung,
Y.L.: Pca-based radiative transfer: Improvements to aerosol scheme, vertical layering and
spectral binning. Journal of Quantitative Spectroscopy and Radiative Transfer 198, 104–111
(2017). https://doi.org/https://doi.org/10.1016/j.jqsrt.2017.05.005
8. Liu, X., Smith, W.L., Zhou, D.K., Larar, A.: Principal component-based radiative transfer
model for hyperspectral sensors: Theoretical concept. Applied Optics 45(1), 201–208 (2006).
https://doi.org/10.1364/ao.45.000201
9. Natraj, V., Jiang, X., Shia, R., Huang, X., Margolis, J., Yung, Y.: Application of the principal
component analysis to high spectral resolution radiative transfer: A case study of the O2 A-
band. Journal of Quantitative Spectroscopy and Radiative Transfer 95(4), 539–556 (2005).
https://doi.org/10.1016/j.jqsrt.2004.12.024
10. del Águila, A., Efremenko, D.S., Trautmann, T.: A review of dimensionality reduction tech-
niques for processing hyper-spectral optical signal. Light & Engineering pp. 85–98 (2019).
https://doi.org/10.33383/2019-017
11. del Águila, A., Efremenko, D.S., Molina Garcı́a, V., Kataev, M.Y.: Cluster low-streams re-
gression method for hyperspectral radiative transfer computations: Cases of O2 A- and CO2
bands. Remote Sensing 12(8), 1250 (Apr 2020). https://doi.org/10.3390/rs12081250
12. Afanas’ev, V., Basov, A.Y., Budak, V., Efremenko, D., Kokhanovsky, A.: Analysis of the dis-
crete theory of radiative transfer in the coupled “ocean–atmosphere” system: Current status,
problems and development prospects. Journal of Marine Science and Engineering 8(3), 202
(Mar 2020). https://doi.org/10.3390/jmse8030202
13. Budak, V., Efremenko, D., Shagalov, O.: Efficiency of algorithm for solution of vector ra-
diative transfer equation in turbid medium slab. Journal of Physics: Conference Series 369,
012021 (Jun 2012). https://doi.org/10.1088/1742-6596/369/1/012021
14. Doicu, A., Trautmann, T.: Discrete-ordinate method with matrix exponential for a pseudo-
spherical atmosphere: Scalar case. Journal of Quantitative Spectroscopy and Radiative Trans-
fer 110(1-2), 146–158 (2009). https://doi.org/10.1016/j.jqsrt.2008.09.014
The Cluster Low-Streams Regression Method... 9
15. Efremenko, D.S., Molina Garcı́a, V., Gimeno Garcı́a, S., Doicu, A.: A review of the matrix-
exponential formalism in radiative transfer. Journal of Quantitative Spectroscopy and Radia-
tive Transfer 196, 17–45 (Jul 2017). https://doi.org/10.1016/j.jqsrt.2017.02.015
16. Molina Garcı́a, V., Sasi, S., Efremenko, D., Doicu, A., Loyola, D.: Radiative trans-
fer models for retrieval of cloud parameters from EPIC/DSCOVR measurements.
Journal of Quantitative Spectroscopy and Radiative Transfer 213, 228–240 (2018).
https://doi.org/10.1016/j.jqsrt.2018.03.014
17. Schreier, F., Gimeno Garcı́a, S., Hochstaffl, P., Städt, S.: Py4cats—PYthon
for computational ATmospheric spectroscopy. Atmosphere 10(5), 262 (2019).
https://doi.org/10.3390/atmos10050262
18. Gordon, I., Rothman, L., Hill, C., Kochanov, R., Tan, Y., Bernath, P., Birk, M., Boudon,
V., Campargue, A., Chance, K., Drouin, B., Flaud, J.M., Gamache, R., Hodges, J., Jacque-
mart, D., Perevalov, V., Perrin, A., Shine, K., Smith, M.A., Tennyson, J., Toon, G., Tran, H.,
Tyuterev, V., Barbe, A., Császár, A., Devi, V., Furtenbacher, T., Harrison, J., Hartmann, J.M.,
Jolly, A., Johnson, T., Karman, T., Kleiner, I., Kyuberis, A., Loos, J., Lyulin, O., Massie,
S., Mikhailenko, S., Moazzen-Ahmadi, N., Müller, H., Naumenko, O., Nikitin, A., Polyan-
sky, O., Rey, M., Rotger, M., Sharpe, S., Sung, K., Starikova, E., Tashkun, S., Auwera, J.V.,
Wagner, G., Wilzewski, J., Wcisło, P., Yu, S., Zak, E.: The HITRAN2016 molecular spec-
troscopic database. Journal of Quantitative Spectroscopy and Radiative Transfer 203, 3–69
(2017). https://doi.org/10.1016/j.jqsrt.2017.06.038
19. Bodhaine, B., Wood, N., Dutton, E., Slusser, J.: On Rayleigh optical depth calcu-
lations. Journal of Atmospheric and Oceanic Technology 16(11), 1854–1861 (1999).
https://doi.org/10.1175/1520-0426(1999)016¡1854:orodc¿2.0.co;2
20. Hess, M., Koepke, P., Schult, I.: Optical properties of aerosols and clouds: The software
package OPAC. Bulletin of the American Meteorological Society 79(5), 831–844 (1998).
https://doi.org/10.1175/1520-0477(1998)079¡0831:opoaac¿2.0.co;2
21. O’Dell, C.W.: Acceleration of multiple-scattering, hyperspectral radiative transfer calcu-
lations via low-streams interpolation. Journal of Geophysical Research 115(D10) (2010).
https://doi.org/10.1029/2009jd012803
22. Kokhanovsky, A.: Cloud Optics. Springer Netherlands (2006). https://doi.org/10.1007/1-
4020-4020-2
23. Budak, V.P., Zheltov, V.S., Lubenchenko, A.V., Freidlin, K.S., Shagalov, O.V.: A fast
and accurate synthetic iteration-based algorithm for numerical simulation of radiative
transfer in a turbid medium. Atmospheric and Oceanic Optics 30(1), 70–78 (Jan 2017).
https://doi.org/10.1134/s1024856017010031