#
Cluster Low-Streams Regression Method for Hyperspectral Radiative Transfer Computations: Cases of O_{2} A- and CO_{2} Bands

^{1}

^{2}

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## Abstract

**:**

_{2}A-band at 760 nm and the CO

_{2}band at 1610 nm for five atmospheric scenarios. The CLSR method is also compared with the principal component analysis (PCA)-based RTM, showing an improvement in terms of accuracy and computational performance over PCA-based RTMs. As low-stream models, the two-stream and the single-scattering RTMs are considered. We show that the error of this approach is modulated by the optical thickness of the atmosphere. Nevertheless, the CLSR method provides a performance enhancement of almost two orders of magnitude compared to the LBL model, while the error of the technique is below 0.1% for both bands.

## 1. Introduction

_{2}A-band, Natraj et al. [12] proposed a fundamentally different PCA-based radiative transfer model, in which the dimensionality of the optical properties data is reduced. A two-stream radiative transfer model was used as an approximate model, and the dependency of the corresponding correction factor on the optical parameters was modeled by a second-order Taylor expansion about the mean value of the optical parameters in the reduced optical data space. This approach was extended to other dimensionality reduction techniques [13] and spectral ranges [14,15,16]; moreover, it was implemented in conjunction with PCA for spectral radiances [17] and with the k-distribution method [18]. The errors of these approaches are usually below 0.1% for the spectral radiances, while the performance enhancement may reach several orders of magnitude depending on the spectral region and the required level of accuracy.

_{2}A-band at 760 nm and the weak CO

_{2}band at 1610 nm. These spectral bands are of great interest for retrieving aerosol and cloud parameters [22], as well as CO

_{2}concentrations [23].

_{2}A- and weak CO

_{2}absorption bands are analyzed and compared with the PCA-based RTM. In addition, the accuracy of the radiance spectra convolved with the slit functions corresponding to GOME-2 [24], TROPOMI on board Sentinel 5-P [25] and GOSAT [26] instruments is examined. The paper concludes with a summary.

## 2. Methodology

#### 2.1. Reference Radiative Transfer Model

_{2}A- and CO

_{2}bands are computed with the LBL model Py4CAtS [34], while the gas absorption cross-sections are taken from the HITRAN 2016 database [35]. The uncertainties in the spectroscopic parameters are not considered because their role is irrelevant for this study [36]. Continuum (also referred to as collision-induced absorption (CIA) [37,38]) contributions to molecular absorption are not taken into account. We note that the CIA process gives a broad and smooth contribution [39] and hence, does not impose difficulties for the regression techniques considered in this study.

#### 2.2. Study Cases

_{2}A-band the spectral sampling is 0.001 nm in the spectral range 755–775 nm, while in the CO

_{2}band the spectral sampling is 0.0015 nm in the spectral interval 1590–1620 nm. Thus, on each band, 20,000 spectral points are considered. The Rayleigh cross-sections and depolarization ratios are computed as in [40], while the pressure and temperature profiles correspond to the US standard model atmosphere [41]. The computations are performed for the unit solar irradiance at the TOA.

#### 2.3. PCA-Based RTM

#### 2.4. Regression Relationship between Multi-Stream and Low-Stream Models

- 1.
- the dependence of $f\left({\tau}_{\mathrm{gas}}\right)$ on ${\tau}_{\mathrm{gas}}$ is non-linear and, therefore, the application of the regression model requires a binning of the ${\tau}_{\mathrm{gas}}$ values;
- 2.
- in a mathematical sense, $f\left({\tau}_{\mathrm{gas}}\right)$ is not a function (for a value of ${\tau}_{\mathrm{gas}}$, there are several values of $f\left({\tau}_{\mathrm{gas}}\right)$) and therefore, even for a fine binning, the regression model will be not accurate.

#### 2.5. Cluster Low-Streams Regression Method

- 1.
- Consider a high-resolution spectrum ${\left\{{I}_{\mathrm{LS}}\left({\lambda}_{i}\right)\right\}}_{i=1}^{N}$ computed at N spectral points ${\left\{{\lambda}_{i}\right\}}_{i=1}^{N}$ by means of a low-stream RTM.
- 2.
- Sort the radiance set ${\left\{{I}_{\mathrm{LS}}\left({\lambda}_{i}\right)\right\}}_{i=1}^{N}$ in ascending order, and let ${\left\{{\widehat{I}}_{\mathrm{LS},i}\right\}}_{i=1}^{N}$, with ${\widehat{I}}_{\mathrm{LS},i}\le {\widehat{I}}_{\mathrm{LS},i+1}$, be the sorted radiance set (Figure 4a).
- 3.
- Consider C clusters in ${\left\{{\widehat{I}}_{\mathrm{LS},i}\right\}}_{i=1}^{N}$ with ${N}_{C}=N/C$ radiance points (Figure 4b), and let the c cluster be defined by the radiance set ${\left\{{\widehat{I}}_{\mathrm{LS},i}^{c}\right\}}_{i=1}^{{N}_{C}}$ for $c=1,\dots ,C$.
- 4.
- Select n equidistant radiance points in the c cluster, i.e., ${\left\{{\overline{I}}_{\mathrm{LS},q}^{c}\right\}}_{q=1}^{n}$, and for the corresponding wavelengths compute the multi-stream radiances ${\left\{{\overline{I}}_{\mathrm{MS},q}^{c}\right\}}_{q=1}^{n}$ (Figure 4c).
- 5.
- Assume that in each cluster c we have the linear relationship$${\widehat{I}}_{\mathrm{MS},i}^{c}={\alpha}^{c}{\widehat{T}}_{i}^{c}+{\beta}^{c}{\widehat{I}}_{\mathrm{LS},i}^{c}+{\gamma}^{c},\phantom{\rule{0.277778em}{0ex}}i=1,\dots ,{N}_{C},$$
- 6.
- Compute the regression coefficients ${\alpha}^{c}$, ${\beta}^{c}$ and ${\gamma}^{c}$ as a solution to the least square problem$$\left({\alpha}^{c},{\beta}^{c},{\gamma}^{c}\right)=arg\underset{{\alpha}^{c},{\beta}^{c},{\gamma}^{c}}{min}\sum _{q=1}^{n}{\left[{\overline{I}}_{\mathrm{MS},q}^{c}-\left({\alpha}^{c}{\overline{T}}_{q}^{c}+{\beta}^{c}{\overline{I}}_{\mathrm{LS},q}^{c}+{\gamma}^{c}\right)\right]}^{2}.$$
- 7.

#### 2.6. Efficiency and Computational Performance Estimation

## 3. Results and Discussion

#### 3.1. Performance of the Low- and Multi-Streams Models

_{2}A- and CO

_{2}absorption bands, respectively. Table 4 shows the computational performance for a different number of discrete ordinates ${N}_{\mathrm{do}}$. The results show that (i) the relative errors of the single-scattering model are higher than those of the two-stream model, (ii) the relative error increases as the optical thickness of the atmosphere increases, and (iii) the speedup factors of the single-scattering model are considerably higher than those of the two-stream model.

#### 3.2. Acceleration Techniques: Accuracy Evaluation

#### 3.2.1. Spectral Residuals

- 1.
- The residuals decrease when increasing the number of PCs and regression points.
- 2.
- The interquartile range for the CLSR method is substantially reduced when switching from 1–2 to 3 regression points for the CO
_{2}band, and from 3 to 4 regression points for the O_{2}A-band. - 3.
- In both spectral bands, the interquartile range for the PCA-based RTM decreases gradually with the number of principal components.
- 4.
- The residuals in the O
_{2}A-band are systematically higher than those in the CO_{2}band. This behaviour is more pronounced when the gas optical thickness is large, thus resulting in larger discrepancies for the PCA-based RTM and almost negligible discrepancies for the CLSR approach. - 5.
- In the O
_{2}A-band, the median values of the residuals Equation (8) are higher than 2%. The median values remain almost constant with the number of principal components and regression points. However, the median values as well as the interquartile range for the PCA method are generally higher than those of the CLSR method. This trend remains coherent using the single-scattering RTM instead of the two-stream RTM, although the residuals are substantially higher for the PCA-based RTM.

#### 3.2.2. Estimation of the Required Parameters for the Acceleration Techniques

#### 3.2.3. Accuracy of the CLSR and PCA-Based Methods

- 1.
- For both low-stream models and cloudy scenarios, (i) the residuals of the PCA-based method in the O
_{2}A-band are substantially larger than those of the CLSR method (on average, for the two-stream model in the O_{2}A-band, the residuals are above 1% for the PCA-based method and below 0.01% for the CLSR method), while (ii) the PCA-based and CLSR methods yield comparable accuracies in the CO_{2}band (on average, for the two-stream model in the CO_{2}band, the residuals are below 0.1% for the PCA-based method and below 0.01% for the CLSR method). Please note similar results were established in previous studies, i.e., the errors of the PCA-based method generally increase with the optical depth [16]. The superiority of the CLSR method is also demonstrated by the interquartile ranges: these are much smaller for the CLSR method. - 2.
- For both O
_{2}A- and CO_{2}bands, (i) the residuals of the single-scattering model with the PCA-based method are higher than those corresponding to the two-stream model, while (ii) the residuals of the two-stream and single-scattering models with the CLSR method are comparable (on average, for the single-scattering model, the residuals of the CLSR method are below 0.2% in the O_{2}A-band and below 0.1% in CO_{2}band, while for the two-stream model the corresponding errors are below 0.01% in both spectral bands).

#### 3.3. Acceleration Techniques: Computational Performance

#### 3.4. Computation of Convolved Spectra

_{2}A-band are convolved with the slit functions corresponding to GOME-2 and TROPOMI instruments, while the radiance spectra in the CO

_{2}band are convolved with the GOSAT slit function. In this paper, slit functions are modeled with a Gaussian function. The corresponding full widths at half maximum (FWHM) are listed in Table 6. The FWHM considered for the O

_{2}A-band are based on pre-launch calibrations [47] and for the CO

_{2}band on [48].

_{2}A-band and the CO

_{2}band, respectively. In addition, the residuals for non-convolved spectra are shown for comparison. For the ‘Clear sky’ and aerosol scenarios the accuracies of both methods are comparable, while for cloud scenarios the CLSR method is more accurate. Please note the residuals estimated for the convolved spectra are very close to those for the non-convolved ones and the value of residuals according to Equation (8) are robust.

## 4. Conclusions

_{2}A- and CO

_{2}absorption bands. The CLSR method exploits a strong close-to-linear relationship between the radiances computed with a low-stream model (which is either the two-stream or the single-scattering model) and the radiances computed with the multi-stream model. The spectral points are grouped in several clusters according to the values of the low-stream radiances. For each cluster, the regression model is established between the low-stream and multi-stream models, where the corresponding regression coefficients are found by using the least-squares method. This approach can be regarded as a variation of the low-streams interpolation method explained in [21], in which the binning is performed in the space of gas optical depths.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CLSR | Cluster Low-Streams Regression |

DOME | Discrete Ordinates with Matrix Exponential |

EOF | Empirical Orthogonal Function |

FWHM | Full Width at Half Maximum |

GOME-2 | Global Ozone Monitoring Experiment–2 |

GOSAT | Greenhouse gases Observing SATellite |

HITRAN | HIgh-resolution TRANsmission molecular absorption database |

IQR | InterQuartile Range |

LBL | Line-By-Line |

LS | Low-Streams |

LSM | Least-Squares Method |

MS | Multi-Streams |

OPAC | Optical Properties of Aerosols and Clouds |

PC | Principal Component |

PCA | Principal Component Analysis |

Py4CAts | Python for Computational ATmospheric Spectroscopy |

RTM | Radiative Transfer Model |

TOA | Top Of the Atmosphere |

TROPOMI | TROPOspheric Monitoring Instrument |

## References

- Clough, S.A.; Rinsland, C.P.; Brown, P.D. Retrieval of tropospheric ozone from simulations of nadir spectral radiances as observed from space. J. Geophys. Res.
**1995**, 100, 16579. [Google Scholar] [CrossRef] - Ambartsumian, V. The effect of the absorption lines on the radiative equilibrium of the outer layers of the stars. Publ. Astron. Obs. Leningr. State Univ.
**1936**, 6, 7–18. [Google Scholar] - Goody, R.; West, R.; Chen, L.; Crisp, D. The correlated k-method for radiation calculations in nonhomogeneous atmospheres. J. Quant. Spectrosc. Radiat. Transf.
**1989**, 42, 539–550. [Google Scholar] [CrossRef] - Fu, Q.; Liou, K. On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres. J. Atmos. Sci.
**1992**, 49, 2139–2156. [Google Scholar] [CrossRef] [Green Version] - Fomin, B.A. A k-distribution technique for radiative transfer simulation in inhomogeneous atmosphere: 1. FKDM, fast k-distribution model for the longwave. J. Geophys. Res.
**2004**, 109. [Google Scholar] [CrossRef] - Fomin, B.A. A k-distribution technique for radiative transfer simulation in inhomogeneous atmosphere: 2. FKDM, fast k-distribution model for the shortwave. J. Geophys. Res.
**2005**, 110. [Google Scholar] [CrossRef] [Green Version] - Hunt, G.E.; Grant, I.P. Discrete space theory of radiative transfer and its application to problems in planetary atmospheres. J. Atmos. Sci.
**1969**, 26, 963–972. [Google Scholar] [CrossRef] [Green Version] - Wiscombe, W.; Evans, J. Exponential-sum fitting of radiative transmission functions. J. Comput. Phys.
**1977**, 24, 416–444. [Google Scholar] [CrossRef] - Moncet, J.L.; Uymin, G.; Lipton, A.E.; Snell, H.E. Infrared radiance modeling by optimal spectral sampling. J. Atmos. Sci.
**2008**, 65, 3917–3934. [Google Scholar] [CrossRef] - Liu, X.; Smith, W.L.; Zhou, D.K.; Larar, A. Principal component-based radiative transfer model for hyperspectral sensors: Theoretical concept. Appl. Opt.
**2006**, 45, 201–208. [Google Scholar] [CrossRef] - Hollstein, A.; Lindstrot, R. Fast reconstruction of hyperspectral radiative transfer simulations by using small spectral subsets: Application to the oxygen A band. Atmos. Meas. Tech.
**2014**, 7, 599–607. [Google Scholar] [CrossRef] [Green Version] - 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 O
_{2}A-band. J. Quant. Spectrosc. Radiat. Transf.**2005**, 95, 539–556. [Google Scholar] [CrossRef] - Efremenko, D.; Doicu, A.; Loyola, D.; Trautmann, T. Optical property dimensionality reduction techniques for accelerated radiative transfer performance: Application to remote sensing total ozone retrievals. J. Quant. Spectrosc. Radiat. Transf.
**2014**, 133, 128–135. [Google Scholar] [CrossRef] - Kopparla, P.; Natraj, V.; Spurr, R.; Shia, R.L.; Crisp, D.; Yung, Y.L. A fast and accurate PCA based radiative transfer model: Extension to the broadband shortwave region. J. Quant. Spectrosc. Radiat. Transf.
**2016**, 173, 65–71. [Google Scholar] [CrossRef] - 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. J. Quant. Spectrosc. Radiat. Transf.
**2017**, 198, 104–111. [Google Scholar] [CrossRef] - Somkuti, P.; Boesch, H.; Natraj, V.; Kopparla, P. Application of a PCA-based fast radiative transfer model to XCO
_{2}retrievals in the shortwave infrared. J. Geophys. Res. Atmos.**2017**, 122, 10477–10496. [Google Scholar] [CrossRef] - del Águila, A.; Efremenko, D.S.; Molina García, V.; Xu, J. Analysis of two dimensionality reduction techniques for fast simulation of the spectral radiances in the Hartley-Huggins band. Atmosphere
**2019**, 10, 142. [Google Scholar] [CrossRef] [Green Version] - Molina García, V.; Sasi, S.; Efremenko, D.S.; Doicu, A.; Loyola, D. Radiative transfer models for retrieval of cloud parameters from EPIC/DSCOVR measurements. J. Quant. Spectrosc. Radiat. Transf.
**2018**, 213, 228–240. [Google Scholar] [CrossRef] [Green Version] - 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 ordinate method. Comput. Phys. Commun.
**2014**, 185, 3079–3089. [Google Scholar] [CrossRef] - Amdahl, G.M. Validity of the single processor approach to achieving large scale computing capabilities. In Proceedings of the Spring Joint Computer Conference, Atlantic City, NJ, USA, 18–20 April 1967. [Google Scholar]
- O’Dell, C.W. Acceleration of multiple-scattering, hyperspectral radiative transfer calculations via low-streams interpolation. J. Geophys. Res.
**2010**, 115. [Google Scholar] [CrossRef] - Fischer, J.; Grassl, H. Detection of cloud-top height from backscattered radiances within the Oxygen A band. Part 1: Theoretical study. J. Appl. Meteorol.
**1991**, 30, 1245–1259. [Google Scholar] [CrossRef] - Kataev, M.Y.; Lukyanov, A.K. Empirical orthogonal functions and its modification in the task of retrieving of the total amount CO
_{2}and CH_{4}with help of satellite Fourier transform spectrometer GOSAT (TANSO-FTS). In Proceedings of the 22nd International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics, Tomsk, Russia, 30 June–3 July 2016. [Google Scholar] - Munro, R.; Lang, R.; Klaes, D.; Poli, G.; Retscher, C.; Lindstrot, R.; Huckle, R.; Lacan, A.; Grzegorski, M.; Holdak, A.; et al. The GOME-2 instrument on the Metop series of satellites: Instrument design, calibration, and level 1 data processing—An overview. Atmos. Meas. Tech.
**2016**, 9, 1279–1301. [Google Scholar] [CrossRef] [Green Version] - Veefkind, J.; Aben, I.; McMullan, K.; Förster, H.; de Vries, J.; Otter, G.; Claas, J.; Eskes, H.; de Haan, J.; Kleipool, Q.; et al. TROPOMI on the ESA Sentinel-5 Precursor: A GMES mission for global observations of the atmospheric composition for climate, air quality and ozone layer applications. Remote Sens. Environ.
**2012**, 120, 70–83. [Google Scholar] [CrossRef] - Butz, A.; Guerlet, S.; Hasekamp, O.; Schepers, D.; Galli, A.; Aben, I.; Frankenberg, C.; Hartmann, J.M.; Tran, H.; Kuze, A.; et al. Toward accurate CO
_{2}and CH_{4}observations from GOSAT. Geophys. Res. Lett.**2011**, 38. [Google Scholar] [CrossRef] [Green Version] - Doicu, A.; Trautmann, T. Discrete-ordinate method with matrix exponential for a pseudo-spherical atmosphere: Scalar case. J. Quant. Spectrosc. Radiat. Transf.
**2009**, 110, 146–158. [Google Scholar] [CrossRef] - Efremenko, D.; Doicu, A.; Loyola, D.; Trautmann, T. Acceleration techniques for the discrete ordinate method. J. Quant. Spectrosc. Radiat. Transf.
**2013**, 114, 73–81. [Google Scholar] [CrossRef] - Efremenko, D.S.; Molina García, V.; Gimeno García, S.; Doicu, A. A review of the matrix-exponential formalism in radiative transfer. J. Quant. Spectrosc. Radiat. Transf.
**2017**, 196, 17–45. [Google Scholar] [CrossRef] [Green Version] - Korkin, S.; Lyapustin, A. Matrix exponential in C/C
^{++}version of vector radiative transfer code IPOL. J. Quant. Spectrosc. Radiat. Transf.**2019**, 227, 106–110. [Google Scholar] [CrossRef] [Green Version] - Waterman, P.C. Matrix-exponential description of radiative transfer. J. Opt. Soc. Am.
**1981**, 71, 410. [Google Scholar] [CrossRef] - Nakajima, T.; Tanaka, M. Matrix formulations for the transfer of solar radiation in a plane-parallel scattering atmosphere. J. Quant. Spectrosc. Radiat. Transf.
**1986**, 35, 13–21. [Google Scholar] [CrossRef] - Afanas’ev, V.P.; Efremenko, D.S.; Lubenchenko, A.V. On the application of the invariant embedding method and the radiative transfer equation codes for surface state analysis. In Light Scattering Reviews 8: Radiative Transfer and Light Scattering; Kokhanovsky, A.A., Ed.; Springer: Berlin/Heidelberg, Germany, 2013; pp. 363–423. [Google Scholar]
- Schreier, F.; Gimeno García, S.; Hochstaffl, P.; Städt, S. Py4CAtS—PYthon for Computational ATmospheric Spectroscopy. Atmosphere
**2019**, 10, 262. [Google Scholar] [CrossRef] [Green Version] - Gordon, I.; Rothman, L.; Hill, C.; Kochanov, R.; Tan, Y.; Bernath, P.; Birk, M.; Boudon, V.; Campargue, A.; Chance, K.; et al. The HITRAN2016 molecular spectroscopic database. J. Quant. Spectrosc. Radiat. Transf.
**2017**, 203, 3–69. [Google Scholar] [CrossRef] - Checa-Garcia, R.; Landgraf, J.; Galli, A.; Hase, F.; Velazco, V.A.; Tran, H.; Boudon, V.; Alkemade, F.; Butz, A. Mapping spectroscopic uncertainties into prospective methane retrieval errors from Sentinel-5 and its precursor. Atmos. Meas. Tech.
**2015**, 8, 3617–3629. [Google Scholar] [CrossRef] [Green Version] - Richard, C.; Gordon, I.; Rothman, L.; Abel, M.; Frommhold, L.; Gustafsson, M.; Hartmann, J.M.; Hermans, C.; Lafferty, W.; Orton, G.; et al. New section of the HITRAN database: Collision-induced absorption (CIA). J. Quant. Spectrosc. Radiat. Transf.
**2012**, 113, 1276–1285. [Google Scholar] [CrossRef] - Mlawer, E.J.; Payne, V.H.; Moncet, J.L.; Delamere, J.S.; Alvarado, M.J.; Tobin, D.C. Development and recent evaluation of the MT_CKD model of continuum absorption. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2012**, 370, 2520–2556. [Google Scholar] [CrossRef] [Green Version] - Tran, H.; Boulet, C.; Hartmann, J.M. Line mixing and collision-induced absorption by oxygen in the A band: Laboratory measurements, model, and tools for atmospheric spectra computations. J. Geophys. Res.
**2006**, 111. [Google Scholar] [CrossRef] [Green Version] - Bodhaine, B.A.; Wood, N.B.; Dutton, E.G.; Slusser, J.R. On Rayleigh optical depth calculations. J. Atmos. Ocean. Technol.
**1999**, 16, 1854–1861. [Google Scholar] [CrossRef] - Anderson, G.; Clough, S.; Kneizys, F.; Chetwynd, J.; Shettle, E. AFGL Atmospheric Constituent Profiles (0–120 km); Air Force Geophysics Laboratory, Hanscom Air Force Base: Bedford, MA, USA, 1986. [Google Scholar]
- Hess, M.; Koepke, P.; Schult, I. Optical properties of aerosols and clouds: The software package OPAC. Bull. Am. Meteorol. Soc.
**1998**, 79, 831–844. [Google Scholar] [CrossRef] - Deirmendjian, D. Electromagnetic Scattering on Spherical Polydispersions; Elsevier: Amsterdam, The Netherlands, 1969. [Google Scholar]
- Natraj, V.; Shia, R.L.; Yung, Y.L. On the use of principal component analysis to speed up radiative transfer calculations. J. Quant. Spectrosc. Radiat. Transf.
**2010**, 111, 810–816. [Google Scholar] [CrossRef] - del Águila, A.; Efremenko, D.S.; Trautmann, T. A review of dimensionality reduction techniques for processing hyper-spectral optical signal. Light Eng.
**2019**, 27, 85–98. [Google Scholar] [CrossRef] [Green Version] - Rogovtsov, N.N.; Borovik, F. Application of general invariance relations reduction method to solution of radiation transfer problems. J. Quant. Spectrosc. Radiat. Transf.
**2016**, 183, 128–153. [Google Scholar] [CrossRef] - Beirle, S.; Lampel, J.; Lerot, C.; Sihler, H.; Wagner, T. Parameterizing the instrumental spectral response function and its changes by a super-Gaussian and its derivatives. Atmos. Meas. Tech.
**2017**, 10, 581–598. [Google Scholar] [CrossRef] [Green Version] - GOSAT Spectral Resolution. Available online: http://www.gosat-2.nies.go.jp/about/spacecraft_and_instruments/ (accessed on 17 March 2020).
- Molina García, V.; Sasi, S.; Efremenko, D.S.; Doicu, A.; Loyola, D. Linearized radiative transfer models for retrieval of cloud parameters from EPIC/DSCOVR measurements. J. Quant. Spectrosc. Radiat. Transf.
**2018**, 213, 241–251. [Google Scholar] [CrossRef] [Green Version] - Butz, A.; Galli, A.; Hasekamp, O.; Landgraf, J.; Tol, P.; Aben, I. TROPOMI aboard Sentinel-5 Precursor: Prospective performance of CH4 retrievals for aerosol and cirrus loaded atmospheres. Remote Sens. Environ.
**2012**, 120, 267–276. [Google Scholar] [CrossRef] - Korkin, S.V.; Lyapustin, A.I.; Marshak, A.L. On the accuracy of double scattering approximation for atmospheric polarization computations. J. Quant. Spectrosc. Radiat. Transf.
**2012**, 113, 172–181. [Google Scholar] [CrossRef] [Green Version] - Kokhanovsky, A.A. Asymptotic radiative transfer. In Light Scattering Reviews. Springer Praxis Books; Kokhanovsky, A.A., Ed.; Springer: Berlin/Heidelberg, Germany, 2006; pp. 253–289. [Google Scholar]

**Figure 1.**Relative errors of the two-stream model as a function of the gas optical depth (${\tau}_{\mathrm{gas}}$) for the O

_{2}A-band (red) and CO

_{2}band (blue). Left panel corresponds to the ‘Aerosol 1’ scenario, while right panel shows the results for the ‘Cloud 1’ scenario. Note that left Y axis refer to CO

_{2}while right Y axis refer to O

_{2}A-band.

**Figure 2.**Spectral radiances for the ‘Aerosol 2’ scenario computed by using the multi-stream (black), the two-stream (gray) and the single-scattering (light gray) RTMs: (upper panel) O

_{2}A-band, (lower panel) CO

_{2}band.

**Figure 3.**Radiance computed with the multi-stream model as a function of radiances computed by using the low-stream models (two-stream and single-scattering models) for the O

_{2}A- and CO

_{2}bands. The figure corresponds to the ‘Aerosol 2’ scenario.

**Figure 4.**Scheme of the Cluster Low-Streams Regression (CLSR) method. (

**a**) Sorted radiance of the low-stream (LS) model in ascending order (blue line). (

**b**) Division of the LS radiance in equal clusters C in the sorted domain. (

**c**) Zoom for one cluster and the selected regression points of the multi-stream (MS) radiance (red crosses). (

**d**) Reconstruction of the MS spectra: the predicted radiance is computed for all the spectral points (dashed red line).

**Figure 5.**Radiance spectra computed by using the multi-stream RTM for three atmospheric scenarios: ‘Clear sky’ (black), ‘Cloud 1’ (purple) and ‘Cloud 2’ (blue). The upper panel corresponds to the O

_{2}A-band, while the bottom panel is for the CO

_{2}weak band.

**Figure 6.**Box plots of the residuals with respect to the continuum in percentage for (upper panels) the PCA model and (bottom panels) the CLSR method, when the low-stream model is (

**a**) the two-stream model for the ‘Cloud 2’ scenario or (

**b**) the single-scattering model for the ‘Aerosol 2’ scenario. The red boxes indicate the O

_{2}A-band while the blue boxes indicate the CO

_{2}band. Box description: the upper and lower limits of the box represent the interquartile range (IQR), which is (${Q}_{3}-{Q}_{1}$) being ${Q}_{i}$ the i-th quartile; the upper and lower whiskers indicate (${Q}_{3}+1.5\xb7$ IQR) and (${Q}_{1}-1.5\xb7$ IQR), respectively; the orange line inside the box represents the median; the orange values on top of each box indicate the median values and the black values correspond to the IQR value.

**Figure 7.**Explained variance ratio in percentage as a function of the number of PCs for all the atmospheric scenarios: ‘Clear sky’, ‘Aerosol 1’, ‘Aerosol 2’, ‘Cloud 1’ and ‘Cloud 2’; and the two spectral bands (O

_{2}A- and CO

_{2}bands). The red color corresponds to the O

_{2}A-band and the blue color to the CO

_{2}weak band. The different dashing of the lines indicates the different atmospheric scenarios.

**Figure 8.**Dependence of the number of clusters and number of regression points with the mean error in percentage for the CO

_{2}band for the ‘Aerosol 2’ scenario. The low-stream model used is the two-stream model.

**Figure 9.**Comparison of the residuals for the methods PCA and CLSR for all the atmospheric scenarios (grey: ‘Clear sky’; blue: ‘Aerosol 1’; red: ‘Aerosol 2’; green: ‘Cloud 1’; yellow: ‘Cloud 2’) and gases (O

_{2}A- and CO

_{2}bands), when the low-stream model is (

**a**) the two-stream model or (

**b**) the single-scattering model. Note the differences in scales for the PCA technique for the O

_{2}A-band with the rest of cases. The orange values on top of each box indicate the median values and the black values correspond to the IQR value.

**Figure 10.**Convolved spectra for the multi-stream model and the two acceleration methods for the ‘Cloud 1’ scenario using PCA and CLSR methods for the sensors GOME-2, TROPOMI and GOSAT. For GOME-2 and TROPOMI the O

_{2}A-band spectra are convolved, while for GOSAT the CO

_{2}spectra are convolved.

**Table 1.**Aerosol optical thickness used in the simulations for O

_{2}A- and CO

_{2}bands at the middle of the corresponding absorption band.

O_{2} A-Band (760 nm) | CO_{2} Band (1610 nm) | |
---|---|---|

Aerosol 1 | 0.2 | 0.08 |

Aerosol 2 | 1.2 | 0.41 |

**Table 2.**Mean relative error $\epsilon $ for the low-stream models (single-scattering and two-stream models) compared with the multi-stream model for the O

_{2}A-band.

Scenario | Single-Scattering Model | Two-Stream Model |
---|---|---|

$\mathit{\epsilon}$ (%) | $\mathit{\epsilon}$ (%) | |

Clear sky | 3.4 | 0.14 |

Aerosol 1 | 36 | 1.0 |

Aerosol 2 | 78 | 4.6 |

Cloud 1 | 93 | 7.1 |

Cloud 2 | 93 | 1.4 |

**Table 3.**Mean relative error $\epsilon $ for the low-stream models (single-scattering and two-stream models) compared with the multi-stream model for the CO

_{2}weak band.

Scenario | Single-Scattering Model | Two-Stream Model |
---|---|---|

$\mathit{\epsilon}$ (%) | $\mathit{\epsilon}$ (%) | |

Clear sky | 0.203 | 0.004 |

Aerosol 1 | 14.7 | 0.40 |

Aerosol 2 | 55.5 | 2.1 |

Cloud 1 | 97.56 | 6.13 |

Cloud 2 | 97.91 | 2.00 |

**Table 4.**Computational time in seconds of the radiative transfer solution as a function of the number of streams, and speedup factors with respect to the case ${N}_{\mathrm{do}}=32$.

${\mathit{N}}_{\mathbf{do}}$ | Time (s) | Speedup Factor |
---|---|---|

0 | 0.4 | 5800 |

1 | 3.2 | 725 |

2 | 12.4 | 187.1 |

4 | 34.4 | 67.4 |

8 | 110 | 21 |

16 | 550 | 4.2 |

32 | 2320 | - |

**Table 5.**Speedup factor of the PCA-based (${S}_{\mathrm{PCA}}$) and CLSR methods with: the two-stream model (${S}_{\mathrm{CLSR}}^{\mathrm{TS}}$) and single-scattering model (${S}_{\mathrm{CLSR}}^{\mathrm{SS}}$).

${\mathit{S}}_{\mathbf{PCA}}$ | ${\mathit{S}}_{\mathbf{CLSR}}^{\mathbf{TS}}$ | ${\mathit{S}}_{\mathbf{CLSR}}^{\mathbf{SS}}$ |
---|---|---|

534 | 505 | 1294 |

**Table 6.**Spectral ranges and FWHM of the Gaussian slit functions of the instruments used in the study: TROPOMI, GOME-2 and GOSAT.

Instrument | Spectral Range | FWHM |
---|---|---|

TROPOMI | 710–775 nm | 0.183 nm |

GOME-2 | 590–790 nm | 0.51 nm |

GOSAT | 1.56–1.69 µm | 0.2 cm^{−1} |

**Table 7.**Mean relative error $\epsilon $ for the convolved spectra compared with the multi-stream spectra for the PCA-based and CLSR methods and for the different atmospheric scenarios considered for the O

_{2}A-band. In all cases, the low-stream model considered is the two-stream model. The instruments analyzed are GOME-2 and TROPOMI, which are compared with the non-convolved values.

Scenario | O_{2} A-Band | |||||
---|---|---|---|---|---|---|

GOME-2 | TROPOMI | Non-Convolved | ||||

${\mathbf{\epsilon}}_{\mathbf{PCA}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{CLSR}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{PCA}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{CLSR}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{PCA}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{CLSR}}(\%)$ | |

Clear sky | 0.021 | 0.004 | 0.021 | 0.004 | 0.022 | 0.006 |

Aerosol 1 | 0.049 | 0.007 | 0.050 | 0.007 | 0.061 | 0.011 |

Aerosol 2 | 0.837 | 0.019 | 0.837 | 0.019 | 0.856 | 0.026 |

Cloud 1 | 1.23 | 0.011 | 1.23 | 0.011 | 1.24 | 0.017 |

Cloud 2 | 2.92 | 0.006 | 2.92 | 0.006 | 2.93 | 0.009 |

**Table 8.**Mean relative error $\epsilon $ for the convolved spectra compared with the multi-stream spectra for the PCA-based and CLSR methods and for the different atmospheric scenarios considered for the CO

_{2}band. In all cases, the low-stream model considered is the two-stream model. The instrument analyzed is GOSAT, which is compared with the non-convolved values.

Scenario | CO_{2} Band | |||
---|---|---|---|---|

GOSAT | Non-Convolved | |||

${\mathbf{\epsilon}}_{\mathbf{PCA}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{CLSR}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{PCA}}(\%)$ | ${\mathbf{\epsilon}}_{\mathbf{CLSR}}(\%)$ | |

Clear sky | 0.0003 | 0.0003 | 0.0006 | 0.0006 |

Aerosol 1 | 0.0075 | 0.0067 | 0.010 | 0.012 |

Aerosol 2 | 0.035 | 0.012 | 0.044 | 0.017 |

Cloud 1 | 0.011 | 0.007 | 0.013 | 0.008 |

Cloud 2 | 0.011 | 0.005 | 0.016 | 0.006 |

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## Share and Cite

**MDPI and ACS Style**

del Águila, A.; Efremenko, D.S.; Molina García, V.; Kataev, M.Y.
Cluster Low-Streams Regression Method for Hyperspectral Radiative Transfer Computations: Cases of O_{2} A- and CO_{2} Bands. *Remote Sens.* **2020**, *12*, 1250.
https://doi.org/10.3390/rs12081250

**AMA Style**

del Águila A, Efremenko DS, Molina García V, Kataev MY.
Cluster Low-Streams Regression Method for Hyperspectral Radiative Transfer Computations: Cases of O_{2} A- and CO_{2} Bands. *Remote Sensing*. 2020; 12(8):1250.
https://doi.org/10.3390/rs12081250

**Chicago/Turabian Style**

del Águila, Ana, Dmitry S. Efremenko, Víctor Molina García, and Michael Yu. Kataev.
2020. "Cluster Low-Streams Regression Method for Hyperspectral Radiative Transfer Computations: Cases of O_{2} A- and CO_{2} Bands" *Remote Sensing* 12, no. 8: 1250.
https://doi.org/10.3390/rs12081250