# A New Algorithm for the Satellite-Based Retrieval of Solar Surface Irradiance in Spectral Bands

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Outline of the Method for the Retrieval of Spectrally Resolved Irradiance

## 3. Clear Sky Approach for the Retrieval of Spectrally Resolved Irradiance

_{2}O) and ozone (O

_{3}), and the reflection by the Earth’s surface. Respective variables belonging to these processes are aerosol optical depth (aod), single scattering albedo (ssa) and asymmetry parameter (gg) of aerosols, the water vapor vertical column and the surface albedo. The analysis of the system’s interaction allows the selection of processes and variables which have to be considered within a basis LUT from those that can be parameterized by simple equations. Atmospheric variables that belong to radiation processes that are not dependent upon the amount/value of other atmospheric variables are not considered within the basis LUT. The basis LUT can be calculated for fixed values of the respective variables. Deviations of the variables from the fixed values can be corrected afterwards as their effect depends not on other variables. The effect of ozone or water vapor amounts, respectively, is pre-dominantly independent of aerosol properties. The effect of the surface albedo can be treated with a simple scaling. These findings are based on radiative transfer model studies [13]. Hence, no RTM calculations are saved in the basis LUT, but their effect is corrected by simple “correction” equations. Water vapor and ozone are pure absorber. The amount of absorption depends on the water vapor amount and the path length through the atmosphere, which is pre-dominantly independent of aerosol properties, amount of ozone and surface albedo.

_{Λ}) are considered within a 3-dimensional basis lookup table. I

_{Λ}is calculated for different values of aod, gg and ssa, and the results are saved in the LUT. The effect of water vapour (H

_{2}O), ozone (O

_{3}) and surface albedo (SAL) is corrected after interpolation to the given aerosol state, defined by aod, ssa and gg, has taken place. Hence, the overall approach to calculate the clear sky surface irradiance used in this paper, can be summarized as:

_{Λ}is the final spectrally resolved irradiance for cloud free skies, ${I}_{\mathrm{\Lambda}(\mathit{aod},\mathit{ssa},\mathit{gg})}^{\mathit{LUT}}$ is the spectrally resolved irradiance for a given aerosol state derived from the basis LUT for fixed water vapour, ozone and surface albedo. I

_{Λ,H2Ocor}and I

_{Λ}

_{,O}

_{3}

_{cor}are the correction of deviations in water vapour and ozone from the fixed values used in the LUT, respectively. Finally SAL

_{Λ}

_{,cor}is the scaling to the given surface albedo relative to 0.2, which has been used in all previous steps.

#### 3.1. Basis LUT: Treatment of Aerosols

_{0Λ}is the optical depth of the vertical column, I

_{Λ}is the solar radiation at ground for a solar zenith angle (SZA) of θ

_{z}and I

_{0}

_{,enh,}

_{Λ}is based on the extraterrestrial irradiance according to Equation (3) at wavelength (wavelengths band) Λ. The cosine of the solar zenith angle accounts for the decrease of the surface flux density due to the increase of the solar zenith angle (the same amounts of photons are distributed over a larger area for increasing solar zenith angles). This relation is identical to the Lambert–Beer function (or MLB function) with exception of an additional “empirical” correction exponent a, hence a correction of the parameter $\frac{\tau}{\mathit{cos}({\theta}_{z})}$ [22]. The correction parameters a

_{Λ}are calculated based on two RTM runs, one at θ

_{z}=0 and the other at θ

_{z}=60°, hence the correction parameters a

_{Λ}can be calculated without the need for a numerical fit. I

_{0}

_{,enh}is based on the extraterrestrial irradiance at the top of atmosphere and estimated using Equation (3)[22].

_{Λ}and B

_{Λ}are the diffuse and direct (beam) component of the solar surface irradiance I

_{Λ}= B

_{Λ}+ D

_{Λ}at a SZA of zero. The application of this equation is needed in order to preserve a good match of the MLB relation with RTM results for high optical depths, where the use of the extraterrestrial irradiance I

_{0Λ}fails. It is important to notice that the Modified Lambert–Beer relation (Equation (2)) is also used for the calculation of the direct (beam) irradiance B

_{Λ}, with the exception that instead of using I

_{0}

_{,enh,}

_{Λ}the original extraterrestrial irradiance I

_{0Λ}must be used. The fitting parameters a

_{Λ}have different values for direct irradiance and (total) solar surface irradiance. Using the MLB function, the calculated direct irradiance, diffuse irradiance as well as the solar surface irradiance can be reproduced very well (see Figure 2 as an example). The MLB function is discussed in detail by Mueller et al.[22], including proof and verification. Further validations are given by Ineichen et al.[23].

_{Λ}, I

_{0}

_{,enh,}

_{Λ}and τ

_{0Λ}are saved in the basis LUT for each wavelength band and different values of aerosol optical depth (aod), single scattering albedo (ssa) and asymmetry parameter (gg). The resulting basis LUT is three dimensional and contains the MLB-parameters to calculate I

_{basis}and B

_{basis}with Equation (2). The effect of aod is significantly larger than the effect of both ssa and gg, so the basis LUT contains MLB parameters for 23 aod values times three ssa values times two gg values (ssa ∈ {0.7, 0.85, 1.0} and gg ∈ {0.6, 0.78}). The spectral effect of aerosols is considered by application of a standard aerosol model [24,25].

#### 3.2. Treatment of Water Vapor and Ozone Variations

_{H2O,Λ}is the difference between the irradiance ${I}_{\mathit{basis},\mathrm{\Lambda}}^{{H}_{2}O}$ for the real amount of water vapor and I

_{basis,}

_{Λ}resulting from interpolation within the basis “aerosol” LUT. ΔI

_{H2O,Λ}is calculated for the following set of input parameters (θ

_{z}=0, a rural aerosol type with aod = 0.2, ssa = 0.94, gg = 0.75, 345 DU ozone, and SAL

_{Λ}= 0.2). b

_{Λ}is a “fitting” parameter applied to match the solar zenith angle dependency of water vapour absorption.

_{H2O,Λ}is pre-calculated for 18 water vapor columns, seven of them in the range of [0...15] mm and eleven within [20...75] mm. Respective values of ΔI

_{H2O,Λ}and b

_{Λ}are saved in an LUT for each wavelength band. The use of this LUT enables the calculation of the irradiance for the real water vapor values given for the specific pixel and time. Only 19 of the 32 Kato bands are influenced by water vapor absorption, consequently only for these the LUT contains non-zero values for ΔI

_{H2O,Λ}and b

_{Λ}. The parameter b

_{Λ}is limited by b

_{Λ}⩽ 1.

_{O3,Λ}is the difference between the irradiance ${I}_{\mathit{basis},\mathrm{\Lambda}}^{{H}_{2}O,{O}_{3}}$ for the real amount of ozone and ${I}_{\mathit{basis},\mathrm{\Lambda}}^{{H}_{2}O}$ derived for the fixed value of ozone. c

_{Λ}is a “fitting” parameter applied to match the solar zenith angle dependency of ozone absorption.

_{O3}values are sufficient. In contrast, for spectrally resolved irradiance eight pre-calculated ΔI

_{O3}values are needed (210 to 525 DU, in steps of 45 DU). The LUT for ΔI

_{O3}and c

_{Λ}contains nonzero values for 13 of 32 Kato bands. For broadband irradiance the parameter c

_{Λ}< 1. This relation also holds for direct irradiance for most wavelength bands, but fails to generate reliable results for the ultraviolet range. Here, appropriate modifications of Equation (5) might resolve the problem.

#### 3.3. Treatment of Surface Albedo

_{Λ}is the variable surface albedo and I

_{Λ}is the solar irradiance after the surface albedo correction has been applied.

_{Λ}. However, the applied equation simplifies the spectral effect of the surface albedo. Increased surface albedo will increase particularly short wavelengths because of the strongly wavelength-dependent backscattering by the atmosphere as discussed, e.g., by Lenoble et al.[26]. This effect is not accounted for by the applied equation.

#### 3.4. Sensitivity on Atmospheric Background Profiles

#### 3.5. Summary of Clear Sky Approach

_{basis}and B

_{basis}with Equation (2). The irradiances are interpolated from this basis LUT for the respective aerosol state, expressed by specific values of aod, ssa and gg. The spectral effect of aerosols is considered by application of a standard aerosol model [24,25]. The effect of deviations in water vapor, ozone and surface albedo relative to the fixed values used in the calculation of the basis LUT are corrected by application of the parameterizations described in detail in Sections 3.2 and 3.3.

## 4. All Sky Approach for the Retrieval of Spectrally Resolved Irradiance

#### 4.1. Retrieval of Effective Cloud Albedo

_{0}is the dark offset, θ is the solar zenith angle and f corrects the variations in the Sun-Earth distance. The effective cloud albedo (CAL) is then derived from the observed normalized reflections by Equation (8):

_{cs}is the clear sky reflection, which is a monthly value derived for every pixel and time slot separately. This is essentially done by using the reflection of the pixel in a cloud free case. This is usually the relative minimum of the reflections during a certain time span (e.g., a month) derived for each pixel of the satellite image. Further details on the method to derive ρ

_{cs}are given in [15]. ρ

_{max}is the “maximum” reflection. It is determined by the 95 percentile of all reflection values at local noon in a target region, characterized by high frequency of cloud occurrence for each month. Changes in the sensitivity of the satellite instrument would lead to a respective change in the 95 percentile. In this manner changes in the satellite brightness sensitivity are accounted for.

#### 4.2. Spectral Correction of Cloud Effect

_{bb}defined as the ratio of solar surface irradiance I to clear sky irradiance I

_{cs}:

_{bb}to a wavelength dependent transmission k

_{Λ}is required. The clear sky state does not predominantly affect the spectral correction discussed hereafter. Hence the standard atmosphere of the basis LUT has been used for the derivation of the spectral correction.

_{bb}= 1, 0.525, 0.244, 0.197, 0.102 and 0.050, respectively.

_{Λ}is the conversion factor, ${k}_{\mathrm{\Lambda}}^{\mathit{RTM}}$ and ${k}_{bb}^{\mathit{RTM}}$ are the clear sky indices for the wavelength bands (denoted by Λ) and the broadband (denoted by bb) calculated with radiative transfer model (denoted by RT M)

_{Λ}is the derived all sky spectrally resolved clear sky index and ${k}_{\mathit{bb}}^{\mathit{sat}}$ is the broadband clear sky index observed by satellite. I

_{Λ}is the all sky and I

_{cs,}

_{Λ}the clear sky spectrally resolved irradiance, respectively. I

_{cs,}

_{Λ}is derived with the clear sky method described in Section 3.

#### 4.3. Used Input on the Atmospheric State

## 5. Irradiance on a Tilted Surface

_{tilt}as a composition of direct beam B

_{tilt}, sky diffuse D

_{tilt}and ground-reflected radiation I

_{reflected}(e.g., [38]):

_{tilt}for each wavelength band.

## 6. Validation of the Retrieved Spectral Solar Surface Irradiance

#### 6.1. Measurements of the Solar Spectral Surface Irradiance

#### 6.2. Validation Results

_{Λ}have been compared with measurements for each wavelength band for

**cloud free skies**. Cloud free cases are classified by the ratio between diffuse and global irradiance, for a ratio below 0.3 clear sky cases are assumed. The applied error measures, Root Mean Square Deviation (RMSE) and Bias, are based on comparison of hourly values and are defined in the Appendix. Please note that the comparison on hourly basis leads to high values of the root mean square deviations. Therefore, the absolute and relative bias values are the main quantities applied for the evaluation of the ability of the method to provide accurate spectral resolved irradiance. The resulting spectra are given for both sites together with the bias in Figure 7.

**cloud-free skies**. SOLIS follows a concept of integrated radiative transfer runs, but takes benefit of the modified Lambert–Beer relation in order to improve the computing performance. The use of identical atmospheric input for SOLIS and SPECMAGIC avoids the occurrence of differences in the comparison that are induced by uncertainties in the atmospheric input. As the MLB approach is applied in both schemes also uncertainties induced by this approach are avoided. Hence, the comparison results provide information about the accuracy of the applied hybrid eigenvector approach and parameterizations for water vapor, ozone and surface albedo corrections relative to explicit radiative transfer modeling for typical atmospheric conditions.

## 7. Discussion

## 8. Conclusions

## Acknowledgments

## References

- Schmetz, J.; Pili, Tjemkes, P.S.; Just, D.; Kerkmann, J.; Rota, S.; Ratier, A. An introduction to Meteosat Second Generation (MSG). Bull. Am. Met. Soc
**2002**, 83, 977–992. [Google Scholar] - GCOS, The Second Report on the Adequacy of the Global Observing Systems for Climate in Support of the UNFCCC; Vol. Rep. GCOS-82; WMO: Geneva, Switzerland, 2003.
- Woick, H.; Dewitte, S.; Feijt, A.; Gratzki, A.; Hechler, P.; Hollmann, R.; Karlsson, K.G.; Laine, V.; Loewe, P.; Nitsche, H.; et al. The satellite application facility on climate monitoring. Adv. Space Res
**2002**, 30, 2405–2410. [Google Scholar] - Schulz, J.; Albert, P.; Behr, H.; Caprion, D.; Deneke, H.; Dewitte, S.; Dürr, B.; Fuchs, P.; Gratzki, A.; Hechler, P.; et al. Operational climate monitoring from space: The EUMETSAT satellite application facility on climate monitoring (CM-SAF). Atmos. Chem. Phys. Discuss
**2008**, 8, 8517–8563. [Google Scholar] - Babst, F.; Mueller, R.; Hollmann, R. Verification of NCEP reanalysis shortwave radiation with mesoscale remote sensing data. IEEE Geosci. Remote Sens. Lett
**2008**, 5, 34–37. [Google Scholar] - Perez, R.; Renne, D.; Seals, R.; Zelenka, A. The strengths of satellite based solar resource assessment. In Production of Site/Time-Specific Irradiances from Satellite and Ground Data; Report 98-3. New York State Energy Research and Development Authority: Albany, NY, USA, 1998. [Google Scholar]
- Möser, W.; Raschke, E. Incident solar radiation over Europe estimated from METEOSAT data. J. Climate Appl. Meteor
**1984**, 23, 166–170. [Google Scholar] - Cano, D.; Monget, J.; Albuisson, M.; Guillard, H.; Regas, N.; Wald, L. A method for the determination of the global solar radiation from meteorological satellite data. Solar Energy
**1986**, 37, 31–39. [Google Scholar] - Bishop, J.K.B.; W.B.R. Spatial and temporal variability of global surface solar irradiance. J. Geophys. Res
**1991**, 96, 839–858. [Google Scholar] - Pinker, R.; Laszlo, I. Modelling surface solar irradiance for satellite applications on a global scale. J. Appl. Meteor
**1992**, 31, 166–170. [Google Scholar] - Darnell, W.; Staylor, W.; Gupta, S.; Ritchey, N.; Wilber, A. Seasonal variation of surface radiation budget derived from ISCCP-C1 data. J. Geophys. Res
**1992**, 97, 15741–15760. [Google Scholar] - Rigolier, M.; Levefre, M.; Wald, L. The method Heliosat-2 for deriving shortwave solar radiation from satellite images. Solar Energy
**2004**, 77, 159–169. [Google Scholar] - Mueller, R.; Matsoukas, C.; Gratzki, A.; Hollmann, R.; Behr, H. The CM-SAFoperational scheme for the satellite based retrieval of solar surface irradiance—A LUT based eigenvector hybrid approach. Remote Sens. Environ
**2009**, 113, 1012–1022. [Google Scholar] - Hammer, A.; Heinemann, D.; Hoyer, C.; R., K.; Lorenz, E.; Mueller, R.; Beyer, H. Solar Energy Assessment Using Remote Sensing Technologies. Remote Sens. Environ
**2003**, 86, 423–432. [Google Scholar] - Posselt, R.; Mueller, R.; Stöckli, R.; Trentmann, J. Spatial and temporal homogeneity of solar surface irradiance across satellite generations. Remote Sens
**2011**, 3, 1029–1046. [Google Scholar] - Stammes, K.; Tsay, S.; Wiscombe, W.; Jayaweera, K. Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl. Opt
**1988**, 27, 2502–2509. [Google Scholar] - Mayer, B.; Kylling, A. Technical note: The libRadtran software package for radiative transfer calculations—Description and examples of use. Atmos. Chem. Phys
**2005**, 5, 1855–1877. [Google Scholar] - Koepke, P.; Bais, A.; Balis, D.; Buchwitz, M.; de Backer, H.; de Cabo, X.; Eckert, P.; Eriksen, P.; Gillotay, D.; Koskela, T.; Lapeta, V.; Litynska, Z.; Lorente, J.; Mayer, B.; Renauld, A.; Ruggaber, A.; Schauberger, G.; Seckmeyer, G.; Seifert, P.; Schmalwieser, A.; Schwander, H.; Vanicek, K.; Weber, M. Comparison of models used for UV index calculations. Photochem. Photobiol
**1998**, 67, 657–662. [Google Scholar] - Mayer, B.; Seckmeyer, G.; Kylling, A. Systematic long-term comparison of spectral UV measurements and UVSPEC modeling results. J. Geophys. Res
**1997**, 102, 8755–8767. [Google Scholar] - Kato, S.; Ackerman, T.; Mather, J.; Clothiaux, E. The k-distribution method and correlated-k approximation for a short-wave radiative transfer. J. Quant. Spectrosc. Radiat. Transfer
**1999**, 62. [Google Scholar] - Mueller, R.; Trentmann, J.; Träger-Chatterjee, C.; Posselta, R.; Stöckli, R. The Role of the Effective Cloud Albedo for Climate Monitoring and Analysis. Remote Sens
**2011**, 3, 2305–2320. [Google Scholar] - Mueller, R.; Dagestad, K.; Ineichen, P.; Schroedter-Homscheidt, M.; Cros, S.; Dumortier, D.; Kuhlemann, R.; Olseth, J.; Piernavieja, G.; Reise, C.; Wald, L.; Heinemann, D. Rethinking satellite based solar irradiance modelling. The SOLIS clear-sky module. Remote Sens. Environ
**2004**, 91, 160–174. [Google Scholar] - Ineichen, P. Comparison of eight clear sky broadband models against 16 independent data banks. Solar Energy
**2006**, 80, 468–478. [Google Scholar] - Shettle, P.; Fenn, R.W. Models of the Atmospheric Aerosols and Their Optical Properties. AGARD Conference Proceedings No. 183, Optical Propagation in the Atmosphere, Lyngby, Denmark; 1976. [Google Scholar]
- Shetlle, E. Models of Aerosols, Clouds and Precipitation for Atmospheric Propagation Studies. AGARD Conference Proceedings No. 454, Atmospheric Propagation in the UV, Visible, IR and MM-Wave Region and Related Systems Aspects, Copenhagen, Denmark, 9–13 October 1989.
- Lenoble, J. Modeling of the influence of snowreflectance on ultraviolet irradiance for cloudless sky. Appl. Opt
**1998**, 37, 2441–2447. [Google Scholar] - Loveland, T.R.; Belward, A.S. The IGBP-DIS Global 1 km Land Cover Data Set, DISCover first results. Int. J. Remote Sens
**1997**, 18, 3289–3295. [Google Scholar] - Brown, J.F.; Loveland, T.; Merchant, J.W.; Reed, B.C.; Ohlen, D.O. Using multi-source data in global land-cover characterization: concepts, requirements, and methods. Photogramm. Eng. Remote Sensing
**1993**, 59, 977–987. [Google Scholar] - Dickinson, R.E. Land surface processes and climate—Surface albedos and energy balance. Adv. Geophys
**1983**, 25, 305–353. [Google Scholar] - Lindfors, A.; Arola, A. On the wavelength-dependent attenuation of UV radiation by clouds. Geophys. Res. Lett
**2008**, 35, L05806. [Google Scholar] - Kinne, S.; Schulz, M.; Textor, C.; Guibert, S.; Balkanski, Y.; Bauer, S.E.; Berntsen, T.; Berglen, T.F.; Boucher, O.; Chin, M.; et al. An AeroCom initial assessment—Optical properties in aerosol component modules of global models. Atmos. Chem. Phys
**2006**, 6, 1815–1834. [Google Scholar] - Holben, B.N.; Eck, T.F.; Slutsker, I.; Tanré, D.; Buis, J.P.; Setzer, A.; Vermote, E.; Reagan, J.A.; Kaufman, Y.J.; Nakajima, T.; et al. AERONET—A federated instrument network and data archive for aerosol characterization. Remote Sens. Environ
**1998**, 66, 1–16. [Google Scholar] - Hess, M.; Koepke, P.; Schult, I. Optical properties of aerosols and clouds: The software package OPAC. Bull. Amer. Meteor. Soc
**1998**, 79, 831–844. [Google Scholar] - Kalnay, E.; Kanamitsu, M.; Kistler, R.; Collins, W.; Deaven, D.; Gandin, L.; Iredell, M.; Saha, S.; White, G.; Woollen, J.; et al. The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Meteor. Soc
**1996**, 77, 437–471. [Google Scholar] - Dee, D.P.; Uppala, S. Variational bias correction of satellite radiance data in the ERA-interim reanalyis. Q. J. Roy. Meteor. Soc
**2009**, 135, 1830–1841. [Google Scholar] - Betts, A.K.; Ball, J.H.; Bosilovich, M.; Viterbo, P.; Zhang, Y.; Rossow, W.B. Intercomparison of water and energy budgets for five Mississippi subbasins between ECMWF re-analysis (ERA-40) and NASA Data Assimilation Office fvGCM for 1990–1999. J. Geophys. Res.
**2003**, 108. [Google Scholar] [CrossRef] - Ziemke, J.R.; Chandra, S.; Labow, G.J.; Bhartia, P.K.; Froidevaux, L.; Witte, J.C. A global climatology of tropospheric and stratospheric ozone derived from Aura OMI and MLS measurements. Atmos. Chem. Phys
**2011**, 11, 9237–9251. [Google Scholar] - Klucher, T. Evaluation of models to predict insolation on tilted surfaces. Solar Energy
**1979**, 23, 111–114. [Google Scholar] - Temps, R.C.; Coulson, K.L. Solar Radiation Incident Upon Slopes of Different Orientations. Solar Energy
**1977**, 19, 179–184. [Google Scholar] - Drews, A.; Beyer, H.; Rindelhardt, U. Quality of performance assessment of PV plants based on irradiance maps. Solar Energy
**2008**, 82, 1067–1075. [Google Scholar]

## Appendix

#### Meaning of Eigenvector Approach

_{H2O}can be defined which fulfills the following equation.

_{0}is the extraterrestrial irradiance and RTM

_{δH2O}is an operator, describing the effect of deviations in water vapour on I

_{0}. For every RTM

_{δH2O}a unique t

_{H2O}exists which depends only on the amount of water vapour and on no other atmospheric variable. The value of t describes the atmospheric transmission. The atmospheric transmission of the operator RT M

_{δH2O}depends only on the amount of water vapour. t

_{H2O}can therefore by interpreted as eigen-value of the operator RT M

_{δH2O}and I

_{o}as “eigenvector”.

_{aod}values exist, as t depends on the value of aod, ssa and gg. The transmission for a given aerosol optical depth depends also on the values for the single scattering albedo and the asymmetry parameter.

#### Applied Error Measures

**Bias:**The bias (or mean error) is simply the mean difference between the two considered datasets. It indicates whether the dataset on average overestimates or underestimates the reference dataset (e.g., ground measurement denoted as o).

**Relative Bias:**is the Bias divided by the absolute value of the reference.

**Standard deviation (SD):**The standard deviation SD is a measure for the spread around the mean value of the distribution formed by the differences between the generated and the reference dataset.

**Root Mean Square Deviation:**The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) results from the bias and the standard deviation and is defined as follows. It measures beside the bias also the scatter of the data.

**Relative Root Mean Square Error:**is the RMSE divided by the absolute value of the reference data set.

**Figure 1.**The principle of an LUT approach. The relation of the transmission to a variety of atmospheric states is pre-calculated with a radiative transfer model (RTM) and saved in a look-up table (LUT). Based on the amount of considered atmospheric states the LUT table is large. Usually, 10

^{5}to 10

^{7}calculations are needed for a classical LUT approach if specific scientific optimizations are not applied. This figure has been previously published in [13].

**Figure 2.**Comparison between RTM calculations and fit using the Modified Lambert–Beer relation. Example for a fit of narrow-band spectral irradiance. Global irradiance is synonymous to solar irradiance at the surface. Direct irradiance is the direct portion (beam) of the solar surface irradiance. This figure has been previously published in [22].

**Figure 3.**Differences between ${I}_{\mathit{basis},\mathrm{\Lambda}}^{{H}_{2}O}$ and I

_{basis,}

_{Λ}estimated by explicit RTM calculations (dots) and by use of the correction formula (lines, Equation (4)) for eight Kato wavelength bands and the solar zenith angle of 20°.

**Figure 4.**Differences between ${I}_{\mathit{basis},\mathrm{\Lambda}}^{{H}_{2}O,{O}_{3}}$ and ${I}_{\mathit{basis},\mathrm{\Lambda}}^{{H}_{2}O}$ estimated by explicit RTM calculations (dots) and by use of the correction formula (lines, Equation (5)) for nine Kato wavelength bands and the solar zenith angle of 20°.

**Figure 5.**Diagram of the new clear sky LUT. SZW is the solar zenith angle. NWP is Numerical Weather Prediction, I is the solar surface irradiance. This procedure is applied for each wavelength band. Figure has been previously published in [13].

**Figure 6.**Conversion from the broadband cloud transmission described by the clear sky index k

_{bb}to a wavelength dependent transmission ${k}_{\mathrm{\Lambda}}={f}_{\mathrm{\Lambda}}\cdot {k}_{\mathit{bb}}^{\mathit{sat}}$. The corresponding conversion factor f is given for different cloud optical depths expressed by different broadband cloud transmissions k

_{bb}.

**Figure 7.**Average spectrum for Kato bands measured at Loughborough (

**left**) and Stuttgart (

**right**) in comparison with SPECMAGIC. The bias is given in addition.

**Figure 8.**Average spectrum for Kato bands retrieved with SPECMAGIC and with SOLIS for Loughborough (

**left**) and Stuttgart (

**right**). The bias is given in addition.

**Figure 9.**Relative bias and RMSE of SPECMAGIC retrieval with respect to ground measurements performed at Loughborough (

**left**) and Stuttgart (

**right**) for all situations, clear sky and cloudy sky.

**Figure 10.**Annual course of normalized irradiance for four wavelength bands for Loughborough. Comparison of measured data and SPECMAGIC retrievals.

**Figure 11.**Annual course of normalized irradiance for four wavelength bands for Stuttgart. Comparison of measured data and SPECMAGIC retrievals.

**Table 1.**Comparison of irradiance calculated with either RTM or the parameterization given in Equation (4) for the worst case scenario of 80 degree SZA and 70 mm water vapour content. The deviations are therefore upper limits, which are only reached in extreme atmospheric conditions. Values are given for three water vapor bands for solar surface irradiance I

_{Λ}and direct irradiance B

_{Λ}.

Katoband Λ[nm] | Width [nm] | Deviation Relative to RTM [W/m^{2}/band] | Dito [W/m^{2}/nm] | |
---|---|---|---|---|

724 | 38.2 | I_{Λ} | −0.74 | −0.019 |

932 | 86.0 | I_{Λ} | −2.11 | −0.025 |

1,335 | 321.8 | I_{Λ} | −0.73 | −0.0023 |

932 | 86.0 | B_{Λ} | −1.93 | −0.022 |

**Table 2.**Comparison of irradiance calculated with either RTM or the parameterization given in Equation (5). Values are given for three ozone bands calculated with high SZA of 80° and extreme ozone content of 525DU for solar surface irradiance I

_{Λ}and direct irradiance B

_{Λ}.

Katoband Λ[nm] | Width [nm] | Deviation Relative to RTM [W/m^{2}/band] | Dito [W/m^{2}/nm] | |
---|---|---|---|---|

317 | 20.9 | I_{Λ} | −0.19 | −0.0091 |

586 | 38.4 | I_{Λ} | −0.20 | −0.0052 |

646 | 41.7 | I_{Λ} | −0.09 | −0.0022 |

317 | 20.9 | B_{Λ} | −0.028 | −0.0013 |

646 | 41.7 | B_{Λ} | −0.12 | −0.0028 |

**Table 3.**Reduction of needed RTM runs, comparison of original prototype CM-SAF algorithm with the new algorithm. RIA: RTM based Impact Analysis, analysis of Principal Components and Symmetries. MLB: Modified Lambert–Beer Relation.

Parameter | RTM Runs broadband prototype | RTM Runs broadband new concept | RTM Runs 32 spectral bands new concept | Method |
---|---|---|---|---|

Background atmosphere | 5 | 1 | 1 | RIA |

Aerosol AOD and type | ×10 × 8 | ×10 × 3 × 2 | ×23 × 3 × 2 | ssa and gg instead of 8 types |

SZA | ×8 | ×2 | ×2 | MLB function |

H_{2}O, O_{3} | ×10 × 5 | +10 +3 | +18 +8 | parameterization |

SAL | ×8 | +0 | +0 | parameterization |

Totel number | 1,280,000 | 133 | 302 |

Dataset | Loughborough | Stuttgart |
---|---|---|

Location | 52.77°N, −1.23°E | 48.75°N, 9.11°E |

Provided by | CREST, Loughborough University | IPE, Stuttgart University |

Period | Sep 2003–Aug 2004 | 2008 |

Apparatus | Horiba, custom build | Stellarnet EPP 2000C UV-VIS, EPP 2000 NIR INGAS |

Temporal resolution | 2 min scan, every 10 min | 1 min scan every min., delivered as 15 min average |

Spectral resolution | 10 nm | 1 nm |

Spectral range | 300 to 1,700 nm | 300 to 1,700 nm |

Orientation | 20° East | South |

Tilt | 52° | 33° |

Completeness original data | 40% | 98% |

Completeness hourly data | 35% | 97% |

## Share and Cite

**MDPI and ACS Style**

Mueller, R.; Behrendt, T.; Hammer, A.; Kemper, A.
A New Algorithm for the Satellite-Based Retrieval of Solar Surface Irradiance in Spectral Bands. *Remote Sens.* **2012**, *4*, 622-647.
https://doi.org/10.3390/rs4030622

**AMA Style**

Mueller R, Behrendt T, Hammer A, Kemper A.
A New Algorithm for the Satellite-Based Retrieval of Solar Surface Irradiance in Spectral Bands. *Remote Sensing*. 2012; 4(3):622-647.
https://doi.org/10.3390/rs4030622

**Chicago/Turabian Style**

Mueller, Richard, Tanja Behrendt, Annette Hammer, and Axel Kemper.
2012. "A New Algorithm for the Satellite-Based Retrieval of Solar Surface Irradiance in Spectral Bands" *Remote Sensing* 4, no. 3: 622-647.
https://doi.org/10.3390/rs4030622