Issue 
Eur. Phys. J. Appl. Phys.
Volume 88, Number 1, October 2019



Article Number  11001  
Number of page(s)  7  
Section  Instrumentation and Metrology  
DOI  https://doi.org/10.1051/epjap/2019190350  
Published online  17 January 2020 
https://doi.org/10.1051/epjap/2019190350
Sensitivity analysis of atmospheric spectral irradiance model
^{1}
Institute of Earth Sciences, University of Évora, Évora, Portugal
^{2}
Science Engineer Laboratory for Energy (LabSIPE), National School of Applied Sciences, University of Chouaib Doukkali, El Jadida, Morocco
^{3}
Energy Laboratory, Solar Energy Unit National Laboratory of Energy and Geology, Lisbon, Portugal
^{*} email: aalbino@uevora.pt
Received:
13
May
2019
Received in final form:
29
August
2019
Accepted:
10
December
2019
Published online: 17 January 2020
Many Radiative Transfer Models (RTM) have been developed to simulate and estimate solar irradiance. Theirs accuracy is well documents in literature nonetheless the effect of the parameters uncertainties on the established models has not been well studied yet. This work focuses on implementing a RTM based on the models found in the literature along with some updates, with the aim to study the sensitivity of the model towards the variations of the input parameters. The parameters studied in this paper are: the day of the year, the solar zenith angle, the local atmospheric pressure, the local temperature, the relative humidity, the height of ozone layer concentration, the ozone concentration, the single scattering albedo, the ground albedo, the Ångström’s exponent and the aerosol optical depth. The sensibility analysis is achieved using the Normalized Root Mean Square Error (NRMSE) as an independent function, calculated with a set of simulated measurements of spectral global solar irradiance and a reference spectrum generated with a group of standard input parameters.
© EDP Sciences, 2020
1 Introduction
Solar irradiance spectrum is important in many fields of life sciences (human health, atmospheric sciences, energy). One of the common features of these sciences topics is to perform studies aiming to quantify the uncertainties of the measured solar irradiance [1,2]. The first step to assess these measurements errors is the use of a radiative transfer models (RTM) to simulate spectral irradiance for a set of atmospheric representative parameters and boundary conditions. Many RTMs have been developed, some of them can be found in [3,4], a couple of the main commonly used ones are libradtran [5] and SMART [6]. RTMs are used for providing information when measurements are not available, to predict the weather conditions, to help with spectral information in calibrations [7] and to retrieve atmospheric proprieties [8,9]. The accuracy of the RTMs is documented in several studies [3,4,10,11]. Nevertheless, few of them take in account the effect of input uncertainties on the output result [12–16].
This work implements a simple RTM and performs the sensitivity analysis of the implemented model towards each of the input parameters. The model’s parameters evaluated are: the day of the year (DOY), the solar zenith angle (θ), the local temperature (T) and atmospheric pressure (P), the relative humidity (RH), the height of ozone layer concentration (z_{3}), the ozone concentration (), the single scattering albedo (w_{0}) as well as the ground albedo (ρ_{g}), the Ångström’s exponent (α) and finally the aerosol optical depth (AOD). This paper seeks to analyze the effects of uncertainties of the model inputs on the spectral global radiation estimated by the RTM, to better understand the response of the model to the input’s variations. Contrasting with the previous published papers [12–16] referring to a specific region or country, this work make use of simulated data since using real data, the uncertainty depends on all the inputs and the sensors used to measure the parameters, while with synthetic data uncertainty over each input parameter can be controlled. The analysis is done modifying only one of the above mentioned parameters at each time, in order to assess the effect of each one. If all the input parameters are analyzed grouped it should be quite impossible to understand the effect of each one. The results also show the importance of the measurement quality of each parameter.
This study is divided into five sections including this introduction. The next section presents the radiative transfer model and its parameters, followed by the methodology presentation where the sensitive analysis of each parameter is explained. The fourth section presents and analyses the results obtained. In the last section the main conclusions are introduced.
2 Radiative transfer model
In this work the implementation of a RTM based on the algorithm established by Iqbal [17] is presented.
The flowchart of the model is depicted in Figure 1. This RTM simulates irradiance for cloudless sky and generates direct, diffuse and global spectral irradiance.
In the geometric part of the RTM, the eccentricity of the earth’s orbit is calculated as [15,17] (1)
With the equation (1), the extraterrestrial solar spectral irradiance (I_{0n} Λ) can be correctedfor the specific day. Regarding the atmospheric composition, the relative optical air mass, the relative optical watervapor mass and the relative optical ozone mass are empirically estimated.
Regarding the atmospheric transmittances, the Rayleigh scattering, the diffusion of uniformly mixed gases, watervapor and ozone are estimated also. The aerosol transmittance is retrieved using Ångström’s turbidity formula. All the transmittances are joint in a unique transmittance due to the combined effects of continuum attenuation and molecular absorption (τ(λ)) and used to calculate the spectral direct solar irradiance (I_{d} (λ)) on the horizontal surface with the LambertBeer Law (Eq. (2)) [17] (2)
Finally, the different components for diffuse and global solar irradiance are calculated.
Using the interpolation method, different wavelength step can be used. Some parts of the model were updated using different data and more detailed equations from recent studies [18–20]. The extraterrestrial solar spectral irradiance was modified using the data from Gueymard [18]. The ozone attenuation coefficients are replaced with the crosssection of ozone determined by Bogumil et al. [19]. The equation presented by Frohlich and Shaw for Rayleigh optical depth (ROD) (Eq. (3)) [20], was also used: (3)
where λ is the wavelength in nm. The RTM use 10 inputs to describe the atmospheric state, reported in Table 1.
Some of the considered quantities are difficult to measure or to obtain, however, they can be retrieved using other proprieties. In this work the values for the precipitable water (w) are calculated with the equation of Leckner [17,21] which is based on the local temperature (T) and the relative humidity (RH) (Eq. (4)) (4)
where p_{s} is the partial pressure of water vapor in saturated air that can be calculated with equation (5) (5)
The Ångström’s turbidity coefficient (β) is calculated from Ångström’s Law [4], knowing the Ångström’s exponent (α) and the aerosol optical depth (AOD) at one wavelength (λ_{0}) as follow: (6)
Fig. 1 Simplified flowchart of the radiative transfer model. 
Inputs parameters used by radiative transfer model.
3 Methodology
To study the effect of the input parameters uncertainty on the established model, a set of values for the 10 inputs listed in Table 1 is fixed to their standard values, presented in Table 2. It can be noted that the number of parameters in Table 2 is not the same of Table 1 due to the fact that the computation of precipitable water (w) and Ångström’s turbidity coefficient (β) values need the inclusion of the local temperature (T) and the relative humidity (RH), plus the aerosol optical depth (AOD) and a reference wavelength (λ_{0}) respectively.The application of these fixed values to the model generates a standard spectral irradiance (I_{std} (λ)) starting in 0.325 μm and ending at 1.075 μm with a step of 0.001μm. The global solar irradiance spectrum generated by the RTM with standard values is presented in Figure 2. Following this step, the selected parameters are sequentially modified. Pressure (P), temperature (T), height of ozone layer concentration (z_{3}), ozone concentration (C_{O3}), Ångström’s exponent (α) and aerosol optical depth (AOD) range from 50% to 150% of the standard values with a step of 0.01%. For the remaining parameters, the full variability range has been analyzed. Single scattering albedo (w_{0}) and ground albedo (ρ_{g}) change between 0 and 1, relative humidity (RH) between 0% and 100%, solar zenith angles (θ) range from 0° to 90° and the day of the year (DOY) from 1 to 365. Using these new values, the output of the model (I(λ)) is evaluated n times with n = 1000, generating a new set of solar irradiances. White noise is added in order to produce a more realistic simulation of solar radiation measurement. The used white noise has mean value sets at zero and standard deviation of 1%. The I_{std} (λ) as well as the new obtained spectral irradiance for each input parameter are then integrated in the frequency domain (I_{std} and I_{p} with p the selected input parameter) and the normalized root mean square error (NRMSE) is calculated as shown in equation (7) (7)
The mean and standard deviation values of NRMSE for each parameter is calculated and plotted versus the normalized variation of the parameter. Through the plots, the sensitivity of the model to each parameter can be observed, i.e., if an error on the input parameter occurs, it is reflected by an increase or decrease in the output of the model.
Standard input values.
Fig. 2 Simulated global solar radiation spectrum generated with the radiative transfer model in the standard condition presented in Table 2. The spectrum ranges from 0.325 to 1.075 μm with a step of 0.001 μm spectral resolution. 
4 Results
This section presents the plots of NRMSE together with its standard deviation (σ) for each input parameter. In some of the plot the σ is not clearly visible due to the low values retrieved. The first parameter analyzed is the day of the year (DOY), and the obtained results for the NMRSE are presented in Figure 3. The influence of DOY on the model has been observed and it is possible to appreciate the seasonality of the irradiance. The day chosen to be the standard value for DOY was 182 – July 1 just at thebeginning of summer season. The nearest days shown a small error of about 0.75% and the higher error value of 4.83% was detected in the winter season, as expected. The standard deviation of NRMSE (gray area in plot) increases with distance of the selected standard value.
The next analyzed parameter is the solar zenith angle (θ), which can be seen in Figure 4. This is the parameter that mostly influences the model. When θ is 90° the mean error goes to 659% with a standard deviation of 3.4%. On the other hand, when θ goes to 0° the error is small than the previous and it stays around 22%  with a deviation of 0.18%. Even so this is the highest error caused by an input parameter of the model. The main reason for this error is the influence of solar zenith anglein the estimation of relative optical air mass. The relative optical air mass is estimated with the formulation of Kasten (Eq. (8)) [17] which exhibits a significant error for angles greater than 86°. (8)
The local atmospheric pressure shown in Figure 5, presents a symmetric NRMSE, i.e., nearly the same error on both sides of the standard pressure value regardless of if the variation is toward greater or lower values than the standard one. The error increases until near 7% at the boundary values. It can also be noticed that the standard deviation value increases with the distance to the reference value, from 0.02% to 0.08%.
The NRMSE of the model output caused by the local temperature variation, can be seen in Figure 6. For the highest temperature studied (37.5 °C) the mean error is 2.05%, slightly inferior than the error of 2.23% obtained for the lowest temperature (12.5 °C) applied to the model. The standard deviation of the NRMSE presents a small negative correlation with temperature. The standard deviation of the error is 0.02% at 25 °C.
The next parameter analyzed was the relative humidity, and its full range (0–100%) was studied. The reference value used is 50% the error for dry condition is almost 7% (Fig. 7). For moist condition the mean error is three and half times less, around 2%. NRMSE standard deviation follow the similar shape, higher for dry and smaller for moist condition.
The NRMSE relative to the height of ozone layer concentration shows that the model is almost not influenced by this parameter (Fig. 8). The change in the mean value of the NRMSE is very small. In this case the noise can be more important than the variation of the parameter.
The effect of the Ozone concentration on the model is depicted in Figure 9. In this plot, it can be seen that an error of 50% in the input ozone concentration relatively to the standard value – 300 DU – will produce a NRMSE of 1.3%. The plot shows a shape of the NRMSE very symmetric. This curve denotes that this parameter does not have a “preferential side”, i.e., the NRMSE varies the same way on either side of the standard value. The error in output increases with the error in input as expected. The standard deviation also increases with the input error; its value at standard ozone concentration is 0.02% and increase to 0.03%.
The next parameter to be analyzed is the single scattering albedo. The maximum error is near 10% when the parameter goes to zero. Thestandard deviation follows the same pattern that the mean value of NRMSE, increasing with the rising of the variation, like it can be seen in Figure 10.
The variation of ground albedo is very symmetric and presents a maximum error of 4% to the limits of its range. The standard deviation has the same shape of the mean value. The standard deviation of the error for the standard value of this parameter – 0.5 – is around 0.02% and reaches the 0.05% at the extreme values. The graph of the NRMSE towards the ground albedo can be seen in Figure 11.
Figure 12 presents the NRMSE versus the variation of the Ångström’s exponent. It is visible that, for the Ångström’s exponent of 2.25, NRMSE will be around 3%. On the other side for a variation of 50% of the standard value – 1.5 – the NRMSE mean value is between 2 and 2.5%. This parameter is more responsive to variations to values higher than the standard value of the Ångström’s exponent. Relatively to the standard deviation, itis, also, increasing faster for the higher values of the parameter analyzed and slower towards the smaller value.
Finally,the last parameter considered is the aerosol optical depth at the 0.675 μm. The aerosol optical depth generates similar mean NRMSE for the higher and lower values of the parameter than the standard value – 0.1 (Fig. 13). The maximum value of the mean NRMSE is around 3.6% to the extreme values, and the minimum value – 0.75% – is obtained for the standard value. Standard deviations are very symmetric and present values near 0.05% for the range limits and 0.02% for the standard values.
Fig. 3 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the day of the year. 
Fig. 4 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the solar zenith angle. 
Fig. 5 Mean value (line) and standard deviation (gray area) of NRMSE for the model output. In the lower axis the local atmosphericpressure variation is present in percentage while the upper axis presents the local atmospheric pressure in k Pa. 
Fig. 6 Meanvalue (line) and standard deviation (gray area) of NRMSE for the model output. In the lower axis the local temperature variation is present in percentage while in the upper axis this presents the local temperature in °C. 
Fig. 7 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the variation of the relative humidity. 
Fig. 8 Mean value (line) and standard deviation (gray area) of NRMSE for the model output. The lower axis reports the percentage variation in height of the ozone layer concentration while in the upper axis the height of ozone layer concentration in km is shown. 
Fig. 9 Mean value (line) and standard deviation (gray area) of NRMSE for the model output. The lower axis shows the percentage of the ozone concentration variation while the upper axis reports the Ozone concentration in DU. 
Fig. 10 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the single scattering albedo. 
Fig. 11 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the ground albedo. 
Fig. 12 Mean value (line) and standard deviation (gray area) of NRMSE of the model output. The lower axis shows the percentage of the Ångström’s exponent variation while the upper axis presents the typical Ångström’s exponent values. 
Fig. 13 Mean value (line) and standard deviation (gray area) of NRMSE of the model output. In the lower axis the percentage variation of aerosol optical depth at the 0.675 μm is presented while in the upper axis the aerosol optical depth values at the 0.675 μm are shown. 
5 Conclusion
The analysis performed in this work allows to appreciate the influence of the errors and uncertainties of the input parameters on the output of the implemented radiative transfer model. This study shows that the main parameters influencing the model simulations are the solar zenith angle with an error higher than 650% and the local single scattering albedo with an error near 10%. For the local atmospheric pressure and the relative humidity the mean error is about 7%. Partially, the error caused to the model output by the solar zenith angle can be explained with the use of the Kasten’s formulation for the relative optical air mass and its error for small solar elevation angles. More information can be withdrawal from this study. Using the obtained graphs, it can now be possible to approximate the initial parameters knowing if they should be underestimated or overestimated. This analysis also shows that the height of ozone layer concentration can be used as a constant, since the effects on the model output can be considered negligible.
Author contribution statement
Conceptualization, A.A., D.B., M.T., A.H. and A.J.; Methodology, A.A, M.T. and D.B.; A.A. planned and carried out the simulations; Formal Analysis and interpretation of the results, A.A., D.B.; Writingoriginal draft preparation, A.A.; WritingReview and Editing, D.B., A.A., M.T., A.H. and A.J.; Visualization, A.A., D.B.; Supervision, D.B., M.T.; Project Administration, D.B., M.T.; Funding Acquisition, M.T., D.B.
Acknowledgements
A. Albino gratefully acknowledges the financial support of “Fundação para a Ciência e Tecnologia” (FCT – Portugal), through the Doctoral Grant SFRH/BD/108484/2015. This work is partially supported by the Portuguese Science Foundation (FCT) within the project TOMAQAPA (PTDC/CTAMET/29678/2017). The work is cofunded by the European Union through the European Regional Development Fund, included in the COMPETE 2020 (Operational Program Competitiveness and Internationalization) through the ICT project (UID/GEO/04683/2013) with the reference POCI010145FEDER432007690.
References
 A. Albino, R. Salgado, M. Tlemçani, D. Bortoli, A. Joyce, in the international conference 9th SpanishPortuguese Assembly of Geodesy and Geophysics, Madrid, Spain (2016) [Google Scholar]
 A. Albino, D. Bortoli, M. Tlemcani, A. Joyce, in the international conference on Advanced Materials for Photonics, Sensing and Energy Applications, Agadir, Morrocos (2017) [Google Scholar]
 J. RuizArias, C.A. Gueymard, Solar Energy 171, 474 (2018) [CrossRef] [Google Scholar]
 V. Badescu, C.A. Gueymard, S. Cheval, C. Oprea, M. Baciu, A. Dumitrescu, F. Iacobescu, I. Milos, C. Rada, Renew. Energy 55, 85 (2013) [Google Scholar]
 B. Mayer, A. Kylling, Atmos. Chem. Phys. 5, 1855 (2005) [Google Scholar]
 F.C. Seidel, A.A. Kokhanovsky, M.E. Schaepman, Atmos. Measur. Technol. 3, 1129 (2010) [CrossRef] [Google Scholar]
 G. Hülsen, J. Gröbner, Appl. Opt. 23, 5877 (2007) [Google Scholar]
 O. Dubovik, M.D. King, J. Geophys. Res. 105, 20673 (2000) [Google Scholar]
 R. Román, J. Bilbao, A. de Miguel, J. Geophys. Res.: Atmos. 119, 4690 (2014) [CrossRef] [Google Scholar]
 D.R. Myers, Energy 30, 1517 (2005) [CrossRef] [Google Scholar]
 S. Ding et al., J. Quant. Spectrosc. Radiat. Transf. 112, 1050 (2011) [Google Scholar]
 G. Belluardo, G. Barchi, D. Baumgartner, M. Rennhofer, P. Weihs, D. Moser, Sol. Energy 132, 558 (2016) [Google Scholar]
 R. Román, J. Bilbao, A. de Miguel, J. Quant. Spectr. Radiat. Transfer 145, 95 (2014) [CrossRef] [Google Scholar]
 R. Román, J. Bilbao, A. de Miguel, Atmos. Environ. 89, 556 (2014) [Google Scholar]
 J. Bilbao, R. Román, C. Yousif, D. Mateos, A. de Miguel, Atmos. Environ. 99, 508 (2014) [Google Scholar]
 A. de Miguel, D. Mateos, J. Bilbao, R. Román, Atmos. Res. 102, 136 (2011) [Google Scholar]
 M. Iqbal, An introduction to solar radiation (Academic Press, 1983) [Google Scholar]
 C.A. Gueymard, Solar Energy 169, 434 (2018) [CrossRef] [Google Scholar]
 K. Bogumil et al., J. Photochem. Photobiol. A 157, 167 (2003) [CrossRef] [Google Scholar]
 C. Fröhlich, G.E. Shaw, Appl. Opt. 19, 1773 (1980) [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 B. Leckner, Solar Energy 20, 143 (1978) [CrossRef] [Google Scholar]
Cite this article as: André Albino, Daniele Bortoli, Mouhaydine Tlemçani, Abdeloawahed Hajjaji, and António Joyce, Sensitivity analysis of atmospheric spectral irradiance model, Eur. Phys. J. Appl. Phys. 88, 11001 (2019)
All Tables
All Figures
Fig. 1 Simplified flowchart of the radiative transfer model. 

In the text 
Fig. 2 Simulated global solar radiation spectrum generated with the radiative transfer model in the standard condition presented in Table 2. The spectrum ranges from 0.325 to 1.075 μm with a step of 0.001 μm spectral resolution. 

In the text 
Fig. 3 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the day of the year. 

In the text 
Fig. 4 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the solar zenith angle. 

In the text 
Fig. 5 Mean value (line) and standard deviation (gray area) of NRMSE for the model output. In the lower axis the local atmosphericpressure variation is present in percentage while the upper axis presents the local atmospheric pressure in k Pa. 

In the text 
Fig. 6 Meanvalue (line) and standard deviation (gray area) of NRMSE for the model output. In the lower axis the local temperature variation is present in percentage while in the upper axis this presents the local temperature in °C. 

In the text 
Fig. 7 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the variation of the relative humidity. 

In the text 
Fig. 8 Mean value (line) and standard deviation (gray area) of NRMSE for the model output. The lower axis reports the percentage variation in height of the ozone layer concentration while in the upper axis the height of ozone layer concentration in km is shown. 

In the text 
Fig. 9 Mean value (line) and standard deviation (gray area) of NRMSE for the model output. The lower axis shows the percentage of the ozone concentration variation while the upper axis reports the Ozone concentration in DU. 

In the text 
Fig. 10 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the single scattering albedo. 

In the text 
Fig. 11 Mean value (line) and standard deviation (gray area) of NRMSE for the modeled irradiances obtained modifying the ground albedo. 

In the text 
Fig. 12 Mean value (line) and standard deviation (gray area) of NRMSE of the model output. The lower axis shows the percentage of the Ångström’s exponent variation while the upper axis presents the typical Ångström’s exponent values. 

In the text 
Fig. 13 Mean value (line) and standard deviation (gray area) of NRMSE of the model output. In the lower axis the percentage variation of aerosol optical depth at the 0.675 μm is presented while in the upper axis the aerosol optical depth values at the 0.675 μm are shown. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.