1. Introduction

In the last years, the UV Index (UVI) has been introduced as a useful vehicle to inform the public about the potentially harmful effects of the UV irradiation and it is directly calculated using erythemal ultraviolet (EUV) [26,30]. Long-term ground-based datasets have been utilized for detecting trends in EUV and for establishing its climatology [13], and references therein). In addition, radiative transfer (RT) and empirical methods are useful tools for estimating EUV where there are no measurements available.

RT models describe the physical interaction between UV irradiation and the atmosphere. These models offer better accuracy in calculated UV irradiances than empirical models, but they need information about meteorological variables which are not commonly measured in ground-based stations. In contrast, empirical models are formulated by simple expressions with the available meteorological variables in each location as independent variables. This type of estimation methods have been broadly used for different goals such as: UVI forecast [4,17,29] estimation of total ozone column [5,27] and reconstruction of UV irradiation series [6, 9, 15, 21].

The independent variables introduced in the empirical model have to characterize the modulation of UV radiation through the atmosphere. In this direction, the most important factors influencing UV radiation reaching ground level are solar elevation, total ozone column, clouds and aerosols. Due to the complexity of the interaction between UV irradiation and the atmosphere, a great number of UV empirical models focus on the relationship between UV and a single factor such as: global solar radiation [7, 8, 10, 12, 23, 31], ozone [32, 33] and the optical air mass [11,18,20,22]. In contrast, few studies analyze the joint relationship between the UV irradiation and its main attenuation factors [4,14,16,17,19].

The objective of this work is proposing empirical models for all-sky conditions in order to estimate the UVI at some Upper Egypt regions, using five independent variables which characterize the main atmospheric attenuation factors of the UV radiation. In addition, an empirical model for calculating UVI over all Upper Egypt regions will be presented.

The studied locations in this paper are described in Sect. 2. Section 3 describes the satellite data followed by methodology and the proposal of the empirical model. Section 4 presents and discusses the results obtained in this work and, finally, Sect. 5 summarizes the main conclusions.

2. Description of The Studied Locations

Upper Egypt is known as southern Egypt because it is upstream, closer to the source of the Nile, extends from the cataract boundaries of Aswan to the area south of Cairo. This work divided Upper Egypt to three geographical regions as western desert region, river Nile region and red sea region. The coordinate of the locations used for this study is presented in table (1) farther more figure (1).

Western desert region, the New Valley Governorate, is one of the governorates in Egypt that covers about 376 km² in the Western Desert. In our study we select the most famous oases. Farafra is the second biggest depression by size and the smallest by population. El-Kharga is the largest of the oases in the Libyan Desert. It is located about 200 km to the west of the Nile valley, and is some 150 km long. It is the capital of New Valley Governorate.

River Nile region, is including six governorates. In our study we selected four locations, as well as famous city Abu-Simbel. The locations from north to south are: El-Minya, which is located approximately 245 km south of Cairo on the western bank of the Nile River. Asyut, which is sandwiched between two mountain ranges of about 600m. Qena is a city in central Egypt, with an area of 1,800 km², situated on the east bank of the Nile in the southern part of Egypt. Aswan is stands on the east bank of the Nile at the first cataract and is a busy market and tourist centre. Abu-Simbel is a small town located on the Western Bank of Lake Nasser, a city in Nubia, about 290 km southwest of Aswan and only 50 km away from the southern borders with Sudan.

In general, these cities lie within the subtropical region and its terrain is semi-desert. The climate is characterized by cold winter, and very hot but non-humid summer, a hot season from March to October and a cold season from November to February. During summertime, temperatures could reach 40^oC and increasing southward, while winter sees temperatures drop to sub-zero levels at night. While hail or snow is extremely rare due to low precipitation averages, frost will occasionally form on cold winter nights. In addition, a phenomenon of the hot spring wind that blows across the all Egypt. The winds, known to Europeans as the Sirocco and to Egyptians as the Khamsin, usually arrive in April.

The Red Sea is one of the coastal governorates that locate between the Nile and the Red Sea in the southeast of the country and its southern border forms part of Egypt's border with Sudan. In our study we select the most famous city. Hurghada is second largest city (after Suez) located on the Red Sea coast. Qusair is located 205 km south of Hurghada, 103 km north of Marsa Alam. Shalatein is the southern most village on Egypt's Red Sea Coast, marks the administrative boundary between Egypt and Sudan. It is located 240 km south of Marsa Alam.

The Red Sea governorate has a subtropical-desert climate, with mild-warm winters and hot to very hot summers. Temperatures in the period December, January and February is moderate warm, while November, March and April comfortable warm. May and October is hot and the period from June to September is very hot. Sunshine hours are average 9 hours of day in December to average 13 hours in July. Average annual temperature of sea is 24°C, from 21°C in February and March to 28°C in August.

Table 1. The coordinate of the studied locations in Upper Egypt Locations Lat. N Long. E Western desert region Farafra 27°00' 28°01' El-Kharga 25°10' 30°35' Red sea region Hurghada 27°15' 33°50' Quseir 26°07' 34°16' Shalatein 23°05' 35°25' River Nile region El-Minya 28°07' 30°33' Asyut 27°11' 31°04' Qena 26°10' 32°43' Aswan 24°04' 32°57' Abu Simbel 21°18' 34°40'

Figure 1. Map of Egypt, illustrate the studied locations (gray area)

3. Description of Data

EUV at noon in mW/m² , total ozone column (TOC) in DU, reflectivity (reflc) in % and aerosol index (AI) from Total Ozone Mapping Spectrometer (TOMS) remote sensing instrument were used, (http://toms.gsfc.nasa.gov). The TOMS remote sensor has been operative on board two satellites: Nimbus-7 (1978–1993) [1] and Earth Probe (EP) (1996–2000) [2]. This satellite instrument measures the backward-scattered Earth solar radiance. The TOMS' data have a daily global coverage over 1° x 1.25° (latitude by longitude) grids. The TOMS’ instruments provide one measurement per day near local noon. For more detailed descriptions of the different sources of uncertainty the reader is referred to [24, 26].

3.1. Erythemal Ultraviolet (EUV)

The erythemal exposure data product is an estimate of the daily integrated UV irradiance, calculated using a model for the susceptibility of Caucasian skin to sunburn (erythema) [3].The Erythemal exposure is defined by the integral

(1)

where d_es is the Earth–sun distance, in A.U.; S is the solar irradiance incident on the top of the atmosphere at 1 A.U. (nWm⁻² nm⁻¹); W is the biological action spectrum for erythemal damage, (see below eq. 2); t_sr and t_ss are the time of sunrise and sunset, in radians; C is the cloud attenuation factor, unitless; τ_cl is the cloud optical thickness, in mbar; Z is the solar zenith angle (function of time, t), in radians; and F is the spectral irradiance at the surface under clear skies, normalized to unit solar spectral irradiance at the top of the atmosphere, unitless.

The weighting function used to approximate the wavelength-dependent sensitivity of Caucasian skin to erythema-causing radiation is the model proposed by McKinlay and Diffey [34], and adopted as a standard by CIE (Commission Internationale de l’clairage, International Commission on Illumination). This model is given by the equations wavelengths λ in nm):

(2)

The UV Index is calculated using the erythemal (CIE) action spectrum. It is non-dimensional, obtained by dividing the EUV by 25 mWm^-2 [15].

3.2. Total Ozone Column (TOC)

A radiative transfer model is used to calculate back scattered radiance as a function of total ozone, latitude, viewing geometry, and reflecting surface conditions. Ozone can then be derived by comparing measured radiance with theoretical radiance calculated for the conditions of the measurement and finding the value of ozone that gives a computed radiance equal to the measured radiance [24, 26].

3.3. Reflectivity (reflc)

Reflectivity is determined from the measurements at 360 nm. For a given TOMS measurement, the first step is to determine calculated radiance at 360 nm for reflection off the ground and reflection from cloud, based on the tables of calculated 360-nm radiance. For reflection from the ground, the terrain height pressure is used, and the reflectivity is assumed to be 0.08. For cloud radiance, a pressure corresponding to the cloud height from the International Satellite Cloud Climatology Project (ISCCP) based climatology is used, and the reflectivity is assumed to be 0.80 [24, 26]. The ground radiance I_ground and cloud radiance I_cloud are then compared with the measured radiance I_measured. If I_ground ≤ I_measured ≤ I_cloud, and snow/ice is assumed not to be present, an effective cloud fraction f is derived using

(3)

3.4. Aerosol Index (AI)

The TOMS technique of aerosol detection and characterization is based on spectral contrast in the UV that results from the interaction between the processes of Rayleigh scattering, particle scattering and absorption. This interaction produces spectral variations of the back scattered radiance at the top of the atmosphere that can be used to separate aerosol absorption from scattering effects discuss in detail the physical basis of the near UV technique of aerosol sensing [25]. The TOMS aerosol index is a measure of how much the wavelength (λ) dependence of back scattered UV radiation from an atmosphere containing aerosols (Mie scattering, Rayleigh scattering, and absorption) differs from that of a pure molecular atmosphere (pure Rayleigh scattering). AI for TOMS is defined as:

(4)

Where subscript meas indicates the measured back scattered radiance at the fixed wavelength and subscript calc indicates calculated using RT model describing a pure Rayleigh atmosphere.

4. Results and Discussion

4.1. Empirical UVI_sat Models in Some Upper Egypt Locations

Empirical models to estimate the ultraviolet index (UVI) for all sky conditions in the studied locations in Upper Egypt have been developed. These models propose a multiple linear regression with the UVI as a dependent variable, and the declination (δ), cosine solar zenith angle at noon (cosSZA_n), total ozone column (TOC) in DU, reflectivity (reflc) in % and aerosol index (AI) as independent variables. A dataset corresponding to the period (November 1978- December 1999) was used to develop the models. The general form of the suggested models is:

(5)

the coefficients e_o, e₁, e₂, e₃, e₄ and e₅ were determined using least mean square method.

4.1.1. Models Constructions and Performance

Models coefficients and statistical analysis for Eq. (5) for different locations are presented in Tables (2). in view of the high values of the R² and F-statistics compared to its critical values, all these models can compute the daily UVI value at noon with a good accuracy. Also the t-ratio for each coefficient has values greater than zero, reflecting the significance of their great contribution and high certainty to the fitting process. It is obvious that there is general similarly of the deduced equations, for all locations, with respect to the parameter's values and signs.

Table 2. Models coefficients and their statistical analysis for Eq. (5) in some Upper Egypt locations Location coefficients Stdev t-ratio F R2 Farafra e0 -0.693 0.169 -4.100 70720.96 98.5 e1 0.004 0.001 2.570 e2 23.704 0.180 131.940 e3 -0.033 0.000 -102.310 e4 -0.073 0.001 -90.640 e5 -0.536 0.008 -63.890 El-Kharga e0 -2.485 0.160 -15.490 73279.02 98.6 e1 -0.032 0.001 -25.920 e2 27.877 0.164 169.660 e3 -0.037 0.000 -102.150 e4 -0.082 0.001 -83.950 e5 -0.587 0.008 -70.710 Hurghada e0 -1.199 0.165 -7.270 73647.8 98.6 e1 0.000 0.001 -0.200 e2 24.913 0.178 140.320 e3 -0.035 0.000 -106.890 e4 -0.075 0.001 -93.330 e5 -0.554 0.010 -57.480 Quseir e0 -1.729 0.155 -11.190 80491.84 98.7 e1 -0.014 0.001 -11.720 e2 26.206 0.163 161.250 e3 -0.036 0.000 -109.620 e4 -0.076 0.001 -91.020 e5 -0.554 0.009 -63.630 Shalatein e0 -3.060 0.188 -16.290 50945.34 98.0 e1 -0.060 0.001 -41.940 e2 31.234 0.187 167.450 e3 -0.043 0.000 -93.840 e4 -0.103 0.001 -102.560 e5 -0.638 0.009 -67.590 El-Minya e0 0.089 0.170 0.520 69124.8 98.5 e1 0.019 0.001 13.310 e2 21.939 0.185 118.37 e3 -0.031 0.000 -103.56 e4 -0.072 0.001 -89.620 e5 -0.520 0.009 -56.610 Asyut e0 -0.220 0.163 -1.350 75044.6 98.6 e1 0.006 0.001 4.530 e2 23.683 0.174 135.80 e3 -0.034 0.000 -107.57 e4 -0.074 0.001 -83.120 e5 -0.547 0.009 -60.810 Qena e0 -0.916 0.147 -6.240 90702.4 98.9 e1 -0.009 0.001 -7.440 e2 25.371 0.153 165.66 e3 -0.036 0.000 -115.35 e4 -0.075 0.001 -90.38 e5 -0.572 0.008 -68.440 Aswan e0 -2.854 0.159 -17.990 72263.8 98.6 e1 -0.047 0.001 -38.890 e2 29.514 0.160 184.85 e3 -0.040 0.000 -105.24 e4 -0.089 0.001 -89.450 e5 -0.607 0.008 -75.750 Abu-Simbel e0 -3.325 0.192 -17.330 40270.4 97.5 e1 -0.083 0.001 -57.390 e2 32.667 0.185 176.68 e3 -0.046 0.001 -90.960 e4 -0.095 0.001 -79.730 e5 -0.638 0.009 -67.590

4.1.2. Models Validation

To verify whether the models, allow a reliable estimate of UVI, Mean Bais Deviation (MBD) in % of the estimated values (UVI_est) from the measured ones (UVI_sat) for a new period (year 2000) are computed, ideally a zero of MBD should be obtain[35]. MBD is an indication of the average deviation of the estimated value from the measured value, if n is the number of observations, MBD is defined by:

(6)

The results were given in figures (2, 3, 4). Most of the points in this figures lie in the range of MBD less than ±10 % while little ones lie in the range from ±10% to ±20%, indicating the accuracy of the calculation in the most days of the year in all studied locations. These results support the accuracy of the calculation. Also, It can be observed from these figures that there is very strong correlation between the estimated (UVI_est) and measured (UVI_sat) values , ranged from (0.97) to (0.99).

Statistical treatment, aims to support the validity of the used model in calculating UVI in new period (year 2000), has been done by calculating the following parameters: the mean bias error (MBE) and the root mean square error (RMSE), which measure systematic and nonsystematic error, respectively. Since the (MBE) cancel significant positive and negative biases, the mean absolute error (MAE), modeling efficiency (ME), modeling index (d), and t-statistics. The results are summarized in Table (3). From this table the following points are summarized:

1. The RMSE values show that the simulations overestimate the measurements by a value vary from 0.282 to 0.303, from 0.310 to 0.393 and from 0.306 to 0.373 at western desert region, river Nile region and red sea region respectively.

2. The MAE values show that the simulations overestimate the measurements by a value vary from 0.214 to 0.221, from 0.221 to 0.256 and from 0.217 to 0.274 at western desert region, river Nile region and red sea region respectively.

3. The models achieved an acceptable level of accuracy; since the MBE vary from -0.017 to 0.023, from -0.090 to -0.021 and from -0.010 to 0.057 at western desert region, river Nile region and red sea region respectively.

4. The models achieved better performance; since the smaller value of t-statistic, ME and d values were closer to 1 in all studied locations.

Table 3. Statistical errors of estimated UVIest Location Statistical errors RMSE MBE MAE ME d t-Stat. Farafra 0.303 0.023 0.221 0.99 0.997 1.278 El-Kharga 0.282 -0.017 0.214 0.97 0.997 0.974 Hurghada 0.353 -0.010 0.258 0.98 0.996 0.451 Quseir 0.306 -0.026 0.217 0.99 0.996 1.422 Shalatein 0.373 -0.057 0.274 0.97 0.993 2.550 El-Minya 0.321 -0.090 0.239 0.99 0.997 4.897 Asyut 0.393 -0.032 0.232 0.98 0.995 1.356 Qena 0.310 -0.021 0.221 0.99 0.996 1.145 Aswan 0.343 -0.032 0.243 0.98 0.995 1.553 Abu-Simbel 0.330 -0.023 0.256 0.97 0.993 1.171

Figure 2. Comparison between the values of UVIest and UVIsat in the left as well as MBD (%) in the right (west desert region)

Figure 3. Comparison between the values of UVIest and UVIsat in the left as well as MBD (%) in the right (red sea region)

Figure 4. Comparison between the values of UVIest and UVIsat in the left as well as MBD (%) in the right (river Nile region)

4.2. General UVI_sat Empirical Model in Upper Egypt

An empirical model to estimate UVI for all sky conditions in Upper Egypt has been developed for the same period. The model takes the form:

(7)

4.2.1. Model Performance

Statistical analysis of the formula (5) has been studied, summerized in table (4). In view of the high values of the R2 ( 96.3 %) and F-statistics (289380) compared to its critical value (54917), formula (7) for Upper Egypt can compute the daily UVI_Egy at noon with good accuracy. Also the t-ratio for each coefficient has values greater than zero, reflecting the significance of their great contribution and high certainty to the fitting process.

Table 4. Empirical equation's coefficients and their statistical analysis for Eq. (5) in Upper Egypt coefficients Stdev t-ratio F R2 e0 0.082 0.08 1.025 289380 96.30 e1 -0.022 0.001 -33.9 e2 27.235 0.085 321.177 e3 -0.045 0 -281.838 e4 -0.083 0 -193.793 e5 -0.518 0.004 -124.628

4.2.2. Model Validation

To verify whether the obtained relationship (7), allows a reliable estimate of UVI_Egy, Mean Bais Deviation (MBD) In % of its estimate values (UVI_Egy) from the measured ones (UVI_sat) for a new period (year 2000) are computed, according to Eq (5), The results were given in figure (5), the most points in this figure lie in the range of MBD less than ±10 % while little ones lie in the range from ±10% to ±20%, indicating the accuracy of the calculation in the most days of the year in all studied locations. These results support the accuracy of the calculation. Also, it can be observed from this figure that there is a very strong correlation between the estimated (UVI_Egy) and measured (UVI_sat) values of ultraviolet index (0.99).

From this figure, it was found agreement between the UVI_Egy value and the observation one UVI_sat, these agreements were checked statistically. Statistical treatment with the aim of supporting the validity of the used model in calculating UVI in the new period has been done by calculating the statistical errors. The RMSE value (0.4919) and MAE value (0.3683) show that the simulations overestimate the measurements. The model achieved an acceptable level of accuracy; since the MBE (0.0525).The model achieved better performance; since the smaller values of t-statistic (6.657), ME (0.964) and d (0.99) were close to 1 in all studied locations.

5. Conclusions

The objective of this work is to propose an empirical model for all sky conditions in order to estimate the UVI_sat at some Upper Egypt regions. In addition, analyzes the use of the empirical model for reporting UVI over Upper Egypt. This model proposes a multiple linear regression with the UVI_sat as a dependent variable, and the declination (δ), cosine solar zenith angle at noon (cosSZA_n), total ozone column (TOC) in DU, reflectivity (reflc) in % and aerosol index (AI) as independent variables which characterize the main atmospheric attenuation factors of the UV radiation. A dataset corresponding to the period (1978-1999) was used to develop the model and an independent dataset (year 2000) was used for validation purposes.

Statistical treatment with the aim of supporting the validity of the used model in calculating UVI in the new period has been done by calculating the following parameters: the mean bias error (MBE) and the root mean square error (RMSE), which measure systematic and nonsystematic error, respectively. Since the (MBE) cancel significant positive and negative biases, the mean absolute error (MAE), modeling efficiency (ME), modeling index (d), and t-statistics. The following points are summarized:

The RMSE values show that the simulations overestimate the measurements by a value vary from 0.282 to 0.303, from 0.310 to 0.395 and from 0.306 to 0.373 at western desert region, river Nile region and red sea region respectively.

The MAE values show that the simulations overestimate the measurements by a value vary from 0.214 to 0.233, from 0.221 to 0.256 and from 0.217 to 0.274 at western desert region, river Nile region and red sea region respectively.

The model achieved an acceptable level of accuracy; since the MBE vary from -0.017 to 0.058, from -0.090 to -0.021 and from -0.010 to 0.048 at western desert region, river Nile region and red sea region respectively.

The model achieved better performance; since the smaller value of t-statistic, ME and d values were closer to 1 in all studied locations.

For Upper Egypt, it was found agreement between the UVI_Egy value and the observation one UVI_sat, these agreements were checked statistically. The RMSE value (0.4919) and MAE value (0.3683) shows that the simulations overestimate the measurements. The model achieved an acceptable level of accuracy; since the MBE (0.0525).The model achieved better performance; since the smaller value of t-statistic (6.657), ME (0.964) and d (0.99) values were closer to 1 in all studied locations.

Figure 5. Comparison between the values of UVIEgy and UVIsat in the left as well as MBD (%) in the right (west desert region)

Figure 6. Comparison between the values of UVIEgy and UVIsat in the left as well as MBD (%) in the right (red sea region)

Figure 7. Comparison between the values of UVIEgy and UVIsat in the left as well as MBD (%) in the right (river Nile region)

Empirical models as well as one single model to estimate the ultraviolet index (UVI) for all sky conditions in ten locations in Upper Egypt have been developed. Multiple linear regression technique has been used for constructing the models. The inputs of the models are the declination (δ), cosine solar zenith angle at noon (cosSZAn), total ozone column (TOC) in DU, reflectivity (reflc) in % and aerosol index (AI). Datasets corresponding to the period (1978-1999) were used to develop the models and an independent dataset (year 2000) was used for validation purposes. The data set for each location, including erythemal ultraviolet (EUV) at noon in mW/m2 , (TOC) in DU, reflectivity (reflc) in % and (AI), was retrieved from Total Ozone Mapping Spectrometer (TOMS) remote sensing instrument.

It was found agreement between the UVIEgy value and the observation one UVIsat, these agreements were checked statistically. The RMSE (0.4919) and MAE (0.3683) show that the simulations overestimate the measurements. The model achieved an acceptable level of accuracy; since the MBE (0.0525). The model achieved better performance; since the smaller value of t-statistic (6.657), ME (0.964) and d (0.99) values were closer to 1 in all studied locations.