An Empirical Model to Estimate UV index in Some Upper Egypt Regions

Empirical models to estimate the u ltraviolet index (UVI) for all sky conditions in ten locations in Upper Egypt have been developed. Multiple linear regression technique has been used for linking each o f the UVI as a dependent variable, and all of the declination (δ), cosine solar zenith angle at noon (cosSZAn), total ozone column (TOC) in DU, reflectivity (reflc) in % and aerosol index (AI) as independent variables. A dataset corresponding to the period (1978-1999) was used to develop the models and an independent dataset (year 2000) was used for validation purposes. The data set for each location, including erythemal u ltraviolet (EUV) at noon in mW/m , (TOC) in DU, reflectiv ity (reflc) in % and (AI), was retrieved from Total Ozone Mapping Spectrometer (TOMS) remote sensing instrument. For all locations, the empirical models exp lain more than 93% of UVI variab ility due to changes in all independent variables. In addit ion, one single model for all the locations has been constructed. Model performance was determined for each location by calcu lating the mean bias error (MBE %) and the root mean square error (RMSE %). In addition, mean absolute error (MAE %), modelling efficiency (ME), modelling index (d ), and t-statistics have been computed. The empirical model provides reliable fo recast UVI in order to inform the public about the possible harmfu l effects of UV rad iation over-exposure.


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 ult raviolet (EUV) [26,30]. Long-term g round-based datasets have been utilized for detecting trends in EUV and for establishing its climatology [13], and references therein). In addition, radiat ive transfer (RT) and emp irical methods are useful tools for estimating EUV where there are no measurements available.
RT models describe the physical interaction between UV irradiat ion and the atmosphere. These models offer better accuracy in calculated UV irradiances than empirical models, but they need informat ion about meteorological variables which are not commonly measured in ground-based stations. In contrast , emp irical models are fo rmu lated by s imp le expressions with the available meteorological variab les in each location as ind ep end ent variab les . Th is typ e o f estimation methods have been broadly used for d ifferent goals such as: UVI fo recast [4,17,29] estimation o f 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 o zone colu mn, clouds and aerosols. Due to the complexity of the interaction between UV irradiation and the atmosphere, a great number of UV emp irical 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 relat ionship 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 so me Upper Egypt regions, using five independent variables which characterize the main at mospheric 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.

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 reg ions 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 mo re 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 o f 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 su mmer, a hot season from March to October and a cold season from November to February. During summertime, temperatures could reach 40 o C and increasing southward, while winter sees temperatures drop to sub-zero levels at night. While hail or snow is extremely rare due to lo w p recipitation 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 sout h 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 comfo rtable warm. May and October is hot and the period fro m 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, fro m 21°C in February and March to 28°C in August.

Description of Data
EUV at noon in mW/ m 2 , total o zone colu mn (TOC) in DU, reflectivity (reflc) in % and aerosol index (AI) fro m 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]. Th is satellite instrument measures the backward-scattered Earth solar rad iance. The TOMS' data have a daily global coverage over 1° x 1.25° (lat itude 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].

Erythemal Ultravi olet (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 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 act ion 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, unit less; 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 irrad iance at the surface under clear skies, normalized to unit solar spectral irrad iance 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 (Co mmission Internationale de l'clairage, International Co mmission on Illu mination). This model is given by the equations wavelengths λ in nm): The UV Index is calculated using the erythemal (CIE) action spectrum. It is non-dimensional, obtained by dividing the EUV by 25 mW m -2 [15].

Total Ozone Column (TOC)
A radiat ive transfer model is used to calcu late back scattered radiance as a function of total ozone, latitude, viewing geometry, and reflecting surface conditions. Ozone can then be derived by co mparing measured rad iance with theoretical radiance calculated for the conditions of the measurement and finding the value of o zone that gives a computed radiance equal to the measured radiance [24,26].

Reflecti vi ty (reflc)
Reflectiv ity is determined fro m the measurements at 360 nm. For a g iven TOM S measurement, the first step is to determine calculated rad iance at 360 n m for reflection off the ground and reflection fro m cloud, based on the tables of calculated 360-n m rad iance. For reflection fro m 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 fro m the International Satellite Cloud Climatology Project (ISCCP) based climatology is used, and the reflectiv ity 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 ground cloud ground measured

Aerosol Index (AI)
The TOMS technique of aerosol detection and characterizat ion 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 variat ions 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 TOM S aerosol index is a measure of how much the wavelength (λ) dependence of back scattered UV radiation fro m an atmosphere containing aerosols (Mie scattering, Rayleigh scattering, and absorption) differs from that of a pure molecular at mosphere (pure Rayleigh scattering). AI for TOMS is defined as: 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.

Empirical UVI sat Models in Some Upper Eg ypt Locations
Emp irical 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 multip le linear regression with the UVI as a dependent variable, and the declination (δ), cosine solar zenith angle at noon (cosSZA n ), total ozone co lu mn (TOC) in DU, reflect ivity (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: the coefficients e o , e 1 , e 2 , e 3 , e 4 and e 5 were determined using least mean square method.

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 R 2 and F-statistics compared to its critical values, all these models can co mpute 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 g reat 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.

Models Validation
To verify whether the models, allow a reliable estimate of UVI, Mean Bais Dev iation (M BD) in % of the estimated values (UVI est ) fro m the measured ones (UVI sat ) for a new period (year 2000) are co mputed, ideally a zero of M BD should be obtain [35]. M BD is an indicat ion of the average deviation of the estimated value fro m the measured value, if n is the number of observations, MBD is defined by: 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 fro m ±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 fro m (0.97) to (0.99).
Statistical treat ment, aims to support the validity of the used model in calcu lating UVI in new period (year 2000), has been done by calculating the following parameters: the mean bias error (M BE) and the root mean square error (RM SE), wh ich measure systematic and nonsystematic error, respectively. Since the (M BE) cancel significant positive and negative biases, the mean absolute error (MA E), modeling efficiency (M E), modeling index (d), and t-statistics. The results are summarized in Table (3). Fro m this table the following points are summarized : 1.
The RMSE values show that the simulat ions overestimate the measurements by a value vary from 0.282 to 0.303, fro m 0.310 to 0.393 and fro m 0.306 to 0.373 at western desert region, river Nile region and red sea region respectively.
2. The MA E values show that the simulat ions overestimate the measurements by a value vary from 0.214 to 0.221, fro m 0.221 to 0.256 and fro m 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 fro m -0.017 to 0.023, fro m -0.090 to -0.021 and fro m -0.010 to 0.057 at western desert region, river Nile reg ion and red sea region respectively. 4. The models achieved better performance; since the smaller value of t-statistic, M E and d values were closer to 1 in all studied locations.

General UVI sat Empirical Model in Upper Eg ypt
An empirical model to estimate UVI fo r all sky conditions in Upper Egypt has been developed for the same period. The model takes the form:

Abu-Simbel
Statistical analysis of the formu la (5) has been studied, summerized in table (4). In view of the high values of the R2 ( 96.3 %) and F-statistics (289380) co mpared to its critical value (54917), formu la (7) fo r 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 g reat contribution and high certainty to the fitting process.

Model Validation
To verify whether the obtained relationship (7), allo ws a reliable estimate of UVI Egy , Mean Bais Deviation (MBD) In % of its estimate values (UVI Egy ) fro m the measured ones (UVI sat ) for a new period (year 2000) are co mputed, 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 fro m ±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 correlat ion between the estimated (UVI Egy ) and measured (UVI sat ) values of ultraviolet index (0.99).
Fro m this figure, it was found agreement between the UVI Egy value and the observation one UVI sat , these agreements were checked statistically. Statistical treat ment with the aim of supporting the valid ity of the used model in calculating UVI in the new period has been done by calculating the statistical erro rs. The RMSE value (0.4919) and MAE value (0.3683) show that the simu lations 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), M E (0.964) and d (0.99) were close to 1 in all studied locations.

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 mult iple linear reg ression with the UVI sat as a dependent variable, and the declination (δ), cosine solar zen ith angle at noon (cosSZA n ), total ozone column (TOC) in DU, reflectiv ity (reflc) in % and aerosol index (AI) as independent variables wh ich characterize the main at mospheric attenuation factors of the UV radiat ion. A dataset corresponding to the period  was used to develop the model and an independent dataset (year 2000) was used for validation purposes.
Statistical t reatment with the aim of supporting the validity of the used model in calculat ing UVI in the new period has been done by calculating the following parameters: the mean bias error (MBE) and the root mean square error (RM SE), wh ich measure systematic and nonsystematic error, respectively. Since the (M BE) cancel significant positive and negative biases, the mean absolute error (MA E), modeling efficiency (ME), modeling index (d), and t-statistics. The following points are summarized: The RMSE values show that the simu lations overestimate the measurements by a value vary fro m 0.282 to 0.303, fro m 0.310 to 0.395 and fro m 0.306 to 0.373 at western desert region, river Nile region and red sea region respectively.
The MAE values show that the simu lations overestimate the measurements by a value vary fro m 0.214 to 0.233, fro m 0.221 to 0.256 and fro m 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 fro m -0.017 to 0.058, fro m -0.090 to -0.021 and fro m -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 MA E value (0.3683) shows that the simu lations overestimate the measurements. The model achieved an acceptable level of accuracy; since the MBE (0.0525).The model ach ieved better performance; since the smaller value of t-statistic (6.657), M E (0.964) and d (0.99) values were closer to 1 in all studied locations.

MBD (%)
Abu-Simbel column (TOC) in DU, reflectiv ity (reflc) in % and aerosol index (AI). Datasets corresponding to the period  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 ultravio let (EUV) at noon in mW/ m2 , (TOC) in DU, reflect ivity (reflc) in % and (AI), was retrieved fro m Total Ozone Mapping Spectro meter (TOM S) remote sensing instrument. It was found agreement between the UVI Egy value and the observation one UVIsat, these agreements were checked statistically. The RMSE (0.4919) and MAE (0.3683) show that the simulat ions overestimate the measurements. The model ach ieved an acceptable level of accuracy; since the MBE (0.0525). The model achieved better performance; since the smaller value of t-statistic (6.657), M E (0.964) and d (0.99) values were closer to 1 in all studied locations.