Validation of Spectral and Broadband UV-B (290-325 nm) Irradiance for Canada

Stratospheric ozone depletion, as a result of increasing chlorofluorocarbons in the stratosphere, allows more UV-B irradiance (290 325 nm) to reach the earth’s surface with possible detrimental biological effects. Because there are few UV-B radiation stations, irradiance models are useful tools for estimating irradiances where measurements are not made. Estimates of spectral and broadband irradiances from a numerical model are compared with Brewer spectrophotometer measurements at nine Canadian stations (Alert, Resolute Bay, Churchill, Edmonton, Regina, Winnipeg, Montreal, Halifax and Toronto) and 26 years of data. The model uses either the discrete ordinate radiative transfer (DISORT) or the delta-Eddington algorithms to solve the radiative transfer equation for a 49-layer, vertically inhomogeneous, plane-parallel atmosphere, with cloud inserted between the 2 and 3 km heights. Spectral calculations are made at 1 nm intervals. The model uses extraterrestrial spectral irradiance, spectral optical properties for each atmospheric layer for ozone, air molecules, and aerosol and surface albedo. A fixed broadband cloud optical depth of 27 was satisfactory for calculating cloudy sky irradiances at all stations except in the arctic. Comparisons are made both for daily totals and for monthly averaged spectral and broadband irradiances. The delta-Eddington method is shown to be unsuitable for calculating spectral irradiances under clear skies, at wavelengths less than 305 nm where absorption by ozone is high, and at large solar zenith angles. The errors are smaller for overcast conditions. The method is adequate for daily total and monthly averaged spectral ( 305 nm) and broadband calculations for all sky conditions, although consistently overestimating irradiances. There is a good agreement between broadband measurements and calculations for both daily totals and monthly averages with mean bias error mainly less than 5% of the mean measured daily irradiance and root mean square error less than 25%, decreasing to below 15% for monthly averages.


Introduction
Human made chemicals, including chlorofluorocarbons and other halocarbons, have damaged the stratospheric ozone layer that protects people, plants, and animals from harmful biologically active ultraviolet (UV-B) irradiance.The effective UV-B waveband is from 290 to 325 nm, which is the wavelength range of the Canadian Brewer spectrophotometer measurements.Even though the UV-B band is biologically important, it contains little energy, constituting only 1.8% of the total solar radiation at the top of the atmosphere, and no more than 1% at the earth's surface [1].
Within the UV-B band the atmosphere becomes more transparent with increasing wavelength since ozone absorption decreases by two orders of magnitude as wavelength increases between 290 -325 nm [2].Over this wavelength range the irradiance at the ground may vary through eleven orders of magnitude.Biological effects are not constant across the waveband.In general, the shorter the wavelength is, the greater the biological effect [3].Therefore, spectral measurements are essential for biological applications.
UV-B irradiance measurements are rare in Canada and the world.Radiative transfer models are potentially very important tools to supplement the spatially sparse network.The DISORT and delta-Eddington algorithms have been used widely to model irradiance [4][5][6][7].Delta-Eddington uses a two-term expansion of the scattering phase function but DISORT allows for any number of expansions of the phase function, therefore, it is potentially an exact solution.Comparisons between both methods for model atmospheres for UV transmittance (290 -400 nm) for various amounts of absorption and scattering have been made by Forster and Shine [8].Here, we present the first extensive comparison of the two methods for real atmospheres in the UV-B waveband.
Forster and Shine [8] showed that the delta-Eddington is not suitable for calculating spectral values for clear skies and at large solar zenith angles but for overcast skies it may be suitable.For thick scattering cloud layers, the two-term expansion is sufficient because multiple scattering is dominant and not too sensitive to detailed phase function structure [9,10].Erlick and Frederick [11] compared the delta-Eddington flux calculations with the 22-stream DISORT model for an isolated optically-thick cloud layer ( = 40) at 290 nm with zero surface albedo.They found that transmission and reflection from these two methods were closely matched except for large zenith angles greater than 60˚ where the delta-Eddington transmissivity and reflectivity were too high and too low (by 10%) respectively.Lubin et al. [12] argued that the uncertainties in spectral irradiance calculations using the delta-Eddington approximation instead of DISORT are less than the uncertainties involved in treating clouds as plane parallel layers.
Validation studies that have compared model calculations with measurements are mostly restricted to data for just a few days and cloudless skies [13][14][15].Few studies have validated surface-based models for all sky conditions [6,16,17].This is the first comprehensive study for Canada.A pilot study was performed by Davies et al. [7] at four Canadian stations (Bedford, Toronto, Winnipeg, Edmonton) using a small amount of data.
Comparison between UV-B irradiance calculated by DISORT model and measurements have been presented by Wang and Lenoble [5], Zeng et al. [14] and Pachart et al. [18] for clear sky conditions.Wang and Lenoble [5] concluded that the variation of the ratio between measurement and model spectral results exceeds ±20%, but the agreement is better than ±6% when the ratio is averaged over intervals of 10 nm.Zeng et al. [14] compared measured spectral irradiances with 8-stream DISORT results.They found that UV-B irradiances could be predicted to within 8% if the input parameters were well known.These differences are due to calibration errors either in the instrument or in the extraterrestrial spectral irradiance.
Our study is important because scientists in Canadahave found that an average ozone depletion of about 6% has been observed over five Canadian monitoring stations (Toronto, Goose Bay, Edmonton, Churchill and Resolute Bay) since the late 1970s [19].In Toronto (43˚47'N, 79˚28'W) Kerr and McElroy [20] reported decreases in the ozone levels between 1989 and 1993 of 4.1% and 1.8% per year in winter and summer, respectively.
This paper evaluates a numerical model for UV-B irradiance for all sky conditions, validates spectral and broadband irradiances using Brewer spectrophotometer measurements, and assesses the relative usefulness of the DISORT and delta-Eddington algorithms in calculating spectral and broadband irradiances.
Section 2 and 3 describe the irradiance and ozone measurements.Section 4 introduces the model and the input parameters.Section 5 presents the model validation results.Section 6 gives conclusions, emphasizes the contributions of this research and details some of the future research needs.

The Brewer Measurements
Spectral UV-B irradiance measurements in Canada began in March 1989 and are made at 13 locations with the Canadian designed single monochromatic Brewer spectrophotometer.Nine of these locations, which have the necessary meteorological data for radiative transfer calculation, are used in this study (Figure 1).The Brewer instrument allows the calculation of daily ozone depth and measures spectral irradiance for wavelengths between 290 and 325 nm at a resolution of 0.5 nm.Each spectral measurement consists of the average of a forward and backward scan across the wavelength range, which takes about 8 minutes to complete [20].Measurements of the radiation intensity that falls on a horizontal diffusing surface are made once or twice each hour throughout the day from sunrise to sunset at irregular times in GMT.These spectral measurements were obtained from the World Ozone and Ultraviolet Radiation Data Centre (WOUDC).
The Brewer instruments have known uncertainties.They receive stray light from longer wavelengths adjacent to the one being measured [21,22] which affects measurements below about 305 nm where the light intensity is very small.Also, they are subject to cosine error such that measurements usually underestimate the horizontal global irradiance by up to 8% depending on louds, aerosols, and solar zenith angle [23,24].Each c instrument has its own cosine error, which can vary from 2% to 20% [e.g., 24,25].Calibration uncertainty for the Brewer instruments ranges from ±5 -7% [26,27].The Brewer instrument is also affected by ambient temperature and humidity variations [21].It is provided with a temperature-stabilized enclosure but this does not totally eliminate the temperature variability.The temperature effect is greater at shorter wavelengths and can produce mean errors ranging from -2% to 2% in winter and summer, respectively over the Brewer spectral range [23].However, Cappellani and Kochler [28] have found that for winter days (temperature range 9.8˚ to 21.7˚C) and for summer days (temperature range 21.7˚ to 42˚C), the Brewer values should be increased by 2% and 8%, respectively.Some quality control procedures are performed by the Meteorological Service of Canada (MSC).These include: calibration with 1000-watt standard lamps that are traceable to the US National Institute of Standards and Technology; daily radiometric stability that is maintained with an internal 20-watt quartz halogen lamp; a wavelength check is made several times per day using a mercury discharge lamp; and a correction for stray light [23].However, corrections for the effect of cosine error on the UV-B spectra and a wavelength-dependent temperature effect are not applied.In this study an increase of 6% was applied to the Brewer data to compensate for the cosine error effect on the basis of research by Krotkov et al. [29] and Wang et al. [27].

Other Measurements
Daily total ozone column measurements from the Brewer instrument were obtained from the WOUDC for the stations shown in Figure 1.Hourly (local standard time) measurements of total cloud opacity, surface temperature, pressure and relative humidity were provided by the MSC.Values were linearly interpolated for the irradiance measurement times in GMT.Solar zenith angles for each measurement time and the ratio of actual to mean ,  Sun-Earth distance were calculated following Michalsky [30].Daily snow depth measurements were provided by the MSC.

Davies Model Description
Surface irradiance is expressed as a cloudiness-scaled combination of cloudless sky irradiance and overcast sky irradiance : where is the fraction of the sky that is cloud covered.
o and  are calculated spectrally at 1 nm intervals using either the DISORT [31] or the delta-Eddington [32] solutions to the radiative transfer equation.

C G G
This model can be applied anywhere where there are daily measurements of column ozone and snow depth and hourly cloud cover observations.Radiative transfer calculations of o and require the spectral UV-B irradiance emitted by the sun and the spectral optical properties for each atmospheric layer for ozone absorption, Rayleigh scattering, aerosol extinction, and cloud scattering and surface albedo.

G G 
In this study, the atmosphere is divided into 49 layers with constant scattering and absorbing properties within each.The layers are thin (1 km) in the lower atmosphere, intermediate (2.5 km) in the middle atmosphere and thick (5 km) in the upper atmosphere.Each layer is regarded as horizontally homogeneous and the curvature associated with sphericity of the earth is ignored.Layer values of spectral optical depths, single scattering albedos and asymmetry factors were calculated as layer averages.These spectral optical properties were combined for each wavelength and layer.The cloud is placed in one layer (between 2 and 3 km) and in this plane-parallel atmosphere radiation transfer is considered only in the vertical.In the calculation, cloud optical properties replace optical properties for the cloudless layer between 2 and 3 km.
The model uses solar spectral extraterrestrial irradiances from the Solar Ultraviolet Spectral Irradiance Monitor (SUSIM) instrument on board the third Atmospheric Laboratory for Applications and Science (ATLAS-3) space shuttle mission launched on Nov. 13, 1994 (D. Prinz, personal communication, 2007), ozone absorption coefficients from Paur and Bass [33], Rayleigh scattering cross sections following Elterman [34], aerosol optical properties from Shettle and Fenn [35].Since the Brewer instrument measures irradiance through a triangular filter with a base of 1.1 nm (full width at half maximum of 0.55 nm), the high spectral resolution (full width at half maximum ~ 0.15 nm, sampled approximately every 0.05 nm) SUSIM data were averaged to mimic the Brewer.SUSIM measurements for average Sun -Earth distance were selected from the 289.45 and 326.55 nm wavelength range at a 0.05 nm interval, and averaged for each nanometer from 290 to 325 nm.
Since there are few measured atmospheric vertical profiles of ozone, temperature, pressure and humidity, standard model atmospheres containing these vertical profiles for 50 atmospheric levels from the surface to 120 km in LOWTRAN 7 [36] were used for the model in this study.Summer and winter midlatitude and subarctic model atmospheres were used to calculate Rayleigh and ozone optical depths.Urban aerosol optical properties for 50 km and 36.5 km visibilities were used for the boundary layer for both Toronto and Montreal and a 50 km visibility rural aerosol was used for all other stations.Ozone concentrations were scaled by the ratio of total measured to total model atmospheric ozone depth.
For this study, Broadband values of cloud single scattering albedo  and asymmetry factor c g were set at 0.999997 and 0.8709, respectively, for equivalent radius of 7 µm (for arctic stations) and 0.999995 and 0.8587, respectively, for equivalent radius of 10 µm (for midlatitude and subarctic stations) at all wavelengths [37].Broadband cloud optical depths c  were calculated iteratively from overcast irradiance measurements for snow free conditions to eliminate irradiance increase from multiple scattering between cloud and snow [37].
Surface albedo measurements for the UV-B band are not available in Canada.Albedo was calculated following Davies et al. [7] as a linear function of daily snow depth measurement between 0.05 for a snow free ground [38] and 0.75 for a snow cover of 30 cm or greater.Albedo is independent of wavelength and the effects of melting and snow contamination are ignored.

Validation of Model Irradiances
The section assesses the model's performance in calculating spectral and broadband irradiances using the extraterrestrial solar spectrum, the calculated spectral optical parameters, and the broadband cloud optical depths given in Binyamin et al. [37].Although the results in Binyamin et al. [37] showed that the DISORT 8 and delta-Eddington algorithms yielded very similar cloud optical depths for all stations in the study it is also important to examine how well irradiances from the two methods compare since the delta-Eddington method is an approximate solution of the radiative transfer equation whereas the DISORT 8 method is close to an exact solution.

Performance Measures
Model performance is assessed using the mean bias error Copyright © 2011 SciRes.ACS

Comparisons of Irradiances from the Delta-Eddington and DISORT Methods
(MBE), which measures systematic error, and the root mean square error (RMSE), which includes both systematic and non-systematic error [39].When MBE is small, the RMSE measures mainly the non-systematic error.If i is the difference between calculated and measured irradiances (daily or monthly), MBE and RMSE are defined from the variance of The numerical experiments by Forster and Shine [8] revealed systematic overestimation by the delta-Eddington method.Here, their analysis has been applied to a real atmosphere (June 24, 1993 at Toronto) for both cloudless and overcast skies.
Figure 2 shows ratios of spectral irradiances calculated by both the delta-Eddington and 8-stream DISORT methods to irradiances calculated by a 16-stream DISORT method for a solar zenith angle of 64.4˚, as used by Forster and Shine [8].The DISORT ratio is close to one at all wavelengths in both cloud cases while, the delta-Eddington values decrease rapidly below 302 nm.The delta-Eddington model agrees to within 2% with DISORT for the overcast case at wavelengths greater than 302 nm but the error increases to 7% for the cloudless sky cases, respectively, at 305 nm.Delta-Eddington values also fall off sharply for wavelengths below about 300 nm in both cloudless and overcast cases.
where is the number of data points.The performance measures are expressed as percentages of the mean measured irradiance for the relevant period.

N
The main source of random error stems from the cloud cover data.Since cloud cover is only reported once an hour, cloudiness variations between hours are missed.Linear interpolation of cloud cover for the Brewer instrument's measurement time only improves the validity of cloud estimates if the real variation of cloudiness between hourly observations is linear.Intuitively, errors arising from interpolation are expected to be random although initial errors in observer cloud estimates are probably systematic since observers tend to overestimate cloud cover because the earth curvature leads to an impression of greater cloudiness toward the horizon in non-overcast sky conditions [40].
Figure 3 compares ratios of delta-Eddington to 8stream DISORT spectral irradiances at seven solar zenith angles for the June 24, 1993 atmosphere and simulated cloudless and overcast skies.In the cloudless case, the delta-Eddington method generally overestimates spectral irradiances at wavelengths greater than 305 nm and unerestimates it at wavelengths below 300 nm at solar ze-d nith angles greater than 60˚.In the overcast case, delta-Eddington estimates are closer to DISORT values except at smaller wavelengths (below 302 nm) at larger solar zenith angles (greater than 50˚).Also, the irradiance drop below unity increases for larger solar zenith angles and with overcast.Figures 2 and 3 show that the delta-Eddington method can be expected to overestimate spectral irradiances at most wavelengths in most cases.
At large solar zenith angles and shorter wavelengths (less than 305 nm) where ozone absorption is high, the delta-Eddington method did not perform well because of the truncation of the scattering phase function to two terms.Forster and Shine [8] showed that this also applies to a two stream DISORT.Although the amount of irradiance is very small at these short wavelengths it may nevertheless be important because this is the portion of the spectrum where biological sensitivities are maximum for many processes.Therefore, the 8-stream DISORT method should be used for spectral irradiances at wavelengths below 305 nm.
Figure 4 shows the variation of the ratio of irradiances of the delta-Eddington and 8-stream DISORT methods with ozone amount and sun angle for cloudless and overcast conditions at 295 nm and 305 nm.At 295 nm, the delta-Eddington error increases strongly with ozone amount, especially at larger solar zenith angles (greater than 50˚).Cloud reduces the range of the ratio values and they never exceed unity.At 305 nm, the delta-Eddington error depends only slightly on ozone except at solar zenith angles larger than 70˚.Under overcast, the ratio range is mainly between 1 and 1.05 except at a zenith angle of 84˚ where it is similar to the cloudless ratio at the same angle.Therefore, for wavelength  305 nm, and when considering daily total spectral irradiances, errors in the delta-Eddington approximation are less important.
his is because the times of day with smaller solar zenith T angles contribute most to the total irradiances.Daily values of spectral and broadband irradiances from the delta-Eddington and DISORT models were compared for all sky conditions using annual values of cloud optical depth for each station showed in Table 1 [37].
Seven wavelengths (295, 300, 305, 310, 315, 320 and 325 nm) were selected to demonstrate model spectral performance for Resolute, Churchill, Winnipeg and Toronto for 1993.Table 2 shows MBE and RMSE for 295 nm and 305 nm.Statistics for 300 nm are similar to those for 295nm and statistics for all other wavelengths are similar to those for 305 nm and therefore are not shown.In general, the delta-Eddington irradiances exceed DI-SORT's values by 3 -7% with the exception of Resolute at 295 nm.RMSE values are mainly within 3 -14%.These differences are within the uncertainty of the Brewer instrument (±10%) and are smaller than the differences between irradiances measured with different instruments [41][42][43].
Resolute is an exception.At 295 nm, delta-Eddington underestimates irradiance by 23%.This is attributed to high cloudiness and large solar zenith angles (Figure 3).orster and Shine [8] have shown that the delta-Edding-F  ton method underestimates the multiple scattering of cloud by up to 14%.The underestimation is not apparent at the lower latitude stations where cloudiness is less and sun angles are higher.For broadband irradiances, Forster and Shine [8] showed for a theoretical atmosphere that the average delta-Eddington transmittance exceeds 16 stream DISORT estimates by 5% at a sun angle of 60˚.We confirm this for real atmospheres for 1993 at Resolute, Churchill, Winnipeg and Toronto (Figure 5) using 8-stream DI-SORT.The overestimates are less than 4%.The irradiances represent the wide range of solar zenith angles and sky conditions found in midlatitude, subarctic and arctic stations.

Spectral Results
Comparison statistics of daily spectral irradiances between models and measurements is given in Table 3 for the two wavelengths (295 nm and 305 nm used previously) for one year at nine stations.For wavelengths ≧ 305 nm the MBE for the two methods is mainly within 5% of the mean measured irradiance.This is well within the uncertainty of the Brewer instrument.Biases for delta-Eddington are mainly positive while those for DISORT are mainly negative.This follows from section 5.2 which showed that the delta-Eddington method, generally, produces larger spectral irradiances than DISORT.The better MBE for delta-Eddington (the inferior model) at longer wavelengths may suggest systematic overestimation by Brewer instruments.The comparisons between model estimates and measurements are poorer at wavelengths below 305 nm although DISORT estimates match measurements closer than the delta-Eddington estimates as expected (Table 3).The larger magnitude of the MBE values at 295 nm at most stations for both models may be attributed in part to the difficulty in measuring within this spectral region.In this range, very low light levels and increased stray light scattering increase the instrumental uncertainty (E.Wu of MSC, personal communication, 2007).
Delta-Eddington's rapid decrease in irradiance at 295 nm, shown in section 5.2, is only detectable at the arctic stations as a result of the greater cloudiness and solar zenith angles.At the other stations, except Halifax, delta-Eddington's MBE values are positive.This follows from flux overestimation in cloudless skies that are more ommon than in the arctic (Figure 3).At Halifax, the c  negative MBE for both models suggests a systematic error in the Brewer instrument.RMSE values for wavelengths greater than 300 nm are mainly within 12% to 25%.These decrease with length of averaging period for both models to below 10% for 30-day averaging periods, which is similar to decreases for broadband solar radiation estimates [44].
Mean monthly measured and calculated spectral irradiances for 295 nm and 305 nm are plotted for four stations (Resolute, Churchill, Edmonton and Toronto) in Figure 6.Both model estimates follow measurements well but at 295 nm the delta-Eddington method consistently overestimates irradiances.
Figure 7 shows the annual variation of measured and Figure 8 shows mean monthly measured spectral irradiance and corresponding model values with both linear and logarithmic plots for three months (January, March and June) for Edmonton in 1994, Halifax in 1993 and Toronto in 1993.The linear plot illustrates more clearly the agreement of measured and calculated irradiances at larger wavelengths while the logarithmic plot is better for showing the agreement at smaller wavelengths.Model values follow measurements well except below 300 nm.This may be attributable as stated earlier to the difficulty of measuring such low irradiance levels and to the light leakage problem even though a correction has been applied to irradiances for wavelengths less than 305 nm [23].Model calculations show the same spectral variation as the Brewer measurements at wavelengths greater than 295 -298 nm.The Halifax and Toronto data show evidence of stray light leakage in the corrected Brewer measurements.

Broadband Results
Performance statistics for daily total and monthly averaged broadband irradiances are given in Table 4 for one year at nine stations.In general, both models perform well for broadband calculations with MBE mainly less than 5% and RMSE less than 25%, which is similar to values obtained from comparisons for global irradiance [45] and the preliminary UV-B irradiance study for Canadian stations by Davies et al. [7].This comparison shows that the delta-Eddington algorithm is adequate for estimating surface broadband UV-B irradiance under all sky conditions from mid-latitudes to the arctic.The method is faster computationally than the DISORT algorithm.Three sets of irradiances were calculated and compared to show the sensitivity of the model to c  .
Daily and monthly broadband irradiances from delta-Eddington algorithm were calculated separately using (a) annual c  for each station, (b) one c  for each station, and (c) one c  for all non-arctic stations.Agreement in all three cases is good and the results are very similar.MBE is less than 7% and daily RMSE less than 25%, ecreasing to less than 15% for monthly averages d   (Table 5).
Binyamin et al. [37] showed that c  values for midlatitudes and subarctic showed little variation with latitude.Therefore, values of c  for non-arctic stations were combined to produce a pooled median value of 27.
Table 5 shows that on average the irradiance changes by less than 0.2% when this pooled median value is used.This agreement suggests that in this range of climate c  variation is similar and possibly representative of other midlatitudes and sub-arctic climates.This obviates the need for extensive computation to retrieve c  for each station and year.An example of the daily variation in model performance is shown in Figure 9 for Halifax (1993)(1994)(1995)(1996).Model irradiances follow the variation of measurements well with no indication of seasonal biases, which implies that a constant c  can be used satisfactorily.Figure 9 also shows the model performance of monthly averaged irradiance for Halifax.Most irradiances compare to within 10%.Larger differences (up to 15%) occur in a few summer months but they change in sign from year to ear.Similar results were found by Norsang et al. [46] in y  Lhasa, Tibet for clear sky irradiances.

Comparisons of Two Different Aerosol Loadings
In this section we show the effect of changing boundary layer aerosol from light (50 km visibility) to heavier (36.5 km visibility), which is the average of 50 km and 23 km models for Montreal (1993 and 1994) and Toronto (1993 and 1995).Separate values of c  were calculated for each station and year.The heavier aerosol reduced c  by an average of 15%.Fluxes from the two urban aerosol loadings are compared with measurements at both stations.The heavier aerosol reduces irradiances by about 2% at Montreal and by 2 -7% at Toronto (Table 6).This agrees well with the findings of Chertock et al. [47] and Wang et al. [48] who found that aerosols could reduce daily solar irradiance up to 3 -5%.For Montreal there is better agreement between the light aerosol model results and measurements with MBE less than 2.5% for daily total and monthly average broadband irradiances (Table 6).For Toronto, the heavier aerosol model shows better agreement with MBE less than 2%.

Conclusions
This study evaluated the relative performance of the delta-Eddington and DISORT algorithms within a numerical model for estimating spectral and broadband UV-B irradiances for Canadian conditions and to validate model results with Brewer spectrophotometer measurements.
The most important findings are: • The delta-Eddington method produces daily total spectral irradiances for all sky conditions, which are generally 3 -7% larger than those from the 8-stream DISORT method.The fractional overestimation decreases as wavelength increases.Irradiances are acceptable for wavelengths ≥ 305 nm.This method is unsuitable for wavelengths below 305 nm where ozone absorption is high due to the truncation of the scattering phase function to two terms.At longer wavelengths its performance varies with solar zenith angle and cloudiness.For clear skies, the method always overestimates irradiances at all sun angles with the error increasing as the solar zenith angle increases.For cloudy skies the errors are much smaller.
• The delta-Eddington method performs very well for broadband calculations for both daily total and monthly averaged irradiances.
• Comparison of spectral estimates from both models with measurements indicate uncertainties in the Brewer measurements at wavelengths < 305 nm.
• At wavelengths ≥ 305 nm better agreement with measurements by the delta-Eddington than by DISORT suggests overestimation by the Brewer spectrophotometer.
• Model estimates for broadband irradiances for both daily totals and monthly averages have a MBE less than 5% and RMSE less than 25% deceases to less than 15% for monthly averages.These statistics compare favourably with those obtained for global radiation [45].
• A constant c  value of 27 is adequate for all stations except the arctic.This is important because it suggests that further estimation of c  is not necessary.
• A light boundary layer aerosol model was suitable for Montreal and a heavy aerosol model for Toronto.
This research is the first to provide extensive evaluation for spectral and broadband irradiances for a large data set, which includes midlatitude, subarctic, and arctic stations.The spectral information is important to biologists who can combine it with various an action spectrum to determine potential biological exposure.
This physically-based model can be applied anywhere.Refinements to the extraterrestrial solar spectrum, Rayleigh scattering cross sections and ozone absorption coefficients are unlikely to be large and the model's lin-ear combination of cloudless and overcast components has been shown to work in a wide range of Canadian conditions.The greatest restriction to its use is the availability of cloud cover information.Future applications of the model should use satellite measurements of ozone and cloud.
The Brewer instrument data sets have not been corrected for the cosine error of the diffuser, temperature errors, as well as absolute radiometric calibration errors.In fact, the 6% increase made to the Brewer data in this study was an approximate correction to remove the systematic cosine error but the actual correction should be made depending on the solar zenith angle and sky illumination conditions [23].

Acknowledgments
We thank Dr. Hanna Maoh of University of Windsor for supplying Figure 1.

Figure 1 .
Figure 1.Location of Canadian stations used in the study.

Figure 2 .
Figure 2. Ratio of spectral irradiance calculated by delta-Eddington and 8-stream DISORT methods to that of 16-stream DISORT method for solar zenith angle of 64.4˚ for clear (C = 0) and overcast (C = 1) sky conditions for Toronto on June 24, 1993 with 302 DU total ozone column and a surface albedo of 0.05.

Figure 3 .
Figure 3. Ratio of spectral irradiance calculated by delta-Eddington and 8-stream DISORT method for solar zenith angle of 64.4˚ for clear (C = 0) and overcast (C = 1) sky conditions for Toronto on June 24, 1993 with 302 DU total ozone column and a surface albedo of 0.05.

Figure 4 .
Figure 4. Ratio of irradiances calculated by delta-Eddington method to 8-stream DISORT method as a function of total column ozone and solar zenith angle for a wavelength of 295 nm(a) and 305 nm (b) for Toronto on June 24, 1993 for cloudless and overcast sky conditions with a surface albedo of 0.05.

Figure 5 .
Figure 5.Comparison of 8-stream DISORT and delta-Eddington daily totals (white circles) and monthly averages (black circles) broadband irradiances using annual values of cloud optical depth for each station (Table 1) at Resolute Bay, Churchill, Winnipeg and Toronto.The dotted lines represent linear regressions constrained to pass through the origin.

Figure 6 .
Figure 6.Mean monthly measured (solid lines) and calculated by delta-Eddington (dotted lines) methods spectral irradiance at 295, 305 nm for Toronto in 1993, Edmonton in 1994 and Resolute Bay in 1995.

Figure 7 .
Figure 7. Mean monthly measured (solid lines) spectral and calculated by the delta-Eddington (dotted lines) and DISORT (dash lines) models for various wavelengths for Toronto 1993.Table gives relative MBE and RMSE values with positive MBE indicating model overestimation.isthe mean monthly measured irradiance.M modeled spectral irradiances for 295 nm and 305 nm for Toronto 1993.The table below Figure7indicates larger MBE and RMSE at 295 nm.Both models perform well at wavelengths 305 nm with greatly reduced MBE and RMSE.Figure8shows mean monthly measured spectral irradiance and corresponding model values with both linear and logarithmic plots for three months (January, March and June) for Edmonton in 1994, Halifax in 1993 and Toronto in 1993.The linear plot illustrates more clearly the agreement of measured and calculated irradiances at larger wavelengths while the logarithmic plot is better for showing the agreement at smaller wavelengths.Model values follow measurements well except below 300 nm.This may be attributable as stated earlier to the difficulty of measuring such low irradiance levels and to the light leakage problem even though a correction has been applied to irradiances for wavelengths less than 305 nm[23].Model calculations show the same spectral variation as the Brewer measurements at wavelengths greater than 295 -298 nm.The Halifax and Toronto data show evidence of stray light leakage in the corrected Brewer measurements.

Figure 8 .
Figure 8. Mean monthly measured (solid lines) and calculated by delta-Eddington (Black circles, triangles and squares) and 8-stream DISORT (white circles, triangles and squares) spectral irradiance on a logarithmic (upper lines, left axis) and linear (lower lines, right axis) scale for January (circles), March (triangles) and June (squares) for Edmonton in 1994, Halifax in 1993 and Toronto in 1993.Table gives N which is the number of days used for each month.

Figure 9 .
Figure 9. (a) Daily total measured (solid lines) and calculated (dotted lines) broadband irradiances for Halifax (1993-1996) and (b) Monthly average broadband irradiances measured (line and white circles) and calculated by delta-Eddington model (dotted and black circles) for Halifax for years 1993-1996.