Atmospheric and Climate Sciences, 2011, 1, 69-85
doi:10.4236/acs.2011.13008 Published Online July 2011 (
Copyright © 2011 SciRes. ACS
Validation of Spectral and Broadband UV-B (290 - 325 nm)
Irradiance for Canada
Jacqueline Binyamin1, John Davies2, Bruce McArthur3
1University of Winnipeg, Department of Geography, Winnipeg, Canada
2McMaster University, School of Geog raphy and Earth Sciences, Hamilton, Canada
3Meteorological Service of Canada, Air Quality Research Branch, Toronto, Canada
Received March 20, 2011; revised April 27, 2011; accepted May 10, 2011
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. Be-
cause 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 mole-
cules, and aerosol and surface albedo. A fixed broadband cloud optical depth of 27 was satisfactory for cal-
culating 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 er-
rors are smaller for overcast conditions. The method is adequate for daily total and monthly averaged spec-
tral ( 305 nm) and broadband calculations for all sky conditions, although consistently overestimating ir-
radiances. 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.
Keywords: Modelling UV-B Radiation, DISORT, Delta-Eddington, Spectral and Broadband Radiation,
Brewer Spectrophotometer, UV-B Measurements in Canada
1. 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 ab-
sorption decreases by two orders of magnitude as wave-
length 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 net-
work. The DISORT and delta-Eddington algorithms have
been used widely to model irradiance [4-7]. Delta-Ed-
dington uses a two-term expansion of the scattering pha-
se function but DISORT allows for any number of ex-
pansions 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 pre-
sent 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 ze-
nith 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 calcula-
tions with measurements are mostly restricted to data for
just a few days and cloudless skies [13-15]. Few studies
have validated surface-based models for all sky condi-
tions [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 meas-
urement and model spectral results exceeds ±20%, but
the agreement is better than ±6% when the ratio is aver-
aged 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 pre-
dicted 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
Our study is important because scientists in Canada-
have found that an average ozone depletion of about 6%
has been observed over five Canadian monitoring sta-
tions (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 de-
creases in the ozone levels between 1989 and 1993 of
4.1% and 1.8% per year in winter and summer, respec-
This paper evaluates a numerical model for UV-B ir-
radiance 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 con-
tributions of this research and details some of the future
research needs.
2. The Brewer Measurements
Spectral UV-B irradiance measurements in Canada be-
gan in March 1989 and are made at 13 locations with the
Canadian designed single monochromatic Brewer spec-
trophotometer. Nine of these locations, which have the
necessary meteorological data for radiative transfer cal-
culation, are used in this study (Figure 1). The Brewer
instrument allows the calculation of daily ozone depth
and measures spectral irradiance for wavelengths be-
tween 290 and 325 nm at a resolution of 0.5 nm. Each
spectral measurement consists of the average of a for-
ward and backward scan across the wavelength range,
which takes about 8 minutes to complete [20]. Measure-
ments 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 ob-
tained from the World Ozone and Ultraviolet Radiation
Data Centre (WOUDC).
The Brewer instruments have known uncertainties.
They receive stray light from longer wavelengths adja-
cent to the one being measured [21,22] which affects
measurements below about 305 nm where the light in-
tensity 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
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Figure 1. Location of Canadian stations used in the study.
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-stabi-
lized 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, respec-
tively 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 val-
ues 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 trace-
able to the US National Institute of Standards and Tech-
nology; 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 dis-
charge lamp; and a correction for stray light [23]. How-
ever, 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 ap-
plied 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].
3. Other Measurements
Daily total ozone column measurements from the Brewer
instrument were obtained from the WOUDC for the sta-
tions 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 irradi-
ance 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.
4. Davies Model Description
Surface irradiance is expressed as a cloudiness- scal-
ed combination of cloudless sky irradiance and
overcast sky irradiance :
 (1)
where is the fraction of the sky that is cloud covered.
oand are calculated spectrally at 1 nm intervals
using either the DISORT [31] or the delta-Eddington [32]
solutions to the radiative transfer equation.
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 absorp-
tion, Rayleigh scattering, aerosol extinction, and cloud
scattering and surface albedo.
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 associ-
ated 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 atmos-
phere 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 irradian-
ces from the Solar Ultraviolet Spectral Irradiance Moni-
tor (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 bound-
ary 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 scat-
tering albedo
and asymmetry factor c
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 midlati-
tude and subarctic stations) at all wavelengths [37].
Broadband cloud optical depths c
were calculated ite-
ratively 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 follow-
ing 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. Al-
bedo is independent of wavelength and the effects of
melting and snow contamination are ignored.
5. Validation of Model Irradiances
The section assesses the model’s performance in calcu-
lating spectral and broadband irradiances using the ex-
traterrestrial solar spectrum, the calculated spectral opti-
cal 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 im-
portant 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 solu-
5.1. Performance Measures
Model performance is assessed using the mean bias error
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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.
5.2. 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 system-
atic 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 de-
fined from the variance of
The numerical experiments by Forster and Shine [8] re-
vealed 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.
 
, (1) Figure 2 shows ratios of spectral irradiances calcu-
lated 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 cloud-
less 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 perform-
ance measures are expressed as percentages of the mean
measured irradiance for the relevant period.
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 in-
strument’s measurement time only improves the validity
of cloud estimates if the real variation of cloudiness be-
tween 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 im-
pression of greater cloudiness toward the horizon in
non-overcast sky conditions [40].
Figure 3 compares ratios of delta-Eddington to 8-
stream 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 un-
erestimates it at wavelengths below 300 nm at solar ze- d
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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.
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-Ed-
dington 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 ir-
radiance 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 wave-
lengths 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 ze-
nith angles larger than 70˚. Under overcast, the ratio ran-
ge 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
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Figure 4. Ratio of irradiances calculated by delta-Eddington method to 8-stream DISORT method as a function of total col-
umn 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.
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
Seven wavelengths (295, 300, 305, 310, 315, 320 and
325 nm) were selected to demonstrate model spectral
performance for Resolute, Churchill, Winnipeg and To-
ronto 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 dif-
ferences between irradiances measured with different
instruments [41-43].
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
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Table 1. Characteristics of the inferred cloud optical depth for the nine Canadian datasets. N is the number of data points.
Station Year Aerosol type Mean and Median cloud optical depth
Mean Median
Mean Median N
Alert (NWT) 1995 Rural + 50 km 13.6 9.0 271
(NWT) 1993 Rural + 50 km 22.9 16.7 22.3 15.9 294
1994 22.9 16.9 271
1995 36.8 22.9 36.3 22.4 313
1996 25.2 16.6 283
Station 27.2 18.0 1161
Churchill (Man.) 1993 Rural + 50 km 39.8 27.5 39.7 26.7 170
1994 78.1 34.2 216
1995 48.3 26.6 216
1996 37.9 23.4 454
Station 48.6 27.1 1056
Edmonton (Alta.) 1993 Rural + 50 km 34.7 24.8 112
1994 40.4 30.6 127
1995 51.0 33.5 149
1996 33.9 24.8 211
Station 39.7 27.9 599
Regina (Sask.) 1994 Rural + 50 km 40.7 25.4 246
1995 54.7 26.4 113
Station 45.1 25.9 359
Winnipeg (Man.) 1993 Rural + 50 km 37.4 26.9 37.6 27.4 405
Montreal (Que.) 1993 Urban + 50 km 53.0 34.7 215
1994 49.5 28.9 239
Station 51.1 29.6 454
1993 Urban + 36.5 km 43.5 29.3
1994 41.7 24.6
Halifax (NS) 1993 Rural + 50km 48.9 31.7 516
1994 39.3
27.6 531
1995 40.5 27.4 609
1996 34.2 25.6 613
Station 40.5 27.8 2269
Toronto (Ont.) 1993 Urban + 50 km 43.9 26.8 43.7 26.7 405
1994 44.7 31.0 765
1995 53.0 29.7 583
1996 53.7 38.4 880
Station 49.4 32.6 2633
1993 Urban + 36.5 km 37.89 22.8 590
1995 47.1 25.2 705
Table 2. Comparison of daily spectral irradiances from the delta-Eddington and 8-stream DISORT methods for the period
indicated for each station. is the number of data points and
NM is the mean annual daily spectral irra diance calculated
by DISORT (J·m–2·day–1·nm–1). Values of MBE and RMSE are given as percentages of M. Positive MBE values indicate
delta-Eddington overestimation.
Statistics Resolute Bay 1993 Churchill 1993 Winnipeg 1993 Toronto 1993
295 nm
N 89 136 136 192
0.07 0.46 0.83 1.37
MBE –23.00 6.92 5.72 6.61
RMSE 26.44 13.94 9.00 10.25
305 nm
N 141 178 228 250
180.45 347.49 370.41 474.93
MBE 7.40 3.71 3.15 3.31
RMSE 11.11 3.26 3.32 4.41
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, Chur-
chill, Winnipeg and Toronto. The dotted lines repres e n t linear regressions constrained to pass through the origin.
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 del-
ta-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 irradi-
ances represent the wide range of solar zenith angles and
sky conditions found in midlatitude, subarctic and arctic
5.3. Comparisons of Model Calculations with
5.3.1. Spec tra l Results
Comparison statistics of daily spectral irradiances be-
tween models and measurements is given in Table 3 for
the two wavelengths (295 nm and 305 nm used previ-
ously) 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 del-
ta-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, pro-
duces larger spectral irradiances than DISORT. The bet-
ter MBE for delta-Eddington (the inferior model) at
longer wavelengths may suggest systematic overestima-
tion by Brewer instruments.
The comparisons between model estimates and meas-
urements are poorer at wavelengths below 305 nm al-
though 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 ze-
nith 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
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Table 3. Comparison of daily spectral irradiances from the delta-Eddington (DE) and 8-stream DISORT (D8) methods
against measurements for the period indicated for each station. is the number of data points and NM is the mean an-
nual measured daily spectral irradiance (J·m–2·day–1·nm–1). Values of MBE and RMSE are given as percentages of M Posi-
tive MBE alues indicate model overest ima tion.
Statistics Alert 1995 Resolute 1995 Churchill 1993Edmonton
1994 Regina 1994Winnipeg
1993 Montreal 1994 Halifax 1993 Toronto 1993
295 nm
N 12 80 136 135 204 136 154 206 192
0.02 0.05 0.46 0.62 0.84 0.84 0.72 0.80 1.25
MBE (DE) –39.93 –3.54 5.25 71.62 1.99 10.50 43.94 –12.31 20.22
MBE (D8) 33.46 24.46 –1.21 51.60 –9.06 –0.42 27.90 –19.67 9.41
RMSE (DE) 85.71 88.09 51.14 87.31 49.19 32.28 70.22 45.18 38.84
RMSE (D8) 64.40 90.81 51.47 75.81 45.56 34.21 51.34 51.89 32.08
305 nm
N 31 154 178 274 211 228 212 231 250
121.09 155.68 355.85 340.43 497.54 405.60 475.38 425.44 494.14
MBE (DE) –1.88 14.37 5.72 5.07 5.23 –1.26 0.28 –6.15 2.85
MBE (D8) –11.58 7.85 –2.35 –2.15 –1.73 –6.68 –6.39 –10.68 –3.89
RMSE (DE) 26.69 23.50 25.11 25.08 20.13 14.18 12.82 13.76 11.83
RMSE (D8) 29.50 25.30 23.28 20.18 17.77 18.40 13.14 17.55 12.15
Figure 6. Mean monthly measured (solid lines) and calculated by delta-Eddington (dotted lines) methods s pectral irradiance
at 295, 305 nm for Toronto in 1993, Edmonton in 1994 and Resolute Bay in 1995.
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 ir-
radiances 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 consis-
tently overestimates irradiances.
Figure 7 shows the annual variation of measured and
Copyright © 2011 SciRes. ACS
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. is the mean monthly measured irradiance.
modeled spectral irradiances for 295 nm and 305 nm for
Toronto 1993. The table below Figure 7 indicates larger
MBE and RMSE at 295 nm. Both models perform well
at wavelengths 305 nm with greatly reduced MBE and
Figure 8 shows mean monthly measured spectral ir-
radiance and corresponding model values with both lin-
ear 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 irradi-
ances 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.
5.3.2 Broadband Results
Performance statistics for daily total and monthly aver-
aged 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 Ca-
nadian 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 algo-
rithm. 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
Copyright © 2011 SciRes. ACS
Figure 8. Mean monthly measured (solid lines) and calculated by delta-Eddington (Black circles, triangles and squares) and
8-stream DISORT (white circles, tr iangles and squares) spectral irr adiance 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.
Table 4. Summary of delta-Eddington (DE) and 8-stream DISORT (D8) performance measures against measurements for
daily total and monthly averaged br oadband irradiances for the period indicated for each station. is the number of data
points and N
M is the mean annual measured daily total irradiance (KJ·m–2·day–1). Values of MBE and RMSE are given as-
percentages (italic) of M. Positive MBE values indicate model overestimation.
Statistics Alert
Daily total
N 31 154 178 274 211 228 212 231 250
38.26 51.28 60.32 49.66 63.23 54.14 62.66 54.80 60.15
MBE (DE) –0.45 3.13 –0.47 5.55 6.30 3.73 –1.19 –1.10 1.92
MBE (D8) –7.59 –0.98 –5.95 –0.50 0.57 –1.52 –7.07 –5.75 –2.96
RMSE (DE) 23.24 11.37 18.09 19.21 16.58 13.34 14.33 17.76 14.78
RMSE (D8) 24.67 11.30 18.78 15.88 14.25 12.60 15.18 18.94 14.61
Monthly average
N 4 8 10 12 10 12 12 12 12
38.27 39.76 49.09 46.33 55.02 47.79 50.17 52.18 55.81
MBE (DE) –2.18 2.80 –1.85 5.10 5.30 3.25 –1.50 –1.20 1.77
MBE (D8) –9.02 –1.51 –7.31 0.91 –0.49 –2.00 –7.53 –5.88 –3.21
RMSE (DE) 10.64 8.13 14.15 10.41 9.37 5.34 6.79 9.28 4.88
RMSE (D8) 15.14 6.55 15.23 7.49 5.86 3.79 10.00 11.59 5.35
Figure 9. (a) Daily total measured (solid lines) and calculated (dotted lines) broadband irradiances for Halifax (1993-1996)
and (b) Monthly average broadband ir radiances measured (line and white circles) and c alculated by delta-Eddington model
(dotted and black circles) for Halifax for years 1993-1996.
(Table 5).
Binyamin et al. [37] showed that c
values for mid-
latitudes and subarctic showed little variation with lati-
tude. 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 perform-
ance is shown in Figure 9 for Halifax (1993-1996).
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
Copyright © 2011 SciRes. ACS
Copyright © 2011 SciRes. ACS
Table 5. Summary of delta-Eddington performance measures for daily total and monthly average irradiances using annual
c, station c and pooled c
τ. is the number of data points and
τ τNM is the mean daily measured broadband irradiance
(kJ·m–2·day–1). Values of MBE and RMSE are given as percentages of M. Positive MBE values indicate model overestima-
Statistics Alert Resolute Churchill Edmonton ReginaWinnipeg Montreal Halifax Toronto
Annual c
Daily total
N 31 574 603 947 373 228 288 855 833
38.26 47.09 58.51 51.47 58.79 54.14 66.12 57.57 66.05
MBE –6.24 –8.35 –1.26 –0.88 –0.46 –3.37 –6.88 –8.15 –7.57
RMSE 25.74 20.73 22.19 19.39 17.01 15.29 16.94 19.34 19.76
Monthly average
RMSE 11.94 14.47 13.8 7.15 4.46 5.98 7.81 10.44 9.27
Station c
Daily total
MBE –8.04 –0.91 –0.69 0.26 –6.35 –8.02 –8.33
RMSE 20.89 21.55 19.42 16.6 16.63 19.19 20.13
Monthly average
RMSE 14.56 12.99 7.53 4.28 7.16 10.27 9.57
Pooled c
Daily total
MBE –1.18 –0.47 –1.42 –3.69 –5.31 –7.9 –5.81
RMSE 21.65 19.31 18.03 15.54 15.06 19.10 18.37
Monthly average
RMSE 13.01 7.49 4.52 6.31 6.24 10.15 7.24
Table 6. Comparisons between measured and calculated (delta-Eddington) daily total and monthly average broadband ir-
radiances for two aerosols loading. is the number of data points and NM is the mean measured irradiance (kJ·m–2·day–1).
Values of MBE and RMSE are given as percentages of M Positive MBE values indicate model overestimation.
Statistics Montreal 1993 Montreal 1994 Toronto 1993 Toronto 1995
Light aerosol
Daily total
N 76 212 250 153
75.72 62.92 60.34 75.20
%MBE –0.63 –1.20 1.94 6.28
%RMSE 14.58 14.31 14.76 18.48
Monthly average
N 8 12 12 10
58.15 53.78 59.53 58.16
%MBE 1.35 –2.36 1.92 5.47
%RMSE 7.49 6.00 4.77 9.61
Heavy aerosol
Daily total
%MBE –2.34 –3.20 0.08 –0.86
%RMSE 14.72 13.99 14.28 15.29
Monthly average
%MBE –0.70 –4.45 –0.01 –1.70
%RMSE 7.30 6.93 4.31 5.61
Lhasa, Tibet for clear sky irradiances.
5.4. 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
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%.
6. Conclusions
This study evaluated the relative performance of the
delta-Eddington and DISORT algorithms within a nu-
merical model for estimating spectral and broadband
UV-B irradiances for Canadian conditions and to vali-
date model results with Brewer spectrophotometer meas-
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 de-
creases as wavelength increases. Irradiances are accept-
able for wavelengths 305 nm. This method is unsuit-
able for wavelengths below 305 nm where ozone absorp-
tion is high due to the truncation of the scattering phase
function to two terms. At longer wavelengths its per-
formance varies with solar zenith angle and cloudiness.
For clear skies, the method always overestimates irradi-
ances 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 spectrophotome-
· 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 favoura-
bly with those obtained for global radiation [45].
· A constant c
value of 27 is adequate for all sta-
tions except the arctic. This is important because it sug-
gests 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 evalua-
tion for spectral and broadband irradiances for a large
data set, which includes midlatitude, subarctic, and arctic
stations. The spectral information is important to biolo-
gists 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 avail-
ability 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 cor-
rected 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 sys-
tematic cosine error but the actual correction should be
made depending on the solar zenith angle and sky illu-
mination conditions [23].
7. Acknowledgments
We thank Dr. Hanna Maoh of University of Windsor for
supplying Figure 1.
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