ATMOSPHERIC SUN PROTECTION FACTOR ON CLEAR DAYS: ITS OBSERVED DEPENDENCE ON SOLAR ZENITH ANGLE AND ITS RELEVANCE TO THE SHADOW RULE FOR SUN PROTECTION

Global irradiances measured in seven 5‐nm bands of UV‐B at Rockville, MD (39.1°N, 77.1oW) on 28 clear days near the summer solstice are convoluted with the erythemal action spectrum of human skin to determine dose rates at various hours of the day. These rates are averaged with respect to solar zenith angle to obtain the diurnal variation of mean dose rate and of the Sun Protection Factor (SPF) of the atmosphere (reciprocal of the normalized atmospheric transmissivity) on a typical clear summer day in Rockville. At a 45o zenith angle the atmospheric SPF is computed to be 2.7 and increases rapidly to greater than 7 at 60o.


INTRODUCTION
The intensity of solar ultraviolet radiation reaching the surface of the earth depends on the sun's UV at the top of the atmosphere, the amount of absorbers, reflectors and scatterers in the atmosphere and on the solar zenith angle. To estimate the time and space distributions of biological effects by radiation in the UV-B band (290-325 nm)? it is necessary to determine the relationships between the irradiance of the effective UV received at the surface of the earth and such factors as the solar zenith angle and the total thickness of atmospheric ozone. This study focuses on the diurnal variation of UV radiation that causes erythema (reddening) of human skin. This type of skin trauma (sunburn) is thought to be related to more serious diseases such as skin aging and skin cancer. The relationships between UV-B dose rates and ozone thickness and cloudiness will be examined in a subsequent paper.
The relative biological effects of UV radiation of different wavelengths varies according to a so-called action spectrum. The convolution of the action spectrum for erythema of human skin and the solar irradiance gives an effective dose rate of the UV radiation for causing sunburn. This dose rate can then be related to the corresponding solar zenith *Retired; present address is 10500 Rockville Pike, Apt. M-10, Rockville angle or the time of day on a given date and location. Furthermore, the dose rates can be integrated over time to give total doses during specific periods. Several researchers have computed effective UV dose rates using theoretical models of atmospheric transmissivity of UV radiation and have used these results to compute doses for different latitudes and at various times of day and year (Henriksen et al., 1990;Dahlback et al., 1989). Scotto et al. (1988) used 12 years (1974-85) of UV-B observations at eight US cities obtained by Robertson-Berger meters which have sensors weighted only very approximately by the erythemal action spectrum for human skin. UV-B doses were computed from these data for periods ranging from a half hour to months and years at all stations.
Recently the Solar Radiation Laboratory of the Smithsonian Environmental Research Center (SERC) at Edgewater, MD (1989MD ( ) published 13 years (1975MD ( to 1988 of preliminary UV-B irradiance observations at 1 h time resolution in eight 5-nm bands measured at Rockville, MD (available on 13 IBM PC compatible floppy disks). These data offer an excellent opportunity to compute UV-B dose rates from observations for any hour throughout the year.

MATERIALS AND METHODS
SERC observed global (direct and diffuse) UV-B irradiance on a horizontal surface in eight 5-nm bands centered approximately on wavelengths 290,295,300,305, 310, 315, 320 and 325 nm at Rockville (39.1°N, 77.1"W).
The observations were collected with high precision scanning radiometers (Goldberg, 1982(Goldberg, , 1986. The data recw n:ta The convolution of the erythema1 action spectrum and UV-B irradiances is performed by a discrete-valued numerical integration illustrated in Table 1. The Commission Internationale de I'Eclairage (CIE) relative action spectrum for erythema of human skin is used in this study (McKinlay and Diffey, 1987). This spectrum is defined by analytic formulas and thus can be evaluated at any desired wavelength. The values used here are computed for nominal wavelengths that are evenly spaced every 5 nm. The observed SERC irradiances are recorded at unevenlyspaced wavelengths that are the means of the half-power points of the individual band filters. These UV-B data must, therefore, be interpolated to the nearby nominal wavelengths before the convolution is performed. Because of the great variation with wavelength of UV-B irradiance at the earth's surface, this interpolation is done logarithmically .
The individual products in each discrete-valued convolution are then summed to obtain an estimate of the total erythemally effective UV-B dose rate for the time and date of the observation in W/m2 on a horizontal surface expressed as equivalent effective irradiance at 297 nm, the wavelength at which the erythemal sensitivity is at a maximum. At the small solar zenith angle (17") used in the example in Table 1 these products peak at 305 nm and fall off by an order of magnitude at either end of the UV-B band. The peak shifts toward longer wavelengths as zenith angles increase. This convolution method has been used by others (Luther, 1985, Parrish et al., 1978. (See Discussion for comments on possible errors of this method.) The analysis of the dose rates obtained by the above procedure is carried out in two steps. First, the mean variation of the UV-B dose rate as a function of the solar zenith angle is determined. Secondly, these dose rates are integrated with respect to time to obtain UV-B doses over chosen periods.
Because cloudiness greatly complicates the analysis of UV-B data, this study is confined to clear days. Perfectly cloud-free days in Rockville are rare, and compromises, therefore, had to be made in selecting clear days to be analyzed. Weather records (NOAA, 1984(NOAA, -1988 at Washington National Airport (about 25 km south of Rockville) provide daily sunshine and cloudiness data for the candidate days. Furthermore, the smoothness of the time variation of observed dose rates gives clues about the clarity In this paper, "summer" is defined as the 3.5month period centered on the June solstice (May-mid August).
The dose rates computed by the convolutions are interpolated to zenith angles at every 5 degrees from 0" to 60" for each of the 28 clear summer days. (Values near the zenith, of course, are extrapolations because the zenith distance of the sun is never less than 15.6" at Rockville). The interpolated dose rates for these 28 days are then averaged with respect to solar zenith angle to obtain the mean diurnal distribution of values for a typical clear summer day at Rockville. The means of these dose rates are listed in Table 3 opposite the corresponding zenith angle.
The solar zenith angles for the hourly SERC observations used in this study are 1 h averages computed from a solar ephemeris derived from the "low precision" (0.01") formulas in the Astronomical Almanac (US Naval Observatory, 1990). The time of the observation is taken to be the middle of the 1 h UV-B collection interval.
For the purpose of integrating dose rates to obtain UV-B exposures for various periods of the day in the preparation of Fig. 2, the independent variable is time in hours before or after local noon rather than zenith angle because the latter is not a linear function of time, especially near local noon (time when sun is due south). Since it is preferable to use days during which the solar declination  Fig. 2(a). This period is hereafter called "mid summer'' in this paper. The bar graphs of mean dose rates in Fig. 2 are based on means of observations symmetric with respect to standard time noon. For example, the dose rate for hour 1.5 is the mean of all observations during both the 10 a.m.-11 a.m. and the 1 p.m.-2 p.m. periods. (The average dose rate bars fall exactly on the half hours because differences between local noon and standard time noon cancel out in the averaging of morning and afternoon observations.) The bar graphs in Fig. 2 are typical of the variation of dose rate against hour either before or after local noon for any location near the latitude of Rockville during the specified season.
The asymmetry of the times of the SERC observations  with respect to local noon provides greater zenith angle resolution for the derivation of the mean dose rates in Table 3, whereas the averaging of data about standard time noon in Fig. 2 sacrifices some of this time resolution for the sake of simplicity.

RESULTS
The mean summer erythemally effective UV-B dose rates at Rockville obtained by the above methods are then normalized by the estimated mean rate at the zenith (0.233 W/m*) to obtain mean UV-B transmissivities relative to a datum of one for an overhead sun. The mean normalized transmissivities are shown in Fig. 1 and listed in Table 3 at 5" intervals of solar zenith angle. Table 3 also lists the zenith-angle-dependent mean dose rates and reciprocals of the mean normalized transmissivities, which I shall call the mean normalized sun protection factors (SPF) of the atmosphere relative to one for an overhead sun. The variation of SPF with zenith angle is also shown in Fig. 1.
This SPF, similar to the numerical rating of a sunscreen lotion, gives the relative protection from UV-B afforded by the atmosphere for various solar zenith angles. In other words, the value of the SPF specifies the magnitude of the attenuation factor for effective UV-B in relation to its overhead intensity.
The value of 2.7 in Table 3 for the mean normalized SPF at a zenith angle of 45 degrees is good justification for the rule of thumb for sun protection that the author proposed in a popular science article (Holloway, 1987) and in The Lancet (Holloway, 1990): "When your shadow is shorter than you are tall, the sun is much more likely to bum you than at other times." Shadows on a horizontal surface outdoors in the sun are shorter than the objects casting them when the solar zenith angle is less than 45 degrees. At those times the mean SPF at Rockville on clear summer days is smaller than 2.7. This mean SPF increases to greater than seven at a 60" zenith angle. On the solstice the solar zenith angle at Rockville is 15.6" at local noon, correspond- ing to a mean SPF of only 1.1. Therefore, the rate of effective erythemal UV-B exposure is significantly less at zenith angles greater than 45" than at midday in summer. The shadow rule is generally more accurate than clock rules for predicting the sunburning potential of solar UV-B. Rules based on time of day, such as the warning to avoid the midday sun between 10 a.m. and 2 p.m., are poor delimiters of periods having the smallest zenith distances associated with highest UV-B dose rates because of seasonal changes, broad time zones, daylight saving time and the great variation of latitude and longitude among different locations.

TIME BEFORE OR AFTER LOCAL NOON (hrs)
The effective UV-B erythemal dose during any period on an average clear day in Rockville in summer can be estimated by a numerical time integration of observed dose rates obtained in this study. Figure 2(a) shows a bar graph of the mean dose rates of 14 mid summer days for each hour (before or after local noon) on the half hour up to 5.5 h. Beyond 6 h from noon the sun's elevation above the horizon is less than 15" and the dose rate is considered to be negligible.
Since the ordinate of the graphs in Fig. 2 is dose rate in Wlmz and the abscissa is time, any area on this diagram represents dose in J/m2. For example, the darkly shaded square in the lower left-hand corner is 90 J/m2. This figure is the product of 0.05 W/m2 and 1800, the number of seconds in a half hour. The total dose in J/m2 from local noon to sunset can be estimated as 90 times the number of squares covered by the bars. The total number of squares covered by the bars in Fig. 2(a) is 26.6. This implies an afternoon dose of about 2400 Jlm2 or a daily dose of 4800 J/m2 on a typical mid summer clear day in Rockville. For argument, arbitrarily assume that an average minimum erythemal dose (MED) for fair-skinned individuals is 360 J/m2 based on the range of 20-50 mJ/cm2 (20G500 JimZ) determined by Pathak et al. (1978) to cause a barely perceptible reddening in skin types 1-111. Therefore, one square on Fig.  2 is one quarter MED by the above definition. Thus the 26.6 squares under the bars in Fig. 2(a) represents 6.7 MED for the mean mid summer afternoon dose. (The mean full-day dose is 13.4 MED.) A truncated vertical line is drawn on this graph at 3.31 h after local noon to delineate the mean mid summer time when the sun has a zenith angle of 45" at 39"N. The area under the bars t o the right of this 45" line gives the mean UV-B dose accumulated in the afternoon from the 45" sun to sunset. This area is about 4.2 squares or equivalent to a little more than one MED as defined above.
Thus near the solstice at Rockville the UV-B dose accumulated between local noon and the time when the sun reaches a 45" zenith angle is 5.6 MED, which is more than five times the dose accumulated during the rest of the afternoon, a period of about 4 h. This shows that the shadow rule for sun protection is confirmed by both this dose rate integration and the diurnal SPF curve derived above for clear summer days in Rockville.
The relatively slow accumulation of UV-B exposure below a solar altitude of 45" at latitudes near that of Rockville is due to two factors; viz., the rapid increase in atmospheric path length with increasing solar zenith angle and the rapid approach of the sun to the horizon at these low solar altitudes.
The reader must be cautioned that the analysis above is based on average conditions of atmospheric transmission and solar path geometry at Rockville in summer. Estimated dose rates from an overhead sun vary from 0.186 to 0.297 W/m2 among the 28 clear summer days in Table 2, and those at other zenith angles vary proportionally. The standard error of the 28-day mean overhead dose rate in Table 2 is 0.006 W/m2.
Near the autumnal (September) equinox, however, at latitudes near Rockville's 39", the sun does not rise much above 45" at local noon, but it spends much more time near this altitude of moderate SPF than in mid summer. In order to examine this situation, 19 additional clear days are selected from dates near the autumnal equinox in the years 1984-1988. These dates are listed in Table 4 along with the corresponding observed dose rates at a 45" zenith angle.
Compare the mean dose rate of 0.099 W/mZ at a 45" sun near the autumnal equinox in Table 4 with the summer mean for this same zenith angle, viz., 0.086 W/m2 in Tables 2 and 3. The higher value at the autumnal equinox may in part be due to the thinner ozone layer that is observed climatically in autumn in mid latitudes (London, 1985).
A bar graph of dose rates at Rockville for the 19 days near the autumnal equinox is shown in Fig.  2(b). On this graph about 13 squares are covered by the bars, representing 1170 J/m2 (3.3 MED) in the afternoon and a mean of 2340 (6.6 MED) for the average September day at Rockville. The average time of the 45" sun is 1.45 h after local noon, and the number of squares to the right of 1.45 h is 6.4, equivalent to 1.6 MED as defined above. In contrast to mid summer, at the autumnal equinox, the total effective UV-B dose at solar zenith distances less than 45" equals that obtained during the remainder of the day.

DISCUSSION
The dose rate estimates obtained in this study are subject to truncation error in the discrete-valued numerical integration of the convolution of UV-B irradiance and relative erythemal action spectra evaluated at 5-nm resolution. This error is aggravated by the great variation of each of these quantities with wavelength. The values of these UV-B intensities and the CIE relative erythemal action spectrum, however, vary in almost an inverse relationship with each other with the result that their products change by only an order of magnitude in the UV-B band (see Table 1). The magnitude of this type of error is probably not more than a few percent. Another source of error is in the tails of the convolution curve that are not computed, but the sum of the products at 295 and 325 nm is only 6% of the total in Table 1. Therefore, the area under any convolution curve beyond these bands is probably much less than that.
The numerical results in this paper are also subject to errors in the assumptions used in the analysis such as the adopted rate of decrease in erythemal effect with increasing wavelength and the estimates of MED values for individuals of various skin types. The ratios computed here (e.g. SPF) are, however, much less sensitive to errors in these assumptions than are the dimensioned values (e.g. dose rates).
Measuring irradiances in the UV-B band is very difficult, and the sources of error are too numerous to mention here. SERC scientists, however, use great care to obtain the most accurate observations possible at the present state of the art. The radiometer's detectors and their interference filters are frequently calibrated and changed if necessary. These radiometers have diffusers that abide by the cosine law to within 2% for zenith angles up to 80" (D. Hayes, Jr., personal communication). Corrections to the data are still being made, such as better determinations of the effective wavelengths of the filters, but it is felt that the accuracy of the preliminary observations used here is adequate for the purposes of the present study.
The estimation of UV-B dose rates can be improved by use of higher resolution U V radiometers that provide data in more ultraviolet bands of narrower bandwidth. SERC is now developing a radiometer that will measure and record UV-B in 18 2-nm half-power bandwidth bands (D. Hayes, Jr., personal communication). This resolution would greatly reduce the truncation error in the convolution of action spectra and UV-B irradiances described above. Finally, the dose rate curves against zenith angle and time of day can be improved by obtaining UV-B observations that are averaged over shorter periods than the 1-h interval used here.