Solar UV Geometric Conversion Factors: Horizontal Plane to Cylinder Model

Solar UV Geometric Conversion Factors: Horizontal Plane toCylinder Model†

Stanley J. Pope1 and Dianne E. Godar*2

1Sun Systems & Svc, Inc., Oak Park, MI2U.S. Food and Drug Admnistration, Silver Spring, MD

Received 30 June 2009, accepted 5 October 2009, DOI: 10.1111 ⁄ j.1751-1097.2009.00679.x

ABSTRACT

Most solar UV measurements are relative to the horizontal

plane. However, problems arise when one uses those UV

measurements to perform risk or benefit assessments because

they do not yield the actual doses people get while they are

outdoors. To better estimate the UV doses people actually get

while outdoors, scientists need geometric conversion factors

(GCF) that change horizontal plane irradiances to average

irradiances on the human body. Here we describe a simple

geometric method that changes unweighted, erythemally

weighted and previtamin D3-weighted UV irradiances on the

horizontal plane to full cylinder and semicylinder irradiances.

Scientists can use the full cylinder model to represent the

complete human body, while they can use the semicylinder model

to represent the face, shoulders, tops of hands and feet. We

present daily, monthly and seasonally calculated averages of the

GCF for these cylinder models every 5� from 20 to 70�N so that

scientists can now get realistic UV doses for people who are

outdoors doing a variety of different activities. The GCF show

that people actually get less than half their annual erythemally

weighted, and consequently half their previtamin D3-weighted,

UV doses relative to the horizontal plane. Thus, scientists can

now perform realistic UV risk and benefit assessments.

INTRODUCTION

Most solar UV measurements for irradiance, dose and UVindex forecasts are relative to the horizontal plane (1).

However, problems arise when one uses those measurementsto do risk or benefit assessments because they do not representrealistic doses; they are taken by stationary spectroradiometers

or radiometers with their detectors horizontally oriented (2,3).Irradiances relative to vertical or arbitrarily oriented tilted flatsurfaces give better estimates, but the human body is not a flatsurface. However, the cylinder is an easily defined geometric

shape widely used as a simplified model for the human body,with its multiple parts in varying orientations (4).

Converting irradiances on a horizontal plane to irradi-ances closer to what people get while they are outdoors hasnot been solved using simple geometric shapes, such as

cylinders, in a universally applicable manner (5–9). Forexample, Sobolewski et al. (9) developed vertical rotatingtwo-dimensional planes while Hoeppe et al. (10) developed a

complex, virtual three-dimensional wireframe human bodysurface model, and Streicher et al. (7) developed a novelmodification of ray tracing algorithms to produce ‘‘patches’’of polygons representing anatomical regions on the human

body. While those sophisticated models have other usefulapplications, such as identifying specific areas at risk for UVdoses higher than the horizontal plane, they have not been

used to convert horizontal plane doses to average cylindricalsurface doses relevant to the complete human form. Becausethey are dependent on proprietary and modified commercial

software, they cannot be used and manipulated by thegeneral public.

Here we describe an approach to get geometric conversionfactors (GCF) that change horizontal plane to cylindrical

surface solar UV irradiances which better represent what thehuman body actually gets while outdoors. We calculatedsolar spectrum every 5� solar zenith angle (SZA) to illuminate

a rotating (every 15� from 0� to 360�) and tilting (every 15�from 0 to 180�) full cylinder or semicylinder model tosimulate the human body or its parts as they are exposed to

solar UV throughout the day. We then calculated averageGCF for each day of the year in the northern hemisphereevery 5� latitude from 20� to 70�N using an albedo of 3%

and an ozone level of 300 Dobson units. We give the averagedaily, monthly and seasonally calculated GCF for unweight-ed, erythemally weighted and previtamin D3-weighted UVfrom 290 to 400 nm for clear and cloudy sky conditions so

that scientists can now perform realistic risk and benefitanalysis.

MATERIALS AND METHODS

To begin calculating the GCF, one first calculates simulated solarspectral irradiances at sea level with specific atmospheric conditionsevery 5� of SZA using the radiative transfer model STARsci (11). Forclear sky conditions, the aerosol optical depth at 550 nm is 0.1 with anaerosol type of continental average, and an ozone level of 300 Dobsonunits. For cloudy sky conditions, the aerosol optical depth at 550 nm is0.38, and an ozone level of 300 Dobson units, with overcast high-cloud

†The mention of commercial products, their sources or their use in connectionwith material reported herein is not to be construed as either an actual or animplied endorsement of such products by the Department of Health andHuman Services. The opinions and ⁄ or conclusions expressed are solely those ofthe authors and do not imply a policy or position of the Food and DrugAdministration.

*Corresponding author email: [email protected](Dianne E. Godar)

� 2010U.S.Government. Journal Compilation. TheAmerican Society of Photobiology 0031-8655/10

Photochemistry and Photobiology, 2010, 86: 457–466

457

conditions set by the program. This model produces the followingoutput data at 1 nm spectral increments from 290 to 400 nm on ahorizontal surface:

(1) Total global irradiance

(2) Direct beam irradiance

(3) Total available diffuse irradiance

We chose to use an adaptation of the Hay and Davies model forirradiance on arbitrarily oriented surfaces (12,13) to calculate theirradiance on a flat plane of known aspect angle to the sun and knownslope angle to the ground (tilt). The total irradiance on a sloped surfaceis the sum of the direct beam, the circumsolar diffuse (forwardscattered), the skylight diffuse and the up well reflected irradiance dueto isotropic ground albedo. The equation adapted from the Hay andDavies (11) model is:

Ets ¼ Eb cos hþ Es ½Eb cos h=ðEed cos/Þþ 0:5ð1þ cosSÞ � Eb=ðEedÞ�þ 0:5Et Rg ð1� cosSÞ

ð1Þ

where Ets is the total global irradiance on surface with slope S,Eb = Ebh ⁄ cosu is the beam irradiance on surface normal to beam,Ebh is the beam irradiance on horizontal surface, u = SZA, Es is thetotal diffuse irradiance, h is the incident angle of plane to directbeam, Eed is the extraterrestrial irradiance at mean earth sun dis-tance, S is the slope angle of plane to horizontal plane, Et is the totalglobal irradiance on horizontal plane and Rg is the ground albedo(isotropic).

One can use other tilted plane models instead of the Hay andDavies (12) model, as long as one meets the following criteria. Theirradiance results must be universally applicable being indepen-dent of latitude, longitude, day of year and time of day. Allorientations are only relative to the SZA and the slope angle fromthe ground.

To represent the position of sloped planes relative to the solarposition, we used a three-dimensional, right-handed coordinate system(see Fig. 1). The solar position, represented by its SZA, is the angle /of a vector with the positive Z-axis passing through the origin andwithin the YZ plane. A set of flat planes creates a cylinder model thatcan substitute for the generally used horizontal plane. The normalvector of two vectors passing through the origin and lying on a planerepresents this flat plane portion of a cylinder. Because one mustdescribe the plane only by its orientation relative to the SZA vectorand to the horizontal XY plane, one must represent the two lines onthe plane using Cartesian coordinates determined by aspect angle Aand slope angle S. Aspect angle A is the angle of the plane with the X-axis as it passes through the XY plane while rotated about the Z-axis,while slope angle S is the angle of the plane with the XY plane.

Each flat plane identifies the orientation of each rectangularsegment in the set of lengthwise rectangles that approximate thelateral area of the cylinder. The cylinder is open-ended with no top orbottom. As the lateral surface is the only portion of the cylindermodeled, the height and radius dimensions of the cylinder areirrelevant because each plane segment has a specific percentage ofthe entire surface area, depending only on the size of the increments ofthe slope angles chosen for the cylinder model.

Its normal vector N in matrix notation identifies the modeled plane:

N ¼Xn

Yn

Zn

24

35 ¼

� sinA sinS� cosA sinS

cosS

24

35 ð2Þ

where Xn, Yn and Zn represent the coordinates of the original surfaceelement normal vectors for a 0� aspect, 0� tilt cylinder.

The average irradiance on the lateral surface of a cylinder isrepresented by the integral of the total global irradiance, En, on planesurfaces represented by the plane normal’s N rotated about thecylinder height axis at slope angle increments ds from 0 to 2p (0–360�).The equation for the average cylinder irradiance, Ec, of a cylinder with0� tilt at 0� aspect to the sun is:

Ec ¼ 1=ð2pÞZ2p

0

EnðSÞds ð3Þ

where En(S) is equal to Ets for each segment of the cylinder. Whencalculated as a modeled approximation using a finite number of slopeangle increments ds and using slope increments of 5�, each lateralsurface segment can either represent 1 ⁄ 72nd of the total lateral surfacearea of a full cylinder model or 1 ⁄ 36th of the total lateral surface areaof a semicylinder model.

To create a model with 72 surface elements that approximates thelateral surface of a full cylinder, the equation for Ec becomes:

Ec ¼ 1=72X72i¼1

fðSiÞDS ð4Þ

where f(Si) is En for slope angles 0 £ S £ 2p and DS are increments of5� (p ⁄ 36).

The semicylinder model has 36 lateral surface elements, so theequation for it is:

A

B

Figure 1. (A) A segment of the horizontal plane (XY) with a certainarea wraps in a circle around the X-axis to form the topless andbottomless full cylinder model with the same area while the solar zenithangle (SZA) changes in the YZ plane. (B) A full cylinder in the proneposition (tilt, T = 0�, aspect, A = 0�) is tilted about the Y-axis on theXZ plane and rotated in a circle in the XY plane. The geometricconversion factors are averages of all orientations determined by tiltand aspect angle during changing SZA.

458 Stanley J. Pope and Dianne E. Godar

Ec ¼ 1=36X36i¼1

fðSiÞDS ð5Þ

where f(Si) is En for slope angles 0 £ S £ p and DS are increments of 5�(p ⁄ 36).

To change the aspect angle and the tilt angle of the cylinder, onerotates the axis of the normal angles of the cylinder surfaces aboutthe original coordinate axis by the aspect angles, A, and by the tiltangles, T. The result is a new normal Nt with a new slope St foreach surface element of the tilted cylinder. One accomplishes thisrotation using three Euler rotations represented by matrix notation,where:

R1 ¼cosA � sinA 0sinA cosA 00 0 1

24

35 ð6Þ

R2 ¼cosT 0 � sinT0 1 0

sinT 0 cosT

24

35 ð7Þ

R3 ¼cosð�AÞ � sinð�AÞ 0sinð�AÞ cosð�AÞ 0

0 0 1

24

35 ð8Þ

The first 3 · 3 matrix (Eq. 6) defines the initial rotation by theaspect angle about the Z-axis, while the second 3 · 3 matrix (Eq. 7)defines the next rotation by the tilt angle of the cylinder, and the third3 · 3 matrix (Eq. 8) defines the final rotation returning the plane to theoriginal aspect angle. These rotations describe the coordinates of arectangular segment of a cylinder with a given aspect angle relative tothe vector of direct beam irradiance and tilt angle relative to thehorizontal axis. Applying all three-matrix rotations to our arbitraryplane normal N yields:

Nt ¼ R3R2R1N ð9Þ

When expressed as coordinates of the new rotated and tilted normal,Eq. (9) becomes:

The scalar product of the tilted normal vector and the solar vectordetermines the incident angle h of the segment plane and the solarvector (path of direct beam). The equation for the incident angle of thesegment plane with the SZA vector is:

h ¼ 90� � arcosð�Ynt sinuþ Znt cosuÞ ð11Þ

Finally, one calculates the new slopes St of the incremental surfaces onthe tilted cylinder. To do this, one calculates Stn, the angle that thetilted and rotated surface element normal vector has with the XYplane. Stn is the arcsine of Znt (from Eq. [10]), which is the Z-coordinate of the rotated and tilted normal vector. The equation forthe angle of the tilted normal Stn is:

Stn ¼ arcsin½ðXn cosA� Yn sinAÞ sinTþ Zn cosT� ð12Þ

Because the limits of the tilt angle of the cylinder are 0–p (0–180�), theslope of the plane of each surface element is defined as:

0� � S � 180�; St ¼ 90� � Stn

180�<S<360�; St ¼ 270� þ Stn

Once one calculates the necessary values for any aspect and tilt angle,one uses Eq. (13) to calculate the average irradiance on a cylinderrotated through all aspect angles:

Eca ¼ 1=2pZ2p

0

Ec da ð13Þ

where da is the increment of aspect angles A for the cylinder as awhole, from 0 to 2p. Likewise, the summation approximation takes theform of Eq. (5):

Eca ¼ 1=24X24i¼1

fðAiÞDA ð14Þ

where f(Ai) is Ec for aspect angles 0 £ S £ 2p and DA are increments of15� (p ⁄ 12).

One then averages the cylinder irradiance for all its tilt angles, withmaximum range 0–180�, because 180–360� is a body tilted upsidedown. The equation for the average cylinder irradiance with a tilt angleincrement of ds is:

Ecat ¼ 1=pZp

0

Eca ds ð15Þ

Alternatively, for a defined range of motion for the tilt of a cylinder,with minimum tilt a and maximum tilt b, Eq. (15) becomes:

Ecat ¼ 1=ðb� aÞZb

a

Eca ds ð16Þ

These equations are valid for one SZA at a time.Using Eq. (16), one calculates the average global irradiance on the

lateral surface of an arbitrarily oriented cylinder for one SZA. If one usesEq. (5) instead of Eq. (4), then Eq. (16) gives the average global

irradiance on the lateral surface of an arbitrarily oriented semicylinderfor one SZA. Note that the semicylinder has no flat back surface, so likethe full cylinder, only its curved lateral surface gets exposure to UV.

One calculates the GCF at 5-min intervals throughout every day ofthe year by interpolating to the SZA present at each time interval. Forhorizontal plane irradiances, one uses an average ozone value of 300Dobson units and calculates every 5-min interval using the samealgorithms used for the calculation of the action spectrum conversionfactors (14) every 5� latitude from 20� to 70�N. Now one calculates theaverage GCF throughout every day of the year by taking into accountthe horizontal plane radiant exposure received during each 5-minperiod. Once one knows the daily GCF average, one can calculatemonthly GCF averages, but similar to the calculation of the diurnalGCF, one must weight the daily GCF by that day’s contribution to thetotal available radiant exposure for the month. In addition to daily andmonthly averages, one can calculate seasonal GCF averages. Thenorthern hemisphere seasons are: spring, from the equinox in March tothe solstice in June; summer, from the solstice in June to the equinoxin September; fall, from the equinox in September to the solstice inDecember; winter, from the solstice in December to the equinox inMarch.

Nt¼Xnt

Ynt

Znt

24

35¼

½ðXn cosA�Yn sinAÞcosT�Zn sinT�cosð�AÞ�ðXn sinAþYn cosAÞsinð�AÞ½ðXn cosA�Yn sinAÞcosT�Zn sinT�sinð�AÞþðXn sinAþYn cosAÞcosð�AÞ

ðXn cosA�Yn sinAÞsinTþZn cosT

24

35 ð10Þ

Photochemistry and Photobiology, 2010, 86 459

These calculations complete one data set. One repeats this processfor overcast, or cloudy sky, conditions as set forth in the STARsciradiative transfer model. In addition, one repeats this entire mathe-matical process for erythemally weighted UV and then for previtaminD3-weighted UV.

For simplicity, one can refer to the GCF for unweighted solar UV(290–400 nm) as ‘‘unweighted GCF’’; likewise, the GCF for erythe-mally or previtamin D3-weighted UV can be referred to as ‘‘erythe-mally weighted’’ or ‘‘previtamin D3-weighted’’ GCF, respectively.

RESULTS

To get accurate GCF, one must create the cylinder and thenperform the tilting and aspect changes in its orientation

(Fig. 1A,B). To get the correct ratios for irradiance or radiantexposure between a cylinder model and the horizontal plane,the dimensions of the cylinder do not matter. However, to

apply the GCF properly for vitamin D3 production, the areasof the receiving bodies do matter. The benefit from exposure toprevitamin D3-weighted UV is dependent on the energy

received, which is dependent on the area of the receivingbody. To compare the radiant energy a segment of the

horizontal plane gets to the radiant energy a cylinder gets, thetotal surface area of the horizontal surface must be equal to thetotal lateral surface area of the cylinder model. Figure 1Aillustrates this concept, where a segment of the horizontal

plane with a specific area wraps around in a circle to form atopless and bottomless full cylinder with the same area (Eq. 3).The total lateral surface area of the full cylinder, with 72 flat

surfaces (or semicylinder with 36 flat surfaces), is equal to the

Figure 2. The geometric conversion factor (GCF) for the semicylindermodel every 15� tilt and an average of all tilt angles with changing solarzenith angle (SZA).

Figure 3. The geometric conversion factor (GCF) for the full cylindermodel every 15� tilt and an average of all tilt angles with changing solarzenith angle (SZA).

A

C

B

Figure 4. The daily clear sky, unweighted (A), erythemally weighted(B) and previtamin D3-weighted (C) geometric conversion factor(GCF) every 5� from 20� to 70�N.


total surface area of the horizontal plane segment. This is animportant concept because the size of the cylinder model doesnot matter when calculating the GCF provided the areas areequivalent. Figure 1B depicts the geometric results of Eqs. (13)

and (15). A cylinder lying flat on its lateral surface has a tiltangle of 0�. As the tilt angle increases, the cylinder rises until itis standing at a tilt angle of 90�. As the orientation to the sun’s

direct beam is arbitrary, the aspect angle also changes, asshown by the rotation of the cylinder model about the Z-axis.These cylinder orientations can represent the human body in

the prone position (tilt 0�) or in the standing position (tilt 90�).Additionally, by applying Eq. (16), these cylinder positionscan also represent different anatomical areas of the human

body, as they are in various inclinations on a body in differentpostures. One can also use the cylinder model to represent anyanimate or inanimate object of similar geometric shape.

We show the semicylinder erythemally weighted GCF for

every 15� of tilt angle (Eq. 14) and the average of all tilt angles(Eq. 15) with changing SZA in Fig. 2. One can use thesemicylinder GCF to calculate exposure to the human face,

shoulders, tops of hands and feet, or any other appropriatelyshaped surface. A semicylinder at a 90� tilt angle, or in theupright position, can get 19–63% of the horizontal plane UV

dose, while in the prone position (tilt 0�), a semicylinder canget 69–87% of the horizontal plane UV dose. The average ofall tilt angles gets 50–79% of the UV dose that the horizontalplane gets, with the minimum ratio at 0� SZA and the

maximum ratio around 75� SZA. Note that the dose, orradiant exposure, is the irradiance (dose rate) multiplied by aspecific period of time when the irradiance is constant, or it is

the time integral of the irradiances when the irradiance ischanging.

The full cylinder GCF in Fig. 3 show every 15� of tilt angle

with the average of all tilt angles as the SZA changes. Anupright cylinder at 90� tilt gets about 19% of the UV dose thatthe horizontal plane gets when the sun is directly overhead (0�SZA). The full cylinder GCF average by orientation (aspect)and position (tilt) gets about 32%, or less than one-third, ofthe horizontal plane dose. When the sun is directly overheadand the cylinder is lying down (0� tilt), it gets about 40% of the

horizontal plane dose. Note that the position of the receivingbody does not matter when the SZA is about 55� or between85� and 90� (during twilight), where it gets about half (49% or

52%, respectively) of the horizontal plane dose. When the SZAis 75�, the full cylinder gets 60 ± 3%, or the highest percent ofthe horizontal plane dose.

We show in Fig. 4 the full cylinder’s daily clear sky,unweighted, erythemally weighted and previtamin D3-weighted GCF for the average of all tilt angles, every 5� from20� to 70�N. We use all tilt angles because while standing,

walking or sitting, some parts of the human body are vertical,some are horizontal, and some parts, such as the arms andlegs, vary in their orientation. The daily clear sky, unweighted

GCF in Fig. 4A show that the dose can reach 40% of thehorizontal plane dose during late spring and early summer forlatitudes between 20� and 35�N, while it can reach 59% of the

horizontal plane dose for latitudes ‡50�N, with dual peaks thatshift with latitude. Figure 4B shows that the daily erythemallyweighted GCF is about 40% during late spring and early

summer, while it is about 55% during the fall and winter.Figure 4C shows the daily previtamin D3-weighted GCF share

almost equal minima with the daily erythemally weighted GCFduring late spring and early summer, and they are only 2%different (53% vs 55%, respectively) during fall and winter.Thus, very little difference exists between these weighted

scenarios and they all share symmetry around the summersolstice.

The three parts of Fig. 5 show the full cylinder’s cloudy sky

daily GCF for each weighted scenario. Cloud altitude andthickness is set at 8.0–9.0 km. Cloudy, overcast conditions

A

B

C

Figure 5. The daily cloudy sky, unweighted (A), erythemally weighted(B) and previtamin D3-weighted (C) geometric conversion factor(GCF) every 5� from 20� to 70�N.


decrease the range of the daily clear sky GCF. Figure 5Ashows the unweighted GCF reaches a low around 43% forabout 1 month on both sides of the summer solstice between20� and 35�N, while it reaches a high around 53% for about

3 months on both sides of the winter solstice between 45� and70�N. Note that all weighted scenarios have different days withthe highest GCF that vary with latitude. Figure 5B shows the

daily erythemally weighted GCF reaches a low around 44% atlatitudes £35�N and a high around 53% at latitudes ‡55�N.Figure 5C shows the daily cloudy sky, previtamin D3-weighted

GCF are almost identical to the daily cloudy sky, erythemallyweighted GCF from 20� to 70�N. These daily GCF hardlydiffer at all during the spring and summer, and they only differ

by about 1% during the fall and winter for latitudes >35�N,where the previtamin D3-weighted GCF is 52% and theerythemally weighted GCF is 53%.

Figure 6 shows the erythemally weighted GCF each month

and season. Figure 6A shows the erythemally weighted GCFeach month, while Fig. 6B shows the erythemally weightedGCF each season every 5� from 20� to 70�N, which are similar

to the unweighted and previtamin D3-weighted GCF. Themonthly erythemally weighted GCF increases from March toSeptember from 20� to 70�N, peaks from October to February

at 45�N and then decreases with increasing latitude. Winter,closely followed by fall, has the highest GCF because theangles of the sun are primarily low in the sky (or higher SZA).Similarly, summer, closely followed by spring, has the lowest

GCF because the angles of the sun are primarily high in thesky (or lower SZA). Winter and fall have almost the sameGCF because the winter GCF are the averages from the wintersolstice (minimum) to the equinox (midpoint), while the fall is

the mirror image from the equinox to the winter solstice.Summer and spring are similar for the same reason onlyaveraged around a different solstice and equinox.

We give the semicylinder’s unweighted erythemallyweighted (Eeff), and previtamin D3-weighted (Deff) GCF inTables 1–3 each season every 5� from 20� to 70�N, for the

average of all tilt angles. Table 1 shows the clear sky GCFwhile Table 2 shows the cloudy sky GCF and Table 3 showsthe percent variances, or the differences between clear and

cloudy sky conditions for all three weighted scenarios. Forexample, during the summer at 50�N, the cloudy sky GCF forerythemally weighted UV is 4% higher than its clear sky GCF,while during the fall its cloudy sky GCF is about 1.7% lower

than its clear sky GCF.In Tables 4–6, we give the full cylinder’s unweighted,

erythemally weighted (Eeff), and previtamin D3-weighted

(Deff) GCF each season every 5� from 20� to 70�N. Table 4shows the clear sky GCF while Table 5 shows the cloudy skyGCF and Table 6 shows the percent variances in the GCF

between clear and cloudy sky conditions. The change in the

A

B

Figure 6. The clear sky, erythemally weighted geometric conversionfactor (GCF) every 5�N from 20� to 70�N by month (A) and season (B)with variance bars showing overcast conditions.

Table 1. The seasonal GCF for every 5� of latitude from 20� to 70�Nusing the semicylinder model. The GCF numbers given are for clearskies for unweighted UV in the bandwidth of 290–400 nm, forerythemally weighted UV and for previtamin D3-weighted UV.

Latitude Weighting Summer Winter

70�N NoneEeffDeff

0.7020.6890.684

0.7630.7260.710

65�N NoneEeffDeff

0.6810.6740.670

0.7630.7280.714

60�N NoneEeffDeff

0.5700.6590.656

0.7560.7250.713

55�N NoneEeffDeff

0.6450.6450.642

0.7490.7200.709

50�N NoneEeffDeff

0.6290.6310.630

0.7370.7110.701

45�N NoneEeffDeff

0.6150.6200.618

0.7200.7000.692

40�N NoneEeffDeff

0.6030.6100.608

0.7000.6870.681

35�N NoneEeffDeff

0.5940.6010.600

0.6800.6580.655

30�N NoneEeffDeff

0.5860.5940.593

0.6610.6580.655

25�N NoneEeffDeff

0.5800.5890.588

0.6440.6450.642

20�N NoneEeffDeff

0.5770.5860.586

0.6280.6320.630

GCF = geometric conversion factor.


GCF from clear to cloudy sky conditions is greater for the fullcylinder than for the semicylinder model. At 50�N during thesummer and fall, the erythemally weighted GCF changes by a

positive 5.6% and a negative 1.4%, respectively. The variancecaused by cloudy sky conditions can be negative when the sunis low in the sky during the fall and winter for latitudes ‡45�N.Note the small differences between the erythemally weighted

and previtamin D3-weighted GCF.Figure 7 displays the percent differences between the

erythemally weighted and the previtamin-D3 weighted GCF

each season every 5� from 20� to 70�N. Spring and summerdifferences are £1.2%, while fall and winter differences are£3.1%.

The full cylinder model, with arbitrary aspect and tiltangles, can represent the human body better than thehorizontal plane. Figure 8 shows the annually averagederythemal UV doses of indoor workers relative to the

horizontal plane (data from Ref. [1]) compared to ourestimates of the annual erythemal UV doses relative to theclear sky full cylinder model used for Figs. 4–6. Variance bars

show the effect of cloud cover conditions for the cylindermodel. The full cylinder model shows that, on average, peopleget less than half the horizontal plane erythemal dose during

their work year (data exclude vacations). We calculated theannual GCF by weighting each seasonal GCF by its percent

dose contribution toward people’s annual dose. For example,the seasonal percent contributions toward the erythemallyweighted annual UV dose of indoor workers around 45�N areabout 51% in summer, 30% in spring, 14% in fall and 5% in

winter. We assumed people in northern European countries(‡45�N) follow a similar outdoor behavioral pattern as peoplein the northern United States at 45�N based on the findings of

Thieden et al. (15) for people in Denmark at 55�N.

DISCUSSION

The GCF are the ratios of the average irradiances on acylinder to the average irradiances on a horizontal plane. Thefull cylinder model can represent the complete human body

or individual body parts: neck, trunk, arms and legs. Thesemicylinder model can also represent individual body parts:face, shoulders, tops of hands and feet. GCF can convert

solar UV irradiances on a horizontal plane to average solarUV irradiances or doses that are closer to what the completehuman body or its parts actually get. Here we describe an

approach to get GCF for each day, month and season of theyear for the full cylinder and semicylinder models. One canalso use this approach to calculate average GCF for severaldays, a week or for any period during an outdoor study,

including a particular time of day, so that one can use the

Table 2. The seasonal GCF for every 5� of latitude from 20� to 70�Nusing the semicylinder model. The GCF numbers given are for cloudcover conditions for unweighted UV in the bandwidth of 290–400 nm,for erythemally weighted UV and for previtamin D3-weighted UV.


70�N NoneEeffDeff

0.6950.6920.690

0.7120.7110.702

65�N NoneEeffDeff

0.6840.6840.682

0.7150.7110.703

60�N NoneEeffDeff

0.6730.6750.673

0.7140.7090.704

55�N NoneEeffDeff

0.6620.6660.665

0.7130.7070.702

50�N NoneEeffDeff

0.6510.6570.656

0.7090.7030.699

45�N NoneEeffDeff

0.6420.6490.648

0.7030.4660.694

40�N NoneEeffDeff

0.6340.6420.641

0.6940.6910.688

35�N NoneEeffDeff

0.6270.6300.630

0.6830.6830.681

30�N NoneEeffDeff

0.6210.6300.630

0.6720.6740.673

25�N NoneEeffDeff

0.6170.6270.627

0.6620.6660.665

20�N NoneEeffDeff

0.6150.6250.625

0.6510.6570.656


Table 3. The variance for every 5� of latitude from 20� to 70�N usingthe semicylinder model. The variance represents the percent differencebetween clear sky conditions and overcast conditions.

Latitude Weighting Summer (%) Winter (%)

70�N NoneEeffDeff

)1.00.50.9

)6.7)2.1)1.2

65�N NoneEeffDeff

0.31.51.8

)6.3)2.4)1.5

60�N NoneEeffDeff

1.62.42.7

)5.6)2.2)1.4

55�N NoneEeffDeff

2.73.33.5

)4.9)1.8)0.9

50�N NoneEeffDeff

3.64.04.2

)3.8)1.2)0.4

45�N NoneEeffDeff

4.44.74.9

)2.4)0.40.3

40�N NoneEeffDeff

5.05.35.4

)0.90.51.1

35�N NoneEeffDeff

5.65.75.9

0.51.51.9

30�N NoneEeffDeff

6.06.16.3

1.72.42.7

25�N NoneEeffDeff

6.36.46.5

2.73.33.5

20�N NoneEeffDeff

6.56.66.7

3.74.04.2


newly calculated GCF for different purposes. Note that onecan also choose from a variety of radiative transfer mod-els, atmospheric conditions and action spectra for weighting

the different scenarios. As altitude above sea level increases,the ratio of diffuse to direct UV decreases. This causes thedaily GCF averages to vary slightly more than seen in

Fig. 4A–C. However, the effect is much less significant thanthat seen from cloud cover, thus the average GCF are notshown.

The accuracy of the cylinder model’s average irradiancedepends on the calculated incident irradiance, En, on each ofits segments. Around 45� slope, 60� (±10�) SZA, and 0�aspect, our sloped plane irradiance reaches a maximum ratio

of 1.2–1.3, in agreement with Wester and Jossefssons’s (16)calculated maximum ratio of 1.2 for a sloped plane and withStick et al.’s (17) maximum ratio of 1.3 for an inclined

cylinder. Our data for sloped planes with various orientationsare within 1.1% of the calculations of Koepke and Mech (3),who performed in-depth modeling of irradiance on arbitrarily

oriented surfaces, and both the simulations and actualmeasurements of Sobolewski et al. (9). They got vertical planeratio minima of about 0.2 for 0� SZA and vertical plane ratiomaxima of about 0.9 for >70� SZA. Moreover, a diurnal

pattern of calculated incident irradiance (En) on our vertical

surfaces follows a pattern similar to the measurements takenby Webb et al. (18).

Although it is possible for a cylinder, or a person, or part of

a person’s body, to get more UV on surface areas that tilttoward the sun than the horizontal plane gets, the averageirradiance or dose received by the full cylinder model is nevermore than 100% of the horizontal plane irradiance or dose.

For this reason, a portion of a person facing the sun canbecome more sunburned or can make more vitamin D3 thanthe horizontal plane irradiances predict. The benefit of

exposure to previtamin D3-weighted UV is dependent on theradiant energy received, which is dependent on the exposedarea of the receiving body. Thus, one mathematically adjusts

for the percent area of a person’s body exposed to UV afteradjusting the dose for geometry. Here we give the averageGCF of all exposures occurring at once on the entire body sothat the radiant energy calculated using GCF is the total

radiant energy received by the cylinder or cylinders represent-ing the entire human body.

The GCF change with season and latitude (Fig. 6B) because

they are dependent on the SZA, while they are primarilyindependent of ozone levels from 230 to 400 Dobson units (3).For all seasons, the average clear sky GCF values are lowest

(40%) at 20�N, varying by about 4.5% from winter ⁄ fall to

Table 4. The seasonal GCF for changing horizontal plane irradianceto cylinder irradiance every 5� latitude from 20� to 70�N using the fullcylinder model. The GCF numbers given are for clear skies forunweighted UV in the bandwidth of 290–400 nm, for erythemallyweighted UV and for previtamin D3-weighted UV.


70�N NoneEeffDeff

0.5170.5050.499

0.5780.5420.528

65�N NoneEeffDeff

0.4970.4900.485

0.5780.5440.529

60�N NoneEeffDeff

0.4780.4750.471

0.5720.5410.529

55�N NoneEeffDeff

0.4600.4610.458

0.5640.5350.524

50�N NoneEeffDeff

0.4450.4480.446

0.5520.5270.517

45�N NoneEeffDeff

0.4310.4360.434

0.5350.5150.508

40�N NoneEeffDeff

0.4200.4260.425

0.5150.5020.496

35�N NoneEeffDeff

0.4100.4180.417

0.4960.4880.484

30�N NoneEeffDeff

0.4030.4110.410

0.4770.4740.471

25�N NoneEeffDeff

0.3970.4060.405

0.4600.4610.458

20�N NoneEeffDeff

0.3940.4030.402

0.4440.4480.446


Table 5. The seasonal GCF for every 5� of latitude from 20� to 70�Nusing the full cylinder model. The GCF numbers given are for cloudcover conditions for unweighted UV in the bandwidth of 290–400 nm,for erythemally weighted UV and for previtamin D3-weighted UV.


70�N NoneEeffDeff

0.5100.5080.509

0.5270.5270.517

65�N NoneEeffDeff

0.4990.4990.500

0.5300.5260.519

60�N NoneEeffDeff

0.4880.4900.490

0.5300.5250.519

55�N NoneEeffDeff

0.4780.4810.481

0.5280.5230.518

50�N NoneEeffDeff

0.4670.4730.472

0.5240.5180.514

45�N NoneEeffDeff

0.4580.4650.464

0.5180.5130.510

40�N NoneEeffDeff

0.4500.4580.457

0.5090.5060.504

35�N NoneEeffDeff

0.4430.4520.452

0.4990.4980.497

30�N NoneEeffDeff

0.4380.4470.447

0.4880.4900.489

25�N NoneEeffDeff

0.4340.4430.443

0.4770.4810.481

20�N NoneEeffDeff

0.4310.4410.441

0.4670.4730.472



summer ⁄ spring, while the average clear sky GCF values arehighest (54%) at 70�N, varying by about 4% from winter ⁄ fallto summer ⁄ spring. For all latitudes, winter and fall have the

highest GCF because during those seasons the solar altitudes

are lower in the sky (or higher SZA) so that the incident anglesare closer to 90� than the incident angle of the horizontal plane

for most of the cylinder, or the human body, yielding moreexposure. Whereas, spring and summer have the lowest GCFbecause the solar altitudes are higher in the sky (or lower SZA)so that the incident angles are almost parallel to most of the

surfaces, yielding less exposure. Although more of the shorterwavelength UV photons are present in the solar emissionduring the summer (and spring), and during the noon hours of

the day, the higher altitude of the sun (or lower SZA) does notallow for optimum body exposures. At 40�N, around themiddle of the United States, a person gets about 43% of the

horizontal plane dose during the spring or summer while theyget about 51% of the horizontal plane dose during the winteror fall. At about 55�N, around the middle of Europe, a persongets anywhere from 46% to 53% of the horizontal plane doses

during the spring ⁄ summer and winter ⁄ fall, respectively. Atmo-spheric conditions, such as overcast skies, that increase thediffuse relative to the direct irradiance increase the GCF when

the SZA is low (summer and spring) and decrease the GCFwhen the SZA is high (fall and winter). This is also true for thediurnal cycle when the SZA is higher during the morning and

evening than during the midday.The GCF dramatically decreases an indoor worker’s total

erythemally weighted annual UV dose relative to the horizon-

tal plane. On average during their work-year, people actuallyget less than half the erythemally weighted UV dose that thehorizontal plane gets (Fig. 8). Because the erythemallyweighted and previtamin D3-weighted GCF are essentially

equivalent, people actually need more than twice the exposuretime to make the amount of vitamin D3 that the horizontalplane predicts. Thus, people actually make less than half the

vitamin D3 calculated relative to the horizontal plane. Theamount of vitamin D3 produced is actually even lower whenone accounts for the action spectrum conversion factors (14)

and the age of the person (19). These factors further decreasethe amount of vitamin D3 produced, so that a senior person(‡60 years) can make less than one-fourth the calculatedamount relative to the horizontal plane.

From the GCF, we can now get good estimates of the UVdoses people actually get while they are outdoors in a varietyof different postures and orientations during each day, month

and season of the year at various latitudes using horizontalplane measurements. Thus, scientists can now perform realisticrisk and benefit analyses.

Table 6. The variance for every 5� of latitude from 20� to 70�N usingthe full cylinder model. The variance represents the percent differencebetween clear sky conditions and overcast conditions. Note that cloudcover lowers the direct irradiance while it increases the diffuseirradiance thus increasing the GCF when the sun is primarily high inthe sky (e.g. summer); however, it has the opposite effect when the sunis low in the sky (e.g. winter).

Latitude Weighting Summer (%) Winter (%)

70�N NoneEeffDeff

)1.40.62.0

)8.7)2.8)1.6

65�N NoneEeffDeff

0.52.03.0

)8.3)3.2)2.0

60�N NoneEeffDeff

2.23.34.0

)7.3)2.9)1.8

55�N NoneEeffDeff

3.74.55.0

)6.4)2.3)1.2

50�N NoneEeffDeff

5.15.66.0

)5.0)1.6)0.5

45�N NoneEeffDeff

6.26.66.9

)3.1)0.50.4

40�N NoneEeffDeff

7.27.57.7

)1.20.81.5

35�N NoneEeffDeff

8.08.28.4

0.72.12.6

30�N NoneEeffDeff

8.78.29.0

2.33.33.8

25�N NoneEeffDeff

9.29.29.4

3.84.54.9

20�N NoneEeffDeff

9.49.49.6

5.25.65.9

Figure 7. The variance or percent difference between the clear sky,erythemally weighted and the clear sky, previtamin D3-weighted GCFduring the spring ⁄ summer and fall ⁄winter.

Figure 8. Indoor workers’ annual erythemally weighted UV dosesrelative to the horizontal plane and to the full cylinder model: UnitedStates around 34� and 44�N (20,21), Netherlands around 52�N (22),Denmark around 55�N (15) and Sweden around 60�N (23).


Acknowledgements—We thank Dr. John Streicher for helpful scientific

consultations and Dr. Frank Samuelson for reviewing this manuscript

prior to submission.

REFERENCES1. Godar, D. E. (2005) UV doses worldwide. Photochem. Photobiol.

81, 736–749.2. Mech, M. and P. Koepke (2004) Model for UV irradiance on

arbitrarily oriented surfaces. Theor. Appl. Climatol. 77, 151–158.3. Koepke, P. and M. Mech (2005) UV irradiance on arbitrarily

oriented surfaces: Variation with atmospheric and ground prop-erties. Theor. Appl. Climatol. 81, 25–32.

4. Mortara, M., G. Patane and M. Spagnuolo (2006) From geo-metric to semantic human body models. Comput. Graph. 30, 185–196.

5. Schauberger, G. (1990) Model for the global irradiance of thesolar biologically-effective ultraviolet-radiation on inclined sur-faces. Photochem. Photobiol. 52, 1029–1032.

6. Schauberger, G. (1992) Anisotropic model for the diffuse biolog-ically-effective irradiance of solar UV-radiation on inclined sur-faces. Theor. Appl. Climatol. 46, 45–51.

7. Streicher, J. J., W. C. Culverhouse Jr, M. S. Dulberg and R. J.Fornaro (2004) Modeling the anatomical distribution of sunlight.Photochem. Photobiol. 79, 40–47.

8. Hess, M. and P. Keopke (2008) Modeling UV irradiances onarbitrarily oriented surfaces: Effects of sky obstructions. Atmos.Chem. Phys. 8, 3583–3591.

9. Sobolewski, P., J. W. Krzyscin, J. Jarosławski and K. Stebel(2008) Measurements of UV radiation on rotating vertical plane atthe ALOMAR Observatory (69�N, 16�E), Norway, June 2007.Atmos. Chem. Phys. Discuss. 8, 21–45.

10. Hoeppe, P., A. Oppenrieder, C. Erianto, P. Koepke, J. Reuder, M.Seefeldner and D. Nowak (2004) Visualization of UV exposure ofthe human body based on data from a scanning UV-measuringsystem. Int. J. Biometeorol. 49, 18–25.

11. Ruggaber, A., R. Dlugi and T. Nakajima (1994) Modeling ofradiation quantities and photolysis frequencies in the troposphere.J Atmos. Chem. 18, 171–210.

12. Hay, J. E. and J. A. Davies (1978) Calculation of the solar radi-ation incident on an inclined surface. In Proceedings of the FirstCanadian Solar Radiation Data Workshop (Edited by John. A.Hay and Thorfne. K. Won) pp. 59–72. Canadian AtmosphericEnvironment Service and the National Research Council ofCanada, Toronto, Ontario, Canada.

13. Bird, R. E. and C. Riordan (1986) Simple solar spectral model fordirect and diffuse irradiances on horizontal and titled planes at theEarth’s surface for cloudless atmospheres. J. Climate Appl. Me-teor. 25, 87–97.

14. Pope, S. J., M. F. Holick, S. P. Mackin and D. E. Godar (2008)Action spectrum conversion factors that change erythemallyweighted to previtamin D3 weighted UV doses. Photochem. Pho-tobiol. 84, 1277–1283.

15. Thieden, E., P. A. Philipsen, J. Heydenreich and H. C. Wulf (2004)UV radiation exposure related to age, sex, occupation, and sunbehavior based on time-stamped personal dosimeter readings.Arch. Dermatol. 140, 197–203.

16. Wester, U. and W. Josefsson (1996) UV-index and influence ofaction spectrum and surface inclination.WMOGlobal AtmosphereWatch No. 127, 63–66.

17. Stick, C., V. Harms and L. Pielke (1994) Measurements of solarultraviolet irradiance with respect to the human body surface.SPIE Ultraviolet Radiat Hazards 2134B, 129–134.

18. Webb, A. R., P. Weihs and M. Blumthaler (1999) Spectral UVirradiance on vertical surfaces: A case study. Photochem. Photo-biol. 69, 464–470.

19. MacLaughlin, J. A. and M. F. Holick (1985) Aging decreases thecapacity of human skin to make vitamin D3. J. Clin. Invest. 76,1536–1538.

20. Godar, D. E., S. P. Wengraitis, J. Shreffler and D. H. Sliney (2001)UV doses of Americans. Photochem. Photobiol. 73, 621–629.

21. Godar, D. E. (2001) UV doses of American children and adoles-cents. Photochem. Photobiol. 74, 787–793.

22. Slaper, H. (1987) Skin cancer and UV exposure: Investigationson the estimation of risks. Ph.D. dissertation, University ofUtrecht.

23. Larko, O. and B. L. Diffey (1983) Natural UV-B radiation re-ceived by people with outdoor, indoor and mixed occupations andUV-B treatment of psoriasis. Clin. Exp. Dermatol. 8, 279–285.


Solar UV Geometric Conversion Factors: Horizontal Plane to Cylinder Model

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