Calibration of the broa ban UV Raiometer Marian Morys an Daniel Berger Solar Light Co., Philaelphia, PA 19126 ABSTRACT Mounting concern about the ozone layer epletion an the potential ultraviolet exposure increase accelerate the nee for an accurate ultraviolet raiation monitoring. To assure the accuracy of measurement the instrument has to be well characterize an the calibration proceure has to be esigne to meet the growing precision an accuracy requirements. In epth error analysis is also necessary to properly estimate the character an amplitue of errors incurre uring the calibration proceure. A metho for calibration of a broa ban UV raiometer is propose. In orer to achieve the highest precision it is base on spectroraiometric transfer from a stanar lamp an a stanar etector in a well controlle laboratory conitions. Ranom an systematic error sources are ientifie an their contribution to the final calibration result is calculate. The sensitivity of the calibration to the measurement errors an the errors of the references varies with wavelength. Statistical epenency of measurements within a spectroraiometric scan contributes significantly to the estimate of calibration precision. The calibration formula is linearly approximate with the first component of its Taylor series an the variance of the calibration is approximate base on the covariance matrices of the error sources. The analytical an numerical methos of error estimation presente can be applie to a broa range of raiometers an spectroraiometers 1. Calibration Proceure 1.1. Assumptions The following assumptions are mae uring the calibration meters are fully characterize both spectrally an in terms of angular response at nominal sensor temperature calibration is correcte for a stanar sun efine as the output of the UV raiation moel 1 uner 2.7mm ozone column 3 solar zenith angle (SZA), at sea level an zero albeo The meter is calibrate in MED/Hr (Minimum Erythema Dose per Hour). The conversion factor M between Erythemally 2 weighte power an MED/Hr is 3 : M=17.1 [(MED/Hr)/(W/m 2 )] calculations an measurement are limite to a 27-4nm range stable 15W Xe arc lamp with 1mm WG35 filter is use as a calibration source; the source is measure before each calibration an measurement is repeate at least every hour 1.2. Principle of calibration. The current I generate by the etector with the absolute spectral response R () uner incient spectral irraiance E() is: I = R E [Amps] (1) With a known calibration factor K [(MED/Hr)/Amp] the meter reas: S = K I = K R E [MED / Hr] (2)
During the calibration proceure the meter s calibration factor K is ajuste so that in front of the calibration UV source with a measure irraiance E Xe (l) the meter reas: S Xe = Xe Sun Ery r E M E R r Sun E where E Sun is the spectral irraiance form a stanar sun, R Ery is the Erythema Action Spectrum an r is a relative spectral response of the etector. The absolute spectral response of the meter R is extremely ifficult to measure irectly so it will be proven that the relative spectral response r is sufficient for the purpose of this calibration an that the meter calibrate accoring to (3) will rea accurately the Erythemal Effectiveness of the stanar sun. From (2) the reaing in a front of the calibration source can be expresse as: Xe Xe S = KR E (4) The calibration factor K can be erive from (3) an (4): K = Sun Ery M E R R Sun E (5) Equation (5) shows, that the result of the calibration proceure oes not epen on the spectrum of the calibration source E Xe proviing it is accurately measure before each calibration an oes not change uring the calibration. Base on (2) an (5) the reaing of the calibrate etector expose to the stanar sun is: Sun Sun Ery S = ME R (6) which by efinition is the erythemal effectiveness of the stanar sun. In other wors, all calibrate meters shoul give the same reaing uner the stanar sun no matter what is the spectral response but with the assumption all have the same angular response. Uner ifferent solar conitions the measure erythemal effectiveness will have a preicable error that epens on the ifference between the R Ery an r. 1.3. Instrumentation. All spectral measurements were performe with ouble grating spectroraiometer Moel 74A/D from Optronic Laboratories. Entrance, mile an exit slits were chosen to provie a 2.5nm half banwith. It gave a goo compromise between the signal to noise ratio an the systematic measurement error. A 2W quartz halogen lamp Moel 22A supplie an calibrate by Optronic Laboratories an traceable to NIST serve as an irraiance stanar. The lamp current was stabilize by Moel 65DS Constant Current Source with a specifie accuracy of.1%. It results in a potential 1% error systematic irraiance error aroun 3nm. For calibration the lamp was positione at a istance of 5cm. (3)
Fig. 1. The calibration setup. The monochromatic output of the system, neee for measurement of the etector spectral response, was etermine with stanar silicon photoetector Moel 73-5C calibrate an supplie by Optronic Laboratories. The specifie 3σ uncertainty in the iscusse wavelength range was estimate at 6%. Super quiet, high pressure, 15W xenon arc lamp type L2274 from Hamamatsu was powere by a very stable xenon lamp Power Supply Moel XPS2 manufacture by the Solar Light Company. After an initial warm-up the current regulation is better than.2%. A 1mm thick WG35 Schott filter is positione in the front of the xenon lamp to absorb the very short, unstable UV raiation emitte by the lamp. 2. Error Analysis 2.1. Ranom calibration errors When calculating the overall calibration ranom error which is a result of the measurement ranom errors as well as stanar etector/source errors the following has to be taken into account: The stanar eviations of the relative measurement error are wavelength epenent (Fig. 2). ue to signal/ noise ratio change. The shape of the measure spectral istribution has great effect. The error of spectroraiometric measurements 1% at ifferent wavelengths oes not form a vector of statistically inepenent ranom variables. The covariance matrix (Fig. 3) contains 1% the variances of ranom variables on the iagonal an the values of 2n egree central moment of their joint istributions elsewhere. Errors at ifferent wavelengths contribute to 1% the overall calibration error with ifferent weight (Fig. 4) The quantities measure uring the calibration % process are: 26 28 3 32 34 36 38 4 Wavelength [nm] Fig. 2. Stanar eviation of the series of spectroraiometric measurements of the calibration source E Xe. Relative st. eviation [%]
27 3 33 Wavelength [nm] 36 39 39 37 35 29 31 33 Wavelength [nm] Fig. 1. Covariance matrix of a series of spectroraiometric measurements of I q. For inepenent processes cov(x i,x j )= for i j, which is not the case here. 27 I c -current from the reference etector uring the monochromatic output calibration I - current from the measure etector uring measurement of its spectral response I q - photoetector current uring the spectroraiometer calibration I Xe - photoetector current uring the calibration source measurement In this notation r I = c R c I an E Xe I = Xe q E q I where R c () is the spectral response of the stanar etector an E q () is the spectral irraiance of the stanar lamp. The calibration formula (Error! Bookmark not efine.) was expane to inclue the system calibrations: c Xe q I R I E Sun Ery M E ( ) R ( ) c q I ( ) I ( ) Xe S = c I R Sun E c I The covariance matrices for all the current measurements were calculate numerically base on a series of measurements. The covariance matrices for the calibrate etectors an sources are not provie. Statistical inepenence of errors at ifferent wavelengths is therefore assume. Consequently, a covariance matrix with variance values on the iagonal an zeros elsewhere was generate for both the stanar etector an stanar lamp. An assumption about normal error istribution was mae both for consecutive realizations of the stanar as well as for the error istribution with wavelength. The systematic component is lost here but there was no ata available to assume otherwise. It was also assume that measurements of I c,i,i q an I Xe are inepenent of each other so that the iniviual contributions can be calculate an their variances ae. Analytical evaluation of the S Xe Relative Sensitivity [% / %].3.2.1 -.1 -.2 variance woul be extremely ifficult. the linear approximation approach was chosen an its principle is presente below. (1) (2) -.3 26 28 3 32 34 36 38 4 Wavelength [nm] Monochromatic output cal. - Ic Detector spectral resp. msrmnt - I Spectroraiometer calibration - Iq Calibration source measurement - Ixe Fig. 2. Sensitivity of the calibration to the measurement errors at various wavelengths.
In general a scalar function y=y(x) of a vector x = [x 1,..., x k ] t can be represente by its Taylor series aroun x : y( ) y( ) t 1 (1) x = x + y( x ) x+ x t H( x ) x+... 2 where x is the vector of input variables. The function's continuity an existence of the ifferentials aroun x is assume. By efinition, x = x - x, the graient vector is efine as: ( ) ( ) ( ) = t y (2) x y x y x,..., x1 x k an H(x ) is a k k matrix of secon egree ifferentials (Hessian): 2 y( x ) Hij( x ) = i j 1 k (3), =... xi xj In close proximity to x the first two components of the Taylor series (1) are a goo approximation of y(x). Introucing ranom variable X such that x = E(X) the expecte value of y is: Ey [ ( X) ] Ey [ ( x )] = y( x ) (4) because E[ t y(x ) X]=. The variance of y can be approximate as: var[ y( X) ] = E[ y( X) y( x 2 )] E[ t 2 y( ) ] t x X = y( x ) K x (5) y( x ) where K x is a covariance matrix of a ranom variable X. The graient vector was calculate numerically at the average point of all measure quantities. The ranom error balance is shown in Table Error! Bookmark not efine...1. Systematic calibration errors Systematic calibration errors are mainly ue to the finite banwith an stray light effects on the spectroraiometer measurement. Other major sources of systematic error are: non-linearity of the raiometer, setup error an error of the stanar lamp current The systematic components of the stanar source an stanar etector are not specifie. The effect of the slit function exhibits itself most prominently when measuring rapily changing spectral characteristics, such as the etector's spectral response or the calibration source output. Figure 1 shows the effect of the slit function on the measurement of logarithmically changing spectral irraiance (there is no slit relate error on linear slopes if the slit function is symmetrical). The slope was normalize in 1 Measurement error [%] 1 1.1.2.4.6.8 1 Normalize slope Fig. 1. Measurement error of the logarithmically changing spectral irraiance cause by the slit function. The slope of the spectral irraiance is normalize in relation to half banwith of the slit function (ecaes/fwhm).
relation to the slit's FWHM (ecaes/fwhm). Table Error! Bookmark not efine. summarizes the systematic error buget in terms of worst case error. The systematic error components of the stanar quartz halogen lamp an the stanar etector were not separately specifie so it was not possible to separate their effects from the ranom error. 1. Conclusions an Recommenations A calibration of the broa ban UV raiometer can be performe in relatively simple setup with a ranom error of 1.6% (1σ) an a systematic error of -6.2... +4.7% worst case; this precision is sufficient to etect trens in the orer of 1%/ecae with 2 calibrations per year In-epth error analysis helps to ientify the biggest error contributors - spectroraiometer calibration an stanar etector uncertainty for the ranom error component It is important to note that the above uncertainty analysis relates to the calibration process only. It means, that if an absoluteley stable etector was repeately calibrate with the above metho the resulting calibration factor series woul have 1.6% stanar eviation an uner the reference conitions (point source, stanar solar spectrum) the average woul be within -6.2 to +4.6% of the accurate erythemal reaing. In the fiel aitional error sources contribute to the overal measurement uncertainty, such as: spectral ifference from erythema, cosine error, pollution of the optics an rifts. The estimate error contribution of the stanar lamp uncertainty (.2%) is lower than the uncertainty of the lamp s output in the UV-B region as a result of an assumption about statistical inepenence of the calibration values. Quite possibly this value is unerestimate. However, to calculate it properly we woul nee a more comprehensive characterization of the lamp uncertainty, namely the covariance matrix, instea just one number typically available from the calibration laboratory. A conservative estimate can be obtaine by assuming that this contribution is equal to the stanar lamp uncertainty at the peek wavelength of the calibrate etector (approx..6% 1σ in this case). It shoul be pointe out that the lamp contribution will have a ranom character for consecutive calibrations only if a new lamp is obtaine for each calibration. The assumption about statistical inepenence within the ata set may cause either uner- or overestimation of the resulting error, epening on the graient vector. The contribution of the stanar etector is overestimate ue to the assume inepenence. Whether this assumption will cause overor uner- estimate epens on the shape of sensitivity functions (Fig. 4). All spectral measurements an ata shoul be checke for statistical inepenence within the scan; if ata are epenent then a covariance matrix shoul be use in final error etermination; using interpolation algorithms for spectral ata introuces epenence to the ata set. Excessive epenence may also inicate a problem with the equipment. Performing the calibration uner well controlle laboratory conitions assures high precision of the calibration, crucial from the point of view of long term stability, at an expense of the absolute accuracy. This calibration metho shoul accompany the typically use transfer from collocate spectroraiometer. It will provie information useful for analysis of calibration coefficients gathere over the years. It is particularly important in situation where the uncertainty of absolute calibration is greater than the meters stability Error! Bookmark not efine., which seem to be the case with the R-B meters. Great care shoul be exercise when classifying the uncertainty components. The nature of the particular component shoul be the criteria, since it etermines the way it propagates, an at the same time the mathematical methos use to estimate it. Combining the ranom, systematic an rift component together rarely simplifies the process, often leaing to unreasonable conclusions.
Table 1. Calibration proceure uncertainty - ranom component summary Uncertainty source Comment Uncertainty contribution (1σ) Stanar lamp uncertainty Error! Bookmark not efine. Provie by calibration lab..2% 1 Stanar etector uncertainty Error! Bookmark not efine. Provie by calibration lab..7% 2 System calibration error Inclues short term lamp current variations, lamp 1.3% instability, mechanical setup error an spectroraiometric measurement error Calibration source E Xe Inclues Xe lamp an filter instability, setup.33% measurement error an spectroraiometric measurement error Monochromatic output <.1% calibration Phosphor measurement error.3% Sensor temperature error The temperature coefficient of the sensor uner the <.1% calibration source is.2%/ C an constant temperature istribution within ±1 C was assume Detector ajustment Inclues the calibration source variations,.4% etector/reaing evice noise, mechanical setup error Total (quarature sum) 1.61% 1 Unerestimate ue to an assumption about statistical inepenence of calibration errors within the wavelength range. See conclusions. 2 Overestimate ue to the same reason as above.
Table 1. Calibration proceure uncertainty - systematic component buget Uncertainty source Comment Uncertainty contribution (worst case %) Spectroraiometer As specifie by manufacturer ±1% non-linearity Stanar lamp current As specifie by manufacturer of the power ±1% error Detector spectral response measurement Calibration source measurement (slit function) Mechanical setup error (combine for system calibration an calibration source measurement) supply Simulate effect of 2.5nm slit with. -2... % Simulate effect of 2.5nm slit with.... +.5% Base on estimate uncertainties of the istances between the sources, spectroraiometer an etector Sensor temperature Assuming.2%/ C temp. coefficient for the ±.2% systematic error spectrum of calibration source. Total (worst case) -6.2... +4.7% ±2% References 1. A.E.S. Green, K.R.Cross an L.A. Smith "Improve analytic characterization of ultraviolet skylight" Photochem. Photobiol. 31, 59-65 (198) 2. A.F.McKinlay an B.L.Diffey A reference action spectrum for ultraviolet inuce erythema in human skin,cie Journal, 6, pp. 17-22, 1987. 3. J.A. Parrish, K.F.Jaenicke, R.R. Anerson Erythema an melanogenesis action spectra of normal human skin Photochem. Photobiol. 36, pp.187-191 (1982) 4. Report of Calibration of One Stanar of Spectral Irraiance Moel 22A, S/N: M- 685, Optronic Laboratories, Inc., July 199 5. Report of calibration of One Silicone Photoetector Moel 73-5C, S/N:948, Optronic Laboratories, Inc., March 1992. 6. A.R. Webb, Instrumentation an implementation of a UV-monitoring network, SPIE Proc. 249, pp.184-193, 1993.