Designation: G214 − 16Standard Test Method forIntegration of Digital Spectral Data for Weathering andDurability Applications1This standard is issued under the fixed designation G214; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This test method specifies a single relatively simplemethod to implement, common integration technique, theModified Trapezoid Rule, to integrate digital or tabulatedspectral data. The intent is to produce greater consistency andcomparability of weathering and durability test results betweenvarious exposure regimes, calculation of materials properties,and laboratories with respect to numerical results that dependupon the integration of spectral distribution data.1.2 Weathering and durability testing often requires thecomputation of the effects of radiant exposure of materials tovarious optical radiation sources, including lamps with varyingspectral power distributions and outdoor and simulated sun-light. Changes in the spectrally dependent optical properties ofmaterials, in combination with exposure source spectral data,are often used to evaluate the effect of exposure to radiantsources, develop activation spectra (Practice G178), andclassify, evaluate, or rate sources with respect to reference orexposure source spectral distributions. Another important ap-plication is the integration of the original spectrally dependentoptical properties of materials in combination with exposuresource spectral data to determine the total energy absorbed bya material from various exposure sources.1.3 The data applications described in 1.2 often require theuse of tabulated reference spectral distributions, digital spectraldata produced by modern instrumentation, and the integratedversion of that data, or combinations (primarily multiplication)of spectrally dependent data.1.4 Computation of the material responses to exposure toradiant sources mentioned above require the integration ofmeasured wavelength dependent digital data, sometimes inconjunction with tabulated wavelength dependent reference orcomparison data.1.5 The term “integration” in the previous sections refers tothe numerical approximation to the true integral of continuousfunctions, represented by discrete, digital data. There arenumerous mathematical techniques for performing numericalintegration. Each method provides different levels ofcomplexity, accuracy, ease of implementation and computa-tional efficiency, and, of course, resultant magnitudes.Hulstrom, Bird and Riordan (1)2demonstrate the differencesbetween results for rectangular (963.56 W/m2), trapezoid rule(962.53 W/m2), and modified trapezoid rule (963.75 W/m2)integration for a single solar spectrum. Thus the need for astandard integration technique to simplify the comparison ofresults from different laboratories, measurementinstrumentation, or exposure regimes.1.6 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.7 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:3E275 Practice for Describing and Measuring Performance ofUltraviolet and Visible SpectrophotometersE424 Test Methods for Solar Energy Transmittance andReflectance (Terrestrial) of Sheet MaterialsE490 Standard Solar Constant and Zero Air Mass SolarSpectral Irradiance TablesE772 Terminology of Solar Energy ConversionE903 Test Method for Solar Absorptance, Reflectance, andTransmittance of Materials Using Integrating SpheresE927 Specification for Solar Simulation for PhotovoltaicTestingE971 Practice for Calculation of Photometric Transmittanceand Reflectance of Materials to Solar Radiation1This test method is under the jurisdiction of ASTM Committee G03 onWeathering and Durability and is the direct responsibility of Subcommittee G03.09on Radiometry.Current edition approved May 1, 2016. Published May 2016. Originallyapproved in 2015. Last previous edition approved in 2015 as G214–15. DOI:10.1520/G0214-16.2The boldface numbers in parentheses refer to a list of references at the end ofthis standard.3For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1E972 Test Method for Solar Photometric Transmittance ofSheet Materials Using SunlightE973 Test Method for Determination of the Spectral Mis-match Parameter Between a Photovoltaic Device and aPhotovoltaic Reference CellG113 Terminology Relating to Natural and Artificial Weath-ering Tests of Nonmetallic MaterialsG130 Test Method for Calibration of Narrow- and Broad-Band Ultraviolet Radiometers Using a SpectroradiometerG138 Test Method for Calibration of a SpectroradiometerUsing a Standard Source of IrradianceG151 Practice for Exposing Nonmetallic Materials in Accel-erated Test Devices that Use Laboratory Light SourcesG173 Tables for Reference Solar Spectral Irradiances: DirectNormal and Hemispherical on 37° Tilted SurfaceG177 Tables for Reference Solar Ultraviolet Spectral Distri-butions: Hemispherical on 37° Tilted SurfaceG178 Practice for Determining the Activation Spectrum of aMaterial (Wavelength Sensitivity to an Exposure Source)Using the Sharp Cut-On Filter or Spectrographic Tech-niqueG197 Table for Reference Solar Spectral Distributions: Di-rect and Diffuse on 20° Tilted and Vertical SurfacesG207 Test Method for Indoor Transfer of Calibration fromReference to Field Pyranometers3. Terminology3.1 Definitions—The definitions given in TerminologiesE772 and G113 are applicable to this test method.3.2 Definitions of Terms Specific to This Standard:3.2.1 first difference, n—the difference, d1i, between adja-cent ordinate values, d1i=yi+1-yi. An approximation of thefirst derivative of the function represented by the tabulateddata.3.2.2 second difference, n—the difference d2i, between ad-jacent first differences (as defined in 3.2.1) in tabulated data;namely d2i=d1i+1–d1i. An approximation of the secondderivative of the function represented by the tabulated data.3.3 For the purposes of this standard, the terms “integral”and “integration” are used in the sense of a computed numeri-cal approximation to a definite integral of continuous functionsrepresented by tabulated or measured numerical (digital) dataas functions of wavelength. The approximations are computedas the summation of discrete magnitudes computed accordingto the method. The data to be integrated may be interpolated toachieve consistent wavelength intervals.4. Summary of Test Method4.1 Given a set of n digital or numerical (tabulated) data yi,1 ≤ i ≤ n, as a function of an independent variable, such aswavelength, λi, compute the area under each trapezoid, Aibounded by λiand λi+1with altitudes (heights) yiand yi+1, for2i n-1, respectively.Ai5 0.5 3 ~λi112 λi! 3 ~yi111 yi! (1)The uniform factor of1⁄2 is needed to compute the area of ageneral trapezoid.4.2 Compute the sum, A0of the n-2 Aiareas over theinterval from i =2toi = n-1.A05 ΣAii 5 2n 2 1 (2)4.3 The total area A, approximating the integral from λ1toλnis computed by adding in the start and end values to A0.Start: Ai5 0.5 30.5 3 ~λ22 λ1! 3 ~y21 y1! (3)End: An5 0.5 30.5 3 ~λn2 λn21! 3 ~yn1 yn21! (4)Eq 1 can be written At, of height h (in this case each h=(λi+1– λi)) and altitudes a= yiand b = yi+1.At5 h 3 ~a 1 b!⁄2 (5)Therefore, for uniform step h, the total area under curve isexpressed as:A 5 0.5 3 h 3 ~y 1 1 2 3 Σ2n21yi1 yn! (6)NOTE 1—For data with variable h, the above calculations must be doneindependently for each segment of the data with the same h.4.4 To compute the integral of the products of two spectraldata sets, such as a reference Spectrum, E(λ), (for examplereference spectra such as Standard Tables G173, G177, andG197), or the spectral content of calibration or other sources(as in Test Methods G207, G130, and G138) and measured ortabulated spectral optical property data, R(λ) such as transmit-tance or reflectance as measured in accordance with TestMethod E903 and E424 and Practice E971, or spectral mis-match errors such as in Test Method E973, it is necessary forall data sets to have identical wavelength (λi) and wavelengthintervals (λi+1 – λi). Then the appropriate products E(λi)·R(λi)are computed and treated using the procedures in 4.1 to 4.3.Ifthe spectral wavelength intervals are different, one data set(usually with the smallest or shortest wavelength interval,should be selected as the data set, M(λ), with which to matchall other data sets wavelength intervals. The other data setsshould be interpolated, using linear interpolation, to obtainvalues at wavelength values and intervals identical to theselected M(λ).4.4.1 When interpolating data sets, it is recommended thatthe data set with the coarsest or largest wavelength step size orinterval be interpolated to the step size of the data set with thesmaller step size or interval.4.5 Compute an estimate for the absolute error in theintegration based on the wavelength limits for the integral, theaverage wavelength interval of the data, and the average of thesecond differences of the spectral data. Compute the estimatedrelative (percentage) error in integral approximation based onthe total integral and absolute error values (see Section 15 onprecision and bias).5. Significance and Use5.1 Weathering and durability testing often requires thecomputation of the effects of radiant exposure of materials tovarious optical radiation sources, including lamps with varyingspectral power distributions and outdoor and simulated sun-light as in Test Methods E972, G130, and G207.5.2 The purpose of this test method is to foster greaterconsistency and comparability of weathering and durability testG214 − 162results between various exposure regimes, calculation of ma-terials properties, and laboratories with respect to numericalresults that depend upon the integration of spectral distributiondata.5.3 Changes in the optical properties of materials such asspectral reflectance, transmittance, or absorptance are often themeasure of material stability or usefulness in various applica-tions. Computation of the material responses to exposure toradiant sources mentioned above requires the integration ofmeasured wavelength-dependent digital data, sometimes inconjunction with tabulated wavelength-dependent reference orcomparison data.5.4 This test method specifies and describes the ModifiedTrapezoid Rule as a single reasonably accurate and easilyimplemented integration technique for computing approxima-tions of spectral source and optical property integrals.5.5 The method includes a procedure for estimating theapproximate absolute and relative (percent) error in the esti-mated spectral integrals.5.6 The method includes a procedure to construct data setsthat match in spectral wavelength and spectral wavelengthinterval, which does not have to be uniform over the spectralrange of interest. Uniform spectral intervals simplify some ofthe calculations, but are not required.6. Interferences6.1 Closed form expressions such as simple functions,spectral properties, and source functions are rarely available,preventing analytical solution to integration of those functions.6.2 Digitized or tabulated data are only approximations tothe continuous spectral property and source functions found innature.6.3 Mismatched spectral abscissae and spectral data inter-vals (steps) for two or more spectral data sets must be adjustedto match at least one of the spectral data sets. Simple linearinterpolation is suggested as a means of putting data sets in aform where they can be multiplied or otherwise combined. Thedata sets should then all match a selected (usually the highestresolution, or smallest step interval) data set. The wavelengthintervals do not need to be uniform, just consistent between themultiple data sets.6.4 Interpolation to produce matching spectral wavelengthsand data intervals can introduce additional uncertainty inintegrated data, above and beyond the error due to theintegration technique and measurement and instrumentationuncertainty.7. Apparatus7.1 A digital computer with computing power, storagecapacity, and capable of ingesting the spectral data in questionand processing it with applications suitable for analyzing data,such as spreadsheet software or mathematical analysis soft-ware.7.2 For applications requiring measurement of spectral dis-tribution of sources (such as Specification E927, PracticeG151, or Test Methods G130 and G207), a spectroradiometercalibrated in accordance with Test Method G138 is required.7.3 For applications requiring measurement of spectralabsorptance, reflectance, and transmittance of materials such asTest Method G138, a spectrophotometer is used.7.3.1 If the measured data alone is to be integrated, thismethod applies directly.7.3.2 If the measured data is to be used in conjunction withother measured or tabulated data, it is recommended that thespectral step interval and data point wavelengths match thedata set with the smallest wavelength interval as closely aspossible.7.3.3 If possible, use the smallest wavelength step intervalavailable for the spectroradiometer measurements that is com-patible with the smallest interval step size in the other data sets.The other data sets (with larger data intervals) can then beinterpolated to the measured data intervals.7.3.3.1 It is recommended that simple linear interpolation, ifneeded, be accomplished in accordance with subsection 12.3.1.8. Hazards8.1 Hazardous levels of ultraviolet or concentrated solar orartificial optical radiation may be encountered in the process ofmeasuring source spectra.8.2 Electrical (high voltage, current) and thermal (hotsurfaces, intense infrared radiation) hazards may beencountered, especially when using high intensity opticalradiation sources.9. Sampling, Test Specimens, and Test Units9.1 Care must be taken to ensure that the units of wave-length and amplitude of the data under analysis are consistent.Any scaling or unit conversion applied to the original data shallbe documented. Examples are conversion from wavelengthunits of microns (10-6m) to nanometres (10-9m) for units ofwavelength; or microwatts per square metre to watts per squaremetre for flux density.9.2 Sampling of data at uniform wavelength intervals orstep sizes will simplify the computations d