Uncertainty in Radiation Measurements

1. Uncertainty in Radiation Measurements • Uncertainty budgets for each type of instrument • Uncertainty in voltage readings according to DMM spec sheet • Uncertainty of the WRR representation by WSG instruments (=uncertainty of WRR reduction factors for WSG instruments) was set to 250 ppm, based on the statistics of WRR factors determined at IPC‘s 1980-2000 Uncertainty in Radiation Measurements and Calibrations

2. Uncertainty – Basics • Formulate a mathematical model of the measurement system  Formula • Identify all sources of uncertainty (basically all input parameters and measurands in that Formula)  Components • Estimate the Individual Uncertainty for each component according to its type (normal, rectangular, U-shape, cosine... distribution) • Calculate the Sensitivity of the measurand to each of the uncertainty components • Sum the squares of each component‘s individual uncertainty multiplied by its respective sensitivity factor  Combined Uncertainty • Multiply the combined uncertainty by the Coverage Factor (according to the effective degrees of freedom of the mathematical model) Expanded Uncertainty Uncertainty in Radiation Measurements and Calibrations

3. (CUI+ctT) Example – PMO6 1) Mathematical Formula 2) Components of Uncertainty (C) , PC (  UC1 , IC1 , UC2 , IC2) , PO ( UO , IO) , T 3) Individual Uncertainties ui Uncertainty in Radiation Measurements and Calibrations

4. Example – PMO6 (contd.) 3) Individual Uncertainties ui U and I are known to within the accuracy of the voltmeter  Rectangular Distribution For rectangular distributions the standard uncertainty is u=a/3 ±a: accuracy of the voltmeter Uncertainty in Radiation Measurements and Calibrations

5. tc  I I Example – PMO6 (contd.) 4) Sensitivity Factors ci Partial Derivatives Uncertainty in Radiation Measurements and Calibrations

6. Example – PMO6 (contd.) 5) Combined Uncertainty u2=(uici)2 6) Expanded Uncertainty Effective degrees of freedom of the system:  Students t table: Coverage factor k=1.96 for a confidence level of 95% U=ku=1.96*u Uncertainty in Radiation Measurements and Calibrations

7. cos(p) A0 Example – PMO6 Calibration Factor p off-pointing A0 „Atmospheric conditions“ Uncertainty in Radiation Measurements and Calibrations

8. Cosine Probability Distribution The off-pointing of the test pyrheliometer with respect to the reference instrument is always positive with an expectation value of zero. This is an example of an asymmetric probability distribution, where the standard procedure for calculating the sensitivity fails because <cos(p)> ≠ cos(<p>). The following expansion is used to calculate cp Uncertainty in Radiation Measurements and Calibrations

9. Uncertainty Budget Uncertainty in Radiation Measurements and Calibrations

10. Uncertainty Budget The contribution of each uncertainty component to the combined and expanded uncertainties Uncertainty in Radiation Measurements and Calibrations