1、Manual of Petroleum Measurement Standards Chapter 13.3 Measurement Uncertainty SECOND EDITION, DECEMBER 2017Special Notes API publications necessarily address pr oblems of a general nature. With respect to particular circumstances, local, state, and federal laws and regulations should be reviewed. N
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16、Washington, DC 20005, standardsapi.org. iiiContents Page 1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Normative References . . . . . . . . . . . . . . . . . . . . . . . . .
17、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 Terms, Definitions, Acronyms, Abbreviations, and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1 Terms and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Acronyms, Abbreviations, and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Measurement Method. . . . . . . . . . . . . .
20、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 True Quantity Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.4 Error . . . . . . . . . .
21、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.5 Statistical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22、 . . . 10 4.6 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.7 Confidence Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23、. . . . . . . . . . . . . . . . . . 11 4.8 Type A vs Type B Uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.9 Representative Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24、 . . . . . . . . . . . . . . . . . . . . . . 11 4.10 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.11 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.12 Standard Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.13 Degrees of Freedom . . . . . . . . . . . . . . . .
26、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.14 Determining Confidence Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.15 Correlation Coefficient . . . . . . .
27、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.16 Independent and Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.17 Combining Uncertainties . . .
28、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.18 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Evaluating Unc
29、ertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30、. . . . 20 5.2 Determine the Measurement Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3 Record Elemental Error Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31、 . . 21 5.4 Create a Primary Uncertainty Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.5 Classify and Evaluate Uncertainty Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.6
32、Determine Sensitivity Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.7 Determine Correlation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.
33、8 Determine Combined Standard Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.9 Determine Expanded Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.10 Reporti
34、ng Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Annex A (informative) Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35、33 Annex B (informative) Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Annex C (normative) Formal Proof of Equation (7a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36、 . . . 54 Annex D (informative) Report Significant Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Annex E (normative) Using a Simplified Estimate of Standard Deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Annex F
37、(informative) Dixons Q-Test for Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38、 . . . . . . . 65 vContents Page Figures 1 Accuracy/Precision Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39、 . . . . . . . . . . . . . . . . . . . . . 18 3 Triangular Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Rectangular Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Relationship between Measurement Model and Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 Relationship between the Uncertainty of the Measurement Model and the Uncertainty of the Inputs. . . . . .
41、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7 Hierarchal Relationship of all Uncertainty Components of a Measurement Model . . . . . . . . . . . . . . . 22 B.1 Vector c, Matrix Diag(c) and Vector u using
42、Data from Example B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 B.2 Matrix R using Data from Example B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 B.3 Result of Multiplying Diag(c) and u from Figure B.1 Diag(c) u. . . . .
43、. . . . . . . . . . . . . . . . . . . . . . . . . . 44 B.4 Result of Transposing Figure B.3 Diag(c) uT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 B.5 Result of Multiplying Figure B.4 by Matrix R Found in Figure B.1 Diag(c) uT R . . . . . . . . . . . . .
44、 44 B.6 Result of Multiplying the Matrix in Figure B.5 by the Matrix in Figure B.3 . . . . . . . . . . . . . . . . . . . . . . 44 B.7 The Matrices Associated with Diag(c) uT RT and Diag(Diag(c) uT RT) . . . . . . . . . . . . . . 45 B.8 The Relative Contribution of Each Component . . . . . . . . . .
45、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 D.1 Uncertainty of the Transmitter Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 D.2 Uncertainty of the Transmitter after Adjustment. . . . . . . . . . . . . . . .
46、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Tables 1 Data with Same Mean but Different SDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Student t Factors for Individual Measurements at Various Confidence Levels . . . . . . .
47、 . . . . . . . . . . . 14 3 Example of a Primary Table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4 Matrix 1Universe of Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48、. . . . . . . . . . 30 5 Matrix 2Correlation Matrix (Informative). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 B.1 Example Conditions and Correction Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49、. . 35 B.2 Primary Uncertainty Table for Meter Factor Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 B.3 Classification and Evaluation of Uncertainty Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 B.4 Sensitivity of Uncertainty Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 B.5 Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
