IAEA SAFEGUARDS TECHNICAL MANUAL

Transcription

1 IAEA-TECDOC-227 IAEA SAFEGUARDS TECHNICAL MANUAL PART F STATISTICAL CONCEPTS AND TECHNIQUES VOLUME 1 SECOND REVISED EDITION A TECHNICAL DOCUMENT ISSUED BY THE INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1980

2 IAEA SAFEGUARDS TECHNICAL MANUAL PART F: STATISTICAL CONCEPTS AND TECHNIQUES, VOL.1 SECOND REVISED EDITION IAEA, VIENNA, 1980 Printed by the IAEA in Austria February 1980

3 PLEASE

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5 PART SAFEGUARDS TECHNICAL MANUAL

6 Chapter Page

7 Chapter

8 Chapter

9

10 As an example of probability, consider the inspection of containers of UOa powder. Suppose that numbers corresponding to containers are written on slips of paper which are mixed up in a bowl, and that slips are drawn out of the bowl one at a time in order to identify the containers to be inspected. If 2.yjo of the containers are empty and the slips of paper are replaced in the bowl after each draw (so that the inspection can continue indefinitely), then we might observe

11 application - 3 -

12 - 4 -

13 - 5 - and E f(x) = 1 or ff(x)dx = 1 (eq ) J R where 'Z X refers to the sum over all possible values of a discrete random variable (e.g. number of defects in

14 Figure - 6 -

15 Normal Density Function There

16 8 - Although the above is a reasonable calculation f the exact calculation, and that more generally used is to write The value Pr (21.45 < x < 21.55) = / f(x)dx 21.45

17 1«5«2 Standard Deviation -9 -

18

19 *2 Sample Mean and Sample Variance In spite of all the statistics that might be used, there are a limited number that are of practical interest. Keeping in mind the basic idea behind statistical inference, namely

20 l.t«3 Sampling Distribution

21 A table

22 We note that the t density functions differ from the standardized normal frequency function by having a lower value at zero and higher values in the tails. The density function for t depends on the single parameter v, called the degrees of freedom, which is v = (n - l) in this application.

23

24 Chapter

25

26 This first revision corrects many typographical errors but does not introduce any major changes in the material. A glossary of statistical terms used

27 n n is

28 quantity

29 null hypothesis:

30

31

32 Chapter

33 One very practical

34 Ejji = 'true' value for the item deviation from \i due to the i 01 for all three measurements) factor t (same deviations deviation from ö due to the h* factor effect on measurement j. i

35 Some authors make the distinction of calling 0 a bias under viewpoint

36 where a is chosen to reflect the assumed knowledge about î. For example

37 As another example, consider

38 3»4»1 Taylor's Series Taylor's series expansion

39 To find the approximate variance of x, the partial derivatives, evaluated

40 Chapter

41 - 32 -

42 The systematic error is smaller in the latter event and it would seem appropriate to not correct for bias in this instance. EXAMPLE (Ú) Over

43 suggest that the bias is in a given direction, is that the safeguards indexes of interest, MUF, D, and MUF-D, are affected by many sources of error. One would anticipate that the statistically significant biases

44 The steps 4»2.2 are followed Step

45

46 Steps 4*

47 -38 - population of all laboratories, ô may be regarded as a random variable with zero mean and variance 01, the 'between laboratories' variance.

48 Step

49 Step 10: M 3 = ( )(77)/ /17 = Step ll If the bias correction were made, the long term systematic error variance would

50 Steps Step

51 4.3 RANDOM ERRORS IN MASS, SAMPLING, ANALYTICAL AND NON-DESTRUCTIVE MEASUREMENTS 4.3«1 Two or More Independent Measurements Although

52 Steps 4*3.1 Step 1: Find d = difference for ith pair of observations;

53 EXAMPLE (b) Three barrels of uranium solid waste are counted on the nondestructive assay measurement system at routine intervals to provide estimates

54 Analysis Analysis Analysis ^ Î Î Steps are followed. The results of Step 1 (dj) are shown in the table. Step

55 Again following steps 4«3«1» -46 -

56 47 - Step 4 If s x - s xy is negative, then the estimate of the random variance

57 Specifically

58 EXAMPLE

59 Step

60 Cylinder Shipper

61 Cylinder s Net Weight R S-R U-235 R S-R R S -ft l ! ! , Step 2 of Steps is now followed for each measurement operation, assuming that

62

63 And, following Step 2 of Steps 4»3«lf the estimate of the random error variance is 8138 x 10~ 8 /12 = x 1ÑÃ 8, and the standard deviation is or 0.26$ relative, this

64

65 Step -56 -

66 S4ep 2s Calculate the k by k matrix with diagonal element n a u = '=1 Ñ Ó W J anc^ ^ d-iagonal element a-ik = n a ki j^ c ig. c kj - Wj Step

67 -58 - In the Example just concluded,the weights were given. More generally,

68 Steps 4*2.3 will be followed to obtain the estimate of the systematic error

69

70

71 Chapter

72 NDA Many

73 -64 - x = g U-235 y = net counts/100 Secs Steps 5»3«

74 Step 1: Sj = (104.29)( ) (417.16)( ) = Step 2: S 2 = (l04-29) (417.16) 2 = Step

75 EXAMPLE (ñ) or Redo example (a) under the assumption tha't the model is Y = a + px

76 Step - y -

77 Step TI S 4 = 3(10.015) 2 + 7(10.OlO) (3.985) 2 = Step

78 Step 5 The variance of 0, denoted by V(p), is 1/S 2 Step 6: At a given measured response, y 0, A the corresponding reported value x 0

79 Step 5: V(p) = 1/ = Step 6: At y 0 = 20,192 counts/100 sees, x 0 = 20,192/2663 = 7.58 g U-235 Step 7 In a = (4)(20,192)xlO~ 5 = GO

80 Step 5! Find the sum n

81 A Step 12: V(p) = 1/ = 518.6? Step 13: V(Y) = 1/ « Step 14: In o = (4)(20,192)xlO* 5 = O 2 o = x 10 5 Vp (x 0 )

82 Step 7* Compute the sample variance among thekaj values of Step

83

84

85 Step

86 Step

87 EXAMPLE (j) Over

88 Steps 5*3*7 Step _ n

89 Note

90 - 8l - Steps 5«4«1 Step 1: Calculate n Step

91

92 Step

93 -84 - In general terms, the error structure may be written symbolically

94 -85 - Note: With tank calibrations, interest is often focussed on transfer amounts

95 86 Transfer Number Manometer Reading Before * After *07 37*66 TOTAL Difference *08 63*06 313*98

96 T 5 -T 6 > 4 /V ST U')V ST (P')

97 Step 30: For all combinations of calibrations i and j, find the sum k-1 k T 1S = Z u-u- ó,- ó,-. There are k(k-l )/2 terras in i-i 3-i + L Step 31; The variance of a" is VCoT ) = 1/T T 1S /T 8 T^4 Step

98 -89

99 Step 14: T 7 = (1.6870)(193-97á) (1.6926)( ) = Step IS: T 8 = = Step 16: I" = / = Step

100 EXAMPLE (ñ) In continuing Example (b),

101 5.6 SUMMARY In this summary,

102 (H) PROBLEM: SOLUTIONi APPLICATION:

103 Chapter 6 CALCULATING THE VARIANCE OP MUF 6.1 INTRODUCTION The variance of the operator's MUF is needed for two reasons. In the planning stage, the evaluation of the adequacy of the inspection plan depends

104 6.2 THE MUF EQUATION The MUF for an element is calculated as the algebraic sum of element weights over

105 VARIANCE OF MUF 6.3«1 Assumptions

106

107 6.3.2 Random Error Variance

108 The average element weight of a given item is 20 kg U. The batch average element factors are each based on five samples

109 Step 2 k=l n k m k r k Ck q p t X ijkqpt X.jkqpt X..kqpt «rq.. r.p. «r..t ë» * 3* l l

110

111 Stratum 1: $» 0%, and O.Q571$ for weighing, sampling, and analytical,respectively. Stratum

112 Step - þç -

113 Then,

114 First, Vr(MUF) is found by following the pertinent Steps of calculation Steps 6.3*1. Only

115 Steps 8 and 9* Same as Steps 8 and 9» Example (b), Section 6.3 Step 14; Mj = 960,000 M ë = 960,000 M ë = 960,000 Ì 2.. = ì 2 = - 955i200 ì.. 2 = f 200 Ì 3 = M ç. = ì..ç = - 4,800 Ì 4..

116 Step - 107

117 In stratum 3» the NDA instrument measures amounts of U-235 directly. Each container is measured individually on the same instrument. A nominal factor of 3.00$ U-235 is used to convert 6.

118 By Step

119 Step 3: Group Number 1 2 S i S 2 S ; (kg U-235) = (120, , S 3 4 S s s S 6 S 7

120 - Ill - Step 8: Vs*(MUF) = (426.15) 2 (0.0008) 2 + ( ) 2 (O.OOIO) 2 +

121

122

123

124 - 115-

125

126 Step

127 Step 1: A table is constructed with Ê column headings corresponding

128 Step 3 k = li Vry(Dj) = 1594 (from Example (a)) k = 2: Vry(D 2 ) = (238,800) 2 {[( ) /100]+ + [( ) 2 /48]} = 1050 k = 3* Vry(D 3 ) = (1,200) 2 ( ) 2 /10 = k = 4: Vry(D 4 ) = (7f200) 2 {[( ) 2 /l8] + [(0 + [(0.0198) 2 /24]} = 2263

129 EXAMPLE (a) Continue Example

130 Step

131 Step 14: The systematic error variance of D is Vs(D) = Vsx(D) -fvsy(d) + Vby(D) Notes

132 Step

133 k t g b g tk 1/4 1/4 1/4 1/4 1/2 1/2 1/2 1/2 Step ll:b n = (1/2)(240,000) = 120,000 B 21

134

135 more knowledgeable about his own errors than about the operator's. Further, the inspector has the means to reduce the effects of his

136

137 Step 8: For each analytical 'method 1 t (which may be a given mass spectrometer or a SAM-2 instrument used by the operator),

138 (7) The analytical error standard deviations for the SAM-2 are I.Cffo relative for the random error and 0»40$» relative for

139 Redo Example (b) of Section 7.5 but use weights of U-235 rather than weights of uranium and delete Stratum 3«Following the steps of this Example givess I, x = o 2 " x =

140 - 131 Step 2: Group Number U (stratum 3) Step 3: From Step 3t Example (a), Section 6.5» 3! = s i v* i a* P t 8f..ty 5 s *.ty

141 Step 8: T lx = S t + S 2 + S 3 + S 5 + S 6 = T 2x =

142

143 134 - ÀÆÅÕ GLOSSARY

144 Bias - a systematic error that can be estimated by comparison of the mean value of a series of measurements to a reference value, in which case, a correction can be applied to remove the effect of the bias on the measurements. Residual Bias - an unknown systematic error that remains even after a bias correction is applied due to the fact that not all systematic errors

145 136- Error propagation - "the determination of the value to be assigned as the uncertainty of a given quantity using mathematical formulae for the combination of measurement errors. Error propagation involves many considerations

146 Significance level - the measure of probability a say (0.05) associated with

147 138- Table 1 Ordinates and Areas of the Normal Distribution / ë C 'ÄÎ /

148 Table

149 Table

150 Table

151 Table 1 (continued)

152 Table - 143

153 Table 2 (continued) '« ' ' J T-7

154 145 - Table 3 Percentage Points

155 Table

156 Table 4(continued) /» «7 8 14! S ] !) !) 3.20

157 Table

158 Table 4 (continued) so es GO es /l dt^rocs !

159 Note: The list of references is not complete. Additional suggestions are welcome for inclusion REFERENCES

160 151 - Chapter 3 [3.1] MANDEL, J. "The Statistical Analysis [3.2]

161 Chapter