Nuclear Instruments and Methods 216 (1983) 455-470 455 :,., North-Holland Publishing Company

CALIBRATION FOR MEASUREMENTS WITH BACKGROUND CORRECTION APPLIED TO URANIUM-235 ENRICHMENT

Walter LIGGETT

National Bureau of Standartl~. Washington. DC 20234. USA

Received 2 May 1983

In enrichment measurement by gamma-ray counting, two spectral regions are observed: In one, the response is due to the enrichment and the background, and in the other, to the background alone. Calibration consists of determining not only the relation between the response and the enrichment but also the relation between the two background levels. A calibration procedure with this property is developed under the assumption that the random errors have constant variance and the assumption that the two background levels are proportional. This procedure provides a consistent estimator for the calibration curve and interval estimates for the unknowns measured after calibration. These intervals have stated percent coverage of the true values when large numbers of measurements are made on the basis of the same calibration experiment. The two assumptions may not adequately fit ,some enrichment measurements. The first assumption is never strictly valid since the random error depends in part on the level. The second assumption is valid for some but not all sources of background radiation. The calibration procedure is applied to enrichment measurements made with the SAM-2 enrichment meter. With these measurements as illustrations, techniques for judging the validity of the assumptions are presented.

I. Introduction

The measurement of 2~s U enrichment is based on counting the 185.7 keV gamma rays emitted by 23~U. The measurement requires observation of two counts, the count in an energy region that includes 185.7 keV and the count in some other energy region where only background radiation is present. If certain conditions are satisfied, the mean number of 185.7 keV gamma rays emitted by the 235U and counted by the detector is proportional to the enrichment [1,2]. These conditions include the requirement that the matrix material, the container, and the counting geometry remain constant from sample to sample. The proportionality is to the enrichment instead of to the total amount of 235U because absorption and scattering allow only 185.7 keV gamma rays emitted very near the surface of the sample to reach the detector. Since the area of the surface that the detector sees is held constant, the amount of 23SU that can contribute to the observed count is limited by the amount of 238U and other uranium isotopes that are present. Thus, the amount of 235U that can contribute is proportional to the enrichment, the part of the total uranium that is 235U.

The counts observed in the 185.7 keV region are due, in part, to background radiation. This background radiation must be modelled if a calibration procedure is to be obtained. In most enrichment measurement procedures, the background level in the 185.7 keV region is assumed to be proportional to the background level in the other region. This model is applicable both to the Compton continuum from sample-related gamma radiation and to ambient radiation not associated with the sample. Kull and Ginaven [I] mention other sources of background radiation for which the assumption of the proportionality of the two backgrounds is not appropriate. An example of this is a gamma-ray peak due to some other radionuclide that is present in one of the energy regions. The observed count rate differs from the true count rate because of random errors from sources such as the random nature of radioactive decay and instrument instabilities. Of course, some sources of random error could also be sources of systematic error that would make the models on which this paper is based invalid. Radioactive decay is assumed to be a Poisson process. Thus, if the random error were due strictly to radioactive decay, the variance of each count would be proportional to the level and thus depend on the enrichment of the particular sample and on the

456 W. Llg'lcett / Measurements wtth hack.e, round correctton

background level of the particular observation. An example of instrument instability is variation in the band of energies that is included in each region [1]. The part of the random error due to sources other than radioactive decay has unknown variance. The presence of both sources complicates the enrichment measurement problem because the resulting random error has both unknown and non constant variance.

Let C l represent the count in the 185.7 keV region, and let C 2 represent the count in the background region. Let E be the enrichment, and let B be the true value corresponding to the count observed in the background region, that is, the true background count rate multiplied by the counting time. The model on which enrichment measurement is based is given by

C I = a E + / 3 B + r a n d o m e r r o r , C z = B + r a n d o m e r r o r . (1)

The coefficients c~ and/3 are estimated as part of the calibration experiment. In this paper, the complicated nature of the random error in eq. (1) is not dealt with fully. Rather, the calibration procedure developed is based on the assumption of constant variance. The assumption of constant variance might be applied to the counts or to the counts scaled by some function of the enrichment. This assumption is discussed further in section 3 in the context of some particular calibration experiments, To deal with the complicated nature of the random error, a more exact calibration procedure might be developed that uses replicate measurements at each enrichment to allow estimation of properties of the part of the random error not due to radioactive decay.

Enrichment measurement consists of the calibration experiment and the measurement of the unknowns. The calibration experiment is based on counting standards samples with known enrichment. The design of this experiment involves the choice of the number of standards and the choice of how many times each standard is counted. The primary purpose of the calibration experiment is to estimate a and B. With these estimates, enrichments can be calculated from counts on unknown samples.

An equally important purpose of the calibration experiment is error assessment. "[he error assessment should be formulated in a way that is appropriate to the application [3]. In particular, it should depend on how many unknowns are to be measured on the basis of the same calibration experiment. The error assessment presented in this paper is appropriate when many measurements are to be based on a single calibration experiment. In this case, interval estimates for the unknowns that cover at least a stated percentage of the true values will often be a good choice. These are the terms in which we formulate the error assessment presented in this paper. When many measurements are made, the procedure proposed gives interval estimates that cover at least 100P% of the true values with confidence 1 - c~. [Note that c~ denotes two different things, a probability here and a model coefficient in eq. (1).] This formulation is the same as the one presented by Lieberman et al. [4] for the linear, single-observation calibration problem.

Eq. (!) shows that the enrichment problem is related to the error-in-variables problem, which has been reviewed by Moran [5]. As discussed further in section 4, the enrichment problem is closely related to one of the special cases that Moran discusses, a special case that is more tractable than most other cases. The enrichment problem shares with the error-in-variables problem the absence of unbiased estimates of c~ and /3 that are similar to those obtained in ordinary linear regression. Thus. regardless of the estimator that is chosen, the bias in that estimator must be considered in the error assessment. The enrichment problem is a tractable case of the error-in-variables problem because data are available at different enrichments.

The first concern in the error-in-variables problem is lack of identifiability. Lack of identifiability means that different values for the unknown parameters can produce data sets that are indistinguishable regardless of how many observations are made. Because observations are made at more than one enrichment, the enrichment problem is identifiable. The second concern is consistency, convergence of the estimates to their true values as the amount of data becomes large. Under reasonable assumptions on the measurement-to-measurement variation of B and under a reasonable interpretation of what is meant by the amount of data becoming large, consistent estimators for the enrichment problem can be obtained.

The most important feature of the calibration procedure developed here is the complete assessment of the effects of the random error on the results. The development of this assessment is closely related to the development of confidence regions for the error-in-variables problem through the use of instrumental variables [6],

14/. L t g g e t t / Measurements wtth ha('kground correction 457

The calibration procedure developed here is one of four procedures that have been proposed for 235U enrichment measurement [1,7,8]. All four procedures have their limitations. As mentioned above, the one developed here does not fully account for the lack of constant variance.

Kull and Ginaven [1] propose a procedure based on ordinary linear regression with the enrichment chosen as the dependent variable and the counts as the independent variables. This procedure gives biased estimates for a and il [5]. Kull and Ginaven provide no analysis of the errors made in estimating a and it.

The procedure suggested by Neuilly that is given in a recent revision of the IAEA Safeguards Technical Manual [8] is also based on ordinary linear regression. This procedure takes the count for the 185.7 keV region as the dependent variable and the enrichment and the background count as the independent variables. This procedure also gives biased estimates of o~ and/3. Further, the error analysis presented does not assess or even recognize this bias.

The procedure recommended by Gotoh [7] is based on the assumption that the random errors are due to the randomness of radioactive decay alone. As shown by the data analysis presented in section 3, this assumption is questionable. Based on this assumption. Gotoh proceeds in his development as though the variance of the random errors were known. With this knowledge, a total least-squares estimate can be obtained [5].

In section 4 of this paper, we investigate maximum likelihood estimators for a and/3. As part of this, we discuss further the relation between Neuiily's estimator, Gotoh's estimator, and the estimator proposed in this paper. In section 2, we propose estimates of a and it. and more important, we develop a complete error assessment for our calibration problem. In section 3. we analyze some data published by Beets et al. [9]. Besides showing that the causes of the random errors extend beyond the random nature of radioactive decay, this analysis shows how to explore data from the calibration experiment to detect deviations from the assumptions such as the proportionality of the two backgrounds.

2. The calibration procedure

2.1. The model

In applying our calibration procedure to enrichment measurements, scaling the counts will sometimes give a closer approximation to the constant variance assumption. For this reason, our development of the calibration procedure includes scaling. The scaling considered is multiplication of the ith pair of counts by a known constant k,. The constant variance assumption applies to the scaled counts.

The primary reason for scaling is to counteract the dependence of the 185.7 keV random error on the enrichment. The component of this random error that depends on the enrichment is the component due to the randomness of radioactive decay. When most of the random error is due to the randomness of radioactive decay and when the 185.7 keV contribution to the observed counts is much greater than the background contribution, properly chosen scaling will give a better approximation to constant variance than k, = 1. Scaling to equalize the variances of the 185.7 keV counts will be appropriate even though it causes the variances of the background counts to be more variable. In practice, a further approximation is usually involved in scaling. The scaling is estimated from the data and then treated as though it were provided beforehand. Such approximations have long been used in gamma-ray spectroscopy [10].

The calibration procedure developed in this section is based on the following formulation. Let n measurements be made in the calibration experiment. Measurement i produces the count C~, from the 185.7 kcV region and count C2, from the background region. The scaled counts are denoted by Y~, and Y2, and are given by

Y,, = k,C,, , Y2, = k,C2,. (2)

The model for the scaled counts, which is obtained from eq. (1), is given by

Y~, = ak,E, + ilk,B, + {,, Yz, = k,B, + 6,. (3)

458 H e. I.iggett ./ Measurements wtth background correction

In eq. (3), E, is the enrichment of the standard measured in measurement i, B, is the expected background count in the background region during measurement i. ~, is the scaled random error in Y~,, and 8, is the scaled random error in Y2,. As discussed above, we proceed as though neither the variance of ~, nor the variance of 8, depends on i. Further. we assume that (, and 8, are independent and normally distributed.

2.2. Poin t estimates"

To obtain point estimates for a and ,8. the procedure starts by regressing Y, on k, E, and k,. Likewise. the procedure regresses Y2, on k, E, and k,. Let the residuals from these regressions be dcnoted by R~, and R2,, respectively. We have

Yn = a l k , E , + b t k , + RI , . Y2, = a 2 k , E , + b2k, + R 2 , . (4)

where a t, hi , a 2, and b 2 are the regression coefficients. Formulas for these cocfficients are given in many texts [1 I]. Also, they can be calculated by most statistical packages. The point estimate of/3 is given by

b = b i b 2,

and the point estimate of a is given by

a = a~ - ha 2.

(5)

(6)

The reasonableness of this pair of estimates can be argued on the basis of some results from the theory of multiple linear regression [11]. First note that only through the coefficient a~ do the expressions for Y~, and Y2, in eq. (4) depend on a. Because Yt, and Y2, are regressed on the same independent variables, k, E, and k,, the expected value of bl is equal to/3 times the expected value of b~. Thus, the estimate of/3 in eq. (5) seems reasonable. The coefficient a~ does depend on the vector (fiB,) unless the B, are constant. However. a 2 depends on the vector (B,) in the same way. Thus. in the estimate of a in eq. (6), u~ is adjusted using a , . Point estimates of a and/3 are discussed further in section 4.

2.3. Con f idence el l ipse

An exact assessment of the errors in estimating a and /3 cannot be made in terms of variances and covariances. However, an exact assessment can be made in terms of a confidence region, a somewhat less familiar concept. Eq. (3) shows that Y~, - flY,_, does not depend on the background. Thus, with Y~, - / 3 Y 2 , as the dependent variable, the usual regression results including the F tests can be applied. Let F"(2. n - 2) be the 100(1 - a) percent point of the F distribution with degrees of freedom 2 and n - 2. The inequality

Y'~ [ (a , - / 3 a 2 - a ) k , E , + ( b~ - / 3 b 2 ) k , ] 2 <~ 2 s ~ F " ( 2 . n - 2). (7)

where

= ( , , - 2 ) - ' Y'. ( R , , - / 3 R : , ) 2 ( s )

is a 100(1 - a)% confidence region for a and/3. This region is analogous to the ellipse customarily obtained in linear regression [11], but as shown below, it is elliptical only under certain conditions.

To facilitate discussion and computation of this confidence region, we let

A,, = E k ? E ? . = E k , 2 , : , .

A22 = ~f'.k~ - 2 r " ( 2 , n - 2)(n - 2) 'Y'.( R 2 , / b 2)2, (9)

")'o = A22 -- A ~ 2 / A I I "

3', = - 2 F"(2, n - 2)(n - 2) l ] ~ ( R , , - b R 2 , ) ( R 2 , / h 2 ) ,

a{ = a ! -- y ; A , 2 / ( y ( ) A , I ) . h; = h, + 7 , / Y o .

14( Ltggett/ Measurements with background correction 459

The confidence region will be elliptical if Y0 > 0. This condit ion will hold if the s tandards used in the calibration experiment have at least two different enrichments and if F " ( 2 , n - 2) is small enough, that is, if the confidence level is low enough. Generally, this condit ion will hold if the range of the enrichments k; is large enough, if the variation in the expected background counts is not too large, if the calibration experiment is reasonably designed, and if the confidence level is not unreasonably high. Since negative values of Yo give unbounded confidence regions, failure of the condit ion ~'0 > 0 is cause for cancelling the measurement of the unknowns or for lowering the confidence level. We will not consider the case of nonposit ive Yo further.

Under the assumption that )'o > 0, we can show that eq. (7) defines an elliptical region by moving the terms involving/3 to the left side and complet ing the square. We obtain

_ , _ _ _ -~ . 2 - , , 2 ) + " Y l ' / Y o A , i ( a l - / 3 a 2 o~)2+ 2 A , 2 ( a , /3a 2 o ~ ) ( b ' , - / 3 h 2 ) + A 2 2 ( h ' , /362)2<..~s,, f (2. n - ~ .

(lo)

where s~, denotes the value of s/~ at the point /3 = b. If we let

ul = a'~ - /3 a 2 - a , u 2 = b'l - f l b 2, (11)

then we have in matrix notat ion an expression equivalent to eq. (10)

u'Au <~ r 2. ( 1 2 )

where u ' = ( u l , u 2) and

r 2 = 2s#~F"(2, n - 2) + Yi'/"{,,- (13)

Another possibility that would lead to cancelling the measurement of the unknowns and to questioning the setup of the calibration experiment is the possibility that the confidence ellipse contains non-positive a values. Using eq. (10), we can show that a is positive everywhere in the confidence ellip:e ,f

a > ' y , a , Ai2 ] r 2 a 2 . A, 2 r 2 "" - - + + - - - - + ( 1 4 )

A,,J yo E + . , , If a > 0, b 2 > 0, Y0 > 0, and the confidence level is low enough, then this condit ion will hold. The possibility of non-positive a in the confidence ellipse will not be considered further.

2. 4. I n t e r v a l e s t i m a t e s f o r t he u n k n o w n s

When an indeterminate number of measurements will be made on the basis of one calibration experiment, an error assessment that treats all of these measurements at once seems appropriate. Consider a long succession of measurements that are all based on the same calibration experiment. In the procedure developed here, each measurement gives an interval that may or may not cover the true value that gave rise to the observation. The percent of the intervals that cover the corresponding true values will converge in the long run. This limiting percentage is used to characterize the intervals produced by the calibration procedure. The procedure is constructed so that we can state with confidence 1 - a that this limiting percentage is greater than 100P. Thus, we can say with confidence I - a that when many unknowns are measured, the intervals cover at least 100P percent of the true values.

These intervals are computed by first finding the confidence ellipse discussed in section 2.3 and then by doing some calculations that depend on the unknowns. The parameters of the confidence ellipse, which are given by eqs. (4), (9), and (13), must be obtained for the confidence level 1 - or/2. Thus, in eqs. (9) and (13), F " ( 2 , n - 2) must be replaced by F " / 2 ( 2 , n - 2 ) . Let C~, C z, and k be the 185.7 keV count, the background count, and the scaling for the unknown. Further, let N ( P ) be the I00[I - (1 - P ) / 2 ] percent point of the normal distribution, and let " ~ 2 x 2 ( n - 2) be the 100(a /2 ) percent point of the chi-squared distribution with n - 2 degrees of freedom. For each unknown, we first compute two values of xj and two

460 W. Ligge t t / M e a s u r e m e n t s wt th b a c k g r o u n d correction

values of x 2 from

x , = C, +_ N ( P ) k ' [ ( n - 2 ) / " / 2 x 2 ( n - 2 ) ] ' / 2 [ s h - b.s"~],

x2 = C2-+ N ( e ) k - ' [(n - 2 ) / " / 2 x2(n - 2)] , / 2 [ _ s ; ] . (15)

where

st',= - ( n - Z ) ' Y , ( R , , - b R 2 , ) R 2 , / . %. (16)

The values of x, and x 2 obtained with the plus sign are used to compute the upper limit of the interval, and the values obtained with the minus sign are used to compute the lower limit. Next. we compute for each ( x , , x 2) pair

v~ = a{ - a 2 x l / X 2 , t; 2 = b' I - b 2 x l / x 2 , (17)

O = [det A(v'Av- r:}] l/:

Finally, the upper limit on the interval estimate for the unknown is given by

b2x , - b'lx 2 + r O ( x 2, O)Av u - (~8)

bza; - b',a 2 + r O ( a 2, bz ) A v "

and the lower limit is given by

b2x , - b'tx 2 - r O ( x 2, 0)Av L = . (19)

bza ' , - b',a 2 - r O ( a2, b z ) A v

2.5. Derivation of the interval est imates

Eqs. (18) and (19) specify the calibration procedure. The derivation of these equations follows very closely the analysis of linear, single-observation calibration discussed in Lieberman et al. [4]. The analysis is based on the Bonferroni inequality. The Bonferroni inequality allows us to combine a 100( 1 - a / 2 ) percent confidence interval for the variance of Y,- f lY2 with a 100(1 - a / 2 ) percent confidence ellipse for a and fl to obtain 100P percent coverage with confidence 1 - a.

We start the derivation by assuming that m/3, and

o~ = Var ( Y,, - f lY>) (20)

are given. For each unknown, consider the interval

C, - tiC 2 N( P ) o R _+ (21)

o~ k s

where C,, C2, k , and N ( P ) are defined above. The probability that this interval will cover the true value is P. Thus, in the long run, these intervals will cover 100P percent of the true values.

Since o R is not given, we replace o R using

o R ~< [(n - 2 ) / " / 2 x 2 ( n - 2)] ' /2s B, (22}

which is the IOO(I - a /2)% confidence interval for o R. The chi-square percent point " ' 2 x Z ( n - 2) is defined above. This gives us the following interval for the unknown enrichment

C , - , C z N ( P ) s p ( n - 2 ) , /2

c~ + ke{ " / : ~ ? n - - 2) " (23)

14: l . t g g e t t / M e a s t l r e m e n t s w t t h b a c k g r o u n d c o r r e c t i o n 461

Since Of and 13 are not given, we expand the interval in eq. (23) to include all the values of a and/3 in the 1 0 0 ( 1 - a/2) percent confidence ellipse. The resulting interval has as its upper bound

U = Max [ C , - f l C , N(P)s ,~ n - 2 )~/2] ~ " + . . . . , ( 2 4 )

.,A.<.,+ a k a a / 2 X 2 ( t l - 2)

and as its lower bound

i = Min C , - f l C 2 N ( P ) S l ~ 'ZZ_ 2 ) ' " :1 u , A u ~ r 2 Of k o [ a '~ ~ , " ' - x ' ( , , - 2) (25)

Note that in eqs. (24) and (25), u, A, and r : are defined as in eqs. (9)-(13) with F"/2(2, n - 2) in place of F"(2, n - 2). Note also that as long as the confidence ellipse does not contain a = 0, the maximum and minimum in eqs. (24) and (25) are finite. Eqs. (24) and (25) specify our calibration procedure since they specify the interval that is computed for each unknown. All that is left is to derive the algorithm for finding the maximum and minimum specified.

In order to avoid an iterative approach to the maximization and minimization, we must make an approximation. We linearize s/~ about /3 = b to obtain

s, = s,, + s ; ( f l - b), (26)

where s;, is defined in eq. (16). Our task is now reduced to finding the extrema of (x~ - / 3 x , ) / a subject to eq. (12), where x z and x 2 are

defined in eq. (15). We use the method of Lagrange multipliers. Differentiating, we obtain

and

- (x, - / 3x , ) / , ~ 2 + 2 x ( - 1.0)Au = 0, (27)

- x 2 / a + 2,X( -a2. -b2)Au = 0. (28)

Eliminating ~ between eqs. (27) and (28) gives

[(x. - f i x , ) / a ] ( - a 2. - b 2 ) A u - x 2 ( - 1, 0)Au = 0. (29)

Using eq. (11), we obtain

v ' A u = r 2, u ' A u = r 2, ( 3 0 )

where v. and v 2 are defined in eq. (17). The solutions to eq. (30) are given by

( t u J ) = [ 6 ) v ' A v ] - ' { r 2 0 ( v ' ) + r A - t ( t'2 )} , (31)

where 6) is defined in eq. (17). One of the signs applies to finding the maximum and the other to finding the minimum. Substituting eq. (31) into eq. (29) and manipulating the result to remove the plus-and-minus sign from the denominator gives

x I - fix 2 (b2xz - b ' jxe)(b2a ~ - b'la2) -T-rOx2b2v'Av+ r26)2(x:, O)Av(a 2, b2)Av (32)

( b2a ' , - b ; a 2 -

Eq. (32) shows that the lower sign [the minus sign in eq. (31) and the plus sign in eq. (32)] applies to the maximum and the upper sign to the minimum. Thus, the upper and lower limits are given by eqs. (18) and (19).

462 W. Liggett / Measurements with background correctton

3. 23SU enrichment

The calibration procedure developed in the last section can be successfully applied to the measurement of-~35U enrichment if the observations involved have the assumed physical characteristics. These character- istics must be determined empirically. Beets et al. [9] have published the data from several enrichment calibration experiments. These data allow us to check assumptions including assumptions about the relation between the background levels in the two regions and assumptions about the random error. Since two of the experiments were made with a SAM-2 enrichment meter, these data allow us to check the applicability of various assumptions to that particular instrument. Of course, assumptions must be checked whenever a calibration experiment is performed. Methods for doing this are illustrated in this section.

The SAM-2 data published by Beets et al. [9] are shown in table 1. These data are given in tables 7 and 8 of ref. 9 and are labeled Experiments 4 and 5. These two experiments involved different cxperimental conditions. Descriptions of the experimental conditions, which are given by Beets et al., will not be repeated here. As shown in table 1, twelve batches of UO 2 powder were used as standards. The enrichments of the batches were measured by mass spectrometry to give the known enrichments E,. The counting time was 720s. Some of the batches were counted more than once in the calibration experiments shown. Thus, each calibration experiment consists of 16 pairs of observations.

23SU enrichment measurements are often based not only on the assumption that the two background levels are proportional but also on the assumption that the random error is strictly due to the random nature of radioactive decay, in other words, is strictly counting error. These assumptions comprise a model of enrichment observations that is more restrictive than the one on which our calibration procedure is based. Nevertheless, we begin by checking the fit of this model.

A two-dimensional confidence interval analogous to the confidence ellipse in eq. (10) can be obtained for the case in which the random error is strictly counting error. Eq. (I) shows that in this case the variance of ( ' l - f l C 2 - a E is a E + 2/3B. Thus, a 100(1 - a) percent confidence interval for a and/3 is given by

Z, (c,,- aE, + 2~B, <~ ~ "X"(n)" (33)

The expected background counts B, are not known so we replace them with an upper bound inferred

Table 1

SAM-2 calibration data from Beets et al. [9l.

Batch Percent identifier enrichment

Counts from Experiment 4 Counts from Experiment 5

185.7 keV Background 185.7 keV Background

I 1.899 214900 91 150 197800 95060

1 1.899 213400 90870 198 100 94480

2 2.320 241 100 90 510 223 500 93 430

3 2.439 248 700 91 120 229 600 94 670 4 2.670 265 700 92 370 243 800 94 650 5 2.800 274400 93260 253 100 96 300

6 2.915 279 300 92 430 259 700 96 890 7 3.020 286 700 92 340 267 200 96 250 8 3.196 296400 88630 275000 92 160 9 3.428 314 700 91 390 289 800 95 760

10 3.602 325 300 89230 299500 93 180

I 0 3.602 325 600 90 320 10 3.602 323 900 90480 I I 3.779 334 700 87 810 307 800 90 930 I I 3.779 308900 92420 I 1 3.779 - - 309 600 92 030

12 3.972 347500 88710 319600 92 110 12 3.972 346800 87910 319300 92610

~ Ltggett/ Measurements with background correction 463

from the background counts. Using (C2,) I,''2 as an estimate of the standard deviation of C2,, we adopt as the upper bound with which we replace B, the quantity C2, + 3(4"2,)~'2. This gives a larger confidence interval than the expected background counts would give.

It may be that no values of a and /3 exist that satisfy eq. (33). In this case, we conclude that the restrictive model on which eq. (33) is based is inadequate. This model may be inadequate in some or all of its assumptions.

For the data from Experiments 4 and 5, we minimized the left side of eq. (33) over a and/3. We obtained 42.28 for Experiment 4 and 32.32 for Experiment 5. These values show that at the 99% level the confidence ellipse in eq. (33) is empty for both Experiments 4 and 5. Thus, we conclude that the model is inadequate.

On the premise that the restrictive model is inadequate due to random error in addition to the counting error and not due to failure of the assumption of the proportionality of the two backgrounds, we now apply the calibration procedure developed in the last section to Experiments 4 and 5. We investigate the proportionality of the backgrounds below.

In applying our calibration procedure, we let k, = 1. If the inadequacy of the restrictive model is indeed due to random error in addition to the counting error, then this additional error would seem to be as large as the counting error. Further, the range of the variance of the 185.7 keV counting error is not very large as can be seen by examining the range of Ct, in table 1. Thus, scaling to deal with the non constant variance does not seem to be worthwhile.

Results of the procedure are given in table 2 and in figs. 1 and 2. The point estimates are given in table 2. Some perspective on the values of b in table 2 can be obtained from the boundaries of the two energy regions. For both experiments, these boundaries were 160-210 keV for the 185.7 keV region and 215-285 keV for the background region. The confidence ellipse given by eq. (10) was computed for each of the experiments. These ellipses, which are at the 95% level, are shown in figs. 1 and 2. They show the range of values that a and /3 might take. These ellipses do not overlap because the two sets of experimental conditions differ.

If in either of the regions there is an energy peak due to some other radionuclide, then the assumption of the proportionality of backgrounds will not be valid. This could be the reason why the restrictive model is not adequate. If this assumption is not valid, then the calibration procedure developed in the last section is not valid. We now investigate this possibility. In figs. 3 and 4. we plot R), versus bR2,. The two sets of residuals appear to be correlated. This correlation is a consequence of variations in the background activity affecting the residuals from both regions. If the two backgrounds are proportional, then the same constant of proportionality will apply to the variation about the mean that applies to the mean. Thus, in figs. 3 and 4, the points should be scattered about a 45 ° line. Figs. 3 and 4 provide no reason to believe that this is not true. Were the background variations larger, the correlation would be stronger, and any deviation from a

O 92 O 94 0 94~ 0 ~ 1 OEI 1 02 O ~ O 86 O 87 0 04B O 60 0 EIO O 91 O 9~

67,~,~- ,L - ' . . . . . - I - 67eee 8 , e ~ , - ~ , ~ , ] , ' , '-] . . . . l . . . . J . . . . 1 . . . . [ . . . . - l - 6 , e e o

L e~lS~ll --~

- p

H ~ ~---555~0 x

645 '09~ - ' - - T - - - ] r l - r l t " r - i - - 645~Jo ,$9008 1- T I i - T [ T I T T [ ' I Y T T [ ' r l - I I ~ "

O 92' 0 94 0 9 6 O 98 IOQ tO2 $ 8S t i n e $7 0 8 8 9 89 8 N 0 9 1 9 92

0ETA B~TA

Fig. I. 95~ confidence ellipse for Experiment 4. Fig. 2 .95~ confidence ellipse for Experiment 5.

464

Table 2 Estimates from the SAM-2 data.

14/. Lzggett / Measurements wzth background correctton

Experiment 4 Experiment 5

a I 64402. 58712. b~ 92250. 87368. a 2 - 1416. - 1451. b 2 94 880. 98 419. a 65 779. 60 000. b 0.9723 0.8877

( n - 2 ) - l Y ' R ~ , 1 .4l X 10 ~' 1 .36 x l 0 ~'

(n -2)- ~7"RhR2, 1.1l x 106 1.38x 10 ~' (n -2}- IER~, 2.00x 10 ~ 2.42x lO ~'

n - lY'('I, 0.29 X 10 ~' 0.27 x 10" n- ~)"('2, 0.09 X l0 ~' 0.09 x 10 ~'

45 ° line would be easier to see. Fig. 3 conta ins two points in the lower right corner that are further from the 45 ° line than the others. These points raise the same quest ions that outliers usually raise.

If we pretend that the variat ion of the background level about its mean is a random variable, then we can model the scatter of the points shown in figs. 3 and 4 in terms of the variances of the two counts and the covariance between them. The variance of the 185.7 keV count is given by flzo z + o~, + o~, where flzo:

reflects the background variation, Oc2t the randomness of radioactive decay, and o~t the other sources of random error. Similarly, the variance of the background count is given by o 2 + o(= + o h. The covariance is

given by/30 :, the cont r ibu t ion from the background variation. Estimates of the variances of the two counts and the covariance between them are shown in table 2. Table 2 also gives average counts which estimate o<-',~

and ' or(': 2 '

Consider o~ and o ~ , the variances that reflect the part of the random error not due to the randomness of radioactive decay. These variances can be estimated by subtract ing from the sample variances of the residuals the part due to the variation in the background and the part due to count ing error. The estimate of 02~ is

( n - 2 ) - t E R ~ , - f l ( n - 2 ) - ' Y ' . R , , R 2 , - n ' Y ' . c , , .

-3~18 - 1888 - 8 ~ 696 I ~ 3880

2 I

× x

o e e e - [-- ~ e eee- -

>. 1 x .

,- - o ~ - d x X x x h _ ~ ,.. - e e e A ., x ' 3

- l e n a - x / - I s e ~ -~eee- ]

,

- ~ e - l ~ - r ~ r- 1 r r ~ ~ l - w ~ r ~ ] ~ ~ ~ r [ - ~ : ~ - , ~ - - 3 e e e - 3 0 ~ - 1 8 ~ -880 680 ~oO8 :3~0~

b • BACK(~OUND RES~OUAL

Fig. 3. Scanerplot of the Experiment 4 residuals.

-39e8 - t s e 9 -oBe oee ~884a

1 1

×

x

x X

X ~ x x

x

x X x ~-

I--eeo

- ~ a e e ~ r ~ r 1 ~ r 1 1 T ' " ~ ; 1 ~ t , , • , r ~ - ~ - 3 ~ - 3 ~ - 1889 -009 006 1 8 ~ :~O~

b • BACKGROUND RESID~IAL

Fig 4. Scatterplot of the Experiment 5 residuals.

I~ Liggett / Measurements wtth background ('orrectton

"/'able 3 Intrinsic Ge calibration data from Beets et al. [91.

465

Batch Percent 185.7 keV Background identifier enrichment

1 1.809 249 500 55 590 2 2.320 292 500 56070 3 2.439 303 700 55 720 4 2.670 328 000 54 560 5 2.800 342 100 56450 6 2.915 352 200 56 290 7 3.020 364 40(.) 57 270 8 3.196 379 100 56 120 9 3.428 404 500 56 840

10 3.602 421 700 57430 I 1 3.779 436 500 56 770 12 3.972 458000 57620

and the estimate of o12 is

( , , - - , , -

The estimates of (o~, o(z) obtained from table 2 are (0.03 x l0 t', 0.76 x 10 ~') for Experiment 4 and ( - 0 .13 × 106, 0.77 x 106) for Experiment 5. The negative value for o(~ in Experiment 5 is a consequence of using a variance estimator that involves subtraction. We must be careful in drawing conclusions from these variance estimates since they are based on few observations. Nevertheless, they suggest that if the assumption of the proportionality of the backgrounds is valid, then o~z is much greater than o~t. If this disparity is not reasonable, then the assumption of the proportionality of the backgrounds must be investigated further.

Beets et al. give a data set obtained with an intrinsic Ge detector which illustrates the case in which the random error is almost entirely counting error and the case in which the background levels in the two regions are not proportional. This data set, which is shown in table 3, is given in table 6 of ref. 9. The counting time was 10000 s. Since the random error is almost entirely counting error as will be shown below, we scale this data set to make the variance of the 185.7 keV counting error equal 1. We obtain the k, values by fitting a simple model to the counts in the 185.7 keV region. We fit a constant plus a term proportional to the enrichment. This fitting gives

k, = (60 101 + 100 182 -E , ) - ' " - ' (34)

We proceed with the analysis ignoring the fact that the k, ' s are estimated. Before applying the calibration procedure, we rejected the pair of observations on batch 4 because this

pair seems unusual. For the other 11 pairs, the calibration procedure developed in the last section gives the estimates shown in table 4. Fig. 5 shows the plot of residuals that is comparable to figs. 3 and 4. The estimates in table 4 provide fitted values for batch 4 and thus a pair of residuals for batch 4. This point is the outlying point in fig. 5. Clearly, the points are not scattered along a 45 ° line. Thus, we reject the hypothesis that the background levels are proportional. How might this be interpreted? The value of b given in table 4 applies to the relation between the means of the two backgrounds. The behavior shown in fig. 5 applies to the relation between the variations about the means. Apparently, these are not equal.

Consider the variance and covariance estimates obtained under the assumption that the background variation is random. These estimates are shown in table 4 along with an estimate of the average variance of the counting error in the background count. The counting error in the 185.7 keV count has been scaled to equal I. As we did above, we can decompose these variance and covariance estimates into components

466

Table 4

Estimates for the intrinsic Ge detector.

W. Ltgget t / Mea~'urements wuh hackwound correctton

a I 100334.

h , 59 64 I.

a 2 882. h 2 53885 . a 99 35X.

h 1 .1068

( n - . - 2 ) ~5"R~; 4 .18

( n - 2 ) - [~.RhR2, l .OI ( n - 2 ) I~ 'R~, 0 .42 n - I ~ ' k , : C 2 , 0 .16

reflecting the background variation, the counting error, and the other random error. This decomposit ion suggests that any random error other than the counting error is small and that fl is close to 3. This value of fl should be compared to !.1068, the value of b given in table 4. Fig. 5 suggests that the cause of this difference is more than just random error.

The characteristics of the random error and the relation between the two backgrounds are not the only two aspects of the model that should be checked. Another aspect is the dependence on the standard (in this case, on which batch of UO 2 powder is being observed). This aspect involves, of course, the dependence on the enrichment. Depending on the design of the calibration experiment, other aspects could and should be checked. For instance, repeated measurements on each standard provide an opportunity for exploratory analysis that is not illustrated here because only four replicate measurements were made in Experiments 4 and 5.

The obvious way to check the dependence on the enrichment is to plot the adjusted residuals R,, - hR2, versus the enrichment as shown in fig. 6. If eq. (1) is an adequate model, the adjusted residuals will show random variation due to the random error but will not show any systematic dependence on the enrichment. The adjusted residuals in fig. 6 do not show any systematic dependence on the enrichment. Thus. fig. 6 provides no reason to question the adequacy of the model given by eq. (1).

The two data sets allow us to ask whether the residuals are related to the batches of UO 2 powder. II1 fig. 7, we plot R , , - bR2, for the data from Experiment 5 versus R,, - hR2, for the data from Experiment 4. Both data sets contain multiple measurements for some batches of UO 2. For fig. 7. we averaged the residuals corresponding to these multiple measurements. Thus, we obtain 12 points, one for each batch. The two outliers identified in Experiment 4 are the two points on the left side. For only one of these is the

- 4 08 .'2 40 -0 80 0 BO 2 40

4 e o ]_ _ . ~ _ '~ , ~ l , . l 3

x 2 40- .

!

o X ~ B 8 0 - ; ×

r- .8 8e~ ~- .j × ×

I × 2 4 0 ~

×

-4e0-- .-.: - I ~ " l r i ~ 1 4 00 -~ 4g -0 8~1 0 88 2 40

b * OACKCROUND RESIDUAl

Fig. 5. Scatterplot of the intrinsic Ge residuals.

4 ~

i

F ~ - 2 40

I • -0 80

-e 8~

l. -2 40 J

~ 2 9 2 5 3 8 3 5 4 8

A D × - J X 0 0 U - 0 -

E × 0 D 0 X X 0 -

x o~

4 0 ~I-

s × [ D ~ r - - ~ U 2 X EXP 4 X A .

- ~ ~ . l ~ - r t - r " l - ~ ' v r - t r ~ x ~ - T t - ~ T - ~

I . ~ 2 0 z 5 3 8 3 S 4 g

ENRICHMENT

Fig. 6. Residuals adjusted for background variation vs. enrichment.

H" Liggett / Measurements with background correction 467

cor re spond ing value from Exper iment 5 extreme, a l though it does not seem to be an outlier. With the except ion of this poin t which comes from batch 6, no relat ion between the two exper iments can be seen. The fact that the r a n d o m error does not seem to be related to the batch suggests that the o ther sources of the r andom error are in the ins t rument . Clearly. more invest igat ion of the sources of r andom error in the SAM-2 ins t rument is needed.

- z ~ -zooe -15oo -IOOO -500 0 54~ tO~Q x ~ I J r J D" L~ J L't~ I x I ~a-a t 1-~ t t-t 1LLx t LLa-~ --zg~

D d X u I x ~- $ 4 T t~Qe~ [-- 1080 E I) X L

X × r E x j L P O ~ X ~ 8 5 F

x × E R E ×

I D - X 7" U J

L j

-L, W0 ~ ] r r F r ] i r l l F r T V r T 1 m i T-r ~n-w 1 T r -

ADdI,~T[D U ~ 4 R[SIDOAI-S

Fig. 7. Residuals adjusted for background variat ion: Exper iments 4 vs. 5.

4. Re lated e s t i m a t e s

The ca l ibra t ion procedure deve loped in sect ion 2 has as its chief advan tage a comple te error analysis. The poin t e s t ima tor for a and /3 that is given by eqs. (5) and (6) has not been shown to be op t imal in terms of some measure of es t imat ion error. Fo r this reason, a discussion of o ther poin t es t imators seems appropr i a t e . The discussion in this sect ion is centered a round the ma x imum l ikel ihood es t imator . In the discussion of this es t imator , the relat ion of the es t imator developed in section 2 to estinaators for the e r ror - in-var iab les p rob lem is clarif ied. Fur ther . the relat ion to other es t imators for the enr ichment ca l ibra t ion p rob lem is examined.

As the first s tep in examin ing a l ternat ive point es t imators , the model for the es t imat ion s i tuat ion must be deve loped further. In par t icular , we model the measu remen t - to -measu remen t var ia t ion in the back- g round level so that we can think in terms of large numbers of observat ions . The basis for our model is the no t ion that many measurements could be made on the same s t andard but that there will usually not be many s tandards avai lable for measurement . Fo r this reason, we divide the true background level into two componen t s , a componen t de te rmined by the s t anda rd itself, B],, and an ambien t componen t . B2,. We assume that Bl, is fixed (the same for every measurement of a par t i cu la r s t andard) and B2, is r andom with var iance independen t of which s t anda rd is being measured. We in terpre t a large number of observa t ions to mean a large number of observa t ions on each s tandard . In discussions of the er ror - in-var iables p rob lem [5], the d is t inc t ion is made between a s t ructural formula t ion (B, r andom) and a funct ional formula t ion ( B, not random) . Our formula t ion is somewhat a mix of the two but closer to the s t ructura l formula t ion .

In the usual formula t ion of the er ror - in-var iables problem, two observat ions are made, one on a + flU and the o ther on U. Both observa t ions are made with error. In M ora n ' s discussion [5], one of the special cases ident if ied is the case of a = 0. This case is closely related to the enr ichment ca l ibra t ion problem. Cons ide r the pro jec t ions of Y], and ~ , on to the subspace or thogonal to k,E, . In this subspace, all d ependence on a d isappears . Thus, we have the a = 0 special case. Moran suggests an es t ima tor for this

468 W Liggett / Measurements with background correctton

special case. This estimator is analogous to eq. (5) as can be seen by rewriting b as

Y'. [Y, ,- (a, + pb, ) k ,E , ] [k , - 0L,E,I t, = Y'. [ r~ , - (a , + W, , )k ,E, ] [k , - p<E,] ' (35)

where

P = E k ~ I £ , / E k 2 F , 2 • (36)

The vectors Y h - (a l + pb~)k ,E , and Y2,- (a2 + Pb2)k ,E , are, respectively, the projections of }'~, and )~, onto the subspace orthogonal to k ,E, . The vector k, - pk ,E , is the projection of k, onto the same subspace. Eq. (35) shows another relation, namely, that b is basically a ratio estimator [12].

Moran [5] notes that the a = 0 problem is identifiable. He presents an estimator, the analog of eq. (35). that he notes is consistent. He also suggests that maximum likelihood might provide a better estimator.

The maximum likelihood estimator has two versions in which we are interested. One version is the case where B2, = 0 and the variances of the random errors are known. "]'his version is of interest because it is essentially the estimator suggested by Ootoh [7]. Actually. Gotoh allows each observation to have a different variance. For the purpose of the qualitative comparison of estimators, this refinement does not make too much difference. The other version is the case where the variance of B2, and the variances of the random errors are unknown. This version is based on essentially the same assumptions as the estimator in section 2.

We now derive some properties of the maximum likelihood estimator. To do this, we change the parameterization slightly from section 2. First, we add another subscript so that the observations arc denoted by Yu: and Y2,/, where i indexes the standard measured and j indexes the measurements on a particular standard. Second, to keep the equations simple, we consider onlv the case in which equal number of measurements are made on each standard. We denote by )'1, and Y2, the averages overj. We have

Yl,: = ak,E, + f lk ,Bi , + 18k,B2, I + ~,/,

Y~,/= k,Bi , + k,B2,: + 8,:. (37)

Note that we are allowing the scaling to depend only on i and not on.j. Consider the parameterizat ion of the variances and covariances of Y~,, and )~,,. Let

ol~ = Var(f lk ,B2, / + ( , , ) ,

o,2 = Cov(Bk,B2,, + c,,, k,B2,, + ,~,, ). (38)

%: = Var( k,B2, , + a , / ) .

In one of the cases we consider, o u and 0"22 are known and 0~2 is zero. In the other case, 0~, 0~2. and 022 would all be unknown even if /3 were known. Thus, in both of these cases, we can use o~, o~2, %2 to parameterize the covariance matrix.

We now derive a pair of equations for a and/3, which are instructive but not the complete solution when the covariance matrix is unknown. Differentiating the log likelihood ratio

_ _ ., ~ , / 2 o ~ ) - , [ _ 1og(2~) log(o, ,o~:-o , : , - ( o , , 0 , 2 - _ o~_=g(r , , , - .k ,E , #kB, , ) :

- 2 o , ~ E (} ' , , , - ,~k,&-/3k B, , ) (Y, , , - k,B,,) + o,, Y'. (F2, ,- k, B,, )"]/2.

with respect to a, ,8, and B~, gives

%2 Y~ ( Y, , . - a k , E , - , S k , B , , ) k , E , - o,2 Y'. ( )2 , . - k,B,, ) / , -E = 0.

o22]~ ( Y , , - ak,E, - / 3 k , B , , ) k , B , , - o , 2 ~ ( ~ , -- k ,B, , )k ,B , , = 0. (39)

(/3o22 - o ,2 ) (Y , , - c~k,h'- /3k,B,,) + (o, , - / 3 0 , 2 )( Y2, - k ,B , , ) = O.

14". L t gge t t / Measurements with background correctton 469

Substi tut ing the third of these equat ions into the first gives

Y~.( r,, - ak ,E, - flY,., )k, E, = 0. (40)

Substi tut ing the third into the second gives

~ ( Y , , - , ~ k , E , - , s r~ , )( r , , - ,~/,, E, + [(o, , - ,8°,: ) / ( ,8o2, - o , , )] r2, ' ) = 0. (41)

We can rewrite eq. (40) in terms of a~. b~, a:. and b_,. We obtain

a = a~ - fla z + p (b , - , s b : ) . (42)

Eq. (42) shows that if the est imate of ,8 is near b then the max imum likelihood est imate of a is near a. Let P~, and P2, be the projections of Yi, and Y:, onto the subspace or thogonal to k,E,. We can rewrite eq. (41) as

S.. ( P,, - BP:, )( P,, + [(o,, - , 8 0 , : - o ,2 ) ] P : , ) = o . ( 4 3 )

where

P, ,= Y . , . - ( a . + p b , ) k , E , , P2,= Y2 - ( a 2 + P b 2 ) k , E , . (44)

The second factor in the summand of eq. (43) is an est imate of the projection of k,B~, onto the subspace or thogonal to k,E,. If we substi tute for this factor k, - pk,E, , then we obtain b as the est imate of ,8, and a as the est imate of a from eq. (42). Thus, we see that the es t imator in section 2 and the m a x i m u m likelihood es t imator are very nearly the same if B, does not vary very much. In the cal ibrat ion exper iments examined in section 3, B, varies little compared to its average.

If we reparameter ize the covariance matrix in terms of

011 --- 02 -I- ,8202, 0"12 = ,802, o22 = 0.? + °2, (45 )

w h e r e

0.? = V a r ( , , , ) . o i = Var (6 , , ) , 0.: = V a r ( k , B , , , ) . (46)

and if we let

bc ~ ,P, ,P2, /~- , ~, b,, ~,P, ' , /~- ,P, ,P2, . (47)

then we obtain

(b H - ,8)/0.~ = ( b ; ' - , 8 - ' ) / o 2 2 . (48)

Eq. (47) gives two est imates of/3, one that is biased low. b t . and one that is biased high, b H. Note that b~ is the est imate presented in the IAEA Safeguards Technical Manual [8]. Eq. (48) shows that the m a x i m u m likelihood est imate lies between these two.

The m a x i m u m likelihood es t imator for the case of o,~, o,2. and 0.z2 unknown involves three more equations, the derivatives with respect to o,,, o~2, and 0.z2. The est imates themselves in this case have to be computed by an iterative procedure. We have not investigated this.

References

[11 L.A. Kull and R.O. Ginaven, Brookhaven National Laboratory Report BNL 50414 (1974). [2] P. Matussek, draft of a Kernforschungszentrum Karlsruhe report (1982). [3] J.R. Rosenblatt and C.H. Spiegelmam Technometrics 23 (1981) 329. [4] G.J. Lieberman, R.G. Miller, Jr. and M.A. Hamilton, Biometrika 54 (1967) 133. [51 P.A.P. Moran. J. Muhivar. Anal. I (1971) 232.

470 W. Liggett / Measurements with background correction

[6] M.G. Kendall and A. Stuart, The advanced theory of statistics, vol. 2 (Hafner, New York, 1961l. [7] H. Gotoh, 2nd IAEA Advisor 3' Group Meeting on Evaluation of the quality of safeguards non-destructive assay measurement

data (November 1980). [8] Safeguards technical manual, vol. 3 (draft) (IAEA, Vienna. 1980). [9] C. Beets, G. Busca, J. Colard. M. Corbellini, M. Cuypers, B. Fontaine, F. Franssen, D. Reilly, F. Van Craenendonck anti P. Van

Loo. C E N / S C K (Studiecentrum Vc×3r Kernenergie) Report 552 (1979). [10] E. Schonfeld. A.H. Kibbey and W. Davis. Nucl. Instr. and Meth. 45 (1966) I. [11] N.R. Draper and H. Smith, Applied regression analysis (Wiley. New York. 1966). [12] W.G. Cochran, Sampling techniques, 2nd ed. (Wiley, New York. 1963).